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Quantum and Semiclassical Optics Journal of the European Optical Society Part B

Quantum chaos in atom optics: using phase noise to model continuous momentum and position measurement

Sigurd Dyrting, Gerard J Milburn

Published Jun 1, 1996• 17 citations
Publisher

Abstract

We study the effect of noise on the quantum nonlinear motion of a two-level atom in a laser standing wave. This system has recently been shown to exhibit dynamic localization in the atomic momentum. Noise is introduced through the phase of the standing wave, and has one of two forms: Weiner noise or Ornstein - Uhlenbeck noise. Using the theory of stochastic Hamiltonians we derive a master equation to describe the quantum mechanical evolution of the momentum statistics. We show that the Weiner noise process is then equivalent to a continuous measurement of the momentum of the atom, and the Ornstein - Uhlenbeck process describes a measurement of the position of the atom up to the period of the optical potential. We study the effect of noise when the classical dynamics shows global chaos.

DOI
10.1088/1355-5111/8/3/017

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We study the effect of noise on the quantum nonlinear motion of a two-level atom in a laser standing wave. This system has recently been shown to exhibit dynamic localization in the atomic momentum. Noise is introduced through the phase of the standing wave, and has one of two forms: Weiner noise or Ornstein - Uhlenbeck noise. Using the theory of stochastic Hamiltonians we derive a master equation to describe the quantum mechanical evolution of the momentum statistics. We show that the Weiner noise process is then equivalent to a continuous measurement of the momentum of the atom, and the Ornstein - Uhlenbeck process describes a measurement of the position of the atom up to the period of the optical potential. We study the effect of noise when the classical dynamics shows global chaos.
This thesis presents an account of experimental and numerical investigations of two quantum systems whose respective classical analogues are chaotic. These are the delta-kicked rotor, a paradigm in classical chaos theory, and the novel delta-kicked accelerator, created by application of a constant external acceleration or torque to the rotor. The experimental realisation of these systems has been achieved by the exposure of laser-cooled caesium atoms to approximate delta-kicks from a pulsed, high-intensity, vertical standing wave of laser light. Gravity’s effect on the atoms can be controlled by appropriate shifting of the profile of the standing wave. Numerical simulations of the systems are based on a diffractive model of the potential’s effect. Each system’s dynamics are characterised by the final form of the momentum distribution and the dependence of the atoms’ mean kinetic energy on the number and time period of the delta-kicks. The phenomena of dynamical localisation and quantum resonances in the delta-kicked rotor, which have no counterparts in the system’s classical analogue, are observed and investigated. Similar experiments on the delta-kicked accelerator reveal the striking phenomenon of the quantum accelerator mode, in which a large momentum is transferred to a substantial fraction of the atomic ensemble. This feature, absent in the system’s classical analogue, is characterised and an analytic explanation is presented. The effect on each quantum system of decoherence, introduced through spontaneous emission in the atoms, is examined and comparison is made with the results of classical simulations. While having little effect on the classical systems, the level of decoherence used is found to degrade quantum signatures of behaviour. Classical-like behaviour is, to some extent, restored, although significant quantum features remain. Possible applications of the quantum accelerator mode are discussed. These include use as a tool in atom optics and interferometry, a technique for measuring gravity, and a method of preparing atoms in a particular region of phase space. This may allow measurement of quantum phase space stability, and hence investigation of quantum chaos and quantum-classical correspondence.
We derive the equations of motion of arbitrary momentum-dependent observables for a fully first-quantized, driven two-level atom in a linear potential under the presence of a stochastic laser phase. In particular, we show that such momentum-dependent observables evolve under stochastic differential equations (SDEs) in the form of geometrical Brownian motion and find that these SDEs reduce to ordinary differential equations (ODEs) for the expectation values under white laser phase noise. As an example, we solve these ODEs numerically and determine the effect of laser phase noise on Rabi oscillations including quantized center-of-mass degrees of freedom of the atom. Moreover, the SDE formalism we derive holds for any phase noise source and can also be applied to extended noise models such as color noise.
We report a direct observation of dynamical localization. This effect is a quantum suppression of diffusion in a system that is classically chaotic. Our experiment measures the momentum transferred from a modulated standing wave of a near-resonant laser to a sample of ultracold atoms and is a realization of a periodically driven rotor. The conceptual simplicity and experimental control over the time dependent Hamiltonian make this system an ideal testing ground for the predictions of quantum chaos.
In this thesis, I study the classical and quantum nonlinear dynamics of two-level atoms in a donut beam, an evanescent beam guiding atoms in a hollow fiber and atoms in an optical standing wave of high-finesse cavity. The laser fields are far-off-resonant beams. The two dimensional classical nonlinear dynamics of atoms in a donut beam and an evanescent-wave in hollow fiber are investigated. Chaotic dynamics exists for classic dynamics of cold atoms in a donut beam and hollow fiber, when the intensity of the beam is periodically modulated. The atoms will accumulate on several annular regions when the system enters a regime of global chaos. The two dimensional quantum nonlinear dynamics of an atom in hollow fiber are investigated. The quantum system of single atom dynamics in cavity QED is investigated. For continuous observation of the nonlinear dynamics of single atom trapped in a high-finesse cavity by a standing wave with intensity modulation, the conditional Hilbert space dynamics of this system, subject to measurement induced perturbations, depends strongly on whether the corresponding classical dynamics is regular or chaotic. The influence of decoherence due to homodyne measurement is investigated. The evolution of Wigner function shows that the higher order quantum correction terms will play an important role together with potential nonlinearity and measurement strength. The decoherence will result in a remarkable improvement for classical-quantum correspondence.
Summary form only given. The recovery of the classical limit in quantum systems that have underlying chaotic dynamics is a key question in current studies of the quantum to classical transition. Interaction with an environment results in decoherence that can cause the quantum Wigner function to approach the corresponding classical phase space distribution by suppressing quantum interference effects, but it fails to extract localized classical trajectories from the quantum dynamics. Such trajectories are crucial for quantifying chaos both theoretically and in experiments through the measure of the Lyapunov exponents. One can recover trajectories from the quantum dynamics through the process of continuous measurements when the record is retained. Ehrenfest s theorem guarantees that quantum systems that are well localized by the measurement effectively obey classical mechanics. We have studied the role of continuous measurement in recovering the classical chaotic dynamics of laser cooled atoms in a magneto-optical lattice of double wells.