Keywords: 1, 1, 2 Dispersion interaction between molecules

In this thesis, after an introduction where we brie y present the general context of Casimir physics, we present the results obtained during the PhD. At rst, we show our work about the van der Waals/Casimir-Polder interactions between two atoms in an out-of-equilibrium condition due to their uniformly accelerated motion. We study the system of two uniformly accelerated atoms in vacuum space, when they are in their ground-state and when they are in a correlated state (one excited and one ground-state atom). We analyze this system both with an heuristic semiclassical model and with a more rigorous method, based on a separation of radiation reaction and vacuum uctuations contributions, that we extend starting from a general procedure known in literature. We nd a change of the distance-dependence of the interaction due to the acceleration. We show that Casimir-Polder forces between two relativistic uniformly accelerated atoms, interacting with the scalar eld, exhibit a transition from the short-distance thermal-like behavior predicted by the Unruh e ect to a long-distance nonthermal behavior, associated with the breakdown of a local inertial description of the system. In addition, we obtain new features of the resonance interaction in the case of atoms interacting with the quantum electromagnetic eld.

Next, we present our work about a new optomechanical coupling of an e ectively oscillating mirror with a Rydberg atoms gas, mediated by the dynamical atom-mirror Casimir-Polder force. We nd that this coupling may produce a near-eld resonant atomic excitation not related to the excitation of atoms by the few real photons expected by dynamical Casimir e ect. In accessible experimental conditions, this excitation probability is signi cant (about 20%) making the observation of this new dynamical Casimir-Polder e ect possible. For this reason, we propose a realistic experimental con guration to realize this system made of a cold atom gas trapped in front of a semiconductor substrate, whose dielectric properties are periodically modulated in time.

Finally, we focus on our results obtained for the Casimir-Lifshitz pressure between two di erent dielectric lamellar gratings. This system is assumed to be in an out-of-thermal-equilibrium con guration, i.e. the two gratings have two di erent temperatures and they are immersed in a thermal bath having a third temperature. The computation of the pressure is based on a method exploiting the scattering operators of the bodies, deduced using the Fourier modal method. In our numerical results we characterize in detail the behavior of the pressure, both by varying the three temperatures and by changing the geometrical parameters of the gratings. In this way we show that it is possible to tune the force from attractive to repulsive or to strongly reduce the pressure for large ranges of temperatures. Moreover, we stress that the interplay between nonequilibrium e ects and geometrical periodicity make this system particularly interesting for the observation of the repulsive Casimir force.

Résumé

Dans cette thèse, après une introduction où nous présentons brièvement la physique des forces de Casimir, nous montrons nos résultats obtenus pendant le doctorat. D'abord, nous montrons notre travail sur les interactions de van der Waals / Casimir-Polder lorsque le système est dans une conguration hors équilibre à cause du mouvement uniformément accéléré des atomes. Nous étudions le système de deux atomes uniformément accélérés dans le vide quantique quand ils sont dans leur état fondamental ou dans un état corrélé (un atome excité et un atome dans son état fondamental). Nous analysons ce système avec un modèle heuristique semi-classique et une méthode plus rigoureuse qui nous avons étendu à partir d'une procédure générale développée dans la littérature. Nous trouvons un changement de la dépendance de l'interaction de la distance en raison de l'accélération. Nous montrons que les forces de Casimir-Polder entre deux atomes uniformément accélérés en mouvement relativiste, qui interagissent avec le champ scalaire, présentent une transition à partir d'un comportement thermique à courtes distances, comme prédit par l'e et Unruh, à un comportement non thermique à longues distances, associé à la rupture de la description inertielle et locale du système. En plus, lorsque le cas d'atomes qui interagissent avec le champ électromagnétique quantique est considéré, on constate que de nouvelles caractéristiques apparaissent dans l'interaction.

Ensuite, nous présentons notre travail sur un nouveau couplage opto-mécanique d'un miroir oscillant de façon e cace avec un gaz d'atomes de Rydberg, médié par la force atome-miroir dynamique de Casimir-Polder. Nous constatons que ce couplage peut produire une excitation de résonance atomique de champ proche, qui n'est pas liée à l'excitation des atomes par les quelques photons réels attendus de l'e et Casimir dynamique. Dans des conditions expérimentales accessibles, cette probabilité d'excitation est importante (environ 20 %) et rend possible l'observation de ce nouvel effet Casimir-Polder dynamique. Don cnous proposons une con guration expérimentale réaliste pour réaliser ce système fait d'un gaz d'atomes froids piégés mis en face d'un substrat semi-conducteur, dont les propriétés diélectriques sont modulées dans le temps.

En n, nous nous concentrons sur nos résultats obtenus pour le calcul de la pression Casimir-Lifshitz entre deux réseaux lamellaires diélectriques di érents. Ce système est supposé dans une con guration hors équilibre thermique. En fait, les deux réseaux présentent deux températures différentes et ils sont immergés dans un bain thermique ayant une troisième température. Le calcul de la pression est basé sur une méthode qui exploite les opérateurs de di usion des réseaux, déduits en utilisant la méthode modale de Fourier. Nous présentons nos résultats numériques caractérisant en détail le comportement de la pression, en faisant varier les trois températures et en modi ant les paramètres géométriques des réseaux. Cette variation des paramètres du système permet de régler la force de répulsive à attractive ou de réduire fortement la pression pour des intervalles de températures. En outre, on montre que la combinaison des e ets de non-équilibre et géométriques rend ce système particulièrement intéressant pour l'observation de la force de Casimir répulsive.

1 Casimir forces 1.1 van der Waals force between neutral molecules . . . . . . . . . . . . . . . 1.1.1 Resonance interaction between molecules . . . .

Contents

Introduction I

Introduction

In this thesis we discuss the work and the results obtained during the PhD in a joint supervision program between Università degli Studi di Palermo, Italy, and Université de Montpellier, France. The topic of the thesis concerns with the Casimir forces in conditions where the system is out of the equilibrium, dynamical or thermal, in the framework of Quantum Electrodynamics. Casimir forces are interactions among neutral arbitrary objects (atoms/molecules or macroscopic bodies) due to the fact that the quantum "vacuum is not empty". The latter statement could seem an oxymoron or a joke of a comedian but it is instead a direct consequence of the quantum nature of the elds. In fact, an important implication of eld quantization is that, also when no external elds are present and the system is at zero temperature, uctuations of the elds exist. In other words, while the quantum average of the elds is zero in the vacuum state, the expectation value of the squared elds is not null and virtual quanta of the eld are always present. The interaction among these virtual quanta (photons in the case of the electromagnetic eld) and the objects in the system generates the Casimir force. Casimir interactions are usually very weak but, nonetheless, they are observable and, recently, they have been measured with remarkable precision in many di erent systems. In the last 60 years many scientists have studied the Casimir forces and there have been many e orts to better understand these interactions and making them detectable in increasingly realistic conditions. Several geometrical con gurations have been studied and the peculiar feature of a strong geometry dependence of these interactions has been revealed as well as a dependence from the magnetodielectric properties of the objects. An important system that we want to mention is when di raction gratings are present. The importance of the geometry on the Casimir force has been showed and experimentally observed in recent years. In particular, in 2013, using a con guration made of a metallic sphere in front of a metallic lamellar grating, a new unexpected regime in the Casimir force has been observed.

In the last 10 years new behaviors of these interactions have been found, speci cally when the systems under scrutiny are in an out of equilibrium, because of dynamic boundary conditions or di erent temperatures among the objects composing the physical system. For example, it has been predicted and observed that when the system is in an out of thermal equilibrium, there is the possibility to observe a repulsive Casimir force, contrarily to typical Casimir interactions at equilibrium. Similarly, when the dynamical atom-wall Casimir-Polder interaction is studied under non-adiabatic conditions, it has been found that the atom-wall force shows an oscillating attractive to repulsive character.

In the work discussed in this thesis, inspired by the results mentioned above, we have in-I vestigated Casimir and Casimir-Polder forces in these out of equilibrium scenarios (thermal and dynamical). The main reasons behind our studies are two: fundamental theoretical research and possible experimental and/or technological applications. The fundamental aim of our work is to increase the knowledge in one of the most important theory in physics, Quantum Electrodynamics, and related e ects. This goal can be well described from the following words of the Nobel Laureate in Physics Claude Cohen-Tannoudji in an interview at the conference "La scienza della materia in Italia: idee e progetti di nuova organizzazione", September 17th 2007, Rome: "... I think that is important the applied research, the research that leads to the development of new things, but the core of the research is in basic research. All modern applications that have changed our lives, from the laser to the transistors, are the result of fundamental research. The problem is that we cannot establish of something before we deal with it. The researchers who worked on the laser did not know that the latter would have been applicable in several important elds, they did not know exactly what have been the appropriate usability, and only later they realized it would have been useful. The important thing, in my opinion, is to do quality research, if applied or fundamental research is only a secondary problem... "

Possible experimental and technological applications also inspired our work presented here. In fact, in relatively recent years, it has been shown that Casimir forces are particularly important in micro and nano-technology. They are relevant for the functioning of technological systems such as micro-electromechanical and nano-electromechanical devices. Moreover, Casimir forces may also have a fundamental role in a problematic phenomenon present in some micro and nano-devices, the so-called "stiction", an e ect appearing when two surfaces, at a very small distance from each other, come into close proximity and are eventually led to permanently adhere determining the breakdown of the device. Then, studying new out of equilibrium conditions and new possible geometries could permit to control such e ects and tune the force by appropriately changing some parameters of the physical system. This thesis is structured as follows. In Chapter 1 we discuss the background for our original work, that is main aspects of Casimir forces. After a brief introduction on Casimir Physics we analyze the microscopic manifestations of these forces, i.e. the van der Waals and the retarded Casimir-Polder interaction between two atoms. After the derivation of their general expressions, and an analysis of the two regimes of near and far zone, we move on discussing these interactions when macroscopic objects are present. We describe the Casimir-Polder atom-wall interaction, and then derive a general formula for the Casimir e ect between two dielectric half-spaces. We also discuss a physical intuitive picture to explain and derive the Casimir force between two perfectly conducting slabs based on vacuum radiation pressure. Finally, we shortly outline the current status of the experimental observations of the Casimir-e ect, mentioning some of the experiments historically important in the literature on Casimir physics.

From Quantum Electrodynamics it is known that the electromagnetic vacuum is equivalent to an in nite ensemble of ground-state harmonic oscillators [1,2]. The so-called zero-point energy, i.e. the energy of the electromagnetic eld when there are not real photons in the system is E 0 = 1/2 k λ ω k where ω k and k are, respectively, the frequency and the wavevector of the photon while λ is its polarization. Since the sum is over an in nite set of possible wavevectors, it is evident that this zero-point energy is in nite. Nevertheless the zero-point energy and its uctuations (which are related to the commutation rules of the electric and the magnetic elds predicted from Quantum Electrodynamics) are responsible for several observable e ects, as example the Casimir forces and the Lamb shift.

Since the late nineteenth century, thanks to the work of van der Waals, it is well known that an attractive force between two neutral and non-polar molecules exists [3]. This force is not an electrostatic force because the molecules does not have neither a net charge nor a permanent multipolar moment. Later, works by Keesom, Debye and London con rmed the results of van der Waals obtaining the same characteristic dependence of the interaction from the intermolecule distance [4]. In 1948, with a fully quantum approach and including also retarded e ects, Casimir and Polder studied the interactions between two neutral atoms giving a more fundamental explanation of the existence of these forces [5]. The conclusion (iv)

Figure 1.1 -Typical Feynman diagrams that must be considered when we treat the electrodynamic interaction between molecules. The arrow indicates that time ows upwards.

that follows is that vacuum uctuations are responsible of the force since it is generated from the common interaction of the atoms/molecules with the zero-point electromagnetic eld. In the same year, Casimir [6] found a related e ect, which can be interpreted as a direct consequence of the vacuum uctuations, describing an attractive force between two conducting neutral perfectly mirrors at zero temperature in the vacuum space. Successively, in 1956, the Casimir e ect was generalized by Lifshitz to bodies having arbitrary optical properties and at nite temperature [7].

In this Chapter we present these e ects, that we encase in the expression Casimir forces [8,9]. In Section 1.1, we describe the expressions for the Casimir-Polder (retarded van der Waals) force between two atoms starting from the resonance interaction between two neutral atoms where one of the two atoms is in an excited state. Next we deal with the case of two atoms in the ground state at zero temperature and we distinguish the two limiting cases of near and far zone. We also consider the case of the thermal force. In Section 1.2, we present the Casimir forces when macroscopic bodies are considered. After a simple derivation of the atom-wall force, we introduce the Lifshitz formula and analyze the force between two perfectly conducting plate, giving an intuitive physical interpretation for this force. We conclude the Chapter giving in Section 1.3 an outline of the current situation of the experiments for measuring Casimir forces.

van der Waals force between neutral molecules

In this Section we focus on the van der Waals forces for atoms/molecules which do not have a permanent dipole moment (possible interactions due to quadrupole or higher multipole moments will be neglected) and we consider retardation e ects (i.e. e ects related to the nite speed of the light). We use the multipolar coupling formalism, which means that the interactions between the molecules are due to the exchange of transverse virtual photons. In Figure 1 Figure 1.1 is not considered in our calculations because there is not any photon exchange between the molecules, it does not describe an interaction between them. Also the diagram (ii) of Figure 1.1 is neglected because no external electromagnetic eld is supposed. As a consequence, we will focus on the diagrams like those in Figure 1.1(iii) and 1.1(iv). When also the electrostatic interactions are considered there are two more kinds of diagrams that describe the interaction between the two molecules (see Figure 1.2). These, in the electric dipole approximation, give the following interaction energy [10] V AB =

µ A i µ B j R 3 δ i j -3 Ri R j (1.1)
where µ µ µ i (i = A, B) are the electric dipole moments and R the intermolecular distance vector.

A CGS system of units (where not otherwise speci ed) is used from now on. Equation (1.1) will be used in Chapter 3 to describe the interaction between a dipole and its image dipole.

Resonance interaction between molecules

We now consider the resonance interaction between atoms/molecules. As we already said, this is the electromagnetic interaction between two neutral polarizable atoms/molecules where one of the two atoms is not in the ground state. We label with A and B the two identical molecules having an energy E g in the ground state E g and an energy E e in the generic excited state |E e . As a consequence, the generic states of the system |E A g , E B e and |E A e , E B g are degenerate with energy E g + E e . Let us suppose that in the initial condition the molecule A is in the excited state while the molecule B is in the ground state. In this system the excitation will oscillate from A to B and vice versa (see for example [11], [10] where a two level system is analyzed). We can consider the following symmetric and antisymmetric states, where the excitation is delocalized between the two atoms,

|Ψ ± = 1 √ 2 |E A e , E B g ± |E A g , E B e . (1.2)
We also consider the interaction Hamiltonian H int in the Coulomb gauge and in the multipolar coupling scheme [10], [12]. In the electric dipole approximation, we have

H int = -µ µ µ(A) • E(R A ) -µ µ µ(B) • E(R B ) (1.3)
where we used E instead of D ⊥ because, outside the atoms, the transverse displacement eld coincides with the total (longitudinal plus transversal) eld. The explicit expression for the electric eld operator is

E(r) = k,λ i 2π ck V 1/2 ǫk λ a k λ e ik•r -ǫ * k λ a † k λ e -ik•r (1.4)
where ǫk λ is the polarization unit vector, while a k λ and a † k λ are, respectively, the annihilation and creation eld operators. We want to calculate the resonance interaction with the help of the perturbation theory. As we can see from Feynman diagrams, the possible diagrams describing the resonance interaction have at least two vertices and then we need at least a second-order expansion in the perturbation theory (M f i is the second-order amplitude for the energy transfer between the two atoms)

M f i = I f |H int |I I|H int |i E i -E I .
(1.5)

In the above equation, assuming the eld in its vacuum state |0 with no photons, we denoted with |i and | f the following initial and nal states

|i = |E A e , E B g ; 0 , | f = |E A g , E B e ; 0 . (1.6)
The intermediate states |I are shown in Figure 1.3. The matrix elements of the interaction Hamiltonian related to the diagram (i) are k, Λ

I|H int |i = 1 k,λ ; E B g , E A g |H int |E A e , E B g ; 0 = i k,λ 2π ck V 1/2 ǫ * i,k λ µ ge i (A) e -ik•R A , (1.7)
f |H int |I = 0; E B e , E A g |H int |E A g , E B g ; 1 k,λ = -i k,λ 2π ck V
Figure 1.3 -Feynman diagrams involved in the calculation of the resonance interaction in the dipole approximation.

where we have introduced, respectively, the electric dipole matrix elements µ ge and µ eg = µ ge * between the excited states and the ground states and vice versa. Then, the contribution to M given from the diagram (i) is

k,λ 2π ck V ǫ * k λ ǫk λ µ ge i (A) µ eg j (B) e ik•R E eg -ck (1.9)
where R = R B -R A is the intermolecular distance and E eg = E e -E g is proportional to the atomic transition frequency. Proceeding similarly for the contribution of the diagram (ii), we nally get

M = k,λ 2π ck V ǫ * i,k λ ǫj,k λ µ ge i (A) µ eg j (B) e ik•R E eg -ck + µ ge j (A) µ eg i (B) e -ik•R -E eg -ck . (1.10)
Exploiting the following property of the polarization unit vectors λ ǫi,k λ ǫ * j,k λ = δ i j -ki k j (1.11) and transforming the sum over k in an integral

k -→ V (2π) 3 k 2 dk dΩ, (1.12)
after some algebra and appropriate regularizations of the frequency integrals, we nally obtain M = µ ge i (A) µ eg j (B) V i j (K e R) (1.13) where V i j (K e R) is the retarded potential tensor (see [13])

V i j (K e R) = 1 R 3 δ i j -3 Ri Rj cos(K e R) + K e R sin(K e R)δ i j -Ri Rj K 2 e R 2 cos(K e R) (1.14) and K e = E eg / c. We want to stress that the equation (1.14) is formally identical to the classical interaction, averaged in an oscillation cycle, between two dipoles oscillating at the same frequency.

Let us analyze the resonance interaction energy given by (1.13). In the limit of small intermolecular distances (near zone), i.e. for K e R ≪ 1 we recover the Coulombian interaction (1.1) between two static dipoles with a dependence with R -3 [14]. The opposite limit of large distance (K e R ≫ 1, far zone) the signi cant term in (1.14) has a dependence with R -1 . This is a quantum e ect leading to a slower decreasing interaction between the atomic dipoles in the far zone, in the case of correlated states.

Dispersion interaction between molecules

We study in this subsection the dispersion interaction between two molecules both in their ground state. The result that we will now show takes into account the e ects related to nite speed of light. The consequence of this is that, at relatively large distances, the interaction changes its dependence with the distance from R -6 (van der Waals) to R -7 (Casimir-Polder regime) [5].

Di erently from the case of the resonance interaction we need a fourth-order expansion in the perturbative theory, as it is possible to see analyzing the Feynman diagrams. The second-order terms do not give contribution to the interatomic energy but only to the Lamb shift of the atoms and related e ects [12], [15], [16]. The fourth-order interaction energy shift is given by

∆E = I,II,III 0|H int |III III|H int |II II|H int |I I|H int |0 E g -E I E g -E II E g -E III - I,II 0|H int |II II|H int |0 0|H int |I I|H int |0 E g -E I 2 E g -E II
.

(1.15)

According to perturbation theory developed by Rayleigh and Schrödinger, the second term in the above equation does not give any contribution for non-polar molecules so we focus only to the rst term. Again, we use the multipolar Hamiltonian (1.3) in the electric dipole approximation. Atoms A and B, as mentioned, are both in their ground state. As an example, the states involved in the above equation are in the form

|0 = |E A g , E B g ; 0 , |I = |E A r , E B g ; 1 k ′ , |II = |E A g , E B g ; 1 k,λ , 1 k ′ ,λ ′ , |III = |E A r , E B g ; 1 k ′ ,λ .
The unperturbed state denoted with |0 describes the state where both molecules are in the ground state g and the electromagnetic eld is in its vacuum state. The possible intermediate states are represented in Figure 1.4. Let us consider, for example, the contribution to the energy shift given by the diagram (i) of Figure 1.4. Starting from

0|H int |III = -i 2π ck V 1/2
ǫa,k λ µ gs a (B) e ik•R B ,

III|H int |II = -i 2π ck ′ V 1/2 ǫb,k ′ λ ′ µ sg b (B) e ik ′ •R B , II|H int |I = i 2π ck V 1/2 ǫ * i,k λ µ gr i (A) e -ik•R A , I|H int |0 = i 2π ck ′ V 1/2 ǫ * j,k ′ λ ′ µ rg j (A) e -ik ′ •R A (1.16)
we obtain, using (1.11), the following contribution

- k,k ′ r, s 2π ck V 2π ck ′ V δ ia -ki ka δ jb -k′ j k′ b × × µ gr i µ rg j µ gs a µ sg b e i(k+k ′ )•R E sg + ck ( ck + ck ′ ) E rg + ck ′ (1.17)
where we omitted the labels (A) and (B) for the electric dipoles µ µ µ (this is because the states r and s are unequivocally related, respectively, to A and B). For the other contributions of the diagrams in Figure 1.4 the procedure is analogous. If we consider that making the substitution k → -k and/or k ′ → -k ′ , the exponential remain e i(k+k ′ )•R , the di erence between the various terms are due only to the denominator in (1.17). These denominators D l are listed in Table 1.1. For each diagram the contribution is

- k,k ′ r, s 2π ck V 2π ck ′ V δ ia -ki ka δ jb -k′ j k′ b µ gr i µ rg j µ gs a µ sg b e i(k+k ′ )•R D l . (1.18)
It is worth to note that the energies of the intermediate states are higher than the energy of the ground state and so the energy shift is negative. This can be interpreted as the molecules take energy from the quantum vacuum for a short period related to the Heisenberg time-energy uncertainty principle (∆E ∆t ≥ ), so generating an attractive interaction.

We now need to sum over all contributions (1.18). As in the resonance case, transforming the sum over k and k ′ into integrals and using (1.11), after some algebra the energy shift can be cast as

∆E = - c πR 2 ∞ 0 α(A; iu)α(B; iu) 1 + 2 uR + 5 u 2 R 2 + 6 u 3 R 3 + 3 u 4 R 4 u 4 e -2uR du (1.19)
which is the result obtained by Casimir and Polder [5]. In the above equation we supposed that the molecules can rotate in every direction with the same probability and we (1.20)

We now analyze the dispersion interaction in the two limit cases of near zone and far zone.

For small distances k eg R ≪ 1 (near zone) the virtual photons which signi cantly contribute to the energy shift are those with energy larger than the typical transition energies of the molecules of the system (that means high k and k ′ ). This statement can be understood by means of the energy-time uncertainty principle. In fact, from this principle it follows that high-energy virtual photons are related to short "life-time"; so these photons do not travel for large distances from the atoms before being reabsorbed; thus they are important for the interaction at small distances and negligible for larger distances. From this analysis we deduce that only some denominators D l (i.e. only some diagrams) contributes signi cantly. These denominators are the smaller ones. In particular the denominators which give relevant contribution to the interaction in near zone are D iii , D iv , D ix , D x because of the presence of the factor (E rg + E sg ). In this way we can rewrite the shift energy obtaining

∆E ≃ -4 k,k ′ r, s 2π ck V 2π ck ′ V δ ia -ki ka δ jb -k′ j k′ b µ gr i µ rg j µ gs a µ sg b × e i(k+k ′ )•R ( ck) ( ck ′ ) E rg + E sg (1.21)
where the factor 4 is due to the fact that the four mentioned denominators under scrutiny give equal contributions. In the continuum limit we get ∆E ≃ -4 π 2 r,s µ gr i µ rg j µ gs a µ sg b E rg + E sg δ ia -ki ka δ jb -k′

j k′ b e i k•R e i k ′ •R d 3 k 4π d 3 k ′ 4π . (1.22)
After the angular integration and the integration over k and k ′ we can write

∆E ≃ - 1 R 6 r,s
µ gr i µ rg j µ gs a µ sg b E rg + E sg δ ia -Ri Ra δ jb -R j Rb (1.23) This result is identical to that deduced considering a second-order perturbative expansion of the interaction Hamiltonian (1.1) which describes the interaction between two static dipoles. If, for example, we consider that the molecules freely rotate in every direction, the equation (1.23) becomes (1.24) which has the typical R -6 dependence of London/van der Waals forces. A similar expression of the above equation but written in terms of atomic polarizabilities can be deduced using the following identity

∆E = - 2 3 R 6 r,s |µ µ µ rg | 2 |µ µ µ sg | 2 E rg + E sg
1 E rg + E sg = 1 π +∞ -∞ E rg E sg E 2 rg + u 2 E 2 sg + u 2 du. (1.25)
As a consequence, using the de nition of the dynamical polarizability in (1.20), introducing the imaginary frequency u = -i ω, we nally get

∆E = - 3 c 2 π R 6 +∞ -∞
α A (iu) α B (iu) du.

(1.26)

Let us consider now the far zone case, i.e. k eg R ≫ 1. In this regime, with the same considerations done before for the near zone regime, the frequencies k e k ′ of the virtual photons which contribute signi cantly to the energy shift are much smaller than the frequency involved in the molecule transitions. Analyzing the denominators in Table 1.1, only the diagrams (and then the relative denominators) (i), (ii), (vii), (viii) give a signi cant contribution to the energy shift in the far zone. The reason is that the presence of the factor ( ck + ck ′ ) implies a more important contribution to ∆E than the remaining terms. The denominators relative to these diagrams are rewritten as E rg E sg ( ck + ck ′ ). In this way we can get the energy shift

∆E ≃ -4 k,k ′ r, s 2π ck V 2π ck ′ V δ ia -ki ka δ jb -k′ j k′ b µ gr i µ rg j µ gs a µ sg b × × e i(k+k ′ )•R E rg E sg ( ck + ck ′ ) (1.27)
where the factor 4 is due to the identical contribution given from each diagram considered. Again,we assume that the molecules are freely rotating, so

∆E = - 4 9 k,k ′ r, s 2π ck V 2π ck ′ V 1 + k • k′ 2 |µ µ µ rg | 2 |µ µ µ sg | 2 E rg E sg e i(k+k ′ )•R ( ck + ck ′ )
.

(1.28)

We introduce the isotropic static polarizability of the two atoms

α(A) = 2 3 r |µ µ µ rg | 2 E rg , α(B) = 2 3 s |µ µ µ sg | 2 E sg (1.29)
and going to the continuum limit, the energy shift becomes

∆E ≃ - c (2π) 4 α(A) α(B) 1 + k • k′ 2 k 3 k ′3 e i(k+k ′ )•R (k + k ′ ) dk dk ′ dΩ dΩ ′ . (1.30)
After some standard algebraic calculation we nally obtain

∆E ≃ - 16 c π 2 α(A) α(B) R 7 ∞ 0 1 η 2 + 1 6 3η 4 -2η 2 + 3 dη ≃ - 23 c 4 π α(A) α(B) R 7 (1.31)
i.e. the standard attractive Casimir-Polder force having the well known R -7 dependence. This shows that, at large distance, the potential decreases more rapidly with the distance compared to the van der Waals potential [5].

Casimir-Polder interaction between two atoms at nonzero temperature

In the previous sections we discussed the Casimir-Polder interaction between molecules at zero temperature. However, these interactions are also present when the atoms/molecules interact with thermal radiation at a temperature di erent from zero. This means that real thermal photons are present in addition to the virtual ones. Thus, considering the Feynman diagrams in Figure 1.1, we must analyze also processes like that depicted in the diagram of Figure 1.1(ii). Using the perturbative approach adopted in the previous sections we can again calculate the Casimir-Polder force and obtain the energy shift due to the atom-eld interaction. A condition to obtain this is the preparation of the uncoupled system in an energy eigenstate. A possibility is an incoherent superposition of energy eigenstates |ψ with probabilities p ψ described by the density matrix

ρ = ψ p ψ |ψ ψ|. (1.32)
The energy shift is then de ned as

∆E = ψ p ψ ψ|H int |ψ + ψ p ψ I ψ ψ|H int |I I|H int |ψ E ψ -E I + . . . (1.33)
Starting from this energy shift it is possible to calculate the Casimir-Polder interaction between two atoms at nonzero temperature [17]. However, here we want to show another approach to perform this calculation. There are many alternative models to explain the dispersion forces (see for example [18]- [23]), but our choice in this section falls on a simple and intuitive model. The key idea of this approach is the following: the eld uctuations induce instantaneous dipole moments in the two atoms, which are correlated because vacuum uctuations are spatially correlated; the Casimir-Polder potential energy then arises from the classical interaction between the oscillating dipoles of the atoms induced and correlated by the vacuum uctuations [24].

The induced dipole moments in the atoms are related to the uctuating quantum eld by the following relation (assuming isotropic atoms)

µ l (k λ ) = α(k)E l (k λ , r) (1.34)
where we used the dynamical polarizability α(k) of the atoms introduced in (1.20). In the above equation E l (k λ , r) is the l component of the k λ Fourier component of the electric eld operator de ned in (1.4). In the multipolar coupling it coincides with the transverse displacement eld D l (k λ , r) [12]. The Casimir-Polder interaction energy in this approach is the following

W AB (R) = k,λ µ A l (k λ )µ B m (k λ ) V lm (R) k,λ α A (k)α B (k) E l (k λ , r A )E m (k λ , r B ) V lm (k, R) (1.35)
where R = |r Br A | is the interatomic distance while V lm (k, R) is the classical electromagnetic potential tensor, identical to that obtained in (1.14), between two oscillating dipoles at frequency ω = ck.

As we can see in (1.35), we need to evaluate the spatial correlation function of the electric eld E l (k λ , r A )E m (k λ , r B ) , that should be calculated on the state of the eld that we are analyzing, i.e. the equilibrium thermal state at temperature T . This means that

a † k λ a k λ = 1 e kc/k B T -1 (1.36)
where k B is the Boltzmann constant. At this stage we have made the assumption k B T ≪ ω 0 , where ω 0 is a typical transition frequency of the atoms. In fact, we are considering situations where the atomic excitation is negligible. In a way similar to the cases previously considered, we go to the continuum limit exploiting the equation (1.36), and after some algebraic calculations we have

λ dΩ k E l (k λ , r A )E m (k λ , r B ) = 8π 2 ck V coth ck 2k B T τ lm (k, R) (1.37)
where we have introduced the tensor

τ lm (kR) ≡ 1 4π δ lm -kl km e ±ik•R dΩ = = δ lm -3 Rl Rm cos(kR) k 2 R 2 - sin(kR) k 3 R 3 + δ lm -Rl Rm sin(kR) kR . (1.38)
Finally, after some algebraic manipulations, we obtain the general expression for the Casimir-Polder interaction energy [25] W

AB (R) = - c πR 3 dk k 3 α A (k) α B (k) coth ck 2k B T × kR sin (2kR) + 2 cos(2kR) -5 sin(2kR) kR -6 cos(2kR) k 2 R 2 + 3 sin(2kR) k 3 R 3 . (1.39)
We notice that the temperature dependence of this interaction energy is inside the hyperbolic cotangent. Let us analyze this expression considering the two limiting cases of near and far zone.

In the near zone, since kR ≪ 1, equation (1.39) reduces to

W AB (R) = - 3 c πR 6 dk α A (k) α B (k) coth ck 2k B T sin (2kR) (1.40)
which is identical to the expression obtained, for example, in [26].

In the far-zone regime we can use the static polarizabilities α A,B = α A,B (0) introduced in (1.29) in place of the dynamical polarizabilities, and after the k integration we nally get

W AB (R) = α A α B k B T D R coth 2πk B T R c (1.41)
where we have de ned the following di erential operator

D r ≡ - 1 16r 2 ∂ 4 ∂r 4 + 1 4r 3 ∂ 3 ∂r 3 - 5 4r 4 ∂ 2 ∂r 2 + 3 r 5 ∂ ∂r - 3 r 6 . (1.42)
From this expression two additional regimes of the far zone can be considered. A lowtemperature regime 2π k B T R/ c ≪ 1 and an high-temperature regime

2π k B T R/ c ≫ 1.
This means that a new distance scale λ = c/2π k B T appears when we consider the far zone case. For distance smaller than this distance scale, the Casimir-Polder energy in (1.41) has a distance-dependence as R -7 , the same of the Casimir-Polder far-zone interaction at zero temperature; for distances larger than the new distance scale, the interaction energy has a R -6 dependence reproducing the same dependence of the Casimir-Polder near-zone interaction at zero temperature [27] - [29].

Macroscopic manifestation of the vacuum uctuations

We have seen that when we consider the atom-eld interaction, even in the absence of real photons in the system, interactions between neutral atoms/molecules are originated. Until now we have studied molecular interactions so we have restricted our analysis to a microscopic point of view. The natural macroscopic consequence of these microscopic interactions is the well known and startling Casimir e ect [6], [30]. According to it, two perfectly conducting slabs, which have null net charge, placed in the vacuum feel an attractive force, as a consequence of the vacuum uctuations. The Casimir e ect was then considered as an evidence of the existence of the quantum vacuum and of the related zero-point energy. However, Milonni and Schwinger obtained the same Casimir results without explicitly using the assumption of the vacuum uctuations, but from the source elds [31], [32], [33] (see also the references [34], [35]). The common conclusion is that these e ects are strictly related to the quantum nature of the system, the quantum nature of the matter for the source-eld picture and the quantum nature of the elds in the vacuum uctuation picture. They are then important cornerstone of Quantum Mechanics. Another point worth stressing is that, even if the Casimir e ect can be considered as a macroscopic manifestation of the microscopic Casimir-Polder forces, it cannot be simply obtained as a sum of the van der Waals forces acting on the atoms composing the macroscopic bodies. The reason is related to the non-additive property of the van der Waals and Casimir-Polder forces [1,36,37].

In this section we present two examples of Casimir forces involving macroscopic bodies: the atom-wall Casimir-Polder interaction and the Casimir force between two parallel plates.

The atom-wall Casimir-Polder interaction

Let us analyze the physical system of an atom in front of a perfectly conducting wall in the vacuum. In the regime of small atom-wall distance d (relative to a typical wavelength transition of the atom) the interaction can be simply obtained considering the dipole-dipole interaction between the dipole and its image on the other side of the wall. This because, when the distance is small, the interaction is instantaneous and the retardation e ects are negligible. Using the potential introduced in (1.1), we nd that the energy interaction has a d -3 dependence. In the far zone, i.e. large atom-wall distance, the distance dependence of the interaction is di erent and the retardation e ects become relevant, as for the far zone case of the two-atom system.

A simple derivation of the interaction energy for this case can be easily deduced from the classical potential energy of an electric dipole U = -p • E. We denoted with p the electric dipole moment related to the electric eld through the relation p = αE, where α is the static polarizability. From this potential energy we can write the interaction energy

W = - 1 2 αE 2 (1.43)
that found a formal quantum support in [38], [39]. We have inserted a factor 1/2 in the above equation because we are considering that the dipole moment is induced rather than permanent [14].

Equation (1.43) can be generalized to the Stark shift in the atom-wall system

W = - 1 2 k,λ α(ω)E 2 k λ (r A ). (1.44)
The quantities k and λ are, respectively, the wave vector and the polarization index and E k λ (r A ) is the k λ Fourier component of the electric eld evaluated at the position r A of the atom. The presence of the perfect conducting wall in the system impose speci c boundary conditions, thus changing the vacuum eld. Let us consider rst a rectangular box with perfectly conducting surfaces, located in the half space opposite to that where the atom is placed. The sides of this parallelepiped are L x = L y = L and L z . In this situation, the components of the electric eld satisfying the speci c boundary conditions are

E x (r) = 16π ω V 1/2 ǫ x cos(k x x) sin(k y y) sin(k z z), E y (r) = 16π ω V 1/2 ǫ y sin(k x x) cos(k y y) sin(k z z), E z (r) = 16π ω V 1/2 ǫ z sin(k x x) sin(k y y) cos(k z z). (1.45)
In the above equation we used that ǫ

2 x + ǫ 2 y + ǫ 2 z = 1, quantization volume V = L 2 L z and k x = lπ L , k y = mπ L , k z = nπ L z (1.46)
where l, m, n ∈ N. Since we are in far zone regime we can replace the dynamical polarizability in (1.44) with the static one α(0). Then, calculating the electric eld at the atom position (L/2, L/2, d), (1.44) can be cast as

W(d) = - 1 2 α k 16πω k V ǫ 2 x cos 2 k x L 2 sin 2 k y L 2 sin 2 (k z z) + ǫ 2 y sin 2 k x L 2 cos 2 k y L 2 sin 2 (k z z) + ǫ 2 z sin 2 k x L 2 sin 2 k y L 2 cos 2 (k z z) .
(1.47)

In the limit L → ∞, the sinusoidal functions related to the components k x and k y can be replaced with their average value 1/2 because they are rapidly oscillating. Then

W(d) = - 2π α V k ω k ǫ 2 x + ǫ 2 y sin 2 (k z d) + ǫ 2 z cos 2 (k z d) . (1.48)
We can de ne the interaction potential V(d) as

V(d) = W(d) -W(∞).
(1.49)

In the limit d → ∞, also the k z component in the sinusoidal functions in (1.48) can be replaced with their average value, obtaining

W(∞) = - 2π α V k ω k 1 2 ǫ 2 x + ǫ 2 y + ǫ 2 z = - 2π α V k 1 2 ω k . (1.50)
We deduce then the potential energy

V(d) = - 2π α V k ω k sin 2 (k z d) - 1 2 ǫ 2 x + ǫ 2 y -ǫ 2 z = π α V k ω k 2k 2 z k 2 cos(2k z d).
(1.51)

In the continuum limit of allowed k, we nally get

V(d) = 2π α V V 8π 3 d 3 k k 2 z k 2 cos(2k z d) = α c 2π +∞ 0 dk k 3 2π 0 dθ sin θ cos 2 θ cos(2k cos θd) = - 3α c 8πd 4 . (1.52)
This is the same identical result obtained originally by Casimir in the far zone [6]. From (1.52), we notice that in the far zone, because of retardation e ects, the distance-dependence of the atom-wall Casimir-Polder potential is as d -4 while in the near zone it is as d -3 . The atom-wall force can then be obtained by taking the derivative of V(d) with respect to d, changed of sign.

The Casimir e ect

We now consider the Casimir e ect, originally obtained for the rst time by Casimir in [6], consisting in an attractive force between two perfectly conducting metallic plates. In this subsection, however, we want to present a more general formula analyzing a more general problem, studied for the rst time by Lifshitz [7], of two dielectric half-spaces, in thermal equilibrium at a nite temperature, placed in the vacuum and separated from a distance d.

The Casimir e ect for ideal metallic plates can be obtained as a limiting case dielectric → metal.

The Lifshitz calculation start from the Maxwell equations in the Fourier space

∇ × E = i ω c B (1.53) ∇ × B = -i ω c ε(ω)E -i ω c K (1.54)
where nonmagnetic dielectric media are considered. The dielectric constant considered in the two Maxwell equations above is in general complex, ε(ω) = ε ′ (ω) + iε ′′ (ω). This means that both absorption and dispersion are possible. Moreover, we introduced a random uctuating current K. These two quantities are strictly related. In fact the use of a complex dielectric permittivity in Lifshitz's theory requires a random current K. This is due to the uctuation-dissipation theorem. From this theorem when an absorbing medium is considered, the dissipative nature of the medium requires a balancing coming from a uctuating source in a way such that the correlations of the uctuating eld K are related to the dissipation. The correlation function that Lifshitz assumed is the following

K i (r, ω)K j (r ′ , ω) = 2 ε ′′ (ω)δ i j δ 3 (r -r ′ ).
(1.55)

The above correlation function was used by Lifshitz to perform the calculation of g i (k)g j (k ′ ) where g i (k) are the Fourier components of K and was used to write the solutions of the Maxwell equations for the eld. After some calculations Lifshitz obtained the following formula for the Casimir force

F(d) A = - 2π 2 c 3 +∞ 1 d pp 2 +∞ 0 dξξ 3 × ×       s 1 + ǫ 1 p s 1 -ǫ 1 p s 2 + ǫ 2 p s 2 -ǫ 2 p e 2ξ pd c -1 -1 + s 1 + p s 1 -p s 2 + p s 2 -p e 2ξ pd c -1 -1       (1.56)
where ǫ j = ǫ j (iω) and s j = p 2 -1 + ǫ j for j = 1, 2, being 1 and 2 the label for the two slabs. Taking the limit ǫ 1 , ǫ 2 → +∞, i.e. perfect conducting plates, the well known Casimir force obtained in [6] is recovered. The power of the formula (1.56) also allows to deduce the Casimir-Polder force between two atoms making the assumption of very rare ed media and exploiting the expression for the dielectric permittivity ǫ(ω) = 1 + 4πNα(ω) (where N denotes the number of atoms per unit volume). We want to discuss brie y a di erent approach than the Lifshitz one, which allows us to write a more general expression compared to the Lifshitz formula [8]. It is well known from classical electromagnetism that the force density f in a medium, when electromagnetic elds are present, is related to the electromagnetic stress tensor T i j

f i = ∂ j T i j , T i j = 1 4π E i D j + H i H j - 1 2 (E • D + H • H) δ i j ⇒ f i = 1 8π (∂ i E j )D j -E j (∂ i D j ) (1.57)
where we assume

1 4πc ∂ ∂t (D × H) = 0 (1.58)
i.e. that, under appropriate equilibrium conditions, the time variation of the momentum density of the eld can be considered zero, and we suppose an isotropic media so that the stress tensor T i j is symmetric. From a quantum point of view, we have to promote E i and D i to operators. Using the Fourier components of the eld, symmetrizing and taking the expectation values, the force density is

f i (r) = 1 8π Re +∞ -∞ dω +∞ -∞ dω ′ ∂ i E j (r, ω) D j (r, ω ′ ) -E j (r, ω) ∂ i D j (r, ω ′ ) .
(1.59)

If we consider the Maxwell equations used by Lifshitz in (1.53) and (1.54), the Fourier component of the transverse displacement eld is D(r, ω) = ε(r, ω)E(r, ω)+K(r, ω), that means

E j (r, ω) D j (r ′ , ω ′ ) = ε(r, ω) E j (r, ω) E j (r ′ , ω ′ ) + E j (r, ω) K j (r ′ , ω ′ ) . (1.60)
We introduce, the dyadic Green function such that

-∇ × ∇ × G(r, r ′ , ω) + ω 2 c 2 ε(r, ω) G(r, r ′ , ω) = -4π ω 2 c 2 δ 3 (r -r ′ ), (1.61)
and we introduce in the rst term of the right side of (1.60) the following form for the correlators [40]

E j (r, ω) E j (r ′ , ω ′ ) = π ℑ G j j (r, r ′ , ω)δ(ω -ω ′ ) . (1.62)
where

E j (r, ω) = 1 4π d 3 r ′′ G ji (r, r ′′ , ω)K i (r ′′ , ω). (1.63)
We also assume that, similarly to (1.55), the operator K(r, ω) has the following property, as a consequence of the uctuation-dissipation theorem,

K † i (r, ω)K j (r ′ , ω ′ ) = 0 (1.64) K i (r, ω)K † j (r ′ , ω ′ ) = 4 ε ′′ (ω)δ i j δ(ω -ω ′ )δ 3 (r -r ′ ), (1.65)
where we denoted with ε ′′ (ω) the imaginary part of ε(ω). Then we have

Re E j (r, ω) D j (r ′ , ω ′ ) = π ℑ ε(r ′ , ω) G j j (r, r ′ , ω)δ(ω -ω ′ ) (1.66)
and nally the density force reads

f i (r) = - 8π 2 ℑ +∞ 0 dω [∂ i ε(r, ω)] G j j (r, r ′ , ω). (1.67)
This is a more general formula for the Casimir force and in the case of two half-space dielectrics, it has been used to deduce the Lifshitz formula (1.56). It also allows, for example, to calculate the force when a third dielectric material is placed between the two half-spaces [31], [41] - [43]. Many other methods have been developed over the years to calculate the Casimir force between macroscopic objects and in particular, in this thesis, in Chapter 4 we will use a very general method, based on the scattering operators, which allows us to derive the force between arbitrary objects with generic optical properties even in out of thermal equilibrium conditions. can guess that the attractive e ects exceed the repulsive ones.

To obtain a quantitative evaluation we should analyze the radiation pressure due to a perpendicular plane wave impinging on a slab. This will be twice the energy density u. More precisely, if we consider an incidence angle θ the radiation pressure under study is P = 2u cos 2 θ. The square-cosine dependence is related to the fact that we have a factor cos θ deriving from the momentum dependence on the incidence angle, and another factor derives because the surface element A and incidence angle are not perpendicular, so that factor (cos θ) -1 arise. Each frequency mode ω = ck contributes to the radiation pressure and then

2 1 2 1 2 V ω cos 2 θ = ω 2V k 2 z k 2 (1.68)
where V represents the quantization volume. In the above equation we have also added a factor 1/2 which takes into account that for each mode k we can have incoming and outgoing propagation from the slabs and the energy is equal for each of these modes. If the slabs have in nite surfaces, we have a continuum spectrum for the x and y components of k while the values of k z are discrete

k z = nπ d (1.69)
with n ∈ N. Taking all the possible modes and summing each contribution to radiation pressure from the virtual photons in the internal region, we get the pressure acting on each slab

P int = c π 2 d ∞ n=1 ∞ 0 ∞ 0 dk x dk y nπ d 2 1 k 2 x + k 2 y + (nπ/d) 2
.

(1.70)

In (1.70) we have added a factor 2 due to the two di erent polarizations. The pressure P int , as we mentioned, generates a repulsive force while the pressure P ext , from the external virtual photons, generates an attraction between the slabs. In the external region all the components of k of the virtual photons have a continuous spectrum

P est = c π 3 ∞ 0 ∞ 0 ∞ 0 dk x dk y dk z k 2 z k 2 x + k 2 y + k 2 z . (1.71)
We notice that both P int and P ext are in nite but only their di erence has a physical meaning (i.e only the di erence is measurable), so we obtain

P int -P ext = π 2 c 4d 4        ∞ n=1 n 2 ∞ 0 dx 1 √ x 2 + n 2 - ∞ 0 du u ∞ 0 dx 1 √ x 2 + u 2        . (1.72)
With the help of the Eulero-Maclaurin formula [45], [46], after some algebra, we nally get the famous result originally obtained by Casimir for the force per unit area

P int -P ext = - π 2 c 240 d 4 . (1.73)
It must be said that, even if this approach allow us to give a simple and intuitive explanation of the Casimir e ect, it gives good results only when rectangular plates are considered. For some other geometries the force deduced by this approach is not satisfactory or even wrong [47] - [50].

Casimir forces and Experiments

In this section we discuss and outline the experiments done to measure the Casimir forces. Accurate measures of the Casimir forces in di erent realistic situations are still under study. In fact, these e ects are quite small when realistic conditions are considered and so the Casimir forces are very di cult to detect experimentally.

The situation is even more di cult when microscopic objects are involved in the experiment. In fact, the control of position, velocity, external forces acting on the atoms, etc... are all conditions that, when dealing with a microscopic system, are extremely di cult to obtain. Then is evident that there exists an intrinsic issue which makes extremely dicult the direct measurement of the van der Waals/Casimir-Polder forces. However indirect manifestations of these forces can be found in the measurement of macroscopic quantities, as for example suggested by van der Waals himself when he studied the modi cations induced by his eponymous force in the equation of state of a gas.

One research eld where accurate measurements of the van der Waals force has been obtained is about scattering experiments. There are two simple groups of experiments that can be made. In the rst it is studied one atomic beam that is scattered from a a stationary gas which serves as target. For this case, the van der Waals interactions between the atoms de ect the atomic beam when this pass through the target gas and an attenuation of the beam is detected [51] - [53]. In the second case the scattering experiment concern two crossed atomic beams. Also in this case, the van der Waals interaction between the atoms of the two beams in uences the scattering and the scattering spectrum of the incident beam as a function of the angle [54], [55].

The Casimir-Polder forces between an atom and a generic macroscopic body can be detected alternatively using scattering techniques. The main experimental con gurations used deal with classical scattering and quantum scattering. In the classical scattering the experimental con guration is similar to that described above for an atomic beam and a target. In this case the target is a macroscopic body and the atomic beam is directed near the target body and its de ection, due to the Casimir-Polder interaction is observed and analyzed [56], [57]. The experimental setup based on quantum scattering exploits the property, which follows from the de Broglie hypothesis, that the matter can have a wave-like behavior. Then, for su ciently small speeds, the re ection against a body of this matter-wave is studied. This re ection is certainly in uenced by the Casimir-Polder interactions which can be then measured [58,59]. The matter-wave behavior is also the basis for measuring the Casimir-Polder force by studying the di raction of these matter-waves by gratings [60] - [62].

Another e ective way to measure the Casimir-Polder forces is to consider trapped atoms. When the atom is trapped it still has a motion, in particular an oscillatory motion. When a surface is near to the trapped atom this motion is modi ed by the Casimir-Polder interaction and the latter can be revealed in appropriate experiments. Using this con guration the Casimir-Polder have been measured, as well as its temperature modi cation or the possible repulsion force in an out of thermal equilibrium situation [63] - [66]. Many other methods to detect and observe indirectly the Casimir-Polder forces have been developed, for example spectroscopy, and we refer to [67] and the references therein for more details.

We now discuss the situation about experiments for the measurement of the Casimir force on macroscopic objects. A simple estimation of the magnitude of the Casimir force between two perfectly conducting slabs gives

F(d) A = π 2 240 c d 4 = 0.013 d 4 dynes/cm 2 . (1.74)
This result gives a force of 10 -2 dyn (10 -7 N in SI units) for plates of area A = 1 cm 2 at distance d = 1 µm and the smallness of this values indicates why it is so di cult to measure such e ects. However, the Casimir e ect was detected for the rst time by Sparnaay, in 1958 [68], only a few years later the theoretical prediction by Casimir. Sparnaay detected an attractive force between the two slabs at a distance between 0.5 µm and 2 µm. The experimental setup was very simple. It was made of a plate attached to a cantilever spring.

Another plate was approached to the rst one with the help of ne screws. The Casimir force was deduced by measuring the bending of the spring via a capacitor connected with the plates. However, because of residual electrostatic forces, the large experimental errors did not allow him to state that the Casimir e ect was demonstrated but, as he said, the attractive force measured by him did not contradict the e ect predicted by Casimir. After that experiment many other attempts was made to demonstrate the Casimir e ect in the subsequent years.

Only in 1997 the Casimir e ect was experimentally demonstrated and accurately measured, almost 50 years later the theoretical prediction! The measurement was given thanks to Lamoreaux and his work [69]. In this experiment Lamoreaux used an electromechanical system based on a torsion pendulum made of a slab and a sphere which reciprocal distances was in the range between 0.6 µm and 6 µm (the plate-sphere geometry was used because it avoids the need for a perfect parallel alignment). He was able to measure the Casimir force achieving a declared accuracy of 5%. The experimental results were in excellent agreement with the theoretical prediction by Casimir.

Just one year later, Mohideen and Roy [70] with an experimental setup based on an Atomic Force Microscope (AFM) measured the Casimir force between a slab and a sphere in a distance range between 0.1 µm and 0.9 µm, achieving a measurement accuracy of 1%.

The measurement of the Casimir e ect in the con guration (parallel plates) proposed by Casimir himself was obtained only in relatively recent years. The reason is that the required perfect parallelism between the two slabs is very hard to achieve with good accuracy, particularly when the two plates are very close eac other In 2001, Bressi and collaborators [71] solved this issue aligning the slabs by means of microresonator with an accuracy un-der the micron. In this way they were able to measure the Casimir force between parallel metallic plates with an accuracy of 15% in a distance range of the slabs of 0.5-3.0 µm.

Measurement of the Casimir forces concerning various geometries have been done. In recent years, for example, the con guration with gratings has been considered. For example in Reference [72] Chan and collaborators measured the Casimir force between a gold sphere and a silicon nanostructured surface (rectangular grating). In this case, in a range of distances between 150 nm and 500 nm, it has been shown that the attractive force is signi cantly di erent from the pairwise additive approximation (the latter approximation is a simple geometric approximation which calculates the modi cation of the force, due to the presence of "imperfections" of the surface, exploiting the force of the slab-slab con guration). This result demonstrates the strong dependence of the Casimir force on the shape of the interacting bodies.

In another example of sphere-grating con guration [73], the strong dependence from geometry of the Casimir force is also more interesting. In fact, using a setup made of a gold sphere in front of a metallic grating instead of a dielectric grating, the Casimir force, at distance scales below the plasma wavelength, shows a new unexpected regime. When a deep metallic lamellar grating, with sub-100 nm features, is considered then, for large inter-surfaces separations, it was found that the force is reduced beyond what would be expected by any existing theoretical prediction. Also in view of the studies about the measurement of the Casimir force in realistic systems keeps on, going hand in hand with the possible technological applications which are, nowadays, always more concrete thanks to the micro-electromechanical (MEMS) devices. An important work concerning the Casimir force in these devices has been done by Chan, Aksyuk, Kleiman, Bishop and Capasso in [74] where they have showed that, when the distance between two surfaces is relatively small, the quantum e ects acting on these devices becomes relevant and not negligible for the proper functioning of the device. The experimental setup is similar to that of Lamoreaux and, also in this case, there is a system made of a sphere in front of a slab. The sphere can rotate because of the Casimir force, thus acting as a torsion pendulum.

Chapter 2 After having introduced the Casimir forces in the previous chapter, we consider some quantum electrodynamic e ects related to the uniform acceleration of atoms in vacuum and show original results that we obtained on Casimir-Polder forces for accelerating atoms. The aim of this chapter is to understand how the dynamical out-of-equilibrium-condition, dictated from the uniform acceleration of the atoms, can modify the Casimir-Polder forces (or equivalently van der Waals forces). The class of physical systems that we are going to analyze are strictly related with one of the most intriguing phenomena in quantum eld theory, the so-called Unruh e ect [1,2,3,4]. According to this e ect, a uniformly accelerated detector in the Minkowski quantum vacuum experiences a thermal bath at a temperature proportional to its acceleration. In qualitative terms, as we will see forward in a dedicated section, this phenomenon originates from time-dependent Doppler shifts of the eld detected by the accelerated detector [5]. Unfortunately, this e ect is very tiny, and there is not yet an experimental evidence of it. In fact, an acceleration of the order of 10 22 cm/s 2 would be necessary to obtain Unruh radiation corresponding to the temperature of 1 K [4]. The question of the perception of the quantum vacuum in accelerated frames remains a widely debated problem. In this chapter, after discussing brie y the physics behind the Unruh e ect, we discuss some e ects related to a uniform acceleration of atoms in the vacuum space, in the framework of quantum electrodynamics. After considering the radiative level shifts of a uniformly accelerated atom in vacuum, we focus on the atom-wall Casimir-Polder interaction for an accelerated atom, as well as on the van der Waals/Casimir-Polder interaction between two accelerating atoms. Also, we will discuss in detail the possibility of detecting the Unruh e ect through these phenomena. Since the Lamb-shift and the Casimir-Polder interactions are directly related to vacuum eld uctuations [6,7,8], we are particularly interested to investigate if thermal e ects due to the acceleration may produce observable e ects in such physical systems. These e ects can suggest new possibilities to detect the Unruh e ect through a measurement of the Casimir-Polder interactions for atoms accelerating in the vacuum space.

E
The chapter is organized as follows. In the introductory Sections 2.1 and 2.2, we present and give a simple physical insight of the Unruh e ect and shortly review some electrodynamic e ects due to accelerated atoms. In Section 2.3 we start to show our results investigating the atom-atom van der Waals/Casimir-Polder force between two accelerating atoms through a physical model based on the spatial correlations of vacuum electromagnetic eld and source eld. In Section 2.4 we discuss and develop in detail a general statistical procedure for the calculation of the Casimir-Polder forces, extending to the fourth-order a method introduced by Dalibard, Dupont-Roc and Cohen-Tannoudji for second order e ects. In Section 2.5 we exploit the new procedure that we developed in Section 2.4 to examine more rigorously the Casimir-Polder force between two ground-state atoms uniformly accelerating in the vacuum space. Finally in Section 2.6 we analyze the resonance interaction for two accelerated atoms, interacting with relativistic scalar eld or with the electromagnetic eld.

The Unruh e ect

We start brie y presenting the Unruh e ect, which plays a fundamental role in our original work presented in the following sections. It is well known, in the literature, that a detector uniformly accelerating in the vacuum space behaves as an identical static detector immersed in a thermal radiation at temperature T proportional to its acceleration. Before Unruh's work, other works in astrophysical problems had suggested thermal e ects of accelerating particles [9] - [12]. Hawking, in his studies on the black holes [13] - [14], predicted that a black hole radiates at a temperature T = g/2πk B c where g is the gravitational acceleration of the black hole and k B is the Boltzmann constant. To nd this result Hawking considered the e ect of the gravity on the vacuum scalar eld. Before the work of Hawking, Fulling had introduced an approach to deal with the quantum eld theory in curved spaces [2]. However Unruh and Davies, with their independent works [1]- [3], were the rst who directly showed the thermal e ects due to acceleration in the background of the quantum eld theory. They found that an observer moving with a uniform acceleration and interacting with the vacuum of the scalar eld in the Minkowski space-time, feels a temperature given by the following expression

T = a 2πck B (2.1)
where a is the observer's acceleration. Here a replaces the gravitational acceleration g in the analogous expression found by Hawking. This acceleration/temperature e ect is called the Unruh e ect or the Unruh-Davies e ect or also the Fulling-Unruh-Davies e ect. This e ect is a striking manifestation of the fact that the quantum vacuum, and its particle content, is observer dependent.

A more general result of the Unruh e ect, concerning not only with the scalar eld but also to more generic spinor and vector eld in a space-time of dimension d, has been found by Takagi [15]. When we will refer to (2.1) we will mean the Hawking-Unruh temperature because the results of Hawking and Unruh are identical on the basis of the equivalence principle and they di er only for the interpretation. In the rst case we have a stationary detector on the event-horizon of a black hole with gravitational acceleration g, while in the second case we have an accelerated detector with uniform acceleration a in the vacuum space.

The methods usually used to deduce (2.1) are not particularly intuitive (see, for example, [1] - [4], [9] - [14], [16] - [18]). The reason is that these works use a formalism related to the quantization of the eld in curved spaces. In this subsection we want to show the same famous result of Unruh using a simple and intuitive method proposed by Milonni and Alsing [5]. The key idea of this method is obtaining the Unruh e ect from considerations on the Doppler shift of the vacuum eld seen by an accelerated observer.

Let us suppose to have a plane wave with frequency ω k and wave vector k which propagates along the ±x direction. We also suppose that the uniformly accelerated Rindler observer moves along the positive direction x. In the latter statement we mean with Rindler observer an observer which is comoving with the particle in an instantaneously inertial reference frame. Since the observer is instantaneously inertial he will perceive the plane wave as having frequency ω ′ k related to ω k through the Lorentz transformation

ω ′ k (τ) = γ(τ) (ω k -k v(τ)) (2.2)
where we have assumed that the plane wave propagates along the positive x direction. Considered the motion of the observer we have

v(τ) = c tanh aτ c , γ(τ) = cosh aτ c (2.3)
and this means that

ω ′ k (τ) = cosh aτ c ω k -k c tanh aτ c . (2.4)
Since ω k = ck, the above equation reads

ω ′ k (τ) = ω k e -aτ/c . (2.5)
For a wave propagating along -x the sign of kv(τ) in (2.2) changes and we get

ω ′ k (τ) = ω k e aτ/c . (2.6)
We notice that if we assume aτ/c ≪ 1, i.e non-relativistic motion, and we expand to rst order the exponential in (2.5), (2.6) we obtain

ω ′ k (τ) ≃ ω k 1 ∓ aτ c (2.7)
which coincides with the non-relativistic Doppler e ect (aτ ≃ v). We want to stress the time-dependence of (2.5), (2.6) which implies, for the accelerated observer, a Doppler e ect which is explicitly dependent on time. Thus, because of the time-dependent Doppler e ect, at each time, the Rindler observer see the wave oscillating with a di erent time-dependent frequency. Then, the accelerated observer will see the spectrum frequency proportional to +∞ -∞ e iΩτ e iϕ(τ) dτ 2 (2.8) where

ϕ(τ) = τ 0 ω ′ k (τ ′ ) dτ ′ .
(2.9)

From equations (2.5) and (2.6) it follows that ϕ(τ) reads

ϕ ± (τ) = τ 0 ω ′ k (τ ′ ) dτ ′ = ω k c a e ±aτ/c
(2.10) (the symbol ± denotes the two di erent cases where the wave propagates along -x or x).

The frequency spectrum S (Ω) is then proportional to

+∞ -∞ e iΩτ exp i ω k c a e ±aτ/c dτ 2 (2.11)
The same spectrum (2.11) can be obtained considering that the plane waves (∝ e i(kx±ω k t) ), in the laboratory frame, are seen by the Rindler observer as e iϕ ± (τ) ≡ e i(kx±ω k t) = exp i ω k c a e ±aτ/c -1 .

(2.12)

where we used

x(τ) = c 2 a cosh aτ c -1 , t(τ) = c a sinh aτ c . (2.13)
Thus, the spectrum S (Ω) is proportional to the squared module of the Fourier transform of (2.12). The term e -iω k c/a in (2.12) is independent from τ, and in the Fourier transform integral gives only a proportionality factor. So we obtain again Eq. (2.11).

After some straightforward algebra we can evaluate the integral (2.11) and we nally get

+∞ -∞ e iΩτ exp i ω k c a e ±aτ/c dτ 2 = 2πc Ωa 1 e 2πcΩ/a -1 . (2.14)
This result shows the Unruh-Hawking e ect. In fact, we have obtained the Planck factor relative to the Bose-Einstein distribution for particles at temperature T = a/2πk B c, which is indeed the Hawking-Unruh temperature. This simple, not formal, method gives us an intuitive physical insight on the origin of the Unruh e ect. Despite its conceptual importance for connections with Hawking radiation [13,14] and for its impact on cosmology, black hole physics, particle physics, and relativistic quantum information [19] - [21], experimental detection of the Unruh e ect remains elusive, since an acceleration of the order of 10 20 m/s 2 would be required in order to measure a temperature of 1K. Many experimental proposals for its measurements in circular accelerators [22], as well as in analogue models of condensed matter physics [23] have been also discussed. Other proposals concern, for example, detecting spin depolarization of accelerated electron [24], or accelerating particles by ultraintense laser pulses [25,26] or laser laments [27,28]. However, despite the intense e orts the problem of detection of Unruh e ect remains an open question because very high accelerations are required to obtain a Unruh temperature of some kelvin. On the other hand, a direct veri cation of the e ect could allow a deeper understanding of some persisting controversies about the interpretation of this e ect [29,30,31]. In this direction, it has been recently argued that interatomic van der Waals interactions between two uniformly accelerated atoms could be good candidates for detecting the Unruh e ect, for reasonable values of the acceleration [32], [AN1].

Among the large number of open fundamental challenges in this eld, a long-standing question concerns whether the e ect of a relativistic acceleration is strictly equivalent to a thermal eld [33]. It has been recently shown, for example, that nonthermal features associated with uniform acceleration manifest on the radiative properties of single accelerated atoms [34] - [40], [AN3]. Also, recent works on entanglement generation or Casimir-Polder interactions between uniformly accelerated atoms, have shown that non-thermal e ects of accelerations arise in a system of two or many particles [41,42,40], [AN3].

In this context, we have recently shown that the Casimir-Polder (CP) force between two uniformly accelerating atoms in their ground state exhibits a cross-over from a short distance thermal behavior to a long distance non-thermal behavior, with respect to a reference length identi ed with z a = c 2 a , the characteristic scale for a breakdown of a local approximate description of the two-body system in terms of a Minkowskian space-time [40], [AN3]. Indeed, Casimir-Polder forces between neutral atoms arise from the retarded interaction among the dipoles induced and correlated by the zero-point quantum eld uctuations, and quanta mediating the interaction between the two atoms cannot disregard, for large distances, the non-inertial character of relativistic acceleration [32,40], [AN1], [AN3]. All these consequences will be presented and discussed in detail in the next sections of this chapter.

Some known e ects of uniform acceleration in quantum electrodynamics

In this Section, we introduce some physical con gurations related to our study concerning with e ects of a uniform acceleration in quantum electrodynamics [38], [AN2]. This will be the basis for our original work on this subject, which will be exposed in the next sections. We consider two di erent physical systems that are already studied in the literature. We rst consider a hydrogen atom moving with a uniform acceleration and interacting with the quantum electromagnetic eld in the vacuum state. We then consider the atom-wall Casimir-Polder interaction when the atom is uniformly accelerated. These two systems are good examples showing physical implications of the acceleration in quantum electrodynamics.

Let us now consider the physical system of a hydrogen atom uniformly accelerated in the quantum electromagnetic vacuum. The Hamiltonian describing the atom-eld interaction in the instantaneous inertial frame of the atom, in the multipolar coupling scheme is [35,34]

H(τ) = H A (τ) + H F (τ) + H I (τ) , (2.15)
with

H A (τ) = n ω n σ nn (τ), (2.16
)

H F (τ) = kj ω k a † k a k dt dτ , (2.17)
H I (τ) = -e mn r mn • E(x(τ))σ mn (τ) , (2.18)
where τ is the proper time and σ ℓm = |ℓ m|, |n being a complete set of atomic states with energy ω n . µ = er is the atomic electric dipole moment. Also, x = (t, x) is the space-time coordinate of the atom and kj the wave vector ( j = 1, 2 is the polarization index). We are interested in investigating the energy level shifts of the uniformly accelerated atom. Exploiting the general procedure in [43,44,46], that we will discuss and use in the following Section 2.4,we can split the energy shift of the atomic level of the accelerated atom in two parts,separating the contributions of vacuum uctuations and radiation reaction eld (indicated respectively with the subscripts vf and rr). These quantities, at second order in e, are [46] (δE

a ) vf = - ie 2 τ τ 0 dτ ′ C F ℓm (x(τ), x(τ ′ ))(χ A ℓm ) a (τ, τ ′ ), (2.19)
(δE a ) rr = - ie 2 τ τ 0 dτ ′ χ F ℓm (x(τ), x(τ ′ ))(C A ℓm ) a (τ, τ ′ ), (2.20)
where C F(A) ℓm and χ F(A) ℓm are the symmetric correlation function and the linear susceptibility of the eld (atom), respectively. Using the appropriate statistical functions of atom and eld [15], after some algebra, we have [35]

(δE a ) vf = e 2 3πc 3 b | a|r(0)|b | 2 ∞ 0 dω ω 3 1 + a 2 c 2 ω 2 × coth πc ω a P 1 ω + ω ab - 1 ω -ω ab , (2.21)
(δE a ) rr = e 2 3πc 3 b | a|r(0)|b | 2 ∞ 0 dω ω 3 × P 1 ω + ω ab + 1 ω -ω ab , (2.22)
where the index a and the relative ket indicate a generic atomic state, ω ab = ω aω b , a is the acceleration of the atom, and the limit ττ 0 to in nity has been taken. We rst note from (2.22) that the radiation reaction contribution to the energy level shift does not depend on the atomic acceleration; it is identical to that obtained in the inertial frame. This result is expected on a physical ground: in fact, the eld emitted by the atom propagates with the velocity of light, and it can act back on the atom only at the same time it is emitted. Thus, radiation reaction contribution is not in uenced by the atomic motion.

As we shall discuss later in this section, the situation radically changes in the presence of a boundary, such as a re ecting mirror. On the other hand, the contribution of vacuum uctuations depends explicitly on the acceleration, through the presence of the thermal term coth(πc ω/a), linked to the Unruh temperature T = a/2πck B , and of an extra term proportional to a 2 . This result indicates that the atomic acceleration induces observable e ects in the energy shifts.

The appearance of a nonthermal term proportional to a 2 is related to a similar term appearing in the correlation function of the electric eld in the accelerated frame. It is possible to show that, for a ground-state hydrogen atom, thermal and non-thermal terms are comparable for a ∼ 10 25 cm/s 2 [35]. This is also the typical acceleration required to detect the Unruh e ect by measuring atomic level shifts.

The same physical arguments given above indicate that also the Casimir-Polder interaction between a uniformly accelerated atom and a perfectly re ecting plate could manifest the Unruh e ect. Corrections to the atom-wall Casimir-Polder force due to the acceleration of the atom, have been calculated in the scalar eld case [36]. It has been shown that such corrections are relevant only for accelerations of the order of 10 24 cm/s 2 , con rming the necessity of extremely high accelerations for a detection of the Unruh e ect. This calculation has been extended to the more realistic case of a uniformly accelerated hydrogen atom interacting with the quantum electromagnetic eld, in the presence of a perfectly re ecting mirror [37,39]. Let us consider an atom moving with uniform acceleration in a direction parallel to the mirror at a distance z 0 from the mirror. In analogy with the case of an atom at rest near a plate, the atom-wall Casimir-Polder interaction can be obtained considering the z 0 -dependent terms in the expression of the energy level shift. As before, we evaluate the contribution of vacuum uctuations and of the radiation reaction eld to the energy shift of the atomic level, in the presence of a conducting plate. After some lengthy algebra, it is found [39]

(δE a ) (b.c.) vf = - 1 8π 2 c 3 b µ ab ℓ µ ba m 1 (2z 0 ) 3 P ∞ 0 dωK ℓm (ω; z 0 , a) × coth πc ω a 1 ω + ω ab - 1 ω -ω ab (2.23)
and

(δE a ) (b.c.) rr = 1 8π 2 c 3 b µ ab ℓ µ ba m 1 (2z 0 ) 3 P ∞ 0 dω × K ℓm (ω; z 0 , a) 1 ω + ω ab + 1 ω -ω ab , (2.24)
where the superscript (b.c.) stands for boundary conditions, z 0 is the atom-wall distance and K ℓm (ω; z 0 , a) is a function containing a combination of sinusoidal functions, which takes into account the presence of the conducting plate [39]

K ℓm (ω; z 0 , a) = σ ℓn (1 + a 2 z 2 0 ) 1/2 (δ nm -n n n m ) (2ωz 0 ) 2 (1 + a 2 z 2 0 )
sin 2 ω a sinh -1 (az 0 )

+ (δ nm -3n n n m ) 2ωz 0 (1 + a 2 z 2 0 ) 3/2 cos 2 ω a sinh -1 (az 0 ) - 1 (1 + a 2 z 2 0 ) 2 sin 2 ω a sinh -1 (az 0 ) +a 2 z 2 0 (δ nm + 3σ nm ) 2ωz 0 (1 + a 2 z 2 0 ) 3/2 cos 2 ω a sinh -1 (az 0 ) + (δ nm -2σ nm ) 2 (1 + a 2 z 2 0 ) 2 sin 2 ω a sinh -1 (az 0 ) -σ nm a 4 z 4 0 4 (1 + a 2 z 2 0 ) 2 sin 2 ω a sinh -1 (az 0 ) + az 0 [(δ lm -k l k m )az 0 + n l k m + k l n m ] × (2ωz 0 ) 2 (1 + a 2 z 2 0 ) sin 2 ω a sinh -1 (az 0 ) - 2ωz 0 (1 + a 2 z 2 0 ) 3/2 cos 2 ω a sinh -1 (az 0 ) + 1 (1 + a 2 z 2 0 ) 2 sin 2 ω a sinh -1 (az 0 ) +a 3 z 3 0 [(δ ℓm -k ℓ k m )az 0 + n ℓ k m + k ℓ n m ] 4ωz 0 (1 + a 2 z 2 0 ) 3/2 cos 2 ω a sinh -1 (az 0 ) + 4 (1 + a 2 z 2 0 ) 2 sin 2 ω a sinh -1 (az 0 ) . (2.25)
We now brie y comment the results obtained. Equation (2.23) clearly shows that the contribution of vacuum uctuations contains not only a thermal correction due to the factor coth(πc ω/a), but also an extra term proportional to the function K ℓm (ω; z 0 , a). This function modulates the Casimir-Polder interaction as a function of the atom-plate distance z 0 and of the atomic acceleration a. On the other hand, Equation (2.24) shows that the radiation reaction term is sensitive to the atomic acceleration. This behavior is not surprising. When a boundary is present, the eld emitted by the atom can act back on the atom after a re ection on the conducting plate. Since the atom accelerates, in the time-interval between the emission and the subsequent absorption of the re ected eld, the atom has moved from its position of a distance depending on its acceleration. This gives rise to a dependence of the radiation reaction contribution on the atomic acceleration. The expression for the total atom-wall Casimir-Polder interaction for the accelerated atom, is obtained by summing (2.23) and (2.24). It is easy to show that in order to reveal e ects of acceleration on the atom-wall Casimir-Polder interaction, accelerations of the order of 10 24 cm/s 2 are necessary, as in the case of the energy shift of an atom in the unbounded space [38], [AN2]. This makes very di cult to observe the e ects of the acceleration through the Lamb shift or the atom-wall Casimir-Polder interactions. As we shall show with the results we obtained in the next sections, the situation seems di erent when we consider the van der Waals/Casimir-Polder interaction between two accelerated atoms.

van der Waals interaction energy between two accelerated atoms

As we have already seen van der Waals dispersion forces between two neutral atoms in the vacuum are related to uctuations of the zero-point electromagnetic eld, and thus they could be a good candidate for detecting an accelerated motion of the atoms and the Unruh e ect. So in this section we will consider the e ect of the acceleration on the dispersion interaction between two atoms, and we will show that new phenomena are present in this case, namely a change of the distance-dependence of the interaction energy and its explicit time-dependence [32], [AN1]. Using a simple model, some hints were already obtained on the e ect of the acceleration on the dispersion force between accelerating atoms, exploiting the relation between acceleration and temperature given by the Unruh e ect [47].

We consider the e ect of a uniform acceleration on the van der Waals interaction energy between two ground-state atoms moving in the vacuum space with uniform acceleration. In particular, we are interested to investigate whether the (uniform) acceleration of the atoms yields a qualitative change of the properties of the force. The two atoms/molecules A and B move, in the laboratory system, with the same uniform acceleration a in the x direction, perpendicular to their distance, so that their separation is constant (see Figure 2.1). In order to obtain their van der Waals interaction, both in the near zone and in the far zone (Casimir-Polder regime) we use the following physical model: the interaction energy arises from the dipolar interaction between the (instantaneous) oscillating dipole moments of the atoms, induced and correlated by zero-point electromagnetic eld uctuations. In this model the dipolar elds are classical elds, and the quantum properties of the radiation are included in the spatial correlations of the electric eld associated to vacuum uctuations. This model has been used and proved valid in many cases: atoms at rest [8,48], three-body dispersion forces [62], when boundary conditions are present [49] or in the presence of external radiation [50]. In the present case we need to generalize this model to the case of accelerating atoms, expressing the eld generated by the atomic dipoles in the accelerated reference frame. An advantage of our method is that, even if the interaction energy is calculated for the accelerating atoms in their co-moving frame (the system in which the atoms are instantaneously at rest), all physical quantities relative to the atoms are given in terms of their known values in the laboratory frame.

We now obtain an explicit expression of the interaction energy and discuss in detail the near-and far-zone limits. Our results show that in the near zone a new term proportional to R -5 adds to the usual R -6 behavior, and in the far zone a term proportional to R -6 adds to the usual R -7 behavior, making the interaction of a longer range. We also nd that the interaction energy has an explicit time dependence. In particular, we show that acceleration-dependent corrections to the R -7 (far zone) and R -6 (near zone) terms, proportional to a 2 t 2 /c 2 , are present. This suggests that signi cant changes to the interaction between the two atoms could be obtained if su ciently long times are considered, even for reasonable values of the acceleration, contrarily to other known manifestations of the Unruh e ect, such as Lamb shift and atom-wall interaction for accelerated atoms previously in the laboratory reference frame at the retarded time t r = tρ(t r )/c. ρ(t r ) is an e ective interaction distance given by the distance traveled by a light signal from its emission by atom A at time t r to the time t when it is received by atom B. We shall evaluate this distance for the speci c case of uniformly accelerated atoms at the end of this Section. In this model, the atoms are assumed having instantaneous oscillating dipole moments and their van der Waals interaction arises from the interaction between the eld emitted by the uctuating dipole of one atom with the dipole moment induced on the second atom. This eld can be expressed as (summation over a repeated index is understood),

Ẽi (kλ, R B , t) = -µ A j Ṽ′ i j (k, R, t r ), (2.26)
where

R B is the position of atom B, R = R B -R A and µ A j is the dipole moment of atom A. Ṽ′ i j (k, R, t r
) is a tensor potential that will be de ned explicitly in the next Subsection. From now onwards, a tilde indicates that the corresponding quantity is evaluated in the co-moving reference frame, where the atoms are instantaneously at rest. In the co-moving frame, the interaction of this eld with the induced dipole moment of atom B is given by

-μB i Ẽi (kλ, R B , t) = μB i µ A j Ṽ′ i j (k, R, t r ), (2.27)
where μB i is the dipole moment of atom B in the accelerated frame. Summation over (kλ) yields the interaction energy.

The Fourier (kλ) component E(kλ; r) of the electric eld, given by

E j (kλ; r) = i 2π ck V 1/2 êj (kλ)a kλ e ik•r -ê * j (kλ)a † kλ e -ik•r
(2.28) (ê j (kλ) is the polarization unit vector), induces a dipole moment in the atom at position r given by

µ ind (kλ; r) = α(k)E(kλ; r), (2.29)
where we are assuming an isotropic atom with dynamic polarizability α(k). As mentioned, the instantaneous dipole moment induced in one atom, let us say atom A, generates an electric eld that then interacts with the other atom (B). This electric eld is the eld generated by atom A with position R A at the retarded time t r = tρ(t r )/c, evaluated at the position of atom B. Because in our case both atoms are accelerating, we need the expression of the electric eld generated by an oscillating dipole in motion. This expression, as well as that of the magnetic eld, is known and it is usually separated in the two components E (pol) and E (Roe) , called the polarization and Röntgen components, respectively. Because we are interested in the interaction between the two accelerating atoms in their co-moving system, that is a locally inertial frame, the electric eld must be Lorentz-transformed to the co-moving system: thus, electric and magnetic elds are both necessary, because Lorentz transformations mix electric and magnetic elds. In the laboratory frame, these elds, for a dipole moving along an arbitrary trajectory x(t), are given in Ref. [51,52] in terms of the retarded time t r = tr/c. We use the general expressions in [51,52] for the polarization and Röntgen components of the electric and magnetic elds in our case of a uniformly accelerated trajectory along x given by [53] x

(t) = c 2 a        a 2 t 2 c 2 + 1 -1        , x(τ) = c 2 a cosh aτ c -1 , (2.30)
where time t is related to the proper time τ by the relation

t = c a sinh aτ c . (2.31)
We also assume x(0) = 0, ẋ(0) = 0, and take into account that the two atoms are moving in a direction orthogonal to their distance, so that their distance does not change with time. We thus obtain the polarization and Röntgen components of the electric and magnetic elds for the uniformly accelerating dipole, evaluated at the position of the other dipole,

E (pol) i (r, t) = - 1 ρ 3 Tij µ j (t r ) + 1 cρ 2 Ti j μ j (t r ) + 1 c 3 ρ Ŝ i j μ j (t r ) , (2.32)
E (Roe) i (r, t) = - 1 c 2 ρ 2 ẋi (t r ) ρj μ j (t r ) + 1 c 2 ρ 2 ẍi (t r ) ρj µ j (t r ) + 1 c 3 ρ ẋi (t r ) ρ j μ j (t r ) + 1 c 3 ρ ẋi (t r ) ρj µ j (t r ) + 2 1 c 3 ρ ẍi (t r ) ρj μ j (t r ) , (2.33)
B (pol) i (r, t) = - ǫ ik j cρ 2 ρk μj (t r ) - ǫ ik j c 2 ρ ρk μ j (t r ), (2.34)
B (Roe) i (r, t) = - 1 cρ 2 Ti j ǫ jkl 1 ρ µ k (t r ) ẋl (t r ) + 1 c µ k (t r ) ẍl (t r ) + 1 c μk (t r ) ẋl (t r ) - 1 c 3 ρ Ŝ i j ǫ jkl µ k (t r ) ẋl (t r ) + 2 μk (t r ) ẍl (t r ) + μk (t r ) ẋl (t r ) , (2.35)
where t r = tr/c is the retarded time, ρ(t) = rx(t) and ǫ ilk is the totally antisymmetric tensor. We have also de ned the following tensors

Ti j ≡ δ i j -3 ρi ρj , (2.36) Ŝ i j ≡ δ i j -ρi ρ j .
(2.37)

Before obtaining the dispersion interaction energy for the two accelerating atoms, we need some considerations about the retarded time and the distance between the atoms to be used in the expressions above for the elds. The "e ective interaction distance" ρ(t r ) is the distance traveled by a light signal from one atom to the other one. For atoms at rest, it coincides with the interatomic distance ρ, while in the case of atoms moving at a constant velocity v, it is easy to show that ρ(t r ) = γρ. In our case the atoms are in an accelerated motion: this makes evident that we should expect an explicit time-dependence of the interaction distance because γ = (1v 2 /c 2 ) -1/2 depends on time. By assuming that at t = 0 the atoms are at rest and have a uniform acceleration a, using (2.30) and simple geometrical considerations, it is possible to show that

ρ(t r ) = ρ + c t - c arctan (at/c) a , (2.38)
showing that indeed ρ(t r ) depends on time and, as expected, it grows with time.

We now evaluate the interaction energy between the uctuating atomic dipoles in accelerated motion. We assume a nonrelativistic motion for the atoms; because their acceleration is given, this assumption limits the timescale of validity of our results, as we shall discuss in more detail in the next Subsection. The potential energy will be evaluated in the co-moving frame of the accelerating atoms. All relevant physical quantities will be however expressed in terms of quantities measured in the laboratory reference frame and thus directly measurable; this makes our approach di erent with respect to results in the literature concerning other radiative processes in accelerated frames (such as Lamb shift, atom-wall interactions, etc), which are expressed in terms of physical quantities measured in the co-moving frame [35,37,39,47].

In our approach, each Fourier component of vacuum eld uctuations induces an oscillating dipole in the atoms, that in the laboratory frame is of the form (in the k space)

µ µ µ A(B),ind (k, λ, t) = µ µ µ A(B),ind (k, λ) cos (ωt) (2.39)
with ω = ck. Using (2.27), the van der Waals interaction energy can be expressed as

∆ Ẽ = k,λ k ′ ,λ ′ μB,ind i (k, λ)µ A,ind j (k ′ , λ ′ ) Ṽij ′ (R, t). (2.40)
We stress that in (2.40) the dipole moment of atom A is in the laboratory frame while that of atom B is still in the co-moving frame. We shall now transform the latter in the laboratory frame, in order to express the energy shift only in terms of quantities in this frame. Under a Lorentz transformation, the dipole moment transforms as a length; because the atoms move along the x direction, we have

μ µ µ = γµ x î + µ y  + µ z k, (2.41)
that shows that only the x component is di erent in the two reference frames. Using the relation (2.29) between the induced dipole moment and the uctuating vacuum eld, we get

∆ Ẽ = k,λ k ′ ,λ ′ α(A, k ′ )α(B, k)E i (k ′ , λ ′ ; R A )E j (k, λ; R B ) Ṽij (R, t).
(2.42)

The Fourier components of the electric eld operator in (2.42) are in the laboratory frame, because they come from relation (2.29) with the induced dipole moment in the laboratory system. The tensor Ṽij (R, t) in (2.42) di ers from the tensor Ṽij ′ (R, t) in (2.40) because the γ factor in (2.41) has been included in it, that is

Ṽx j (R, t) = γ Ṽ′ x j (R, t), Ṽyj (R, t) = Ṽ′ y j (R, t), Ṽz j (R, t) = Ṽ′ z j (R, t). (2.43)
In (2.42) a factor 2 should be added, taking into account that we should also consider an equal interaction energy obtained by exchanging the role of the two atoms. We shall include this factor 2 in the expression of the potential tensor Ṽi j (R, t) given in the following of this Section. We now take the vacuum expectation value of (2.42), taking into account that the electric eld operators are in the laboratory frame. Thus we have

0|E i (k ′ , λ ′ ; R A )E j (k, λ; R B )|0 = 2π ck V êi (k, λ)ê * j (k, λ) e -ik•(R B -R A ) δ kk ′ δ λλ ′ .
(2.44)

In the continuum limit, V → ∞, k → V/(2π) 3 k 2 dkdΩ; performing polarization sum and angular integration,

λ êi (k, λ)ê * j (k, λ) = δ i j -ki kj , (2.45
)

1 4π δ i j -ki kj e ±ik•R dΩ = δ i j -Ri R j sin(kR) kR + δ i j -3 Ri R j cos(kR) k 2 R 2 - sin(kR) k 3 R 3 , (2.46)
we obtain

∆ Ẽ = 2 c π Ŝ i j sin(kR) kR + Ti j cos(kR) k 2 R 2 - sin(kR) k 3 R 3 Ṽi j (R, t) k 3 dk. (2.47)
In the approximation of a nonrelativistic motion, we have ẋ(t) = at, ẍ(t) = a and ẋ(t) = 0. Using these expressions in (2.32-2.35), we obtain the electric and magnetic elds generated by the uniformly accelerating dipole in the laboratory frame. In order to obtain the expression of the tensor Ṽij (R, t) in (2.47), we need the electric eld in the co-moving frame. Thus we Lorentz-transform the elds according to the well-known relations

Ẽx = E x Ẽy = γ(E y -βB z ) Ẽz = γ(E z + βB y ).
(2.48) [54]. Using these transformations, the potential tensor Ṽi j (R, t) in (2.47) is obtained as

Ṽ1j (R, t) = - 2γ(t) R      T1j R - 1 R A(R, t) + ω c B(R, t) + Ŝ 1 j ω 2 c 2 A(R, t) + Rj a c 2 1 R + ω 2 t c A(R, t) + ωt R + 2ω c B(R, t) , (2.49)
Ṽ2j (R, t) = - 2γ(t) R      T2 j R - 1 R A(R, t) + ω c B(R, t) - β(t) c Rl ε 3l j ω ω c A(R, t) + 1 R B(R, t) + T3l ε l j1 a R - 1 c + t R A(R, t) + tω c B(R, t) + Ŝ 3l ε l j1 aω c 2 ωtA(R, t) + 2B(R, t) + Ŝ 2 j ω 2 c 2 A(R, t) , (2.50) Ṽ3j (R, t) = - 2γ(t) R      T3 j R - 1 R A(R, t) + ω c B(R, t) + β(t) c Rl ε 2l j ω ω c A(R, t) + 1 R B(R, t) + T2l ε l j1 a R - 1 c + t R A(R, t) + tω c B(R, t) + Ŝ 2l ε l j1 aω c 2 ωtA(R, t) + 2B(R, t) + Ŝ 3 j ω 2 c 2 A(R, t) , (2.51)
where

β(t) = v(t)/c, γ(t) = (1 -β 2 (t)) -1/2 .
We have used (2.36) and (2.37) with R in place of ρ, and de ned the functions

A(R, t) = cos(ωt) cos ω t - R c , (2.52)
B(R, t) = cos(ωt) sin ω t - R c . (2.53)
Some considerations about the time-dependence of the potential tensor Ṽi j (R, t) are now necessary. In the case of atoms at rest in the laboratory system, discussed in [8], the potential tensor is calculated, for each mode (k, λ), after a time average on an oscillation period 2π/ω of the dipoles. In that case, this is equivalent of taking a time-average of the quantities A(R, t) and B(R, t) in (2.52) and (2.53), respectively. In our case of accelerating atoms, extra time dependence is contained in the factors β(t) and γ(t) appearing in equations (2.49-2.51). We take the time-average of Ṽij (R, t ′ ) on a time t much larger than ω -1 (that is we take ωt ≫ 1 for a given ω) and keep the leading term in t only, which gives the main contribution to the time average. We thus consider the quantity

Ṽi j (R, t) = 1 t t 0 V i j (R, t ′ )dt ′ . (2.54)
We take a nonrelativistic approximation; then

β(t) ≃ at c ; γ(t) ≃ 1 + a 2 t 2 2c 2 (2.55)
and keep only terms up to the second order in at/c. In order to evaluate (2.54) we need to calculate integrals of A(R, t ′ ) and B(R, t ′ ) and integrals of these functions multiplied by t ′ or t ′2 , keeping only leading terms in t. After lengthy straightforward algebraic calculations, we nally obtain

Ṽij (R, t) = 1 + a 2 t 2 6c 2 1 R 3 Ti j [cos(kR) + kR sin(kR)] Ŝ i j k 2 R 2 sin(kR) + Z i j , (2.56)
where R = (0, 0, R) is along the z axis, and Tij = diag(1, 1, -2) and Ŝ i j = diag(1, 1, 0) are diagonal 3x3 matrices. The 3x3 matrix Z i j is de ned below. Substituting (2.56) into (2.47), we obtain the van der Waals interaction energy shift of the two accelerating atoms

∆ Ẽ = 1 + a 2 t 2 6c 2 ∆E r + 2 c π Ŝ i j sin(kR) kR + Tij cos(kR) k 2 R 2 - sin(kR) k 3 R 3 Z i j k 3 dk (2.57)
where

∆E r = - c πR 3 ∞ 0 k 3 dk α(A; k)α(B; k) × kR sin(2kR) + 2 cos(2kR) -5 sin(2kR) kR -6 cos(2kR) k 2 R 2 + 3 sin(2kR) k 3 R 3 = - c πR 2 ∞ 0 du α(A; iu)α(B; iu) 1 + 2 uR + 5 u 2 R 2 + 6 u 3 R 3 + 3 u 4 R 4 u 4 e -2uR
(2.58)

is the well-known van der Waals potential energy for two atoms at rest [7,8]. In our result (2.57), t is the observation time and ∆ Ẽ is the interaction energy averaged between times 0 and t, as it follows from our averaging in (2.54); however, for sake of simplicity, we shall call it as the interaction energy at time t.

The result in (2.57) clearly shows that one e ect of the uniform acceleration of the atoms is a correction to the potential energy proportional to a 2 t 2 /c 2 and a new term (that with the k integral), that we are now going to evaluate explicitly. We will show that this new term gives also a change of the R-dependence of the van der Waals potential energy when the two atoms are subjected to a uniform acceleration. From (2.57) and taking into account that Tij and Ŝ i j are diagonal matrices, we notice that only diagonal elements of the matrix Z i j appearing in (2.56) are relevant. Their values are

Z 11 = 0 (2.59) Z 22 = T33 a 2 t 2c 3 R 2 cos(kR) + a 2 t 2 3c 2 R 3 cos(kR) + a 2 t 2 3c 2 R 2 k sin(kR) , (2.60)
Z 33 = T22 a 2 t 2c 3 R 2 cos(kR) + a 2 t 2 3c 2 R 3 cos(kR) + a 2 t 2 3c 2 R 2 k sin(kR) + Ŝ 22 a 2 t c 3 R k sin(kR) - a 2 t 2 3c 2 R k 2 cos(kR) . (2.61)
Substitution of (2.59-2.61) into (2.57), nally yields

∆ Ẽ =∆E r + a 2 t 2c 3 c πR 3 ∞ 0 α(A; iu)α(B; iu) 3 + 4 uR + 2 u 2 R 2 u 2 e -2uR du + + a 2 t 2 6c 2 c πR 2 ∞ 0 α(A; iu)α(B; iu) -1 + 4 uR + 8 u 2 R 2 + 8 u 3 R 3 + 4 u 4 R 4 u 4 e -2uR du (2.62)
The equation (2.62) is our main result for the van der Waals/Casimir-Polder interaction energy between uniformly accelerating atoms. It shows two terms correcting the van der Waals potential energy due to the atomic uniform acceleration: both are proportional to the square of the acceleration, and they explicitly depend on time as t and t 2 , within our approximations. Because the potential for inertial atoms ∆E r is negative (attractive interaction), Equation (2.62) shows that the e ect of the acceleration is to reduce the interaction energy, and this reduction grows with time. This is consistent with the fact that the "effective interaction distance" ρ(t r ) in (2.38) grows as time goes on, yielding a decrease of the interaction energy between the accelerating atoms. However, as we shall discuss in more detail in Subsection 2.3.2, these corrections cannot turn the potential energy from attractive to repulsive, at least within our approximations.

We can consider two limiting cases of the van der Waals dispersion energy, the so-called near zone and far zone.

In the near zone, the interaction energy ∆E r for atoms at rest is as R -6 . In this zone, we can approximate uR ≪ 1 in (2.62), obtaining

∆ Ẽ ≃ -1 - 4a 2 t 2 9c 2 3 c 2 πR 6 ∞ 0 α(A; iu)α(B; iu) du + a 2 t π c 2 R 5 ∞ 0 α(A; iu)α(B; iu) du.
(2.63)

In the far zone we can approximate the atomic dynamical polarizabilities to their static value α A,B (0), obtaining

∆ Ẽ =∆E r -α A (0)α B (0) c πR 3 ∞ 0 a 2 t 2c 3 k 3 sin(2kR) + 4 cos(2kR) kR -2 sin(2kR) k 2 R 2 + a 2 t 2 6c 2 kR sin(2kR) -2 cos(2kR) + 3 sin(2kr) kR + 2 cos(2kR) k 2 R 2 - sin(2kR) k 3 R 3 k 3 dk , (2.64)
where in this case (far zone) the dispersion energy ∆E r behaves as R -7 . Performing the k integrals, we nally get

∆ Ẽ ≃ - c π α A (0)α B (0) R 7 23 4 - 7 24
a 2 t 2 c 2 + 11 a 2 t 8π c 2 α A (0)α B (0) R 6 .
(2.65)

These results clearly show the two new main features of the van der Waals interaction energy for accelerating atoms: a change of the dependence on the distance and an explicit time-dependence. In fact, from Equation (2.65) we can see that in the far zone an e ect of the acceleration is to add a new (time-dependent) term behaving as R -6 , which has a longer range than the usual R -7 van der Waals energy in the Casimir-Polder regime for atoms at rest. A R -6 term in the atom-atom dispersion energy is known to occur when the interaction is calculated for atoms at rest at nite temperature [55], and this indicates the deep connection between our results and the Unruh e ect. The near-zone result (2.63) also shows corrections giving an explicit time-dependence of the interaction energy proportional to the acceleration squared, and a new term proportional to acceleration and time, and decreasing as R -5 . The explicit time dependence as a 2 t 2 /c 2 in the rst line of equations (2.63) and (2.65), for the near and far zone respectively, gives corrections to the interaction energy which grows with time and may become signi cant even for a reasonable value of the acceleration. In fact, it is possible to nd time intervals such that, from one side the nonrelativistic approximation is still valid (a 2 t 2 /c 2 ≪ 1), and on the other side the corrective term, although relatively small, is not negligible. For example, if a 2 t 2 /c 2 ≃ 0.2, we can still consider reasonable our approximation of a nonrelativistic motion of the atoms, and the correction to the van der Waals interaction energy from (2.63) and (2.65) is around ten percent in the near zone and one percent in the far zone. These changes are small, but not negligible. Because only the product of acceleration and time is relevant for our correction to the dispersion energy (and not the absolute value of the acceleration, as in the correction to the Lamb shift or the atom-wall interaction energy [35,37,39,56]), this should be achievable even with reasonable accelerations, provided a su ciently long time is taken. Also, the corrections as R -6 and R -5 in the second lines of (2.63) and (2.65), respectively, give a change to the van der Waals interaction of a few percent, using the same value of the acceleration considered above and an interatomic distance R such that aR/c 2 ∼ 0.1, for which our use of a locally inertial system is valid (see also the discussion at the end of next Subsection). These new results we found suggest a new possibility for detecting the Unruh e ect, or in general e ects related to accelerated motion in quantum electrodynamics, without necessity of extremely high accelerations as in the case of other quantum-electrodynamic e ects recently discussed in the literature [4,35,37,39].

Concluding remarks on the results

In the previous subsection we have considered the van der Waals interaction energy between two ground-state atoms (or polarizable bodies) moving in the vacuum with the same uniform acceleration. The acceleration is assumed orthogonal to the separation between the atoms, so that their distance is constant. We have shown that the main e ects of the acceleration are twofold: an explicit time-dependence of the the interatomic interaction and a qualitative change of its dependence from the interatomic distance, which depends on the acceleration squared, making the interaction of longer range. In particular, in the near zone a new term as R -5 adds to the usual R -6 behavior, while in the far zone a R -6 term adds to the usual R -7 van der Waals energy in the Casimir-Polder regime.

We now discuss some physical consequences of our results as well as the limits of our approximations.

Our result (2.62) for the van der Waals dispersion interaction energy for two uniformly accelerating atoms, and approximated in (2.63) and (2.65) for the near-and far-zone respectively, clearly shows how the accelerated motion of the atoms a ects the interaction energy and changes its distance dependence. The latter is an important point showing that the e ect of the accelerated motion is not only a correction to the strength of the potential energy, but also a qualitative change of its properties. This also suggests, in perspective, the intriguing possibility of detecting signatures of the Unruh e ect in interacting atomic systems, in particular when their properties, even at the macroscopic level, may critically depend on the form of the interaction among the atoms. The time dependence of the interaction in (2.62) is related to the e ective interaction distance given by (2.38), which grows with time for the accelerated atoms, making larger the "e ective distance" traveled by the virtual photons exchanged between the atoms, as time goes on. A similar e ect is not present in cases previously considered for the Lamb shift of an accelerated hydrogen atom [56,35] or the atom-surface Casimir-Polder interaction for an atom accelerating parallel to an in nite conducting plate [39]: in these cases, the eld uctuations perceived by the atom are time-independent and the atom-surface "e ective distance" is constant, and thus a time-dependence is not expected and the corrections depend on the absolute value of the acceleration only. We have also shown that taking appropriate values of the product of acceleration and time, the relative change of the van der Waals interaction, with parameters such that all our approximations are valid, can be in the range 1-10 percent, and thus not negligible.

In our model, we have neglected the possibility that the atoms are excited due to their acceleration. It is known that accelerated atoms have a nite probability of being spontaneously excited [56,57,58]. In principle, this could add another source of change of the distance dependence of the dispersion interaction between the atoms, because this interaction behaves di erently if one or both atoms are excited [59]. The excitation probability, however, behaves as 1/(e 2πcω 0 /a -1), ω 0 being a main atomic transition frequency [56,57,58]. It is thus very small (exponentially) when a ≪ cω 0 . Taking a typical value for ω 0 ∼ 10 15 s -1 , we expect that this contribution be negligible for a ≪ 10 23 m/s 2 . Since we can obtain a signi cant change of the van der Walls energy for much smaller accelerations (making negligible the excitation probability, which decreases exponentially with decreasing accelerations), provided we consider a su ciently long time (see discussion above), atomic excitation induced by acceleration can be neglected in our case. Moreover, the contribution of the atomic excitation to the interatomic potential energy is a higher-order e ect. In fact, the van der Waals interaction is a fourth-order e ect, both for ground-and excited-state atoms [59]. Because the atomic excitation probability due to acceleration is a second-order e ect, its contribution to the van der Waals interaction starts from sixth-order in the atom-eld interaction.

Finally, we wish to make some considerations on the sign of the interaction energy of the accelerated atoms, which determines the attractive or repulsive character of the electric van der Waals force between two ground-state atoms (for atoms at rest it is always attractive). Equations (2.63) and (2.65) show that the accelerated motion reduces the potential energy between the atoms; this reduction grows with time, in agreement with the increasing e ective interaction distance given by (2.38). One interesting question is to investigate whether the terms related to the acceleration in (2.63) and (2.65) can turn the van der Waals force to a repulsive character, thus making the interaction energy positive. In the near zone, analyzing equation (2.63), we see that the R -6 term changes sign when at/c is of the order of one; however, this is not compatible with our nonrelativistic approximation. On the other hand, the new (positive) R -5 term becomes comparable with the usual (negative) R -6 term for an interatomic distance R ∼ c 3 /(a 2 t) and, due to our nonrelativistic approximation at/c ≪ 1, this would require R ≫ c 2 /a. This situation, however, would require a di erent treatment of our problem, by quantizing the eld in a curved space-time; in fact, our use of a locally inertial system for the accelerated atoms is valid only when the dimension of the system is much less than c 2 /a [17,60,61]. In other words, an interatomic distance larger than c 2 /a cannot thus be considered by adopting the locally inertial frame we have used. Similar considerations hold for the far-zone potential energy in (2.65), too. In our model there is not relative motion between the two atoms, so it seems that our results have no relation to the quantum friction (for discussion on quantum friction see Chapter 3). We can then conclude that, within our approximations, the attractive character of the van der Waals interaction is preserved also for the accelerated atoms. However, our results show that the van der Waals interaction between the two atoms is signi cantly a ected by their uniformly accelerated motion and the time-dependence of the interaction energy could allow to detect the accelerated motion without necessity of the extremely high accelerations. As a possible future perspective it could be very interesting analyze if the same e ects can rise for atoms at rest under the in uence of gravity and then if the equivalence principle can be invoked.

A fourth order method for the calculation of the Casimir-Polder force

In this section we present and extend a more rigorous approach to study the same problem faced in the previous section. As for the works presented in Section 2.2 we exploit the general procedure by Dalibard, Dupont-Roc and Cohen-Tannoudji (DDC) [43,44]. However here we develop and extend, for the rst time, the method up to order four in the coupling constant [45], [AN6]. This allows us to calculate the interaction energy between two atoms separating it into vacuum uctuations and radiation reaction eld contributions. These general equations then can be used to several physical situations and, in particular, we will use them to evaluate the interaction energy between the atoms in three very di erent cases: atoms at rest, atoms at rest in a thermal bath and atoms accelerated in the same direction in the vacuum. Let us start describing the general method. The total Hamiltonian of a physical system interacting with a reservoir (see Figure 2.2) can be simply written as

H = H S + H R + V (2.66)
where H S and H R are, respectively, the Hamiltonian of the system and of the reservoir. The system and the reservoir interact through a coupling potential that we indicated with V and it can be written in the form

V = g i R i S i . (2.67)
In this equation g is a coupling constant and S i (R i ) are Hermitian operators of the system (reservoir). Using the Heisenberg equations we can calculate the rate of variation of an arbitrary observable G and in particular the contribution given by the coupling is

dG(t) dt coupling = i g i [R i (t)S i (t), G(t)] = i g R i (t) i [S i (t), G(t)]. (2.68)
Let us now write the operator of the reservoir R i (t) as sum of a free part and a source part

R i (t) = R f i (t) + R s i (t) where R f i (t) represents its free evolution between time t 0 to time t R f i (t) = e i(t-t 0 ) H R (t 0 )/h R i (t 0 )e -i(t-t 0 ) H R (t 0 )/h . (2.69)
dissipation that, from now on, will be neglected. Then, for the two e ective Hamiltonians, we have

H e rf (t) = g 4 i R f i (t), S i (t) H e sr (t) = g 4 i R s i (t), S i (t) . (2.72)
We are interested, in this section and in the following ones, to calculate the Casimir-Polder interaction energy between atoms. To obtain it we will make an average of the two e ective Hamiltonians in the reservoir state. If the coupling constant g is relatively small we can perform a perturbative expansion in g. In our case we need a perturbative expansion up to fourth-order in the coupling constant. We thus need to extend this method to the fourth-order because the van der Waals/Casimir-Polder interaction is a fourth-order e ect [59] (in [44] the rates are calculated up to order two in the coupling constant).

Perturbative calculation of the variation rates

We are interested to evaluate the average of the two rates of an arbitrary system observable, given by (2.70), using a perturbative expansion. At rst we suppose that the reservoir and the system start to interact at time t 0 . In mathematical terms, this means that the density operator at t = t 0 is factorized in a part relative to the system and in a part relative to the reservoir.

Considering the general Hamiltonian (2.66), let us write the Heisenberg equation for an arbitrary operator O(t):

d dt O(t) = i [H(t), O(t)] = i [H S (t) + H R (t), O(t)] + i [V(t), O(t)].
(2.73)

Since the interaction term V depends on the coupling constant we can use this equation for our perturbative expansion in g. Let |α and β| (|a , b|) indicate eigenstates of H R (H S ) with eigenvalues E α , E β (ε a , ε b ). A basis of operators for the spaces of R and S is

Q αβ = |α β| (2.74)
q ab = |a b|. (2.75)
These operators have a free evolution Bohr frequency given by Ω αβ = (E α -E β )/ for Q αβ and ω ab = (ε aε b )/ for q ab . For example, the Heisenberg equation for the operator

Q αβ is d dt Q αβ (t) = iΩ αβ Q αβ (t) + ig i S i (t)[R i (t), Q αβ (t)]. (2.76)
We can write the general solution of this equation as

Q αβ (t) = Q f αβ (t) + Q s αβ (t) (2.77)
where

Q f αβ (t) = Q αβ (t 0 )e iΩ αβ (t-t 0 ) (2.78)
is the part of Q αβ (t) having a free evolution (zeroth-order in g), while

Q s αβ (t) = ig t t 0 dt ′ e iΩ αβ (t-t ′ ) × i S i (t ′ )[R i (t ′ ), Q αβ (t ′ )] (2.79)
is the source part; it is of rst order or higher in the coupling constant g. An analogous solution can be written for the operator q ab (t):

q s ab (t) = ig t t 0 dt ′ e iω ab (t-t ′ ) × i R i (t ′ )[S i (t ′ ), q ab (t ′ )] (2.80)
Now, starting from (2.79) we can perform our perturbative expansion in g by iteration. In fact, if we replace in (2.79) each operator with its free part, we get the rst-order approximation of Q s αβ (t). In a similar way we get the expansion of the operator q ab (t).

Q s αβ (t) ≃ ig t t 0 dt ′ i S f i (t ′ )[R f i (t ′ ), Q f αβ (t)] (2.81) q s ab (t) ≃ ig t t 0 dt ′ i R f i (t ′ )[S f i (t ′ ), q f ab (t)]. (2.82)
Where we have used (2.78) and an analogous expression for q f ab (t). The expansion can be generalized to any reservoir (system) operator R (S ), by exploiting its expansion in terms of Q αβ (t) (q ab (t)). In fact, we can write

R(t) = αβ Q αβ (t) α|R|β = R f (t) + R s (t), S (t) = ab q ab (t) a|S |b = S f (t) + S s (t), (2.83)
where

R f (t) = αβ Q f αβ (t) α|R|β , R s (t) = αβ Q s αβ (t) α|R|β , S f (t) = ab q f ab (t) a|S |b , S s (t) = ab q s ab (t) a|S |b .
(2.84) Thus, at rst order we have

R s (t) ≃ ig t t 0 dt ′ i S f i (t ′ )[R f i (t ′ ), R f (t)] (2.85) S s (t) ≃ ig t t 0 dt ′ i R f i (t ′ )[S f i (t ′ ), S f (t)].
(2.86)

After we have found the expansion for each reservoir and system operator at the rst order, we can use it to nd the higher-order terms of the perturbative expansion by a recursive calculation.

The vacuum uctuations contribution

The equation obtained in (2.72) and (2.84) are very general and can be used for many physical systems. Because we are interested in the Casimir-Polder interaction energy between two accelerating atoms, for the sake of simplicity, we will rst consider a pair of two-level atoms (with the same transition frequency ω 0 ) interacting with the massless scalar eld in its vacuum state. Then the free Hamiltonians of the two-level atoms A and B, that evolve with respect to the proper time τ, are (from now on in this section we use units such that c = = 1)

H A = ω 0 σ A 3 (τ), H B = ω 0 σ B 3 (τ) (2.87)
where we introduced the operator σ

A/B 3 = (1/2) |e A/B e A/B | -|g A/B g A/B | (Dicke notation [66]).
The free Hamiltonian of the quantum scalar eld is

H F (τ) = d 3 k ω k a † k a k dt dτ (2.88)
where a k and a † k are the bosonic annihilation and creation operators of the scalar eld φ(x). The atoms and the eld are coupled in a linear way through the following interaction Hamiltonian

H int A = λσ A 2 (τ)φ[x A (τ)], H int B = λσ B 2 (τ)φ[x B (τ)] (2.89)
where λ is the atom-eld coupling constant and we have introduced the operators

σ A/B 2 = (i/2)(σ A/B - -σ A/B + ), indicating with σ A/B - = |g A/B e A/B | and σ A/B + = |e A/B g A/B
| the atomic lowering and raising operators.

We now want to expand the e ective Hamiltonian previously introduced up to order four in λ. Let us start calculating the vacuum uctuations contribution. The vacuum uctuations Hamiltonian as a function of the proper time τ relative to atom A is (see equation (2.72))

H A vf (τ) = λ 4 φ f [x A (τ)], σ A 2 (τ) (2.90)
(we get the same term for the atom B by interchanging the label A and B). To expand this vacuum uctuations term up to order four, we must expand the operator σ A 2 (τ) up to order three. Since in the physical problem we are considering the system S is composed by two subsystems, that relative to atom A and that relative to atom B, we have S = S A ⊗ S B . Then we de ne two basis for the operators relative to the system S , one for the subsystem S A and one for the subsystem S B , in a way similar to (2.75)

q aa ′ = |a a ′ |
(2.91)

q bb ′ = |b b ′ |. (2.92)
Now we expand σ A 2 . Let us rst write q s aa ′ (τ):

q s aa ′ (τ) = iλ τ τ 0 dτ ′ φ[x A (τ ′ )][σ A 2 (τ ′ ), q f aa ′ (τ)]. (2.93)
Using the (2.84) we can write for σ A,s 2 (τ)

σ A,s 2 (τ) = iλ τ τ 0 dτ ′ φ[x A (τ ′ )][σ A 2 (τ ′ ), σ A 2 f (τ)]. (2.94)
We can nd a similar expression for σ B,s 2 (τ) exchanging the label A with the label B in (2.94) while for the eld operator we can write

φ s [x A/B (τ)] = iλ τ τ 0 dτ ′ σ A 2 (τ ′ )[φ[x A (τ ′ )], φ f [x A/B (τ)]] +σ B 2 (τ ′ )[φ[x B (τ ′ )], φ f [x A/B (τ)]] . (2.95)
Looking at (2.94), we have three possibilities to work out the third-order term of σ A 2 (τ). We can use σ A 2 (τ ′ ) with his second-order term and replace the free part for the eld operator; we can replace the two operator, σ A 2 (τ ′ ) and φ[x A (τ ′ )], with their respective rst-order term; we can replace the eld operator in (2.94) with his second-order term and use the free part for σ A 2 (τ ′ ). Let us start with the rst case mentioned. Keeping in mind the (2.83) and using recursively the (2.94) we can write the following contribution

λ T (τ) + λ 2 τ τ 0 dτ ′ T ′ (τ ′ ) + O(λ 3 ).
(2.96)

where T (τ) is a function of free operators only (so the rst term on the right side of (2.96) is a term at rst order in λ) and

T ′ (τ ′ ) ∝ τ ′ τ 0 τ τ 0 dτ ′ dτ ′′ φ[x A (τ ′ )]φ[x A (τ ′′ )] × [σ A 2 (τ ′′ ), σ A 2 (τ ′ )], σ A, f 2 (τ) .
(2.97)

As we already said, the Casimir-Polder interaction is a fourth-order e ect then we can neglect the term with T (τ) in (2.96). For this reason, from now on, we will neglect all the terms such that the e ective Hamiltonian is of order di erent from the fourth. Then, to expand the term in (2.96) of σ A 2 (τ) up to third order we have to take the rstorder expansion of T ′ (τ ′ ). Since we will average the Hamiltonian (2.90) in the ground state of the atoms A and B we have to keep in mind that

g a/b | σ A/B 2 (τ) 2n+1 |g a/b = 0 (2.98) g a/b | σ A/B 2 (τ) 2n |g a/b 0 (2.99)
where n ∈ N and |g a/b is the ground state of the atom A (B). Also considering that we are interested to calculate an interaction energy between the atoms A and B our expression for the energy must have an interatomic-distance dependence and contain both operators relative to atom A and atom B. Terms which have atomic operators relative only to one of the two atoms could be considered as describing Lamb shift. So any expansion up to rst-order of (2.97) will not give a contribution for the interaction energy. We consider now the second case. That is we take the two operators at rst order. Let us rst consider the rst-order expansion of σ A 2 (τ ′ ). We can obtain it similarly to what we have done in (2.82), i.e. replacing all the operators in the integral with the respective free operators. In this way we can easily see that the contribution given in (2.94) by this term would have three operators σ A 2 (at three di erent times) and therefore it is zero. This because of the average on the ground states of the atoms given by (2.98).

We now consider the third case where we do an expansion up to order two of the eld operator and show that it gives a nonzero contribution. Using (2.95), we can cast the equation (2.94) as follows

σ A,s 2 (τ) = (iλ) 2 τ τ 0 τ ′ τ 0 dτ ′ dτ ′′ σ B 2 (τ ′′ ) φ[x B (τ ′′ )], φ[x A (τ ′ )] σ A 2 f (τ ′ ), σ A 2 f (τ) . (2.100)
where we have neglected a term containing three operators σ A 2 for the reasons before discussed. Analyzing (2.100), we have three possible expansions to get the third-order term. We start expanding σ B 2 (τ ′′ ) and using their free part for the other operators, to obtain the contribution

(iλ) 3 τ τ 0 τ ′ τ 0 τ ′′ τ 0 dτ ′ dτ ′′ dτ ′′′ ×φ f [x B (τ ′′′ )] σ B 2 f (τ ′′′ ), σ B 2 f (τ ′′ ) φ f [x B (τ ′′ )], φ f [x A (τ ′ )] σ A 2 f (τ ′ ), σ A 2 f (τ) .
(2.101) Considering the Hamiltonian (2.90), we must calculate the following expectation value of the energy shift relative to the term in (2.101)

φ f [x A (τ)], φ f [x B (τ ′′′ )][φ f [x B (τ ′′ )], φ f [x A (τ ′ )]] .
(2.102)

In order to compute expectation values as that in (2.102), Wick's theorem is very useful.

According to this theorem it can be shown that, in our case, if we have four operators Â, B, Ĉ, D, the expectation value for the product of the four operators on a state |ψ can be expressed as

ψ| Â B Ĉ D|ψ = ψ| Â B|ψ ψ| Ĉ D|ψ + ψ| Â Ĉ|ψ ψ| B D|ψ + ψ| Â D|ψ ψ| B Ĉ|ψ -2 ψ| Â|ψ ψ| B|ψ ψ| Ĉ|ψ ψ| D|ψ . (2.103)
From this expression we can easily deduce the following identities

[ Â, B] Ĉ D = Ĉ D Â B -B Â , Â[ B, Ĉ] D = Â D B Ĉ -Ĉ B , Â B[ Ĉ, D] = Â B Ĉ D -D Ĉ . (2.104)
With the help of the identities (2.104), we can then write the expression in (2.102) as follows

{φ f [x A (τ)], φ f [x B (τ ′′′ )]} [φ f [x B (τ ′′ )], φ f [x A (τ ′ )]] =2 C F AB (τ, τ ′′′ )(-2)χ F AB (τ ′ , τ ′′ ) (2.105)
where we have introduced, respectively, the eld symmetric correlation function and the eld susceptibility

C F AB (τ, τ ′ ) = 1 2 0|{φ f [x A (τ)], φ f [x B (τ ′ )]}|0 , (2.106)
χ F AB (τ ′ , τ ′ ) = 1 2 0|[φ f [x A (τ)], φ f [x B (τ ′ )]]|0 . (2.107)
The other possible contributions given by (2.100) can be obtained doing the expansion up to order one of the eld operator φ[x B (τ ′′ )] and replacing the others operators with their free part, or expanding the eld operator φ[x A (τ ′ )] at rst order and substituting the others operators with their free part. In this way we obtain the following two contributions

φ f [x A (τ)], φ f [x B (τ ′′′ )], φ f [x B (τ ′′ )] , φ f [x A (τ ′ )] , φ f [x A (τ)], φ f [x B (τ ′′′ )], φ f [x B (τ ′′ )], φ f [x A (τ ′ )] . (2.108)
It is easy to see that, using the identities (2.104), the terms above vanish. We are thus able to write the fourth-order energy shift given by vacuum uctuations contribution

δE A a vf = 4iλ 4 τ τ 0 dτ ′ τ ′ τ 0 dτ ′′ τ ′′ τ 0 dτ ′′′ C F AB (τ, τ ′′′ ) χ F AB (τ ′ , τ ′′ ) χ B b (τ ′′ , τ ′′′ ) χ A a (τ, τ ′ ) (2.109)
where we introduced the linear susceptibilities of the two atoms

χ A a (τ, τ ′ ) = 1 2 a|[σ A 2 f (τ), σ A 2 f (τ ′ )]|a , χ B b (τ, τ ′ ) = 1 2 b|[σ B 2 f (τ), σ B 2 f (τ ′ )]|b . (2.110)
deduce that the operators σ A 2 in the rst addend of (2.112) can be replaced with their relative free parts. Thus to obtain the contribution of the rst part of (2.112) we must expand φ[x A (τ ′ )] to order two. Using (2.95), we can write

H A,1 rr (τ) = λ 4 (iλ) 2 τ τ 0 dτ ′ σ A 2 f (τ), σ A 2 f (τ ′ ) σ B 2 (τ ′′ ) φ[x B (τ ′′ )], φ f [x A (τ ′ )] , φ f [x A (τ)] (2.113)
where we have neglected the part of (2.95) with σ A 2 because it would give Lamb shift terms or vanishing terms, due to (2.98). Now, to obtain the fourth-order expansion we have two possibilities, expanding σ B 2 or expanding φ(x B ). When we expand φ[x B (τ ′′ )], it is easy to show that for the eld operators we have two terms of the form

[ Â, B], Ĉ D -D [ Â, B], Ĉ . (2.114)
With the help of the identities (2.104) we obtain that the two addends in (2.114). This means that only the expansion of σ B 2 (τ ′′ ) gives a nonzero term for equation (2.113). So, using (2.104), we nally get

δE A,1 a rr (τ) = 4iλ 4 τ τ 0 dτ ′ τ ′ τ 0 dτ ′′ τ ′′ τ 0 dτ ′′′ χ F AB (τ, τ ′′′ ) χ F AB (τ ′ , τ ′′ ) χ B b (τ ′′ , τ ′′′ ) C A a (τ, τ ′ ) (2.115)
where we have introduced the symmetrical correlation functions for the atom

A/B C A a (τ, τ ′ ) = 1 2 a|{σ A 2 f (τ), σ A 2 f (τ ′ )}|a , (2.116) C B b (τ, τ ′ ) = 1 2 b|{σ B 2 f (τ), σ B 2 f (τ ′ )}|b . (2.117)
We should remember that (2.115) is only one contribution from the radiation reaction Hamiltonian (2.112). We still must compute the terms given by the expansion of σ B 2 in (2.112). However, the procedure is very similar to the previous ones yielding the vacuum uctuation energy shift and (2.115). After lengthy but straightforward algebra, we can obtain this second contribution of the radiation reaction Hamiltonian. Adding the two contributions of H A rr , we can nally write the complete radiation reaction energy shift

δE A a rr = 4iµ 4 τ τ 0 dτ ′ τ ′ τ 0 dτ ′′ τ ′′ τ 0 dτ ′′′ C A a (τ, τ ′ ) χ B b (τ ′′ , τ ′′′ )χ F AB (τ, τ ′′′ ) χ F AB (τ ′ , τ ′′ ) -4iµ 4 τ τ 0 dτ ′ τ ′ τ 0 dτ ′′ τ ′′ τ 0 dτ ′′′ C A a (τ ′′′ , τ) χ B b (τ ′′ , τ ′ )χ F AB (τ ′′′ , τ ′′ ) χ F BA (τ ′ , τ) -4iµ 4 τ τ 0 dτ ′ τ ′ τ 0 dτ ′′ τ ′′ τ 0 dτ ′′′ C B b (τ ′ , τ ′′′ ) χ A a (τ ′′ , τ)χ F BA (τ ′′′ , τ ′′ ) χ F BA (τ ′ , τ) +4iµ 4 τ τ 0 dτ ′ τ ′ τ 0 dτ ′′ τ τ 0 dτ ′′′ C F AB (τ ′′′ , τ ′′ ) χ F AB (τ, τ ′ ) χ B b (τ ′′ , τ ′ ) χ A a (τ ′′′ , τ). (2.118)
eld in a particular trajectory. In this section, to show the utility of the method we have developed and of our results (2.109) and (2.118) we study the van der Waals force between two atoms at rest. The atom A is xed at the origin while the B atom is placed at a distance z. In this physical system, the statistical functions of the eld can be calculated starting from (2.106) and (2.107); we get

C F AB (τ, τ ′ ) = 1 8π 2 z dω sin(ωz) e -iω(τ-τ ′ ) + e iω(τ-τ ′ ) , χ F AB (τ, τ ′ ) = 1 8π 2 z
dω sin(ωz) e -iω(τ-τ ′ )e iω(τ-τ ′ ) .

(2.120)

For the statistical functions of the atoms instead we have

C A a (τ, τ ′ ) = 1 8 e -iω 0 (τ-τ ′ ) + e iω 0 (τ-τ ′ ) , χ A a (τ, τ ′ ) = 1 8 e iω 0 (τ-τ ′ ) -e -iω 0 (τ-τ ′ ) (2.121)
where we assume for simplicity that the atoms A and B are identical. We rst evaluate the vacuum uctuations contribution. Using equations (2.120) and (2.121) we can cast equation (2.109) as follows

δE A a vf = - iλ 4 1024 π 4 z 2 ∞ 0 dω ∞ 0 dω ′ sin(ωz) sin(ω ′ z) T 0 du ′ T -u ′ 0 du ′′ T -u ′ -u ′′ 0 du ′′′ × e iω 0 u ′ -e -iω 0 u ′ e iω(u ′′′ +u ′′ +u ′ ) + e -iω(u ′′′ +u ′′ +u ′ ) e iω ′ u ′′ -e -iω ′ u ′′ e iω 0 u ′′′ -e -iω 0 u ′′′ (2.122)
where we did the substitutions

u ′′′ = τ ′′ -τ ′′′ , u ′′ = τ ′ -τ ′′ , u ′ = τ -τ ′ , T = τ -τ 0 . (2.123)
Moreover, we will consider the limits τ → +∞ and τ 0 → -∞, so that T → +∞. We start performing the integration over dτ ′′′ . All time integrands are integrals of exponential functions of the form

lim η→0 D 0 du e ±(Ω±iη)u = e ±(Ω±iη)D -1 ± 1 i(Ω ± iη) (2.124)
where Ω is in general sum of two of the three frequencies ω, ω ′ , ω 0 , and D is one of the three values T, Tu ′ , Tu ′u ′′ . We added a pure imaginary part η in the argument of the exponentials; we will let it go to zero after the integration. The sign of η is chosen in a way such that when the u goes to in nity we have a decaying exponential that regularizes

(i) (ii)
Figure 2.5 -Integration contours used to evaluate the integral over ω ′ . The contour (a) has been used for the term in the rst row of (2.126) and the contour (b) for the term in the second row of (2.126).

our integral. This allows to easily deal with the poles in the frequency integration. Using (2.124), after the rst time integration we get

e -iω(u ′ +u ′′ ) 1 i(ω -ω 0 -iη) - 1 i(ω + ω 0 -iη) + e iω(u ′ +u ′′ ) 1 i(ω -ω 0 + iη) - 1 i(ω + ω 0 + iη) .
(2.125)

In the equation above we neglected the terms coming from the exponentials in the right hand side of (2.124), because in the integrand of (2.122) they yield the exponential factor exp(±iωT ). Thus, making the integration over ω and considering that T goes to ∞, from the Riemann-Lebesgue lemma we can assert that these terms are vanishing. We want to stress that, consequently, we have no more explicit time dependence. We can do the same analysis for the integrations over τ ′′ and τ ′ getting the expression

1 i(ω -ω 0 -iη) - 1 i(ω + ω 0 -iη) 2 1 i(ω -ω ′ -iη ′ ) - 1 i(ω + ω ′ -iη ′ ) + + 1 i(ω -ω 0 + iη) - 1 i(ω + ω 0 + iη) 2 1 i(ω -ω ′ + iη ′ ) - 1 i(ω + ω ′ + iη ′ ) . ( 2

.126)

We can now work out the ω ′ integral in the above equation. Analyzing the poles of the integrand we can use the two integration contours in Figure 2.5, respectively for the term in the rst row and for the second row of (2.126), and then the limit of η ′ → 0. The path is chosen according to the sign of η which suggests how avoid the pole. The integral over ω ′ is where we have exploited the parity of the function that multiply sin(ω ′ z) and the residue theorem.

+∞ 0 dω ′ sin(ω ′ z) 2ω ′ ω 2 -ω ′2 = 1 2i +∞ -∞ dω ′ e iω ′ z 2ω ′ ω 2 -ω ′2 = π cos(ωz) for z > 0 (2.
For the last integral over dω we have to deal with the poles at ±ω 0 . These poles are closely related to the dynamic polarizability of the two atoms α(ω) ∝ 1/(ω 2 0ω 2 ). When we consider the generalization that includes the possibility of absorption we can see that a linewidth parameter appears in the de nition of the polarizability [65]. This means that the poles of the polarizability are always in the lower complex plane (the imaginary part is always negative). When we perform our integrations and the limit η → 0 we must take into account that the poles have a negative imaginary part. So the energy shift for the vacuum uctuations contribution becomes

E (vf) CP = λ 4 1024 π 3 z 2 +∞ 0 dω sin(2ωz) 4ω 2 0 (ω 2 -ω 2 0 ) 2 = λ 4 1024 π 3 z 2 ℑ        i +∞ 0 dωe i2ωz 4ω 2 0 (ω 2 -ω 2 0 ) 2        .
(2.128)

To calculate the integral above it is useful to transform it in an integration in the complex plane with the path shown in Figure 2.6. It is simple to see that the integral over I 3 is null. Since there are not singularities inside our closed integration contour, we have

I 2 = -I 1 .
Thus we calculate only the integral over I 1 (that is done by the substitution ω = i u in (2.128)). We will evaluate this integral in the two regimes of near and far zone, introduced in Chapter 1. In the far zone we know that only photons with low frequency (ω ≪ ω 0 ) contributes signi cantly to the interaction and then we simply get

E (vf) CP ≃ - 1 512π 3 λ 4 ω 2 0 z 3 .
(2.129)

In the near zone case, we know that only the high frequency photons are relevant for the interaction (ω ≫ ω 0 ), so

I 1 = +∞ 0 dωe -2ωz 4ω 2 0 (ω 2 -ω 2 0 ) 2 ≃ +∞ 0 dω 4ω 2 0 (ω 2 -ω 2 0 ) 2 (2.130)
and we obtain

E (vf) CP ≃ - 1 1024π 3 λ 4 ω 0 z 2 .
(2.131)

To compute the complete interaction energy it remains to work out the radiation reaction contribution in (2.118). We have already seen that the last row vanishes for symmetrical statistical correlation functions of the eld and this is our case. The term in the rst row is pretty similar to that from vacuum uctuations that we have already calculated. After analogous analysis and time integrals calculations, we can cast this term as follows

iλ 4 1024 π 4 z 2 ∞ 0 dω ∞ 0 dω ′ sin(ωz) sin(ω ′ z) × 1 (ω + ω 0 -iη) 2 - 1 (ω -ω 0 -iη) 2 1 i(ω + ω ′ -iη ′ ) - 1 i(ω -ω ′ -iη ′ ) + + 1 (ω + ω 0 + iη) 2 - 1 (ω -ω 0 + iη) 2 1 i(ω + ω ′ + iη ′ ) - 1 i(ω -ω ′ + iη ′ )
.

(2.132)

The integral over ω ′ is the same in (2.127). Considering again that the poles of the polarizability are in the lower complex plane and performing the integral over ω in the complex plane using the contour of Figure 2.6, we get

+∞ 0 dω sin(2ωz) ωω 0 (ω 2 -ω 2 0 ) 2 =ℑ        +∞ 0 dω e i2ωz ωω 0 (ω 2 -ω 2 0 ) 2        =ℑ        +∞ 0 i du e -2uz i uω 0 (u 2 -ω 2 0 ) 2        = 0. (2.133)
The last integral above is zero because the integrand is a real function so the imaginary part of the integral will be null. It means that the rst term in (2.118) is zero.

We still need to calculate the other two remaining terms of the radiation reaction contribution in equation (2.118). Their calculation is similar and for brevity we will show only the computation for the term in the second row of (2.118). Starting with the integration over τ ′′′ we get terms of the following form

e iω 0 (u ′′ +u ′ ) 2ω ω 2 -ω 2 0 , e -iω(T -u ′ ) e ±iω 0 T i(ω ∓ ω 0 ) e i(ω+ω 0 )u ′′ -e i(ω-ω 0 )u ′′ , e iω(T -u ′ ) e ±iω 0 T i(ω ± ω 0 ) e -i(ω-ω 0 )u ′′ -e -i(ω+ω 0 )u ′′ . (2.134)
For terms as the rst in (2.134), using previous considerations on the poles of the polarizability, it is possible to perform the integration over ω without performing the time integrations rst. Hence we have

+∞ 0 dω sin(ωz) 2ω ω 2 -ω 2 0 = ℑ +∞ 0 dω e iωz 2ω ω 2 -ω 2 0 = ℑ +∞ 0 i du e -uz i 2u u 2 -ω 2 0 = 0. (2.135)
For the other terms in (2.134), the ones contain the parameter T , after the time integrations, we obtain expressions containing e ±iωT , e ±iω ′ T and other terms proportional to

± sin(ωz) sin(ω ′ z) (ω ± ω 0 ) 2 2ω ′ ω ′2 -ω 2 0 .
(2.136)

Terms containing e ±iωT and e ±iω ′ T vanish because of the Riemann-Lebesgue lemma, while the others are null when we integrate over ω ′ (see equation (2.135)). Thus the second and third terms of (2.118) are zero.

We can nally conclude that in the case considered here the radiation reaction contribution to the Casimir-Polder interaction energy is zero. The interaction energy is given exclusively by the vacuum uctuations contribution; in the regimes of far and near zone the expressions of this energy are, respectively, the (2.129) and (2.131). This last result it is not too surprising if we think to the physical interpretation of Casimir-Polder forces given by Power and Thirunamachandran in [8] (equivalent to that given for the vacuum contribution in (2.109)), and often used to calculate consistently the Casimir-Polder forces for many physical situations.

Thermal and non-thermal signatures of the Unruh e ect in Casimir-Polder forces

In this Section, we present our original results aiming at bridging the Casimir forces and the Unruh e ect, showing that both thermal and non-thermal features associated to a relativistic uniformly accelerated motion can be probed through the Casimir-Polder force between two accelerating atoms. In order to inspect the hallmarks of relativistic accelerations on Casimir-Polder forces, we start using the general formula (2.72), which allows for the computation of Casimir-Polder forces in generic stationary conditions from rst principles, extended to fourth order in perturbation theory (see equations (2.109) and (2.118)) [45], [AN6]. In particular, we consider the interaction energy, arising from quantum vacuum uctuations, among two atoms moving with a uniform proper acceleration a in the same direction and separated by a constant distance z, perpendicular to their trajectories, and linearly coupled to a scalar eld [40], [AN3]. We show that the Casimir-Polder force between the two accelerating atoms displays a novel transition in its distance dependence at a new length scale, z a , given by the inverse of the atomic acceleration (hereafter we use natural units = c = k B = 1.). Such a transition is a cross-over in the interaction energy, from a Casimir-Polder potential for az ≪ 1 , where the static zero-temperature interaction (as z -2 and z -3 in the nonretarded and retarded regimes, respectively) receives a small thermal-like correction due to acceleration at the Unruh temperature T U = a/2π, to a nonthermal interaction energy for az ≫ 1, characterized by a z -4 power law decay. This result should be compared with the Casimir-Polder force between two static atoms interacting with the scalar eld at temperature T , where at the thermal wavelength λ th. ∼ 1/T the interaction shows a transition from the z -3 quantum regime to the z -2 thermal classical regime. The new characteristic length z a ∼ 1/a is associated with the breakdown of the approximate description of the system in terms of a local inertial frame, and it indicates that the Casimir-Polder interaction is strongly reshaped by the presence of the non-inertial space-time background, associated to the relativistic accelerated motion of the two atoms. This phenomenology is a non-trivial extension of the Unruh thermal response detected by a single accelerated observer to a system of two accelerated particles [40], [AN3].

Thermal Casimir-Polder interactions

We consider the same Hamiltonian of a pair of two-level atoms (A,B), interacting with the scalar eld, that we introduced in Section 2.4, characterized by the same transition frequency ω 0 and linearly coupled to a massless scalar eld φ(x) by the coupling constant λ. This Hamiltonian, in the Dicke notation [66] and in natural units ( = c = 1), is

H = ω 0 σ A 3 (τ) + ω 0 σ B 3 (τ) + d 3 kω k a † k a k dt dτ + λσ A 2 (τ)φ(x A (τ)) + λσ B 2 (τ)φ(x B (τ)), (2.137)
where σ i (i = 1, 2, 3) are the Pauli matrices, a k , a † k are the annihilation and creation operators of the massless scalar eld φ(x) with the linear dispersion relation ω k = |k|. The Hamiltonian (2.137) is expressed in terms of the same proper time τ of the two atoms (assuming a background at spacetime), and the interaction term is evaluated on a generic stationary trajectory x(τ) of the two atoms. The distance z between the atoms, perpendicular to their acceleration, is constant. Quantum uctuations of the eld, as well as radiation source elds, can induce an e ective interaction among the two atoms at fourth order in the atom-eld interaction. Following the procedure developed in the previous section, we use (2.109) and (2.118) as our starting point.

Since we want to focus on the relation between Casimir-Polder interactions and Unruh e ect and to consider the connection between the latter and thermal e ects we start our analysis calculating the Casimir-Polder energy in the case of a nite temperature T . For the thermal Casimir-Polder interaction, following procedures and considerations analogous to that done in Section 2.4.4, it is possible to show that the radiation reaction contribution is negligible compared to the vacuum uctuation contribution for all the cases considered in this section. Speci cally, at small temperatures, i.e. for T ≪ ω 0 , or, in the case of two uniformly accelerating atoms, for a ≪ ω 0 . Thus we concentrate on the vacuum uctuations contribution only.

A simple generalization of (2.109) allows us to obtain the scalar Casimir-Polder force at nite temperature T , in terms of the thermal correlation function and susceptibility for a scalar eld

C F th. (φ f (x A (τ)), φ f (x B (τ ′ ))) = 1 8π 2 1 z ∞ 0 dω sin(ωz) coth ω 2T (e -iω(τ-τ ′ ) + e iω(τ-τ ′ ) ), χ F th. (φ f (x A (τ)), φ f (x B (τ ′ ))) = 1 8π 2 1 z ∞ 0
dω sin(ωz)(e -iω(τ-τ ′ )e iω(τ-τ ′ ) ).

(2.138)

The explicit computation is performed in the limit of small temperatures, T ≪ ω 0 , following a general method originally introduced by Lifshitz [68,69,70]. In view of the comparison with the Casimir-Polder force between two accelerated atoms, which is the main point of this section, it is important to stress that at nite temperatures, the massless thermal wavelength λ th. ∼ 1/T separates a quantum regime from a classical thermal regime. Indeed, for distances z ≪ λ th. we nd the expression for the static scalar Casimir-Polder force in near and far zone plus subleading thermal corrections proportional toλ 4 z ( T ω 0 ) 2 ; on the other hand, for distances larger than the typical length scales associated to quantum e ects, i.e. for z ≫ λ th. , the Casimir-Polder force manifests again a classical thermal behavior similar to that in the near zone

E th. CP = - 1 512π 3 λ 4 ω 2 0 T z 2 ,
(2.139)

as it has been already noticed for the electromagnetic case [55,70].

Unruh corrections to Casimir-Polder interactions

We now apply the method we have developed in Section 2.4 to the case of two atoms moving with the same uniform acceleration, perpendicular to their separation. In this case, a modi cation of their Casimir-Polder interaction is expected, because the two atoms perceive modi ed vacuum uctuations, as the Unruh e ect would suggest [47,32]" [AN1]. It is important to stress that for the accelerated case here analyzed we will focus our attention only on vacuum uctuations contribution because it can be shown, with a calculation similar to that performed in Section 2.4.4 that the radiation reaction contribution is negligible compared to the vacuum uctuations contribution. An atom moving with uniform relativistic acceleration a in the x direction follows the worldline (we remember that we are using natural units = c = k B = 1)

t(τ) = 1 a sinh(aτ) x(τ) = 1 a cosh(aτ) y(τ) = z(τ) = 0. ( 2

.140)

We are now going to show how interatomic Casimir-Polder interactions allow to distinguish the e ect of a relativistic acceleration from a thermal behavior. Even if such a thermal character have been envisaged in a large number of situations [4,56,71,72], departures from thermal predictions for accelerating atoms have been shown in the Lamb shift and in the spontaneous excitation of accelerating atoms, coupled to the electromagnetic eld, in vacuum space [15,35] or in front of a conducting plate [36,39,57,73].

In such a situation it is convenient to introduce a new set of coordinates, necessary to cover the Minkowski spacetime (t, x) accessible to accelerated observers. They are de ned in two regions, the Rindler wedges, which are causally disconnected, and where a Rindler metric can be de ned accordingly [4,16].

We consider two uniformly accelerating atoms, moving along the worldlines (2.140) with the same uniform acceleration a ≪ ω 0 , and separated by a distance z orthogonal to the acceleration direction x. We now show that, at short distances, Casimir-Polder interactions can probe thermal Unruh-like e ects, while at larger distances they reveal a nonthermal behavior due to the intrinsically non-inertial nature of the Rindler metric. As done in (2.138) for the thermal Casimir-Polder force, we rst obtain the correlation function and susceptibility of the scalar eld in the accelerated background

C F acc. (φ f (x A (τ)), φ f (x B (τ ′ ))) = 1 8π 2 1 N(z, a) ∞ 0 dω f (ω, z, a) coth πω a (e -iω(τ-τ ′ ) + e iω(τ-τ ′ ) ) , χ F acc. (φ f (x A (τ)), φ f (x B (τ ′ ))) = 1 8π 2 1 N(z, a) ∞ 0 dω f (ω, z, a)(e -iω(τ-τ ′ ) -e iω(τ-τ ′ ) ) , (2.141)
where f (ω, z, a) = sin( 2ω a sinh -1 ( az 2 )) and N(z, a) = z 1 + (az/2) 2 . A close comparison between (2.141) and (2.138) shows that for az ≪ 1 the correlation function (2.141) has a thermal-like behavior set by the Unruh temperature T U . Hence, the vacuum uctuations contribution (2.109) to the Casimir-Polder interaction exhibits, at the lowest order in az, the same thermal-like correction ∼ -λ 4 z ( T U ω 0 ) 2 , found for the Casimir-Polder interaction at nite temperature. At higher orders in az, equation (2.141) shows that the correction due to the accelerated atomic motion starts to di er signi cantly from a thermal-like correction. (A similar behavior of the correlation functions (2.141) can be extrapolated from the Lamb shift and spontaneous emission corrections of an accelerated atom near a conducting plate obtained in [36,39,73]) This discrepancy suggests a strong breakdown of the common analogy between acceleration and nite temperature for the Casimir-Polder potential at distances z ≫ z a ∼ 1/a (z a ∼ c 2 /a when units with c 1 are considered), resulting in a novel power law behavior of the Casimir-Polder interaction,

E acc. CP = - 1 512π 4 λ 4 ω 2 0 z a z 4 .
(2.142)

Our result (2.142) shows that the Casimir-Polder interaction energy between two accelerated atoms decreases faster with the distance than in both near and far zones [40], [AN3]. This can be guessed from the following heuristic argument: since both atoms are accelerating, the distance traveled by a scalar photon emitted by one atom to reach the other atom increases with time, and this results in an overall decrease of the interaction strength among them (see Section 2.3). A more precise comparison between our result in (2.142) and the results obtained in 2.3 is not straightforward because in the latter case the interaction energy is time-dependent and valid in a well-de ned time interval, while the present result involves a time average of the interaction energy, as it is evident from (2.109). In fact, considerations similar to those used after (2.125) show that in the present case there is not time dependence of the force, due to our time-averaging procedure. We can consider the behavior described by (2.142) as a new quantum regime, as opposed to the classical thermal regime given by (2.139) which, on the contrary, destroys the quantum retarded z -3 Casimir-Polder interaction. Also, we wish to stress that the distance z a is the characteristic length scale for the breakdown of the local inertial frame approximation [60]: for distances smaller than z a , it is possible to nd a local inertial frame where the correlation functions of the scalar eld are fairly well described by the their thermal Minkowski analogue, and the only net e ect of acceleration is embodied in the Unruh thermal analogy; on the other hand, signals spreading over distances larger than z a must take into account the non-inertial character of relativistic acceleration, encoded in the non-Minkowskian metric. Consequently, eld quantization in Rindler spacetime will strongly a ect the nature of vacuum uctuations (C F ) and eld susceptibility (χ F ), ultimately leading to the novel power law behavior of the Casimir-Polder potential (2.142). This phenomenology is in sharp contrast with the classical e ect outlined above for the Casimir-Polder interaction at nite temperature (see equation (2.139)) and it is summarized in Fig. 2.7. It should be noted that such an e ect cannot be detected by a single uniformly accelerated point-like detector in the unbounded space, as in [4,56], since in this case it is always possible to nd a local set of Minkowski coordinates in the neighborhood of a point-like detector. With this respect, our result can be seen as a non-trivial extension of the Unruh thermal response detected by a single accelerating observer, to a system of two relativistic accelerated systems. Finally, we wish to point out that the qualitative change of Casimir-Polder force described by equation (2.142) is ultimately grounded on the noninertial character of the accelerated background and it is expected to manifest ubiquitously also for other elds, as well as for multi-level atoms.

In conclusion, we have shown how Casimir-Polder forces among two uniformly accelerating atoms can probe non-thermal e ects beyond the Unruh analogy between uniform acceleration and nite temperature. We have shown that for interatomic distances above the characteristic length scale associated to a local inertial description of the system, the Casimir-Polder energy shows a di erent power law dependence with the distance, compared to the corresponding potential at nite temperature [40], [AN3].

A qualitative change of the interatomic potential may also a ect some macroscopic properties of an accelerated many-atoms system, as the following example would suggest (analogous ideal experiments were envisaged in [74] for an accelerating box lled with photons). Let us consider a box lled with atoms with a given proper density and moving with nite acceleration a. The qualitative change of the interaction between the atoms from a marginal long-range z -3 to a short-range z -4 at the acceleration-dependent scale z a given by (2.142), could manifest in a change of its thermodynamical properties (for example in the equation of state of the gas), if the average interatomic distance is larger than ∼ 1/a, since the thermodynamics of long-range and short-range interacting systems is sharply di erent. (We adopt the de nition of long-range interacting systems, U(z) ≃ 1/z α , with α ≤ d where d is dimensionality of the system, relevant for thermodynamics [75]). This density/acceleration cross-over is of quantum origin, and it could have also consequences on the thermodynamics of the Universe during the stages of its evolution.

Also, we wish to stress that our new expressions for the fourth-order vacuum uctuations and radiation reaction contributions to energy shifts have a general validity (see [45], [AN6]), and they could be straightforwardly applied to investigate electromagnetic dispersion interactions involving accelerating atoms [76,15,35,39,32], [AN1] or atoms in circular motion, which could be relevant to detect the Unruh e ect [24]. Furthermore, they can be easily employed to compute dispersion forces between two atoms outside a Schwarzschild black hole or in de Sitter spacetime, where Casimir forces could provide new physical insights into problems of cosmological interest (similarly to recent calculations of the Lamb Shift in curved backgrounds [77,78]).

The resonance interaction between two uniformly accelerated identical atoms

In this section we show our results obtained in the investigation of the resonance interaction between two uniformly accelerated identical atoms, one excited and the other in its ground state, prepared in a correlated (symmetric or antisymmetric) state [79], [START_REF] Rizzuto | quindi per ultimi i ringraziamenti a voi appartenenti a questa terra piena di contraddizioni. Grazie a tutti gli amici di Palermo, agli storici coinquilini, agli ancor più storici amici fraterni di[END_REF]. We show that this interaction exhibits, similarly to the ground-state accelerated atoms considered in the previous section, a pure non-thermal behavior, carrying no signature of Unruh thermal uctuations on the interatomic force. Nevertheless, we show that the relativistic acceleration still causes a qualitative change of the distance-dependence of the interaction between the two atoms, as a consequence of the metric e ects associated to relativistic accelerations.

We have seen in Chapter 1 that resonant interactions between atoms occur when one or more atoms are in their excited state and an exchange of real photons between the atoms is involved [START_REF] Craig | Thirunamachandran Molecular Quantum Electrodynamics[END_REF][START_REF] Salam | Molecular Quantum Electrodynamics[END_REF]. If the two atoms are prepared in an uncorrelated state, resonant Casimir-Polder interaction requires a fourth-order perturbation theory. In this case, the interaction scales as R -2 for large interatomic separations, R ≫ λ (R being the interatomic distance and λ the wavelength associated to the atomic transition). These e ects have been recently discussed in the literature mainly focusing on the spatially oscillating behavior of the force [START_REF] Berman | [END_REF]83,84]. On the other hand, resonant interactions can occur also when the two atoms are prepared in a correlated (symmetric or antisymmetric) state, and in this case they manifest as a second-order e ect in the electric charge. Such interactions are of very long range, showing a R -1 dependence in the far-zone limit, and can be much larger than the usual dispersion interactions. It should be noted that decoherence e ects induced by the environment can destroy the entanglement between the atoms, and this poses serious limits on the observability of such resonant e ects. Recently, the possibility to control (for example to enhance) resonant forces between atoms placed in nano-structured materials (such as a photonic crystal) has been discussed [85]. Also, such e ects have been investigated in relation to the resonant energy transfer between molecules, and it has been discussed that they can play a fundamental role in biological systems [86,87,88].

We shall now consider two identical atoms, one in the excited state and the other in the ground state, prepared in a correlated (symmetric or antisymmetric) state and uniformly accelerating in vacuum, and investigate the e ect of the atomic acceleration on the resonance interaction between the two atoms. We rst consider the atoms interacting with a relativistic scalar eld and then we generalize our investigation to the electromagnetic eld case. Since the atoms are prepared in a correlated state, our calculation requires only a second-order perturbation theory. We shall show that also for this system that new features appear as a consequence of the acceleration; speci cally a di erent scaling of the interaction energy with the distance and a new dependence on the acceleration, if compared to the "Unruh-thermal" Casimir-Polder interaction case. Following the procedure adopted in Section 2.4, we separate the contributions of vacuum uctuations and radiation reaction to the resonant energy shift of the two atoms. We develop the method at second order in perturbation theory as in [43,44,56,46]. We show that, in both cases considered, the resonance interaction is exclusively related to the radiation reaction contribution. Thus, Unruh thermal uctuations do not a ect the interatomic interaction. Beyond a characteristic length associated to the breakdown of a local inertial description of the system of two atoms, non-thermal e ects, present in the radiation reaction corrections, change qualitatively the distance-dependence of the resonance interaction. Thus our approach permits to highlight non-thermal signatures of the atomic acceleration, through the second order resonant interaction between atoms.

The section is organized as follows. In Section 2.6.1 we introduce the model and we discuss the resonance interaction between two accelerated atoms interacting with the scalar eld in the vacuum state. In Section 2.6.2 we generalize our procedure to the more realistic case of atoms interacting with the electromagnetic eld. Section 2.6.3 is nally devoted to conclusions and perspectives. Details of some calculation are given in the Appendix A

The scalar eld case

Let us consider two identical atoms A and B modeled as point-like systems with two internal energy levels, ± 1 2 ω 0 , associated with the eigenstates |g and |e , respectively, and separated by a distance z. We assume the two atoms accelerating with the same uniform acceleration along two parallel trajectories, x A (τ) and x B (τ), and interacting locally with a real massless scalar eld in its vacuum state. Also we assume that ω 0 includes any direct modi cation of the atomic transition frequency due to the accelerated motion. The Hamiltonian describing the atom-eld interacting system in the instantaneous inertial frame of the two atoms is (τ is a proper time) (2.137)

H(τ) = ω 0 σ A 3 (τ) + ω 0 σ B 3 (τ) + k ω k a † k a k dt dτ + λ σ A 2 (τ)φ x A (τ) + σ B 2 (τ)φ x B (τ) (2.143)
where σ i (i = 1, 2, 3) are the atomic pseudospin operators" and a † k , a k are the bosonic operators of the scalar eld

φ(x, t) = k 2Vω k a k (t)e ik•x + a † k (t)e -ik•x . (2.144)
We want to calculate the resonant energy shift of the system of the two accelerated atoms, using the procedure of Section 2.4. Since the resonant interaction is a second-order e ect in the coupling constant, we use our expansion of the (2.72) simply up to second order. Then we can derive, after some algebra, the e ective Hamiltonians H v f,sr at second order in the coupling for one of the two atoms. We obtain

H e f f A v f = -i λ 2 2 τ τ 0 dτ ′ C F (x A (τ), x A (τ ′ )) σ f 2,A (τ), σ f 2,A (τ ′ )
and

H e f f A sr = -i λ 2 2 τ τ 0 dτ ′ χ F (x A (τ), x A (τ ′ )) σ f 2,A (τ), σ f 2,A (τ ′ ) -i λ 2 2 τ τ 0 dτ ′ χ F (x A (τ), x B (τ ′ )) σ f 2,A (τ), σ f 2,B (τ ′ ) (2.145)
where the statistical functions for the scalar eld have been introduced in (2.106) and (2.107).

The resonant interaction between two atoms moving on the stationary trajectories x A (τ) and x B (τ), is then obtained evaluating the expectation value of e ective Hamiltonians (2.145) and (2.145), on the correlated state of the two atoms, and taking into account only the terms depending on the atomic separation.

In order to do that, we suppose the system prepared in one of the correlated states

| ψ ± = 1 √ 2 (| g A , e B ; 0 k ± | e A , g B ; 0 k ) , (2.146)
where, as mentioned before, g (e) indicates the ground (excited) state of the atom, and | 0 k the scalar eld vacuum state. In such states the atomic excitation is delocalized among the two atoms. The symmetrical state is called a superradiant state, because in the Dicke model its decay rate is larger than that of the individual atoms, yielding a collective spontaneous decay [66]. On the contrary, the antisymmetric combination is called a subradiant state, because its decay rates is inhibited. Di erent methods to obtain super-and sub-radiant states for two-level systems, have been proposed (see for example [89,90]). On the other hand, it has been recently shown that entanglement between accelerated systems can be induced by the Unruh bath [41,42]. This is relevant for our problem, because resonance interactions require the atoms prepared in a correlated (symmetric or antisymmetric) state, and decoherence induced by the environment could pose serious limits on the observability of such e ects.

To obtain the resonant energy shift for the system considered, we now evaluate the expectation values of H e v f and H e sr on the state (2.146). After some algebra, we get

(δE) v f = (δE A ) v f + (δE B ) v f + lim (τ-τ 0 )→∞ - iλ 2 τ τ 0 dτ ′ C F (x A (τ), x A (τ ′ ))χ A (τ, τ ′ ) + (A ⇆ B terms) (2.147) (δE) sr = (δE A ) sr + (δE B ) sr = lim (τ-τ 0 )→∞ - iλ 2 τ τ 0 dτ ′ χ F (x A (τ), x A (τ ′ ))C A (τ, τ ′ ) + lim (τ-τ 0 )→∞ - iλ 2 τ τ 0 dτ ′ χ F (x A (τ), x B (τ ′ ))C A,B (τ, τ ′ ) + (A ⇆ B terms) (2.148)
were we have introduced the atomic statistical functions

C A,B (τ, τ ′ ) = 1 2 ψ ± | σ f 2,A (τ), σ f 2,B (τ ′ ) |ψ ± (2.149) χ A,B (τ, τ ′ ) = 1 2 ψ ± | σ f 2,A (τ), σ f 2,B (τ ′ ) |ψ ± . (2.150) (2.151)
It is clear that, at the order considered, the only contribution to the resonant interaction between the two atoms arises from the second term of (2.148). In fact, the (second order) vacuum uctuations contribution (2.147) is exclusively related to nonlocal eld correlations (expressed by C F (x A (τ), x A (τ ′ ))) evaluated on the trajectory of each atom (A or B), as if the other were absent; therefore (2.147) describes only the contribution of vacuum uctuations to the Lamb-shift of each atom. Similarly, the rst term of (2.148) describes the self-reaction contribution to the Lamb shift of each atom. On the contrary, the second term of (2.148) is the only relevant for the resonance interaction; it describes the interaction of each atom with the eld it emits modi ed by the presence of the other atom (as expressed by χ F (x A (τ), x B (τ ′ ))). It depends on the distance between the two atoms, and therefore it contributes to the interatomic interaction. It turns out that the resonance interaction is entirely due to the radiation reaction contribution, δE sr , vacuum eld uctuations having no role at the order considered. This is indeed expected on a physical ground, because resonant interactions are second order e ects, originating from the exchange of photons between two correlated atoms. Atomic correlations are not induced by the (correlated) vacuum uctuations, as in the dispersive interactions, but they are given by the atomic state considered.

The procedure outlined above is general and valid for any arbitrary stationary trajectory. We now focus on the speci c situation of the atoms moving along the trajectories (2.152) with the same uniform acceleration a,

t(τ) = c a sinh aτ c , x A/B (τ) = c 2 a cosh aτ c , y A/B (τ) = 0, z A (τ) = z A , z B (τ) = z B . (2.152)
We rst evaluate the the linear susceptibility of the scalar eld and the atomic correlation function. We have

χ F (x A (τ), x B (τ ′ )) = - 8π 2 c 2 1 z √ N(z, a) ∞ 0 dωg(ω, z, a), e iω(τ-τ ′ ) -e -iω(τ-τ ′ ) (2.153) and C A,B (τ ′ , τ) = ± 1 8 e iω 0 (τ-τ ′ ) + e -iω 0 (τ-τ ′ ) (2.154)
where we have de ned

N(z, a) = 1 + (za/2c 2 ) 2 , g(ω, z, a) = sin 2ωc a sinh -1 za 2c 2 .
(2.155) Now, substituting (2.153) and (2.154) in (2.148) and taking into account only z-dependent terms, we nally obtain the resonance interaction between the two accelerated atoms

δE = (δE A ) sr + (δE B ) sr = ∓ λ 2 16π 2 c 2 1 z √ N(z, a) ∞ 0 dωg(ω, z, a) 1 ω + ω 0 + 1 ω -ω 0 (2.156)
where ∓ signs refer, respectively, to the energy shift for the symmetric and antisymmetric states. The integral above can be computed analytically, giving

δE = ∓ λ 2 16πc 2 1 z √ N(z, a) cos 2ω 0 c a sinh -1 za 2c 2 .
(2.157) Equation (2.157) is the main result of this section. Since the interaction is entirely due to the radiation reaction contribution, the e ect of relativistic acceleration will leave no thermal signatures on the resonance interaction; its only e ect comes from the normalization factor N(z, a) and the function g(ω, z, a). Most importantly, the factor N(z, a) yields a deviation from the inertial character of the metric. Indeed, as discussed in previous section, we can identify a new characteristic length scale, z a = c 2 /a, associated to the breakdown of the approximate description of the system in terms of a local inertial frame. For distances smaller than z a , it is possible to nd a local inertial frame where the linear susceptibility of eld is fairly well described by its static counterpart; on the other hand, signals spreading over distances larger then z a , cannot disregard the non-inertial character of relativistic acceleration, encoded in the non-Minkowskian Rindler metric. Accordingly, we expect that relativistic accelerations can deeply modify the qualitative behaviour of the resonant interaction energy. In fact, in the limit z ≫ c 2 /a, we get

δE ≃ ∓ λ 2 8π 1 z 2 a cos 2ω 0 c a log za c 2 , (2.158)
while for z ≪ c 2 /a we recover the static result

δE = ∓ λ 2 16πc 2 1 z cos ω 0 z c (2.159)
Thus, the resonance interaction strongly bears signatures of the relativistic acceleration, resulting in a new power law decaying as z -2 compared to the usual z -1 of the static case (see equation (2.159)). This result should be compared with that obtained in Section 2.5, where it was shown that, as a consequence of the metric e ects, the scalar Casimir-Polder interaction between two uniformly accelerated atoms, was characterized by a new z -4 power law decay, for distances z ≫ c 2 /a. Also, equation (2.158) exhibits a global overall pre-factor depending on the inverse of the acceleration, while the "thermal-Unruh" analogy would have suggested the presence of a Unruh term at temperature T U = a/2π, directly proportional to acceleration [40], [AN3]. Therefore, our result shows that it is possible to single out metric e ects associated to relativistic accelerations from the usual "Unruh thermal-like" e ects. The limit z ≪ c 2 /a gives back the expression of the resonant interaction in the static case. For typical interatomic distances (z ∼ 10 -6 m), this is valid also for high acceleration values, thus suggesting that the resonance interaction is almost insensitive to the atomic acceleration in the limit z ≪ z a . Actually, such a behavior can be expected from the following qualitative considerations. Resonance interactions arise from the exchange of real photons between the atoms. Then, if the distance between the atoms is much smaller then z a , in the time spent by the real photon emitted by one atom to reach the other atom (t ∼ z/c), the accelerating atoms move of a distance (x) smaller than their interatomic distance z. The photon mediating the interaction then cannot discern the atomic motion, and the interaction appears to be the static one.

Finally, it is worth to note from (2.158), that the static resonance interaction decreases as z -1 for any interatomic distance. This is a consequence of the scalar model we have considered and of the fact that the resonance interaction is essentially the interaction of an atom with the eld emitted by the other atom. The situation is quite di erent in the case of electromagnetic eld, as we are going to discuss in the next subsection.

The electromagnetic eld case

We now extend our previous investigation to the case of two uniformly accelerated atoms interacting with the electromagnetic eld in the vacuum state. To describe our system, we adopt the Hamiltonian in the multipolar coupling scheme and in the dipole approximation

H = H A + H B + k j ω k a † k j a k j dt dτ -µ A (τ) • E(x A (τ)) -µ B (τ) • E(x B (τ)) (2.160)
where E(x(τ)) is the electric eld operator and µ = er the atomic dipole moment operator. As already discussed, the resonant interaction energy is related only to the radiation reaction contribution and it is obtained from the expectation value of the e ective Hamiltonian (H e f f A ) sr + (H e f f B ) sr on the states

|ψ ± δE = - e 2 2 τ τ 0 dτ ′ χ F ℓm (x A (τ), x B (τ ′ ))C A/B ℓm (τ, τ ′ ) + (A ⇆ B terms) (2.161)
In order to calculate this quantity, we rst obtain the eld and atomic statistical functions. The susceptibility of the electromagnetic eld in the accelerated frame can be obtained from the two-point eld correlation function in the proper reference frame of the two accelerated atoms (Rindler noise) [15]. After lengthy calculations involving Lorentz transformations of the electromagnetic eld, we obtain

g ℓm (x A (τ), x B (τ ′ )) = 0|E ℓ (x A (τ))E m (x B (τ ′ ))|0 = a 4 4πc 7 1 sinh 2 a(τ-τ ′ -iǫ) 2c -za 2c 2 2 3 × δ ℓm - za 4c 2 n ℓ k m sinh 2 a(τ -τ ′ ) 2c + za 2c 2 2 [δ ℓm -2n ℓ n m ] 1 + 2(δ ℓm -k ℓ k m ) sinh 2 a(τ -τ ′ ) 2c (2.162)
(ℓ, m = x, y, z). n = (0, 0, 1) is the unit vector along the z direction and k = (1, 0, 0) is the unit vector along the direction x of acceleration. A simple calculation shows that the only nonzero components of g ℓm are the xx, yy, zz, and xz components. In particular, g ℓℓ (x A (τ), x B (τ ′ )) g mm (x A (τ), x B (τ ′ )) (for ℓ m); Therefore, the Rindler noise evaluated on the atomic trajectories of the two accelerated atoms, is not isotropic and displays a nondiagonal component. A similar anisotropy is not present in the case of a single uniformly accelerated atom in the unbounded space, where it is possible to show that the Rindler noise is isotropic. Actually, we have two distinct spatial directions, namely the direction of the acceleration and the direction of distance vector between the two atoms; in this sense, the anisotropic aspect of the Rindler function can be ascribed to the spatial extent of the two-particles system considered. From Eq (2.162), we can obtain the linear susceptibility of electromagnetic eld in the proper reference frame. Its expression, as an integral over frequencies, is

χ ℓm (x A (τ), x B (τ)) = ia 4 πc 7 1 √ N(z, a) M ℓm c 1 (z, a)] d 2 dT 2 + c 2 (z, a) d dT + c 3 (z, a) F(T, u) + Q ℓm c 4 (z, a) d dT + c 5 (z, a) F(T, u) (2.163)
where M ℓm (z, a), Q ℓm (z, a) and the coe cients c i (z, a) (i = 1, ...5) are given in the Appendix A, and we have de ned

F(T, u) = ∞ 0 dω sin(ωT )(e iωu -e -iωu ), u = τ -τ ′ T = 2c a sinh -1 za 2c 2 (2.164)
The symmetric correlation function for the atoms is

C A/B ℓm (τ, τ ′ ) = 1 2 ψ ± | r A ℓ (τ), r B m (τ ′ ) |ψ ± = ± 1 4 e iω 0 (τ-τ ′ ) + e -iω 0 (τ-τ ′ ) (r A 12 ) ℓ (r B 21 ) m (2.165)
The resonance interaction between the two accelerated atoms is now obtained substituting Eqs. (2.163) and (2.165) into Equation (2.161), and taking the limits τ 0 → -∞, τ → ∞. After some algebraic manipulation (details are given in the Appendix A), we obtain

δE = ±(µ A 21 ) ℓ (µ B 12 ) m V ℓm (ω 0 , z, a) + ( za 2c 2 ) 2 1 N(z, a) U ℓm (ω 0 , z, a) (2.166)
where the explicit expressions of V ℓm (ω 0 , z, a) and U ℓm (ω 0 , z, a) are given in the Appendix A. These quantities explicitly depend on the atomic acceleration and can be interpreted as a generalization of the static interaction potential to the case of accelerated atoms. The expression given above is valid for any value of az/c 2 . As before, we now investigate the two cases, z ≪ c 2 /a and z ≫ c 2 /a.

It is easy to show that, for za ≪ 1, the linear susceptibility (2.163) is fairly well described by its static counterpart. Therefore, at the lowest order in za, we recover the usual expression of the resonance interaction in the static case [START_REF] Craig | Thirunamachandran Molecular Quantum Electrodynamics[END_REF]

δE = ±(µ A 21 ) ℓ (µ B 12 ) m V ℓm (ω 0 , z) (2.167)
where V ℓm (ω 0 , z) is the well-known tensor potential

V ℓm = 1 z 3 (δ ℓm -3n ℓ n m ) cos ω 0 z c + ω 0 z c sin ω 0 z c -(δ ℓm -n ℓ n m ) z 2 ω 2 0 c 2 cos ω 0 z c (2.168)
In particular, in the far-zone limit (R ≫ λ), the resonance interaction decreases as ∼ 1/z, while in the near-zone (R ≪ λ), δE ∼ z -3 [START_REF] Craig | Thirunamachandran Molecular Quantum Electrodynamics[END_REF].

On the other hand, at higher orders in az, we expect that the corrections due to the atomic acceleration cause a qualitative change of the resonance interaction, scaling with a di erent power law. In particular, it is interesting to consider the situation when one of the two dipoles is oriented along the x direction and the other along the z direction. A similar situation was not present in the case of scalar eld considered before, since it is clearly related to the peculiar anisotropic nature of the Rindler noise of the electromagnetic eld.

In this case, the only non-zero contribution to the resonant interaction comes from the anisotropic term W ℓm (ω 0 , z, a) in (2.166), and we nd

δE CP = ±µ A z µ B x 1 2z 3 cos 2cω 0 a ln az c 2 (2.169)
Our result (2.169) shows that, for speci c orientations of the two dipole moments, the resonance interaction between two accelerated atoms decreases with the distance as z -3 . This also shows that it is possible to control the e ect of atomic acceleration by an appropriate choice of the orientation of the dipole moments. A comparison with the case of the scalar eld discussed in the previous section shows that the emergence of a qualitative change of the resonance interaction behavior comes not only from the presence of the metric factor, N(z, a), but also from the anisotropic structure of the eld susceptibility, related to the vector nature of the electromagnetic eld and to the fact that our two-atom system is spatially extended.

Conclusions and Future Perspectives

In this section we have investigated the resonance interaction between two uniformly accelerated atoms, one excited and the other in ground state, prepared in correlated (symmetrical or anti-symmetrical) state [79], [START_REF] Rizzuto | quindi per ultimi i ringraziamenti a voi appartenenti a questa terra piena di contraddizioni. Grazie a tutti gli amici di Palermo, agli storici coinquilini, agli ancor più storici amici fraterni di[END_REF]. We have considered the contributions of vacuum uctuation and radiation reaction eld to the resonance interaction, and shown that the Unruh thermal uctuations do not a ect the interatomic interaction, which is exclusively modi ed by radiation-reaction corrections. We have discussed that beyond a characteristic length scale associated to the breakdown of the approximate description by a local inertial frame of the system of the atoms, non-thermal e ects change qualitatively the distance-dependence of the resonance interaction. Speci cally, we have shown that these non-thermal e ects related to the non-inertial character of acceleration result in a di erent scaling with the distance and a di erent dependence on acceleration, if compared to the usual thermal case. The merit of our approach is the simplicity in highlighting non-thermal features, exploiting atomic entanglement as condition to have resonant second-order interactions. Our results open the way for new future developments. First of all, it is known that the Unruh bath does not induce full decoherence and can even create entanglement during the time evolution of a pair of accelerating atoms [42]; this limits the description presented in this work up to time scales of the order of inverse of acceleration but it poses the intriguing question on the fate of the resonant interaction in the long-time limit, since for its existence a minimum of entanglement among the two atoms is required. Moreover, since the basic set-up we have considered in this work is the pillar to build the Dicke model, one might ask which is the impact of the change of interatomic potential on its phase transition (sub or superradiant transition phase). Understanding it the qualitative change of interatomic interactions in accelerated systems, can modify macroscopic phenomena, such as thermodynamics phase transitions, is one promising direction to highlight non-thermal signatures of relativistic accelerations in a many-body context. Zero-point uctuations are among the most striking consequences of the quantum description of the electromagnetic eld. They are at the origin of the Casimir-Lifshitz force, a long-range quantum electromagnetic interaction between neutral polarizable bodies, and they are also responsible for the Casimir-Polder forces between an atom and a surface or between two atoms (van der Waals forces) [1] - [8].

When boundary conditions are set in motion with nonuniform acceleration in the vacuum, or when material properties are changed nonadiabatically, a dynamical Casimir effect is realized, and a parametric excitation of vacuum uctuations may lead to the emission of real photons [9] - [15]. Similarly, a dynamical Casimir-Polder e ect occurs when physical parameters of an atom near a conducting plate are rapidly changed [16].

Another rapidly growing research eld is that of quantum optomechanics, which deals with systems where mechanical degrees of freedom are coupled to cavity elds [17]. Such systems have been experimentally and theoretically investigated, for example, for realizing sensitive force detectors, cooling macroscopic mirrors or obtaining quantum superposition states for macroscopic objects [18]. Signi cant experimental progress has been obtained in precision trapping of cold atoms near a nanoscale optical cavity, allowing to probe cavity near elds [19]. The e ect of quantum uctuations of the position of a cavity mirror on Casimir and Casimir-Polder interactions has been also demonstrated [20,21].

In this Chapter, after an overview of the Dynamical Casimir e ect that we discuss brie y, we propose a new optomechanical Rydberg atoms-surface coupling based on a novel aspect of the dynamical Casimir-Polder (CP) e ect, able to a ect the internal atomic state [22], [AN4]. It is a new near-eld e ect, not related to the excitation of atoms by the few real photons expected in the dynamical Casimir e ect [23] - [27]. Speci cally, when considering a gas of dilute Rydberg atoms trapped in front of a substrate whose refractive index is changed in time (dynamical mirror) at a frequency corresponding to one of atomic transition frequencies. Due to the e ective periodical change of the atom-mirror distance, the optomechanical coupling between the wall and the Rydberg atoms yields a periodic perturbation on the atoms, which can be excited to upper levels [22]. On the experimental side, this scheme may largely pro t from recent progresses in the realization of dynamical mirrors [28], and in the cigar-shape trapping of Rydberg atoms and their preparation in long-lived excited states [29].

It is worth saying that recently a micromechanical atom-wall system has been realized with a trapped Rb Bose Einstein Condensate (BEC) close to a dielectric substrate, and the collective oscillations of the gas have been used to measure the CP force [30,31]. In particular, this allowed measurement of the more elusive thermal component of this interaction [32]. Theoretical predictions of the thermal component of the Casimir-Polder force can be found in [33,34]. Di erently from that case, where the external degrees of freedom of an atomic gas have been used to detect the CP force, here we use the CP force to couple a substrate to the internal atomic degrees of freedom.

This Chapter has the following structure. The rst two sections are introductory and pave the way to the original results that we show in the last section. In particular, in Section 3.1 we present and discuss two e ects due to moving bodies and vacuum uctuations, the dynamical Casimir e ect and the quantum friction. After the analysis on the physical origin of these e ects we present in Section 3.2 some experiments, presented in literature, related to them. Finally, in the last Section 3.3, we present and discuss our original work about a new e ect between an atom and an oscillating mirror due to the dynamical atom-mirror Casimir-Polder force. We discuss the possibility to detect this new e ect and we also present an original experimental proposal.

The dynamical Casimir e ect and the quantum friction

The Dynamical Casimir E ect (DCE) is an e ect related to photon generation from the vacuum due to accelerating nonuniformly neutral bodies in free space, as suggested for the rst time by Moore [9]. In general this e ect is related to quick changes of the system geometry or of the optical properties of neutral macroscopic objects. Although Casimir did not write anything about this e ect, the presence of the Casimir's name in the DCE is due to the close connection between the DCE and vacuum uctuations. This e ect can be physically explained stating that it is the parametric ampli cation of the electromagnetic quantum uctuations in systems having non adiabatic time-dependent parameters.

From Quantum Electrodynamics we know that any quantum observable associated to the electromagnetic eld has uctuations. Also the vacuum radiation pressure uctuates. Thus, if we consider a body at rest in the vacuum, we can nd a uctuating force acting on it which generates a quantum Brownian motion. Due to the uctuation-dissipation theorem this yields dissipative e ects too [35]. The force on a body at rest is zero as well as we consider a uniform motion in vacuum, due to the relativity principle. If we consider a mirror moving with a 1D nonuniform motion, it has been found [36] that the Casimir dissipative force is proportional to the second-order derivative of its velocity. In this way we can understand that there is a relation between the Casimir dissipation and the emission of photons by the accelerated neutral mirror. Exploiting energy conservation, we nd that the power dissipated because of the motion of the mirror is equal to the total radiated power in the DCE.

Other simple considerations can be made to understand the DCE. These considerations start from the analogy between the quantum electromagnetic eld and a set of quantum harmonic oscillators. Let us write the Hamiltonian of the free eld in a cavity

H 0 = ∞ n ω n 1 2 + a † n a n (3.1)
where a n and a † n are, respectively, the annihilation and creation bosonic operators of the eld mode at the given frequency ω n . When we consider the possibility of the interaction between the system and the environment, we must add an interaction term H I (t) (in general function of time). Usually we can consider that this interaction is relatively weak and we can expand the H I (t) term in series with respect to powers of the bosonic operators a n , a † n . The linear term of this expansion can be interpreted as describing eld excitations due to external currents and/or charges. The second-order terms, proportional to a n a † n , a † n a n , a n a n , a † n a † n with time-dependent coe cients, can be considered as terms arising from possible time-dependent changes of the system geometry or optical properties of the cavity. The latter can generate eld excitations that can be interpreted as an ampli cation of the initial uctuations of the eld. So the photon creation from vacuum of the DCE can be read as an excitation of the vacuum due to the time-dependent dependence of the interaction Hamiltonian.

It must be said that the DCE and its analogues, when realistic situations are considered, are very tiny e ects due to the small number of emitted photons. Nevertheless (as we will see in Section 3.2) they have been recently observed in superconducting circuits [37], in Josephson metamaterials [38] and in Bose-Einstein condensates [39] and many experimental setups have been proposed and realized to measure them.

We discuss now a simple 1D model to show the dynamical Casimir e ect. We consider an electrically neutral point-like mirror in nonuniform motion and coupled to a massless scalar eld Φ(x, t). The motion of the mirror is described by its position Q(t). Following [7], the idea to show the DCE in this model is to calculate the vacuum radiation-pressure-force F(t) by making simple considerations. If we consider a non-relativistic motion, we know that the force must be proportional to some velocity derivative of the moving mirror. At the same time this force is a quantum e ect, so we expect that it is proportional to . In addition its expression should be consistent with the Lorentz invariance of the vacuum led state. In a 1D space, after a dimensional analysis, we deduce that

F(t) ∝ Q(t) c 2 (3.2)
where the dot indicates a time derivative. We see that the third time derivative of the position is involved: it follows that a nonuniform motion is necessary. We now calculate the dimensionless factor in the above equation. We treat the motion of the mirror as a small perturbation and we use the Dirichlet boundary conditions on the mirror

Φ(Q(t), t) = 0 (3.3)
to rst order in Q(t). In the frequency domain we can write the Fourier transform φ(x, ω) of the eld as a sum of an unperturbed eld φ 0 (x, ω), corresponding to a static mirror xed at x = 0, plus a perturbation δφ(x, ω) φ(x, ω) = φ 0 (x, ω) + δφ(x, ω).

(3.4)

The boundary condition for φ 0 (0, ω) = 0 at x = 0 is automatically ful lled, while for δφ(x, ω) we make a Taylor expansion around x = 0. At rst order in Q we get

δφ(0, ω) = - +∞ -∞ dω i 2π q(ω 0 -ω i )∂ x φ 0 (0, ω i ) (3.5)
where we have introduced the Fourier transform q(Ω) of Q(t). The key element generating the photon emission is already in this equation. This element is the frequency Ω = ω 0ω i .

In fact, as we are going to show, since in the integral in (3.5) the function q(Ω) is translated by ω 0 , the only frequencies that contribute to the dissipative Casimir force are the negative frequencies in the range [-Ω, 0]. In order to show this point we must calculate the force. We take the Fourier transform of the component T 11 of the energy-momentum tensor

T 11 = 1 2 1 c 2 (∂ t Φ) 2 + (∂ x Φ) 2 (3.6)
and consider the expansions given by (3.4) and (3.5). After averaging over the vacuum state, the force is given by

f (Ω) = χ(Ω)q(Ω) (3.7)
where

χ(Ω) = 2i c 2 +∞ -∞ dω i 2π (Ω + ω i )|ω i |. (3.8)
Analyzing the above expression for χ(Ω), using an appropriate regularization of the integral, we can see that, due to the "translation" of the frequency Ω i , the contribution given

We have seen the close connection between the DCE and the dissipative Casimir force. The latter seems acting as a radiation reaction force (generated from the vacuum!) and this can be seen from the equation (3.2) which has the same form of the radiation reaction force of classical electrodynamics.

We can give an order of magnitude for the rate of photon emission in the DCE in a 3D space. We expect that, when we analyze a plate in the xy plane with surface A and moving along the z direction, the 3D dissipative Casimir force should depend linearly on the surface A. Following the same method described above and with appropriate changes for the electromagnetic case, we get

F(t) = - A 30π 2 c 4 d 5 Q(t) dt 5 . (3.10)
Since from the energy conservation we know that the total radiated energy is the negative of the work done by the dissipative Casimir force on the moving mirror, we can write

E = - +∞ -∞ dtF(t) Q(t). (3.11)
For simplicity we examine one oscillating mirror with frequency Ω and amplitude Q 0 , whose motion is exponentially damped over a time scale T ≫ 1/Ω. For this simple case we obtain

E = T AQ 2 0 Ω 6 120π 2 c 4 .
(3.12)

In order to obtain the number of emitted photons N we can exploit the fact that the spectrum is symmetrical with respect to the frequency Ω/2, writing E = N Ω/2. We nally obtain

N T = 1 15 A λ 2 0 ν max c 2 Ω. (3.13)
In the above equation we introduced ν max ≡ ΩQ 0 and λ 0 ≡ 2πc/Ω. If, for examples, we use in (3.13) physical parameters compatible with possible actual experimental setups, such as A ∼ λ 2 0 ∼ 10 cm 2 ν max /c ∼ 10 -7 and Ω/2π ∼ 10 GHz, we nd that the photon emission rate is of the order of 10 -5 , which means one photon pair every about two days. Of course this is a very small e ect. However, many situations have been studied to amplify the photon emission e ect [10], [41], [42]. Typically, a second mirror or a cavity with moving walls are considered to signi cantly enhance the photon emission rate by a resonance e ect [43] (see also [42] and reference therein).

In the context of quantum e ects due to motion of bodies in the electromagnetic vacuum there is another e ect, stronly related to the DCE, that we wish to mention, the quantum friction. It predicts the possibility that a force, which tends to work against the relative motion, can be experienced between two electrically neutral and polarizable bodies in parallel relative motion, then a non-contact "friction " is originated. This e ect is expected to be present even at zero temperature and when the surfaces of the moving bodies are at and perfectly smooth. It has been also predicted for an atom moving parallel in front of a surface [44]. Di erently to what we have seen with the DCE, here the friction e ect due to the quantum vacuum can take place also when the motion of the bodies is uniform. The reason is that, even if the two bodies are in uniform motion, the relative shear motion can not be removed changing the frame of reference. From an energy conservation point of view, the energy lost during the quantum friction is balanced by the external energy necessary to keep the body a constant velocity.

A simple qualitative explanation of the quantum friction can be given thinking that the electric dipoles generated from the quantum vacuum in one of the surfaces (a similar picture to that given before for the Casimir-Polder interaction) induce image electric dipoles on the other surface which lags behind because of the relative motion. We have that when the shear motion of the two bodies is considered, the van der Waals-like attraction experienced between the two bodies is related to the relative velocity (roughly speaking the photons exchanged for the interaction carry the motion information) and the generation of a friction force follows. Many scientists consider the quantum friction as a purely quantum e ect with no classical analogue and they do not unanimously agree on the interpretation. This e ect is still debated in the literature [45,46].

Experimental setups for DCE

In this section we present a short overview about the possibility to observe the DCE in a laboratory experiment. We already discussed in Section 3.1 that the rate of photon emission by DCE in the presence of an accelerating mirror is small. The experimental observation of such phenomenon is a formidable technological and scienti c challenge. The dynamical Casimir e ect is considered as a direct manifestation of the existence of the vacuum uctuations. As a consequence, the detection of the photon emission due to the DCE is also considered an important contribute to fundamental physics and a milestone which provides further relevance to Quantum Electrodynamics.

Many e orts have been done recently to nd promising systems to detect the DCE. In particular the photon emission generated in a resonant time-dependent cavity has been widely considered. For these con gurations the photon emission rate is strongly ampli ed compared to the simple and ideal situation of a single accelerating mirror [47]. In these setups, the average number of photons created is

N = sinh 2 (ηωεt) (3.14)
so it grows exponentially with time [23,48]. In the above expression, ω represents the frequency of the resonant mode of the cavity, η is a parameter related to the geometry of the cavity (its order of magnitude is 1) and ε indicates the relative amplitude of the oscillations involved, i.e. ε = a/λ = aω/2πc where a is the amplitude of the mirror oscillation. Notwithstanding this exponentially growing time dependence of the average number of emitted photons, the observation of these photons in realistic laboratory conditions is arduous, due to cavity losses and the presence of thermal photons.

Let us rst analyze other important aspects involved in realistic laboratory situations. If, for example, we consider a cavity of length L 0 ≃ 10 -2 m, in order to exploit the resonance condition, and thus amplify the photon emission, the order of magnitude of the oscillation frequencies (either mirror oscillation or conductivity oscillation) is of about 10 GHz. Achieving these frequencies from a mechanical point of view is not feasible. An alternative method could be dealing with surface vibrations without having a motion of the mirror center-of-mass. These kind of systems lead to amplitude values ε which need a minimum quality factor Q of 10 8 . In spite of the fact that greater values of the Q-factor has been achieved experimentally, the presence of oscillating walls complicates a lot the situation decreasing the value of the Q-factor, making then very di cult the observation of the dynamical e ect.

We now report the main aspects of recent experiments on the DCE. Considering all these di culties related to mechanical oscillations of the mirror, some scientists thought to other experimental setups for the DCE revelation where a mobile boundary condition is somehow simulated. The rst observation of photon emission related to the DCE has been made in 2011 in a superconducting circuit [37]. The experimental setup was made by an open transmission line terminated by a superconducting quantum interference device (SQUID). In the SQUID two Josephson junctions were present and connected in parallel to form a loop. The key idea was to use a time-dependent magnetic ux to control the e ective inductance of the SQUID, placed at the end of a transmission line. This generates a time-dependent boundary condition for the phase eld (the time integral of the electric eld) which can be seen as a transmission line with a variable length. The change in the electrical length of the transmission line simulates a time-dependent boundary condition, as the idealized moving mirror already discussed. The e ective velocity of the simulated e ective mirror, de ned as the rate of change of the electrical length, can be very large, thus increasing the rate of photon emission. This means that a non-relativistic approach as that in Section 3.1 it is not possible and a fully relativistic approach is necessary. The theoretical prediction for this case gives a photon production rate several orders of magnitude larger than in other systems. The involved photons exhibited two-mode squeezing correlations, which are characteristic of photons generated in correlated pairs.

Similarly to this experiment in a superconducting circuit, a recent work in 2013 has demonstrated the DCE with the help of Josephson metamaterials [38]. The considered setup was a Josephson metamaterial, made of an array of 250 SQUIDs, forming the signal line of a superconducting coplanar waveguide. This metamaterial is embedded into a low-quality-factor cavity. Here the DCE was produced by modulating, through an external magnetic eld, the refraction index of this material near its quantum ground state. This setup, compared with the superconducting circuit setup described above, allows for an enhancement of the DCE. This is possible because the presence of a cavity in this system produces a resonance e ect avoiding the problem of uncontrolled resonances which a ect the detection of photon emission in [37].

Another indirect observation of the DCE is that presented in [39]. In this work an acoustic analog of the dynamical Casimir e ect is described. Through the modulation of the speed of sound in a Bose-Einstein Condensate, correlated pairs of elementary excitations are produced, both phonon-like and particle-like. The idea inspiring this work starts from the study of an interacting Bose gas. Keeping in mind that the change in the interaction strength of the gas is analogous to an optical index change (for example the speed of sound changes) an acoustic analog of the dynamical Casimir e ect is realized by changing the scattering length in the gas. From a microscopic point of view, the ground state of such a gas is the vacuum of Bogoliubov quasi-particles. When the interaction strength is changed the old vacuum state is mapped into a new state where pairwise excitations appears and it can be seen as pairs of the new quasi-particles. Also, the con ning potential of the BEC can be modi ed (it means changing of the density of the BEC) and the same e ect occurs. However there is a substantial di erence between the results related to this experiment and the standard con guration of the DCE. In fact, in this acoustic analog of the DCE the e ect of the temperature is not negligible. It seems that the pair generation does not arise from the vacuum but rather from the thermal noise marking a fundamental di erence with the DCE which is predicted also at zero temperature.

We now brie y describe the motion induced radiation (MIR) experiment [49,50,51], which is related to the our work we will describe in the next section. In the MIR experiment, the di culty to reach su ciently high frequencies of mechanical oscillation is bypassed using an e ective dynamical mirror. The main idea in this approach is to change the length of the cavity by modifying the re ectivity of one of the slabs present in the cavity walls. This is possible exploiting the property of some semiconductors that, when irradiated with short laser pulses, change their optical properties becoming a re ector or a transparent medium. With the help of this kind of system it is possible to reach greater oscillation amplitudes ε ≃ 10 -4 mm and high oscillation frequency of the order of tens of GHz, allowing much smaller values of the Q-factor. However, in this system dissipative e ects can be signi cant. The change of the optical properties by laser impulses is due to creation of electron-hole pairs in the semiconductor slab. Thus, the charge carriers of the irradiated semiconductor must have a very short recombination time and a very high mobility. This kind of material has been built [49] and then the photon emission by DCE for this experiment could be measured in a pure quantum electrodynamics framework. It must be said that the MIR experiment is still in the making.

Optomechanical Rydberg atoms excitation via dynamic Casimir-Polder coupling

In this section, we focus on the main issue of this Chapter. We describe our original work concerning with the study of a new dynamical Casimir-Polder e ect. It is based on the optomechanical coupling of an oscillating mirror (dynamical mirror) with a diluite gas of Rydberg atoms, mediated by the dynamical atom-mirror Casimir-Polder force in the nonretarded near-eld regime [22], [AN4]. This coupling may produce a near-eld resonant atomic excitation whose probability scales as ∝ (d 2 a n 4 t) 2 /z 8 0 , where z 0 is the average atom-surface distance, d the atomic dipole moment, a the mirror's e ective oscillation amplitude, n the initial principal quantum number of the Rydberg atoms, and t the time. We propose an experimental con guration to realize this system with a cold Rydberg atoms gas trapped at a distance ∼ 2 • 10 µm from a semiconductor substrate, whose dielectric constant is periodically driven by an external laser pulse, hence realizing an e ective mechanical mirror motion due to the periodic change of the substrate from transparent to re ecting. For a parabolic gas shape, this e ect is predicted to excite about ∼ 10 2 atoms of a dilute gas of 10 3 trapped Rydberg atoms with initial principal number n = 75 after about 0.5 µs, hence high enough to be detected in typical Rydberg gas experimental conditions.

To describe our system we now consider a xed Rydberg atom near a perfectly conducting plate; the plate is forced to move harmonically around its equilibrium position. We model the atom as a two-level system. The mirror's position coincides with the plane z = 0 at t = 0, and the atom-mirror distance is z(t). We rst analyze the case of a xed mirror at a distance z from the atom; we assume that this distance is much smaller than a main transition wavelength λ 0 = 2πc/ω 0 of the Rydberg atom, ω 0 being the corresponding transition angular frequency. The atom-mirror Casimir-Polder (CP) interaction energy is thus in its near-zone nonretarded regime, where electrostatic (longitudinal eld) contributions are dominant [1,30]. We assume the atom prepared in a long-lived Rydberg state and treat it as a stable state, assuming to study the system for times shorter than its lifetime (Rydberg atoms in circular states can have a very long lifetime). As we saw in Chapter 1, for typical short wavelength atomic transitions, there is a thermal regime dominating at large separations. For long wavelength transitions (molecular or Rydberg states, as in this section) and for the conditions considered in this section, this thermal regime is absent [34,52]. The CP interaction energy between a ground-state atom and a xed perfectly conducting mirror, within dipole approximation and in the nonretarded regime, is (cgs units) [53,54]

V(z) = - d 2 x + d 2 y + 2 d 2 z 16z 3 = - 1 16
σ i j d i d j z 3 , (3.15)
where the sum over repeated indices is used, and the average of the squared components of the atomic dipole moment operator d are taken on the atomic state considered. The atom-mirror distance z is along the ẑ direction. We have also de ned the diagonal matrix σ = diag(1, 1, 2). The expression (3.15) of the atom-wall interaction for an ideal conductor is a very good approximation in our case, since we shall consider atom-wall distances of the order of 2 • 10 -3 cm, much larger than the plasma wavelength of a typical metal (of the order of λ ∼ 10 -5 cm), where real-conductor corrections are known to be negligible [55].

It is well known that in the near-zone limit, the atom-wall nonretarded interaction (3.15) is well described by the interaction between the atomic dipole and its image [54]. In order to describe the interaction of the atom with the oscillating mirror, we adopt a semiclassical model: we obtain the atom-wall interaction as the interaction energy between the atomic dipole and an e ective classical eld due to the image atom (Figure 3.2). Using the method of image charges [56], we can describe the near-zone atom-wall interaction by the coupling term H I = -d•E(r)/2, where d is the atomic dipole moment operator and E(r) is the electric eld generated by the image dipole d = (-d x , -d y , d z ) at the atom's position r = (0, 0, z). The factor 1/2 takes into account that, when the atomic dipole is moved from in nity to its nal term yielding the usual z -3 0 near-zone static Casimir-Polder potential (3.15), and

V I (t) ≃ -d i d j 3σ i j 16z 3 0 a z 0 sin ωt , ( 3
P e (z 0 , t) ≃ 9 2 10 2 a z 0 2 | σ i j (d i d j ) eg | 2 z 6 0 t 2 . (3.20)
We now estimate the order of magnitude of the excitation probability P e (t). For a Rydberg atom, the matrix element of the product of components of the atomic dipole moment appearing in (3.20) is related to the electron charge e, the Bohr radius a 0 and the principal quantum number n through the relation | σ i j d i d j |∼ e 2 a 2 0 n 4 [58,59]. Thus the excitation probability approximately becomes

P e (z 0 , t) ≃ 3 • 10 -19 cm 6 s -2 a 2 z 8 0 n 8 t 2 . ( 3.21)
The condition P e (z 0 , t) ≪ 1 must be satis ed for the validity of our perturbative approach, and this sets an upper limit to the acceptable values of time t, once the other parameters have been xed. For n = 75, yielding a frequency of about 30 GHz for the transition n = 75 → 77, a/z 0 ≃ 10 -1 and z 0 ≃ 2 • 10 -3 cm, the single-atom excitation probability is P e (t) ≃ 5 • 10 10 s -2 t 2 . This shows that, by taking a time of the order of 2 µs (well compatible with achievable trapping times of Rydberg atoms [60,61]), the probability is of the order of 20%.

If we consider now a trapped Rydberg gas instead of a single atom, if the trap size is comparable with the atom-mirror distance, equation (3.21) could not be su ciently accurate, and the actual pro le of the atomic trap should be taken into account. If ρ(z) is the atomic linear density in the direction z orthogonal to the surface, the number of excited atoms, neglecting interactions among them, can be written as

N e (t) = ∞ 0 dzρ(z)P e (z, t) . (3.22)
If the gas pro le is cigar-shaped, as shown in Figure 3.3, parallel to the mirror, as a rst approximation we may use a parabolic pro le in the three dimensions. Let N be the number re ectivity at all eld frequencies relevant for the Casimir-Polder interaction at the considered atom-mirror distance z 0 . In the gure a parallel-line structure mesh is shown, even if more complex patterns can be designed. To optimize the dynamical mirror, the size of the non-metallic areas in the mesh could be further reduced well below the incident laser wavelength. The laser beam would then be transmitted through the rear mirror by exploiting the EOT (extraordinary optical transmission) e ect [63]. In the devised scheme, a is comparable with the thickness of the semiconductor layer, of the order of a few micrometers, which is in turn comparable with α -1 , the absorption length of near infrared light in direct band gap semiconductors. For example, α -1 ∼ 1 µm in GaAs excited at the band gap photon energy that corresponds to λ ∼ 800 nm. The Rydberg atoms are prepared in an initial state characterized by a principal quantum number n, which determines the required oscillation frequency of the mirror in order to obtain a resonance e ect for the transition to a higher energy level. We could assume that the initial state of the Rydberg atom is a circular state (maximum angular momentum quantum numbers), yielding a very long lifetime of the initial state. Angular momentum selection rules for the transition give ∆ℓ = 0, ±2. A possible transition worth to consider in our case is that with ∆ℓ = 2. By tuning the mirror oscillation frequency we can set also ∆n = 2. Such a transition brings the Rydberg atom to a nal state with maximum azimuthal quantum number, that is a long-lived state too, because only one decay channel by a dipole transition is allowed [59]. This should make easier the detection of the atomic excitation. If, as mentioned, an initial n = 75 state is prepared, the atom should be promoted to the upper level with n = 77 when the moving mirror oscillates at a frequency of approximately 30 GHz. A MOPA (master oscillator power ampli er) laser system [64] with a seed oscillator operating at 30 GHz delivers the pulses at the required repetition rate, and with an energy/pulse (few µJ ) su cient to excite a plasma mirror in the semiconductor layer. Lower mirror oscillation frequencies (thus lower repetition rates) would be allowed for initial states with a higher principal number. A limitation to higher values of n is however set by the detection technique of the excited states, which would rely on the selective eld ionization [59] (in experiments using Rydberg states with n = 30 -85, a eld control at the mV/cm level has been demonstrated feasible [65]). On the other hand, the di culties in obtaining higher frequencies of the wall vibration become ever more stringent for decreasing initial quantum numbers.

The non-harmonicity of the atom-wall distance in the presented experimental scheme can be easily included by using the atom-mirror distance

z(t) = z 0 [1 -a z 0 f (t)],
where f (t) is the appropriate function describing the mirror's motion. In this case, equation (3.19) becomes

c e (z 0 , t) = - 1 √ 2π 3σ i j (d i d j ) eg 16z 3 0 a z 0 × ∞ -∞ dωg(ω) e -i(ω-ω 0 )t -1 ω -ω 0 , (3.25)
where g(ω) is the Fourier transform of f (t). Once its form is obtained for the speci c experimental setup considered, the squared modulus of equation (3.25) gives the corresponding Figure 3.4 -Schematic gure representing the system: The trapped Rydberg atoms interact with the dynamical mirror (M), made by a semiconductor layer whose rear surface is covered by a metallic mesh. The semiconductor layer is illuminated by a multigigahertz repetition rate laser that induces a periodical variation of its dielectric properties. This dynamical mirror is sandwiched between a transparent bulk dielectric (S) that acts as thermal sink, and a Bragg high re ectivity mirror (HR).

atomic excitation probability, generalizing equation (3.20).

In our experimental proposal, the optomechanical coupling with the moving surface could be optimized with a quasi-one-dimensional Rydberg gas prepared in a magnetooptic trap. Such a Rydberg gas has been recently obtained in a trap with a density of 10 10 atoms/cm 3 and number of atoms reaching the order of 10 6 , whose distance from a surface could be controlled with a few micrometers precision [29,60]. Lower densities are preferable in our case, in order to reduce possible interactions among the atoms. The average atom-wall distance we consider, in the range 20 -50 micrometers, is much less than a typical transition wavelength of the Rydberg atom (of the order of 1 cm for the transition mentioned above), so that the near-zone Casimir-Polder interaction between the atoms and the mirror is relevant in this case, as we have assumed.

Trapping a sample of about N ∼ 10 3 87 Rb atoms in a cigar-like shape with R ⊥ ∼ 5 • 10 -2 cm and R z ∼ 10 -3 cm at a distance z c = 2 • 10 -3 cm from a surface for times up to 10 µs, is also realistic using actual Rydberg atoms trapping techniques.

From (3.24), with the values given above and for a = 2 • 10 -4 cm, we obtain that about 100 atoms in the sample of 10 3 are excited after 0.5 µs (atoms in the sample closer to the wall are more likely excited, due to the strong dependence of the excitation probability with the atom-wall distance). Hence a considerable number of trapped atoms can be excited in a quite short time interval, making possible to detect the dynamical e ect we have proposed.

The atom density considered in the estimate above is such that the long-range atomatom van der Waals interaction, which scales as r -6 with the interatomic distance r, can be neglected. In fact, the closest atom-atom distance is around 10 -3 cm and the average atom-wall distance is 2 • 10 -3 cm. Using known expressions for atom-atom and atom-wall interactions [66], it is possible to show that the atom-wall interaction energy is some three orders of magnitude larger than the interaction energy between one atom and its closest atom. Interatomic interactions can be thus neglected for the gas density we have considered. Similarly, it is easy to check that quadrupolar interactions [66,67] are several orders of magnitude smaller than dipolar ones, and can be therefore neglected. This is also expected on a physical basis, because in our case the size of the Rydberg atoms, ∼ 10 -5 cm, is much smaller than the average relevant atom-wall and atom-atom distances.

Finally, we can compare our excitation probability of the Rydberg atoms with that due to absorption of the real photons emitted by dynamical Casimir e ect. Using known results for the number of emitted photons by an oscillating wall [36], [68], with the same parameters given above for our proposed experiment, the number density (for unit area) of real photons emitted is ∼ 10 -1 /cm 2 . Then the number of photons that can excite our trapped atomic sample, which has a front area of ∼ 10 -5 cm 2 , is ∼ 10 -6 . This is an upper limit for the single-atom excitation probability by the real photons (far eld) emitted by the oscillating wall under resonance condition; it is thus negligible compared to our near-eld excitation probability (∼ 10%). The number of emitted photons could be increased by a resonant cavity [23], but also in this case the atomic excitation probability by photon absorption is order of magnitudes smaller than our near-eld e ect.

Summing up, we proposed a new optomechanical dynamical coupling between Rydberg atoms and a substrate, based on the dynamical Casimir-Polder e ect [22], [AN4]. In particular, we have analyzed a dilute sample of Rydberg atoms trapped in the proximity (tens of micrometers) of an oscillating re ective mirror. This e ect could be observed using currently available experimental techniques.

It shows how quantum vacuum uctuations may be used to realize an optomechanical coupling between a macroscopic body and an elementary or mesoscopic quantum system, and to change the internal state of that system.

Chapter 4

Casimir-Lifshitz force out of thermal equilibrium between dielectric gratings Casimir-Lifshitz force is an interaction between polarizable bodies originating from the uctuations of the electromagnetic eld. We already saw that it was rst theoretically derived by Casimir in 1948 for an idealized con guration of two in nite conducting plates at zero temperature and generalized later by Lifshitz and collaborators in the case of bodies having arbitrary optical properties and nite temperature [1] - [3]. The Casimir-Lifshitz interaction, experimentally veri ed for several geometries [4], results from two contributions, one originating from vacuum uctuations and present also at zero temperature, the other one from purely thermal uctuations. The latter becomes relevant when the distance separating the bodies is larger than the thermal wavelength λ T = c/k B T , of the order of 104 8 µm at ambient temperature. This explains why it has been only very recently experimentally observed at thermal equilibrium [5].

Nevertheless, the situation completely changes for systems out of thermal equilibrium. It was rst theoretically predicted in 2005 that the atom-surface interaction (usually referred to as Casimir-Polder force) is qualitatively and quantitatively modi ed with respect to thermal equilibrium [6,7]. New power-law behaviors appear: the force, which depends on the temperatures involved in the systems, can turn into repulsive (being only attractive at thermal equilibrium) and it is strongly tunable by modifying the temperatures. This prediction was veri ed in 2007, providing the rst experimental observation of thermal e ects [8]. These results paved the way to a renewed interest in Casimir-Lifshitz e ects out of thermal equilibrium. In fact, these e ects have been studied for two slabs [9,10] and in presence of atoms [11] - [16], and more recently several di erent approaches have been developed to deal with the problem of the force out of thermal equilibrium and heat transfer between two [17] - [24] or more [25] - [28] arbitrary bodies. The physics of the electromagnetic eld out of thermal equilibrium has also stimulated the study of other e ects, such as the manipulation of atomic populations [29,30] and entanglement [31,32].

In parallel with the interest in the absence of thermal equilibrium, Casimir-Lifshitz interactions have been studied in several di erent geometries, with particular interest to the sphere-plane con guration, the most studied experimentally. More recently, nanostructured surfaces have been theoretically considered in the contexts of both force [33] - [43] and heat transfer [44,45,46]. Experimentally, the force have been measured between a sphere and a dielectric [47,48] or a metallic grating [49].

The problem we discuss in this Chapter concerns with the results obtained from the calculation of the Casimir-Lifshitz force out of thermal equilibrium in presence of dielectric gratings, computation made in our work [39], [AN5] for the rst time in literature. This physical system has been analyzed in order to study the combination of non-equilibrium and geometrical e ects, that we show in detail. In particular we consider a system made of two di erent gratings having di erent temperatures, immersed in an environmental bath at a third temperature. Our calculations can be relevant both to imagine new experiments measuring the Casimir-Lifshitz force out of thermal equilibrium and in the more general context of the manipulation of the force in micro-and nano-electromechanical systems [50,51].

The Chapter is structured as follows. In the rst two sections we present the methods, already developed in literature, that we have used to get our original results presented in the last section of this chapter. In particular, in Section 4.1 we introduce the method used to calculate the force between two arbitrary objects out of thermal equilibrium and we also show, for example, its application to the case of two slabs. In Section 4.2 we focus our attention on the issue of nding the scattering operators for the gratings, necessary to calculate the Casimir-Lifshitz force in our approach. Here we discuss shortly some usual methods developed for the scattering problem in gratings and we aim to develop and solve the problem of the scattering upon a single 1D lamellar dielectric grating by means of the Fourier Modal Method. In Section 4.3, we apply the results of the previous sections in order to calculate the force out of thermal equilibrium between two di erent gratings. We explore the behavior of the force as a function of the three temperatures and of the geometrical parameters of the gratings, with a speci c attention to the appearance and features of repulsion and nally we give some conclusive remarks.

4.1 A scattering-matrix approach to Casimir-Lifshitz force out of thermal equilibrium

In this section we show how to derive an expression for the Casimir-Lifshitz force out of thermal equilibrium between two bodies at two di erent temperatures immersed in external radiation, an environment, characterized by another temperature. We see that this derivation, deduced in [19], is also strictly related to the heat transfer problem. In particular we focus our attention to cast the average values of the electromagnetic eld in terms of the scattering operators. Finally we numerically apply the general expression found to the simple and well known con guration of two slabs, by calculating the heat transfer and the force out of thermal equilibrium [19].

Let us start introducing in detail the general system considered. We analyze two bodies, labeled with indexes 1, 2, with arbitrary geometries and material properties. The geometry of the system is represented in Figure 4.1. We assume that the two bodies are separated by a geometrical planar surface. This assumption is useful because it allows us to use a planewave decomposition and can be in principle relaxed by an appropriate change of basis. In addition, it is an assumption veri ed in most experimental con gurations. So we can de ne three di erent zones A, B, C, enclosed respectively in

z 1 ≤ z ≤ z 2 , z 2 ≤ z ≤ z 3 , z 3 ≤ z ≤ z 4 .
Our physical system is considered in a con guration out of thermal equilibrium (OTE). This means that each body is supposed to be in local thermal equilibrium with a constant temperature T i . We also assume that the two bodies are immersed in a radiation bath coming from bounding walls far from the system and having temperature T e , in general di erent from the temperatures of the two bodies (see also [19]). The whole system is considered in a stationary regime so that the three temperatures involved are constant in time.

The procedure that we use is based on a mode decomposition of the elds. Each mode (ω, k, p, φ) is identi ed by the direction of propagation φ = {+, -} along the z axis, by the polarization index p (assuming the values p = 1, 2 which respectively correspond to transverse electric (TE) and transverse magnetic (TM) modes), by the frequency ω and the transverse wave vector k = (k x , k y ). In this description, the z component of the wave vector k z is a dependent variable de ned as

k z = ω 2 c 2 -k 2 . ( 4.1)
while the complete wave vector K is

K φ = (k, φk z ) = (k x , k y , φk z ). (4.2)
We see that when k ≤ ω c , k z is real so the corresponding wave is propagative. Instead when k > ω c , k z becomes imaginary and the corresponding wave is evanescent. In the latter case φ represents the direction along which the amplitude of the evanescent wave decays. We consider now the expression for the electric eld. We rst decompose it with respect to frequency and we work only with positive frequencies

E(R, t) = 2 Re +∞ 0 dω 2π exp(-iωt)E(R, ω) . (4.3)
Then we decompose the single-frequency component E(R, ω) with respect to the parallel wave vector k, the direction of propagation φ and the polarization p

E(R, ω) = φ,p d 2 k (2π) 2 exp(iK φ • R)ǫ φ p (k, ω)E φ p (k, ω). (4.4)
In general, from now on, the sum on φ runs over the values {+, -} and the sum on p over the values {1, 2}. The polarization vectors ǫφ p (k, ω) appearing in (4.4) are de ned in the following standard way

ǫφ TE (k, ω) = ẑ × k = 1 k (-k y x + k x ŷ) ǫφ TM (k, ω) = c ω ǫφ TE (k, ω) × K φ = c ω (-kẑ + φk z k) (4.5)
where x, ŷ and ẑ are, respectively, the unit vectors along the directions x, y and z while k = k/k. Using Maxwell's equations we can easily nd the expression of the single-frequency component of the magnetic eld and we get

B(R, ω) = 1 c φ,p d 2 k (2π) 2 exp(iK φ • R) βφ p (k, ω)E φ p (k, ω) (4.6)
where

βφ p (k, ω) = (-1) p ǫφ S (p) (k, ω) (4.7)
and the function S is de ned as S (1) = 2 and S (2) = 1.

The starting point for our calculation of the OTE Casimir-Lifshitz force and the heat transfer on each body are the following integrals

F = Σ T(R, t) sym • dΣ H = - Σ S(R, t) sym • dΣ. (4.8)
These are surface integrals of the quantum symmetrized average of the Maxwell stress tensor T (having cartesian components T i j , with i, j = x, y, z) and the Poynting vector S through a closed surface Σ enclosing the body under study. In classical electromagnetism, for bodies immersed in vacuum, the de nitions of these two quantities in SI units are the following

T i j (R, t) = ǫ 0 E i (R, t)E j (R, t)+c 2 B i (R, t)B j (R, t) - 1 2 E 2 (R, t) + c 2 B 2 (R, t) δ i j , S(R, t) = ǫ 0 c 2 E(R, t) × B(R, t). (4.9)
and the quantum symmetrized average value AB sym is de ned as

AB sym = 1 2 AB + BA (4.10)
being A an ordinary quantum average value. From the expressions in (4.9) we see that, for the calculation of the integrals in (4.8), the following products of the elds are involved

E i (R, t)E j (R, t), B i (R, t)B j (R, t) (4.11)
and, keeping in mind the passage to quantum operators, as a further step their symmetrized quantum average. In this last stage the terms like EE and E † E † have null quantum average and only the terms EE † are relevant. So it is convenient to introduce the following expression of the operator C φφ ′ , de ned in terms of the correlation functions of the eld amplitudes. With φ and φ ′ we indicate the propagation direction of the eld in region where z is located

E φ p (k, ω)E φ ′ † p ′ (k ′ , ω ′ ) sym = 1 2 E φ p (k, ω)E φ ′ † p ′ (k ′ , ω ′ ) + E φ ′ † p ′ (k ′ , ω ′ )E φ p (k, ω) = 2πδ(ω -ω ′ ) p, k|C φφ ′ |p ′ , k ′ . (4.12)
In the above equation, we stress that, generally, two modes of the eld propagating in opposite directions do not necessarily commute and the conservation of frequency has been inserted. The latter is due to the time invariance since no dynamics is considered in the system under scrutiny.

From now on, the calculation concerns the heat transfer and the force acting on body 1. We have chosen as enclosing surface Σ a closed box as that pictured in Figure 4.2. This box has one side of length D and a square base of side L parallel to the xy plane. According to the de nition of T i j (S i j ), the m component of the force (m = x,y,z) (the heat transfer) is given in this case by the ux of T mz (S z ) through the two surfaces orthogonal to the z axis, plus the uxes of T mx (S x ) and T my (S y ) through the surfaces of the parallelepiped orthogonal to the x and y axes, respectively. Taking the limit L → +∞, we can see how the surface of the two faces of the box which are orthogonal to the z axis diverges more rapidly (as L 2 ) than the other four surfaces (as L). Then we see that for the calculation of the uxes in (4.8) one simply needs to calculate the ux of T mz (S z ) on the surface (which is now a plane with in nite area) in region A and to subtract this result from the ux of T mz (S z ) through the plane in region B. In addition, as a consequence of the arbitrariness of the box, the two uxes calculated in region A and the region B, must not depend on the z coordinates of the respective planes, in spite of the fact that the stress tensor depends, in general, on z.

Then we only need the ux of T zz and S z through two planes z = z on the two sides of the body. From now on, we use a more compact notation for the expressions relative to force and heat ux and we introduce an index m whose value m = 1 is associated to the heat ux and m = 2 to the force. In this way we have

ϕ m (z) = z=z d 2 r        S z sym m = 1 T zz sym m = 2 (4.13)
Using the expressions of the elds given in (4.4) and in (4.6), after some simple algebra (see [19] for details), the generalized ux reads

ϕ m (z) = -(-1) m 2ǫ 0 c 2 p d 2 k (2π) 2 φ=φ ′ +∞ ck dω 2π + φ φ ′ ck 0 dω 2π φk z ω m p, k|C φφ ′ |p, k . (4.14)
We observe that with the knowledge of the generalized ux above and thanks to the explicit expression of the correlators in any region of our system, we can calculate the force and the heat transfer for our considered system. We also note that C φφ ′ is the only quantity in (4.14) depending on the position z of the plane where we calculate the ux. Moreover, analyzing the equation (4.14) we can see that the ux have two separate contributions, coming from the propagative and evanescent sectors. The rst is given by the correlators between the eld propagating in a direction φ and itself. The second involves the correlators of counterpropagating elds.

If we introduce the following de nitions p, k|P

(pw/ew) n |p ′ , k ′ = k n z p, k|Π (pw/ew) |p ′ , k ′ (4.15)
where Π (pw) (Π (ew) ) are the projector on the propagative (evanescent) sector and de ning the trace operator

Tr A = p d 2 k (2π) 2 +∞ 0 dω 2π p, k|A|p, k (4.16)
we have

ϕ m (z) = -(-1) m 2ǫ 0 c 2 φφ ′ Tr φ ω m C φφ ′ δ φφ ′ P (pw) m + (1 -δ φφ ′ )P (ew) m . (4.17)
where δ φφ ′ is the Kronecker delta.

Correlation functions of the total eld

To go on with the computation of the generalized ux (4.17) we need to nd the correlation functions of the total eld in any region of the system. This results from the source elds present in our system, i.e the elds E (i)φ emitted by the body i and propagating in direction φ and the counterpropagating elds emitted by the environment E (e) φ, (Figure 4.1). We have to consider then all the possible scattering processes undergone by our system. In this way we can make explicit the connection between total and source elds by introducing the re ection and transmission operators associated to each body. To de ne these scattering operators let us analyze a body placed in the region z 1 < z < z 2 . We assume that an external eld is impinging on our body, either from its left or from its right side. The body will scatter the incoming eld. In this way new components of the eld on both sides of the body are produced. The eld coming from the left (right) will produce a re ected eld propagating toward the left (right) on the left (right) side and a transmitted eld that propagates toward the same direction of the incoming one (see Figure 4.3). We will use the notations R (i)φ and T (i)φ to indicate, respectively, the re ection and transmission operators of body i, where φ describes the direction of propagation of the outgoing eld.

For example, let us consider an incoming eld coming from the left side. In this way we have

E (in)+ (R, t) = E (re)-(R, t) + E (tr)+ (R, t). (4.18)
We already noted the conservation of the frequency for the system here considered. We get a de nition of the operators R -and T + that connect each mode of the outgoing elds to all modes of the incoming eld at the same frequency ω

E (re)- p (k, ω) = p ′ d 2 k ′ (2π) 2 p, k|R -|p ′ , k ′ E (in)+ p ′ (k ′ , ω) (4.19) E (tr)+ p (k, ω) = p ′ d 2 k ′ (2π) 2 p, k|T + |p ′ , k ′ E (in)+ p ′ (k ′ , ω). (4.20)
We want to remark that the frequency dependence is implicitly contained in the scattering operators. Obviously an analogous de nition of the operators R + and T -can be given in a similar way. Making use of the scattering operators we introduced, it is possible to get the selfconsistent system of equations giving the total eld in each region. We simply need the amplitudes E (1)+ p (k, ω), E (2)- p (k, ω) and E (3)± p (k, ω). In each region (A,B) of Figure 4.1 the eld propagates in both directions. We gather all the modes and denote the amplitudes with the symbol E (γ)φ , where γ = A, B, and we write the system of equations

                     E (A)+ = E (e)+ E (A)-= E (1)-+ R (1)-E (e)+ + T (1)-E (B)- E (B)+ = E (1)+ + R (1)+ E (B)-+ T (1)+ E (e)+ E (B)-= E (2)-+ R (2)-E (B)+ + T (2)-E (e)-. (4.21)
In the equations above the products between scattering operators and elds must be considered as matrix-vector products. Besides, we used a more compact notation where the ω dependence of all operators and eld amplitudes is omitted. We also introduce the following operators

U (1,2) = +∞ n=0 R (1)+ R (2)-n = (1 -R (1)+ R (2)-) -1 , U (2,1) = +∞ n=0 R (2)-R (1)+ n = (1 -R (2)-R (1)+ ) -1 . (4.22)
describing the in nite series of multiple re ections in the cavity formed by bodies 1 and 2. Now, using the scattering operators de ned above and performing simple algebraic manipulations on (4.21), we obtain the total elds in regions A and B E (A)+ = E (e)+ , (4.23)

E (A)-= T (1)-U (2,1) R (2)-E (1)+ + E (1)-+ T (1)-U (2,1) E (2)-+ R (1)-E (e)+
+ T (1)-U (2,1) R (2)-T (1)+ E (e)+ + T (1)-U (2,1) T (2)-E (e)-, (4.24)

E (B)+ = U (1,2) E (1)+ + R (1)+ E (2)-+ T (1)+ E (e)+ + R (1)+ T (2)-E (e)-, (4.25)
E (B)-= U (2,1) R (2)-E (1)+ + E (2)-+ R (2)-T (1)+ E (e)+ + T (2)-E (e)-. (4.26)
In this way we have found an expression of the total eld in each region as a function of the source elds E (i)φ (i = 1, 2, φ = +, -) and the environmental eld E (e)φ .

We can now return to develop an explicit expression for the correlators related to the scattering operators. If we consider an identical system to that we are considering here but at thermal equilibrium, the correlators of the total electromagnetic eld outside the body can be deduced from the uctuation-dissipation theorem [27]. This expression relates the correlators with the Green function of the system and with the thermal population density

N(ω, T ) = ω 2 coth ω 2k B T = ω 1 2 + n(ω, T ) , n(ω, T ) = 1 e ω k B T -1 . (4.27)
However, when we consider the out of thermal equilibrium con guration, the uctuationdissipation theorem, strictly speaking, is not applicable. Though, if we assume that a local temperature can be de ned for each body and that it remains constant in time, the correlators of the eld emitted by each body can still be deduced using the uctuation-dissipation theorem at the local temperature. This is possible because the assumption made leads to think that the part of the total eld emitted by each body is the same as if the body were at thermal equilibrium with the environment at its temperature. It is like saying that the emission of each body is not signi cantly in uenced by the modi cation due to the external radiation impinging on the body. It should be said, however, that the limits of validity of this hypothesis, already used in literature, require further experimental and theoretical investigations. Now it remains to nd the connection between the Green function and the scattering operators. We will not go into the details of the calculation; it is discussed in [19] and here we present only the result. Similarly to what we have done in (4.12) we de ne the following matrices

E (i)φ p (k, ω)E (i)φ ′ † p ′ (k ′ , ω ′ ) sym = 2πδ(ω -ω ′ ) p, k|C (i)φφ ′ |p ′ , k ′ , E (e)φ p (k, ω)E (e)φ ′ † p ′ (k ′ , ω ′ ) sym = 2πδ(ω -ω ′ ) p, k|C (e)φφ ′ |p ′ , k ′ . (4.28)
In addition we introduce the useful function

f α (R) =                            P (pw) -1 -RP (pw) -1 R † + RP (ew) -1 -P (ew) -1 R † α = -1 P (pw) m + (-1) m R † P (pw) m R + R † P (ew) m +(-1) m P (ew) m R α = m ∈ {1, 2} (4.29)
. In this way we can write the correlation functions of the source elds as follows

C (i)φφ = ω 2ε 0 c 2 N i f -1 (R (i)φ ) -T (i)φ P (pw) -1 T (i)φ † , (4.30)
C (i)φ,-φ = ω 2ε 0 c 2 N i -R (i)φ P (pw) -1 T (i)-φ † -T (i)φ P (pw) -1 R (i)-φ † + T (i)φ P (ew)
-1 -P (ew) -1 T (i)-φ † , (4.31)

C (e)φφ ′ = δ φφ ′ ω 2ε 0 c 2 N e P (pw) -1 , (4.32)
where for α ∈ {1, 2, e} we have de ned N α = N(ω, T α ).

At this stage we have found the expression of the total eld in each region in terms of the source elds. Since this relation is always linear, we can write it under the general form

E (γ)φ = 3 i=1 α=+,- A (γ)φ iα E (i)α + α=+,- B (γ)φ α E (e)α , (4.33)
where γ ∈ {A,B,C}. From this general expression we can derive the general expression of the correlation functions of the total eld in the region γ

C φφ ′ γ = 3 i=1 α,α ′ =+,- A (γ)φ iα C (i)αα ′ A (γ)φ ′ † iα ′ + α=+,- B (γ)φ α C (e) B (γ)φ ′ † α , (4.34)
where C (e) = C (e)φφ . Since A (γ)φ iα and B (γ)φ α are the coe cients of the decomposition derived in (4.23)-(4.26), knowing the source correlation functions obtained in (4.12), we are able to derive the correlators of the total eld in any region. We are now ready to calculate the uxes in the region A and B and derive our nal expression of the OTE force and of the heat transfer of our system in terms of the scattering operators.

Finally exploiting the results obtained in the limit case of thermal equilibrium (see [19] and the more general case in [28] for more details), it is convenient to separate the force as the sum of two terms

F 1z = F (eq) 1z (T 1 ) + ∆ 2 (4.35)
where

F (eq) 1z (T 1 ) = -2 Re Tr k z ω -1 N(ω, T 1 ) U (1,2) R (1)+ R (2)-+ U (2,1) R (2)-R (1)+ , (4.36)
is the force acting on body 1 if the system were at the thermal equilibrium at temperature T 1 , while ∆ 2 is a nonequilibrium term that depends on the three temperatures of the system. The latter is null when all the three temperatures of the system are the same. It is important to note that the equilibrium force contains both the zero-temperature term and the thermal correction. In addition it has a dependence only on the intracavity re ection operators R (1)+ and R (2)-, i.e., the operators that describe the re ection produced by each body on the side of the other one. With this separation of the force and indicating the heat transfer as H = ∆ 1,1 , after lenghty algebraic manipulations, we can write the expression for heat transfer and the OTE Casimir-Lifshitz force

∆ m = -(-1) m Tr ω 2-m n e1 U (2,1) T (2)-P (pw) -1 T (2)- † U (2,1) † f m (R (1)+ ) -T (1)- † P (pw) m T (1)- + (-1) m U (1,2) T (1)+ P (pw) -1 T (1)+ † U (1,2) † -P (pw) -1 f m (R (2)-) + R (2)-P (pw) -1 R (2)- † -R (12)-P (pw) -1 R (12)- † P (pw) m + n 21 U (2,1) f -1 (R (2)-) -T (2)-P (pw) -1 T (2)- † U (2,1) † f m (R (1)+ ) -T (1)- † P (pw) m T (1)-
where we introduced

R (12)-= R (1)-+ T (1)-U (2,1) R (2)-T (1)+ (4.37)
and n i j = n(ω, T i )n(ω, T j ), i,j=1, 2, e. (4.38) This expression gives the heat transfer (m = 1) and the non-equilibrium contribution to the force (m = 2) on body 1 for an arbitrary set of the two bodies having arbitrary geometry, dielectric properties and temperatures and immersed in a thermal bath having a third different temperature. The equation (4.35) and the equation (4.37) are the starting point for our original calculation discussed in this chapter. We want to conclude this derivation saying that it is also possible to generalize the method developed here to the case of three body in an OTE con guration [28].

The force between two slabs

In this subsection we shortly analyze, as example for an application of the expressions found above, the OTE Casimir-Lifshitz force and the heat transfer for the case of two slabs. These cases, already studied in literature (see references [9], [52] and [19]), are presented here to show the power and the applicability of the method introduced in this Section. Moreover, since the two slabs case has been deeply studied in Casimir physics [53], this system is a very good example to understand some aspects of the OTE Casimir-Lifshitz force and make a comparison with the thermal con guration. We consider two parallel homogeneous dielectric slabs, labeled i (i = 1, 2) of nite thickness δ i placed at a distance d (see Figure 4.4). At rst we analyze the re ection and transmission operators R (1)+ and T (1)+ . For the case of homogeneous slab, as that we are considering here, these operators are diagonal and can be written as p, k|R (1)

+ |p ′ , k ′ = (2π) 2 δ(k -k ′ )δ pp ′ ρ 1p (k, ω), (4.39)
p, k|T (1)

+ |p ′ , k ′ = (2π) 2 δ(k -k ′ )δ pp ′ τ 1p (k, ω), (4.40)
where we introduced the modi ed Fresnel coe cients, respectively the re ection and transmission ones, considering the nite thickness of the slabs:

ρ 1p (k, ω) = r 1p (k, ω)
1e 2ik z1 δ 1 1r 2 1p (k, ω)e 2ik z1 δ1 , (4.41)

τ 1p (k, ω) = t 1p (k, ω) t1p (k, ω)e i(k z1 -k z )δ 1 1 -r 2 1p (k, ω)e 2ik z1 δ1 .
(4.42)

In the equations above we de ned the z component of the wave vector k inside the slab 1 and we used the standard Fresnel coe cients for the vacuum-medium con guration

k z1 = ε 1 (ω) ω 2 c 2 -k 2 , (4.43) r 1,TE = k z -k z1 k z + k z1 , r 1,TM = ε 1 (ω)k z -k z1 ε 1 (ω)k z + k z1 , (4.44
)

t 1,TE = 2k z k z + k z1 , t 1,TM = 2 √ ε 1 (ω)k z ε 1 (ω)k z + k z1 , (4.45)
and we have also introduced the Fresnel coe cients for the medium-vacuum transmission

t1,TE = 2k z1 k z + k z1 , t1,TM = 2 √ ε 1 (ω)k z1 ε 1 (ω)k z + k z1 . (4.46)
The transmission operator T (1)-and the scattering operators R (2)-, T (2)-of body 2 can be found from geometrical considerations of the system and solving the problem of the scattering operators change when we perform a translation from a frame of reference O → Õ. Let us discuss this translation transformation and let us introduce the operator S φφ ′ in the following way

S ++ = T + , S +-= R + , (4.47)
S -+ = R -, S --= T -. (4.48)
This operator simply connects the outgoing modes propagating in direction φ to the incoming modes propagating in direction φ ′ . If we indicate with R S the translation vector, from the de nition of the electric eld (4.4), we can write

E(R, ω) = φ,p d 2 k (2π) 2 exp(iK φ • (R -R S ))ǫ φ p (k, ω)E φ p (k, ω) exp(iK φ • R S ) (4.49)
and we obtain the amplitude Ẽφ p (,ω) in the translated frame

Õ Ẽφ p (k, ω) = exp(iK φ • R S )E φ p (,ω). (4.50)
We deduce then

Ẽ(out)φ p (k, ω) = exp(iK φ • R S ) p d 2 k ′ (2π) 2 p, k|S φφ ′ |p ′ , k ′ exp(-iK ′φ ′ • R S ) Ẽ(in)φ ′ p ′ (k, ω) (4.51)
and from this we nally have

p, k| Sφφ ′ |p ′ , k ′ = exp -i(K φ -K ′φ ′ ) • R S p, k|S φφ ′ |p ′ , k ′ . (4.52)
So if we consider a translation of d along the z axis, as in the case of two slabs we are considering, we obtain the following relations

p, k| R+ |p ′ , k ′ = exp i(k z + k ′ z )d p, k|R + |p ′ , k ′ , (4.53)
p, k| R-|p ′ , k ′ = exp -i(k z + k ′ z )d p, k|R -|p ′ , k ′ , (4.54)
p, k|

T + |p ′ , k ′ = exp i(k z -k ′ z )d p, k|T + |p ′ , k ′ , (4.55) p, k| T -|p ′ , k ′ = exp -i(k z -k ′ z )d p, k|T -|p ′ , k ′ (4.56)
and we are able to get all the scattering operators to which we are interested. We note that for the physical system that we are considering, the matrix element of the scattering operators between two generic states is proportional to a Dirac delta function, as we can see from (4.39) and (4.40). This delta function comes from the translational invariance along x and y dictated by the particular geometry under scrutiny. We can deduce that the force acting on the body 1 is formally divergent as well as the heat transfer. However this is not an issue of concern because the body 1, a slab, has an in nite surface so this

(i) (d) (e) (f) (ii)
Figure 4.5 -In this gure, coming from [19], it is plotted the pressure acting on a fused silica slab of thickness δ 1 = 2 µm placed in front of a second silicon slab of thickness δ 2 = 1000 µm. Lines: pressures of the equilibrium state at T = 0 K (black solid), 300 K (blue dashed), and 600 K (red dash-dotted). Symbols: pressures of the nonequilibrium state where T 3 = 0 K (blue circles), 300 K (green diamonds), and 600 K (magenta plus). T 1 = 300 K and T 2 = 0 K in (a), T 1 = 0 K and T 2 = 300 K in (b), and T 1 = T 2 = 300 K in (c). (d), (e) and (f) are the same plots of those on the left but with linear pressure scale. divergence is not surprising. In fact, we can consider the force density or the heat transfer per unit area which are instead nite quantities. We can then calculate the pressure and heat transfer density on body 1 and make our physical considerations on these quantities.

We have now all the ingredients for the calculation of the pressure and of the heat transfer density. We present and discuss the case of a slab made of fused silica with a thickness of 2µm placed in front of a silicon slab with thickness of 1000 µm. It is numerically evaluated and studied in reference [19].

Let us start with the results of the pressure, for di erent con gurations of temperatures, shown in Figure 4.5 (see the caption for the temperature con gurations). In these three plots, the pressure as a function of the distance (range from 1 to 10 µm) is shown. For each of them both the pressure of the non-thermal-equilibrium state and the pressure of the thermal-equilibrium state (at the environment temperature) are reported. The remarkable point of these group of plots is that the transition from an equilibrium to a nonequilibrium con guration change signi cantly the qualitative and quantitative behavior of the interaction. Moreover, the environmental temperature can be considered as a parameter to tune the interaction. In particular in the plot (c) of Figure 4.5, where the two bodies are in relative thermal equilibrium (T 1 = T 2 ), the environmental temperature assumes an important [19] where is plotted the radiative heat transfer per unit surface between a fused silica slab of thickness δ 1 = 2 µm placed in front of a second silicon slab of thickness δ 2 = 1000 µm. The temperatures of the slabs are T 1 = 300 K and T 2 = 0 K in (a), T 1 = 0 K and T 2 = 300 K in (b), and T 1 = T 2 = 300 K in (c). Symbols: T e = 0 K (black squares), 300 K (green plus), 400 K (blue crosses), 500 K (brown diamonds), and 600 K (red circles). role since it can modify the force. This point stresses the fact that the environmental temperature should be carefully controlled also in experiments where the force is measured at the relative equilibrium between the two bodies. In the second group of plots of Figure 4.5 (plots (d), (e), (f)) we see the same features already described but the transition of the force from attractive to repulsive becomes evident thanks to the linear scale used; also, it can be seen more easily what are the distances where the transition occurs (around 6 µm).

We turn now to the results of the heat transfer density pictured in Figure 4.6. We have three di erent con gurations, corresponding to the three di erent choices of the slab temperatures T 1 and T 2 listed in the caption. IN each case we have a plot for the heat transfer when the environmental temperature T e has the values (0,300,400,500,600) K. We can immediately stress the importance of the environmental radiation, as we already seen for the pressure. In fact, the heat transfer h1 shows an oscillating behavior with an amplitude that increases when the temperature T e takes higher values. It can be shown (see [19]) that these oscillations are due to the propagative sector. We also wish to stress that the occurrence of a change of sign in the heat transfer. The higher is the value of T e , the smaller is the distance at which this change of sign occurs.

Methods for the di raction by gratings

In this section we focus our attention on the description of the di raction of light in presence of periodic media, in particular di raction gratings. Gratings are optical devices having a periodic structure (1D or 2D periodicity) which, when impinged by electromagnetic radiation, split and di ract the incident rays of the radiation in several beams. There are several applications in science and in technology related to di raction by gratings and also in nature we nd many examples of these systems (butter y wings for example). For this reason several methods have been developed over the years, each of these more suitable in some speci c physical situations [54]. We dwell on describe the Fourier Modal Method which, thanks to its simplicity, versatility, numerical stability and rapidity has been chosen for the calculation of our original work showed in this chapter, the OTE Casimir-Lifshitz force between two dielectric gratings. We also brie y discuss other two methods strictly related to the Fourier Modal Method, the adaptive spatial resolution method and the Cmethod. The latter is important for possible future applications to the calculation of the force between two gratings when they are made of metal. The C-method instead is interesting because, besides the FMM and the modal method that for brevity is not described here, is a one of the few methods used to calculate the Casimir-Lifshitz force between gratings with continuous pro le.

The Fourier Modal Method

We start presenting and discussing the Fourier Modal Method (FMM), the method we have used to obtain our original results showed in the next section. It is the most used and popular method to describe the di raction by gratings. With this approach, based on a Rayleigh expansion of the elds in the homogeneous zones, the problem of the search for solutions of the elds in the grating region simply translates in an eigenvalue problem. This is possible after an appropriate expansion of the elds into Floquet-Fourier series and the periodic permittivity in the grating region into Fourier series. The method is relatively simple. We need to know the Maxwell equations, the Fourier series and basic concepts of geometry and algebra. It is also a quick method with respect to numerical convergence and the code necessary to implement it in a computer calculation requires relative few steps in programming and it can be easily written in several di erent programming languages (Matlab, Mathematica, C, Fortran). Moreover, it is a versatile method because the range of systems that can be treated with it is very wide.

Before discussing the FMM we want to state the paradigm of the di raction problem. We want to nd the distribution, in near and far eld, of the electromagnetic waves after an incident electromagnetic wave irradiates a grating (with known geometrical and optical properties). This means that we must solve a boundary value problem. The FMM faces this problem through the following steps. At rst the geometrical space where the grating is placed is divided in three zones (see Figure 4.7). In the upper semi-in nite transparent region with dielectric constant ε 1 the incident plane wave comes from in nity; in the middle region, i.e. the grating region, the dielectric constant ε 2 (x) is periodic and in the lower semi-that ε 1 = ε b while ε a = ε 3 . The zones with ε a and ε b are called, respectively, ridges and grooves of the grating. The geometrical parameters describing the grating are the period, indicated with D, the groove depth h and the ridge width l. It is also useful to introduce the lling factor de ned as f = l/D. From now on we will consider that the wave vector k (in) stays on the xz plane

k (in) = k 0 √ ε 1 (sin θ x + cos θ ẑ) (4.58)
where k 0 is the wave number of the vacuum. The conical case (i.e. the case where the incident wave vector stays in a plane di erent from the xz plane) is easily found with a generalization and it is treated in the next section. As a consequence of this y invariance of k (in) we have that also the elds E and H are independent of y. The grating problem, from the properties of the Maxwell equations thus, becomes scalar and it is necessary to nd only one component of the electric/magnetic eld, the other ones are then easily determined from it with the help of the Maxwell equations (for our system we have

∂ t = -iω)              ✟ ✟ ✟ ∂ y E z -∂ z E y = iωµ 0 H x ∂ x E y -✟ ✟ ✟ ∂ y E x = iωµ 0 H z , TE case ∂ z H x -∂ x H z = -iωεε 0 E y (4.59)              ✟ ✟ ✟ ∂ y H z -∂ z H y = -iωεε 0 E x ∂ x H y -✟ ✟ ✟ ∂ y H x = -iωεε 0 E z , TM case ∂ z E x -∂ x E z = iωµ 0 H y (4.60)
This means that we can consider only two di erent cases, the transverse electric (TE) case and the transverse magnetic TM case (see Figure 4.8). For the rst case we must determine the y-component of the electric eld while for the second one we must determine the ycomponent of the magnetic eld.

Let us now introduce the following useful compact notation:

u (i) (x, z) =        E (i) y (x, z) TE H (i) y (x, z) TM (4.61)
where i = 1, 2, 3 indicates one of the three regions. In this notation we get from the equations (4.59) and (4.60) the following wave equations valid in the grating region

∂ 2 u (2) ∂x 2 + ∂ 2 u (2) ∂z 2 + k 2 ε (2) (x) u (2) (x, z) = 0 TE, (4.62)
∂ ∂x

1 ε (2) ∂u (2) ∂x + 1 ε (2) ∂ 2 u (2) ∂z 2 + k 2 ε (2) (x) u (2) (x, z) = 0 TM. (4.63)
Since the electric permittivity is periodic in the grating region, from the Floquet-Bloch theorem we deduce that the elds are pseudo-periodic

u (2) (x + D, z) = e i k x D u (2) (x, z) (4.64)
where we introduced the de nition

k (m) z,n = ε (i) (ω) ω 2 c 2 -k 2 x,n , (4.71)
Re k (m) z,n ≥ 0 and Im k (m) z,n ≥ 0. (4.72)

Finally we obtain the well known Rayleigh expansion for the eld in the homogenous zones

u (m) (x, z) = n∈Z
A n e ik x,n x+ik z,n z + B n e ik x,n x-ik z,n z . (4.73)

In particular we use the following notation for the Rayleigh expansion

u (1) (x, z) = n∈Z
I n e ik (1) z,n z + R n e -ik (1) z,n z e ik x,n x , (4.74)

u (3) (x, z) = n∈Z
T n e ik (3) z,n (z-h) e ik x,n x (4.75)

where we added a phase factor e -ik z,n h in the zone 3, useful for the following calculations and that we can remove later on, and we indicated with I n the known coe cients of the incident wave. R n , T n are, respectively, the unknown re ection and transmission coe cients.

We can turn now to analyze the eld in the grating region. Let us start with the TE case. Exploiting the expansions in Fourier series made in (4.66) for the elds and for the electric permittivity

ε p = 1 D D 0 dx ε (2) (x)e -i 2π D p x (4.76)
we can cast the wave equation (4.62) as follows

∂ 2 ∂x 2        n u n (z)e ik x,n x        + ∂ 2 ∂z 2        n u n (z)e ik x,n x        + k 2         p ε p e i 2π D p x                n u n (z)e ik x,n x        = 0. (4.77)
The above equation can be rewritten exploiting the rules of Fourier Factorization. If we consider a function h(x) = g(x) f (x), where g(x) and f (x) are periodic piecewise-continuousfunctions, and assuming that all three functions have the same period, to compute the Fourier coe cients of h(x) in terms of the Fourier coe cients g n and f m , we use the following rules [55], [56]:

(1) h n,Ω = m∈Ω g n-m f m , n ∈ Ω, with g(x) discontinuous and f (x) continuous (2) h n,Ω = m∈Ω 1 g -1 nm
f m , n ∈ Ω, with g(x) and f (x) discontinuous and h(x) continuous at some point x 0 (4.78)

where Ω indicates a set of integers in the range m 1 < m < m 2 for some xed m 1 and m 2 . We have also introduced the Toeplitz matrix a , de ned by the relation a i j = a i-j , a n being the n-th Fourier component of a.

ε =                      . . . ε -1 ε -2 ε -3 . . . ε 1 ε 0 ε -1 ε -2 ε -3 ε 2 ε 1 ε 0 ε -1 ε -2 ε 3 ε 2 ε 1 ε 0 ε -1 . . . ε 3 ε 2 ε 1 . . .                      (4.79)
If we are not in one of the two cases mentioned in (4.78), there is not a general rule to compute the Fourier coe cients of h(x). Usually the approach used is to rearrange g(x), f (x) as a sum of terms that ful ll one of the two cases listed above. Having in mind these rules we obtain from (4.77) that

        p ε p e i 2π D p x                n u n (z)e ik x,n x        = n         p ε n-p u p e ik x,n x         (4.80)
and we can deduce the equation

n d 2 u n (z) dz 2 -k 2 x,n u n (z) + k 2 p ε n-p u p (z) e ik x,n x = 0. (4.81)
Using a matrix notation

U = [. . . , u -2 , u -1 , u 0 , u 1 , u 2 , . . .] T α = diag . . . , k x,-1 , k x,0 , k x,1 , . . . (4.82)
we get the following di erential equation

d 2 U dz 2 = α 2 -k 2 ε U(z) ≡ AU(z). (4.83)
If we look for solutions of the form We see from the equation (4.83) that the problem to nd the solutions of the elds in the grating region has been reduced to an eigenvalue problem. Numerically the matrix A has to be truncated to N × N. The number of eigensolutions is N, but there are 2N eigensolutions for the physical problem when we take the square root of the eigenvalues of A. The truncation number N determines the accuracy of the numerical results. When we increase N, the results become more accurate but at the expense of more memory and computer time. As a consequence N should be chosen large enough to include all propagating orders and su ciently many evanescent orders on both side of the propagating orders. The choice of how large N can be, depends on the speci c problem and a general criterion cannot be given. However, in general, a metallic grating requires a larger N than a dielectric grating does. Then, we conclude that it is necessary to run a few convergence tests for the speci c problem under study. In general a good and reasonable criterion for a convergence test is to consider the minimum N showing stable numerical results for the quantities of interest.

U(z) = e Xz U ′ 0 (4.84) then X 2 e Xz U ′ 0 = Ae Xz U ′ 0 ⇒ X = ± A 1/2 = ± PD 2 P -1 1/2 = ± PDP -1 , (4.85
The last step is to impose the boundary conditions at the interfaces of the system, in this case at z = 0 and z = h. This is realized by matching the Fourier coe cients of the tangential components of the total elds. For the TE case it means to impose the continuity of E y and H x , that in our notations means u(x, z) and ∂ z u(x, z). At the z = 0 interface we have

∀ 0 ≤ x ≤ D,        E (1)
y (x, 0) = E (2) y (x, 0) H (1) x (x, 0) = H (2) x (x, 0)

(4.90)                      n δ 0n e ik (1) z,n 0 + R n e -ik (1) z,n 0 e ik x,n x = n         p P np a p e D p 0 + b p e -D p 0         e ik x,n x n ik (1) z,n δ 0n e ik (1) z,n 0 -R n e -ik (1) z,n 0 e ik x,n x = n         p P ′ np a p e D p 0 -b p e -D p 0         e ik x,n x (4.91)
and for the z = h interface It is convenient to write these boundary conditions in a more compact form through the matrix notation. In this way we can write

∀ 0 ≤ x ≤ D,        E (2) y (x, h) = E (3) y (x, h) H (2) x (x, h) = H (3) x (x, h) (4.92)                      n         p P np a p e D p h + b p e -D p h         e ik x,n x = n T n e ik (3) z,n (h-h) + 0 e -ik (3) z,n (h-h) e ik x,n x n         p P ′ np a p e D p h -b p e -D p h         e ik x,n x = n ik (3) z,n T n e ik (3) z,n (h-h) -0 e -ik (3) z,n (h-h) e ik x,n x .
z = 0;        I + R = P(A + B) i K 1 (I -R) = P ′ (A -B) (4.94) and z = h;        P(φA + φ -1 B) = 0 + T P ′ (φA -φ -1 B) = i K 3 (-0 + T ) (4.95)
where we have introduced the following vectors I = [. . . , 0, 0, I 0 , 0, 0, .

. .] T , R = [. . . , R -1 , R 0 , R 1 , . . .] T , T = [. . . , T -1 , T 0 , T 1 , . . .] T , (4.96)
and operators

K i = diag k (i) z,n , φ = e Dh . ( 4

.97)

We exploit the well known S-matrix formalism to nd the re ection and transmission matrices of the grating. We de ne the S-matrix which relates the electric eld amplitudes of the incident wave to the amplitudes of the re ected and transmitted electric eld

R T = S I 0 = R T ′ T R ′ I 0 . ( 4

.98)

S is a block matrix composed by the re ection and transmission matrix R and T. The matrices R ′ and T ′ are respectively the re ection and transmission matrices for the analogous problem of di raction when the incident wave is coming from the region 3. If we analyze equation (4.94) and we consider the convention de ned in Figure 4.9 for the signs used for A and B, we can cast the boundary conditions at z = 0 as

R A = -1 P i K 1 P ′ -1 1 -P i K 1 P ′ I B ≡ S 1 I B . ( 4
P ′ i K 1 -1 P -1 P ′ i K 3 φ 0 0 1 A 0 ≡ S 2 A 0 . ( 4

.100)

De ning the associative operation A = B ⊛ C for square matrices as follows

A 11 = B 11 + B 12 (1 -C 11 B 22 ) -1 C 11 B 21 , (4.101)
A 12 = B 12 (1 -C 11 B 22 ) -1 C 12 , (4.102)
A 21 = C 21 (1 -B 22 C 11 ) -1 B 21 , (4.103)
A 22 = C 22 + C 21 (1 -B 22 C 11 ) -1 B 22 C 12 , (4.104)
we get the nal expression of the S-matrix as S ≡ S 1 ⊛ S 2 .

We have found the S-matrix, therefore the re ection and transmission matrices, in the transverse electric case. We turn now to the transverse magnetic case and discuss it brie y. The procedure to nd the solutions of the elds in the grating region and the S-matrix are quite similar to the TE case. It is not di cult to see that, starting from (4.63) and using the compact matrix notation, we obtain the following di erential equation

d 2 U dz 2 = 1 ε -1 α 1 ε α -k 2 1 U(z) ≡ AU(z) (4.105)
where we used the second of the rules de ned in (4.78) for the Fourier factorization. From this di erential equation we nd the solutions

           U(z) = P e Dz A + e -Dz B Re(D) ≤ 0 ∂ z U(z) = 1 ε PD e Dz A -e -Dz B Re(D) ≤ 0. ( 4

.106)

To nd the scattering operators, i.e. the S-matrix, for the TM case we apply again the boundary conditions at the interfaces z = 0 and z = h. In this particular situation, we impose the continuity of the elds H y = u(x, z) and E x = 1 ε ∂ z u(x, z). Proceeding analogously to the TE case, we obtain the same formally identical matrix S where we have to do the substitutions

K i → K i ε i , P ′ → 1 ε P ′ . (4.107)

Some other methods for the di raction by gratings

To conclude this section we brie y present some other methods to study the di raction by gratings. At rst we discuss concisely the possible application of the FMM to the case of gratings di erent from the rectangular one and one possible improvement of the FMM, especially for the case of metallic gratings. We start discussing the common extension of the FMM for gratings with smooth pro le, i.e. the so-called staircase approximation. The main idea of this approach is to divide the grating region in many subregions as, for example, in the situation represented in the Figure 4.11. In each subregion we have a rectangular grating and then we can apply the FMM. With this idea we replaced the resolution of the original grating problem with this new "approximated" problem of the modi ed grating. This staircase approximation seems reasonable because, as the number of layers increase and tends to in nity, the layer thickness goes to zero, and we recover the initial original grating. Unfortunately we must also consider the physical problems introduced with this approximation. Several numerical evaluations show that, for 1D periodic gratings, the approximation is quite good for TE polarization but not for TM polarization, especially if we consider highly conducting metallic gratings [57]. We can explain this because when we analyze, at the sharp edge of a wedge made of nonmagnetic media with di erent dielectric constants, the transverse component of the electric eld to the edge direction we see that it is, if nonzero, in nite, as it is known from classical electromagnetism. When we then do the staircase approximation we introduce many arti cial edges. For the TE case we do not have particular numerical problems because the electric eld remains nite everywhere. Instead in TM polarization the electric eld, that in the real situation without the staircase approximation is nite near the smooth grating pro le, becomes in nitely large at the edges and numerical convergence and the computation accuracy problems arise. One of the major known issue of the FMM is the poor convergence in the TM case. This problem is partially solved when the right factorization rules (4.78) are applied. Even if these improvements are adopted, the problems of convergence can remain. We have already said that, when highly conducting metallic gratings are studied, numerical problems arise. Other numerical issues arise because of the use of Fourier expansions. In fact, it can be shown that the convergence is strictly related to the gradient of permittivity in the grating region. In this way also the so-called Gibbs phenomenon at the discontinuity points plays a role. An e ective solution can be the adaptive spatial resolution (ASR) proposed by Granet [58]. The key idea is to use a new coordinate system that increases the spatial resolution where the discontinuities of the grating are present by stretching the coordinate around them. However, the cost of this change of coordinates is solving eigenvalue problems also in the homogeneous regions (the boundary conditions are written in the new space) and possibility to write Maxwell equations in a covariant form. Another powerful characteristic of the C-method is the possibility to apply it to a wide variety of physical systems (many di erent surface pro les, multi-layer coated gratings). The importance of the C-method is relevant in literature and for example it inspired the ASR, that we previously discussed, and matched coordinates [60].

Let us consider the general con guration in Figure 4.12. The grating has a surface prole described by the function a(x) and it is in nite along the x and y direction. When the electromagnetic eld is present, the boundary conditions impose that the tangential components of the electric eld vector and the normal component of the displacement eld vector are continuous at the surface. This means that the boundary conditions depend on the position on the surface. The coordinate transformation (u, v, z) before mentioned is such that

u = x, v = y, w = z -a(x).
(4.115)

It maps the surface that separates the two region 1 and 2 to the surface w = 0 making the boundary conditions in this new coordinate system as that on a planar surface. At this stage we can write the covariant Maxwell equations

ξ i jk ∂ j E k = iωµ 0 √ gg i j H i (4.116) ξ i jk ∂ j H k = -iωε 0 ε √ gg i j E i (4.117)
where g i j is the metric tensor, g = det g i j and ξ i jk is the Levi-Civita tensor. In the case of the transformation considered we can write the line element

ds 2 = dx 2 + dy 2 + dz 2 = du 2 + dv 2 + (dw + ȧdu) 2
= (1 + ȧ2 )du 2 + dv 2 + dw 2 + 2ȧdu dw = g i j dx i dx j . (4.118)

We deduce

g i j =           1 + ȧ2 0 ȧ 0 1 0 ȧ 0 1           , g i j =           1 0 -ȧ 0 1 0 -ȧ 0 1 + ȧ2           , g = 1. ( 4
            ∂ y E w -∂ w E y = iωµ 0 (H x -ȧH w ) ∂ x E y -∂ y E x = iωµ 0 (1 + ȧ2 )H w -ȧH x ∂ w E x -∂ x E w = iωµ 0 H y , (4.120)              ∂ y H w -∂ w H y = -iωεε 0 (E x -ȧE w ) ∂ x H y -∂ y H x = -iωεε 0 (1 + ȧ2 )E w -ȧE x ∂ w H x -∂ x H w = -iωεε 0 E y . ( 4

.121)

From them, using the expansion in Fourier series and the matrix notation, in a similar way as in the FMM, it is possible to see that again the problem to nd the eld can be reduced to an eigenvalue problem.

OTE Casimir-Lifshitz force between two dielectric gratings

In this section we study the problem of the OTE Casimir-Lifshitz force between two gratings. This problem has been studied in our original work [39] and represents the most important result of this Chapter. We address the Casimir-Lifshitz force between two dielectric gratings immersed in vacuum (ε = 1) in the geometrical con guration shown in Figure 4.13. We label the two gratings with an index i taking values 1 and 2. The gratings are in nite in x and y directions, with periodicity along the x axis. Their distance d is de ned in Figure 4.13 and can only take positive values (i.e. a plane z = z must exist separating the two bodies).

The gratings share the same period D and have corrugation depth h i , permittivities ε i (ω) in the homogeneous zone, permittivities ε i (x, ω) along the grating zone having thickness δ i , and lling factors f i = l i /D (l i is de ned as in Figure 4.13).

The physical system, similarly to the system of two slabs studied previously, is out of thermal equilibrium. It means that the gratings have a constant temperature T i that could be di erent from each other. The two gratings are supposed to be immersed in a radiation bath having temperature T e , in general di erent from the temperatures of the two gratings. The three temperatures involved can be considered constant in time.

As we discussed in Section 4.1, the latter assumption has been used in literature to characterize the properties of the source elds (the elds emitted by the two bodies and coming from the surrounding walls) in terms of eld correlation functions. Now we use the same procedure of Section 4.1 to calculate the OTE force of our system. Our starting point is the decomposition of the force as a sum of an equilibrium term and a nonequilibrium term obtained in (4.35) (where the distance dependence is implicit)

F 1z = F (eq) 1z (T 1 ) + ∆(T 1 , T 2 , T e ), (4.122)
where F (eq) 1z (T 1 ) is the force acting on grating 1 at thermal equilibrium at its temperature T 1 given in (4.36) and the non-equilibrium term is given in (4.37).

In order to calculate the force, we now need to compute the re ection and transmission operators associated to a lamellar 1D grating. This will be achieved in the framework of the Fourier Modal Method [61]. In the following, we implement this method, di erently than Subsection 4.2.1, in a conical mounting (it means that the incident wave vector stays in a plane di erent from the xz plane). Moreover we apply it to a grating of nite size along the z axis (see Figure 4.14) in order to take into account nite-size e ects on the Casimir-Lifshitz force. We also solve the scattering problem directly in TE and TM components, in order to be coherent with the formalism presented in Section 4.1.

Let us consider a system composed of a grating like that in Figure 4.14. The space is divided in four zones: zone 1 (z < 0), zone 2 (0 < z < h), zone 3 (h < z < h + δ) and zone 4 (z > h + δ). While zones 1, 3 and 4 are homogeneous with dielectric permittivities ε i (ω) (i = 1, 3, 4), zone 2 represents the grating, with a dielectric function ε 2 (x, ω), periodic in x with period D. In each zone, every physical quantity is independent of y.

We rst decompose the electric eld in any zone with respect to frequency (only positive where the function S is de ned as S (1) = 2 and S (2) = 1.

We now move to the periodic region (zone 2) where we write an arbitrary frequency component of the eld as

E (2) (R, ω) = π D -π D dk x 2π n∈Z +∞ -∞ dk y 2π e ik n •r E (2) (z, k n , ω), (4.132)
where R = (r, z).

We now write Maxwell's equations

             ∂ y E z -∂ z E y = iωµ 0 H x = ik 0 H x ∂ z E x -∂ x E z = iωµ 0 H y = ik 0 H y ∂ x E y -∂ y E x = iωµ 0 H z = ik 0 H z              ∂ y H z -∂ z H y = -iωεε 0 E x ∂ z H x -∂ x H z = -iωεε 0 E y ∂ x H y -∂ y H x = -iωεε 0 E z (4.133)
where we used ω = ck 0 , ωµ 0 = k 0 Z 0 , ωε 0 = k 0 /Z 0 , Z 0 = µ 0 /ε 0 and de ned H i = Z 0 H i . From (4.133) we can easily obtain

∂ z        E x E y        =              - i k 0 ∂ x 1 ε(x) ∂ y ik 0 + i k 0 ∂ x 1 ε(x) ∂ x -ik 0 - i k 0 ∂ y 1 ε(x) ∂ y i k 0 ∂ y 1 ε(x) ∂ x                     H x H y        (4.134)
and

∂ z        H x H y        =              i k 0 ∂ x ∂ y -ik 0 ε(x) - i k 0 ∂ x ∂ x ik 0 ε(x) + i k 0 ∂ y ∂ y - i k 0 ∂ y ∂ x                     E x E y        . ( 4

.135)

We now employ a Fourier factorization for the elds E and H. Correspondingly, the operator ∂ y is replaced by iβ, β being a scalar, whereas the operator ∂ x is replaced by iα, where α = diag(k x,n ) n . These replacements allow us to rewrite Maxwell's equations of our system in a more compact form:

∂ z E =              iβ k 0 α ε -1 ik 0 1 - iα k 0 ε -1 α -ik 0 1 + iβ 2 k 0 ε -1 - iβ k 0 ε -1 α              H = F H, (4.136)
∂ z H =                - iβ k 0 α -ik 0 ε + iα 2 k 0 ik 0 1 ε -1 - iβ 2 k 0 iβ k 0 α                E = GE, (4.137)
where for an arbitrary eld U we have introduced the decomposition Of course, in order to exploit numerically the FMM, a truncation has to be made, limiting the number of di raction orders taken into account. For a given truncation M, this corresponds to keeping 2M + 1 scattering orders

U = {U x (z, k n , ω)} n , {U y (z, k n , ω)} n T , ( 4
{A n } n = A -M , . . . , A M , (4.139)
and the size of the corresponding column vector U is thus 2(2M + 1). Based on this truncation, we obtain

∂ 2 z E = FGE = PD 2 P -1 E, (4.140)
where P and D 2 are respectively the eigenvectors and eigenvalues 2(2M + 1) × 2(2M + 1) matrices of the matrix FG P = P (11) P (12) P (21) P (22) , D = D (11) 0 0 D (22) . A and B being arbitrary constant vectors, and where P ′ = F -1 PD.

Based on the knowledge of the electric and magnetic elds in the four regions, we can now impose the continuity of the x and y components of both elds at the three interfaces z = 0, z = h and z = h + δ. In the following boundary conditions the values of k x , k y and ω are given. Exploiting this fact we use the generic simpli ed expression A p,n to refer to the amplitude A p (k n , ω). Before proceeding in the calculation, we introduce an additional phase factor in the expression of the elds in zones 3 and 4. In particular, in zone 3 we replace exp

[ik (i)φ z z] with exp[ik (i)φ z (z -h)], while in zone 4 we replace exp[ik (i)φ z z] with exp[ik (i)φ z (z - h -δ)]
. These factors make the calculation easier and can be simply recovered at the end. At the rst interface z = 0 we have for the x and y components of the electric eld (repeated indices are implicitly summed over) 11) nm (A x,m + B x,m ) + P (12) nm (A y,m + B y,m ) P (21) nm (A x,m + B x,m ) + P (22) nm (A y,m + B y,m )

         - k y k n I 1,n + R 1,n + c √ ε 1 ω k (1) z,n k x,n k n (I 2,n -R 2,n ) k x,n k n (I 1,n + R 1,n ) + c √ ε 1 ω k (1) z,n k y k n (I 2,n -R 2,n )          =        P (
       , (4.143)
while for the magnetic eld we get 11) nm (A x,m -B x,m ) + P ′ (12) nm (A y,m -B y,m ) P ′ (21) nm (A x,m -B x,m ) + P ′ (22) nm (A y,m -B y,m )        .

        -c ω k (1) z,n k x,n k n I 1,n -R 1,n - √ ε 1 k y k n I 2,n + R 2,n -c ω k (1) z,n k y k n I 1,n -R 1,n + √ ε 1 k x,n k n I 2,n + R 2,n         =        P ′(
(4.144) The boundary conditions at z = h give us the following equations for the electric eld 11) nm e D (11) mm h A x,m + e -D (11) mm h B x,m + P (12) nm e D (22) mm h A y,m + e -D (22) m h B y,m P (21) nm e D (11) mm h A x,m + e -D (11) mm h B x,m + P (22) nm e D (22) mm h A y,m + e -D (22) mm h B y,m

         - k y k n (C 1,n + C ′ 1,n ) + c √ ε 3 ω k (3) z,n k x,n k n (C 2,n -C ′ 2,n ) k x,n k n (C 1,n + C ′ 1,n ) + c √ ε 3 ω k (3) z,n k y k n (C 2,n -C ′ 2,n )          =          P (
         , (4.145)
and the following ones for the magnetic eld 11) nm e D (11) mm h A x,me -D (11) mm h B x,m + P ′( 12) nm e D (22) mm h A y,me -D (22) mm h B y,m P ′( 21) nm e D (11) mm h A x,me -D (11) mm h B x,m + P ′( 22) nm e D (22) mm h A y,me -D (22) mm h B y,m

        -c ω k (3) z,n k x,n k n (C 1,n -C ′ 1,n ) - √ ε 3 k y k n (C 2,n + C ′ 2,n ) -c ω k (3) z,n k y k n (C 1,n -C ′ 1,n ) + √ ε 3 k x,n k n (C 2,n + C ′ 2,n )         =          P ′(
         . (4.146)
Finally, the boundary conditions at z

= h + δ read          - k y k n e ik (3) z,n δ C 1,n + e -ik (3) z,n δ C ′ 1,n + c √ ε 3 ω k (3) z,n k x,n k n e ik (3) z,n δ C 2,n -e -ik (3) z,n δ C ′ 2,n k x,n k n e ik (3) z,n δ C 1,n + e -ik (3) z,n δ C ′ 1,n + c √ ε 3 ω k (3) z,n k y k n e ik (3) z,n δ C 2,n -e -ik (3) z,n δ C ′ 2,n          =          - k y k n (T 1,n + I ′ 1,n ) + c √ ε 4 ω k (4) z,n k x,n k n (T 2,n -I ′ 2,n ) k x,n k n (T 1,n + I ′ 1,n ) + c √ ε 4 ω k (4) z,n k y k n T 2,n -I ′ 2,n )          , (4.147)
and the ones for the magnetic eld are given by  The system of equations (4.149) has to be solved for the unknowns R, T , A, B, C, and C ′ . The expression of R and T as a function of I and I ′ will provide us the desired re ection and transmission operators. The fact that for A and B we solve in cartesian components and not in polarization is not an issue since these appear as mute variables not participating to the scattering operators. The explicit expression of the S matrices appearing in (4.149) can be obtained by means of algebraic manipulation of (4.143)-(4.148). The nal result is In these expressions we have de ned

       -c ω k (3) z,n k x,n k n e ik (3) z,n δ C 1,n -e -ik (3) z,n δ C ′ 1,n - √ ε 3 k y k n e ik (3) z,n δ C 2,n + e -ik (3) z,n δ C ′ 2,n -c ω k (3) z,n k y k n e ik (3) z,n δ C 1,n -e -ik (3) z,n δ C ′ 1,n + √ ε 3 k x,n k n e ik (3) z,n δ C 2,n + e -ik (3) z,n δ C ′ 2,n         =         -c ω k (4) z,n k x,n k n (T 1,n -I ′ 1,n ) - √ ε 4 k y k n (T 2,n + I ′ 2,n ) -c ω k (4) z,n k y k n (T 1,n -I ′ 1,n ) + √ ε 4 k x,n k n (T 2,n + I ′ 2,n )         . ( 4
S 1 = K ′ 1 -P L ′ 1 -P ′ -1 K 1 P L 1 -P ′ , ( 4
K ′ 3 K 4 L ′ 3 L 4 -1 K 3 K ′ 4 L 3 L ′ 4 σ (3)
K ′ i = -A y -B x,i A x -B y,i , L ′ i = √ ε i B x,i -A y B y,i A x , K i = A y -B x,i -A x -B y,i , L i = √ ε i B x,i A y B y,i -A x , (4.154)
where

A x = diag k x,n k n n , A y = diag k y k n n , B x,i = c √ ε i ω diag k x,n k n k (i) z,n n , (4.155)
B y,i = c √ ε i ω diag k y k n k (i)
z,n n .

The symbol diag(a n ) n denotes a (2M+1)×(2M+1) diagonal matrix having diagonal elements a -M , a -M+1 , . . . , a M . We have also de ned the square matrices of dimension 2(2M + 1) σ (2) h ≡ e Dh = e D (11) h 0 0 e D (22) h , (4.156) We now need to calculate the re ection and transmission operators associated to the two gratings represented in Figure 4.13. As far as grating 1 is concerned, the problem we need to solve is exactly the one presented in Section 4.2.1, with the appropriate values of the geometrical parameters. Concerning grating 2, we need to take into account the fact that its interface is the plane z = d and not z = 0. The modi cation of the scattering operators with respect to translations has been discussed in [19] (see also Section As we will show later, this operator is the only one associated to grating 2 appearing in the expression of the force for our con guration.

σ (3) δ ≡       diag(e ik

Numerical results

We now present a numerical application concerning the force between two di erent gratings and we show our results [39], [AN5]. Being both gratings in nite in the xy plane, we actually calculate the pressure acting on any of them, as discussed in the case of two slabs in [19]. In the rst con guration we have chosen both gratings to have period D = 1 µm, corrugation depth h = 1 µm and lling factor f = 0.5. As shown in Figure 4.13, the transition points of the two gratings are aligned, i.e. there is no shift along the x axis. Grating 1 is made of Fused Silica (SiO 2 ) and has thickness δ 1 = 10 µm, while grating 2 is made of Silicon and has in nite thickness. In order to take into account this point we have imposed ε 3 = ε 4 in the FMM relative to grating 2 (see Subsection 4.2.1) and removed in (4.37) all the terms proportional to the transmission operators of body 2. Physically, this can be explained by observing that because of the in nite thickness all the radiation coming from the upper side of body 2 is absorbed and does not reach the cavity between the gratings. Both Silicon and Fused Silica have been described by means of optical data taken from [62].

As anticipated in Subsection 4.2.1, the numerical use of FMM demands to choice a truncation order, problem that will be addressed in this Subsection. We noted before that by choosing a truncation order M in the FMM we obtain as a result re ection operators which are square matrices of dimension 2(2M + 1), that is two polarizations times 2M + 1 di raction orders. Their typical structure is thus

TE TM TE TM            A 1,1 [n, n ′ ] A 1,2 [n, n ′ ] A 2,1 [n, n ′ ] A 2,2 [n, n ′ ]            , (4.162)
where each block A i, j [n, n ′ ] is a (2M + 1) × (2M + 1) matrix, the indices n and n ′ running from -M to M.

It is worth stressing that, for a given M, only the elements closer to the center of each block of the matrix (i.e. close to n = 0 for each couple of polarizations) are at convergence. Thus, for a given m, we can increase the value of M starting from M = m in order to extract a 2(2m + 1) × 2(2m + 1) (m < M) scattering operator whose elements are at convergence with a given accuracy (in our case of the order of one percent). The operators obtained following this procedure can be used to compute the force using (4.36) and (4.37). Since these equations imply a trace containing also a sum over the di raction orders n, the series has to be replaced with a nite sum fromm to m. The value of m has to be found by imposing the convergence of the series at a chosen accuracy. Also in this case, we required an accuracy smaller than one percent.

The calculation of the pressure at a given distance requires the evaluation of the traces (4.36) and (4.37) at several di erent values of the wave vector k and the frequency ω, in order to reach the convergence on the integral on the three variables. We have observed that a single calculation of the trace requires values of m of the order of 2 (with peaks going up to 7) and corresponding values of M of the order of 5 (with peaks around 20). Moreover, we have that the values of m and M depend on the ratio of the distance to the grating period. This is not unexpected and we have, in fact, that when this ratio increase the values of m and M decrease since in this case we have that our systems approximate that of two slabs. A single value of the pressure required a computation time of the order of 16 hours on three 3 GHz CPUs.

In the con guration described above, we have calculated the pressure acting on grating 1. To point out the features of our OTE con guration we present in Figure 4.15 the pressure as a function of distance for di erent sets of the temperatures (T 1 , T 2 , T e ).

We clearly see that the modi cation of the three temperatures strongly a ects the value of the force. In particular, three of the four curves show a transition from an attractive to a repulsive behavior, not realizable at thermal equilibrium for this con guration. This qualitative di erence is a well-known consequence of the absence of thermal equilibrium and it has already been predicted in the case of two parallel slabs [10,19]. We stress that the transition point between attraction and repulsion is a function of the temperatures. For the values chosen, it roughly varies from 3 to 5 µm.

To underline even more the richness of our OTE con guration, we focus on the temperatures (T 1 , T 2 , T e ) = (200, 400, 10) K and compare the pressure to its equivalent at thermal equilibrium at the temperature of body 1, i.e. T 1 = 200 K. This comparison is presented in Figure 4.16. In the same gure we also plot the pressure, both at and out of thermal equilibrium, for lling factors f 1 = f 2 = 1 (corresponding to lled gratings, that is a 11 µm-thick SiO 2 slab at distance d from an in nite Si slab) and for f 1 = f 2 = 0 (corresponding to empty gratings, that is a 10 µm-thick SiO 2 slab at distance d + 2 µm from an in nite Si slab).

Besides the transition to a repulsive behavior, Figure 4.16 shows that the pressure in presence of a grating always lies between the two results corresponding to lled and empty ones. As a check of our numerical calculations, we have veri ed that the asymptotic behavior of the equilibrium pressure correctly reproduces the corresponding analytical result for two slabs. Finally, a comparison between Figs. 4.15 and 4.16 shows that the asymptotic value of the pressure can be tuned by varying the three temperatures of the system to values comparable to the pressure at thermal equilibrium (apart from the sign) at much smaller distances, of the order of 3 µm.

To conclude this Subection, we compare the Casimir-Lifshitz pressure between the two gratings obtained using FMM to the result coming from the PFA (Proximity Force Approximation), typically used to deal with complex geometries such as sphere-plane and nanostructured surfaces. In the case of two aligned gratings with equal lling factors f 1 = f 2 = f the pressure in the PFA reduces to the following weighted sum of the pressures of simple where P (ss) 1 (δ 1 , δ 2 , d) is the pressure acting on a δ 1 -thick slab at a distance d from a δ 2 -thick slab.

In Figure 4.17 we plot the ratio between the exact pressure and the PFA results for the four temperature con gurations used in Figure 4.15. We observe that PFA provides in our range of distances a description of the pressure with a relative error typically well below 20%. The fact the PFA predicts a change of sign not exactly at the position predicted by the exact calculation results in the existence of a vertical asymptote of the ratio P/P PFA , clearly indicated in the blue and orange curves in Figure 4.17.

Dependence on geometrical parameters

It is now interesting to understand how a modi cation of the geometrical parameters of the gratings is able to tune the value of the pressure. To this end we have chosen as a reference the pressure at a distance d = 4 µm for (T 1 , T 2 , T e ) = (200, 400, 10) K, for which the pressure is around P 0 = -10 -6 N m -2 (see Figure 4.15). Starting from this result, we have modi ed one by one the values of the lling factor f , period D, corrugation depth h and calculated the ratio between the modi ed pressure and the reference P 0 .

The results are shown in Figure 4.18, where the pressure ratio is plotted as a function of the ratio between the modi ed parameter and the reference ones ( f 0 = 0.5, D 0 = 1 µm and h 0 = 1 µm). First, we observe that geometrical modi cations can tune the pressure by a factor going from 0.5 to 1.6. In particular, this range can be fully explored by varying the lling factor between the two extreme values f = 0 and f = 1, i.e. between the two limiting slab-slab con gurations. Concerning the depth h, its variation also allows a wide variation of the pressure. We remark that for h going to zero we recover the result corresponding to f = 1, that is a lled grating. On the contrary, for increasing values of h, we see that we approach to a pressure approximately equal to half the value of the pressure for f = 1. This can be interpreted by noticing that roughly speaking at some point the corrugation is so deep that only the upper part (half of the total surface, being f = 0.5) contributes to the pressure. Di erently, the dependence of the pressure on the period D is less pronounced, and absent within our accuracy in the case of a lateral shift between the gratings, not reported here. As we have shown, the lling factor is a promising tool to tailor the behavior of the pressure. This is further pointed out in Figure 4.19, where the distance-dependent pressure is plotted for three di erent values of f . Whereas the asymptotic value of the pressure is practically the same, we note that for small distances the three curves di er visibly. More interestingly, the attractive-repulsive transition can be tuned approximately from 2.5 to 3.5 µm by changing f from 0.15 to 0.85. Let us focus now on the spectral properties of the pressure, by analyzing the quantity ∆(ω), de ned as the spectral component at frequency ω of the non-equilibrium contribution to the force (4.37), that is and compare its spectral distribution with the two slab-slab cases ( f = 0 and f = 1) as well as with some variations of one of the three parameters discussed above. The result is shown in Figure 4.20. We see that no striking spectral di erence is present between the con gurations compared. Roughly speaking, no new modes (such as the spoof plasmons observed in metal gratings [63,64]) are observed in the spectral region of interest, that is up to ω of the order of 3 × 10 14 rad s -1 . The spectral properties for any considered value of the geometrical parameters show small di erences with respect to the ones of the two slab-slab con gurations. this aim, we implemented the Fourier Modal Method in order to derive the scattering operators associated to each individual grating. Using the general formalism for Casimir-Lifshitz force based on scattering matrices, we calculated the pressure acting on a nite Fused Silica grating in presence of an in nite Silicon grating, and also compared our results to those obtained using the Proximity Force Approximation.

∆(T 1 , T 2 , T e ) =
We showed that the combination of geometrical structuring of the surface and absence of thermal equilibrium o ers an extremely rich domain of variation both with respect to thermal equilibrium and with respect to planar slabs out of thermal equilibrium [39], [AN5]. In fact, as in the case of two slabs, non-equilibrium is able to produce a repulsive pressure, whose intensity can be tuned by varying the temperatures. In addition, the several geometrical parameters associated to each grating add more tools to tune the pressure. We also pointed out the presence of regimes in which the pressure is close to zero and almost independent of the environmental temperature. Remarkably, the variations of all the parameters strongly a ect the distance at which the transition between attractive and repulsive pressure occurs, allowing to obtain transition distances as low as 2.5 µm. This feature is indeed promising for the experimental observation of a repulsive force. Moreover, our results can be relevant in the context of force manipulations on micro-mechanical systems [51]. Finally, an extension of this study to three-body con gurations is also promising toward the manipulation of heat transfer [65,28].

Conclusions

Casimir forces are one of the most startling consequences of the existence of vacuum uctuations and of the quantum nature of the electromagnetic eld. They are quantum interactions of electromagnetic origin between neutral microscopic or macroscopic objects in the vacuum and are the main topic of this thesis. These interactions, although they are weak, are measurable and have been observed in many di erent con gurations. It is well known that they are strongly related to the geometry of the system, to the boundary conditions and magnetodielectric properties of the objects involved, and to the temperature of the system. We have been inspired by these remarkable features of Casimir forces for the research project summarized in this thesis. In particular, the theoretical e ects occurring when out of equilibrium conditions are considered, and the possibility to tune and control the force in speci c systems, have led our research to study the following new congurations: atoms uniformly accelerated, atoms in front of an oscillating mirror, dielectric gratings out of thermal equilibrium. These are the main original problems studied during the PhD. We now summarize the original results obtained.

The rst part of our research concerned with the Casimir-Polder interaction between atoms uniformly accelerating. This system is in an out of dynamical equilibrium, due to the uniform accelerated motion of the atoms. In a rst work we studied the problem with the help of an heuristic semiclassical model, based on the spatial correlations of vacuum uctuations, already used in the literature to describe the Casimir-Polder forces in the static case. We showed signi cant qualitative modi cations of these interactions due to the acceleration, and in particular we stressed that their distance dependence is modi ed both in the near and in the far zone (respectively, new terms as R -5 and R -6 appear). For this physical system it is also important to consider the connection with the Unruh e ect. In fact, the qualitative change of the Casimir-Polder interaction suggests a new possibility to detect the Unruh e ect indirectly, measuring the modi cations of the Casimir-Polder force induced by the accelerated motion of the atoms. Moreover, we have found a temporal dependence of the force for the system considered. This adds a further possibility to detect the Unruh e ect through the Casimir-Polder interaction, without the extremely large accelerations necessary for other radiative processes such as the Lamb shift (it allows to decrease the acceleration by increasing the observation time). We have also investigated the same problem with a more rigorous approach, extending a known general statistical procedure to separate the contributions from vacuum uctuations and radiation reaction. Using this approach we again found a signi cant change of the scalar Casimir-Polder force induced by acceleration.

In addition to the previous case, we showed a novel transition in the distance-dependence of the force. We noticed that this new transition occurs at a characteristic length scale, where there is a breakdown of the local inertial frame approximation. For distances larger than this characteristic length scale, the non-inertial character of relativistic acceleration must be taken into account. We have then considered the Casimir-Polder resonant force between accelerated atoms, when the two atoms, one in the ground state and the other in an excited state, are prepared in a correlated state. We have considered both cases of interaction with a scalar and an electromagnetic eld. Also in this case, we found a qualitative modi cation of the distance-dependence of the interaction compared to the case of inertial atoms, and we have discussed the close connection of our results with the Unruh e ect. Possible developments on this topic could aim to envisage realistic experimental setups to detect the Unruh e ect through Casimir-Polder interactions.

In the second part of our work, we considered the Casimir-Polder interaction between a Rydberg atom and an oscillating e ective mirror (dynamical mirror). We showed that this optomechanical coupling may produce a new near-eld resonant atomic excitation. We have found that the excitation probability, consistently with our approximation, can be of the order of 20 per cent using physical parameters currently achievable in the laboratory. To detect this new e ect, we also proposed an experimental con guration composed of a cold Rydberg atoms gas trapped at a distance of about 2 × 10 µm from a dynamical mirror. The latter is made of a semiconductor layer whose dielectric constant is periodically driven by an external laser pulse train. Since the system we studied is related to the dynamical Casimir e ect, we also compared the atom-excitation probability due to our near-eld e ect to that related to the dynamical Casimir e ect. We found that our near-eld excitation process is much more e ective than the atomic excitation due to the absorption of the photons emitted by dynamical Casimir e ect. We nally stressed how our dynamical e ect can be used to realize an e ective optomechanical coupling between a macroscopic body and an elementary quantum system and how the quantum uctuations may change the internal state of this kind of system. Future perspectives of our work involve re nements of the model used in view of possible experiments, for example: e ect of dielectric properties of the oscillating wall and of the patch potentials, temperature e ects and possible dissipative e ects.

Finally, in the third part of our research, we studied and analyzed in detail a system in an out of thermal equilibrium condition. The system is composed of two 1D lamellar dielectric gratings. In order to study such a system, we used a general formalism for the Casimir-Lifshitz force based on scattering matrices. The necessary scattering operators of the lamellar gratings were deduced with the implementation of the Fourier Modal Method (FMM). We then numerically evaluated the Casimir-Lifshitz pressure acting on anite Fused Silica grating in the presence of an in nite Silicon grating when the temperature of the bodies and of the environment were in general di erent. We stressed the possibility to observe a repulsive force and we showed that, for the system we studied, the combination of geometrical structuring of the gratings surface and the temperatures involved in the system give the possibility to tune and control the force. For example, we showed that the distance between the gratings at which the transition between attractive and repulsive pressure occurs can be easily controlled. In our system the minimum transition distance is around 2.5 µm, a value smaller than the transition distances of the analogous case of two slabs. This feature makes the system we studied promising for the detection of the repulsive Casimir-Lifshitz force. Future planned work on this subject is relative to the extension to metallic gratings, as well as the Casimir-Lifshitz lateral force between gratings and the study of di erent grating pro les.

Figure 1 . 2 -

 12 Figure 1.2 -Typical Feynman diagrams describing the electrostatic interaction between molecules. The horizontal line denotes instantaneous photon exchange.

1 / 2 ǫj

 12 ,k λ µ eg j (B) e ik•R B (1.8)

Figure 1 . 4 -

 14 Figure 1.4 -Feynman diagrams involved in the calculation of the dispersion interaction between two ground-state molecules.

127) I 1 I 3 I 2 zFigure 2 . 6 -

 13226 Figure 2.6 -Integration contour in the complex plane used to calculate the frequency integral in (2.128).

Figure 2 . 7 -

 27 Figure2.7 -Comparison between the Casimir-Polder interaction between two atoms (scalar case) moving with relativistic uniform acceleration a and constant separation z (red continuous line), and the static interaction for atoms at rest at temperature T = a/2π and same distance (blue dashed line), in far zone, z ≫ 1/ω 0 . While for short distances, z ≪ 1/a, both potentials display the same thermal-like behavior, at distances larger than the characteristic length scale 1/a, the thermal and the accelerated Casimir-Polder potentials exhibit a sharply di erent power law decay with the interatomic distance.

1

 1 The dynamical Casimir e ect and the quantum friction . . . . . . . 85 3.2 Experimental setups for DCE . . . . . . . . . . . . . . . . . . . . . . . 90 3.3 Optomechanical Rydberg atoms excitation via dynamic Casimir-Polder coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 3.3.1 Experimental proposal . . . . . . . . . . . . . . . . . . . . . . . . 96 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

Contents 4 .

 4 1 A scattering-matrix approach to Casimir-Lifshitz force out of thermal equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 4.1.1 Correlation functions of the total eld . . . . . . . . . . . . . . . 110 4.1.2 The force between two slabs . . . . . . . . . . . . . . . . . . . . . 114 4.2 Methods for the di raction by gratings . . . . . . . . . . . . . . . . . 119 4.2.1 The Fourier Modal Method . . . . . . . . . . . . . . . . . . . . . 119 4.2.2 Some other methods for the di raction by gratings . . . . . . . . 128 4.3 OTE Casimir-Lifshitz force between two dielectric gratings . . . . . 132 4.3.1 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 4.3.2 Dependence on geometrical parameters . . . . . . . . . . . . . . 143 4.3.3 Modulation of the attractive-repulsive transition . . . . . . . . . 146 4.3.4 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . 147 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

Figure 4 . 1 -

 41 Figure 4.1 -The geometry of the system. We de ne three di erent zones A, B, and C useful when we consider the scattering. The width of the zone B de nes a distance between the two bodies z 2 ≤ z ≤ z 3 .

Figure 4 . 2 -

 42 Figure 4.2 -The enclosing box chosen for the calculation of the uxes.

Figure 4 . 3 -

 43 Figure 4.3 -A schematic gure to de ne the scattering operators.

Figure 4 . 4 -

 44 Figure 4.4 -A gure that represents the geometry of the system of two slabs studied in this subsection.

Figure 4 . 6 -

 46 Figure 4.6 -Figure from[19] where is plotted the radiative heat transfer per unit surface between a fused silica slab of thickness δ 1 = 2 µm placed in front of a second silicon slab of thickness δ 2 = 1000 µm. The temperatures of the slabs are T 1 = 300 K and T 2 = 0 K in (a), T 1 = 0 K and T 2 = 300 K in (b), and T 1 = T 2 = 300 K in (c). Symbols: T e = 0 K (black squares), 300 K (green plus), 400 K (blue crosses), 500 K (brown diamonds), and 600 K (red circles).

) givinge

  Xz = e ±PDP -1 z = Pe ±Dz P -1 (4.86) where P and D 2 are respectively the eigenvectors and eigenvalues matrices of the matrix X. For the elds we then get        U(z) = P e Dz A + e -Dz B Re(D) ≤ 0 ∂ z U(z) = PD e Dz Ae -Dz B Re(D) ≤ 0 (4.87) using the following de nitions PD ≡ P ′ , A = [. . . , a -1 , a 0 , a 1 , . . .] T (4.88) B = [. . . , b -1 , b 0 , b 1 , . . .] T . (4.89)

Figure 4 . 9 -

 49 Figure 4.9 -Schematic gure that depicts the de ned directions for A, B at the interface z = 0.

Figure 4 . 10 -

 410 Figure 4.10 -Schematic gure that depicts the directions for A ′ , B ′ at the interface z = h.

  (4.136) and (4.140), we obtain that elds are        E(z) = P e Dz A + e -Dz B H(z) = P ′ e Dz Ae -Dz B (4.142)

  .148)In the following, we are going to cast (4.143)-(4.148) under the form A and B appearing in this equation gather two vectors de ned as in equation (4.138). On the contrary, all the six other column vectors gather the two polarizations of the eld under the formV = {V 1 (z, k n , ω)} n , {V 2 (z, k n , ω)} n T .(4.150)

S = S 1 ⊛ S 2 ⊛

 12 S 3 , (4.159) having introduced the associative operation A = B ⊛ C, which for three square matrices A, B and C of dimension 4(2M + 1) is de ned in (4.101) -(4.104), where each matrix have been decomposed in four square blocks of dimension 2(2M + 1).Equation (4.158) allows to identify the four blocks of S as the re ection and transmission operators associated to the two sides of the grating. For example, the block S 11 is the coe cient linking the re ected amplitudes R to the incident ones I : it then coincides with the re ection operator R -for a wave impinging on the grating of Figure (4.14) from z < 0. By analog reasoning, we write the full S matrix as S = R -T - T + R + . (4.160)

  4.1). Based on these results, and using the mode expansion used in this work (see Subsection 4.1.2), the R - 2 operator of grating 2 can be expressed as a function of the R - 2 derived from FMM asp, k, n, ω|R - 2 |p ′ , k ′ , n ′ , ω ′ (4.161) = exp[i(k z,n + k ′ z,n ′ )d] p, k, n, ω| R - 2 |p ′ , k ′ , n ′ , ω ′ .

Figure 4 .

 4 Figure 4.15 -Pressure acting on grating 1 (Fused Silica, having h 1 = 1 µm, δ 1 = 10 µm, D = 1 µm and f 1 = 0.5) in front of grating 2 (Silicon, having h 2 = 1 µm, in nite thickness, D = 1 µm and f 2 = 0.5) as a function of distance d. The four curves correspond to di erent choices of the three temperatures (T 1 , T 2 , T e ) as shown in legend.

Figure 4 .

 4 Figure 4.16 -Non-equilibrium (OTE) pressure [(T 1 , T 2 , T e ) = (200, 400, 10) K, solid lines] compared to equilibrium pressure (T = 200 K, dashed lines) for two gratings (black squares), and two slab-slab con gurations corresponding to lled gratings ( f = 1, green circles) and an empty ones ( f = 0, red triangles).

Figure 4 . 17 -

 417 Figure 4.17 -Ratio between the exact pressure and the PFA counterpart (see equation (4.163)), for the same distances and choices of temperatures of Figure (4.15).

Figure 4 .

 4 Figure 4.18 -Variation of the pressure between two gratings at d = 4 µm [temperatures (T 1 , T 2 , T e ) = (200, 400, 10) K] as a function of the geometrical parameters. The reference point (black circle) corresponds to the set of parametersf 1 = f 2 = 0.5, h 1 = h 2 = 1 µm, δ 1 = 10 µm, in nite δ 2 , D = 1 µm.The three curves show the variation of pressure when changing one parameter at a time (red diamonds for the lling factor, green triangles for the period, blue squares for the corrugation depth). Axis scales are normalized to reference value (P 0 , f 0 = 0.5, D 0 = 1 µm, h 0 = 1 µm). Note the scale break on the x axis.

Figure 4 . 19 -

 419 Figure 4.19 -Pressure on grating 1 as a function of distance [temperatures (T 1 , T 2 , T e ) = (200, 400, 10) K] for three di erent values of lling factor, all the other geoemtrical parameters being the reference ones.

Figure 4 . 20 -

 420 Figure 4.20 -Spectral density of the OTE contribution to the force (de ned in (4.164)) at d = 4 µm [temperatures (T 1 , T 2 , T e ) = (200, 400, 10) K]. The solid black line corresponds to lled gratings ( f = 1), the dot-dot-dashed red line to empty ones ( f = 0), the dotted blue line to our reference gratings, having f = 0.5. In the other curves we vary the geometrical parameters one by one with respect to our reference case: dot-dashed violet line for f = 0.75, short-dashed green line for D = 4 µm, long-dashed brown line for h = 2 µm.

  case, we consider our reference point d = 4 µm and (T 1 , T 2 , T e ) = (200, 400, 10)

Table 1 .

 1 E sg + ck ( ck + ck ′ ) E rg + ck ′ (ii) E sg + ck ′ ( ck + ck ′ ) E rg + ck ′ (iii) E sg + ck E rg + E sg E rg + ck ′ (iv) E sg + ck E rg + E sg E sg + ck ′ (v) E sg + ck ′ E rg + E sg + ck + ck ′ E rg + ck ′ (vi) E sg + ck ′ E rg + E sg + ck + ck ′ E sg + ck (vii) E rg + ck ( ck + ck ′ ) E sg + ck ′ (viii) E rg + ck ( ck + ck ′ ) E sg + ck ′ (ix) E rg + ck E rg + E sg E sg + ck ′ (x) E rg + ck E rg + E sg E rg + ck ′ (xi) E rg + ck E rg + E sg + ck + ck ′ E sg + ck ′ (xii) E rg + ck E rg + E sg + ck + ck ′ E rg + ck

	Diagram	Denominator
	(i)

1 -The D l denominators appearing in

(1.18)

.

  The Unruh e ect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

	ects of uniform acceleration on
	atom-eld interactions
	Contents
	2.1 2.2 Some known e ects of uniform acceleration in quantum electro-
	dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
	2.3 van der Waals interaction energy between two accelerated atoms .
	2.3.1 The model for the van der Waals interaction in the accelerated
		frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
	2.3.2 Concluding remarks on the results . . . . . . . . . . . . . . . . .
	2.4 A fourth order method for the calculation of the Casimir-Polder
	force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
	2.4.1 Perturbative calculation of the variation rates . . . . . . . . . . .
	2.4.2 The vacuum uctuations contribution . . . . . . . . . . . . . . .
	2.4.3 The radiation reaction contribution . . . . . . . . . . . . . . . . .
	2.4.4 The van der Waals force between two atoms at rest . . . . . . . .
	2.5 Thermal and non-thermal signatures of the Unruh e ect in Casimir-
	Polder forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
	2.5.1 Thermal Casimir-Polder interactions . . . . . . . . . . . . . . . .
	2.5.2 Unruh corrections to Casimir-Polder interactions . . . . . . . . .
	2.6 The resonance interaction between two uniformly accelerated identical
	atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
	2.6.1 The scalar eld case . . . . . . . . . . . . . . . . . . . . . . . . .
	2.6.2	The electromagnetic eld case . . . . . . . . . . . . . . . . . . .
	2.6.3	Conclusions and Future Perspectives . . . . . . . . . . . . . . . .

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  = E e -E g and (d i d j ) eg = e | d i d j | g . Under resonance conditions (ω ≃ ω 0 ), the atomic excitation probability P e (z 0 , t) = |c e (z 0 , t)| 2 is given by

							.18)
	is a time-dependent term linked to the plate oscillation. The approximation 1 z 3 (t) ≃ 1 z 3 0 3a z 0 sin ωt) has been used. The perturbation (3.18) can induce transitions in the atom. Ap-(1 + plying time-dependent perturbation theory with V I (t) as the perturbation operator, we can
	easily obtain the probability amplitude for the transition |g → |e from the atomic initial state g to a more excited state e,
	c e (z 0 , t) =	i 3σ i j (d i d j ) eg 16z 3 0	a z 0	0	t	dt ′ e iω 0 t ′ sin ωt ′ ,	(3.19)
	where ω 0

  .138) gathering x and y components and denoting with {. . .} n a set of scattering orders. We have also introduced the Toeplitz matrix a , de ned by the relation a i j = a i-j , a n being the n-th Fourier component of a. We remark that going from (4.134)-(4.135) to (4.136)-(4.137) we have used the modi ed factorization rule introduced in Subsection 4.2.1.

Acknowledgements

To conclude this thesis, please, let me thank the people that kept me company during the journey of my PhD. They induced and represented, in this period, a very important evolution in my professional and human life.

At rst, I want to thank my supervisors, they showed me the direction to take for my PhD research. I thank Roberto Passante for his scienti c and human support and for the con dence he always gave me. I thank Mauro Antezza for the opportunity he o ered me to work in Montpellier and learn more about research. I thank Brahim Guizal for his advices on numerical simulations and for his funny explanations about the etymology of French, English and Arabic words.

I wish to thank Riccardo Messina for the many stimulating scienti c discussions made working shoulder to shoulder and for his very important contribution in my professional growth.

I thank Lucia Rizzuto, Jamir Marino, Salvo Spagnolo and Margherita Lattuca for their willingness to help me and for their encouragement. I wish to thank Antonio Emanuele, Stefan Buhmann and Marco Pettini for their perusal of this thesis and for their important and useful comments. Now, pardon me, I want to say something in Italian. Il mio periodo del dottorato ha visto davvero tante persone che hanno arricchito le mie giornate. Non vi nominerò tutte una ad una ma credo che ognuno potrà riconoscersi nelle seguenti criptiche de nizioni e sentire il mio davvero sentito ringraziamento.

Beh, voglio cominciare con un profondo grazie a chi mi ha permesso di fare questo percorso, a coloro che sono stati sempre presenti, coloro che mi hanno sempre sostenuto in maniera incondizionata e che stanno alla base di tutto: grazie a mia madre, mio padre, mio fratello, mia cognata, mia zia ed i miei cugini.

Parlando di dottorato inevitabile per me è e sarà pensare a Montpellier. Voglio ringraziare tutti gli amici incontrati in questa città che mi ha segnato. Fra questi, un ringraziamento particolare va all'ami-maître-pianiste-chef-macaronier e al randomico saggio-burlone mate KB (amici del fantomatico ma onnipresente Fabio), al mio omonimo virtuale ma non reale il cui nome è andato LOST, al sostenitore di Antonellina (non la Clerici) e della sua pasta. Un grazie particolare a voi per i vostri consigli, la vostra guida e per aver portato allegria nelle mie giornate d'oltralpe.

Appendix A

Linear susceptibility

In this Appendix we give the expression of the linear susceptibility of the electromagnetic eld in the proper reference frame . It can be obtained from the correlation function (2.162).

We have

For our purposes, it is more convenient to express Equation (A.0.2) as integrals over frequencies. Using the same procedure as in the reference [39] of Chapter 2, after some calculation, we obtain

where we introduced the tensors

taking into account the dependence of the anti-symmetric correlation function on the spatial directions, and

Also, we de ned the coe cients c i (z, a) (i = 1, ...5)

(for simplicity, we have omitted in N the explicit dependence on z and a). Equation (A.0.3) shows that the symmetric correlation function of the electromagnetic eld depends on the atomic acceleration, through the presence of the coe cients c i and the factor N. In particular, these quantities depend on the interatomic distance and are responsible of the qualitative change of the resonant interaction energy, due to the atomic acceleration, as we have discussed in Section 2.6.2.

We now evaluate the resonance interaction between the two accelerated atoms. Putting Eqs. (A.0.3) and (2.165) into Equation (2.161), we get

)) e iω 12 (τ-τ ′ ) + e -iω 12 (τ-τ ′ ) (A.0.7)

We now perform integration over time taking the limits τ 0 → -∞, τ → ∞; after some algebra, we obtain

The integral above can be easily evaluated and after some algebraic manipulations the resonance interaction assumes the form given in Equation (2.166)

where we introduced the tensor potentials