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arXiv:quant-ph/0303148

Invariant Operators vs Heisenberg Operators for Time-Dependent Generalized Oscillators

Sang Pyo Kim

Published Mar 25, 2003• 0 citations
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Abstract

We investigate the relation between the invariant operators satisfying the quantum Liouville-von Neumann and the Heisenberg operators satisfying the Heisenberg equation. For time-dependent generalized oscillators we find the invariant operators, known as the Ermakov-Lewis invariants, in terms of a complex classical solution, from which the evolution operator is derived, and obtain the Heisenberg position and momentum operators. Physical quantities such as correlation functions are calculated using both the invariant operators and Heisenberg operators.

DOI
10.48550/arxiv.quant-ph/0303148

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We investigate the relation between the invariant operators satisfying the quantum Liouville-von Neumann and the Heisenberg operators satisfying the Heisenberg equation. For time-dependent generalized oscillators we find the invariant operators, known as the Ermakov-Lewis invariants, in terms of a complex classical solution, from which the evolution operator is derived, and obtain the Heisenberg position and momentum operators. Physical quantities such as correlation functions are calculated using both the invariant operators and Heisenberg operators.
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.
An exact invariant operator of time-dependent coupled oscillators is derived using the Liouville-von Neumann equation. The unitary relation between this invariant and the invariant of two uncoupled simple harmonic oscillators is represented. If we consider the fact that quantum solutions of the simple harmonic oscillator is well-known, this unitary relation is very useful in clarifying quantum characteristics of the original systems, such as entanglement, probability densities, fluctuations of the canonical variables, and decoherence. We can identify such quantum characteristics by inversely transforming the mathematical representations of quantum quantities belonging to the simple harmonic oscillators. As a case in point, the eigenfunctions of the invariant operator in the original systems are found through inverse transformation of the well-known eigenfunctions associated with the simple harmonic oscillators.
Quantum dynamical properties of a general time-dependent coupled oscillator are investigated based on the theory of two-dimensional (2D) dynamical invariants. The quantum dynamical invariant of the system satisfies the Liouville–von Neumann equation and it coincides with its classical counterpart. The mathematical formula of this invariant involves a cross term which couples the two oscillators mutually. However, we show that, by introducing two pairs of annihilation and creation operators, it is possible to uncouple the original invariant operator so that it becomes the one that describes two independent subsystems. The eigenvalue problem of this decoupled quantum invariant can be solved by using a unitary transformation approach. Through this procedure, we eventually obtain the eigenfunctions of the invariant operator and the wave functions of the system in the Fock state. The wave functions that we have developed are necessary in studying the basic quantum characteristics of the system. In order to show the validity of our theory, we apply our consequences to the derivation of the fluctuations of canonical variables and the uncertainty products for a particular 2D oscillatory system whose masses are exponentially increasing.
The Ermakov Lewis quantum invariant for the time dependent harmonic oscillator is expressed in terms of number and phase operators. The identification of these variables is made in accordance with the correspondence principle and the amplitude and phase representation of the classical orthogonal functions invariant. The relationship between the number and phase operators is established through this invariant as the system evolves from one frequency to another. In the specific case where the excitations represent the photon number, these relations are equivalent to the power density transport equations derived in nonlinear optical processes.