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Physical Review A

Phase fluctuations of an electron cyclotron: Nondissipative decoherence in a quantum stochastic oscillator

S. Brouard, Jesús Plata

Published May 8, 2002• 2 citations
Publisher

Abstract

The phase damping induced in the cyclotron mode of a trapped electron by the coupling to the axial fluctuations is studied analytically. This system, described as a nonlinear oscillator stochastically driven in frequency, allows testing the generality of some elements present in the phenomenology of decoherence. In our approach, the reduced density matrix is obtained by performing a statistical average from the propagator for each noise realization. For short times, the decay of the coherences presents a nonexponential form, rooted in the non-Gaussian character of the stochastic driving. For large times, the decay becomes purely exponential, the rate showing a complex dependence on the difference between the Fock indices. As the populations do not change, the asymptotic state corresponds to a nonthermalized statistical mixture.

DOI
10.1103/physreva.65.053412

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The phase damping induced in the cyclotron mode of a trapped electron by the coupling to the axial fluctuations is studied analytically. This system, described as a nonlinear oscillator stochastically driven in frequency, allows testing the generality of some elements present in the phenomenology of decoherence. In our approach, the reduced density matrix is obtained by performing a statistical average from the propagator for each noise realization. For short times, the decay of the coherences presents a nonexponential form, rooted in the non-Gaussian character of the stochastic driving. For large times, the decay becomes purely exponential, the rate showing a complex dependence on the difference between the Fock indices. As the populations do not change, the asymptotic state corresponds to a nonthermalized statistical mixture.
The effect of a driving field on the cyclotron mode of a relativistic electron in a Penning trap is studied analytically. The Hamiltonian dynamics of this driven nonlinear oscillator is analyzed by using linearization techniques and displaced squeezed-state formalism. With the approximate analytical expressions obtained for the eigenstates in this approach, a simplified treatment of the dissipative dynamics is carried out and some of the nontrivial features found in a recent numerical study [D. Enzer and G. Gabrielse, Phys. Rev. Lett. 78, 1211 (1997)] are unraveled. The emergence of different time scales and the generation of a metastable statistical mixture are understood in terms of the changes induced in the structure of the master equation by the nonuniform characteristics of the eigenstates; the partial revivals of specific coherent states are accounted for by the evolution of particular coherences. The control of these effects by a proper choice of the driving parameters is discussed.
We address the dynamics of nonclassicality for a quantum system interacting with a noisy fluctuating environment described by a classical stochastic field. As a paradigmatic example, we consider a harmonic oscillator initially prepared in a maximally nonclassical state, e.g., a Fock number state or a Schr\"odinger-cat-like state, and then coupled to either a resonant or a nonresonant external field. Stochastic modeling allows us to describe the decoherence dynamics without resorting to approximated quantum master equations and to introduce non-Markovian effects in a controlled way. A detailed comparison among different nonclassicality criteria and a thorough analysis of the decoherence time reveal a rich phenomenology whose main features may be summarized as follows: (i) Classical memory effects increase the survival time of quantum coherence and (ii) a detuning between the natural frequency of the system and the central frequency of the classical field induces revivals of quantum coherence.
We describe an analytically solvable model of quantum decoherence in a nonequilibrium environment. The model considers the effect of a bath driven from equilibrium by, for example, an ultrafast excitation of a quantum chromophore. The nonequilibrium response of the environment is represented by a nonstationary random function corresponding to the fluctuating transition frequency between two quantum states coupled to the surroundings. The nonstationary random function is characterized by a Fourier series with the phase of each term starting initially with a definite value across the ensemble but undergoing random diffusion with time. The decay of the off-diagonal density matrix element is shown to depend significantly on the particular pattern of initial phases of the terms in the Fourier series, or equivalently, the initial phases of bath modes coupled to the quantum subsystem. This suggests the possibility of control of quantum decoherence by the detailed properties of an environment that is driven from thermal equilibrium.
Abstract We derive a non-Markovian master equation for a charged particle in a magnetic field coupled to a bath and study decoherence by analyzing the temporal decay of the off-diagonal elements of the reduced density matrix in the position basis. The coherent oscillations characterized by the cyclotron frequency get suppressed as a result of decoherence due to coupling with the environment. We consider an Ohmic bath with three distinct models for the high-frequency cutoff for the spectral density of the bath and compare the three cases. As expected, the three cutoff models converge in the limit of the uppermost frequency of the bath tending to infinity. We notice a dramatic slowing down of loss of coherence in the low-temperature limit dominated by zero point quantum fluctuations compared to the high-temperature classical limit dominated by thermal fluctuations. We also go beyond the Ohmic model and study super-Ohmic and sub-Ohmic baths with the spectral densities deviating from a linear dependence on the frequency. Our results are testable in a state of the art cold atom laboratory.
We analytically investigate the fidelity dynamics of a Jaynes--Cummings qubit coupled to a single-mode cavity in the presence of stochastic fluctuations of the qubit--cavity coupling. Such fluctuations naturally arise in cavity- and circuit-QED platforms due to fabrication imperfections, control-field noise, mechanical vibrations, and environmental disturbances. Since the qubit--cavity coupling governs the rate of coherent energy and information exchange, understanding the effects of its fluctuations is essential for assessing the performance of quantum communication, quantum sensing, and quantum information-processing protocols. We consider both Gaussian white noise and Ornstein--Uhlenbeck noise with finite correlation time as models for the coupling fluctuations. Exploiting the closed dynamics within each excitation manifold, we derive exact analytical expressions for the noise-averaged density matrix and the fidelity relative to the ideal noiseless evolution. We show that the fidelity is determined by a single decoherence factor governed by the integrated noise correlation function, establishing a direct connection between microscopic coupling fluctuations and macroscopic coherence loss. In the white-noise limit, the fidelity decays exponentially, whereas finite noise correlation times produce nonexponential decay and slower coherence loss. We further show that the crossover between Markovian-like and non-Markovian regimes is controlled by the ratio of the noise correlation rate to the coherent coupling strength, and that decoherence increases with photon number because of the enhanced effective qubit--cavity coupling. These results provide a unified analytical framework for quantifying decoherence induced by stochastic coupling fluctuations and for evaluating the robustness of cavity-QED protocols under correlated noise.
An oscillator with stochastic frequency is discussed as a model for evaluating the quantum coherence properties of a physical system. It is found that the choice of jump statistics has to be considered with care if unphysical consequences are to be avoided. We investigate one such model, evaluate the damping it causes, the decoherence rate and the correlations it results in, and the properties of the state for asymptotically long times. Also the choice of initial state and its effect on the time evolution of the correlations are discussed.