Dynamical Localization: Classical vs Quantum Oscillations in Momentum Spread of Cold Atoms

Abstract

We investigate the classical and quantum dynamics of atoms moving in a phase-modulated standing light field. In both cases the width of the momentum distribution exhibits characteristic oscillations as a function of the modulation amplitude. We argue that at the maxima of these oscillations the system is chaotic, whereas in the valleys it is almost regular. Quantum localization appears only in the chaotic regime. We connect our analysis with a recent experiment [F. Moore et al. , Phys. Rev. Lett. 73, 2974(1994].PACS numbers: 05.45.+b, 42.50.Lc, 42.50.Vk Most recently the description of cold atoms [1] in the framework of atom optics [2] has opened [3 -5] a new avenue in the search for fingerprints of classical chaos in quantum systems: A strongly detuned atom in a standing light field moves like a particle in a spatially periodic potential [6,7]. An atom in a phase-modulated light field additionally experiences a time dependent force [3]. Classically, the resulting motion can be chaotic [8]. But how does classical chaos manifest itself in the quantum dynamics of the atom and how to reach the quantum domain'l The landmark experiment [4,5] by the Austin group answers these questions: The measured momentum transfer of atoms in a phase-modulated light field shows the characteristic signature of quantum chaos: dynamical localization. Moreover, the appropriate choice of the experimental parameters such as the wave number of the standing light field or the modulation frequency allows one to step from regimes which ask for a classical description to regimes which ask for a quantum description.In the present paper we predict a new effect in this playground of atom optics and quantum chaos: The width of the atomic momentum distribution shows characteristic oscillations as a function of the modulation depth A.These oscillations appear in the classical as well as in the quantum description. In the domains of A where maxima appear the classical system is chaotic, whereas in the valleys the motion is almost regular. In the quantum case the peaks are lower, that is the quantum mechanical momentum distribution is narrower than the classical one. Moreover, it is an exponential rather than a Gaussian a signature of quantum localization [9,10]. The motion of a strongly detuned atom of mass M, position x, and momentum p along a standing light field of wave number k, is described by the Hamiltonian H = P /2M -Vpcos(2kx), where Vp is the height of the periodic potential determined by the coupling of the atom to the light field [6]. The motion of the atom in a phase-modulated light field, as created, for example, by an oscillating mirror, is described [3] by the Hamiltonian H = P /2M -Vp cos[2kx -csin(tot)], where A denotes -+ p --a sin(x -Asinr) P(x, p;r) = 0 (2) l97 Bx Bp and periodic boundary conditions P(x + L, p; r) = P(x, p; r), where L is a multiple of 2~. The solution of Eq. (2) satisfying the initial condition P(xp, pp', r = 0) = Po(xp, pp) has the form P(x, p;r) = dxo dpp 6[x g(r; xp, pp)] X 6[p -7r(r;xp, pp)] && Po(xo, po)...

Homoclinic signatures of dynamical localization

“…While no classical or semi-classical explanation for DL has as yet been proposed, a formal analogy with the Anderson localization appearing in disordered systems has been discussed [7]: DL occurs over the time coordinate with localization in momentum space, while Anderson localization refers to spatial coordinates. Additional evidence of the DL phenomenon has been provided by the momentum transfer from a modulated standing light wave to a sample of ultracold atoms [8][9][10][11][12]. It was found experimentally [9,11] and numerically [10,11] that the width of the atomic momentum distribution exhibits oscillations as a function of the modulation amplitude, while a theoretical analysis [8] predicted its overall monotonous decrease for sufficiently large modulation amplitudes (quantum regime).…”

“…Additional evidence of the DL phenomenon has been provided by the momentum transfer from a modulated standing light wave to a sample of ultracold atoms [8][9][10][11][12]. It was found experimentally [9,11] and numerically [10,11] that the width of the atomic momentum distribution exhibits oscillations as a function of the modulation amplitude, while a theoretical analysis [8] predicted its overall monotonous decrease for sufficiently large modulation amplitudes (quantum regime). Also, a preliminary qualitative analysis of such oscillatory behavior was previously discussed in terms of the zeros of Bessel functions [10,11].…”

“…It was found experimentally [9,11] and numerically [10,11] that the width of the atomic momentum distribution exhibits oscillations as a function of the modulation amplitude, while a theoretical analysis [8] predicted its overall monotonous decrease for sufficiently large modulation amplitudes (quantum regime). Also, a preliminary qualitative analysis of such oscillatory behavior was previously discussed in terms of the zeros of Bessel functions [10,11]. However, no theoretical approach has as yet been proposed to explain in detail the waveform of the aforementioned oscillations or the amplitude values at which the maximum effect of DL had been found experimentally and numerically.…”

Corrigendum to “Classical and quantum chaos in atom optics”

“…The study of atomic dynamics in a phase modulated standing wave by Graham, Schlautman and Zoller (Graham 1992), made atom optics a testing ground for quantum chaos as well. Their work got experimental verification as the dynamical localization of cold atoms was observed later in the system in momentum space (Moore 1994, Bardroff 1995, Amman 1997.…”

Dynamical localization and signatures of classical phase space

“…Rapid developments in atom optics [2][3][4] have made this subject a testing ground for the dynamical localization and hence for quantum chaology. Atomic dynamics in periodically driven systems, such as, an hydrogen atom in micro-wave field [5][6][7] an atom in modulated standing wave field [8][9][10], and the motion of an ion in Paul trap in presence of standing wave [11][12][13], have manifested the phenomenon of dynamical localization. Latest work on the dynamics of an atom in Fermi accelerator [14][15][16] has established the presence of dynamical localization in the system.…”