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Journal of Optics B Quantum and Semiclassical Optics

Decoherence and transfer of quantum states of field modes in a one-dimensional cavity with an oscillating boundary

V. V. Dodonov, Mauro Antonio ANDREATA, Salomon Sylvain Mizrahi

Published Nov 4, 2005• 14 citations
Publisher

Abstract

We study the evolution of Wigner functions of arbitrary initial quantum states of field modes in a one-dimensional ideal cavity, whose boundary performs small harmonic oscillations at the frequency ω W = p ω 1 (where ω 1 is the fundamental field eigenfrequency). Special attention is paid to the case of initial even and odd coherent states, which serve as models of the 'Schrödinger cat states'. We show that the strong intermode interaction (due to the Doppler upshift of the fields reflected from the oscillating mirror) results in the decoherence of initial quantum superpositions in selected modes, even in the absence of any external 'environment'. Different quantitative measures of decoherence are discussed. The analytical solutions obtained show that any initial state of the field goes asymptotically to a highly mixed and moderately squeezed state in the 'principal resonance case' p = 2 and to the vacuum state in the 'semiresonance case' p = 1. It is shown that the decoherence process has several stages. In the first one, the interference between the components of the initial superposition is rapidly destroyed during the time of the primary decoherence , which is inversely proportional to the first power of the initial distance between the components, as opposed to the second power in the case of usual dissipative reservoirs. However, some weak traces of coherence (quantumness of states), such as the regions of negativity of the Wigner function, survive for much longer times, which do not depend on the size of the initial superposition.

DOI
10.1088/1464-4266/7/12/006

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We study the evolution of Wigner functions of arbitrary initial quantum states of field modes in a one-dimensional ideal cavity, whose boundary performs small harmonic oscillations at the frequency ω W = p ω 1 (where ω 1 is the fundamental field eigenfrequency). Special attention is paid to the case of initial even and odd coherent states, which serve as models of the 'Schrödinger cat states'. We show that the strong intermode interaction (due to the Doppler upshift of the fields reflected from the oscillating mirror) results in the decoherence of initial quantum superpositions in selected modes, even in the absence of any external 'environment'. Different quantitative measures of decoherence are discussed. The analytical solutions obtained show that any initial state of the field goes asymptotically to a highly mixed and moderately squeezed state in the 'principal resonance case' p = 2 and to the vacuum state in the 'semiresonance case' p = 1. It is shown that the decoherence process has several stages. In the first one, the interference between the components of the initial superposition is rapidly destroyed during the time of the primary decoherence , which is inversely proportional to the first power of the initial distance between the components, as opposed to the second power in the case of usual dissipative reservoirs. However, some weak traces of coherence (quantumness of states), such as the regions of negativity of the Wigner function, survive for much longer times, which do not depend on the size of the initial superposition.
We study the dynamics of the coefficient of quantum separability between resonantly coupled modes of the massless scalar field in a one-dimensional ideal cavity, whose boundary performs small harmonic oscillations at the frequency ω w = p ω 1 (where ω 1 is the fundamental field eigenfrequency). A qualitative difference between the cases p = 1 and p = 2 is discovered. In the first case, initially factorized thermal states remain separable in the process of evolution, whereas the dynamics of entanglement of initial entangled states (modelled by two-mode squeezed states ) turns out to be more intricate, showing sudden changes from ‘classical’ to ‘nonclassical’ regimes. In contrast, in the ‘principal resonance case’ p = 2 the degree of entanglement changes with time monotonically, and any initially factorized thermal states go asymptotically to mixed quantum states with maximal degree of entanglement. Moreover, for highly thermalized states, the time dependence of the separability coefficient exhibits sharp jumps, which are, nonetheless, not correlated with time dependences of the energies of the field modes.
We study the decoherence of superpositions of displaced quantum states of the form (where | g ⟩ is an arbitrary 'fiducial' state and is the usual displacement operator) within the framework of the standard master equation for a quantum damped or amplified harmonic oscillator interacting with a phase-insensitive (thermal) reservoir. We compare two simple measures of the degree of decoherence: the quantum purity and the height of the central interference peak of the Wigner function. We show that for N > 2 'mesoscopic' components of the superposition, the decoherence process cannot be characterized by a single decoherence time. Therefore, we distinguish the 'initial decoherence time' and 'final decoherence time' and study their dependence on the parameters α k and N . We obtain approximate formulae for an arbitrary state | g ⟩ and explicit exact expressions in the special case of | g ⟩ = | m ⟩, i.e., for (symmetrical) superpositions of displaced Fock states of occupation number m . We show that the superposition with a large number of components N and rich 'internal structure' ( m ∼ |α| 2 ) can be more robust against decoherence than simple superpositions of two coherent states (with m = 0), even if the initial decoherence times coincide. Also, we show how initial pure quantum superpositions are transformed into highly mixed and totally classical superpositions in the case of a phase-insensitive amplifier.
We study time dependence of various measures of entanglement (covariance entanglement coefficient, purity entanglement coefficient, normalized distance coefficient, entropy coefficients) between resonantly coupled modes of the electromagnetic field in ideal cavities with oscillating boundaries. Two types of cavities are considered — a three-dimensional cavity possessing eigenfrequencies ω3 = 3ω1, whose wall oscillates at the frequency ωw = 2ω1, and a one-dimensional (Fabry–Perot) cavity with an equidistant spectrum ωn = nω1 where the distance between perfect mirrors oscillates at the frequencies ω1 and 2ω1. The behavior of entanglement measures in these cases turns out to be completely different, although all three coefficients demonstrate qualitatively similar time dependences in each case (except some specific situations where the covariance entanglement coefficient based on traces of covariance submatrices seems to be essentially more sensitive to entanglement than other measures, which are based on determinants of covariance submatrices). Different initial states of the field, namely, vacuum, squeezed vacuum, thermal, Fock, and even/odd coherent states, are considered.
ABSTRACT: The processes of quantum decoherence in QED cavities have been extensively studied in the last decade, both experimentally by Haroche, Raimond et. al., as theoretically by Nemes, Davidovich et. to the. We present here the dynamics of the Wigner function for a system consisting of a two-level atom and a cavity in a dissipative environment, using a numerical integration scheme that allows us to consider initial Squeezed and Schrodinger's Cat States. The interaction between the atom and the single-mode field is considered by a dispersive model, that is, the interaction in the Hamiltonian of Jaynes-Cummings appears as a small disturbance. The interest is focused both on the quantum characteristics of the dispersive approach (preparation of mesoscopic states) and on the aspects that characterize dynamics and dissipation. In particular, the gradual loss of quantum coherence is observed through the use of the Wigner function, where the disappearance of the interference regions of the initial quantum states is evidenced.