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arXiv:quant-ph/9708013 · Journal of Physics A Mathematical and General

Fidelity for displaced squeezed thermal states and the oscillator semigroup

Horia Scutaru

Published Apr 17, 1998• 172 citations
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Abstract

The fidelity for two displaced squeezed thermal states is computed using the fact that the corresponding density operators belong to the oscillator semigroup. A novel calculation technique is developed for the computation of the traces of product of Gaussian density operators and of square roots of Gaussian density operators. The method is exemplified for the one-mode case but it is also applicable to the multimode case.

DOI
10.1088/0305-4470/31/15/025

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•Jan 1, 2000•1 citation
The fidelity for two displaced squeezed thermal states is computed using the fact that the corresponding density operators belong to the oscillator semigroup. A novel calculation technique is developed for the computation of the traces of product of Gaussian density operators and of square roots of Gaussian density operators. The method is exemplified for the one-mode case but it is also applicable to the multimode case.
In the theory of quantum transmission of information the concept of fidelity plays a fundamental role. An important class of channels, which can be experimentally realized in quantum optics, is that of Gaussian quantum channels. In this work we present a general formula for fidelity in the case of two arbitrary Gaussian states. From this formula one can get a previous result [H. Scutaru, J. Phys. A 31, 3659 (1998)], for the case of a single mode. The concept of fidelity used in this paper is the standard one [D. Bures, Trans. Am. Math. Soc. 135, 199 (1969); A. Uhlmann, Rep. Math. Phys. 9, 273 (1976); R. Josza, J. Mod. Opt. 41, 2314 (1994)]. It can be defined by $F({\ensuremath{\rho}}_{1},{\ensuremath{\rho}}_{2}{)}^{\underset{=}{\mathrm{def}}}{\mathrm{max}}_{|{\ensuremath{\psi}}_{1}〉,|{\ensuremath{\psi}}_{2}〉}|〈{\ensuremath{\psi}}_{1}|{\ensuremath{\psi}}_{2}〉{|}^{2},$ where $|{\ensuremath{\psi}}_{i}〉,$ i=1,2i=1,2i=1,2 are purifications of the density matrices ${\ensuremath{\rho}}_{i}.$
We calculate the quantum mechanical, temporal second-order coherence function for a single-mode, degenerate parametric amplifier for a system in the Gaussian state, viz. a displaced–squeezed thermal state. The calculation involves first dynamical generation at time t of the Gaussian state from an initial thermal state and subsequent measurements of two photons a time τ≥0 apart. The generation of the Gaussian state by the parametric amplifier insures that the temporal second-order coherence function depends only on τ, via τ/t, for the given Gaussian state parameters, Gaussian state preparation time t, and average number n¯ of thermal photons. It is interesting that the time evolution for displaced thermal states shows a power decay in τ/t rather than an exponential one as is the case for general, displaced–squeezed thermal states.
In the coordinate representation of thermofield dynamics, we investigate the thermalized displaced squeezed thermal state which involves two temperatures successively. We give the wavefunction and the matrix element of the density operator at any time, and accordingly calculate some quantities related to the position and particle number operator, special cases of which are consistent with the results in the literature. The two temperatures have different correlations with the squeezing and coherence components. Moreover, different from the properties of the position, the average value and variance of the particle number operator as well as the second-order correlation function are time independent.
In the coordinate representation of thermofield dynamics, we investigate the thermalized displaced squeezed thermal state which involves two temperatures successively. We give the wavefunction and the matrix element of the density operator at any time, and accordingly calculate some quantities related to the position, momentum and particle number operator, special cases of which are consistent with the results in the literature. The two temperatures have diffenent correlations with the squeeze and coherence components. Moreover, different from the properties of the position and momentum, the average value and variance of the particle number operator as well as the second-order correlation function are time-independent.