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Europhysics Letters (EPL)

Time-dependent formulation of the Bogoliubov transformation and time-evolution operators for time-dependent quantum oscillators

W.-S. Liu, X.-P Li

Published Jun 1, 2002• 5 citations
Publisher

Abstract

The time-dependent formulation of the Bogoliubov transformation (BT) for single-mode boson operators is derived in matrix multiplication. The combination of the BT differential equation with the time-evolution equation for time-dependent quantum oscillators yields general formulas to get the time-evolution operator of the systems. Examples are given.

DOI
10.1209/epl/i2002-00397-8

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The time-dependent formulation of the Bogoliubov transformation (BT) for single-mode boson operators is derived in matrix multiplication. The combination of the BT differential equation with the time-evolution equation for time-dependent quantum oscillators yields general formulas to get the time-evolution operator of the systems. Examples are given.
In this paper,the exact expressions of the evolution operator and wave function are obtained by means of the combination of the inhomogeneous Bogoliubov transformation(IBT) and the evolution equation associated with a su(1,1)h(4).As a model example,we have got the exact expressions of the evolution operator and Schrdinger wave function of the driven time-dependent harmonic oscillator.These results agree well with those derived in general refrences,but the present method is more simple and definite.what's more,it is easily extended to other su(1,1)h(4) quantum system
A general dynamical invariant operator for three coupled time-dependent oscillators is derived. Although the obtained invariant operator satisfies the Liouville-von Neumann equation, its mathematical formula is somewhat complicated due to arbitrariness of time variations of parameters. The parametric conditions required for formulating this invariant are definitely specified. By using the unitary transformation method, the invariant operator is transformed to the one that corresponds to three independent simple harmonic oscillators. Inverse transformation of the well-known quantum solutions associated with such a simplified invariant enables us to identify quantum solutions of the coupled original systems. These solutions are exact since we do not use approximations not only in formulating the invariant operator but in the unitary transformation as well. The invariant operator and its eigenfunctions provided here can be used to characterize quantum properties of the systems with various choices of the types of time-dependent parameters.