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Canadian Journal of Physics

Density operator for describing driven damped harmonic oscillator in the diffusion-limited channel

Hong-Yi Fan, Qian-Fan Chen

Published Jan 23, 2014• 2 citations
Publisher

Abstract

In the present work, the Kraus-operator sum representation of the exact solution to the master equation describing a harmonic oscillator (driven by an external source) damping in the diffusion-limited channel is obtained by virtue of the thermo-entangled state representation and the technique of integration within an ordered product of operators. According to this solution, the initial coherent state will evolve into a displaced chaotic state that manifestly exhibits quantum decoherence.

DOI
10.1139/cjp-2013-0488

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In the present work, the Kraus-operator sum representation of the exact solution to the master equation describing a harmonic oscillator (driven by an external source) damping in the diffusion-limited channel is obtained by virtue of the thermo-entangled state representation and the technique of integration within an ordered product of operators. According to this solution, the initial coherent state will evolve into a displaced chaotic state that manifestly exhibits quantum decoherence.
We solve various master equations to obtain density operators' infinite operator-sum representation via a new approach, i.e., by virtue of the thermo-entangled state representation that has a fictitious mode as a counterpart mode of the system mode. The corresponding Kraus operators from the point of view of quantum channel are derived, whose normalization conditions are proved. Miscellaneous characters possessed by different quantum channels, such as decoherence, phase diffusion, damping, and amplification, can be shown explicitly in the entangled state representation of the density operators. Squeezing transformation is applied to the density operator for decoherence to generate a master equation for describing the phase sensitive process. Partial trace method for deriving new density operators is also introduced. Throughout our discussion, the technique of integration within an ordered product (IWOP) of operators is fully used.
In the present work, the exact formal solution of the Caldeira–Leggett quantum master equation for the reduced density matrix of an oscillator is obtained by virtue of the thermo entangled state representation and the technique of integration within an ordered product (IWOP) of operators. Also, the solution for high temperature approximation is derived. When the initial state is a coherent state, it will evolve into a displaced-squeezed chaotic state (a mixed state) which shows that decoherence is involved in the Caldeira–Leggett equation due to the interactions between the system and environment.
This paper solves the newly constructed nonlinear master equation dρ/d t = κ[2 f ( N ) a ρ (1/ f ( N − 1)) a † − a † a ρ−ρ a † a ], where f ( N ) is an operator-valued function of N = a † a , for describing amplitude damping channel, and derives the infinite operator sum representation of quasi-Kraus operators for the density operator. It also shows that in this nonlinear process the initial pure number state density operator will evolve into the binomial field (a mixed state) when
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Abstract Using the operator correspondence of the real and fictious modes in the thermo entangled state representation, we solve the quantum master equation describing the diffusion channel and obtain the Kraus operator-sum representation of its analytical solution. we find that the pure coherent states evolve into the new mixed thermal superposed states in the diffusion channel. Also, we investigate the statistical properties of the initial coherent states and their entropy evolutions in the diffusion channel, and find that the entropy evolutions are only related to the decay time and without the amplitudes of the initial coherent states.