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Modern Physics Letters B

THE MIXED COHERENT STATE REPRESENTATION OF THE DENSITY OPERATOR IN THERMO-FIELD DYNAMICS

Heng Fan, Hai-liang Lu

Published Feb 10, 2007• 20 citations
Publisher

Abstract

In the framework of thermo-field dynamics, invented by Umezawa et al., we construct a mixed coherent state representation of density operator ρ. This new representation is useful because it provides an approach to retrieve ρ from its c-number solution of master equations in the entangled state representation.
DOI
10.1142/s0217984907012694

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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.
According to the thermo field dynamics proposed by Takahashi–Umezawa, the Hamiltonian h = a†a - ã†ã and h|n, ñ〉 = 0 of the original optical field excitation state |n〉 and its tilde state |ñ〉 in the Hilbert space cannot embody the eigenvalue spectrum of h. To avoid the weakness, new thermo excitation states are successfully constructed by introducing appropriate thermo excitation operators. It is noted that they are entangled states. Applications of the new thermo excitation state in constructing new thermo squeezed state and phase state are also presented.
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.
By analyzing the characters of thermo field dynamics, which is proposed by Takahashi and Umezawa, we find that a new physical state, named the thermo-invariant coherent state, should exist in thermal equilibrium. This state not only maintains the whole energy of the system and the reservoir but is also a coherent state. We find its explicit form, its relation to the thermal coherent state and the thermal entangled state. When the whole system is in this state, annihilating a quantum of the system and in the meanwhile annihilating a hole with negative energy in the reservoir will not change the whole energy.