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ENTANGLEMENT OF RESONANTLY COUPLED FIELD MODES IN CAVITIES WITH VIBRATING BOUNDARIES

Mauro Antonio ANDREATA, Alexandre V. Dodonov, V. V. Dodonov, Via Washington Luiz

Published Sep 25, 2002• 9 citations
OpenAlex

Abstract

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.

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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.
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.
The entanglement of the two movable mirrors in a single mode cavity field is studied in an ideal situation without thermal noise. When gn/w <<1 where g is the coupling coefficient between the mirror and the field, w is the mirror' vibration frequency and n is photon number of the cavity field, we work out Wootters concurrence of the system initially with the field in Fock number state and the two mirrors in vacuum states, and find that the states of the mirrors are entangled into space spanned by |0>,|1> via radiation pressure, though the concurrence is very small.
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.
The coherence and correlation properties of effective bosonic modes of a nanomechanical cavity composed of an oscillating mirror and containing an optical lattice of regularly trapped atoms are studied. The system is modeled as a three-mode system: two orthogonal polariton modes, representing the coupled optical lattice and the cavity mode, and one mechanical mode, representing the oscillating mirror. We examine separately the cases of two-mode and three-mode interactions, which are distinguished by suitable tuning of the mechanical mode to the polariton mode frequencies. In the two-mode case, we find that the occurrence of entanglement in the system is highly sensitive to the presence of first-order coherence between the modes. In particular, the creation of the first-order coherence among the polariton and mechanical modes is achieved at the expense of entanglement between them. In the three-mode case, we show that no entanglement is created between the independent polariton modes if both modes are coupled to the mechanical mode by the parametric interaction. There is no entanglement between the polaritons even if the oscillating mirror is damped by a squeezed vacuum field. The interaction creates first-order coherence between the polaritons, and the degree of coherence can, in principle, be as large as unity. This demonstrates that the oscillating mirror can establish first-order coherence between two independent thermal modes. Further analysis shows that two independent thermal modes can be made entangled in the system only when one of the modes is coupled to the intermediate mode by a parametric interaction and the other is coupled by a linear-mixing interaction.
We study the entanglement dynamics between two cavities when one of them is harmonically shaken in the context of quantum information theory. We find four different regimes depending on the frequency of the motion and the spectrum of the moving cavity. If the moving cavity is three dimensional only two modes inside get coupled and the entanglement can either degrade asymptotically with time or oscillate depending on the driving. On the other hand, if the cavity has an equidistant spectrum the entanglement can either vanish asymptotically if it is driven with its fundamental frequency or have a sudden death if it is driven with an uneven harmonic frequency.
Characterizing and quantifying quantum entanglement in states of open quantum systems are at the heart of quantum information theory. In this work, we study the mixedness and entanglement of a two-mode symmetrical Gaussian state evolving in a thermal Markovian environment. The considered state is that of two movable end-mirrors of two distant Fabry–Pérot cavities coupled via a two-mode squeezed light and driven by coherent lasers. We use the logarithmic negativity to consistently quantify the degree of entanglement between the two mechanical modes. The purity is employed as a witness of mixedness and characterize of entanglement in the two-mode mechanical bi-partition. By comparing the behavior of the logarithmic negativity with that of the purity under the effect of various parameters and based on the entanglement classification established in [G. Adesso, A. Serafini and F. Illuminati, Phys. Rev. Lett. 92, 087901 (2004)], we found that the purity can be used as a characterize and measure of entanglement in the two-mode symmetrical state when the squeezing parameter is fixed. The effects of thermal noise, squeezing and optomechanical coupling are also discussed in detail.