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AIP conference proceedings

A Generalized Time-Dependent Harmonic Oscillator at Finite Temperature

Hiroki Majima

Published Jan 1, 2006• 1 citation
Publisher

Abstract

We show how a generalized time‐dependent harmonic oscillator (GTHO) is extended to a finite temperature case by using thermo field dynamics (TFD). We derive the general time‐dependent annihilation and creation operators for the system, and obtain the time‐dependent quasiparticle annihilation and creation operators for the GTHO by using the temperature‐dependent Bogoliubov transformation of TFD. We also obtain the thermal state as a two‐mode squeezed vacuum state in the time‐dependent case as well as in the time‐independent case. The general formula is derived to calculate the thermal expectation value of operators.

DOI
10.1063/1.2204561

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We show how a generalized time‐dependent harmonic oscillator (GTHO) is extended to a finite temperature case by using thermo field dynamics (TFD). We derive the general time‐dependent annihilation and creation operators for the system, and obtain the time‐dependent quasiparticle annihilation and creation operators for the GTHO by using the temperature‐dependent Bogoliubov transformation of TFD. We also obtain the thermal state as a two‐mode squeezed vacuum state in the time‐dependent case as well as in the time‐independent case. The general formula is derived to calculate the thermal expectation value of operators.
We formulate the thermofield dynamics for time-dependent systems by combining the Liouville-von Neumann equation, its invariant operators, and the basic notions of thermofield dynamics. The new formulation is applied to time-dependent bosons and fermions by using the time-dependent annihilation and creation operators that satisfy the Liouville-von Neumann equation. It is shown that the thermal state is the time- and temperature-dependent vacuum state and a general formula is derived to calculate the thermal expectation value of operators.
Abstract We develop a wave‐function‐based method for the simulation of quantum dynamics of systems with many degrees of freedom at finite temperature. The method is inspired by the ideas of Thermo Field Dynamics (TFD). As TFD, our method is based on the doubling of the system's degrees of freedom and thermal Bogoliubov transformation. As distinct from TFD, our method implements the doubling of thermalized degrees of freedom only, and relies upon the explicitly constructed generalized thermal Bogoliubov transformation, which is not restricted to fermionic and bosonic degrees of freedom. This renders the present approach computationally efficient and applicable to a large variety of systems.
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.
Although the thermofield dynamics (TFD) was developed specifically for systems to which the ladder canonical operators 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. 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.
The dynamical invariant for a general time-dependent harmonic oscillator is constructed by making use of two linearly independent solutions to the classical equation of motion. In terms of this dynamical invariant we define the time-dependent creation and annihilation operators and relevantly introduce even and odd coherent states for time-dependent harmonic oscillator. The mathematical and quantum statistical properties of these states are discussed in detail. The harmonic oscillator with periodically varying frequency is treated as a demonstration of our general approach.