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Coordinate-representation Calculation on Observables of Thermalized Coherent States

Wen-Fa Lu

Published Mar 26, 1999• 0 citations
OpenAlex

Abstract

The coordinate representation, a usual one in quantum mechanics, is seldom adopted for calculations of non-classical states with thermal noises. In this paper, we provide a coordinate-representation calculation for the thermalized coherent state. Within the framework of thermofield dynamics, the density matrix element in the coordinate repersentation is calculated with the help of the position wavefunctions. Furthermore, we give the position and momentum probability densities. Besides, we also calculate the average values and variances of the position, momentum and particle number, which are consistent with results obtained by other methods.

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The coordinate representation, a usual one in quantum mechanics, is seldom adopted for calculations of non-classical states with thermal noises. In this paper, we provide a coordinate-representation calculation for the thermalized coherent state. Within the framework of thermofield dynamics, the density matrix element in the coordinate repersentation is calculated with the help of the position wavefunctions. Furthermore, we give the position and momentum probability densities. Besides, we also calculate the average values and variances of the position, momentum and particle number, which are consistent with results obtained by other methods.
Within the framework of thermofield dynamics, we construct a thermalized coherent thermal state, which is a general type of the coherent state with the thermal effects and can be presumably produced experimentally. The wavefunction and the density matrix element in the coordinate repersentation are calculated, and furthermore we give the probability densities, average values and variances of the position, momentum and particle number, which in special cases are consistent with those in the literature. All calculations are performed in the coordinate representation.
In the coordinate representation of thermofield dynamics, we investigate the thermalized displaced squeezed thermal state which involves two temperatures successively. We give the wavefunction and the matrix element of the density operator at any time, and accordingly calculate some quantities related to the position, momentum and particle number operator, special cases of which are consistent with the results in the literature. The two temperatures have diffenent correlations with the squeeze and coherence components. Moreover, different from the properties of the position and momentum, the average value and variance of the particle number operator as well as the second-order correlation function are time-independent.
Within the framework of thermofield dynamics, the wavefunction of the thermal vacuum is given in terms of the position coordinates, and a thermal coordinate representation is introduced through the thermal unitary transformation. Exploting the representation, we obtain position-coordinate wavefunctions of thermal coherent and squeezed states. In the thermal coordinate representation, these wavefunctions are correspondingly the same forms as vacuum, coherent and squeezed states in the coordinate representation.
In the coordinate representation of thermofield dynamics, we investigate the thermalized displaced squeezed thermal state which involves two temperatures successively. We give the wavefunction and the matrix element of the density operator at any time, and accordingly calculate some quantities related to the position and particle number operator, special cases of which are consistent with the results in the literature. The two temperatures have different correlations with the squeezing and coherence components. Moreover, different from the properties of the position, the average value and variance of the particle number operator as well as the second-order correlation function are time independent.
A generalized Wigner function W T(α, ) including the thermal degrees of freedom is presented to calculate the ‘reduced’ Wigner functions for all kinds of coherent and squeezed states with thermal noise by the formalism of thermofield dynamics. The results show that the different definitions of coherent and squeezed state with the same corresponding density matrix give rise to the same reduced Wigner function. By our method, the reduced Wigner function can be calculated for an arbitrary vector in the thermal Fock space, of which the density matrix is too complicated to use.