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Journal of Optics B Quantum and Semiclassical Optics

New quantum squeezed states for the time-dependent harmonic oscillator

Antonio Boulhosa Nassar

Published Apr 5, 2002• 6 citations
Publisher

Abstract

New generalized squeezed states for the time-dependent harmonic oscillator are found through the theory of invariants. Our method gives a comparatively clearer picture than methods using evolution operators because we can establish a direct connection between the classical and quantum solutions. The additional significance of our method is that it is possible to find new generalized quantum squeezed states from just one particular solution to the classical time-dependent oscillator. Accordingly, more general results for the variance of x (squared uncertainty) are found for the cases of linear sweep of the restoring force and compared to recent results encountered in the literature.

DOI
10.1088/1464-4266/4/3/379

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New generalized squeezed states for the time-dependent harmonic oscillator are found through the theory of invariants. Our method gives a comparatively clearer picture than methods using evolution operators because we can establish a direct connection between the classical and quantum solutions. The additional significance of our method is that it is possible to find new generalized quantum squeezed states from just one particular solution to the classical time-dependent oscillator. Accordingly, more general results for the variance of x (squared uncertainty) are found for the cases of linear sweep of the restoring force and compared to recent results encountered in the literature.
In this paper we generalize our previous results of the generalized one-mode harmonic oscillator to the generalized two-mode case. Systematic use is made of the SO(3,2) dynamical group and we are able to write a general form for the exact time evolution operator in terms of squeezing operators of one and two modes. A complete classification of the exact solutions is made and we derive them explicitly whenever possible. The relevant results on algebraic decomposition, coherent state generators and classification of the solutions are shown in tables. A plethora of soluble Hamiltonians already treated in the literature, which appear to be particular cases of the general formalism presented herein, are analysed as well as new cases, which to the authors' knowledge, have not yet been considered.
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.
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.