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Matter wave soliton bouncer

Dynamics of a matter wave soliton bouncing on the reflecting surface (atomic mirror) under the effect of gravity has been studied by analytical and numerical means. The analytical description is based on the variational approach. Resonant oscillations of the soliton's center of mass and width, induced by appropriate modulation of the atomic scattering length and the slope of the linear potential are analyzed. In numerical experiments we observe the Fermi type acceleration of the soliton when the vertical position of the reflecting surface is periodically varied in time. Analytical predictions are compared with the results of numerical simulations of the Gross-Pitaevskii equation and qualitative agreement between them is found.

Dynamics of a matter wave soliton bouncing on the reflecting surface (atomic mirror) under the effect of gravity has been studied by analytical and numerical means. The analytical description is based on the variational approach. Resonant oscillations of the soliton's center of mass and width, induced by appropriate modulation of the atomic scattering length and the slope of the linear potential are analyzed. In numerical experiments we observe the Fermi type acceleration of the soliton when the vertical position of the reflecting surface is periodically varied in time. Analytical predictions are compared with the results of numerical simulations of the Gross-Pitaevskii equation and qualitative agreement between them is found.

I. INTRODUCTION
A particle bouncing on the reflecting surface under the effect of gravity represents one of the analytically solvable models in quantum mechanics [1,2]. Gibbs introduced the name "quantum bouncer" [3] for the object, and it was extensively studied in many articles of pedagogical orientation [4,5] and original research papers (for a recent review see [6]). The practical interest in this model has emerged from recent experiments aimed at probing the coherence properties of Bose-Einstein condensates falling under gravity and bouncing off a mirror formed by a far-detuned sheet of light [7], quantum reflection of matter waves [8], and measuring the Casimir-Polder force acting upon the atoms near solid surfaces [9].
Another important result linked to the quantum bouncer problem has been the experimental observation of quantum bound states of neutrons in the Earth's gravitational field [10][11][12]. In these pioneering experiments the quantum states of matter formed by a gravitational field were observed for the first time. Also, the model is of particular interest from the viewpoints of the physics and applications of quantum states of nanoparticles in the vicinity of surfaces [13]. An optical analogue of the quantum bouncer, a photon bouncing ball, was experimentally demonstrated using the circularly curved optical waveguide [14]. Significance of the model for the study of the dynamics of particles in quantum-classical interface was pointed out in [15].
In this work we extend the quantum bouncer model to the nonlinear domain by considering the dynamics of a matter wave soliton governed by the Gross-Pitaevskii equation (GPE). The linear potential entering the GPE represents the Earth's gravitational field acting on the soliton in vertical direction, while the horizontal atomic mirror [16] created by a laser beam or magnetic field stands for the reflecting surface. The matter wave soliton performs bounded motion in such a gravitational cavity. The effect of nonlinearity, originating from the atomic interactions in Bose-Einstein condensate (BEC), shows up as an ability of the bouncing wave packet to remain localized during the evolution, behaving like a rigid ball, rather than a deformable wave packet. The possibility of tuning the atomic interactions in the condensate by external magnetic [17] and optical [18] fields opens perspectives in exploring the bouncer problem in both the quantum and classical limits.
Our main objective is to develop analytical description of the soliton's dynamics above the atomic mirror under the effect of gravity. As an illustration of the developed model we consider the resonant oscillations of the soliton's center of mass position under periodically varying strength of nonlinearity and the slope of the quasi-1D trap with respect to the horizontal reflecting surface. The strength of nonlinearity can be tuned using the Feshbach resonance technique [17], or alternatively, by changing the strength of the radial confinement. In numerical simulations we demonstrate the Fermi type acceleration of the soliton when the vertical position of the mirror is periodically varied in time. It should be noted that Fermi acceleration of matter wave packets was previously considered in [19] for the case of non interacting BEC, in the setting where matter wave solitons do not exist. In these works non-dispersive acceleration of the wave packet was reported to take place under certain conditions, when the modulation strength and frequency provide the dynamical localization of the matter wave.
The advantage of the present setting is that, the bouncing matter wave packet preserves its integrity due to the focusing nonlinearity of the BEC, which counteracts the dispersive spreading. Another interesting approach to acceleration of a single quantum particle, also feasible in the context of matter waves, was reported in [20]. The mechanism consists in binding the wave packet by a delta function potential well and involving in accelerated motion along with the potential. In the linear case and ideal mirror potential our model reduces to the equation which has analytic solution in terms of Airy functions. The dynamics of Airy beams currently represents one of the actively explored topics motivated by important applications in optical communications and nonlinear optics [21].
The paper is structured as follows. In Sec. II we introduce the mathematical model and illustrate the distinctive features of the nonlinear model as compared to its linear counterpart. In Sec. III a variational approach for analytical treatment of the nonlinear model has been developed and its predictions are compared with numerical simulations of the original GPE. Sec. IV is devoted to exploring the resonant oscillations of the wave packet above the mirror, and Fermi type of acceleration of matter wave solitons when the vertical position of the reflecting surface is periodically varied in time. In Sec. V we summarize our findings.

II. THE MODEL AND MAIN EQUATIONS
The Bose-Einstein condensate is a giant matter wave packet which is strongly affected by gravity. In particular, a matter wave packet released from the trap falls towards Earth like a bunch of coherent atoms. The effect of gravity is essential for the operation of atom lasers [22].
In the present model the gravitational field acting on atoms in the vertical direction and a horizontal atom mirror which reflects them back, form a cavity for the matter wave packet. Below we consider the motion of a matter wave soliton within such a gravitational cavity. The model is based on the following one dimensional GPE where ψ(x, t) is the wave function of the condensate trapped in a tight quasi-1D trap, x is the spatial coordinate of the wave packet above the horizontal atomic mirror, represented by the reflecting potential U (x), g is the strength of the gravitational potential, ω ⊥ is the trap frequency in the tightly confining radial direction, m, a s are the atomic mass and s -wave scattering length, respectively. The gravitational units of space and time, defined as allow to rewrite the Eq. (1) in the dimensionless form where the new variables are defined as Here we took into regard that for BEC with attractive atomic interactions, a s < 0. In Eq. (3) the linear potential term (∼ x) accounts for the effect of gravity, while the atomic mirror is represented by V (x). We introduced an additional parameter α = sin(β) to account for the possibility of altering the effect of gravity by changing the angle β formed by the axis of the quasi-1D waveguide and the horizontal reflecting surface. For vertical position (β = π/2) of the waveguide α = 1, at smaller angles 0 < β < π/2, then 0 < α < 1. Such a setting is of interest in view of recent research on the behavior of BEC in microgravity [23] and the quantum reflection of matter waves [8], where the cold atoms should approach the attractive potential at very low velocity. Similarly, the additional parameter γ can be used for nonlinearity management γ(t) = a s (t)/a 0 s , then in the normalization for ψ in Eq. (3) the background value of a 0 s should be assumed. The following two cases will be relevant to our further analysis where δ(x) is the Dirac delta function which has been multiplied by the strength V 0 . A detailed study of the wave packet dynamics described by Eq. (3) in the linear model (γ = 0) for ideal mirror was reported in Ref. [4]. Before proceeding to analytical description of the nonlinear model (γ = 1) it is instructive to compare these two limits by numerical simulations of the governing equation (3). Such a preliminary study will help to elucidate the effect of nonlinearity on the dynamics of a wave packet bouncing above the atomic mirror.
In Fig. 1 we illustrate the features of the linear and nonlinear models for the dynamics of the wave packet dropped from the height x 0 = 10 above the mirror positioned at x = 0. The main difference appears to be enhanced spreading of the wave packet and strong interference with reflected waves in the linear model, as compared to the nonlinear case, where these phenomena do not show up. The distinctions between the two models is clearly observed in Fig. 2, where we compare the corresponding wave profiles at different times during one period of bouncing T b , which is estimated from classical equation d 2 x/dt 2 = −g. In dimensionless units introduced for Eq. (3) we need to set g = 1. Then a classical particle dropped from the height x 0 reaches the ground at t b = √ 2x 0 , therefore the classical bouncing period To estimate the parameters of the model we consider the 85 Rb condensate, for which a s = −20 nm, l g ≈ 1.3 µm, t g = 0.36 ms. At the strength of radial confinement ω ⊥ = 10 3 rad/s we have γ = 1. For N = 4 the soliton contains ≈ 720 atoms. Similar estimates for 7 Li condensate with a s = −1.6 nm gives l g ≈ 7µm, t g = 0.84 ms, ω ⊥ = 10 4 rad/s, the soliton contains ≈ 1400 atoms.

III. VARIATIONAL APPROXIMATION
For arbitrary forms of the reflecting potential V (x), the governing Eq. (3) cannot be analytically investigated. One of the efficient approaches to the problem in such cases is the variational approximation (VA), first developed for pulse propagation in optical fibers [24], and later applied to many other areas of nonlinear physics [25].
Below we develop the VA for the governing equation using the second choice (b) for the potential Eq. (4). It is well known from quantum mechanics textbooks that the wave packet falling on the delta potential barrier is always partially transmitted. However, by increasing the strength of the barrier (V 0 ) the transmission coefficient can be reduced to negligible level. This allows us to consider the norm of the wave packet above the mirror as a conserved quantity and develop the VA using an appropriate ansatz for the pulse shape.
Eq. (3) can be generated from the following Lagrangian density An important step in the development of VA is the proper choice of the trail function. We shall consider the following hyperbolic secant ansatz where A(t), a(t), ζ(t), ξ(t), b(t), ϕ(t) are variational parameters representing the amplitude, width, center of mass position, velocity, chirp parameter and phase of the wave packet, respectively. This choice is motivated by the fact that when the wave packet is sufficiently far from the reflecting potential V (x) (and therefore its effect can be neglected), Eq. (3) has the exact accelerated soliton solution of the hyperbolic secant form [26]. Substituting the ansatz (6) into Eq. (5) and integrating over the space variable we get the averaged Lagrangian where we have taken into account that the velocity is equal to the time derivative of the center mass position ξ = ζ t and A 2 = N/(2a), with the norm of the wave packet N = ∞ −∞ |ψ(x)| 2 dx being the conserved quantity. Now the usual procedure of the VA, applied to Eq. (7) leads to the following set of equations for the width and center of mass position of the wave packet The coupled system of equations (8)-(9) represents the main result of this paper. Its fixed points provides the stationary width of the soliton (a 0 ) and its distance from the mirror (ζ 0 ), where the actions of the gravity and repulsive potential V (ζ) cancel each other. As a result of this balance, the soliton placed at a fixed point remains at rest (levitates) above the mirror. Small amplitude dynamics of the soliton's width and center of mass position near the stationary state can be described as motion of a unit mass particle in the anharmonic potentials U 1 (a) and U 2 (ζ), respectively, In Fig. 3 the shapes of the potentials in Eqs.  (ζ 0 , a 0 ). The corresponding period for V 0 = 1, ζ 0 = 1.213 and a 0 = 0.468 is T 0 = 2π/ω 0 ≃ 9.7. This is in quite good agreement with numerical simulations of the GPE (3), shown in the middle panel of Fig. 3. An expression similar to Eq. (12) can be derived for the frequency of the soliton's width.

IV. FERMI TYPE ACCELERATION OF A MATTER WAVE SOLITON
The capability of the matter wave soliton to perform bouncing motion above the atomic mirror, preserving its integrity, suggests to consider the Fermi type acceleration (FA) in this system. FA is the energy gain by a particle exposed to periodic or random driving forces. It was proposed by Enrico Fermi [27] to explain why cosmic rays have so high energy. For the mechanical analogue, the possibility of unbounded growth of energy by an elastic ball bouncing vertically on a single periodically oscillating plate, under the effect of gravity, was rigorously proven in Ref. [28]. Most studies of FA of matter waves are concerned with dynamical localization and chaotic behavior. In our model localization of the matter wave naturally arises from the nonlinearity of the condensate, and the parameter space does not contain chaotic regions.
Although the matter wave soliton does not have all necessary properties to demonstrate true FA (due to non elastic collision with the mirror, leakage of energy via tunnel effect, etc.), nevertheless some features of FA can be observed, as we have revealed in numerical experiments. At first we need to prepare the initial stationary state of the matter wave packet levitating above the atomic mirror. The prediction of VA for parameters of the soliton and stationary state distance above the mirror (where the forces of gravity and repulsion of the mirror balance out) is approximate, as we have seen in the previous section. The inaccuracy leads to small amplitude oscillations of the soliton near the equilibrium state in the GPE simulations (see middle panel in Fig. 3). In order to create a truly stationary initial state of the soliton above the reflecting surface we consider the first choice (a) for V (x) in Eq. (4). For this ideal mirror potential, the Eq. (3) in the linear limit (γ = 0), with boundary condition ψ(0, t) = 0, has analytic stationary solutions in terms of the Airy functions [29], where N is some normalization constant. Below we shall be concerned with the ground state (n = 0) of the wave packet in the gravitational cavity. The first root, given by Ai[(2α) 1/3 x] = 0 for α = 0.1, is found to be equal to x 0 = −3.998. The corresponding normalization factor is To produce the initial state for numerical simulations of the Fermi acceleration we insert the ground state wave function (14) with appropriate norm into the GPE (3)  nonlinearity (via atomic scattering length) is periodically changed in time. It is evident that nonlinear resonance takes place at the frequency of small amplitude oscillations ω 0 estimated from the VA Eq. (12). Similar behavior was observed when the slope of the linear potential (strength of gravity) is changed with appropriate frequency (see the right panel of Fig. 3). Since the resonant frequencies are different for the center of mass (ω 0 ) and width (Ω 0 ) of the soliton, periodic modulation of the parameter α or γ with frequency ω 0 does not induce resonant oscillations of the width, and vice versa. Characteristic feature inherent to both cases is that, oscillations show notable phase shift as compared to predictions of VA, which can be explained by asymmetric deformation of the soliton at the impact with the reflecting surface. In the VA we deal with the dynamics of a unit mass particle in the anharmonic potential. Nevertheless the VA provides qualitatively correct description of the system.
The focusing nonlinearity, inherent to BEC with negative s-wave scattering length, provides the wavepacket's robustness against dispersive spreading and different kinds of perturbations. Due to this property matter wave solitons keep their integrity after reflection from the atomic mirror. Below we consider the possibility of Fermi type of acceleration in the system. In numerical simulations we take the stationary state of the wave packet, predicted by VA as initial condition for Eq. (3) and periodically change the vertical position of the reflecting surface or the slope of the linear potential.
The figure 5 illustrates the progressive gain of energy by the soliton when the position of the reflecting delta potential is periodically varied in time at a parametric resonance frequency. As the amplitude of oscillation above the mirror increases, de-tuning from the resonance occurs and further gain of energy stops. A proper synchronization would allow more increase of the kinetic energy of the soliton. Also there is a contribution of tunnel loss of the wave packet through the reflecting delta potential barrier.
The corresponding predictions of the VA for the center of mass position ζ and width a are also shown on the right panel of Fig. 5. Note that the space coordinate in GPE and VA equations are designated by x and ζ respectively. Variation of the vertical position of the delta function mirror V (x) = V 0 δ(x + f (t)), where f (t) = ε sin(Ωt) is a periodic function with amplitude ε and frequency Ω, leads to the VA equations, similar to Eqs. (8) -(9), but with replaced space variable on the right hand side ζ → ζ + f (t). The frequency of small amplitude oscillations of the width, measured at upper turning point is T 0 ≃ 2.1, which is close to the estimation from Eq. (13).

V. CONCLUSIONS
The model of a "quantum bouncer" has been extended to a nonlinear domain of Bose-Einstein condensates. The analytical description is based on the variational approach. It has been revealed that a matter wave soliton bouncing above the reflecting surface (or atomic mirror) better preserves its integrity compared to a linear wave packet due to the focusing effect of the nonlinearity. This feature of the bright matter wave soliton allows to develop a variational approach, using appropriate trial function, which provides a qualitatively correct description of its dynamics. A particle like behavior of the matter wave soliton bouncing above the atomic mirror suggested to consider the Fermi type acceleration in the system. In numerical experiments we observed the progressive energy gain by the soliton when the vertical position of the mirror is periodically varied in time. Further development of the proposed model may include the stochastic variation of the nonlinearity, the slope of the linear potential, and vertical position of the reflecting surface.