A Laliotis

I Maurin

P Todorov

I Hamdi

G Dutier

A Yarovitski

S Saltiel

M.-P Gorza

M Fichet

M Ducloy

D Bloch

Testing the distance-dependence of the van der Waals interaction between an atom and a surface through spectroscopy in a vapour nanocell

Keywords: atom-surface, reflection spectroscopy, van der Waals, nanocell, fundamental test, Casimir-Polder, Dicke narrowing

This paper presents our current measurements in a vapor nanocell aiming at a test of the distance-dependence of the atom-surface interaction, when simple asymptotic descriptions may turn to be not valid. A state-of-the-art of atomsurface interaction measurements is provided as an introduction, along with the comparison with the theory of the van der Waals (or Casimir-Polder) interaction; it is followed by a presentation of the most salient features of nanocell spectroscopy.

From van der Waals to Casimir interaction and the state-of-the-art in the measurements

The interaction between neutral bodies can be seen as the universal attraction at the scale of atomic, molecular and solid-state physics. It is indeed at the origin of covalent interaction, and other type of "van der Waals" crystals. If interaction between "particles" was recognized by van der Waals (1873) to be a major cause for gases not to follow the equation of state of an ideal gas, the effective origin of these interactions were identified in the 1930 with the works of London and Lennard-Jones and coined as "van der Waals" (hereafter vW) type (for references, see [START_REF] Bloch | Atom-wall interaction[END_REF][START_REF] Israelachvili | Proc. Roy. Soc. A[END_REF]). It is indeed the instantaneous correlation between (quantum) fluctuations of particles that is responsible for their attraction. For an atom in front of a reflecting surface, the concept of electrostatic image implies that an electric image of each of the elementary charges in the atomic system atom is created. Although the atom is neutral, and even without a permanent dipole, the atom is attracted by its image through the non-zero dipole-dipole attraction resulting from its quantum dipole fluctuations. These fluctuations have to be developed over the (virtual) dipole transitions. Naturally, the idea of an "instantaneous" correlation of fluctuation is not acceptable in regards to light propagation, in particular when the distance of separation is non negligible relatively to the wavelength of the considered transition. As early as 1948, the Casimir-Polder work [START_REF] Casimir | [END_REF], triggered by the "mundane" problem of interaction between colloids, has shown that retardation effect should be considered when the atom-surface distance is non negligible relatively to the wavelength of the virtual transitions responsible for the quantum dipole fluctuations. In particular, the predicted 1/z 3 potential (z: the atom-surface distance) should be replaced by an interaction decaying even faster with the distance, up to a long range regime in 1/z 4 . In the same year, Casimir discovered as a consequence of Quantum Electrodynamics that two reflecting surfaces should attract each other, with a 1/d 4 dependent pressure (d : the distance between the two surfaces) [4].

Although a very tiny effect in most cases, the Casimir interaction (for reviews, see [5]) has been the topic of some experiments in the 50's, that were merely able to evidence an attraction compatible with the prediction. In the same time, an alternative theoretical approach was developed by Lifchitz based upon thermodynamics arguments [6]. Some experiments on the vW interaction (see [START_REF] Israelachvili | Proc. Roy. Soc. A[END_REF] and references therein) between two plates should also be included in these early attempts to measure the Casimir interaction. Only recently the Casimir interaction has become a hot spot of fundamental physics, following the experimental advances of Lamoreaux in 1997 [7] -with a torsion balanceand of Mohideen et al. [8] with an atomic force microscope (since 1999), and the claim of very high accuracy ( ~ 1%) and of a succesful comparison between the experiment and the theory. In these experiments, the measurement includes a demonstration of the distance dependence on a 0.6µm-6µm range in the Lamoreaux experiment, in the 100-900 nm range for the Mohideen work, a range that has been now pushed down to 62 nm. These experiments have stimulated refinements in the theoretical description (QED at non zero temperature, corrections for non ideal reflectivity, ... influence of the exact shape of the surface, ... ). It is now recognized that the evaluation of the cosmological constant may depend on the exactness of the QED predictions among which the Casimir interaction is a major test. Also, the measurement of a possible deviation to the Newtonian gravitation ("fifth force") with an additional short distance gravity can be attempted only if the unavoidable Casimir interaction is rightly taken into account [5]. Apart from this fundamental interest, the Casimir interaction being responsible for a pressure comparable to the atmospheric pressure for d~10 nm, is also nowadays considered as a limitation, or a possible tool, for the development of nano-actuators (NEMS).

Measurements of the atom-surface interaction had remained scarce for a long time owing to the extreme weakness of the force. If the vW interaction for an atom-atom potential can be of the order of a few tenths of eV, and can be estimated, but with a limited accuracy, as the asymptotic tail of an interatomic potential for distances exceeding several atomic units, the current distance of observation for atom-surface interaction is considerably larger, leading to extremely tiny effects. Remarkably, the vW interaction is supposed to span over ~10 orders of magnitude in energy on a distance scale that approximately covers the 1nm-1 µm distance. The first observations were conducted in the Columbia group in the late 60's [9]. They demonstrated a deflection of an atomic (and molecular) beam by a cylinder, and an effective impact parameter covering a range ~50-100 nm (a too strong attraction makes the atoms caught by the surface, a too weak one is undetectable). A deflection compatible with a z -3 law could be found, that looked to be relatively more satisfying than a z -2 law or a z -4 law. About 20 years later, a second generation of experiments [10] was conducted by the Yale group based upon the measurement of the transmission rate of a beam of Rydberg atoms through a system of two metal-coated plates. These experiments allowed a precise observation of the distance-dependence in the 500-3000 nm range -distance between plates-, and also evidenced the predicted dependence of the vW interaction with the level of atomic excitation. Indeed, one expects for a state i> a vW interaction governed by a potential: V vW (i>)= [or i i 12

1 2 3 D = C if the atom is isotropic]. A notable point is that for an atom, one has 0 i i 2 ≠ D
, in spite of the absence of a permanent dipole (< iD i>=0) : the dipole fluctuations, of quantum origin, corresponds to a summing over all virtual dipole transitions

∑ = i D D 2 2 j i i i
, which are much stronger in the far IR, as found for the couplings relevant for a Rydberg level, than for a ground state or a low excited level. In these experiments on Rydberg levels, that combine a mechanical detection with a spectroscopic excitation, energy shifts up to 500 MHz were obtained. A derived set-up was used to obtain the first evidence of the retarded Casimir-Polder regime [11], that can be seen only for ground state atoms, because excited atoms tend to undergo a real -and not virtual-emission process: indeed, excited states radiate according to the far field regime of an oscillating dipole, and largely exceeds the retarded vW attraction.

Contemporarily to the Yale's group experiments, our Paris13 group entered into the field of measurement of vW atom-surface interaction [12,13], through a spectroscopic technique (for a review, see [START_REF] Bloch | Atom-wall interaction[END_REF]) of selective reflection in a vapor that typically probes the vapor on a depth ~λ/2π (~ 100 nm) from the surface. Two peculiar features should be mentioned for these types of experiments: (i) spectroscopy is intrinsically sensitive to the energy of a transition between atomic levels, so that what is measured is the difference between the vW interaction for an excited level, and the vW level of another level (ground state for a one-photon experiment, another excited level on more complex setup); (ii) as a major difference with mechanical detection, spectroscopy allows to study the vW interaction undergone by short-lived atoms, instead of being intrinsically limited to ground states, or long-lived Rydberg levels. Naturally, reflection spectroscopy at an interface requires transparent surfaces, so that dielectric materials are investigated. Among the variety of results [START_REF] Bloch | Atom-wall interaction[END_REF] that we have demonstrated on the basis of this selective reflection technique, one should mention as far as the atom-surface interaction is concerned: (i) the observation of weak and strong vW regimes [13] that bring respectively either a perturbation (shift and distortion) to the lineshape expected in the absence of the vW interaction, or a severe change in the lineshapes, and (ii) the influence of the nature of the dielectric surface, with the possibility to resonantly enhance the vW attraction, or to turn it into a vW repulsion when the surface mode structure coincides with a virtual atomic emission [14]. As will be reported in the next sections, we have essentially upgraded our selective reflection spectroscopic technique, providing a fixed spatial resolution, to a technique of spectroscopy in a vapor nanocell, allowing us to study the spatial dependence of the vW interaction in a sub-100 nm range.

To complete this description of the state-of-the art, it is needed to mention the recent blossom of experimental techniques for the measurement of the vW attraction, that has notably been triggered by the importance of the problem of surface interaction for the storage and manipulation of cold atoms, and by development in atom interferometry as well. Cold atom techniques have been used to measure the vW interaction through a combination with an applied repulsive potential [15]. Very recently, the repulsion from a strong magnetic field allowed a test of the distance dependence in the 20-100 nm range (for Cs ground state [16]), and showed a C 3 value remaining constant within 15-20%. Interestingly, this verification of a C 3 value remaining constant with the atom-surface distance seems to disagree with refined theoretical predictions [17], because at the shorter distances, the transitions from the atomic core, often negligible due to the severe retardation effects owing to their short wavelengths, cannot be ignored. In the other measurements based upon a cold atom technology, it is rather the long distance range that is conveniently probed, as can be inferred from the comparison between the thermal energy k B T and the vW potential V vW = -C 3 z -3 . Indeed, for a ground state atom, the vW interaction exceeds the thermal energy only for z < 1nm at room temperature, but for z ~ 100 nm one has already an interaction potential ~300 µK, corresponding to a temperature much hotter than the one commonly attained with cold atoms. This long-distance probing is a favorable situation for the observation of the Casimir-Polder regime, probably observed in [15], and that influences the quantum reflection of very cold atoms [18]. A record long distance -6µm-has now been attained in a BEC-type experiment [19]. This paves the way to observation of temperature effects, and enables to rule out some coupling strength for an hypothetical non-Newtonian gravity whose range would span over the micronic range. In addition to these long-distance measurements enabled with cold atoms, experiments connected to atom interferometery [20] enable the exploration of the interaction at a very short distances (~5 nm), in a manner that globally integrates the spatial variation of the vW potential, not allowing to resolve details concerning its shape or spatial dependence.

Spectroscopy in micro-and nano-cells of resonant vapor : from sub-Doppler spectroscopy to atom-surface interaction

Selective reflection spectroscopy under near normal incidence has been known from early times [21] to lead to sub-Doppler resonances, recognized to originate in the transient interaction regime undergone by those atoms leaving the surface. Indeed, in selective reflection, only the atoms of a slice of vapor of a thickness λ/2π emit backward coherently, so that under normal incidence, only slow atoms (insensitive to the Doppler shift) reach an efficient steady-state of interaction in the ascribed length. This sub-Doppler nature of selective reflection spectroscopy, that helps to make this technique sensitive to the atom-surface interaction [START_REF] Bloch | Atom-wall interaction[END_REF] -because the response of atoms spending a long time in a region of strong atom-surface interaction is enhanced-has inspired us to look for an equivalent signature in the transmission regime [22,23], expected to be independent of the cell length -as opposed to regular volume absorption. Our earliest experimental attempts were conducted on 10-1000 µm thin vapor cells [24,25], in order to maximize the ratio between the region sensitive to atoms departing from the surface, and the isotropic "bulk" of the vapor. The pressure of the (dilute) vapor was kept low enough so that most atomic trajectories are wall-to-wall, with the light beam diameter largely exceeding the cell thickness. This offers the key condition of an atomic free path that becomes anisotropic. With such microcells, we observed indeed a sub-Doppler signal (whose contrast is enhanced with a frequency derivation technique, as provided by demodulating an applied FM), yielding an original sub-Doppler method with a single beam, that originated in a velocity-selective non linear effect (enhanced optical pumping for slow atoms). Such a sub-Doppler spectroscopic technique had been independently predicted by Izmailov [26]. Soon after, in an effort to keep incident intensities well below the (quite low) intensities required for optical pumping, we obtained in a linear regime a different sub-Doppler signature [27], purely relying on a two-level model (with no need of considering an "open" two-level model, i.e. a model where at least a third level influences the atomic lifetime). Several differences can be pointed out with the sub-Doppler regime of optical pumping : in particular, the sub-Doppler signature, even in FM technique, is reduced if the cell length exceeds the mean distance needed for a thermal atom to reach the steady state. Also, because 2-level absorption is a coherent process, the sub-Doppler spectral properties, initially calculated in [23] for a situation limited to a thickness between λ/2 and λ, can be shown to depend interferometrically on the cell length, in a way consistent with the predictions by Romer and Dicke [28], established 50 years ago for microwave experiments : for a λ/2 thick cell, the transient response of all velocity groups interfere constructively at line center, yielding a narrow signal, while for a λ-thick cell, the successive absorption and gain corresponding to a full Rabi oscillation would lead to a null signal in the absence of absorption relaxation. This is why successive revivals -with a λ periodicity-of these spectroscopic features is predicted [25,27].

If some interesting features for sub-Doppler spectroscopy could be revealed with experiments in a 10 µm-thick cell, such a cell was nevertheless not convenient to observe the vW atom-surface interaction.

As will be established in the next section, the production of sub-micrometric thin vapor cells by the D. Sarkisyan group [29] has been the key to apply realistically spectroscopy in a thin cell to the monitoring of the atom-surface interaction. A remarkable point with these cells, that are described in more detail in the presentation given by D. Sarkisyan [30], is that after evacuation (before proceeding to the filling with a metallic vapor), and in spite of their construction with two thick and well-polished windows separated by a spacer of a constant thickness, the cell undergoes local deformations of the windows, leading to a variable local thickness of the thin cell, usually spanning from a few 10's of nm to one or a few microns [31]. This provides a key ingredient, when using a focused beam, to study the thickness dependence of the spectrum and ultimately to monitor the distance dependence of the vW interaction. Practically, it is indeed sufficient to simply translate the cell, that is truly a "nanocell" in some regions. Before being able to perform an evaluation of the atom-surface potential, it has been needed to demonstrate a full understanding of the spectroscopy in these sub-micrometric cells. A notable point is that, because, the two windows of a nanocell are intrinsically extremely parallel -in spite of the minute deformation after evacuation-a nanocell is a genuine Fabry-Perot cavity [32], although of a low finesse : on the one hand, this provides a very suitable tool to measure the local cell thickness with an interferometric accuracy [31]; on the other hand; all nanocell spectra are unavoidably the result of a mixing [32] between absorption-like transmission, and reflection (which would be a pure dispersive signal only for an infinite cell length (see appendix of ref. 13). Our initial studies have enabled the observation of the Dicke revival as predicted in [25,27] with the corresponding lineshapes both in transmission and reflection detection. The easiest interpretations are naturally obtained in non saturating conditions, that are more convenient to reach than in longer cells (even micro-cells): indeed the interaction time is very limited, and the smaller interaction region is usually compensated by a stronger atomic density, implying an increase in the saturation intensity because of collision broadening. Note that a fluorescence detection can be used in nanocell spectroscopy, as it eventually provides narrower linewidths than transmission/reflection spectra [29,33]. However, fluorescence is an incoherent second-order process: apart from being insensitive to the periodical coherent Dicke narrowing [33], this makes fluorescence spectra intrinsically much more delicate to interpret when the evaluation of the atom-surface interaction is at stake.

The monitoring of atom-surface interaction in nanocells

The general theory of spectroscopy in a thin cell or a nanocell [32] resembles by many aspects the theory of selective reflection spectroscopy in the presence of a potential, as developed in [34]. There are additional spectroscopic effects due to the finite length of the cell (i.e. integrations are performed within finite limits, and not within infinite boundaries [22]), and eventually the Fabry-Perot mixing of transmission and reflection. Moreover, a dual detection of reflection and transmission is possible. The basic ingredients of the theory include : (i) evaluation of the atomic response at a given point, as resulting from the integration of the transient response from the departing wall to the actual position. This response, that would be elementary to calculate -and time integrate-in the absence of a vW potential, depends on the interaction potential (for a 2 wall-interaction). An interesting variety of situations has to be considered if the dispersive nature of the surface response is included, that implies a lot of complexity to describe the shape of the interacting potential [35,36]. In a pure electrostatic approach, that neglects the dispersive properties of the surface, the potential should include the equivalent of multiple electrostatic images, and the interacting potential is then described with a Lerch function. Although not rigorous, an elementary model simply adding the elementary vW contribution of each wall (i.e. V(z)= -C 3 [(1/z) 3 + [(1/d-z) 3 ], with d the cell thickness) can be most often sufficient, as the difference with the exact Lerch potential does not exceed 5 % when dispersive properties are neglected.

(ii) spatial integration all over the cell of the instantaneous response of an atom at a given position; for the reflective contribution, or in some terms of the transmission response when a Fabry-Perot-like response has to be considered, this integration has to includes an exp(2ikz) phase factor (with k the wave vector).

(iii) a velocity integration, over the (usually) thermal distribution of atoms. Instead of a family of curves depending on a unique parameter that describes in dimensionless units all the features of vW, as was obtained for selective reflection spectroscopy (at least in the approximation of the infinite Doppler width), one predicts for thin cell spectroscopy independent sets of spectral lineshapes when varying the cell thickness, or the vW strength, or also the velocity distribution (in a regime of small thickness, the vW interaction is not negligible relatively to the Doppler broadening, and the "infinite Doppler width" cannot hold).

Figure 1 illustrates some predicted lineshapes, when the strength of the van der Waals is varied for a given thickness, and when the cell thickness varies for a given vW coefficient. The potential is an approximate Lerch function, but similar results would actually be obtained for the elementary summing of two single-wall potentials. Experimentally, we have already mentioned in [36] that by using the above described modeling of the transmission lineshapes in a nano-cell, we observe on the Cs D 1 resonance line (λ=894 nm) some deviations relatively to the lineshape predicted in the absence of an atom-surface interaction for a 225 nm (λ/4)-thick cell. Also, in even earlier experiments [START_REF] Dutier | Laser Spectroscopy, Proceedings of the XVI International Conference[END_REF], we had shown through a stepwise two-photon excitation that allows to reach the high-lying 6D level of Cs, that considerable shifts and broadenings could be observed in extremely thin regions. We had even attempted to verify a z -3 dependence through the position of the transmission peak. However, these preliminary experiments were not conducted in a systematic way: the pressure effects were not controlled, and there was at that time no lineshape analysis. Indeed, the major goal was an attempt to check for resonant 2-wall atom-surface coupling, as our observation of a vW repulsion had been performed on Cs(6D 3/2 ) in front of a sapphire surface [14].

Here our report is on a systematic series of experiments [START_REF] Fichet | [END_REF] performed on the Cs(6P 3/2 )-Cs(6D 5/2 ) transition (λ= 917 nm), as reached after a pumping step (λ pump = 852 nm) to the resonant Cs(6P 3/2 ) level. Such an experimental choice has two important conveniences : on the one hand, the detection sensitivity approaches the quantum noise sensitivity in a modulated pump-probe scheme, as the detection is not performed at a null frequency (when all low-frequency noises are present), but at the frequency of the modulation applied to the pump (50 kHz in our case) : the sensitivity issue is obviously an important one when the measurements of interest should deal with the minimal atomic density; on the other hand, the vW interaction is predicted to be larger by about an order of magnitude for Cs(6D) than for Cs(6P), so that larger interactions should be observed if there is a minimal nanocell thickness that can be technically reached. The principle of the experiment, schematized in figure 2, is very simple. One monitors, through lock-in detectors, the changes in the transmission (T) and reflection (R) signals in the nanocell when the pump beam is on. The incident pump and probes are focused (diameter~100 µm). The pump and probe beam are nearly counterpropagating (to avoid unwanted pump scattering light) and sent at normal incidence on the nanocell. The local nanocell thickness is measured through the reflection coefficient of the pump beam (once detuned offresonance) and of an auxiliary He-Ne beam : in the range of very small thickness ( <150 nm) that we consider, a dual wavelength measurement is sufficient to provide unambiguously a reliable thickness measurement. Moreover, being interested in the smaller thickness, we explore a region of the cell that is intrinsically flat, and the accuracy in the thickness measurement, currently ~ 5 nm, can occasionnally reach a 1-2 nm level. An auxiliary reference set-up of volume saturation spectroscopy is implemented on the 917nm line (with prior 852 nm pumping), while the 852 nm frequency itself is kept fixed on one hyperfine component of a Doppler-free saturated absorption line. A typical experimental result is shown in figure 3 for a thickness 50 nm, with a simultaneous recording of transmission (T) and reflection (R) spectra. The spectra are quite broad (the Doppler width is only ~250 MHz) and strongly red-shifted relatively to the volume reference. The influence of the pressure on these features reveals to be quite weak (e.g. spectra recorded with a higher Cs pressure remain unchanged). Such spectra can be fitted with one set (T and R) of the calculated lineshapes for d=50 nm. In the fitting adjustment, only adjustable amplitudes have been used (one parameter for T, one for R) . In the velocity integration, the resonant Cs(6P 3/2 ) level has been assumed to be pumped in a thermal manner. Even if such an assumption may be problematic, the influence of this velocity distribution on the global spectrum remains quite weak -in particular, the choice of a smaller Doppler width, that would correspond to a more efficient pumping of the slow atoms, provides very similar results-. A notable result is hence that T and R lineshapes can be fitted simultaneously with the same C 3 parameter. In this regime in which the vW interaction dominates over the pressure or Doppler broadening, it is easily seen (see e.g. fig . 1) that any attempt to fit a given lineshape with notably different C 3 values will lead to a differing frequency shift at the peak, and to different lineshapes. It is only for marginal changes of C 3 around an acceptable value that a least square fitting is a help to determine the C 3 value with an increased accuracy.

Reflection Transmission

To study systematically the distance dependence of the C 3 value, shown above to be measurable with our technique, we had studied various Cs densities (pressure) conditions for each considered cell thickness. As shown in figure 4, we obtain constant C 3 values independent of the pressure, except for the larger thickness, when the vW interaction can turn out to be smaller than the pressure broadening -for high Cs densities, or temperature, as an increase of 20°C is roughly a doubling in Cs density-Also, for the larger thickness, the unresolved hyperfine structure of the pumped state Cs (6P 3/2 ) -350 or 450 MHz depending on the ground state hyperfine component -makes more difficult the frequency referencing to the volume resonance. This constant value of the estimated C 3 vW interaction constant is illustrated in the spectra of figure 5, when the conditions for Cs vapor are identical, and when a single set of parameter for C 3 and for the transition width is considered. Only the amplitudes are adjustable by an arbitrary factor, because the density of active Cs -in the resonant 6P 3/2 level -is expected to change with the thickness, owing to the limited time available for the excitation. Also, for a given thickness, slight disagreements between the relative amplitudes for R and T can be simply due to the (local) scattering properties of the surfaces, which makes the non resonant reflection and transmission factor apparently disagree with the Fresnel formulae. Before comparing the experimental findings and the predictions, it is worth discussing the reproducibility of the results presented here. The surface that we probe is typically a spot ~(100µm)² size, as a result of the laser focusing. Although this is a small size relatively to most comparable studies, this remains very large in comparison to the atom-surface distance. The defects in planarity, roughness, or parallelism of the surfaces, naturally affect the measurement of the distance-dependence that we are interested in. Moreover, it is known that various stray fields can severely affect the measurement of atom-surface interaction, notably for a dielectric surface. Indeed, local domains generate residual fields. This is why we have repeated each measurement performed on a spot of a given "nominal" (average) thickness (as determined by our 2-wavelength interferometric measurement) by measurements on other spots of a similar nominal thickness, either through a systematic density analysis, or at least in the most significant conditions (i.e. low density). We have found that for all experiments for d ≥ 65 nm or above, the spectra are identical within our accuracy. For d = 50 nm, some variations are observed from spot to spot, although some spots are able to provide identical spectra. These spectra only are retained, because any broadening or shift induced by a residual field would be strongly specific of the considered spot, not being susceptible to produce the same spectrum as another spot. For d = 40 nm, the vW shift is so considerable that minor changes (e.g. in the distribution of the local thickness over the spot size) can induce visible changes on the lineshape. However, the analysis of the lineshapes reveal that these minute changes have a negligible influence on the parameters that can be estimated in the fitting procedure. This reproducibility may appear unexpectedly good relatively to what can be inferred from other experimental results. Among the favorable points in our experiments, is the fact that the vW energy is pretty strong, owing to the investigated range of small distances, and to the relatively low excitation of the Cs (6D) (notably in comparison with experiments on Rydberg levels that are very sensitive to residual Stark shifts)

Discussion and prospects

Our experiments evidence the possibility to fit the lineshapes for the Cs(6P 3/2 )-Cs(6D 5/2 ) transition with lineshapes governed by a C 3 value ~ 14 (±3) kHz/µm 3 for a cell thickness comprised between 40 nm and 130 nm, and hence for a maximal atom-surface distance ranging from 20 nm to 65 nm. This corresponds to a surface-induced shift obeying a 1/z 3 law for an energy range spanning from ~200 MHz to 6 GHz, i.e. a factor ~30. Higher energies have been demonstrated to be observable [36,[START_REF] Dutier | Laser Spectroscopy, Proceedings of the XVI International Conference[END_REF], although not fully analyzed, and hence perhaps not to be attributed entirely to the surface interaction. Nevertheless, these experiments represent a bridge between the asymptotic limit of long distance Casimir-polder measurements, and the tail of a chemical potential. It shows also that in spite of a strong interacting potential, the atomic structure is still recognizable in spite of important shifts and broadening. Nevertheless, it is clear that in the explored range, with the observation of record energy shifts exceeding several GHz, mechanical effect should have considerable effects for the slowest atoms.

In addition to the experimental findings, the comparison with the theoretical prediction, detailed in [START_REF] Fichet | [END_REF], is remarkably good. The theoretical predictions is indeed evaluated to ~15 kHz./µm 3 [-i.e. 17 kHz./µm 3 for Cs(6D), and 2 kHz./µm 3 for Cs(6P)], once taken into account the dielectric coefficient, and a summing over the virtual transitions, assumed to be isotropically pumped. There are anyhow various pitfalls susceptible to affect the theoretical predictions: the modeling assumes the population to be thermal and spatially homogeneous, while it is here provided through a pumping step in a confined environment. The modeling also assumes an isotropic preparation of substates, allowing all orientations of the fluctuating atomic dipole to contribute [35]. The interaction between the highly-excited Cs atom, and the YAG windows, may include some resonant contributions, hard to determine in view of various uncertainties concerning the detailed spectrum of YAG surface resonances [START_REF] Saltiel | [END_REF].

Preliminary experiments on the Cs D 1 resonance line ( 894 nm) are now in progress. Their interpretation, and the possible comparison with theory, should be easier than for an interaction mostly affecting a highly-excited state. However, the additional difficulties are that shorter distances are needed to reach a comparable strength of the vW interaction. Also, the sensitivity is affected by the fact there is no possibility to modulate an external (pump) beam, so that the detection has to be operated with all the amplitude noise of the source in the low-frequency range. Preliminary measurements enable us to record reproducible spectra down to a 30 nm thickness -i.e. all atoms are at less than 15 nm from the wall-but the observed vW shift seems larger (by a factor up to 2) than expected.

Our measurements clearly show that reaching an effective atom-surface distance below 10 nm is a realistic objective. This contrasts with the measurements based upon the reflection of slow atoms [15,16] when the minimal distance of approach of the observed atomic trajectories is limited by the considerable force needed to compensate for the vW attraction. In the range of small distances that we explore here, we have evidenced an interaction energy largely exceeding the ones obtained for all previous investigations (up to ~5 GHz, or 0.25K). The corresponding acceleration is considerable (~ 8.10 7 g for a Cs(6D) atom 20 nm away from one of the wall), notably with respect to the one obtained in laser cooling techniques. Extrapolating to even smaller distance, one may even wonder if some kind of "atmospheric" gradient of density could not be found for an atomic gas in the extreme vicinity with a surface. The specificity of a spectroscopic measurement implies that we are essentially sensitive to a difference between the interaction undergone by the different states: on the one hand, this may make our measurements insensitive to an hypothetical non Newtonian gravity, on the other hand, this allows a simpler test of the validity of the van der Waals interaction at small distances, nearly-free of the influence of core transitions, as they are expected to be nearly independent of the excitation degree of the peripheral electron [START_REF] Bloch | Atom-wall interaction[END_REF]. At last, our measurements on excited states could be pushed up to the situation of resonantly coupled excited states, with atoms confined between two walls by a repulsive potential [36].

Figure 1 :

 1 Figure 1 : Theoretical transmission lineshapes for a Doppler width ku = 250 MHz : (a) for a 80 nm thickness and various C 3 values, (b) for C 3 = 14 kHz.µm 3 and indicated thicknesses.

Figure 2 :

 2 Figure 2 : principal scheme of the experimental set-up

Figure 3 :

 3 Figure 3 : Transmission and reflection spectra recorded in a 50 nm nanocell, with a discreetly tunable laser 917 nm laser. The full lines are a theoretical fitting. The vertical dashed line is a frequency reference as determined by an auxiliary saturated absorption (volume) reference

Figure 4 :

 4 Figure 4 : optimal fittings C 3 values for various thicknesses, and in differing pressure conditions

Figure 5 :

 5 Figure 5 : Reflection and Transmission 917nm spectra recorded in a Cs nanocell heated-up at 200°C for different local thickness. The dotted lines are a fitting obtained with a single C 3 value 14 kHz.µm 3 , and a single width for the optical transition 300 MHz. The vertical dashed line is a reference frequency for the volume transition

Acknowledgments

Work supported by the European Union through the research training network FASTnet (contract HPRN-CT-2002-00304), and by the French-Bulgarian programme of co-operation (RILA 09813UK)


P Fraundorf

Stephen Wedekind

Taylor Savage

David Osborn

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Although general multi-slice calculations remain too slow, web-browsers on many platforms now make possible real-time single-slice (strong phase/amplitude object) simulation, with live image, diffraction, image power-spectrum, and darkfield-image modes, including specimen rotation e.g. for atomic-resolution images with specimens having several tens of thousands of atoms. Moreover, a wide range of qualitative phenomena emerge that include diffraction-contrast effects associated with thickness, orientation changes, and defect strain. Hence students with no math background can get a visceral feel for the way 2-D lattice-projections, diffraction-patterns, image power-spectra, aperture size/position, and darkfield images relate to a specimen's structure & orientation, as well as microscope contrast-transfer, well before access to a real electron microscope is available.

CONTENTS

I. The problem 1

II. The strategy 1

III. Sample implementation 1

References 2

I. THE PROBLEM

Artifacts from specimen-preparation aside, transmission electron-microscopy (TEM) involves the effective usage of electron-optics plus an understanding of beamspecimen interactions as well as insight into the structure of your particular specimen. Computer support for these characterization tasks is also evolving rapidly. However instrument-time for both operator-training and interfacedevelopment is limited and quite expensive.

II. THE STRATEGY

Here we discuss the potential of strong phase/amplitude-object simulations, with help from JavaScript (JS) on modern (HyperText Markup Language or HTML version 5) devices, to serve up an online, mobile-friendly platform for TEM operator-training and analytical TEM software-interface development. The simulator that we make available has already been put to use with students in two NanoScience Practicals classes, and makes available real-time digital-darkfield tools useful with both deBroglie phase-contrast (e.g. high-resolution TEM) and incoherent amplitude-contrast (e.g. high-angle annular-darkfield scanning TEM or HAADF-STEM) lattice-images.

A surprising range of contrast-mechanisms (like thickness-fringes) manifest themselves (phenomenologically at least) without the need for slower multi-slice calculations. Microscopy-community ideas for new "un-knowns" to add, as well as new analytical-features for the software interface, are invited in this context.

III. SAMPLE IMPLEMENTATION

Fig. 1 is screenshot of two live-captures, with phasecontrast brightfield at left and amplitude-contrast darkfield at right, of an "unknown" in our online simulator 1 . The electron-diffraction pattern in the top-right panel of each capture is found "upstream" in the electron wavefield as (to first order) the log-intensity power-spectrum of the specimen's projected-potential. Microscopists might think of it as a parallel-illumination selected-area diffraction-pattern of the region of the specimen in the field of view.

The centerpiece of the simulator (from which later panels are obtained) is the electron-optical image in the topleft panel, which is recorded "downstream" of the microscope's imaging lenses. It is affected by the microscope's damping-envelope and spherical-aberration coefficient (presently fixed), as well as by lens focus-settings 2 which are adjustable with buttons above the top row of panels.

When the page is loaded, the focus setting of the system is randomized (in the same way for all "unknowns"), so that a (one-time) exercise in correcting astigmatism and optimizing focus may be needed. All panels of course are affected by specimen orientation, which along with unknown choice and darkfield aperture size are selected with buttons below the top row of panels.

The panel at bottom-right is a (log-intensity) powerspectrum of the electron-optical image at top-left. In this panel the spherical-aberration zeros in the contrasttransfer function are useful for focus and astigmatism correction, as well as for understanding lens effects on the images. The panel also contains the moveable digitaldarkfield aperture, which behaves like a physical aperture in the diffraction pattern. The resulting digitaldarkfield image 3 is shown in logarithmic complex-color in the bottom-left panel.

P

  . Fraundorf, * Stephen Wedekind, Taylor Savage, and David Osborn Physics & Astronomy/Center for NanoScience, U. Missouri-StL (63121) (Dated: September 8, 2016)

FIG. 1 .

 1 FIG.1. Brightfield high-resolution TEM (phase) and darkfield scanning-TEM (amplitude) screenshots of an "unknown" specimen in an online simulator, with the same defocus, orientation, and darkfield-aperture.


Florian Bouguet

A Probabilistic Look at Growth-Fragmentation Equations

Keywords: growth-fragmentation, Markov process, stationary measure, tails of distribution, Foster-Lyapunov criterion. MSC 2010: 60J25, 60B10, 45K05, 92D25

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Introduction

In this work, we consider the growth and fragmentation of a population of microorganisms (typically, bacteria or cells) through a structured quantity x which rules the division. For instance, one can consider x to be the size of a bacterium. The bacteria grow and, from time to time, split into two daughters. This behavior leads to an integro-dierential equation, which can also model numerous phenomena involving fragmentation, like polymerization, network congestions or neurosciences. In the context of a dividing population, we refer to [Per07, Chapter 4] for background and biological motivations, and to [Mic06, DJG10] for motivations in determining the eigenelements of the equation, which correspond to the Malthusian parameter of the population (see [Per07]). Regardless, if we denote by u(t, x) the concentration of individuals of size x at time t, such dynamics lead to the following growth-fragmentation equation:

∂ t u(t, x) + ∂ x [τ (x)u(t, x)] + β(x)u(t, x) = 2 ∞ x
β(y)κ(x, y)u(t, y)dy,

(1.1) for x, t > 0, where τ and β are the respective growth rate and fragmentation rate of the population, and κ is the fragmentation kernel (here we adopt the notation of [CDG12]).

The evolution of this population, or rather its probabilistic counterpart, has also been widely studied for particular growth and division rates. In the context of network congestions, this is known as the TCP window size process, which received a lot of attention recently (see [LvL08, CMP10, BCG + 13b, ABG + 14]). Let us provide the probabilistic interpretation of this mechanism. Consider a bacterium of size X, which grows at rate τ and randomly splits at rate β following a kernel κ, as before. We shall denote by Q(x, dy) := xκ(yx, x)dy to deal with the relative size of the daughters compared to the mother's, so that x 0 f (y)κ(y, x)dy = 1 0 f (xy)Q(x, dy).

We shall naturally assume that, for any x > 0, x 0 κ(y, x)dy = 1 0 Q(x, dy) = 1.

If we dismiss one of the two daughters and carry on the study only with the other one, the growth and fragmentation of the population can also be modeled by a Piecewise deterministic Markov process (PDMP) (X t ) t≥0 with càdlàg trajectories a.s. The dynamics of X are ruled by its innitesimal generator, dened for any function f in its domain D(L):

Lf (x) := τ (x)f (x) + β(x) 1 0 [f (xy) -f (x)]Q(x, dy).
(1.2)

We shall call (X t ) t≥0 a growth-fragmentation process. It is a Feller process, and we denote by (P t ) t≥0 its semigroup (for reminders about Feller processes or PDMPs, see [START_REF] Ethier | Markov processes[END_REF]Dav93]). If we denote by µ t = L (X t ) the probability law of X t , the Kolmogorov's forward equation

∂ t (P t f ) = LP t f is the weak formulation of ∂ t µ t = L µ t ,
where L is the adjoint operator of L in L 2 (L) where L stands for the Lebesgue measure. Now, if µ t admits a density u(t, •) with respect to L, then the equation above writes

∂ t u(t, x) = L u(t, x) = -∂ x [τ (x)u(t, x)] -β(x)u(t, x) + ∞ x β(y)κ(x, y)u(t, y)dy. (1.3) Note that (1.
3) is the conservative version of (1.1), since for any t ≥ 0, ∞ 0 u(t, x)dx = 1, which comes from the fact that there is only one bacterium at a time.

Remark 1.1 (Link with biology): Working with the probabilistic version of the problem allows us not to require the absolute continuity of µ t nor Q(x, •). This is useful since many biological models set Q(x, •) = δ 1/2 (equal mitosis) or Q(x, •) = U ([0, 1]) (uniform mitosis). Note that biological models usually assume that 1 0 yQ(x, dy) = 1/2, so that the mass of the mother is conserved after the fragmentation, which is automatically satised for a symmetric division, but we do not require this hypothesis in our study. Without going into further details, we highlight that it is possible to study both daughters with a structure of random tree, as in [START_REF] Bansaye | Limit theorems for Markov processes indexed by continuous time Galton-Watson trees[END_REF]DHKR15], the latter also drawing a bridge between the stochastic and deterministic viewpoints. ♦

In the articles [DJG10, CDG12, BCG13a], the authors investigate the behavior of the rst eigenvalue and eigenfunction of (1.1), with a focus on the dependence on the growth rate τ and the division rate β. Although it has been previously done for specic rates, they work in the setting of general functions τ and β. The aim of this paper is to provide a probabilistic counterpart to the aforementioned articles, by studying the Markov process (X t ) t≥0 , and to explain the assumptions for the well-posedness of the problem. We provide a probabilistic justication to the links between the growth and fragmentation rates, with the help of the renowned Foster-Lyapunov criterion. We shall also study the tails of distribution of the stationary measure of the process when it exists. We will see that, although the assumptions are similar, there is a dierence between the tails of the stationary distribution in the conservative case and in the non-conservative case. This paper is organized as follows: in Section 2, we study the Harris recurrence of X as well as the existence and uniqueness of its stationary distribution π, and we compare our conditions to those of [CDG12]. In Section 3, we study the moments of π and obtain precise upper bounds for its tails of distribution in the neighborhoods of both 0 and +∞.

To investigate the assumptions used in [CDG12], we turn to the study of the Markov process generated by (1.2). More precisely, we will provide a justication to the balance between τ and β with the help of a Foster-Lyapunov criterion. Note that we shall not require the fragmentation kernel Q(x, dy) to admit a density with respect to the Lebesgue measure L(dy). Moreover, in order to be as general as possible, we do not stick to the biological framework and thus do not assume that 1 0 Q(x, dy) = 1/2, which will be (technically) replaced by Assumption 2.2.i) below.

We start by stating general assumptions on the growth and fragmentation rates.

Assumption 2.1 (Behavior of τ and β). Assume that:

i) The functions β and τ are continuous, and τ is locally Lipschitz. ii) For any x > 0, β(x), τ (x) > 0.

iii) There exist constants γ 0 , γ ∞ , ν 0 , ν ∞ and β 0 , β ∞ , τ 0 , τ ∞ > 0 such that

β(x) ∼ x→0 β 0 x γ0 , β(x) ∼ x→∞ β ∞ x γ∞ , τ (x) ∼ x→0 τ 0 x ν0 , τ (x) ∼ x→∞ τ ∞ x ν∞ .
Note

= |γ ∞ | or µ = |ν ∞ |, and r 0 = |ν 0 | therein).
The following assumption concerns the expected behavior of the fragmentation, and is easy to check in most cases, especially if Q(x, •) does not depend on x. For any a ∈ R, we dene the moment of order

a of Q(x, •) by M x (a) := 1 0 y a Q(x, dy), M (a) := sup x>0 M x (a).
Assumption 2.2 (Moments of Q). Assume that: i) There exists a > 0 such that M (a) < 1.

ii) There exists b > 0 such that M (-b) < +∞.

Note that, in particular, Assumption 2.2 implies that, for any x > 0,

Q(x, {1}) = Q(x, {0}) = 0.
Let us make another assumption, concerning the balance between the growth rate and the fragmentation rate in the neighborhoods of 0 and +∞, which is fundamental to obtain an interesting Markov process.

Assumption 2.3 (Balance of β and τ ). Assume that

γ 0 > ν 0 -1, γ ∞ > ν ∞ -1.
Let us mention that Assumptions 2.1.iii) and 2.3, could be replaced by integrability conditions in the neighborhoods of 0 or +∞, see Assumptions (2.21) and (2.22) in [CDG12]. However, we make those hypotheses for the sake of simplicity, and for easier comparisons of our results to [CDG12].

Remark 2.4 (The critical case): This remark concerns the whole paper, and may be omitted at rst reading. Throughout Section 2, we can weaken Assumption 2.3 with the following: i) Either

γ 0 > ν 0 -1, or γ 0 = ν 0 -1 and b M (-b) -1 < β 0 τ 0 . (2.1) ii) Either γ ∞ > ν ∞ -1 or γ ∞ = ν ∞ -1 and a 1 -M (a) < τ ∞ β ∞ . (2.2)
Indeed, a careful reading of the proof of Theorem 2.5 shows that computations are similar, and the only change lies in the coecients in (2.9) and (2.10), which are still negative under (2.1) and (2.2). This corresponds to the critical case of the growth-fragmentation equations (see for instance [BW16, DE16]).

However, the behavior of the tail of the stationary distribution changes radically in the critical case. As a consequence, Section 3 is written in the framework of Assumption 2.3 only. Indeed, it is crucial to be able to choose a as large as possible (which is ensured in Assumption 3.3), so that π admits moments of any order. This is not possible under (2.2), since then

lim a→+∞ a 1 -M (a)
= +∞, so that the Foster-Lyapunov criterion does not apply and we expect the stationary measure to have heavy tails. ♦ Dene V as a smooth, convex function on (0, ∞) dened by

V (x) = x -b if x ∈ (0, 1], x a if x ∈ [2, ∞). (2.3)
We can now state the main result of this article.

Theorem 2.5 (Recurrence of growth-fragmentation processes). Let X be the PDMP generated by (1.2). If Assumptions 2.1, 2.2 and 2.3 are in force, then X is irreducible and aperiodic, compact sets are petite for X, and the process possesses a unique (up to a multiplicative constant) stationary measure π.

Moreover, if b ≥ ν 0 -1, a ≥ -γ ∞ ,
then X is positive Harris recurrent and π is a probability measure. Furthermore, if

ν 0 ≤ 1, γ ∞ ≥ 0, then X is exponentially ergodic in (1 + V )-norm.
Note that counterexamples of existence of the stationary measure are provided in [DJG10], when β is constant and τ is ane. Before proving Theorem 2.5, let us shortly present the Foster-Lyapunov criterion, which is the main tool for our proof (the interested reader may nd deeper insights in [MT93a] or [MT93b]). The idea is to nd a so-called Lyapunov function V controlling the excursions of X out of petite sets. Recall that a set K ⊆ R + is petite if there exists a probability distribution A over R + and some non-trivial positive measure ν over R + such that, for any x ∈ K, ∞ 0 δ x P t A (dt) ≥ ν. We produce here three criteria, adapted from [MT93b, Theorems 3.2, 4.2 and 6.1], which provide stronger and stronger results. Recall that, for some norm-like fonction V , we dene the V -norm of a probability measure µ by

µ V := sup |f |≤V |µ(f )| = sup |f |≤V f dµ .
Theorem 2.6 (Foster-Lyapunov criterion). Let X be a Markov process with càdlàg trajectories a.s. Let V ≥ 1 be a continuous norm-like real-valued function. Assume that compact sets of (0, +∞) are petite for X.

i) If there exist a compact set K and a positive constant α such that LV ≤ α 1 K , then X is Harris recurrent and possesses a unique stationary measure π.

ii) Moreover, if there exist a function f ≥ 1 and a positive constant α such that

LV ≤ -αf + α 1 K ,
then X is positive Harris recurrent, π is a probability measure and π(f ) < +∞.

iii) Moreover, if f ≥ V , then X is ergodic and there exist C, v > 0 such that

µ t -π 1+V ≤ C(1 + µ 0 (V ))e -vt .
Note that the exponential rate v provided in Theorem 2.6 is not explicit; if one wants to obtain quantitative speeds of convergence, it is often useful to turn to ad hoc coupling methods (see [BCG + 13b] for instance). Also, note that Assumption 2.2 is sucient but not necessary to derive ergodicity from a Foster-Lyapunov criterion, since we only need the limits in (2.9) and (2.10) to be negative. Namely, we only ask the fragmentation kernel Q(x, •) to be not too close to 0 and 1, uniformly over x.

Remark 2.7 (Construction of V ): If we are able to prove a Foster-Lyapunov criterion with a normlike function V , we want to choose V as explosive as possible (i.e. such that V (x) goes quickly to +∞ when x → 0 or x → +∞) to obtain better bounds for the tail of π, since π(V ) is nite: this is the purpose of Section 3. If we dene V with (2.3), this choice brings us to choose a and b as large as possible in Assumption 2.2. However, the larger a and b, the slower the convergence (because of the term µ 0 (V )), so there is a balance to nd here.

For many particular growth-fragmentation processes, it is possible to build a Lyapunov function of the form x → e θx , so that π admits exponential moments up to θ. We shall use a similar function in Section 3 to obtain bounds for the tails of the stationary distribution. ♦

Proof of Theorem 2.5: We denote by ϕ z the unique maximal solution of ∂ t y(t) = τ (y(t)) with initial condition z, and let a, b > 0 be as in Assumption 2.2. Firstly, we prove that compact sets are petite for

(X t ) t≥0 . Let z 2 > z 1 > z 0 > 0 and z ∈ [z 0 , z 1 ]. Since τ > 0 on [z 0 , z 2 ], the function ϕ z is a dieomorphism from [0, ϕ -1 z (z 2 )] to [z, z 2 ]; let t = ϕ -1 z0 (z 2
) be the maximum time for the ow to reach z 2 from [z 0 , z 1 ]. Denote by X z the process generated by (1.2) such that L (X 0 ) = δ z , and T z n the epoch of its

n th jump. Let A = U ([0, t]). For any x ∈ [z 1 , z 2 ], we have ∞ 0 P(X z s ≤ x)A (ds) ≥ 1 t t 0 P(X z s ≤ x|T z 1 > ϕ -1 z (z 2 ))P(T z 1 > ϕ -1 z (z 2 ))ds ≥ P(T z 1 > ϕ -1 z (z 2 )) t t 0 P(ϕ z (s) ≤ x)ds ≥ P(T z 1 > ϕ -1 z (z 2 )) t ϕ -1 z (x) 0 ds ≥ P(T z 1 > ϕ -1 z (z 2 )) t x z (ϕ -1 z ) (u)du.
(2.4) Since β and τ are bounded on [z 0 , z 2 ], the following inequalities hold:

P(T z 1 > ϕ -1 z (z 2 )) = exp - ϕ -1 z (z2) 0 β(ϕ z (s))ds = exp - z2 z β(u)(ϕ -1 z ) (u)du ≥ exp -(z 2 -z 0 ) sup [z0,z2] β(ϕ -1 z ) ≥ exp -(z 2 -z 0 ) sup [z0,z2] β inf [z0,z2] τ -1 , since inf [z0,z2] (ϕ -1 z ) = sup [z0,z2] τ -1
. Hence, there exists a constant C such that, (2.4) writes, for

x ∈ [z 1 , z 2 ], ∞ 0 P(X z s ≤ x)A (ds) ≥ C(x -z 1 ),
which is also

∞ 0 δ z P s A (ds) ≥ CL [z1,z2] ,
where L K is the Lebesgue measure restricted to a borelian set K. Hence, by denition, [z 0 , z 1 ] is a petite set for the process X.

Now, let us show that the process

(X t ) is L (0,∞) -irreducible with similar arguments. Let z 1 > z 0 > 0 and z > 0. If z ≤ z 0 , E ∞ 0 1 {z0≤X z t ≤z1} dt ≥ P(T z 1 > ϕ -1 z (z 1 ))E ∞ 0 1 {z0≤X z t ≤z1} dt T z 1 > ϕ -1 z (z 1 ) ≥ exp -(z 1 -z 0 ) sup [z0,z1] β inf [z0,z1] τ -1
ϕ -1 z0 (z 1 ).

(2.5)

If z > z 0 , for any t 0 > 0 and n ∈ N, the process X z has a positive probability of jumping n times before time t 0 . Recall that 1 0 y a Q(x, dy) ≤ M (a) < 1. For any n > (log(z) -log(z 0 )) log(M (a) -1 ) -1 , let 0 < ε < z a 0 -(zM (a) n ) a . By continuity of (x, t) → ϕ x (t), there exists t 0 > 0 small enough such that,

∀(x, t) ∈ [0, z] × [0, t 0 ], ϕ x (t) a ≤ x a + ε n + 1 , E[(X z t0 ) a |T z n ≤ t 0 ] ≤ (zM (a) n ) a + ε < z a 0 .
Then, using Markov's inequality

P(X z t0 ≤ z 0 |T z n ≤ t 0 < T z n+1 ) ≥ 1 - E[(X z t0 ) a |T z n ≤ t 0 < T z n+1 ] z a 0 > 0.
Then, P(X z t0 ≤ z 0 ) > 0 for any t 0 small enough, and, using (2.5)

E ∞ 0 1 {z0≤X z t ≤z1} dt ≥ E ∞ t0 1 {z0≤X z t ≤z1} dt X z t0 ≤ z 0 P(X z t0 ≤ z 0 ) ≥ exp -(z 1 -z 0 ) sup [z0,z1] β inf [z0,z1] τ -1 ϕ -1 z0 (z 1 )P(X z t0 ≤ z 0 ) > 0.
Aperiodicity is easily proven with similar arguments.

We turn to the proof of the Lyapunov condition. For x ≥ 2, V (x) = x a and

LV (x) = a τ (x) x V (x) + β(x) 1 0 V (xy)Q(x, dy) -β(x)V (x) ≤ a τ (x) x -β(x) V (x) + β(x) 1/x 0 (xy) -b Q(x, dy) + β(x) 2/x 1/x 2 a Q(x, dy) + β(x) 1 2/x (xy) a Q(x, dy) ≤ a τ (x) x -β(x) V (x) + β(x) x -b M x (-b) + 2 a + x a M x (a) ≤ a τ (x) x -β(x) 1 -M x (a) - M x (-b) x b V (x) - 2 a V (x) V (x).
(2.6)

For x ≤ 1, V (x) = x -b and LV (x) = -b τ (x) x + β(x)(M x (-b) -1) V (x).
(2.7)

Combining γ ∞ > ν ∞ -1 with Assumption 2.2.i), for x large enough we have

a τ (x) x -β(x) 1 -M x (a) - M x (-b) x b V (x) - 2 a xV (x) ≤ a τ (x) x -β(x) (1 -M (a) + o(1)) ≤ 0. Likewise, combining γ 0 > ν 0 -1 with Assumption 2.2.ii), -b τ (x) x + β(x)(M x (-b) -1) ≤ -b τ (x) x + β(x) (M (-b) -1) ≤ 0
for x close enough to 0. Then, Theorem 2.6.i) entails that X is Harris recurrent, thus admits a unique stationary measure (see for instance [START_REF] Kaspi | On Harris recurrence in continuous time[END_REF]).

Note that (2.6) writes

LV (x) ≤ -β ∞ (1 -M (a) + o(1))x a+γ∞ ,
so that, if we can choose a ≥ -γ ∞ , then

lim x→∞ -β ∞ (1 -M (a) + o(1))x a+γ∞ < 0.
Likewise, (2.7) writes

LV (x) ≤ -(bτ 0 + o(1))x ν0-1-b , (2.8) so, if b ≥ ν 0 -1, we get lim x→0 -(bτ 0 + o(1))x ν0-1-b < 0.
Then, there exist positive constants A, α, α

LV ≤ -αf + α 1 [1/A,A] ,
where f ≥ 1 is a smooth function, such that f (x) = x ν0-1-b for x close to 0, and f (x) = x a+γ∞ for x large enough. Then, Theorem 2.6.ii) ensures positive Harris recurrence for X.

Now, if we assume γ ∞ ≥ 0 and ν 0 ≤ 1 in addition, then there exists α > 0 such that

lim x→+∞ a τ (x) x -β(x) 1 -M x (a) - M x (-b) x b V (x) - 2 a xV (x) ≤ lim x→+∞ a τ (x) x -β(x) (1 -M (a) + o(1)) ≤ -α, (2.9) and lim x→0 -b τ (x) x + β(x)(M x (-b) -1) ≤ lim x→0 -b τ (x) x + β(x) (M (-b) -1) ≤ -α.
(2.10) Combining (2.6) and (2.7) with (2.9) and (2.10) respectively, and since V is bounded on [1, 2], there exist positive constants A, α such that LV ≤ -αV

+ α 1 [1/A,A] .
The function V is thus a Lyapunov function, for which Theorem 2.6.iii) entails exponential ergodicity for X.

Tails of the Stationary Distribution

In this section, we use, and reinforce when necessary, the results of Theorem 2.5 to study the asymptotic behavior of the tails of distribution of the stationary measure π. We will naturally divide this section into two parts, to study the behavior of π(dx) as x → 0 and as x → +∞. Hence, throughout this section, we shall assume that X satises Assumptions 2.1, 2.2 and 2.3. The key point is to use the fact that π(f ) < +∞ provided in the second part of Theorem 2.6. We recall that L stands for the Lebesgue measure on R.

In order to compare our results to those of [BCG13a, Theorem 1.8], we consider the same framework and make the following assumption: Assumption 3.1 (Density of Q and π). Assume that: i) For any x > 0, Q(x, •) L and Q(x, dy) = q(y)dy, and there exist constants q 0 , q 1 ≥ 0 and µ 0 , µ 1 > -1 such that

q(x) = x→0 q 0 x µ0 + o(x µ0 ), q(x) = x→1 q 1 (1 -x) µ1 + o((1 -x) µ1 ).
ii) π L and π(dx) = G(x)dx, and there exist constants

G 0 , G ∞ , G ∞ > 0 and α 0 , α ∞ , α ∞ ∈ R such that G(x) ∼ x→0 G 0 x α0 , G(x) ∼ x→+∞ G ∞ x α∞ exp -G ∞ x α∞ .
We do not require the coecients q 0 , q 1 to be (strictly) positive, so that this assumption can also cover the case Q(x, dy) = δ r (dy) for 0 < r < 1, which is widely used for modeling physical or biological situations. For the sake of simplicity, the hypotheses concerning the density of π (resp. Q) in the neighborhood of both 0 and +∞ (resp. 0 and 1) are gathered in Assumption 3.1, but it is clear that we only need either the assumption on the left behavior or on the right behavior to precise the fractional moments or the exponential moments of the stationary distribution. In the same spirit, we could weaken

Q(x, •) L into Q(x,
•) admitting a density with respect to L only in the neighborhoods of 0 and 1, bounded above by q.

Theorem 3.2. Let X be the PDMP generated by (1.2). If Assumptions 2.1, 2.2 and 2.3 hold, and if

b ≥ ν 0 -1, (3.1) then 1 0 x ν0-1-b π(dx) < +∞.
Moreover, if Assumption 3.1 holds and µ 0 + 2 -ν 0 > 0, then

α 0 ≥ µ 0 + 1 -ν 0 .
Proof: The rst part of the theorem is a straightforward consequence of (2.8).

Combining Assumption 2.2.ii) with Assumption 3.1.i), we naturally have to take b < µ 0 + 1. Thus, for any ε ∈ (0, µ 0 + 1), we take b = µ 0 + 1 -ε. Dene V with (2.3) as before, so that, for x ≤ 1, V (x) = x -µ0-1+ε and

LV (x) ≤ -b τ (x) x + β(x)(M (-b) -1) V (x) ∼ x→0 -bτ 0 x ν0-µ0-2+ε .
Applying Theorem 2.6.ii) with f (x) = x ν0-µ0-2+ε , which tends to +∞ when µ 0 + 1 -ν 0 > -1, we have π(f ) < +∞ so

1 0 x α0+ν0-µ0-2+ε dx < +∞, α 0 > 1 + µ 0 -ν 0 + ε, for any ε > 0. Thus α 0 ≥ µ 0 + 1 -ν 0 .
Now, we turn to the study of the tail of distribution of π(dx) as x → +∞. Since choosing a polynomial function as a Lyapunov function can only provide the existence of moments for π, we need to introduce a more coercive function to study in detail the behavior of its tail of distribution and get the existence of exponential moments. We begin with the following assumption.

Assumption 3.3. Let θ = γ ∞ + 1 -ν ∞ . Assume there exists 0 < C < 1 such that, for any ε > 0 and 0 < η < β ∞ (θτ ∞ ) -1 , there exists x 0 > 0 such that

sup x≥x0 1 0 y -ε exp ηx θ (y θ -1) Q(x, dy) < 1 -C.
It is easy to understand this assumption if

V (x) = x -ε e ηx θ (3.2)
and if L (Y (x) ) = Q(x, .); then, Assumption 3.3 rewrites

sup x≥x0 E[ V (xY (x) )] V (x) ≤ 1 -C.
Once again, this is asking the fragmentation kernel to be not too close to 1. As we will see, this is quite natural when Q has a regular behavior around 0 and 1.

Proposition 3.4. Assumption 3.3 holds for any C ∈ (0, 1) whenever Assumption 3.1.i) holds.

Proof: Dene V as in (3.2) for x ≥ 1, larger than 1, and increasing and smooth on R. For any (large)

x > 0, for any (small) δ > 0,

E[ V (xY (x) )] = E[ V (xY (x) )|Y (x) ≤ 1 -δ]P(Y (x) ≤ 1 -δ) + E V (xY (x) )1 {Y (x) >1-δ} ≤ V ((1 -δ)x) + E V (xY (x) )1 {Y (x) >1-δ} . (3.3) It is clear that, for δ < 1, lim x→+∞ V ((1 -δ)x) V (x) = lim x→+∞ (1 -δ) -ε exp -η(1 -(1 -δ) θ )x θ = 0.
On the other hand, using Hölder's inequality with q > max(1, -1/µ 1 ) and p -1 + q -1 = 1, as well as a Taylor expansion, there exists some constant C δ ≥ 1 such that

1 1-δ V (xy)q 1 (1 -y) µ1 dy = q 1 V (x) δ 0 exp ηx θ ((1 -y) θ -1) y µ1 (1 -y) -ε dy ≤ q 1 (1 -δ) δ 0 y qµ1 dy 1/q V (x) δ 0 exp ηpx θ ((1 -y) θ -1) dy 1/p ≤ C δ V (x) δ 0 exp -ηpθx θ y dy 1/p ≤ C δ V (x) 1 -exp -ηpθx θ δ ηpθx θ 1/p .
The term (ηpθx θ ) -1 (1 -exp -ηpθx θ δ ) converges to 0 as x → +∞, so that there exists x 0 > 0 such that, for any x ≥ x 0 and C ∈ (0, 1),

1 -exp -ηpθx θ δ ηpθx θ 1/p ≤ 1 -C 2C δ , V ((1 -δ)x) V (x) ≤ 1 -C 2 .
Plugging these bounds into (3.3) achieves the proof. Now, we can characterize the weight of the asymptotic tail of π and recover [BCG13a, Theorem 1.7].

Theorem 3.5. Let X be the PDMP generated by (1.2). If Assumptions 2.1, 2.2, 2.3 and 3.3 hold, then

+∞ 1 x ν∞-1-ε exp ηx θ π(dx) < +∞, θ = γ ∞ + 1 -ν ∞ , η = Cβ ∞ θτ ∞ , ε > 0.
Moreover, if Assumption 3.1 is also in force, then either:

• α ∞ > γ ∞ + 1 -ν ∞ ; • α ∞ = γ ∞ + 1 -ν ∞ and G ∞ > Cβ ∞ ((γ ∞ + 1 -ν ∞ )τ ∞ ) -1 ; • α ∞ = γ ∞ + 1 -ν ∞ , G ∞ = Cβ ∞ ((γ ∞ + 1 -ν ∞ )τ ∞ ) -1 and α ∞ ≥ -ν ∞ .
Remark 3.6 (Link with the estimates of [BCG13a]): Note that the hypotheses (2.3) and (3.1) corresponds to the assumptions required for [BCG13a, Theorem 1.8] to hold, with the correspondence

µ 0 ↔ µ -1, ν 0 ↔ α 0 , µ 0 + 2 -ν 0 > 0 ↔ µ + 1 -α 0 > 0.
Actually, the authors also assume this strict inequality to prove the existence of the stationary distribution, which we relax here, and we need it only in Theorem 3.1 to provide a lower bound for α 0 , which rules the tail of the stationary distribution in the neighborhood of 0. By using Lyapunov methods, there is no hope in providing an upper bound for α 0 , but we can see that this inequality is optimal by comparing it to [BCG13a, Theorem 1.8] so that, in fact,

α 0 = µ 0 + 1 -ν 0 .
If ν 0 > 1, we do not recover the same equivalents for the distribution of G around 0. This is linked to the fact that there is a phase transition in the non-conservative equation at ν 0 = 1, since the tail of G relies deeply on the function

Λ(x) = x 1 λ + β(y) τ (y) dy,
where λ is the Malthusian parameter of the equation, which is the growth of the proles of the integrodierential equation. However, we deal here with the conservative case, for which this parameter is null. The bounds that we provide are indeed consistent with the computations of the proof of [BCG13a, Theorem 1.8] in the case λ = 0.

Concerning the estimates as x → +∞, as mentioned above, we can not recover upper bounds, and then sharp estimates, for α ∞ , G ∞ , α ∞ with Foster-Lyapunov methods. From the proof of Theorem 3.5, it is clear that the parameters η and θ are optimal if one wants to apply Theorem 2.6. Under secondorder-type assumptions like [BCG13a, Hypothesis 1.5], it is clear that

Λ(x) = x 1 β(y) τ (y) dy ∼ x→+∞ β ∞ τ ∞ (γ ∞ + 1 -ν ∞ )
x γ∞+1-ν∞ .

This explains the precise value of η, but we pay the price of having slightly more general hypotheses about Q than [BCG13a] with a factor C arising from Assumption 3.3, which leads to have no disjunction of cases for α ∞ . Also, since we deal with the case λ = 0, the equivalent of the function Λ is dierent from the aforementioned paper when γ ∞ < 0, so that max{γ ∞ , 0} does not appear in our computations. ♦

Proof of Theorem 3.5: Let V as in (3.2), that is

V (x) = x -ε e ηx θ ,
with η, θ given in Theorem 3.5. Then, following the computations of the proof of Theorem 2.5, we get, for x > x 0 ,

L V (x) ≤ ηθτ ∞ x θ-1+ν∞ -Cβ ∞ x γ∞ -ετ ∞ x ν∞-1 (1 + o(1)) V (x) ≤ -ετ ∞ x ν∞-1 (1 + o(1)) V (x) ≤ - ετ ∞ 2 x ν∞-1 V (x).
Using Theorem 2.6.ii) with f (x) = x ν∞-1 V (x), the last inequality ensures that x ν∞-1-ε+α∞ exp ηx θ -G ∞ x α∞ dx < +∞.

(3.4)

It is clear then that the disjunction of cases of Theorem 3.5 is the only way for the integral on the right-hand side of (3.4) to be nite.

+∞ 1 f

 1 (x)π(dx) < +∞. Now, in the setting of Assumption 3.1, the following holds: +∞ 1 f (x)π(dx) < +∞ ⇐⇒ +∞ 1

Acknowledgements: The author wants to thank Pierre Gabriel for fruitful discussions about growthfragmentation equations, as well as Florent Malrieu, Jean-Christophe Breton and Eva Löcherbach for their precious help and comments. This work was nancially supported by the ANR PIECE (ANR-12-JS01-0006-01) and the Centre Henri Lebesgue (programme "Investissements d'avenir" ANR-11-LABX-0020-01).

Contact address: florian.bouguet@inria.fr.


René Schott
email: schott@loria.fr

George Stacey Staples
email: sstaple@siue.edu

Nilpotent Adjacency Matrices, Random Graphs, and Quantum Random Variables

Keywords: AMS subject classification: 60B99, 81P68, 05C38, 05C50, 05C80, 15A66 fermions, random graphs, cycles, paths, quantum computing

For fixed n > 0, the space of finite graphs on n vertices is canonically associated with an abelian, nilpotent-generated subalgebra of the 2n-particle fermion algebra. Using the generators of the subalgebra, an algebraic probability space of "nilpotent adjacency matrices" associated with finite graphs is defined. Each nilpotent adjacency matrix is a quantum random variable whose m th moment corresponds to the number of mcycles in the graph G. Each matrix admits a canonical "quantum decomposition" into a sum of three algebraic random variables: a = a ∆ +a Υ +a Λ , where a ∆ is classical while a Υ and a Λ are quantum. Moreover, within the algebraic context, the NP problem of cycle enumeration is reduced to matrix multiplication, requiring no more than n 4 multiplications within the algebra.

Introduction

Links between quantum probability and graph theory have been explored in a number of works. Hashimoto, Hora, and Obata (cf. [START_REF] Hashimoto | Central limit theorems for large graphs: Method of quantum decomposition[END_REF], [START_REF] Obata | Quantum probabilistic approach to spectral analysis of star graphs[END_REF]) obtained limit theorems for increasing sequences of graphs G n whose adjacency matrices admit a quantum decomposition A n = A n + + A n -. Examples include Cayley graphs, Johnson graphs, and distance-regular graphs.

Obata [START_REF] Obata | Quantum probabilistic approach to spectral analysis of star graphs[END_REF] uses this approach to focus on star graphs, which are obtained by gluing together the common origins of a finite number of copies of a given graph. The adjacency matrices of star graphs admit a quantum decomposition of the form

A n = A n + + A n -+ A n • .
Star graphs are of particular interest because they are related to Boolean independence in quantum probability.

Homogeneous trees are also of interest in quantum probability. These are related to the free independence of Voiculescu [START_REF] Voiculescu | Free Random Variables, CRM Monograph Series[END_REF].

Comb graphs, which provide models of Bose-Einstein condensation, are related to monotone independence discovered by Lu [START_REF] Lu | An interacting free Fock space and the arcsine law[END_REF] and Muraki [START_REF] Muraki | Noncommutative Brownian motion in monotone Fock space[END_REF]. Accardi, Ben Ghorbal, and Obata [START_REF] Accardi | Monotone independence, comb graphs and Bose-Einstein condensation[END_REF] computed the vacuum spectral distribution of the comb graph by decomposing the adjacency matrix into a sum of monotone independent random variables.

Another work of interest is that of Franz Lehner [START_REF] Lehner | Cumulants in noncommutative probability theory III. Creation and annihilation operators on Fock spaces[END_REF], who investigated the relationships among non-crossing partitions, creation and annihilation operators, and the cycle cover polynomial of a graph. In that work, the cycle indicator polynomials of particular digraphs are used to understand the partitioned moments and cumulants occurring in Fock spaces associated to characters of the infinite symmetric group of Bozejko and Gut ¸ȃ [START_REF] Bozejko | Functors of white noise associated to characters of the infinite symmetric group[END_REF].

In contrast to the works cited above, the philosophy of the current work is to begin with an arbitrary finite graph and then to construct an associated algebraic probability space in which the moments of random variables reveal information about the graph. The graph need possess no particular relationship to notions of independence or Fock spaces.

Letting n = |V |, the vertices of G can be represented by unit coordinate vectors in R n . When A is the adjacency matrix of G, a well-known result in graph theory states that x 0 , A k x 0 corresponds to the number of closed k-walks based at vertex x 0 ∈ V (G).

We are interested in the related problem of recovering the k-cycles based at any vertex x 0 . This can be done for any finite graph with the methods described herein.

The algebras used in this paper were originally derived as subalgebras of Clifford algebras. Clifford algebras and quantum logic gates have been discussed in works by W. Li [START_REF] Li | Clifford algebra and quantum logic gates[END_REF] and Vlasov [START_REF] Vlasov | Clifford algebras and universal sets of quantum gates[END_REF]. The role of Clifford algebras in quantum computing has been considered in works by Havel and Doran [START_REF] Havel | Geometric algebra in quantum information processing[END_REF], Matzke [START_REF] Matzke | Quantum computation using geometric algebra[END_REF], and others.

In the Cℓ context, i.e., in terms of the number of multiplications performed in the algebra, the computational complexity of enumerating the Hamiltonian cycles in a graph on n vertices is O(n 4 ). In this context, some graph problems are moved from complexity class NP into class P.

Graphs and Adjacency Matrices

A graph G = (V, E) is a collection of vertices V and a set E of unordered pairs of vertices called edges. A directed graph is a graph whose edges are ordered pairs of vertices. Two vertices v i , v j ∈ V are adjacent if there exists an edge e = {v i , v j } ∈ E. Adjacency will be denoted v i ∼ v j . A graph is finite if V and E are finite sets, that is, if |V | and |E| are finite numbers. A loop in a graph is an edge of the form (v, v). A graph is said to be simple if 1) it contains no loops, and 2) no unordered pair of vertices appears more than once in E.

A k-walk {v 0 , . . . , v k } in a graph G is a sequence of vertices in G with initial vertex v 0 and terminal vertex v k such that there exists an edge (v j , v j+1 ) ∈ E for each 0 ≤ j ≤ k -1. A k-walk contains k edges. A self-avoiding walk is a walk in which no vertex appears more than once. A closed k-walk is a k-walk whose initial vertex is also its terminal vertex. A k-cycle is a self-avoiding closed k-walk with the exception v 0 = v k . A Hamiltonian cycle is an n-cycle in a graph on n vertices; i.e., it contains V. An Euler circuit is a closed walk encompassing every edge in E exactly once.

When working with a finite graph G on n vertices, one often utilizes the adjacency matrix A associated with G. The adjacency matrix is defined by

A xy = 1 if x ∼ y, 0 otherwise. (1.1)
A graph in which multiple edges exist between a given pair of vertices is called a multigraph. If a vertex is allowed to be self-adjacent, the edge joining the vertex to itself is referred to as a loop. Graphs containing loops are known as pseudographs.

Adjacency matrices as operators on Hilbert space

Let G = (V, E) be a graph on n vertices. Let Γ = ∞ n=0 ⊕ CΦ n be the Hilbert space with complete orthonormal basis {Φ n }. Here Φ 0 represents the "vacuum vector." Identifying the vertices of G with the vectors {Φ k }, (1 ≤ k ≤ n), the adjacency matrix of G is seen to be a bounded linear operator on Γ.

When G is an undirected graph, its adjacency matrix A is symmetric, i.e., Hermitian. Bounded Hermitian operators are random variables in quantum probability theory.

More generally, one considers the Hilbert space ℓ 2 (V ) of all C-valued functions f on the vertex set of G satisfying

|f (x)| 2 < ∞. (1.2)
When G is a simple graph, it is well-known [START_REF] Hashimoto | Central limit theorems for large graphs: Method of quantum decomposition[END_REF] that the adjacency matrix of G acts on ℓ 2 (V ) according to

Af (x) = y∈V A xy f (y) = y∼x f (y). (1.3)
This notion can also be extended to multigraphs and pseudographs. In an undirected graph with multiple edges or loops, the edge multiplicity of a pair (x, y) of adjacent vertices is defined as the number of edges incident to both x and y. In this case, denoting the edge multiplicity of x ∼ y by ̟(x, y), the adjacency matrix of G acts on ℓ 2 (V ) according to

Af (x) = y∈V A xy f (y) = y∼x ̟(x, y)f (y).
(1.4)

The degree of a vertex x ∈ V , denoted κ(x), is defined for any undirected graph by κ(x) = y∼x ̟(x, y).

(1.5)

For simple graphs, the degree takes the simpler form

κ(x) = |{y ∈ V : y ∼ x}|. (1.6)
Related to the adjacency matrix is another operator called the combinatorial Laplacian, which is denoted by ∆. The combinatorial Laplacian associated with G is defined by its action on ℓ 2 (V ) according to

∆f (x) = κ(x)f (x) - y∼x ω(x, y)f (y) = κ(x)f (x) -A f (x).
(1.7)

The degree operator κ is defined by

κf (x) =   y∈V A xy   f (x) = κ(x)f (x). (1.8)
The combinatorial Laplacian can then be written as ∆ = κ -A.

An edge-weighted graph G = (V, E) is a graph together with a real-valued function ω : V × V → R. The adjacency matrix of G is then defined by

A xy = ω(x, y) if x ∼ y, 0 otherwise. (1.9)
In this case, the action of A on ℓ 2 (V ) is defined by replacing ̟ by ω in (1.2).

Quantum probability: operators as random variables

Let A be a complex algebra with involution * and unit 1. A positive linear * -functional on A satisfying ϕ(1) = 1 is called a state on A. The pair (A, ϕ) is called an algebraic probability space.

If A is commutative, the probability space is classical. If A is non-commutative, the probability space is quantum.

Elements of A are random variables. A stochastic process is a family (X t ) indexed by an arbitrary set T . Given a stochastic process X = (X t ), the polynomial * -algebra P(X) is a * -subalgebra of A. The restriction of ϕ to this * -subalgebra gives a state ϕ X on P(X) called the distribution of the process X.

If T = {1, 2, . . . , n} is a finite set, then ϕ X is called the joint distribution of the random variables {X 1 , . . . , X n }.
Of particular interest in quantum probability theory are interacting Fock spaces. Let λ 0 = 1, and let {λ i } 1≤i be a sequence of nonnegative real numbers

Type

Relation

Realized by Boson such that

[B -, B + ] = 1 λ n = n!. Fermion {B -, B + } = 1 λ 0 = λ 1 = 1, λ n = 0 for n ≥ 2. Free B -B + = 1 λ n = 1 for all n ≥ 0.
λ m = 0 ⇒ λ m+1 = λ m+2 = • • • = 0.
In the case that λ n > 0 for all n, one defines the linear operators B + and B -by

B + Φ n = λ n+1 λ n Φ n+1 , n ≥ 0, (1.10)
and

B -Φ n =    λ n λ n-1 Φ n-1 , n ≥ 1 0 n = 0.
(1.11)

On their natural domains, B + and B -are mutually adjoint and closed. Operators B + and B -are called the creation and annihilation operators, respectively. The number operator is defined by

N Φ n = nΦ n , n ≥ 0. (1.12)
Recalling the definitions of the commutator and anti-commutator, respectively,

[a, b] = ab -ba, (1.13) {a, b} = ab + ba, (1.14)
direct calculation shows

B + B -Φ 0 = B -B + Φ 0 = 0, (1.15)
{B + , B -} = λ n 2 + λ n+1 λ n-1 λ n λ n-1 , n ≥ 1, (1.16)
B + , B -= λ n 2 -λ n+1 λ n-1 λ n λ n-1 , n ≥ 1, (1.17)
B + n Φ 0 = λ n Φ n , n ≥ 0, (1.18)
B -n Φ n = λ n Φ 0 , n ≥ 0. (1.19)
Realizations of commutation relations are summarized in Figure 1.

When there exists some m ≥ 1 such that λ m > 0 but λ n = 0 for all n > m, one defines the finite dimensional Hilbert space

Γ = m n=0 ⊕ CΦ n .
The finite-dimensional operators B + and B -are then defined in the obvious way. In this case, B + Φ m = 0.

In either the finite-or infinite-dimensional case, the Hilbert space Γ(C, {λ n }) = (Γ, {λ n }, B + , B -) is called an interacting Fock space associated with {λ n }.

One can begin with a classical random variable and construct a stochastically equivalent quantum random variable. Given a probability measure µ having finite moments of all orders; that is,

R |x| m µ(dx) < ∞, m = 0, 1, 2, . . . ,
let {P n } be the associated orthogonal polynomials. The polynomials are assumed to be monic via an appropriate normalization. By orthogonality,

P 0 (x) = 1,
(1.20)

P 1 (x) = x -α 1 , (1.21) (x -α n+1 )P n (x) = P n+1 (x) + ω n P n-1 (x), n ≥ 1, (1.22)
(1.23)

where the numbers {α n , ω n } ∞ n=0 are called are the Szegö-Jacobi parameters of µ.

The book by Szegö [START_REF] Szegö | Orthogonal Polynomials[END_REF] provides a detailed treatise on orthogonal polynomials. The work of Asai, Kubo, and Kuo [START_REF] Asai | Generating function method for orthogonal polynomials and Jacobi-Szegö parameters, Quantum probability and infinite-dimensional analysis[END_REF] details a generating function method for deriving the orthogonal polynomials and Szegö-Jacobi parameters associated with a probability measure µ.

Letting Γ(C, {λ n }) be the interacting Fock space associated with λ 0 = 1, λ 1 = ω 1 , and λ n+1 λ n = ω n+1 , Accardi and Bozejko [START_REF] Accardi | Interacting Fock spaces and Gaussianization of probability measures[END_REF] proved the existence of an isometry U from Γ(C, {λ n }) into L 2 (R, µ) uniquely determined by

U Φ 0 = P 0 , U B + U * P n = P n+1 , and
Q = U (B + + B -+ αn+1 )U * ,
where Q is the multiplication operator by x densely defined in L 2 (R, µ) and αn+1 is the operator αn+1

= α n+1 n N, so that αn+1 Φ n = α n+1 Φ n . Then, R x m µ(dx) = Φ 0 , (B + + B -+ αn+1 ) m Φ 0 Γ , m = 0, 1, 2, . . . . ( 1

.24)

Let A denote the * -algebra generated by Q, and let X be a classical random variable with probability distribution µ having moments of all orders. Then, an algebraic probability space (A, ϕ) is defined with the state ϕ determined by ϕ(a) = P 0 , aP 0 L 2 (R,µ) , a ∈ A.

(1.25)

In light of the results of Accardi and Bozejko, the following relation becomes evident:

E(X m ) = R x m µ(dx) = ϕ(Q m ), m = 1, 2, . . . . (1.26)
In other words, X and Q are stochastically equivalent. The classical random variable X then admits a quantum decomposition of the form

B + + B -+ αn+1 .
That is, X admits a decomposition as a linear combination of creation, annihilation, and number operators.

Operators as adjacency matrices

Graphs can be interpreted as operators on Hilbert spaces. In quantum probability, bounded Hermitian operators on Hilbert spaces are quantum random variables. It is not surprising that quantum random variables can then be interpreted as graphs.

Given an interacting Fock space (Γ, {λ n }, B + , B -) associated with {λ n },the operator A = B + + B -+ N can also be interpreted as the adjacency matrix associated with an edge-weighted pseudograph. The associated finite graph is constructed with vertex set {Φ 0 , Φ 1 , . . . , Φ n }, edge weights {

λ k λ k-1 }, and k loops based at vertex Φ k .
In this context, the adjacency matrix A is the sum of the upper-triangular matrix B -, the lower triangular-matrix B + , and the diagonal matrix N . Visualizations of graphs associated with finite-dimensional realizations of the commutation relations from Figure 1 appear in Figures 2, 3, and4.

A Bernoulli random variable X admits an expression of the form

X = 0 1 0 0 + 0 0 1 0 , (1.27)
commonly referred to as "quantum coin-tossing." Labeling the matrices of the right-hand side of (1.27) as f -and f + , respectively, X is decomposed into a sum of upper-and lower-triangular nilpotent matrices. It is apparent that by defining Φ 0 = 1 0 and Φ 1 = 0 1 , one obtains

f + Φ i = Φ 1 if i=0 0 otherwise, f -Φ i = Φ 0 if i=1 0 otherwise. (1.28)
The corresponding number operator has the form

f N = 0 0 0 1 . ( 1

.29)

The matrix A = f + + f -+ f N is the adjacency matrix of the pseudograph appearing in Figure 4. The adjacency matrix A = B + + B -+ N corresponding to Figure 2 is

A =            
0 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 1 2 1 0 0 0 0 0 0 1 3 1 0 0 0 0 0 0 1 4 1 0 0 0 0 0 0 1 5 1 0 0 0 0 0 0 1 6 1 0 0 0 0 0 0 1 7 where

            , ( 1
B + =            
0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0

            , B -=            
0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 and

            , ( 1
N =            
0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 7

            . (1.32)
Furthermore, with the convention that loops contribute two to the degree of a vertex, the degree mapping is defined by

κ =            
1 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 14 0 0 0 0 0 0 0 0 16

           
.

(1.33)

Nilpotent adjacency matrices

The philosophy of this paper is to begin with an arbitrary graph and use it to construct a quantum random variable whose moments reveal information about the structures contained within the graph. Let Cℓ n nil be the associative algebra generated by the unit scalar ζ ∅ = 1 ∈ R and the commuting nilpotents {ζ {i} }, where 1 ≤ i ≤ n.

For n > 0 and nonnegative integers p, q satisfying p + q = n, the Clifford algebra Cℓ p,q is defined as the associative algebra generated by the unit scalar e ∅ = 1 and the collection {e i } 1≤i≤n subject to multiplication rules

e i e j + e j e i =      0 if i = j, 2 if i = j ≤ p, -2 if i = j > p. (2.1)
It can be shown that for n > 0, the Clifford algebra Cℓ 2n,2n is canonically isomorphic to the 2n-particle fermion algebra. Using this correspondence, the algebra Cℓ n nil is generated within Cℓ 2n,2n by ζ ∅ = 1 along with the set {ζ {i} } 1≤i≤n , where

ζ {i} = f i + f n+1 + .
Here f + i denotes the i th fermion creation operator.

In terms of Pauli matrices, the generators of Cℓ n nil can be written as

ζ {i} = σ ⊗(i-1) 0 ⊗ (σ x + iσ y ) ⊗ σ ⊗(n-i) 0 . (2.2)
Remark 2.1. Writing the generators {ζ {i} } 1≤i≤n in terms of fermion annihilation operators is an acceptable alternative.

For each n > 0, the algebra Cℓ n nil has dimension 2 n and is spanned by unit multi-vectors, or blades, of the form

ζ i = ι∈i ζ {ι} , (2.3)
where i ∈ 2 [n] is a canonically ordered multi-index in the power set of [n] = {1, 2, . . . , n}. Let u ∈ Cℓ n nil , and observe that u has the canonical expansion

u = i∈2 [n] u i ζ i , (2.4)
where u i ∈ R for each i. Define the dual u ⋆ of u by

u ⋆ = i∈2 [n] u i ζ [n]\i . (2.5)
The scalar sum evaluation of u is defined by

ϕ(u) = i∈2 [n] u i . (2.6)
Remark 2.2. The scalar sum evaluation is equivalent to the Clifford 1-norm u 1 = i∈2 [n] |u i | when all the coefficients in the expansion are nonnegative.

ϕ(α u + v) = α ϕ(u) + ϕ(v).
(2.8)

Moreover, u ∈ Cℓ n nil + ⇒ ϕ(u ⋆ u) ≥ 0.
(2.9)

Proof. The result follows immediately from the definitions and is omitted.

Let {ζ {i} } 1≤i≤n denote the orthonormal nilpotent generators of Cℓ n nil . Associated with any finite graph G = (V, E) on n vertices is a nilpotent adjacency matrix a n defined by

(a n ) ij = ζ {j} if (v i , v j ) ∈ E, 0 otherwise. ( 2
a ℓi ≡ s.a. 1-walk v ℓ → v i , (2.15)
where s.a. is short for self-avoiding. Hence, the product of these terms corresponds to 2-cycles are nonzero if and only if they correspond to self-avoiding m + 1-walks

v i → v i . v 1 v 2 v 3 v 4 v 5 v 6 v 7 a 0 0 0 Ζ 4 0 0 Ζ 7 0 0 0 0 0 0 0 0 0 0 Ζ 4 Ζ 5 Ζ 6 Ζ 7 Ζ 1 0 Ζ 3 0 Ζ 5 Ζ 6 0 0 0 Ζ 3 Ζ 4 0 0 0 0 0 Ζ 3 Ζ 4 0 0 0 Ζ 1 0 Ζ 3 0 0 0 0
v i → v ℓ → v i .
Summing over all vertices v ℓ gives the total number of m+1-cycles based at v i . The correction factor ϑ = 2 in the undirected case accounts for the two possible orientations of each cycle. An orientation correction is not needed when the graph is directed.

Additional properties of nilpotent adjacency matrices over Cℓ n nil can be found in [START_REF] Staples | Norms and generating functions in Clifford algebras[END_REF] and [START_REF] Staples | Clifford algebras and random graphs[END_REF]. (2.20)

Nilpotent Adjacency Matrices as Quantum Random Variables

Let M denote the * -algebra generated by n × n nilpotent adjacency matrices with involution defined by a * = (a ⋆ ) † . Here, ⋆ denotes the dual defined by linear extension of

ζ i ⋆ = ζ [n]
\i , and † denotes the matrix transpose. The norm of a ∈ M is defined by

a 2 = tr(a * a) dζ n • • • dζ 1 . (3.1)
Remark 3.1. This is the Clifford-Frobenius norm constructed in [START_REF] Staples | Norms and generating functions in Clifford algebras[END_REF].

Let ϕ(u) denote the scalar sum evaluation of u ∈ Cℓ n nil , and define ψ :

M → R by ψ(a) = ϕ (tr(a)) n . (3.2)
Direct computation shows that ψ is a linear mapping and that ψ(1 M ) = 1. The positivity requirement for states, ψ(a * a) ≥ 0, does not generally hold in the * -algebra, however.

The collection of nilpotent adjacency matrices A ⊂ M under matrix multiplication defined in terms of the algebra product constitutes a semigroup with involution * . On this semigroup, the map ψ satisfies the positivity requirement, and (A, ψ) is a quantum probability space.

The proofs of the following propositions are virtually identical and follow naturally from Proposition 2.4. Proposition 3.2 (Cycles in finite graphs). Let a ∈ A. Letting X m denote the number of m-cycles in the graph associated with a,

ψ(a m ) = ϑ m n X m , (3.3)
where ϑ is defined as in (2.12). Thus, the nilpotent adjacency matrix a ∈ A is a quantum random variable whose m th moment in the state ψ corresponds to the number of m-cycles in the graph associated with a.

Proposition 3.3 (Time-homogeneous random walks on finite graphs). Let M denote a stochastic matrix corresponding to an n-state Markov chain, and let τ denote the nilpotent stochastic matrix defined by

τ ij = M ij ζ {j} , (3.4)
where ζ {j} is a nilpotent generator of the abelian algebra Cℓ n nil . Let the state ψ be defined as in (3.2), and let ϑ be defined as in (2.12). Define the random variable w m on the space of m-step walks by

w m =
1 if the m-walk forms a cycle, 0 otherwise.

(3.5)
Then,

ψ(τ m ) = ϑ m n E(w m ). (3.6)
In other words, τ is a quantum random variable whose m th moment corresponds to the expected number of m-walks forming cycles in the n-state Markov chain associated with matrix M . Proposition 3.4 (Cycles in random graphs). Consider a random directed graph on n vertices, corresponding to edge-existence matrix A, defined by

A ij = P ((v i , v j ) ∈ E) . (3.7)
It is assumed that the probabilities are pairwise-independent. Let ξ denote the nilpotent adjacency matrix defined by

ξ ij = A ij ζ {j} , (3.8)
where ζ {j} is a nilpotent generator of the abelian algebra Cℓ n nil . Let the state ψ be defined as in (3.2), let ϑ be defined as in (2.12), and define the random variable z m as the number of m-cycles occurring in the graph. Then,

ψ(ξ m ) = ϑ m n E(z m ). (3.9)
That is, ξ is a quantum random variable whose m th moment corresponds to the expected number of m-cycles occurring in the graph.

Decomposition of Nilpotent Adjacency Matrices

In the work of Hashimoto, Hora, and Obata (cf. [START_REF] Hashimoto | Central limit theorems for large graphs: Method of quantum decomposition[END_REF], [START_REF] Obata | Quantum probabilistic approach to spectral analysis of star graphs[END_REF]), fixing a vertex v 0 in a finite graph induces a stratification of all the vertices by associating each vertex with the length of the shortest path linking it with v 0 . This stratification is then used to define a quantum decomposition of the graph's adjacency matrix.

The nilpotent adjacency matrix of a graph also admits a quantum decomposition as the sum of three algebraic random variables: one classical, and two quantum. The decomposition considered here differs from that of Hashimoto, et al.

The nilpotent adjacency matrix a of any finite graph has the canonical quantum decomposition a = a ∆ + a Λ + a Υ , (

where

a ∆ ij = δ ij a ij , (4.2)
a Υ ij = θ ij a ij , (4.3)
a Λ ij = (1 -θ ij )a ij . (4.4)
Here δ ij is the Kronecker delta function, and θ ij is the ordering symbol defined by

θ ij = 1 if i < j 0 otherwise. (4.5)
The matrix a ∆ is an element of the multiplicative semigroup D of diagonal nilpotent adjacency matrices, which is commutative. Hence, a ∆ is a classical algebraic random variable in the space (D, ψ).

Elements a Λ and a Υ , respectively, reside in the semigroups L and U of lowerand upper-triangular nilpotent adjacency matrices with matrix multiplication, which are noncommutative. Hence, a Λ and a Υ are quantum algebraic random variables in the spaces (L, ψ) and (U, ψ), respectively.

In particular, let γ represent the diagonal matrix defined by

γ =    ζ {1} • • • 0 . . . . . . . . . 0 • • • ζ {n}    . (4.6)
Then, recalling the (finite) adjacency matrix realizations of the commutation relations of Table 1, the following correspondence is seen: . The sequence {a n } will be said to converge to the operator a if for each k ≥ 0 and any coordinate basis vector x 0 , the following equation holds: lim n→∞ ϕ x 0 , a n k x 0 = ϕ x 0 , a k x 0 .

B + γ = a Λ (4.7) B -γ = a Υ (4.8) N γ = a ∆ . ( 4
(5.8)

Denote this convergence by a n w → a, and notice that this implies

lim n→∞ ϕ ( x 0 , exp (a n ) x 0 ) = ϕ ( x 0 , exp (a) x 0 ) . (5.9)
An immediate consequence of this convergence is found in the following proposition.

Proposition 5.3 (Ascending chains). Let {a n } ⊂ A be a sequence of operators such that for each n > 0, a n represents the nilpotent adjacency matrix associated with a finite graph on n vertices. Let t be a real-valued parameter.

Let X k (n) denote the number of k-cycles based at vertex x 0 in the n th graph of the sequence. Then,

ϑX k (n) = ϕ ∂ k ∂t k x 0 , exp (ta n ) x 0 t=0 . ( 5

Applications

Letting X k denote the number of k-cycles contained in a simple random graph on n vertices, the nilpotent adjacency matrix method has been used to recover higher moments of X k [START_REF] Staples | Clifford algebras and random graphs[END_REF].

The nilpotent adjacency matrix method has been used to give a graphtheoretic construction of iterated stochastic integrals [START_REF]Graph-theoretic approach to stochastic integrals with Clifford algebras[END_REF]. In particular, stochastic integrals of L 2 ⊗ Cℓ p,q -valued processes are defined and recovered with this approach.

A graph-free algebraic construction of iterated stochastic integrals of classical processes has also been developed using sequences within a growing chain of fermion algebras [START_REF]Multiple stochastic integrals via sequences in geometric algebras[END_REF].

Algebraic generating functions for Stirling numbers of the second kind, Bell numbers, and Bessel numbers have been defined [START_REF] Schott | Partitions and Clifford algebras[END_REF]. These lemmas emphasize the potential power of a computer architecture based on geometric algebra. Given a device in which the product of any pair u, v ∈ Cℓ nil could be computed in polynomial time, the NP class problem of cycle enumeration would be reduced to class P. Remark 6.4. Using the Coppersmith-Winograd algorithm, multiplying two n×n matrices can be done in O(n 2.376 ) time [START_REF] Coppersmith | Matrix multiplication via arithmetic progressions[END_REF]. Applying this result to nilpotent adjacency matrices further reduces the Cℓ time complexity [START_REF] Schott | How can NP problems be moved into P?[END_REF].

Figure 1 :

 1 Figure 1: Realizations of commutation relations.

Figure 2 :

 2 Figure 2: Graph with adjacency matrix B + + B -+ N for free commutation relations.

Figure 3 :

 3 Figure 3: Graph with adjacency matrix B + + B -+ N for Boson commutation relations.

.31) 0 1 Figure 4 :

 14 Figure 4: Graph with adjacency matrix B + + B -+ N for Fermion commutation relations.

Lemma 2 . 3 .

 23 Let u, v ∈ Cℓ n nil , and let α ∈ R. Let Cℓ n nil + denote those elements of Cℓ n nil having strictly nonnegative coefficients. Then, the scalar sum evaluation ϕ : Cℓ n nil → R satisfies ϕ(1) = 1 , and (2.7)

. 10 ) 2 . 4 .

 1024 Proposition Let a n denote the nilpotent adjacency matrix of a graph G on n vertices. Let x 0 represent any fixed vertex in the graph. Let X m denote the number of m-cycles based at x 0 in G. Then,ϕ ( x 0 , a n m x 0 ) = ϑX m ,(2.11)where ϑ = 2 if the graph is undirected, 1 otherwise.(2.12)Proof. Proof is by induction on m. When m = 2,a 2 ii = (a × a) ii = n ℓ=1 a iℓ a ℓi . (2.13) By construction of the nilpotent adjacency matrix, a iℓ ≡ s.a. 1-walks v i → v ℓ , and (2.14)

Figure 5 :

 5 Figure 5: A graph and its nilpotent adjacency matrix.

Example 2 . 5 .

 25 The two 5-cycles contained in the 7-vertex graph of Figure 5 are recovered by examining the trace of a 5 . Computations performed using Mathematica reveal tr(a 5 ) 10 = ζ {1,3,4,5,7} + ζ {1,3,4,6,7} .

. 10 )

 10 Assuming a n w → a as n → ∞ and fixing k ≥ 1, one findsϑ lim n→∞ X k (n) = ϕ ∂ k ∂t k x 0 , exp (ta)

Example 6 . 1 .

 61 In Figure6, the number of Hamiltonian cycles contained in a randomly generated, 10-vertex graph are recovered by examining the trace of a 10 .

  v

Figure 6 :Lemma 6 . 2 .Lemma 6 . 3 .

 66263 Figure 6: A 10-vertex graph containing 176 Hamiltonian cycles.

  Cartesian product of Cℓ n nil , i.e. the space of n-tuples from the algebra. Let {x i } 1≤i≤n denote the collection of basis Observe that ϕ is a state on the semigroup A.Define a sequence of operators {a n }, n ≥ 1, in A such that for each n, a n

	5 Extending to Infinite Dimensions Denote by Cℓ n nil n the n-fold is an operator on Cℓ n nil n	.9)

coordinate vectors of the form x i = (0, . . . , 0, 1 i th pos. , 0, . . . , 0) such that for

, one has u, x i = u i .

(5.1)

The following finite-dimensional result follows from the previous results and definitions.

Proposition 5.1 (Generating function method). Fix n > 0, and let G be a finite graph on n vertices with associated nilpotent adjacency matrix a. Coordinate basis vectors x 1 , . . . , x n are also associated with the vertices of G by construction of a. Let X k denote the number of k-cycles based at an arbitrary vertex x 0 in G. Then,

M is a Hilbert space.

Denote by A n the semigroup of nilpotent adjacency matrices associated with finite graphs on n vertices under matrix multiplication. Denote by A the collection of operators in M that represent nilpotent adjacency matrices. In other words, A = {a ∈ M : (∃ξ ∈ A n )(∀i ∈ {1, . . . , n}) [ x i , ax i = x i , ξx i ]}.

(5.7)