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Physical Review A

Wigner phase-space representation of thermal excitations

Michael R. Berman

Published Aug 1, 1990• 5 citations
Publisher

Abstract

The Wigner phase-space representation of thermal excitations used in thermo-field-dynamics (TFD) is investigated. The phase-space description is obtained by introducing a reduced density operator in which the fictitious degrees of freedom of TFD are traced over. A new closed-form expression for the Wigner function of the harmonic oscillator thermal excitation is derived, thereby showing that the thermal excitation may be described as a temperature-dependent radial spreading of the coordinates and momenta in phase space. The population distribution in these stationary excitations is determined and shown to reduce to a Boltzmann distribution for the thermal vacuum only.

DOI
10.1103/physreva.42.1863

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The Wigner phase-space representation of thermal excitations used in thermo-field-dynamics (TFD) is investigated. The phase-space description is obtained by introducing a reduced density operator in which the fictitious degrees of freedom of TFD are traced over. A new closed-form expression for the Wigner function of the harmonic oscillator thermal excitation is derived, thereby showing that the thermal excitation may be described as a temperature-dependent radial spreading of the coordinates and momenta in phase space. The population distribution in these stationary excitations is determined and shown to reduce to a Boltzmann distribution for the thermal vacuum only.
A new Wigner phase-space distribution function is introduced for thermal quasiparticles currently in use in thermo field dynamics. The properties of this phase-space distribution are studied and compared with those of the corresponding zero-temperature distribution. Generalized uncertainty relations are implemented in order to study the explicit spreading in phase space as a function of temperature and quantum number of the thermal excitation, thereby permitting the investigation of quantum and thermal fluctuations.
Thermofield dynamics (TFD) is a powerful framework for accounting for thermal effects in a wave function setting and has been extensively used in physics and quantum optics. TFD relies on a duplicated state space and creates a correlated two-mode thermal state via a Bogoliubov transformation acting on the vacuum state. However, a very useful variant of TFD uses the vacuum state as the initial condition and transfers the Bogoliubov transformation into the propagator. This variant, referred to here as the inverse Bogoliubov transformation (iBT) variant, has recently been applied to vibronic coupling problems and coupled-oscillator Hamiltonians in a chemistry context, where the method is combined with efficient tensor network methods for high-dimensional quantum propagation. In the iBT-TFD representation, the mode expectation values are clearly defined and easy to calculate, but the thermalized reduced particle distributions, such as the reduced 1-particle densities or Wigner distributions, are highly non-trivial due to the Bogoliubov back-transformation of the original thermal TFD wave function. Here, we derive formal expressions for the reduced 1-particle density matrix (1-RDM) that use the correlations between the real and tilde modes encoded in the associated reduced 2-particle density matrix. We apply this formalism to define the 1-RDM and the Wigner distributions in the special case of a thermal harmonic oscillator. Moreover, we discuss several approximate schemes that can be extended to higher-dimensional distributions. These methods are demonstrated for the thermal reduced 1-particle density of an anharmonic oscillator.
When a quantum field is in contact with a thermal bath, the vacuum state of the field may be generalized to a thermal vacuum state, which takes into account the thermal noise. In thermo field dynamics, this is realized by doubling the dimensionality of the Fock space of the system. Interestingly, the representation of thermal noise by means of an augmented space is also found in a distinctly different approach based on the Wigner transform of both the field operators and density matrix, which we pursue here. Specifically, the thermal noise is introduced by augmenting the classical-like Wigner phase space by means of Nosé–Hoover chain thermostats, which can be readily simulated on a computer. In this paper, we illustrate how this may be achieved and discuss how non-equilibrium quantum thermal distributions of the field modes can be numerically simulated.