A realization of a quasi-random walk for atoms in time-dependent optical potentials

We consider the time dependent dynamics of an atom in a two-color pumped cavity, longitudinally through a side mirror and transversally via direct driving of the atomic dipole. The beating of the two driving frequencies leads to a time dependent effective optical potential that forces the atom into a non-trivial motion, strongly resembling a discrete random walk behavior between lattice sites. We provide both numerical and analytical analysis of such a quasi-random walk behavior.


I. INTRODUCTION
Over the past few decades, the optical control of motional degrees of freedom has seen great progress both on the experimental and theoretical fronts. As a particular example of such achievement, the cavity QED setting provides a paradigm for the observation and manipulation of motion of atoms, ions or atomic ensembles via tailored cavity modes [1][2][3][4][5]. In such a system, the time delay between the action of the field onto the atomic system's motion via the induced optical potential and the back-action of the particle's position change onto the cavity field leads to effects such as cooling [6][7][8] or selfoscillations (for a recent review see [9]). Addressing of the particle's motion works best for quantum emitters with sharp transitions such as ions or atoms but manipulation via the effective polarizability is also possible in the case of molecules in standing wave [10] or ring cavities [11], or macroscopic particles such as levitated dielectric micron-sized spheres [12][13][14]. Typically, the driving is done either by direct pumping into the cavity mode via one of the side mirrors (longitudinal pumping), or indirectly via light scattering off the atom into the field mode (transverse pumping). For transverse pumping, an exotic phenomenon dubbed as self-organization can occur in many atoms case [15]. Combinations of the two techniques have been theoretically investigated in the limit of equal frequencies [16]; in such a case, in a properly chosen rotating frame the effective combined optical potential can be rendered time independent and the analysis greatly simplifies. Here we depart from this scenario to consider frequency beatings between the two pumps. While dynamics in the regime where either pump acts alone is well understood, extra forces arise from the interference between photons of different frequencies: i) scattered from the transverse light field and ii) entering the cavity mode from the longitudinal pump. The immediate effect of this interference force is to generate a time-dependent optical potential with a time modulation leading to a sign change that effectively induces the particle into undergoing jumps along the cavity sites in a quasi-random walk fashion. We analyze such a regime both numerically and via simplified analytical models. The mechanism is similar to the ones exploited in the creation of artificial potentials in optical lattices [17]. By discretizing the trajectories, we analyze Schematics. Illustration of the model considered involving an atom moving inside an optical cavity with longitudinal driving of strength ηL and frequency ωL and transverse driving ηT and frequency ωT . The interference between the longitudinally pumped field and the field scattered from the transverse pump into the cavity mode leads to an effective optical potential that has a time dependent part oscillating at the frequency difference δT = ωL − ωT .
the emerging discrete process via its correlation function and find that it corresponds to an environment with a very short memory. The goal of this analysis is to provide a quantum optical setting in which the classical random walk can be observed and which would constitute a starting point into further generalizations into the quantum regime. This is similar to previous works on implementation of the quantum random walk with photons [18], atoms in optical lattices [19] and ions in traps [20]. Our analysis is mainly based on a single two-level system but we discuss as well an extension involving doped microspheres where the field addresses a collective atomic variable (along the lines of hybrid optomechanics with doped mechanical resonators [21]).
The paper is organized as follows: in Sec. II we introduce the model. In Sec. III we present numerical evidence showing the occurrence of a quasi-random walk behavior and compute correlations of the engineered process that map close to those expected from a true random walk. In Sec. IV, we present a simplified analytical model that allows us to derive the different forces acting on the particle and identify different regimes and associated scalings for the occurrence of the random walk. In Sec. V, we discuss possible extensions of the model involving a tailored driving via a frequency comb laser. We conclude and present an outlook in Sec. VI.

II. MODEL
We consider an optical cavity mode at ω c , decaying at rate κ (which in the following we set to unity as a reference for all the other frequencies) and driven through a side mirror by a laser of amplitude η L and frequency ω L . Transversally, a second laser drives the atom directly with effective amplitude η T at ω T . The longitudinal mode spatial variation inside the cavity is f (x) = cos(kx) (k is the corresponding wave-vector for the light mode with wavelength λ = 2π/k) and the atom-photon coupling is specified by g(x) = gf (x) (with g being the maximum coupling). The total system is described by the HamiltonianĤ consisting of a free partĤ 0 , a pumping termĤ p and the Jaynes-Cummings interactionĤ JC . The free evolution Hamiltonian describes the dynamics of a free particle of mass m and momentum operatorp plus that of the cavity mode (annihilation operatorâ) and the two-level atom The atom dynamics is described with the help of the Pauli operatorsσ ± andσ z satisfying [σ + ,σ − ] =σ z and [σ ± ,σ z ] = ±2σ ± . The atom's interaction with the cavity field is included as a Jaynes-Cummings photon-excitation exchange process quantified by the position dependent coupling strength gf (x): Finally driving is included in the pump termŝ including direct pumping into modeâ and atom driving of the dipole operatorσ − . We proceed in a standard way to derive equations of motion for classical quantities. First, we make a set of transformations to dimensionless normalized position x = k x and momentum p = p ( k) −1 . We denote the field amplitude by α = â and by β = σ − the atomic averaged polarization (in a frame rotating at ω L ). In a first stage we consider finite saturations of the population difference operatorσ z whose classical average we denote by β z . We furthermore assume that the build-up of quantum correlations between the atom and the photon field can be neglected so that we can replace the nonlinear terms such asâσ z by their factorized classical averages αβ z . The complete equations of motion for atom and field are: Frequencies. Illustration of the frequencies and detunings as defined/used in the equations of motion. The detunings are taken with respect to the longitudinal driving frequency such that ∆c < 0 corresponding to the stable regime of cavity QED with moving atoms and ∆a < 0 that corresponds to U0 = g 2 /∆a < 0 as for high-field seekers. Fig. 2 with corresponding sign conventions). However, for the moment, we restrict our treatment to the low saturation case, where |β| 2 1 which allows us to linearize theâσ z term by setting σ z → −1. Such a linearized regime allows one to analytically derive the forces acting on the particle. Numerical evidence points out that this simplified limit provides similar effects with the finite saturation case and we will base our analytical treatment on the following simplified system of equations: The motion of the atom is described bẏ where we have condensed the particle's properties into the recoil frequency ω r = k 2 /(2m).

III. THE QUASI-RANDOM WALK -NUMERICAL RESULTS
Before obtaining insight from analytical considerations, we start by simulating the dynamics of the system. This is achieved by fixing the set of parameters to: ∆ a = −20, ∆ c = −2.4, η L = 0.5, δ T = 0.04, γ = 0.1, g = 1.2 and recoil frequency ω r = 20. We treat η T as a free varying parameter. We choose a regime described in the next analytical section as 'trapping via longitudinal pump', where we first tune the parameters such that trapping of the particle is ensured in the absence of the transverse driving. We then increase η T , and notice that past a given threshold, the particle starts jumping out of its trapping site to the neighboring left/right sites in an apparently random way. In Fig. 3 we exemplify such a trajectory obtained for a particle initialized with p 0 = 0 around the origin at x 0 = 10 −1 and for η T = 1.9.
With these parameters, we then analyze the dynamics as a function of randomized initial conditions; we initialize the particle in a state of zero momentum and an initial position drawn from a Gaussian distribution with zero mean and standard deviation of σ = 0.1, and follow the evolution over time. We identify the jump time as half the period of the potential time oscillations T = π/δ T at a given site and plot the evolution for 10 3 initial values as a function of time normalized to T . First, we illustrate the mixing of trajectories, as shown in Fig. 4, by color coding the trajectories starting with x 0 > 0 in green and those starting with x 0 < 0 in black. The mixing is evident and can be taken as a first indicator for randomness.

A. Discrete process -single trajectories
We then discretize the process by choosing time steps in units of T : at t n = nT the particle finds itself in one of the trapping sites. This is illustrated in Fig. 5 as the transition from the continuous trajectories in the upper plot to the discrete plots of site number in the middle plot. The discrete positions (the locations of the sites) are defined as: where the square brackets stand for the rounding of the integral to π times the nearest integer, corresponding to the location of the site where trapping occurs. One can introduce the jump sequence J N = (j 1 , j 2 , ..., j N ), where the jump indicators are defined as j n = x n+1 − x n , and according to their sign show either left or right jump behavior. One can define the autocorrelation function for this process as which characterizes the joint probability of the jump occurring with a time delay τ T . The behavior of this function for a single trajectory is shown in Fig. 5

B. Discrete process -many trajectories statistics
We then numerically simulate a large number of trajectories (with starting point around the origin) and plot the final occupancy at neighboring sites. As Fig. 6 illustrates, the expected binomial distribution arising from a random walk process is reproduced already for N = 65 trajectories. The fitting is done with a Gaussian distribution of standard deviation ∆x = √ N . The main result of the numerical section is however the behavior of the jump correlation function averaged over many trajectories (see Fig. 7). For a perfectly random walk process, the correlation function for jumps sepa- Correlations. Numerical data showing the variation of the correlation function with the delay time between jumps and corresponding fitting function. As a basis of comparison, the correlations of a pure random walk would correspond to a function reaching unity for τ = 0 and zero elsewhere. On the given numerical example, the particle is subjected to an effective reservoir with a non-vanishing memory that allows for anticorrelations for jumps close to each other. rated by τ > T would be vanishing. This corresponds to a reservoir having no memory. In our case, however, short time delays (below 5 jumps) are anti-correlated while after around 5 jumps correlations occur. To gain some physical understanding, one can inspect Eq. (11)b where the right-hand side represents the effective optical force. In some limit (revealed by the numerical results) this force shows effective quasi-random kicks whose correlations map onto the correlation function for the discrete process. For perfectly uncorrelated kicks the effect would be a random walk. However, in the realistic case some correlations between jumps remain. One can consider the following argument: the momentum kick at one site is the integral of the force over a time T during which the force varies non-trivially. In the continuous limit the process is deterministic. However, in the limit of many oscillations inside a single site, the phases of the momentum kicks occurring at different sites are randomized.

IV. ANALYTICAL RESULTS
The dynamics numerically derived in the previous section can be explained at least in some particular limits by a simplified model where the atomic and field degrees of freedom are eliminated and an effective set of equations are derived for the particle motion only. Let us first rewrite the effective force acting of the particle [from Eq. (11)] in the following form: and proceed with finding analytical expressions in the adiabatic limit.

A. Elimination of the atomic dipole
Under the assumption of purely dispersive coupling brought by the weak, far off-resonance driving ∆ a γ, η T one can eliminate the atomic variable and compute where the per photon dispersive coupling is given by U 0 = g 2 /∆ a and the effective transverse pump is defined as η T = gη/∆ a . Replacing the steady state value of β in the force expression we obtain: Notice that the first term is the well-known force arising from the longitudinal pump into the cavity. Combined with time-delay effects coming from the finite ring-down time of the cavity field, such a force can lead to cavity cooling, heating,bistability or self-oscillations [8,9]. The second term is of more importance in our treatment as it shows the interference between the two pumps and it contains the time-modulation needed for the potential sign change.

B. Elimination of the field variable
Since the time-delay effects do not play a role in the occurrence of jumps, we proceed by considering the limit of small U 0 where we eliminate the cavity field. Replacing β from Eq. (15), leads tȯ Under the assumption that η L ,η T max(1, ∆ c ) and defining a position dependent cavity detuning ∆ c − U 0 f 2 (x) ≡ ∆(x), one obtains: We now can compute the cavity photon number as well as and replace these expressions in Eq. (16).

C. Optical forces
We can now group the different contributions contributing to the total optical force acting on the particle as follows: i) arising from the longitudinal field, ii) from the transverse field and iii) a time dependent interference term. Explicitly writing the three terms as which describes the standard cos 2 (x) optical potential induced by the longitudinal pump. The next term is and it shows the effect of the time-independent interference between transverse pump photons and the particle scattered photons filling the cavity mode. The most interesting term is showing modulation at δ T . Notice that the time independent limit can be reached by setting δ T = 0, and this force reduces to 2f (x)η T η L /(1 + ∆ 2 (x)).

D. Trapping by interference
Let us consider the limit of small g and tune the driving field amplitudes such that η T gη L while at the same time η T η L ∆ a /(g∆ c ). To satisfy both conditions simultaneously one has to require U 0 ∆ c 1. Under these conditions, the time interference force is dominant and gives rise to an effective time-modulated trapping potential. We neglect the spatial modulation U 0 f 2 (x) as well with respect to the larger ∆ c and obtain the total force on the particle simplified to: where the phase is defined as: φ ∆ = arctan ∆ c . The equations of motion for the particle are those of a frequency modulated pendulum and similar to the ponderomotive force with an important difference in that the amplitude of the driving changes sign periodically.
In the limit of good localization (where we can expand sin(x) x) we identify this as a restoring force for a harmonic oscillator with a time dependent normal frequency. We can readily compute the maximum trapping frequency ω tr as The quasi-random walk behavior arises from the periodic force sign change which, after a number of oscillations at a given site (roughly proportional to T /T tr,LT where T tr,LT = 2π/ω tr,LT ) forces the particle to settled itself inside an adjacent well.

E. Trapping by longitudinal pumping
A different regime is reached when the F L and F LT contributions are of the same order of magnitude. We achieve this by first turning on the longitudinal pump and see that a trapping time-independent potential is established. Along the equilibrium points, the longitudinal trapping frequency is around: Tuning up the transverse pump, a time modulation of the trap frequency is achieved and a threshold emerges for η T (for a rough estimate one can equate the maxima of F L and F LT ) after which the particle starts jumping to adjacent sites. Given the two different spatial modulations of the forces, sin(2x) vs. sin(x), a double well structured potential arises with frequencies: A. Frequency comb driving In the limit where the pump interference provides the sole trapping mechanism, the force acting on the particle as in Eq. (24) reminds of a parametrically driven pendulum. The dynamics consists of pendulum motion inside a given trap (which reduces to harmonic motion in the limit of good localization) with a time modulated frequency. However, one can engineer a multiple frequency transversal pumping scheme where the time modulation becomes a periodic kick instead of a sinusoidal function. To this end we consider a frequency comb driving (with 2N f + 1 frequecies) with the minimum frequency separation δ and centered at ω L , such that the pump frequencies are ω T,j = ω L − jδ. If we neglect spontaneous atomic decay and the coupling between atom and cavity field, Eq. (9)b with the total pump term inserted giveṡ In the limit N f → ∞ the driving on the right-hand side becomes a Dirac comb function. Inserting the steady state solution of Eq. (28) together with α η L into Eq. (11)b we get an equation of motion that maps onto the kicked rotor dynamics: where T δ = 2π/δ. Together with Eq. (11)a the system can be exactly described via a transformation to the discrete and can be shown to exhibit chaotic motion past a threshold characterized by the tuning up of the force amplitude factor.

B. Hybrid optomechanics with doped nano-spheres
We can extend our treatment to consider a hybrid optomechanical system where we replace the two level system with a doped nano-sphere containing a collection of N such systems. We assume the nano-sphere transparent to light except for the doped part where the cavity mode excites a transition close to resonance. Let us consider the nano-particle of mass M with radius much smaller than specific length in which the cavity mode changes considerably (the cavity mode wavelength). The lightmatter interaction takes place via the Tavis-Cummings Hamiltonian, that changes from the single atom picture in thatσ ± is replaced by jσ ± j ≡ S ± . In the bosonic limit, where the saturation is very low, we can assume that [S − , S + ] = −N and proceed to write equations for averages (β N = Ŝ − ): The immediate gain in this approach from the single atom approach is the relaxation of the requirement of β 1 that turns into β N N . One has however have to pay the price of a reduced recoil frequency owing to an increased mass at least N times larger. The upshot is that a cavity QED regime where a quasi-random with a macroscopic particle can be unraveled.

C. Implementation considerations
To experimentally observe the proposed quasi random walk, we advance a possible two step experimental procedure: i) turn on longitudinal pumping detuned by −κ from the resonance such that cavity cooling takes place in the absence of transverse pumping and ii) turn on η T past the threshold such that jumps are initiated. The typical cooling procedure can ideally cool the particle towards a thermal wavepacket with minimum energy κ divided between the position and momentum quadratures (according to the thermal equipartition principle). In terms of normalized momentum and position initial variations this corresponds to δp 0 = 1/ √ 2ω r and δx 0 = 1/( √ 2U 0 η L ). For localization within a site notice that we require δx 0 1 which results in U 0 η 2 L 1. This can be still fulfilled in the polarizable particle regime by tuning η such that g 2 /∆ a 1 while gη L 1.

VI. CONCLUSIONS AND OUTLOOK
We considered the dynamics of particles (either as single two-level systems or sub-micron spheres doped with multiple emitters) inside time-dependent potentials resulting from interference in a two-color, two directional pump scheme. Past a given threshold, chaotic-like behavior can be observed, with correlations very close to those of a typical random walk. While the treatment here is in the classical regime, an immediate generalization into the quantum realm can be made by either: i) treating motion classically and considering the effect of the quantum nature of the two-level system onto the dynamics or ii) treating motion quantum mechanically and analyzing the dynamics of an initial wave packet, with direct connection to matter-wave interferometry applications. Another future direction aims to extend the 1D treatment to 3D dynamics where the beating of the two pumps give rise to an effective ponderomotive force. Investigations will be carried out on the possibility to exploit such a force for all optical trapping of polarizable particles