Cold Atoms bouncing on modulated atomic mirror exhibits acceleration and dynamical localization subject to modulated strength. We explain characteristics of acceleration and define control parameters in terms of effective Planck's constant. We show that the effective Planck's constant plays a vital role in limiting classical-like overall linear growth of the variance of accelerated atoms with time. For large values of the effective Planck's constant the atomic quantum acceleration is seized as localization window overlaps the accelerated window.

1. Introduction

Fermi Accelerator is a system well investigated to study Hamiltonian chaos and its manifestations in quantum mechanical systems [1, 2]. In 1961, Ulam suggested the accelerator as a two-wall system, one fixed and the other oscillating with a frequency, named as Fermi-Ulam Accelerator. At the end of the seventies, PustylΕ„ikov replaced fixed wall by gravity and discussed unbounded acceleration for it. The classical dynamics in Fermi accelerator is described by the Standard Map which explains that the stochasticity in the phase space increases with the driving strength, and when the latter is sufficiently strong global diffusion takes place. In latter accelerator model, for particular set of initial data in phase space and modulation strength [3], onset of unbounded acceleration modes takes place [4–7]. In this paper we explain Fermi-PustylΕ„ikov accelerator and show that by increasing effective Planck’s constant the acceleration of a bouncing particle in the accelerator system is modified, due to the onset of dynamical localization [8].

A classical system subject to time-periodic modulation, in general, becomes globally chaotic for increasing modulation strengths and absorbs energy from the external field in a diffusive way. However, in corresponding quantum domain the diffusive dynamics may be suppressed by quantum interference effect. It is a manifestation of dynamical localization phenomenon in the system, which is an analogous to Anderson localization of solid state physics. The phenomenon has been discussed in model systems in quantum chaos, such as, kicked rotator [9], modulated quantum bouncer [10], atoms in modulated standing wave fields [11], ion in a Paul trap [12, 13], and molecular systems in the presence of electric and magnetic fields [14]. Dynamical localization is a general phenomenon in periodically driven systems [15, 16]. The π‘‘π‘’π‘™π‘œπ‘π‘Žπ‘™π‘–π‘§π‘Žπ‘‘π‘–π‘œπ‘› in such quantum systems is a purely quantum effect since the long-time unbounded propagation is not related to the corresponding classical diffusion [2, 17].

In Section 2, we describe the physical system and develop its Hamiltonian. In Section 3, we discuss the diffusion versus acceleration in Fermi accelerator with the help of acceleration windows. In Section 4, we discuss quantum control on accelerator dynamics, discuss results, and explain numerical data based on analytical calculations in Section 5.

2. The Experimental System

Thirty years after the first suggestion of Fermi, PustylΕ„ikov provided detailed study of the accelerator model, which we call Fermi Accelerator or modulated quantum bouncer [4, 18] in this paper. In his work, PustylΕ„ikov proved that a particle bouncing in the accelerator system attains modes, where it always gets unbounded acceleration. The feature makes the Fermi-PustylΕ„ikov model richer in its dynamical beauty. In the atomic Fermi accelerator, atoms move under the influence of gravitational field towards an atomic mirror made up of an evanescent wave field. The atomic mirror is provided a spatial modulation by means of an acousto-optic modulator which provides intensity modulation to the incident laser light field [10]. Hence, an ultra-cold two-level atom, after a normal incidence with the modulated atomic mirror, bounces off and travels in the gravitational field, as shown in Figure 1. In order to avoid any atomic momentum along the plane of the mirror the laser light which undergoes total internal reflection is reflected back. Therefore, we find a standing wave in the plane of the mirror which avoids any specular reflection [19]. The periodic modulation in the intensity of the evanescent wave optical field may lead to the spatial modulation of the atomic mirror as 𝐼𝑧,𝑑=𝐼0π‘’βˆ’2πœ…π‘§+π‘Žcos(πœ”π‘‘).(2.1) Thus, the motion of the atom in 𝑧-direction follows effectively the Hamiltonian 𝐻=𝑝𝑧22π‘š+π‘šπ‘”π‘§+ℏΩeο¬€π‘’βˆ’2πœ…π‘§+π‘Žcos(πœ”π‘‘),(2.2) where Ξ©eff denotes the effective Rabi frequency [5]. Moreover, π‘Ž and πœ” express the amplitude and the frequency of the external modulation, respectively.

3. Diffusion versus Acceleration in Fermi Accelerator

In case the decay constant πœ… of the evanescent wave field is large, simplified Hamiltonian of our system in moving coordinates becomes 𝐻=𝑝𝑧22π‘š+π‘šπ‘”Μƒπ‘§+𝑉̃𝑧cosπœ”π‘‘,̃𝑧β‰₯0,(3.1) where ̃𝑧=π‘§βˆ’(πœ–/2πœ…)cosπœ”π‘‘. The Hamiltonian given in (3.1) describes a particle of mass π‘š bouncing off an oscillating hard surface in the presence of gravitational field. We proceed onward by introducing the dimensionless position and momentum coordinates. Here, we define 𝐻=𝐻(πœ”2/π‘šπ‘”2) as dimensionless Hamiltonian, 𝑉=β„πœ”2Ξ©/4π‘šπ‘”2 potential of the external field and then other parameters for this Hamiltonian system as, πœ–=πœ”2π‘Ž/2πœ…π‘”, 𝑑=πœ”π‘‘, and dimensionless Planck’s constant, kβˆ’=ℏ(πœ”3/π‘šπ‘”2). Hence, the Hamiltonian takes the dimensionless form as 𝑝𝐻(𝑧,𝑝,𝑑)=22+𝑧+πœ–π‘§cos𝑑,𝑧β‰₯0.(3.2)

The Hamiltonian system is integrable in the absence of time dependent term. We may express the time development of the particle moving in time dependent system by the impact map which gives the evolution immediately after a bounce in terms of immediately after the previous bounce [1, 2], that is, β„˜π‘–+1=β„˜π‘–ξ€·πœ™+𝐾sin𝑖,πœ™π‘–+1=πœ™π‘–+β„˜π‘–+1.(3.3) The map obtained in (3.3) is the Standard Map, where β„˜π‘–=2𝑝𝑖 and πœ™π‘–=πœ”π‘‘π‘–. The onset of diffusive excitation in the system takes place as the chaos parameter 𝐾=4πœ– takes a value larger than πΎπ‘π‘Ÿβ‰ˆ0.96, or when the perturbation amplitude exceeds the critical value πœ–π‘π‘Ÿ=0.24 [20], while the quantum evolution remains localized until a larger value πœ†π‘’ of the modulation. Above that point both the classical and the quantum dynamics are diffusive. However, for specific set of initial conditions that originate within phase space disks of radius 𝜌, accelerating modes appear for values of the modulation strength πœ– within the windows [4–6] ξ‚΅π‘ πœ‹1βˆ’πœ‰ξ‚ΆξƒŽ1+πœ‰β‰€πœ–<1+(π‘ πœ‹)2ξ‚΅1βˆ’πœ‰ξ‚Ά1+πœ‰2,(3.4) where, 𝑠 can take integer and half-integer values for the sinusoidal modulation of the reflecting surface considered here. As the fundamental requirement for the acceleration is met by choosing a modulation strength within the acceleration windows and the bouncing particle from supporting areas of phase the particle displays acceleration discussed in Figure 2. We found numerically that for a modulation strength outside the windows of (3.4) the dynamics is dominantly diffusive as shown in Figure 3(a). However, for the ensemble originating from areas of phase space of radius 𝜌, exhibit unbounded acceleration for modulation strength from acceleration windows, as shown in Figure 3(b). A small diffusive background results from a small part of the initial distribution which is residing outside the area of phase space supporting acceleration. This coherent acceleration restricts the momentum space variance Δ𝑝 which then remains very small indicating the absence of diffusive dynamics.

4. Quantum Control on Acceleration Dynamics

As a matter of fact, the variables β„˜ and πœ™ in (3.3) do not form a conjugate pair in the full Hamiltonian formulation of the model. The variable conjugate to the phase πœ™ is the quantity 𝑁=𝐸, where 𝐸 is unperturbed energy [16] and its value can be determined as 𝐸=(3πœ‹πΌ)2/3/2. By introducing β„˜π‘–+1=√2(𝑁𝑖+1βˆ’π‘§) and β„˜π‘–=√2(π‘π‘–βˆ’π‘§) in (3.3), the impact map in conjugate variables (𝑁,πœ™) takes the form 𝑁𝑖+1=π‘π‘–βˆš+2πœ–2𝑁𝑖sinπœ™π‘–,πœ™π‘–+1=πœ™π‘–βˆš+22𝑁𝑖+1+𝑂(πœ–),(4.1) here we neglected higher-order terms. The preservation of phase space volume for Hamiltonian systems has a consequence that there are no attractors, that is, no subregions of lower phase-space dimension to which the motion is confined asymptotically. The map given in (4.1) is the analogue of the Kepler map which was found very helpful in the hydrogen atom problem [21]. A classical analysis of mapping given in (4.1) leads to predict the onset of chaos under the same conditions found for the Standard Map description. Above the chaotic threshold a diffusive growth of 𝑁 is observed. Consider the phase space defined in conjugate coordinates (𝑁,πœ™) and introduce the initial distribution of phase-space points 𝑓(𝑁,πœ™,𝑑=0). The time evolution of 𝑓 is described by a Fokker-Planck equation πœ•π‘“=1πœ•π‘‘2πœ•ξ‚΅π·πœ•π‘π‘πœ•π‘“ξ‚Άπœ•π‘,(4.2) where 𝑑 is time measured in the number of iterations of the map, that is, in the numbers of the bounces. The Fokker-Planck (4.2) can be solved by the method of characteristics [22]. We consider the conjugate pair mapping as in (4.1), switching parameters and with the help of (4.2), we get the interesting relation between, Δ𝑁, and kβˆ’, that is, Δ𝑁2=4π‘π‘—πœ–2kβˆ’.(4.3) Here, Δ𝑁 is the dimensionless energy and 𝑗 describes the number of bounces. The diffusion coefficient 𝐷𝑁 is 𝐷𝑁=4πœ–2𝑁kβˆ’.(4.4) We plot Δ𝑁 versus time for different values of kβˆ’, shown in Figure 4, which illustrates that, as we increase the value of the kβˆ’, we expand the dynamical localization window, which is √0.24<πœ–<kβˆ’/2. In this window, the lower boundary πœ–π‘™=0.24 is set by classical dynamics, we can find the πœ–π‘™ by evaluating Lyapunov exponents and from the standard map. For a modulation amplitude πœ–<0.24, the Lyapunov exponent converges to zero in a vast range of initial conditions, expect in small regions near separatrices. In the simulation, we initially take our wave packet as Gaussian distribution. A comparison between quantum mechanically and classical values of Δ𝑁 versus kβˆ’ is shown in Figure 5. We take the average value of Δ𝑁 for last 400 to 1000 times. For different values of effective Planck’s constant kβˆ’, we extend localization window towards accelerated window. We note that Δ𝑁 displaying a decaying behavior for the higher values of effective Planck’s constant, kβˆ’.

5. Results and Discussions

In this contribution, we derive a quantum mechanical map for an atom bouncing off a modulated mirror under the influence of gravity by reducing standard map into conjugate pair map. We write the standard mapping as conjugate pair mapping in quantum domain which provides dispersion law and its dependence on system parameters, such as modulation strength, energy, effective Planck’s constant, and number of bounces. Our numerical result show a very good agreement with analytically obtained results. We show the dependence of Δ𝑁 on kβˆ’ by numerically and by deriving the Fokker Planck equation for our system as shown in Figure 5.

As discussed in Section 4 the localization occurs in the quantum modulated bouncer in a window defined by classical and quantum dispersion laws. The latter is a function of effective Planck’s constant and follows a square root law. Whereas the accelerated dynamics takes place in the system for another window on modulation strength in the presence of initial areas in phase space [2]. As we increase the value of the kβˆ’, we extend localization window towards the accelerated window. For larger value of kβˆ’ variance in energy decreases and the two windows may overlap, where we do not find any accelerated dynamics, but dynamical localization as shown in Figure 4.


Authors thank Rameez-Ul-Islam, M. Ayub, and T. Abbas for stimulating discussions and useful suggestions. F. Saif thanks Higher Education Commission for financial support under Grant no. R&D03143. F. Saif submits his thanks to Professor E. D. Leonel for his invitation to contribute in the special issue of Mathematical Problem in Engineering.