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Journal of Mathematical Physics

Coherent and thermal coherent state

A. Mann, M. Revzen, Kazutaka G. Nakamura, Hiroomi Umezawa, Y. Yamanaka

Published Dec 1, 1989• 27 citations
Publisher

Abstract

The characterization of coherent states as the quantum states that split into two uncorrelated beams is considered. The characterization leads to the study of coherent states at finite temperature—thermal coherent states (TCS’s). These TCS’s are defined within the formalism of thermo field dynamics (TFD). TFD allows a generalization of the uncertainty relation that accounts for both thermal and quantum fluctuations. The TCS is shown to be a minimal state for the generalized uncertainty relation.

DOI
10.1063/1.528470

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The characterization of coherent states as the quantum states that split into two uncorrelated beams is considered. The characterization leads to the study of coherent states at finite temperature—thermal coherent states (TCS’s). These TCS’s are defined within the formalism of thermo field dynamics (TFD). TFD allows a generalization of the uncertainty relation that accounts for both thermal and quantum fluctuations. The TCS is shown to be a minimal state for the generalized uncertainty relation.
By making use of thermo-field dynamics(TFD),higher-order fluctuations and higher-order squeezing of quadrature components for coherent states and squeezed states with thermal noise have been studied.The higher-order fluctuations of the phase for these thermalized states have been investigated using the concept of measured phase operators.Thus the dependence of the lower-and higher-order fluctuations and squeezing on temperature in these states at finite temperature can be seen.It is significant to discuss the effect of temperature on the fluctuations and squeezing,since the states of real field are always at finite temperatures.
Abstract By making use of thermo field dynamics(TFD),higher order fluctuations and higher order squeezing of quadrature components for coherent states and squeezed states with thermal noise have been studied.The higher order fluctuations of the phase for these thermalized states have been investigated using the concept of measured phase operators.Thus the dependence of the lower and higher order fluctuations and squeezing on temperature in these states at finite temperature can be seen.It is significant to discuss the effect of temperature on the fluctuations and squeezing,since the states of real field are always at finite temperatures.
In this paper, we investigate a thermal coherent state defined with the Lie–Trotter product formula under the formalism of the thermo field dynamics. In the definition of our thermal coherent state, we treat the thermalizing operator and the displacement operator symmetrically. We examine its uncertainty relation and quasiprobability distributions. Although this thermal coherent state is equivalent to the conventional ones except for different parameterizations and a phase factor, it is convenient for describing an experimental setup of the optical parametric oscillator laser.
Within the framework of thermo-field-dynamics, we define the reduced probability densities, and investigate the sum of information-theoretic entropies in quantum mechanics associated with the measurements of the position and momentum of a particle surrounded by the thermal environment. It is found that this quantity cannot be made arbitrarily small but has a universal lower bound dependent on the temperature. We develop the finite-temperature variational technique for the reduced system, and show that this relation, termed the thermal information-entropic uncertainty relation, is exactly saturated by the thermal coherent state. We also show that the Heisenberg uncertainty relation at finite temperature can be derived from this relation.
Although the thermofield dynamics (TFD) was developed specifically for systems to which the ladder canonical operators and are associated, in the paper we showed that this approach can also be formulated for systems of deformed bosons, associated with a pair of deformed operators and . In this context we built sets of temperature dependent coherent states, both of the Barut-Girardello and of the Klauder-Perelomov type, these two types being dual to each other. In calculations involving ladder operators, both independent and temperature-dependent, we used the Diagonal Operator Ordering Technique (DOOT), which is a generalization of the Integration Within an Ordered Product technique (IWOP) applicable to canonical operators associated with the one-dimensional harmonic oscillator. We also refer to the connection between TFD and thermodynamic quantities.