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Modern Physics Letters A

ON THE THERMO-INVARIANT COHERENT STATE IN THERMO FIELD DYNAMICS

Xuefen Xu, Hong-Yi Fan

Published Apr 10, 2006• 8 citations
Publisher

Abstract

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.

DOI
10.1142/s0217732306018901

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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.
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.
We introduce coherent mixed states (or thermal coherent states) associated with the displaced harmonic oscillator at nonzero temperature (T ≠ 0), as a "random" (or "thermal" or "noisy") basis in Hilbert space. A resolution of the identity for these states is proven and is used to generalize the usual pure (T = 0) coherent state formalism to the mixed (T ≠ 0) case. This new formalism for thermal coherent states is then further generalized to a broader class of so-called negative-binomial mixed states. It is known that the negative-binomial distribution is itself intimately related to the discrete series of SU(1,1) representations. We consider the pure SU(1,1) coherent states in the two-mode harmonic oscillator space, and show how our negative-binomial mixed states arise from taking the partial trace with respect to one of these two modes. This observation is then used to show how the formalism of thermo-field dynamics may be generalized to a correspondingly much broader negative-binomial-field dynamics, which we expect to have many uses.