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arXiv:quant-ph/0001056 · Physical Review A

Sensitivity to measurement perturbation of single-atom dynamics in cavity QED

X. M. Liu, M. Hug, Gerard J Milburn

Published Sep 5, 2000• 3 citations
arXivHTMLPDF

Abstract

We consider continuous observation of the nonlinear dynamics of single atom trapped in an optical cavity by a standing wave with intensity modulation. The motion of the atom changes the phase of the field which is then monitored by homodyne detection of the output field. We show that 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. If the classical dynamics is chaotic, the distribution of conditional Hilbert space vectors corresponding to different observation records tends to be orthogonal. This is a characteristic feature of hypersensitivity to perturbation for quantum chaotic systems.

DOI
10.1103/physreva.62.043801

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We consider continuous observation of the nonlinear dynamics of single atom trapped in an optical cavity by a standing wave with intensity modulation. The motion of the atom changes the phase of the field which is then monitored by homodyne detection of the output field. We show that 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. If the classical dynamics is chaotic, the distribution of conditional Hilbert space vectors corresponding to different observation records tends to be orthogonal. This is a characteristic feature of hypersensitivity to perturbation for quantum chaotic systems.
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We study the coupled translational and internal dynamics of a two-level atom in a spatially periodic optical potential created by a single-mode field in a high-finesse Fabry-Perot cavity. The strongly coupled atom-cavity system with atomic center-of-mass motion included is modeled by semiclassical nonlinear Heisenberg equations of motion with two degrees of freedom. At exact resonance between the internal atomic transition and the cavity eigenfrequency, the internal and translational degrees of freedom are separated from each other resulting in periodically modulated Rabi oscillations and regular translational motion. Near the atom-field resonance, the coupled dynamics is found to be irregular and even chaotic in the sense of extremal sensitivity to initial conditions. It is manifested as a chaotic motion of an atom in an absolutely regular spatially periodic potential in the absence of any random fluctuations. This effect may be explained as a result of the interaction of nonlinear resonances in the atom-field nonlinear pendulum whose low-frequency translational motion is modulated by the high-frequency Rabi oscillations. The irregular motion is studied in detail in numerical experiments using the arsenal of methods of the dynamical systems theory. The results obtained are compared with recent experiments with ultracold single atoms interacting with single photons in the strong-coupling regime. We estimate the ranges of the system's control parameters, the detuning, the recoil frequency, and the mean number of intracavity photons, in which the interaction of nonlinear resonances may lead, under realistic conditions, to local instability of the center-of-mass motion and some manifestations of dynamical chaos, which is the effect of interest in studying the quantum-classical correspondence and quantum chaos in atomic optics.
This thesis makes a unique contribution to the field of quantum chaos by theoretically demonstrating the effect that measurement has on the emergence of chaos from the quantum world and demonstrating a means to control the onset of chaos in the quantum system using adaptive measurements. Here we investigate how the choice of the continuous measurement strategy for an open quantum system affects the emergence of chaos in the transition from the quantum limit to the classical limit when the system is dissipative. We consider two models in our research. The Duffing oscillator is classically chaotic and also dissipative( ie. an open quantum system), whereas the driven top is classically a closed system; adding dissipation via continuous measurement therefore changes the behaviour in from the classical limit. The first half of this thesis presents the investigation of a dissipative system whose classical limit is chaotic. We explore the emergence of chaos from the open quantum system that is continuously monitored and investigate the dependence on the choice of monitoring by changing a single parameter in a homodyne measurement scheme, effectively changing the information gained by the measurement. We show that the emergence of chaos in the regime where quantum effects are still present can be determined solely by changing the measurement parameter. This is a result of the interplay between the quantum interference effects induced by the nonlinear dynamics and the localisation and decoherence that occurs due to the measurement. We also investigate the case where the classical limit is regular for the Duffing oscillator, and demonstrate the semiclassical effect of chaos induced by the measurement back-action. A certain choice of measurement leads to a noise which drives the system to large spread in the dimensionless position enabling a non-classical transition mechanism that is classically forbidden, inducing chaos in the system. These results are verified by the numerical calculation of the maximal Lyapunov exponent in the quantum regime. The second part of this thesis investigates the possibility of controlling the degree of chaos with quantum control. We design an effective control scheme to control the degree of chaos using the measurement dependency of the state. We propose an adaptive measurement scheme which changes the homodyne measurement angle in real time depending on the direction of the state's interference fringes in phase space. This is done using the knowledge gained by the measurement signal. We show that this control scheme can enhance or suppress chaos. By enhancing the degree of chaos we are also able to push the onset of chaos further into the quantum regime than was possible before. By suppressing chaos we generate highly non-classical states and regular motion. The feasibility of experimentally realising this control technique is discussed in detail. The final section of this thesis considers a chaotic system that is not dissipative in the classical limit: the driven top. We investigate the effect that opening the quantum system to decoherence has on the degree of chaos when we continuously measure the system. We demonstrate that the presence of decoherence suppresses the chaos and alters the dynamics of the quantum system. This is seen to worsen as the strength of the measurement is increased unless a particular measurement is chosen that perfectly cancels out the decoherence resulting in the Hamiltonian evolution in addition to noise from the measurement. These results are verified by the separation time between classical and quantum dynamics.
Cold atoms in optical potentials provide an ideal test bed to explore quantum nonlinear dynamics. Atoms are prepared in a magneto-optic trap or as a dilute Bose–Einstein condensate and subjected to a far detuned optical standing wave that is modulated. They exhibit a wide range of dynamics, some of which can be explained by classical theory while other aspects show the underlying quantum nature of the system. The atoms have a mixed phase space containing regions of regular motion which appear as distinct peaks in the atomic momentum distribution embedded in a sea of chaos. The action of the atoms is of the order of Planck’s constant, making quantum effects significant. This tutorial presents a detailed description of experiments measuring the evolution of atoms in time-dependent optical potentials. Experimental methods are developed providing means for the observation and selective loading of regions of regular motion. The dependence of the atomic dynamics on the system parameters is explored and distinct changes in the atomic momentum distribution are observed which are explained by the applicable quantum and classical theory. The observation of a bifurcation sequence is reported and explained using classical perturbation theory. Experimental methods for the accurate control of the momentum of an ensemble of atoms are developed. They use phase space resonances and chaotic transients providing novel ensemble atomic beamsplitters. The divergence between quantum and classical nonlinear dynamics is manifest in the experimental observation of dynamical tunnelling. It involves no potential barrier. However a constant of motion other than energy still forbids classically this quantum allowed motion. Atoms coherently tunnel back and forth between their initial state of oscillatory motion and the state 180 ° out of phase with the initial state.
We study dynamics of the atom-photon interaction in cavity quantum electrodynamics, considering a cold two-level atom in a single-mode high-finesse standing-wave cavity as a nonlinear Hamiltonian system with three coupled degrees of freedom: translational, internal atomic, and the field. The system proves to have different types of motion including Lévy flights and chaotic walkings of an atom in a cavity. The corresponding equations of motion for expectation values of the atom and field variables have two characteristic time scales: fast Rabi oscillations of the internal atomic and field quantities and slow translational oscillations of the center of the atom mass. It is shown that the translational motion, related to the atom recoils, is governed by an equation of a parametric nonlinear pendulum with a frequency modulated by the Rabi oscillations. This type of dynamics is chaotic with some width of the stochastic layer that is estimated analytically. The width is fairly small for realistic values of the control parameters, the normalized detuning delta and atomic recoil frequency alpha. We consider the Poincaré sections of the dynamics, compute the Lyapunov exponents, and find a range of the detuning, |delta| less, similar 3, where chaos is prominent. It is demonstrated how the atom-photon dynamics with a given value of alpha depends on the values of delta and initial conditions. Two types of Lévy flights, one corresponding to the ballistic motion of the atom and the other corresponding to small oscillations in a potential well, are found. These flights influence statistical properties of the atom-photon interaction such as distribution of Poincaré recurrences and moments of the atom position x. The simulation shows different regimes of motion, from slightly abnormal diffusion with <x(2)> approximately tau(1.13) at delta=1.2 to a superdiffusion with <x(2)> approximately tau(2.2) at delta=1.92 that corresponds to a superballistic motion of the atom with an acceleration. The obtained results can be used to find new ways to manipulate atoms, to cool and trap them by adjusting the detuning delta.
Nonlinear dynamics in the fundamental interaction between a two-level atom with recoil and a quantized radiation field in a high-quality cavity is studied. We consider the strongly coupled atom-field system as a quantum-classical hybrid with dynamically coupled quantum and classical degrees of freedom. It is shown that at given initial conditions the regime of atomic motion is governed mainly by the detuning of the atom-field resonance /spl delta/. We show that, even in the absence of any other interaction with environment, the interaction of a purely quantum atom-field system with the external atomic degree of freedom provides the emergence of classical dynamical chaos from quantum electrodynamics (QED). It manifest itself in the classical degree of freedom as a random walking of an atom inside the cavity and in the quantum degrees of freedom as a sensitive dependence of atomic inversion on small variations in initial conditions. It is shown that dependence of variance of entanglement on /spl delta/, correlates strongly with the dependence of maximum Lyapunov exponent on /spl delta/. This result provides a quantum-classical correspondence in a closed physical system.