M Bhattacharya

P.-L Giscard

P Meystre

Entanglement of a Laguerre-Gaussian cavity mode with a rotating mirror

Keywords: numbers: 03.65.Ud, 42.50.Pq, 42.50.Lc, 45.20.dc

Entanglement of a Laguerre-Gaussian cavity mode with a rotating mirror

I. INTRODUCTION

The optomechanical driving of optical cavities with laser radiation is an experimentally promising method for eliciting quantum mechanical behavior from classical objects [1][2][3][4][5]. In the case of a two-mirror cavity with a perfectly reflecting end-mirror mounted on a cantilever and allowed to vibrate along the cavity axis, laser radiation entering through the slightly transmissive fixed input mirror can both trap and cool the mirror motion, depending on whether it is detuned slightly above or below a cavity resonance [5]. Radiation pressure can thus decrease the number of phonons in the vibrating mirror,

n m = k B T eff ω eff , (1)
since cooling lowers the effective temperature of the mirror T eff while trapping increases its effective vibration frequency ω eff . A key goal is to achieve n m < 1, i.e. to place the mirror near its quantum mechanical ground state. Evidence of reaching that regime could come from measuring n m , and a strategy for doing so has recently been proposed [6].

Radiation pressure can play an additional important role as a mechanism for generating entanglement. In addition to its basic interest, quantum entanglement -the existence of correlations disallowed by classical physics between two objects -has been demonstrated to be a valuable resource for quantum information processing tasks such as teleportation and superdense coding (see Ref. [START_REF] Nielsen | Quantum Computation and Quantum Information[END_REF] and references therein).

It has been shown theoretically that radiation pressure can entangle a light field to a vibrating mirror both in the absence [START_REF] Pirandola | [END_REF] and in the presence [9] of a cavity. It can also entangle two vibrating mirrors, both when they form a cavity [10,11] and when they do not [12,13]. A single vibrating mirror can also entangle two [14,15] or more [16] light fields. The teleportation of quantum states from a light field to a vibrating mirror has been proposed in Ref. [17].

In these proposals the physical mechanism behind the generation of entanglement is the exchange of linear momentum between the photons in the cavity and the mir-ror. In recent work, we have proposed a rotational analog of mirror cooling and trapping where the light-mirror interaction relies on the exchange of angular momentum between a Laguerre-Gaussian cavity mode and a spiral phase element used as a cavity end-mirror [18]. This article further investigates that system and shows how the radiation torque can generate entanglement between the optically charged light field and a rotating mirror. We investigate the entanglement as a function of ambient temperature, detuning of the laser radiation, and orbital angular momentum carried by the cavity mode.

The rest of the article is organized as follows. Section II briefly reviews the cavity configuration proposed in Ref. [18]. Section III discusses the steady state of the system, discusses the linear fluctuations of the field and mirror motion about that steady state, and examines the dynamical stability of the rotating mirror. Section IV demonstrates the onset of entanglement between the intra-cavity Laguerre-Gaussian mode and the rotating end-mirror and discusses its dependence on ambient temperature, detuning, and angular momentum of the light field. We also briefly discuss the experimental measurement of the entanglement. Section V provides a summary and a conclusion. Appendix A derives and solves the Lyapunov equation for the correlation matrix determining the entanglement between the mirror and the cavity mode.

II. CAVITY MODEL

The optomechanical system under consideration is described in detail in Ref. [18] and illustrated in Fig. 1. Both cavity mirrors are spiral phase elements [19]. The input mirror is slightly transmissive but does not change the orbital angular momentum of any beam passing through it. In reflection, however, it removes a charge 2l from the beam. Likewise the end mirror, which is assumed to be perfectly reflecting, is designed to add a charge 2l to a beam on reflection. With these specifications it can be seen that a Gaussian beam (with no charge) incident on the cavity can transfer a torque ξ φ = cl /L per photon to the end mirror, where c is the Arrangement for entangling the rotation of a mirror with a Laguerre-Gaussian (LG) mode using angular momentum exchange. A Gaussian (G) field of optical charge 0 enters the cavity through a fixed partially transparent input coupler (IC) that does not affect its charge. A perfectly reflecting rear mirror (RM) rotating about the cavity axis on a support S adds a charge 2l to the beam on reflection. Both the IC and the RM are spiral phase elements. The angular deflection of the RM from equilibrium (φ0 = 0) is indicated by the angle φ. The charge on the beams at various points is also indicated. velocity of light and L is the length of the cavity [18].

In dimensionless units, the Hamiltonian that describes this system is [18]

H = ω c a † a + ω φ 2 L 2 z + φ 2 -ga † aφ. (2)
where a and a † are the bosonic annihilation and creation operators for the cavity mode of frequency ω c , L z and φ are the angular momentum and angular displacement, respectively, of the rear mirror about the cavity axis, with [L z , φ] = -i. Finally ω φ is the angular rotation frequency of the rear mirror and

g = cl L Iω φ (3)
is the optorotational coupling parameter, with I = M R 2 /2 the moment of inertia of the mirror of mass M and radius R about the cavity axis.

III. QUANTUM DYNAMICS

The evolution of the operators describing the dynamics of the intracavity field and the mirror motion is given by quantum Langevin equations accounting for noise and damping due to the vacuum fluctuations entering the cavity field as well as to the Brownian noise coupling to the moving mirror [18],

ȧ = -i(δ -gφ)a - γ 2 a + √ γa in , φ = ω φ L z , Lz = -ω φ φ + ga † a - D φ I L z + in . ( 4
)
where δ = ω c -ω L is the detuning of the laser frequency ω L from the cavity resonance, γ is the cavity damping rate and D φ is the intrinsic damping constant of the rotating mirror. The noise operator a in describes the laser field incident on the cavity, of mean amplitude a in (t) = a in s . Its fluctuations are taken to be deltacorrelated, δa in (t)δa in, † (t ) = δ(t -t ) and add vacuum noise to the cavity modes. The Brownian noise operator in describes the mechanical noise that couples to the mirror from the environment. Its mean value is zero and its fluctuations are correlated at temperature T as [START_REF] Gardiner | Quantum Noise[END_REF]

δ in (t)δ in (t ) = D φ ω φ I ∞ -∞ dω 2π e -iω(t-t ) ω 1 + coth ω 2k B T , (5)
where k B is Boltzmann's constant. For high mechanical quality, (ω φ I/D φ 1), the fluctuations become deltacorrelated and Eq. ( 5) simplifies to

δ in (t)δ in (t ) = D φ I (2n + 1)δ(t -t ), (6)
where

n = e ω eff /k B T -1 -1 (7)
is the mean number of thermal phonons available at the mirror of effective rotation frequency ω eff and temperature T [see Eq. ( 29) for an expresion for ω eff ]. The thermal phonon spectrum peaks approximately at the temperature

T c ∼ ω eff 4k B , (8)
We show later on that it is possible to maintain a mirrorfield entanglement close to its maximum value for T < T c , but that that entanglement decreases sharply beyond T ∼ T c (Fig. 3). This can be understood intuitively by noting that for T > T c we obtain ω > ω eff and the mirror can be rotationally excited by thermal phonons, but this ceases to be the case for T < T c , in which case ω < ω eff .

A. Steady state

The semiclassical, steady-state solution of Eq.( 4) is

a s = √ γa in s [γ/2 -i(δ -gφ s )] (γ/2) 2 + (δ -gφ s ) 2 φ s = g|a s | 2 ω φ , L z,s = 0. ( 9
)
The steady state field amplitude a s is complex in general, but in simple situations where the incident field is kept constant throughout, it is possible to chose its phase such that a s is real,

a s = √ γ|a in s | ( γ 2 ) 2 + (δ -gφ s ) 2 1/2 . ( 10
)
We restrict our discussion to that situation here. The steady-state is bistable for high enough incident field [START_REF] Meystre | [END_REF]22], but we assume in the following that an electronic feedback loop maintains the length of the cavity and hence removes bistability. This is a standard practice in current experiments on optomechanical cooling [1][2][3][4].

B. Fluctuations

To account for the quantum fluctuations about the semiclassical steady-state we expand the Heisenberg operator a about the a s state in the usual way as a = a s +δa, and similarly for the other operators. Linearizing the quantum Langevin equations of motion in the fluctuations gives then

δ X = - γ 2 δX + ∆δY + √ γδX in a , δ Y = - γ 2 δY -∆δX + Gδφ + √ γδY in a , δ φ = ω φ δL z , δL z = -ω φ δφ + GδX - D φ I δL z + δ in , (11)
where ∆ = δ -gφ s is the effective cavity detuning, G = ga s √ 2 is the effective optorotational parameter, and we have redefined the field fluctuations in terms of their quadratures as

δX a = (δa + δa † )/ √ 2, δY a = (δa -δa † )/i √ 2.
Eq. ( 11) can be recast compactly as

u(t) = Bu(t) + n(t), (12)
where u(t) = (δφ, δL z , δX a , δY a ) is the vector of fluctuations,

n(t) = (0, δ in , √ γδX in a , √ γδY in a )
is the input noise vector, and

B =    0 ω φ 0 0 -ω φ -D φ /I G 0 0 0 -γ/2 ∆ G 0 -∆ -γ/2    . (13)
This matrix determines the dynamic stability of the physical system, and also a measure of the entanglement between its two subsystems, the intracavity field and the rotating mirror.

C. Dynamic stability

According to the Routh-Hurwitz criterion, the stability of the steady-state solution is assured if none of the eigenvalues of the matrix B has a positive real part. This requirement can be reframed as a series of inequalities that have to be obeyed by the matrix elements of B [23]. These inequalities, given by

D φ I 2 γ 16ω 4 φ + 32G 2 ω φ ∆ + 8ω 2 φ (γ 2 -4∆ 2 ) + (γ 2 + 4∆ 2 ) 2 ) + 16G 2 I 3 ω φ γ 2 ∆ + 4D 3 φ (γ 3 + 4γ∆ 2 ) + 4D 2 φ I(4ω 2 φ γ 2 + γ 4 + 4G 2 ω φ ∆ + 4γ 2 ∆ 2 ) > 0, ω φ (γ 2 + 4∆ 2 ) -4G 2 ∆ > 0. ( 14
)
have been numerically verified to hold for the parameters of this article.

IV. ENTANGLEMENT

We quantify the entanglement between the cavity mode and the rotating end mirror in terms of the logarithmic negativity E N , a measure of entanglement that was proposed in Ref. [24] to describe continuous Gaussian variables and has been evaluated in a number of examples [9,25,30,31]. The logarithmic negativity can be expressed in terms of the elements of the correlation matrix

C ij = [ u i (∞)u j (∞) + u j (∞)u i (∞) ]/2. ( 15
)
For the problem at hand, C is a 4 × 4 matrix that can be cast in the form

C = R F F T S (16)
where R, F and S are 2 × 2 matrices. In terms of these matrices the logarithmic negativity is [24]

E N = max[0, -ln(2η -)], (17)
where

η -= 2 -1/2 σ -[σ 2 -4|C|] 1/2 1/2 , (18)
and σ = |R|+|S|-2|F |, |R| being the determinant of the matrix R. Quantum entanglement occurs for E N > 0, i.e. for η -< 1/2. The bipartite entanglement between the cavity mode and the rotating mirror can be evaluated directly from the knowledge of the matrix B [9]. Integrating Eq. ( 12) formally we have

u(t) = M (t)u(0) + t 0 dsM (s)n(t -s), (19)
where M (s) = exp(Bs). When the inequalities in Eq.( 14) are obeyed the system is stable, as we have seen, hence M (∞) = 0 so that

u(∞) = lim t→∞ t 0 ds M (s)n(t -s), (20)
and the steady state is independent of the initial conditions u(0) as it should be.

The stationary quantum fluctuations are zero-mean Gaussian processes fully characterized by the 4 × 4 correlation matrix C ij which reads, with Eq. [START_REF] Gardiner | Quantum Noise[END_REF], [START_REF] Meystre | [END_REF] where

C ij = k,l ∞ 0 ds ∞ 0 ds M ik (s)M jl (s )Φ kl (s -s ),
Φ kl (s -s ) = ( n k (s)n l (s ) + n l (s)n k (s ) )/2
characterizes the stationary noise correlations. In the limit of a large mechanical quality factor, Eq. ( 6) yields readily Φ kl (s -s ) = D kl δ(s -s ), where

D =    0 0 0 0 0 D φ (2n + 1)/I 0 0 0 0 γ/2 0 0 0 0 γ/2    , (22)
and Eq. ( 21) simplifies to

C = ∞ 0 dsM (s)DM (s) T , (23)
which can be integrated by parts to give

BC + CB T = -D, (24)
see Appendix A. Using Eqs. ( 13) and ( 22), Eq. ( 24) can be solved analytically to yield the correlation matrix C, see Appendix A. The full solution is quite complicated and its form is not particularly instructive, hence it will not be reproduced here.

We have evaluated numerically the logarithmic negativity E N for various parameters. Our main results are summarized in Figs. 234for the parameter values of Table. I, where the mechanical quality is

Q φ = ω φ D φ /I , ( 25
)
and F is the optical finesse. Our parameters are somewhat similar to the case of the linearly vibrating mirror of Ref. [9], hence we make comparisons between specific figures in that article and the present one when appropriate.

A. High temperature regime of the mirror environment. Our results, which resemble those reported in Fig. 2 of Ref. [9], indicate that quantum entanglement persists until roughly 100K, that is, it should therefore be measurable in experiments in cryogenic environments.

We remark that the temperature in Fig. 2 T is not the temperature of the mirror; as is well known the presence of red-detuned laser light in the cavity can increase the mirror damping from D φ to D eff and thus lower the effective temperature of the mirror T eff significantly below T . More precisely,

T eff = D φ D eff T. (26)
For the case of T = 50K, for example, we find T eff = 10µK for the parameters of Table I, where D eff is (see Eq. ( 7) of Ref. [18]),

D eff = D φ + 2ξ 2 φ γP in ω c ∆ ∆ 2 + (γ/2) 2 × γ [(γ/2) 2 + (ω -∆) 2 ] [(γ/2) 2 + (ω + ∆) 2 ] . (27)
Here ω is the response frequency of the system, taken in our case to be ω eff [see Eq.( 29)].

B. Low temperature regime

Figure 3 is a blown-up view of the portion of Fig. 2 for low ambient temperatures. The entanglement is maximum at T ∼ 0, as would be intuitively expected. It remains at that value as the temperature is increased until a temperature T c where it undergoes a sharp decrease. At that temperature the peak frequency of the thermal phonons becomes larger than ω eff and they are capable of randomly exciting the rotating mirror, thus degrading its entanglement with the cavity mode.

This point is further illustrated in Fig. 3, which shows the mean number of thermal phonons n at frequency ω eff , I.

[Eq.( 7)] as a function of T . It increases noticeably for temperatures above T c and is clearly anti-correlated to E N . This is consistent with an approximate calculation that shows that to lowest order in n we have

E N E 0 -κn, (28)
where E 0 is the maximum value of the entanglement and κ is a constant whose value is a function of the parameters of the system. The analytical forms of E 0 and κ are involved. For the parameters of Table I their values are E 0 ∼ 0.5 and κ ∼ 3.1 × 10 -6 .

C. Detuning

Fig. 4 shows the logarithmic negativity as a function of the detuning ∆ = ω c -ω L -φ s = ω c -ω L -g 2 |a s | 2 /ω φ for three masses of the rotating mirror. This result may be compared to Fig. 1 of Ref. [9]. A careful examination of the detuning results reveals that for each mass the entanglement is maximized when the detuning is equal to the mirror's effective angular frequency of rotation ∆ = ω eff , As is known, the presence of red-detuned radiation (∆ > 0) causes anti-trapping and thus lowers the effective mechanical frequency of the mirror [5]. The effective frequency ω eff is given explicitly by

ω 2 eff = ω 2 φ - 2ξ 2 φ γP in Iω c ∆ ∆ 2 + (γ/2) 2 × ((γ/2) 2 -(ω 2 -∆ 2 ) [(γ/2) 2 + (ω -∆) 2 ] [(γ/2) 2 + (ω + ∆) 2 ] , (29)
hence ω eff scales as 1/ √ I, that is, as the inverse square root of the mass. Note however that for the parameters of Fig. 4, ω eff ∼ 0.8ω φ for all choices of mass. Thus the maximum entanglement occurs for ∆/ω φ ∼ 0.8.) Physically, the condition ∆ = ω eff corresponds to the optimum detuning for optomechanical cooling [34] and can be understood in terms of cavity enhanced scattering of the anti-Stokes photons by the rotating mirror [4]. Thus it is not surprising that E N reaches a maximum at this detuning.

D. Angular momentum

As pointed out in Ref. [18], in the case of optical cooling of a vibrating mirror the linear momentum of the photons is difficult to change experimentally, and essentially entails replacement of the laser radiation source. For example to increase the linear momentum just by a factor of two would require an ultraviolet laser in place of an infrared laser. In contrast in the case of rotational cooling it is comparatively easier to achieve higher l. The laser does not neeed to be changed since the model requires a Gaussian input beam. In light of this fact we also studied the entanglement as a function of the orbital angular momentum carried away from the mirror by the Laguerre-Gaussian mode,. The results of this analysis are summarized in Fig. 5. For the temperature T = 10K considered in the figure, entanglement appears only beyond the threshold value l c ∼ 21. This is consistent with Eq. ( 3) which shows that for sufficiently low angular momentum, i.e. l < l c , the optorotational coupling between radiation and the rotating mirror may not be high enough for entanglement to occur. The value of l c depends on the other parameters that contribute to the optorotational coupling and of course also on the ambient temperature T . In general we found that entanglement appears when the effects of radiation torque coupling g are strong enough to counter the thermal noise, which increases perceptibly with temperature beyond T = T c .

E. Experimental considerations

In this section we make some remarks on the experimental aspects of our proposal. First we comment on the possible measurement techniques, then we discuss some state of the art parameters relevant to our scheme.

The measurement of the entanglement investigated in this paper can be carried out in a manner analogous to that proposed in the case of a linearly vibrating cavity and illustrated in Fig. 3 of Ref. [9]. In the present case the mirrors are replaced by spiral phase elements, and a second optical cavity can be formed by placing a third transmissive spiral phase element beyond the rotating mirror of Fig. 1. By using a weak intracavity field for this second cavity all the elements of the correlation matrix C can be obtained as proposed in Ref. [9] and demonstrated in a measurement of the entanglement between two optical fields [25].

From Table I we can see that our proposal requires an end mirror of 100ng mass, 10µm radius and mechanical quality factor 2×10 6 . This is reasonably close to the state of the art as can be seen from Ref. [2] which describes a recent mirror cooling experiment performed using a 25ng mirror of 15µm radius and mechanical quality factor 1.3 × 10 5 . We also require a torsional frequency of 1 MHz, which has been demonstrated for nanomechanical oscillators [26].

Also important is the use of high l spiral phase elements, as this would imply a larger opto-rotational cou-pling. Currently spiral phase elements are manufactured by etching [27] or micromachining [28] a spiral surface into a substrate such as fused silica. It is difficult to generate continuously varying surfaces as shown in Fig. 1. A more practical approach is to divide the azimuthal ramp into discrete steps. This leads to the presence of additional vortex modes, i.e. apart from l other charges are also imparted to the reflected light. However with the use of a large number of steps, most of the light can be sent into the chosen l mode. The ability to generate a pure Laguerre-Gaussian mode is therefore restricted by the number of steps or equivalently by the resolution of the etching stylus. Presently reflective spiral phase elements with up to l = 4 have been produced [27]. Consultation with an experimentalist indicates that with current technology a reflective spiral phase element can be etched to yield 95% power into the l = 20 Laguerre-Gaussian mode [29]. This value of l is very close to that at which entanglement begins to appear (see Fig. 5), and with either improvements in the technology or the use of more stringent parameters (i.e. higher laser power) the experiment we have proposed can be realized.

V. CONCLUSION

To conclude, we have demonstrated theoretically that radiation torque can serve to entangle a Laguerre-Gaussian cavity field with a rotating spiral phase element. We have shown that the physical basis of entanglement is the exchange of angular momentum between the cavity mode and the rotating mirror. The mirror-field entanglement was investigated as a function of temperature and shown to be measurable at the temperatures routinely achieved by cryogenic apparatus.

The entanglement was also shown to remain at its maximum value until a critical temperature at which the peak of the thermal spectrum occurred at the frequency corresponding to the rotor quanta, and beyond which the random excitation due to the thermal photons degraded the entanglement. The behavior of the entanglement with respect to cavity detuning was shown to be consistent with the mechanism of optomechanical cooling, i.e. the entanglement was maximum at a detuning known to correspond to optimal cooling of the mirror. Further, the fact that the angular momentum of the light can be changed relatively easily justified examining the entanglement with respect to changes in angular momentum. The main result here is that the mirror-radiation coupling only becomes strong enough to overcome the thermal noise and yield entanglement beyond a critical value of the angular momentum. Here we derive Eq.( 24), which is a typical Lyapunov equation encountered in the study of dynamical systems. Also, we show our method for solving the equation.

Derivation

Beginning from Eq.( 23)

C = ∞ 0 dsM (s)DM (s) T , (A1)
where M (s) = e Bs and the matrices B, D are given by Eqs.( 13) and ( 22) respectively, we integrate by parts using f (s) = e Bs and g(s) = De B T s in the rule where in the second step we have used the fact if the Routh-Hurwitz criterion is satisfied then lim s→∞ e Bs = 0, since all the eigenvalues of B have negative real parts.

We then pre-multiply the last line of Eq. (A3) by B and rearrange to obtain

BC + CB T = -D, (A4)
which is Eq. ( 24).

Solution

Here we record our method of solving Eq. (A4) for the correlation matrix C. By definition [see above Eq. ( 21)] C is a 4 × 4 real symmetric matrix. Thus we can write

C =    λ 1 a 1 a 2 a 3 a 1 λ 2 b 1 b 2 a 2 b 1 λ 3 c 1 a 3 b 2 c 1 λ 4    (A5)
where we have denoted the diagonal elements with Greek letters and the off-diagonal with Roman letters. We now define a matrix C where

C = BC + CB T + D, (A6)
which equals zero as per Eq. ( 24). Evaluating the RHS of Eq. (A6) we find that at least one matrix element of C involves a single non-diagonal element of C. This observation, which is central to our method, turns out to be true at every iteration, where the process of iteration is defined below. We now solve for the non-diagonal element of C mentioned above by equating the relevant matrix element of C to zero. We then use that value of the non-diagonal element in C and re-evaluate C . This process is performed iteratively until all the off-diagonal elements of C are expressed in terms of the diagonal elements of C. At this stage C has the form

ΦFIG. 1 :

 1 FIG.1:(Color online). Arrangement for entangling the rotation of a mirror with a Laguerre-Gaussian (LG) mode using angular momentum exchange. A Gaussian (G) field of optical charge 0 enters the cavity through a fixed partially transparent input coupler (IC) that does not affect its charge. A perfectly reflecting rear mirror (RM) rotating about the cavity axis on a support S adds a charge 2l to the beam on reflection. Both the IC and the RM are spiral phase elements. The angular deflection of the RM from equilibrium (φ0 = 0) is indicated by the angle φ. The charge on the beams at various points is also indicated.

Fig. 2 FIG. 2 :

 22 Fig.2shows E N as a function of the temperature T

FIG. 3 :

 3 FIG.3:(Color online) Solid line: Deviation ∆EN of EN from its maximum value E0 ∼ 0.5 as a function of T . The graph shows the low temperature end of the curve. Dashed line: mean number of thermal phonons n [Eq.[START_REF] Nielsen | Quantum Computation and Quantum Information[END_REF]] at frequency ω eff as a function of T . The vertical dotted line corresponds to the temperature Tc at which thermal phonons of energy larger than ω eff start exciting the rotor, degrading the entanglement. Parameters of TableI.

FIG. 4 :

 4 FIG.4:The logarithmic negativity EN plotted as a function of the detuning ∆ (normalized by the angular frequency ω φ of the rotating mirror) of the laser radiation from the cavity resonance. The three different curves correspond to different masses of the rotating mirror. The maximum of the entanglement occurs practically at ∆ = ω eff ∼ 0.8ω φ (see the discussion in Section IV C) for all three curves. For this plot we used the parameters from Table I except for the following changes : L = 0.1mm, R = 250µm, Q φ = 10 7 , F = 10 4 , and T = 400mK.

FIG. 5 :

 5 FIG. 5: Logarithmic negativity EN as a function of the orbital angular momentum carried by the Laguerre-Gaussian mode. The threshold angular momentum lc ∼ 21 at which entanglement begins to appear is indicated. Parameters of Table I, temperature T = 10K.

C = B - 1 e 0 - ∞ 0 ds B - 1 e

 1001 Bs De B T s ∞ Bs De B T s B T = -B -1 D -B -1 ( ∞ 0 ds e Bs De B T s )B T = -B -1 D -B -1 CB T , (A3)

γ 2 -γ 2

 22 (λ3+λ4-1)-4∆λ2ω φ -4λ1(G 2 -∆ω φ) 4Gω φ (λ1-λ2)ω φ G (λ3+λ4-1)-4∆λ2ω φ -4λ1(G 2 -∆ω φ ) C =0 yields 4 equations in the diagonal elements λ 1,2,3,4 of C, and these can be easily solved, giving us analytical expressions for all the elements of the C matrix.

TABLE I :

 I Definitions and approximate values of some of the parameters used in the text.by the US Army Research Office. We thank Prof. G. Swartzlander for discussions regarding spiral phase element manufacture. We acknowledge fruitful conversations with H. Uys, O. Dutta and M. Leclerc.

	No. Parameter	Description	Value Units
	1.	L	Cavity length	1	mm
	2.	λ	Laser wavelength	810	nm
	3.	ωc	Cavity resonance frequency	2π10 14 Hz
	4.	ω φ	Rotating mirror angular frequency 2π10 MHz
	5.	M	Rotating mirror mass	100	ng
	5.	R	Rotating mirror radius	10	µm
	7.	Q φ	Mechanical quality factor	2×10 6	-
	8.	F	Optical finesse	5×10 3	-
	9.	l	Orbital angular momentum	100
	10.	Pin	Input laser power	50	mW
	11.	∆/ω φ	Normalized laser detuning	1	-

Appendix A: Lyapunov Equation

Acknowledgments

This work is supported in part by the US Office of Naval Research, by the National Science Foundation and


M Bhattacharya

S Singh

P.-L Giscard

P Meystre

Optomechanical control of atoms and molecules

Keywords: numbers: 42.50.Pq, 04.80.Nn, 42.65.Sf, 03.67.Lx

In this article we present recent and ongoing developments in our group related to nanoscale optomechanics, an emerging area at the confluence of atomic, condensed matter and gravitational wave physics. The basic paradigm of optomechanics is the placement of a mechanical harmonic oscillator in its quantum ground state. First we discuss how the motion of such a macroscopic quantum oscillator can be squeezed, using the laser excitation of a high finesse optical cavity. Next we show how the placement of a ferroelectric tip on the oscillator allows it to manipulate polar molecules. Lastly we discuss how the role of the mechanical oscillator can be played by a quantum degenerate gas placed inside an optical cavity, giving rise to rich optomechanical physics.

I. INTRODUCTION

Vladilen Letokhov's recent monograph "Laser Control of Atoms and Molecules" reviews some of the most important developments in atomic and molecular physics following the invention of the laser. He personally pioneered an astoundingly large fraction of these developments. His ground-breaking contributions are too numerous to list in any detail. They cover areas ranging from the laser control of atomic and molecular processes, including ionization and dissociation, to the detection of single atoms and molecules, from isotope separation to laser velocity selection and to laser trapping and cooling, and much more. According to the ISI Web of Knowledge he has published well over 700 papers. As just one illustration of his breadth of interests, in the last two years he has published work on astrophysical lasers, on scanning near-field optical microscopes, on the quantum theory of radiation by atoms placed near a microresonator, and on the excitation and dissociation of polyatomic molecules under the action of femtosecond laser pulses.

One of us (PM) first met Vladilen in the mid-1970s during one of his frequent visits to his close friend the late Peter Franken and to Marlan Scully at The University of Arizona. At the time, laser isotope separation was a major research topic and he was an undisputed leader in the field. Everybody wanted to learn his latest thoughts on the subject. We later met again at a number of occasions, sometimes in Europe, in particular at the Max-Planck Institute for Quantum Optics, or back in Tucson, where our interactions increased significantly following the break-up of the Soviet Union, and also in Moscow and in Troisk. It is therefore an honor and a pleasure to contribute an article to the Festschrift celebrating his seventieth birthday.

Returning to Dr. Letokhov's recent book, we remark that an important requirement to achieve further progress in the manipulation and control of atoms and molecules is the availability of neutral particle detectors that can operate at the single atom or single molecule level. With this in mind, this paper briefly reviews some of our recent work toward that goal, concentrating on the emerging area of cavity optomechanics, a field that combines ideas of AMO physics, nanoscience, and condensed matter physics. It promises to open up new routes to the coherent control of atoms and molecules, relying on their coupling to nanomechanical oscillators laser-cooled close to their ground state of vibrations by techniques directly inspired from those originally developed and demonstrated in the laser cooling of neutral atoms and ions.

With the detailed understanding of mirror cooling gained in the last few years, we are now rapidly moving toward the longer-term goal of coupling nanoscale cantilevers to atomic or molecular samples. For instance the coupling of a cantilever carrying a magnetic domain with a Bose-Einstein condensate was recently described in Ref. [1]. In another example we have considered the coupling of a nanomechanical oscillator in the quantum regime with molecular (electric) dipoles [2]. We found theoretically that the cantilever can produce single-mode squeezing of the center-of-mass motion of an isolated trapped molecule and two-mode squeezing of the phonons of an array of molecules. This work opens up possibilities for manipulating "crystals" of ultracold dipolar molecules, which have been recently proposed as quantum memories [3]. More broadly, it indicates the promise of nanoscale cantilevers for the quantum detection and control of atomic and molecular systems.

II. CAVITY OPTOMECHANICS

A basic arrangement that can be used to understand optomechanics theory and experiment consists of a linear two-mirror optical cavity (2MC) driven by laser radiation tuned close to a resonance of the cavity. One of the cavity mirrors is movable, and the goal is to optically cool its vibrational motion to a point as close to its quantum mechanical ground state as possible. This procedure normally involves the use of two laser beams, the first one detuned above a cavity resonance so as to increase the intrinsic oscillation frequency of the cantilever to a value ω eff ; and the second one detuned below the cavity resonance, so as to (almost) independently increase the damping constant of the oscillating mirror from its field-free value Γ m to Γ eff [4]. Cooling of moving mirrors has now been achieved for a remarkably broad range of systems, from nanogram or picogram mirrors attached to nano-cantilevers to LIGO-class massive mirrors.

The laser fields change the properties of the moving mirror in two ways: they increase the energy level spacing of the harmonic mirror trap from ω M to ω eff , and cool the mirror from its initial equilibrium temperature T e to a lower value given by

T eff = Γ m Γ eff T e . (1)
The resultant trapping and cooling thus lowers the number of quanta of vibrational excitation of the oscillating mirror to

n m = k B T eff ω eff = k B T e ω eff Γ m Γ eff , (2)
where k B is Boltzmann's constant. The goal is to achieve n M < 1, i.e. to place the mirror in its quantum mechanical ground state.

In recent work we proposed an alternative to the "traditional" two-mirror geometry that allows one to reach and detect lower n m 's for comparable parameters [5] by suspending a nearly perfectly reflecting mirror inside a high-finesse two-mirror cavity. This three-mirror cavity (3MC) arrangement was shown to possess at least three advantages. First, it provides a higher value of ω eff for the mirror [6][7][8][9], leading to fewer quanta of excitation, see Eq. (2). Second it removes bistability problems completely for the trapping laser fields, and partially for the cooling fields. Lastly, it increases the time available for observing the quantum dynamics of the suspended mirror before thermal decoherence sets in. In addition, the coupling of the mirror to the intracavity field can be either linear (or dissipative) or quadratic (or dispersive) in the mirror position. This has several important implications, including the possibility to measure an energy eigenstate of the mirror and that of preparing nonclassical squeezed states of mirror motion.

An experiment that demonstrates the working of the 3MC and points out some of its additional virtues was recently reported in Ref. [10]. In that work a 50nm thick dielectric membrane placed inside a high-finesse optical cavity was cooled down from room temperature (294K) down to 6.82mK, i.e. by a factor of 4.4 × 10 4 .

III. SQUEEZING THE MIRROR MOTION

There is little doubt that the laser cooling of vibrating mirrors will soon be able to lower one or several modes of vibration to their quantum mechanical ground state [5,[11][12][13]. In addition to metrological applications, this will represent an important first step in exploring characteristic features of quantum mechanics such as superposition [14] and entanglement in macroscopic systems [15]. This section discusses a possible way to realize a squeezed state of a mechanically moving mirror in a high finesse optical cavity.

Squeezed states of the harmonic oscillator have attracted much attention, due to their favorable quantum noise properties [START_REF] Dodonov | Theory of Nonclassical States of Light[END_REF]. Squeezed states of light are expected to find applications in precision measurements [START_REF] Caves | [END_REF] and optical communications [START_REF]Advances in Communication Systems[END_REF][START_REF] Yuen | [END_REF]. The squeezing of classical noise in mechanical oscillators has been demonstrated in optomechanical cavities [21], ion traps [22,23] optical lattices [24] and other systems [25,26]. Quantum squeezing of phonons has been achieved in ion traps [27] and in crystals [28]. Proposals to realize squeezed states of nanomechanical oscillators in the quantum regime have been made involving two-mirror cavities [29], parametric mixing in solid state circuits [30][31][32], microwave coupling to a charge qubit [33], quantum non-demolition measurements [34] and the parametric modulation of a mechanical spring [35,36]. Their application to gravitational interferometry has also been discussed [37].

Previous proposals to achieve squeezing in optomechanical systems have relied on a mathematical analogy between an optical resonator with a moving mirror and light propagation in Kerr media, and the mechanism of squeezing has been parametric driving. Here we invoke compression as an alternative route to squeezing [16]. In that scheme squeezing of the mirror motion relies on coupling it dispersively with the cavity. More precisely, we consider a moving mirror coupled dispersively to an optical field and the modes and dissipatively to a second one, a configuration that was recently proposed as an efficient cooling and trapping configuration for semi-transparent mirrors [39].

This system is described by the model Hamiltonian [39]

H = H f + H m + V (3)
where the free field Hamiltonian H f is

H f = ω D a † a + ω S b † b, (4)
the oscillating middle mirror Hamiltonian H m is

H m = p 2 2m + 1 2 mω 2 m q 2 , (5)
and the interaction Hamiltonian V is

V = ξ D a † aq + ξ S b † bq 2 . ( 6
)
Here ω m is the oscillation frequency of the middle mirror of transmittivity T and equilibrium position q 0 ,

|ξ D | = sin 2k n q 0 (1 -T ) -1 -cos 2 2k n q 0 ξ, (7)
ω n = nπc/L, ξ = ω n /L, k n = ω n /c, and
|ξ S | = τ ξ 2 2 1 -T T 1/2 , (8)
where τ = 2L/c. The frequencies of the optical fields can be chosen such that ξ D,S are either positive or negative. For ξ S this corresponds to the use of trapping and anti-trapping modes, respectively [39]. For low values of ξ S the spring potential energy dominates the anti-trapping due to radiation pressure, hence the middle mirror still behaves as a harmonic oscillator. However, for ξ s negative, increasing |ξ S | leads to a point

C S = -ω m /2, (9)
where the mirror behaves as a free particle. For even higher values of |ξ S | radiation pressure-induced antitrapping dominates and the mirror behaves like an inverted harmonic oscillator [40]. We do not consider that regime here.

In the following we consider a semiclassical version of the Hamiltonian H valid for situations where the optical fields can be treated classically. In that case

a → α, b → β. (10)
Expressing the mirror displacement in terms of raising and lowering operators,

q = 2mω m (c † + c), (11)
H becomes

H = C D (c + c † ) + 2 C R K 0 + C S (K -+ K + ) (12)
where we have removed a constant energy

E 0 = ω D |α| 2 + ω S |β| 2 ,
and

C D = ξ D |α| 2 2mω m / , C S = ξ S |β| 2 mω m , C R = C S + ω m . (13)
We have also introduced the operators

K 0 = (c † c + cc † )/4, K -= c 2 /2 , K + = c †2 /2. ( 14
)
which together with c and c † form the basis of a Lie algebra [41] with

[K 0 , K ± ] = ±K ± , [K -, K + ] = 2K 0 , [K -, c] = K + , c † = 0, K -, c † = c, K 0 , c † = c † /2. ( 15
)
The Hamiltonian (12) was previously studied in the context of molecular translational-vibrational interactions [42], of laser-plasma scattering [43], and of atomic vapors inside resonators [44].

Using the Lie-algebraic symmetries of H, the associated evolution operator can be disentangled as [41]

U = exp[-iHt/ ] = e iδ D(ν)R(φ)S(κ), ( 16
)
where δ is an unimportant overall phase. The operator

D(ν) = e νc † -ν * c , (17)
is the displacement operator, with [41]

ν = C D χ ω m χ (cos χt -1) -i sin χt (18)
and

χ = C 2 R -C 2 s = [ω m (ω m + 2C S )] 1/2 . ( 19
)
In the bound oscillator regime, i.e. for C S > -ω m /2, we have χ 2 > 0, and we can choose χ > 0 without loss of generality. That parameter largely determines the time scale of the mirror dynamics; in the absence of squeezing (C S = 0) it is just the harmonic oscillator period. The second operator,

R(φ) = e iφK0 (20)
is a rotation operator, and the third operator,

S(κ) = e κ * K--κK+ (21)
is a squeezing operator, with

|κ| = sinh -1 C S χ sin χt . ( 22
)
As expected that operator does not depend on C D , i.e. displacement does not affect squeezing. It turns out that the rotation angle φ in Eq. ( 20) is exactly opposite the angle at which the squeeze operator tilts the error ellipse of the moving mirror in phase space [START_REF] Mandel | Optical coherence and quantum optics[END_REF], i.e.

φ = - 1 2 [phase(κ) + π] . (23)
The two rotations therefore cancel each other out and φ effectively plays no role in the dynamics. It is in fact intuitively clear that the effects of rotation should cancel, i.e. the axes of the final error ellipse should be aligned along p and q in phase space. This is because Eq. ( 3) stipulates that position is the only quadrature of the oscillating middle mirror that can be squeezed or anti-squeezed, the latter situation corresponding to momentum squeezing. Figure 1 shows |κ| versus time for typical experimental parameters. The squeezing first grows linearly in time, as can be confirmed by analytically expanding Eq. ( 22) for the case of t If the middle mirror is initially prepared in its quantum mechanical ground state, the squeezing operator ( 21) produces a squeezed vacuum [START_REF] Scully | Quantum Optics[END_REF]. From Fig. 1, the maximum value of κ is approximately 5, which implies a maximum squeezing of R ∼ e -5 ∼ 0.007, or Log 10 (0.007) ∼ 22dB of squeezing.

1/χ |κ| C S t - (ω m + C S ) 2 6 t 3 + O t 5 . ( 24
) 0 Π Χ 2 Π Χ 3 Π Χ 4 Π Χ
Any realistic model should also include the damping of the mirror. We estimate this effect by including noise and damping in the Heisenberg equations of the mirror in a manner consistent with the fluctuation-dissipation theorem. Setting α = 0 for simplicity and concentrating therefore solely on the squeezing part of the Hamiltonian H this leads to the quantum Langevin equations

q = p/m, ṗ = -mχ 2 q - Γ m m p + (t). ( 25
)
Here Γ m = mω m /Q is the damping constant of the mirror of mechanical quality factor Q and (t) represents brownian noise with average zero and fluctuations correlated as

δ (t)δ (t ) = Γ m ∞ -∞ dω 2π e -iω(t-t ) ω 1 + coth ω 2k B T e . (26)
Linearizing all operators in Eq. ( 25) as sums of a semiclassical steady-state value and small quantum fluctuations (i.e. q = q s +δq) we obtain linear dynamical equations for the fluctuations. Following Ref. [START_REF] Grabert | [END_REF], we find the equaltime correlation function for the position

δq 2 = (2n T + 1) ω m 2mχ 2 , (27)
which is independent of time since the noise process is stationary [Eq. (26)]. In the absence of squeezing (C S = FIG. 2: (Color online). Nanomechanical oscillator coupled to A) a single polar molecule B) a one-dimensional dipolar crystal made of many such molecules. In both cases the molecules are weakly confined by a harmonic trap along the x axis and oriented along the y axis by a strong uniform electric field.

0), and at high temperatures, (n T ∼ k B T e / ω m 1),

δq 2 = k B T e /mω 2 m ( 28
)
while at low temperatures (n T 1),

δq 2 = /2mω m , (29)
which is just the square of the oscillator length of the ground state of the moving mirror.

IV. MOTIONAL SQUEEZING OF DIPOLAR MOLECULES

We mentioned already that ultracold mirrors and cantilevers operating close to or in the quantum regime provide a novel class of quantum sensors that may find applications in matter-wave optics and in new approaches to the coherent and quantum control of ultracold atomic and molecular samples. A broad perspective on the subject can be gained from related successes in atomic physics where laser cooling and trapping techniques have enabled impressive coherent control of microscopic systems. A particularly promising research direction at the interface between atomic and solid state physics, is the interaction of cold atomic systems with quantum nanomechanical oscillators. Proposals along these lines have included the coupling of cantilevers to cold ions [48], Bose-Einstein condensates [1], and cold atomic gases [49], etc. In this section we review some recent work on the dipole-dipole coupling between ultracold nanomechanical cantilevers with a ferroelectric tip and ultracold polar molecules [2].

Consider first the case of a single molecule, see Fig. 2. Its coupling with the cantilever is described by the Hamiltonian

H = H c + H mol + V I , (30)
where

H c = ω c a † a (31)
describes a vibrational mode of the cantilever of effective mass m c and frequency ω c ,

H mol = ω t b † b, (32)
accounts for the center-of-mass motion of the trapped molecular dipole of mass m oscillating at the trap frequency ω t , and the cantilever-molecule interaction is

V I = d mol d c 4π 0 r 3 1 - 3(R + y c ) 2 r 2 . ( 33
)
Here

y c = 2ω c m c (a + a † ), ( 34
)
d c is the dipole moment of the ferroelectric domain situated a distance r = (R + y c ) 2 + x 2 mol 1/2 , and R is the the equilibrium molecule-cantilever separation. In practice R x mol , y c and in this case the dipolar interaction is approximately

V I ≈ d mol d c 2π 0 R 6 (-R 3 + 3y c R 2 + 3x 2 mol R -15y c x 2 mol ) (35)
with

x mol = 2ω t m (b + b † ). ( 36
)
For small R the presence of the cantilever results in a tightening of the trap, with ω t → ω t . In addition, it produces a parametric squeezing of the molecular motion. This can be seen by transforming to an interaction picture with respect to the free Hamiltonian H c + H mol , choosing ω c = 2ω t . Performing the rotating-wave approximation and assuming that the cantilever amplitude can be treated classically, a → α, the potential V I reduces to

V I = -C b 2 + b †2 , (37)
where

C = L c 15d mol d c 4π o mω t R 6 , (38)
and L c is the classical amplitude of oscillation of the cantilever given by

L c = N 2m c ω c 1/2 . ( 39
)
Equation ( 37) is the familiar quantum optics squeezing Hamiltonian of the degenerate parametric amplifier, see e.g. Ref. [58]. The squeezing resulting from the interaction in Eq. ( 37) is of course degraded by thermal noise, which affects the system in the form of phase fluctuations of the cantilever field. As in the previous section these fluctuations are related to the cantilever damping rate Γ m by the fluctuation-dissipation theorem. For times t such that Γ m < t -1 < 2C, the variance in the dimensionless molecule position quadrature is given by [58] (∆x 1 )

2 t = 1 4 e -2u + 1 8 e 2u Γ m t, (40)
where the squeezing parameter is u = 2Ct. We now extend these considerations to a onedimensional chain of N heteronuclear molecules contained in a harmonic potential V t , see Fig. 2. A uniform polarizing electric field is once again applied along the y-axis to align the dipoles. The dipole-dipole interaction being repulsive in that case, the molecules self-organize into a linear chain of period l described by the Hamiltonian

H chain = N i p 2 i 2m + d 2 mol 4π o N i<j 1 |x i -x j | 3 + V t , (41)
where x i , p i are the position and momentum, of the ith molecule and V t is the external trapping potential.

For small molecular oscillations the Hamiltonian (41) can be reexpressed in terms of acoustic phonon modes of momentum k and energy ω k [59],

H chain = k ω k b † k b k , (42)
where b k , b † k are bosonic phonon annihilation and creation operators, ω k = 2ω 0 |sin(kl/2)|, and ω 0 = d m 3/2π 0 ml 5 1/2 . Only purely harmonic terms have been retained in arriving at Eq. ( 42) from Eq. ( 41). Higher order terms represent phonon-phonon interactions and in particular determine the lifetime of the phonons in the crystal [60].

As in the case of a single molecule, a zero-temperature description is adequate in the regime k B T ω 0 . The nanomechanical cantilever energy is still correctly described by Eq. (31), and the interaction between the chain of molecules and the oscillator is given by

V I = i d mol d c 4π 0 r 3 i 1 - 3(R + y c ) 2 r 2 i . ( 43
)
Here y c is the displacement of the cantilever along the y axis, R is now its distance from the center of the dipolar crystal and r i = (R + y c ) 2 + (il + x i ) 2 1/2 is its distance from the i-th molecule.

Typically x i l, N l R, and we use these relations to keep only the lowest order terms in Eq. (43). As was the case for a single molecule, they result in a slight frequency shift ω k → ω k -now in the phonon frequency -and to a squeezing interaction, now in the form of a two-mode squeezing Hamiltonian given in the interaction picture by 44) where

V I = - k C k a + a † b k b -k + b † k b † -k + b † k b k + b -k b † -k (
C k = - 17d m d c 16π o mω k R 6 2m c ω c 1/2 . ( 45
)
We select for concreteness the cantilever frequency ω c = 2ω k , with k = π/l, so that it couples mainly to excitations near the edge of the first Brillouin zone, where the density of phonon states is largest. Assuming as in the single-molecule case that the motion of the nanomechanical cantilever can be described classically, in the rotating wave approximation we then obtain the two-mode squeezing hamiltonian

V I = -C k b k b -k + b † -k b † k . ( 46
)
where C k = √ N C k and N is the average occupation number of the cantilever, related to the classical amplitude of oscillations L c by Eq. ( 39). Two-mode squeezing is conveniently described in terms of the two dimensionless quadratures

s 1 = 1 √ 2 (b k + b -k + b † k + b † -k ) ( 47
)
s 2 = 1 √ 2i (b k -b † k -b -k + b † -k ). ( 48
)
The sum of variances in the two quadratures, taking into account the phase fluctuations in the cantilever motion resulting from thermal noise, is then [61]:

(∆s 1 ) 2 + (∆s 2 ) 2 = e -Γmt 2 C k0 {Γ m sinh(C k0 t) + 2C k0 cosh(C k0 t)} - i,j,k,i =j,j =k e λit 2C k0 (λ i + 4Γ m ) (λ i -λ j )(λ i -λ k ) , ( 49
)
where u is the phonon squeezing parameter and is equal to 2C k0 t, with

C k0 = 1 2 4C 2 k -Γ 2
m and the λ i 's are the roots of the cubic equation: N =100, and a ferroelectric domain with dipole moment d c = 2.1 × 10 -23 C-m attached to the cantilever and placed at R = 2µm from a linear chain of dipolar molecules with inter-molecular distance l ≈ 200nm. These parameters give a phonon frequency ω o = 4MHz, and thus an interaction C k = 4.4Hz. Figure 4 gives the sum of variances of s 1 and s 2 as a function of the squeezing parameter.

λ 3 + 5Γ m λ 2 + (4Γ 2 m -C 2 k0 )λ -2C 2 k0 Γ m = 0 (

V. CANTILEVER-BASED COHERENT CONTROL

Building upon these results, we now show that it is possible to use nanoscale cantilevers to engineer nonclassical center-of-mass states of an atomic or molecular system. These results are preliminary and involve two dipolar molecules only. An extension to larger systems will be the object of a future publication.

The Hamiltonian describing the coupling of the cantilever to the two molecules is

H = H c + H m + V dd , (51)
where as before

H c = ω c c † c, (52)
describes a single mode of vibration of the cantilever. We assume that the two molecules are trapped tightly in the y and z-direction, such that they align themselves in the x-direction. As before, we also assume that a strong polarizing DC field freezes out the rotational freedom of the dipoles so that they align along the y-axis.

The vibrations of the two-molecule system are conveniently described in terms of their center-of-mass (COM) and relative modes of motion (see inset in Fig. 5). Ignoring the trap potential, the two-molecule Hamiltonian is then

H m = p 2 com 4m + p 2 rel m + 6d 2 m π 0 l 5 x 2 rel . (53)
where p com and p rel are the center-of-mass and relative momenta, and the third term in the Hamiltonian describes the dipole-dipole interaction between molecules in the harmonic approximation. Here m is the mass of each molecule, d m its electric dipole moment, and l their equilibrium separation. For x 1 , x 2 l R the dipole-dipole interaction between the ferroelectric domain on the cantilever of dipole moment d c and the molecules is given approximately by

V dd = d m d c 8π 0 R 7 24R 2 (x 2 com + x 2 rel + x 2 c ) -48R 2 x c x com -140lx 3 rel + 60lx rel x 2 com -60lx rel x 2 c + 120lx c x rel x com , (54)
where

x c = 2ω c m c (c + c † ). ( 55
)
The design of appropriate fields for the internal control of systems of interacting atoms and molecules on their natural timescales is a difficult task. A ground-breaking approach using Genetic Algorithms and a pulse-shaping element inside a feedback loop to let atoms design their own optimal field was first proposed by Rabitz and Judson as early as 1992 [50]. In the present example we apply similar ideas to the center-of-mass motion of the pair of molecules. We proceed by modulating the frequency of oscillations of the cantilever so as to achieve specific populations of the center-of-mass and relative modes of vibration of the molecular system. We proceed by decomposing that frequency into a constant term and a time-dependent part to be determined iteratively via a genetic algorithm,

ω c = ω 0 + δω(t). (56)
In an interaction picture with respect to the molecular Hamiltonian and the ω 0 part of the cantilever Hamiltonian and assuming once more that the cantilever motion can be treated classically the interaction potential V cm reduces in the rotating wave approximation to 7 (ab † e i(ω rel -ωcom-ωc)t + h.c.), (57) where ω com and ω rel have been redefined to include their shifts due to the presence of the cantilever. Here a and a † are the annihilation and creation operators of the COM mode, and b and b † are the annihilation and creation operators of the relative mode of vibration. The distances R and l are defined in Fig. 5, and L c is defined in Eq. (39). We have also assumed that the dc cantilever frequency, ω 0 is close to ω rel -ω com .

V dd = 15 L c 2π 0 √ ω com ω rel d m d c l mR
Equation (57) indicates that the interaction between the molecules and the cantilever enables the transfer of population from one vibrational mode to the other. By changing the frequency of the cantilever in an iterative fashion determined by a genetic algorithm, it is then possible to drive any desired population transfer.

Figure 6 illustrates the design of a pulse that can be used to cool one of the normal modes of the molecular system. In this example two SrO molecules are trapped 200nm apart from each other, and 2µm from the cantilever the specifications for which are the same as in the previous section. The center-of-mass and relative modes of the two-molecule system are both initially in a thermal state at a temperature of 100 µK. The specific time dependence of the cantilever frequency is chosen to minimize the average occupation number in the COM mode. The average occupation number for each genetic algorithm iteration is plotted in Fig. 7. We remark that in this example the total center-of-mass energy of the molecular system is decreased at the end of the pulse produced by the last generation of the algorithm (as seen in Fig. 8), and hence the method can demonstrably cool the molecular system.

VI. OUTLOOK: CONDENSATES IN HIGH-Q CAVITIES

As we have seen, a central tenet of cavity optomechanics is the cooling of mechanical oscillators close to their ground state of vibration, and we have shown in a few examples that these systems have fascinating potential applications in basic and applied physics. In recent work, the groups of Esslinger [51] in Zurich and of Stamper-Kurn at Berkeley [52] have proposed and demonstrated an alternative approach to cavity optomechanics that uses Bose condensates instead of "traditional" mechanical oscillators. In these systems, the optical length of the cavity is controlled and modified via the dispersive properties of the condensate, just as in usual optical bistability. In contrast to the moving mirrors and membranes discussed in this paper, the mechanical oscillator is no longer based on the presence of an external spring. Rather, it is provided by condensate density excitations resulting from photon recoil, a situation closely related to CARL amplification [53][54][55][56].

Esslinger et al have shown that in the simplest case there is a one-to-one mathematical correspondence be- tween this system and the moving mirror situation. This can be seen simply by expanding the order parameter of the condensate as

ψ(x, t) = c 0 (t) + √ 2c 2 (t) cos(2kx), (58)
where the second term on the right-hand side accounts for the photon recoil of the atoms due to virtual transitions involving the absorption and reemission of cavity photons of momentum k. Assuming that the light-atom interaction does not significantly deplete the initial condensate one can replace the operator c 0 by a complex number, in which case the Hamiltonian describing the interaction between the atoms and the intracavity field reduces approximately to

H = ∆ + g(c 2 + c † 2 ) a † a, (59)
where a and a † are the annihilation and creation operators of the optical field, g is a coupling constant, and ∆ is the detuning between the frequency of the intracavity optical field and the cavity resonant frequency. At this point, the analogy between this system and the case of a cavity with a suspended mirror is fully apparent:

The condensate side-mode is the analog of the moving mirror, and the intracavity field acts on this side mode amplitude exactly in the same way radiation pressure acts on the moving mirror.

Several experiments exploiting that analogy have already been performed, demonstrating in particular the appearance of optical bistability in these systems. But it is known from general principles that generically, bistability ceases to exist as systems become truly microscopic, due to quantum tunneling between the coexisting metastable state and true ground state that are responsible for the bistable behavior. In classical systems, this tunneling time is typically exceedingly long, certainly much longer than the duration of typical experiments. This results in the observed bistability. But as the systems become more microscopic this tunneling time decreases, leading to a behavior that, in the open systems at hand, is somewhat reminiscent of the Maxwell construction familiar from statistical physics. It will be interesting to study this transition by reducing the size of the initial condensate to a point where its approximate classical description ceases to be valid and quant um fluctuations dominate the system.

It should also be possible to study quantum phase transitions in ultracold atomic systems in cavity optomechanical environments. For example, it is known that radiation pressure induced bistability can occur in these sys-tems. (This closely related to the more familiar bistability that occurs when a Kerr nonlinear medium is confined inside a Fabry-Pérot resonator. The difference is that instead of an intensity-dependent index of refraction, we are now in a situation where it is the true, physical length of the resonator that is changed when the light intensity inside the resonator is varied.) Radiation pressure induced optical bistability is generally considered as detrimental when trying to cool nanoscale cantilevers. However it can also result, at least in principle, in fascinating new effects such as a bistable quantum phase transition between a Mott insulator state and a superfluid state of the atoms [62].

These examples, and numerous other fascinating aspects of cavity optomechanics and of the optomechanical control of atomic and molecular systems, promise to lead to exciting developments in future years, further contributing to the merging of AMO physics, nanoscience and condensed matter physics.

FIG. 1 :

 1 FIG. 1: Modulus of |κ| from Eq. (22) as a function of time for L = 5mm, λ = 514nm, m = 1µg, ωm = 2π × 1kHz, mechanical quality factor Q = 5 × 10 6 , end mirrors transmissivity T = 10 -4 , q0 = λ/10, background temperature Te = 100mK. and incident power 10mW. The period of the oscillations is 2π/χ 5ms.

FIG. 3 :

 3 FIG.3: Variance in the molecule position quadrature component x1 as a function of the squeezing parameter u. The parameters are for a SrO molecule interacting with a cantilever. Dotted curve: single mode squeezing obtained in the absence of any thermal noise. Solid curve: effect of a cantilever damping rate Γm =1 Hz. Motional squeezing of the molecular motion occurs for (∆x1) 2 < 1/4, and is degraded by phase noise in the cantilever.

50 )FIG. 4 :

 504 FIG.4: Sum of the variances in s1 and s2 [see Eqs.(47- 49)], as a function of the squeezing parameter u, for a dipolar crystal of SrO molecules interacting with a cantilever. Dotted curve: two-mode squeezing in the absence of thermal noise. Solid curve: effect of thermal noise producing a cantilever damping rate Γm =1 Hz. Two-mode squeezing and phonon entanglement occurs for a variance less than 2.

FIG. 5 :

 5 FIG.5:(Color online). Schematic of the coupling a nanomechanical oscillator to two dipolar molecules. The molecules are along x-axis and interact with both the cantilever and with each other via dipole-dipole interaction. A weakly confining harmonic trap for the dipoles is shown along the x axis. The two insets show the center-of-mass and relative modes of vibration of the molecular system.

FIG. 6 :

 6 FIG. 6: (Color online) Frequency δω(t) corresponding to the final generation of the genetic algorithm. That pulse shape results in an average occupation number of 0.98 in the COM mode, and 1.6 in the relative mode.

FIG. 7 :

 7 FIG. 7: (Color online) Average occupation number of the COM and relative modes as functions of the number of generations of the genetic algorithm.

FIG. 8 :

 8 FIG. 8: (Color online) Excitation probability P (n rel , ncom) of the molecule pair (a) initially (upper figure) and after the final iteration of the genetic algorithm (lower picture.

Acknowledgments

This work is supported in part by the US Office of Naval Research, by the National Science Foundation, and by the US Army Research Office.