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Journal of Physics A Mathematical and General

New approach for solving master equations of density operators by virtue of the thermal entangled states

Hong-Yi Fan, Yue Fan

Published Jul 30, 2002• 24 citations
Publisher

Abstract

Based on the thermal field dynamics, by introducing the thermal entangled state we find a new approach for converting density operators' quantum master equations into c-number equations. This approach, in many cases, is concise and convenient for deriving the solutions of these equations, because the thermal entangled states possess favourable properties.

DOI
10.1088/0305-4470/35/32/307

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Based on the thermal field dynamics, by introducing the thermal entangled state we find a new approach for converting density operators' quantum master equations into c-number equations. This approach, in many cases, is concise and convenient for deriving the solutions of these equations, because the thermal entangled states possess favourable properties.
By introducing a fictitious mode to be a counterpart mode of the system mode under review we introduce the entangled state representation ?|, which can arrange master equations of density operators ?(t) in quantum statistics as state-vector evolution equations due to the elegant properties of ?|. In this way many master equations (respectively describing damping oscillator, laser, phase sensitive, and phase diffusion processes with different initial density operators) can be concisely solved. Specially, for a damping process characteristic of the decay constant ? we find that the matrix element of ?(t) at time t in ?| representation is proportional to that of the initial ?0 in the decayed entangled state ?e??t| representation, accompanying with a Gaussian damping factor. Thus we have a new insight about the nature of the dissipative process. We also set up the so-called thermo-entangled state representation of density operators, ? = ?(d2?/?)?|?D(?), which is different from all the previous known representations.
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.