CONTROLLED ACCELERATION OF A MODULATED ... - Springer Link

Journal of Russian Laser Research, Volume 30, Number 2, 2009

CONTROLLED ACCELERATION OF A MODULATED QUANTUM BOUNCER

Javed Akram,∗ Khalid Naseer, and Farhan Saif Department of Electronics Quaid-i-Azam University 45320 Islamabad, Pakistan ∗ Corresponding

author, e-mail: jvdakramqau @ gmail.com e-mail: Farhan.Saif @ fulbrightmail.org Abstract

Cold atoms bouncing on a modulated atomic mirror exhibit dynamic localization and acceleration subject to modulated strength. We explain the characteristics of acceleration using accelerated mapping and define control parameters. We show that the effective Planck’s constant plays a vital role in limiting overall linear growth of accelerated atoms with time. For large values of the effective Planck’s constant, the atomic quantum accelerated is seized if the localization window overlaps the accelerated window.

Keywords: Fermi accelerator, quantum bouncer, accelerating mapping, dynamical localization, quantum mapping and unbounded acceleration, dynamical delocalization.

1.

Introduction

A classical system subjected to time-periodic modulation, in general, becomes globally chaotic at increasing modulation strengths and absorbs energy from the external field in a diffusive way. However, in the corresponding quantum domain the diffusive dynamics may be suppressed by quantum interference effects. It is a manifestation of the dynamical localization phenomenon in the system, which is an analog of the Anderson localization of solid state physics. The phenomenon has been discussed in model systems in quantum chaos, such as kicked rotator [1], modulated quantum bouncer [2], atoms in modulated standing wave fields [3], ion in a Paul trap [4, 5], and molecular systems in the presence of electric and magnetic fields [6]. Dynamical localization is a very general phenomenon in periodically driven systems [7, 8]. The delocalization in such quantum systems is a purely quantum effect since the long-time unbounded propagation is not related to the corresponding classical diffusion [9, 10]. The Fermi accelerator or modulated quantum bouncer is a system well-investigated in studying the classical Hamiltonian chaos and its manifestations in quantum-mechanical systems [11]. Its dynamics is described by the standard map, which explains that the size of the stochasticity in the phase space of the map increases with the driving strength, and when the latter is sufficiently strong, the global diffusion takes place. However, for a particular set of initial data in the phase space and conditions on modulation strength [12], the onset of accelerator modes takes place [13, 14]. In recent experiments [15, 16], atoms move in a direction parallel to the gravitational acceleration in a quantum bouncer. A fraction of the atoms may accelerate in the presence of a driving field, at a rate Manuscript submitted by the authors in English first on February 18, 2009 and in final form on March 19, 2009. c 1071-2836/09/3002-0157 2009 Springer Science+Business Media, Inc. 157

Journal of Russian Laser Research

Volume 30, Number 2, 2009

that is faster or slower than the gravitational acceleration [10]. Such atoms are exempt from the diffusive spread [11]. Acceleration of atoms that takes place in a modulated standing wave field in the presence of gravity is suggested to resemble the accelerator modes in the standard map [17]. The accelerating parts of the distributions were hence termed quantum accelerator modes. In this paper, we discuss the quantum accelerator mapping for acceleration of atoms in a modulated quantum bouncer. We explain the control of acceleration and its transition to dynamic localization with increasing Planck’s constant. In Sec. 2, we describe the physical system and its Hamiltonian. In Sec. 3, we derive the quantummechanical form of conjugate-pair mapping and also discuss the controlling conditions; moreover, we discuss the Fokker–Planck equation for the given system. In Sec. 4, we discuss the mapping suitable for explaining the accelerated dynamics. In Sec. 5, we discuss the results and explain our numerical data based on analytical calculation.

2.

The Experimental System

Thirty years after the first suggestion of Fermi, Pustyl’nikov provided a detailed study of another accelerator model, which is presently called the Fermi accelerator or modulated quantum bouncer [18,19]. In his work, Pustyl’nikov proved that a particle bouncing in the accelerator system attains modes, where it always gets unbounded acceleration. The feature makes the Fermi–Pustyl’nikov model richer in its dynamic beauty. In the atomic Fermi accelerator, atoms move under the influence of the gravitational field towards an atomic mirror made up of an evanescent wave field. The atomic mirror provides a spatial modulation by means of an acousto-optic modulator which, in turn, provides the intensity modulation of the incident laser- Fig. 1. A cloud of atoms is trapped and cooled in light field [2]. Hence, an ultracold two-level atom, after a magneto-optical trap (MOT) up to a few microkelvin. MOT is placed at a height at the start of the a normal incidence with the modulated atomic mirror, experiment. On switching off the MOT, atoms move bounces off and travels in the gravitational field as with constant gravitational acceleration towards the shown in Fig. 1. In order to avoid any atomic mo- exponentially decaying field. mentum along the mirror plane, the laser light, which undergoes the total internal reflection, is reflected back. Therefore, we find a standing wave in the mirror plane, which avoids any specular reflection [20]. The periodic modulation in the intensity of the evanescent wave of the optical field may lead to the spatial modulation of the atomic mirror as follows: ¯ z , t¯) = I0 exp [−2κ¯ I(¯ z + cos(ω t¯)] . (1) Thus, the atom’s motion in the z¯ direction follows effectively the Hamiltonian 2 ¯ = p¯z + mg¯ H z + ~Ωeff exp [−2κ¯ z + cos(ω t¯)] , (2) 2m where, Ωeff is the effective Rabi frequency and and ω denote the amplitude and frequency of the external modulation, respectively.

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3.


Quantum Mapping

In the case where the decay constant κ of the evanescent wave field is large, the simplified Hamiltonian of our system in moving coordinates becomes 2 ¯ = p¯z + mg˜ H z + V z˜ cos ω t¯, 2m

z˜ ≥ 0,

(3)

~ω 2 Ω cos ω t¯. Here, V = is the potential of the external field. 2κ 4mg 2 We proceed by introducing the dimensionless position and momentum coordinates and define H = 2 ω V ¯ H as the dimensionless Hamiltonian. The other parameters for this Hamiltonian system are = , 2 mg mg ω3 t = ω t¯ and, finally, dimensionless Planck’s constant is − k = ~ . The Hamiltonian given in Eq. mg 2 (3) describes a particle with mass m bouncing on an oscillating hard surface in the presence of the gravitational field. Hence, the Hamiltonian takes a dimensionless form, namely, where z˜ = z¯ −

H(z, p, t) =

p2 + z + z cos t, 2

z ≥ 0.

(4)

This Hamiltonian is integrable in the absence of time-dependent terms. One can express the time evolution of the particle moving in a time-dependent system by the impact map, which gives the evolution immediately after a bounce expressed through the parameters of the previous bounce [11], that is, ( pi+1 = −pi + 4ti + 2 sin(ti ), (5) ti+1 = ti + 4ti . The mapping given in Eq. (5) takes a simple form when the amplitude of the motion between the subsequent kicks is much larger then the amplitude of the surface oscillations, that is, ( ℘i+1 = ℘i + K sin(φi ), (6) φi+1 = φi + ℘i+1 . The map obtained in Eq. (6) is the standard map, where ℘i = 2pi and φi = ωti . The onset of diffusive excitation in the system appears when the chaos parameter K = 4 takes a value larger than 0.96, i.e., when the perturbation amplitude exceeds the critical value cr = 0.24 [21]. The fact that the standard map provides a good description of the classical model does not imply any similarity of the quantum model with the quantum kicked rotator (which is a quantization of the classical standard map). As a matter of fact, the variables ℘ and φ in Eq. (6) do not create a conjugate pair in the full Hamiltonian formulation of the model. The variable conjugate to the phase φ is the quantity N = E, where E is the (3πI)2/3 unperturbed energy [8] and its value can be determined as E = . In the conjugate variables 2 (N, φ), the impact map given in Eq. (6), neglecting the second-order terms in , takes the form ( √ Ni+1 = Ni + 2 2Ni sin φi , (7) √ φi+1 = φi + 2 2Ni+1 + O().

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Preserving the phase-space volume for Hamiltonian systems has a consequence that there has no attractors, that is, no subregions of lower phase-space dimensions to which the motion is confined asymptotically. The map given in Eq. (7) is an analog of the Kepler map which was found very helpful in the hydrogen-atom problem [22]. A classical analysis of mapping given in Eq. (7) leads to the prediction of the onset of chaos under the same conditions that have been found for the standard-map description. Above the chaotic threshold, a diffusive growth of N is observed, which we discuss below.

Fig. 2. Expectation value of the position z versus time for = 1.7 and 4p=0.5.

Consider the phase space defined in conjugate coordinates (N, φ) and introduce the initial distribution of phase-space points f (N, φ, t = 0). The time evolution of f is described by the Fokker–Planck equation ∂f 1 ∂ = ∂t 2 ∂N

∂f DN ∂N

,

(8)

where t is time measured in the number of iterations of the map, i.e., in the number of bounces. The diffusion coefficient DN is 42 N DN = − . (9) k The Fokker–Planck equation (8) can be solved by the method of characteristics [23].

4.

Accelerated Mapping We define a new type of mapping as (

℘i+1 = α℘i + S(1 + α) cos φi , φi+1 = φi + ℘i+1 ,

α ≥ 1,

(10)

where S = 2 . If we put α = 1, we obtain the standard map. We can determine the value of α from the numerics, as shown in Fig 2. In our calculations, the average value of α is 5.9. Here, α is a dimensionless quantity; it is the ratio of momenta before and after the kick. From Eq. (10), we find that by increasing or decreasing the value of α, we can increase or decrease the momentum gain of the accelerated particle.

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We consider the conjugate-pair mapping similar to Eq. (7) containing the switching parameters and with − , that is, the help of Eq. (8) obtain the interesting relation between 4N and k 4N 2 =

4N j2 , − k

(11)

where 4N is the dimensionless energy and j describes the number of bounces. We plot 4N versus time − (see Fig. 4). In Fig. 4, we show that with increase in the value of k − , we expand for different values of k √ − the dynamic localization window, which is 0.24 < < k /2. In this window, the lower boundary l is set by classical dynamics; we can find l by evaluating the Lyapunov exponent. For a modulation amplitude < 0.24, the Lyapunov exponent converges to zero in a vast range of the initial conditions (expect in small regions near separatrices). In the simulation, we initially take our wave packet as the Gaussian; in the position distribution, it reads ψ(z, 0) =

1 2π 4 z 2

1/4

z − z0 2 exp − , 24z

(12)

where 4z describes the width of the atomic wave packet. In order to calculate the wave function of the system after a certain time of the evolution, we integrate the time-dependent dimensionless Schr¨ odinger wave equation. Information on the system at a later time t is stored in the wave function ψ(z, t), which leads to the probability distribution in the position space, that is, P (z, t) =| ψ(z, t) |2 . A comparison − is shown in Fig. 3. We take the between the quantum-mechanical and classical values of 4N versus k − . We note that 4N displays average value of 4N for the last 400 to 1000 times for different values of k − a decaying behavior when the effective Planck’s constant k increases.

Fig. 3. 4N versus − k at = 1.7, N = 1235, and j = 15 Fig. 4. 4N versus time for different values of effective with numerical results of 4N (squares on curve) and the Planck’s constant − k = 1 (circles on curve), 7 (stars on corresponding analytical results obtained in Sec. 3. curve), and 16 (triangles on curve). The rest parameters are the same as in Fig. 2. One can see that an increase in the value of − k leads to a decrease in the dispersion.

161


5.


Results and Discussions

Here we derive a quantum-mechanical map for an atom bouncing off a modulated mirror under the influence of gravity by reducing the standard map into the conjugate-pair map. The mapping explains the accelerated modes and provides an average momentum gain in terms of the parameter α. The accelerated mapping explains the accelerated dynamics of the particles when α is positive and greater than unity. Earlier, we write the standard mapping as a conjugate-pair mapping in the quantum domain, which provides the dispersion law and its dependence on the system parameters, such as the modulation strength, energy, effective Planck’s constant, and number of bounces. Our numerical results show a good agreement with analytically obtained results. We obtained the dependence of 4N on − k numerically and derived the Fokker–Planck equation for our system, as shown in Fig. 3. As was discussed in Sec. 4, the localization occurs in the quantum modulated bouncer in a window defined by the classical and quantum dispersion laws. The latter is a function of the effective Planck’s constant and follows a square-root law, whereas the accelerated dynamics takes place in the system for another window of modulation strength in the presence of initial areas in the phase space [10]. With an increase in the value of − k . the localization window extends towards the accelerated window. For larger − values of k the two windows can overlap, where we do not find any accelerated dynamics, but the dynamic − the energy localization is shown in Fig. 4, from which we conclude that with an increase in the value k dispersion decreases.

6.

Acknowledgments

The authors thank Rameez-ul-Islam, M. Ayub, T. Abbas, and A. Khosa for stimulating discussions and useful suggestions. F.S. thanks the Higher Education Commission for financial support under Grant No. R&D03143.

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