Quantum slow motion

Abstract

We investigate the center-of-mass motion of cold atoms in a standing amplitude modulated laser field. We use a simple model to explain the momentum distribution of the atoms after any distinct number of modulation cycles. The atoms starting near a classical phase-space resonance move slower than we would expect classically. We explain this by showing that for a wave packet on the classical resonances we can replace the complicated dynamics in the quantum Liouville equation in phase space by its classical dynamics with a modified potential. To have an intuitive picture of the quantum-mechanical dynamics of a wave packet we are usually confined to the semiclassical regime, that is, to orbits with action large compared to Planck's constant ͓1,2͔, or to special systems like the harmonic oscillator, where the quantum evolution equations in phase space are identical to the classical ones ͓3͔. In this paper we analyze the center-of-mass motion of cold atoms in an amplitude-modulated standing laser field in the limit of large detuning. In this limit we can describe the dynamics by a sinusoidally modulated cosine potential.In terms of this physical system we propose a scheme which enables us to describe a wave packet, localized near a resonance of a classical mixed phase space, by classical dynamics in a modified potential. We apply the theory of Henriksen et al. ͓4͔ to replace the potential in the high-order quantum Liouville equation by an effective potential in such a way that we obtain a classical Liouville equation. Hence we describe the quantum motion as a classical motion in this modified potential. We are then able to characterize the quantum effect by comparing the modified dynamics with the dynamics in the original potential. This method is applicable well beyond the semiclassical regime for many different potentials.Usually quantum effects on wave packets express themselves in the revival and fractional revival properties ͓5͔ or in the occurrence of tunneling phenomena ͓6͔. Both take place on a comparatively long-time scale so that we intuitively do not expect quantum effects to be visible on a short-time scale. We disprove this intuitive assumption in our model where we use the center-of-mass motion of cold atoms in a standing amplitude-modulated laser field. Here we demonstrate that the momentum distribution after each cycle of the modulation is peaked at smaller momenta than we would expect classically. This shows that the atoms are traveling slower than we would expect from classical simulations and we can give a very simple explanation of this ''quantum slow motion'' phenomenon.We investigate a cloud of two-level atoms situated in a standing laser field with a periodic modulated amplitude. This system has been the subject of several experiments ͓7,8͔. The Hamiltonian of the center-of-mass motion in the limit of large detuning is ͓9͔ H͑t ͒ϭ p 2 2 Ϫ͑1Ϫ2⑀ cos t ͒cos q, ͑1͒where p and q denote scaled dimensionless momentum and position, t time, and and ⑀ are the parameters defining the depth of the standing w...

Experimental tests of quantum nonlinear dynamics in atom optics

“…However, if one assumes the mean of maximum and minimum momenta occupied by the region of regular motion (when positioned on the momentum axis) as a classical momentum approximation for the experimentally observed momentum peak, the classical resonance momenta are still significantly faster than the experimentally measured values. These results suggest a possible explanation in terms of the theory of quantum slow motion [44], but more rigorous and detailed investigation would be needed to confirm this. The theory of quantum slow motion shows that the complicated quantum dynamics of the atomic wavepacket on a classical resonance given by the quantum Liouville equation can be replaced by the wavepacket's (in fact, the corresponding classical probability distribution's) classical dynamics with a modified potential within certain limits.…”

“…For most tunnelling phenomena which have been observed in the past, for example α-particle decay, conservation of energy forbids the process classically. Furthermore, Hug and Milburn [44] showed that quantum mechanical velocity predictions for phase space resonances disagree by up to 20% with classical predictions in the atomoptics-driven pendulum. Experimental results which may possibly be linked to this phenomenon will be discussed in section 5.…”

“…The different density matrices are evolved and the results are combined to gain a more accurate result. The final momentum distribution after an interaction time t is obtained by calculating n|ρ(t)|n (44) which is equal to the number of atoms having momentum p 0 +hkn. To obtain smoother results a new set of momentum states is chosen, still spaced byhk, but shifted in momentum by fractions ofhk compared to the original set.…”

Atoms in an amplitude-modulated standing wave - dynamics and pathways to quantum chaos

“…To date, we know of two kinds of quantum effects for this system. Firstly, the velocity of the resonances predicted in quantum and classical models differs significantly for values of large k. A velocity shift was predicted theoretically by Milburn et al in the theory of quantum slow motion [14].…”

Macroscopic Quantum Self-Trapping in Dynamical Tunneling

“…The behaviour of such systems has provided important insights into the quantum-classical transition [7][8][9][10][11][12][13][14]: in particular, the period of the dynamical tunnelling is strongly affected by a number of subtle effects [11][12][13][14]. Dynamical tunnelling has mostly been studied in the single-particle regime [12,[15][16][17][18][19], and has been demonstrated experimentally with ultra-cold atoms in modulated optical lattice potentials [20,21]. Recently it has been shown in [14] that atomic interactions in trapped Bose-Einstein condensates (BECs) can have a detectable effect for experimentally realistic parameters.…”