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Acta Physica Sinica

The evolution and two-mode squeezed states of the time-dependent two coupled harmonic oscillators

Jiangfan Li, Huang Chun-Jia, Jiang Zong-Fu, Huang Zu-hong, 长沙理工大学物理与电子科学系长沙; 长沙理工大学物理与电子科学系长沙; 国防科技大学理学院长沙; 国防科技大学理学院长沙

Published Jan 1, 2005• 3 citations
Publisher

Abstract

A closed solution of the Schrdinger equation for a timedependent two coupled harmonic oscillators including a driving part and a twomode coupled part is derived by adopting the LewisRiesenfeld invariant theory and properly chosen Hermitian invariant operators. The evolution operator of the system and the condition for generating twomode squeezed states are obtained. It is shown that the quantum fluctuation of squeezed states of the system is independent of the driving part, but depends on the initial states of the system.

DOI
10.7498/aps.54.522

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A closed solution of the Schrdinger equation for a timedependent two coupled harmonic oscillators including a driving part and a twomode coupled part is derived by adopting the LewisRiesenfeld invariant theory and properly chosen Hermitian invariant operators. The evolution operator of the system and the condition for generating twomode squeezed states are obtained. It is shown that the quantum fluctuation of squeezed states of the system is independent of the driving part, but depends on the initial states of the system.
By using the Lewis–Riesenfeld quantum invariant theory and properly choosing Hermitian invariant operator, a closed solution of the Schrödinger equation is derived for two forced quantum oscillators with mixing of two modes, and the quantum fluctuations in the output fields are evaluated. For the initial two-mode squeezed number or squeezed coherent state, in some particular conditions, the time evolution of the oscillators can not only preserve the initial two-mode squeezing, but also produce squeezing in the individual modes; and exhibit a periodical squeezing behaviour. For the initial two-mode number state or coherent state, there is no squeezing in the individual and mutual quadrature phases of the two-mode fields. Furthermore, regardless of which state above being initially considered, the quantum fluctuations of all the quadrature phases in the output fields are all independent of the driving parameters. In particular, for the initial two-mode coherent state, the variances of the output fields are also independent of other parameters in the Hamiltonian, and always preserve their initial values 1/4.
By properly selecting the time-dependent unitary transformation for the linear combination of the number operators, we construct a time-dependent invariant and derive the corresponding auxiliary equations for the degenerate and non-degenerate coupled parametric down-conversion system with driving term. By means of this invariant and the Lewis–Riesenfeld quantum invariant theory, we obtain closed formulae of the quantum state and the evolution operator of the system. We show that the time evolution of the quantum system directly leads to production of various generalized one- and two-mode combination squeezed states, and the squeezed effect is independent of the driving term of the Hamiltonian. In some special cases, the current solution can reduce to the results of the previous works.
Abstract In quantum mechanics courses, students often solve the Schrödinger equation for the harmonic oscillator with time-independent parameters. However, time-dependent quantum harmonic oscillators are relevant in modeling several problems as, for instance, the description of quantum motion of particles in traps, shortcuts to adiabaticity, generation of squeezed states, as well as quantum scalar fields evolving in expanding universes. In the present paper, we discuss, with a pedagogical approach, the quantum harmonic oscillator with time-dependent frequency via the Lewis–Riesenfeld dynamical invariant method, revisiting the main steps to obtain the wave function associated with this model, and briefly discussing the relation between this oscillator and the generation of squeezed states. As examples of didactic applications of time-dependent harmonic oscillators and the Lewis–Riesenfeld method in quantum mechanics courses, we solve the following problems: the calculation of the transition probability associated with a harmonic oscillator which undergoes jumps in its frequency, and the analysis of the dynamics of a quantum particle in a Paul trap.
The two quantum invariant operators are found from the time-dependent Hamiltonian of the harmonic oscillator with an auxiliary condition. The solution of the Schr\"odinger equation for the system, such as the eigenfunctions, eigenvalues, and minimum uncertainty, is derived by utilizing these invariant operators. The coherent states of this system are not the squeezed states, and the eigenfunction of the invariant operator is not the eigenfunction of the Hamiltonian of the system unless it is in the invariant representation. The squeezing function, which is an eigenfunction of the Hamiltonian of the system in the invariant representation and which also gives the minimum uncertainty, is obtained by a set of unitary transformed operators, i.e., squeezing operators.