Quantum Coherence and Decoherence by Spontaneous Emission in a Quantum Optical Realization of a Driven Pendulum

Abstract

A quantum version of the periodically driven pendulum can be realized by a system of laser-cooled two-level atoms subject to an off-resonance standing wave laser field periodically modulated in time. In the fully chaotic state the classical angular momentum of the driven pendulum diffuses, like in a kicked rotor, via a random walk, while coherence effects give rise to dynamical localization of the quantized angular momentum. For the quantum optical realization this implies that classically, in the fully chaotic state, the momentum transferred to the atoms from the periodically modulated laser field grows diffusively in time, limited only by the size of the chaotic domain, while quantum mechanically the momentum transfer is limited by dynamical localization, i.e. a quantum coherence effect analogous to Anderson localization, but in momentum space rather than in real space and without any randomness in the Hamiltonian. A classical to quantum cross-over of the observed momentum transfer as a function of the modulation strength or frequency is therefore predicted to occur as soon as the localization length becomes smaller than the width in momentum of the classically chaotic domain. The experiment has recently been carried out in Mark Raizen’s group in Austin. Even though the experiment still operates in a domain where sizeable regular islands are embedded in the chaotic domain there is reasonable agreement with the theory, and dynamical localization can be observed for some parameter values of the modulation strength. We also discuss the influence of spontaneous emisssion on the quantum coherence effects influencing the atomic momentum transfer. Cases where the momentum transfer is limited classically by regular islands, and quantum mechanically by dynamical localization are affected in very different ways. For dynamically localized states we predict a quantum diffusion of the momentum transfer due to decoherence by spontaneous emission, which is fundamentally different from and much weaker than the classical diffusion due to chaos for experimentally relevant parameter values. Our numerical and analytical results indicate nevertheless that quantum diffusion due to decoherence by spontaneous emission may actually be visible in the existing experimental data and some simple tests to check this claim are proposed.