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Acta Physica Sinica (Overseas Edition)

Squeezing properties of q -deformed oscillator

Guo Guang-Can, Liang Jiu-Qing

Published Feb 1, 1994• 1 citation
Publisher

Abstract

The q -coherent states are indeed minimum uncertainty states for corresponding q -deformed operators. The time evolution of q -coherent states are given by introducing the q -boson Hamiltonian analogous to the one for normal bosons. It is shown that initially prepared q -coherent states are no longer the eigenstates of q -annihilation operator during time evolution and eventually become squeezing states. The squeezing properties are studied numerically.

DOI
10.1088/1004-423x/3/2/002

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The q -coherent states are indeed minimum uncertainty states for corresponding q -deformed operators. The time evolution of q -coherent states are given by introducing the q -boson Hamiltonian analogous to the one for normal bosons. It is shown that initially prepared q -coherent states are no longer the eigenstates of q -annihilation operator during time evolution and eventually become squeezing states. The squeezing properties are studied numerically.
Coherent states and the extension to the even and odd coherent states, and further to the eigenstates of the higher power of the annihilation operator are basic and important macroscopic quantum states in quantum field theory. Recently, these investigations have been extended to the aharmonic oscillator and the quantum algebra with q -deformed or qs-deformed commutation relations. In this paper,the orthogonal eigenstates of the higher order of the annihilation operator of the q -deformed aharmonic oscillator are also derived. It is pointed out that these different macroscopic quantum states have similar squeezing properties of higher order amplitude. The reason is found that the orthogonal eigenstates of the higher order of the annihilation operator of the harmonic oscillator, the q -deformed oscillator, the qs -deformed oscillator, the Enharmonic oscillator and the q -deformed anharmonic oscillator all have even and odd structure, which is similar to the simple even and odd coherent states. For other systems like the system with SU(1 ,1) Lie algebra or q -deformed quantum algebra also have such properties.
We obtained the uncertainty relation in squeezed states for a time-dependent oscillator. The uncertainty relation in coherent states is same as that of the number states with n=0. However, the uncertainty relation in squeezed states does not satisfy this property and depends on squeezing parameter c. For instance, the uncertainty relation is ℏ/2 which is the minimum value as far as quantum mechanics permits for c=1, same as that in coherent state for c=±∞, and infinity for c=-1. If the time-dependency of the Hamiltonian for the system vanishes, the uncertainty relation in squeezed states will no longer depend on c and becomes the same as that in number state with n=0, like the uncertainty relation in coherent states.
Time evolution of squeezed coherent states for a quantum parametric oscillator with the most general self-adjoint quadratic Hamiltonian is found explicitly. For this, we use the unitary displacement and squeeze operators in coordinate representation and the evolution operator obtained by the Wei-Norman Lie algebraic approach. Then, we analyze squeezing properties of the wave packets according to the complex parameter of the squeeze operator and the time-variable parameters of the Hamiltonian. As an application, we construct all exactly solvable generalized quantum oscillator models classically corresponding to a driven simple harmonic oscillator. For each model, defined according to the frequency modification in position space, we describe explicitly the squeezing and displacement properties of the wave packets. This allows us to see the exact influence of all parameters and make a basic comparison between the different models.