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Physical Review A

Temperature changes and squeezing properties of the system of time-dependent harmonic oscillators

Jeong‐Young Ji, Jae Kwan Kim

Published Feb 1, 1996• 20 citations
Publisher

Abstract

The density operator is defined as a function of a generalized invariant for the system of time-dependent harmonic oscillators. Applying this to the Hamiltonian with two asymptotic regions, we calculate the temperature change driven by a time-varying Hamiltonian. Furthermore, the squeezing properties of an eigenstate and a coherent state of the initial Hamiltonian are discussed. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreva.53.703

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The density operator is defined as a function of a generalized invariant for the system of time-dependent harmonic oscillators. Applying this to the Hamiltonian with two asymptotic regions, we calculate the temperature change driven by a time-varying Hamiltonian. Furthermore, the squeezing properties of an eigenstate and a coherent state of the initial Hamiltonian are discussed. \textcopyright{} 1996 The American Physical Society.
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.
It is shown that squeezing will inevitably be generated in a harmonic oscillator with time-dependent frequency for any nonvanishing rate of frequency change, sudden or smooth. The general form of the time-dependent transition probabilities among the eigenstates of the initial Hamiltonian is obtained with an exact operator approach. Results for the case of a sudden frequency jump are explicitly given as an illustration. It is also pointed out that to generate squeezing in a harmonic oscillator, the Hamiltonian does not have to be a unitary squeeze transformation of the free Hamiltonian.
The author examines the possibility of generating squeezed states out of coherent states by external changes of the oscillator frequency. Using the evolution operator method developed by Cheng and Fung, he investigates the evolution of a coherent state of a time-dependent harmonic oscillator as well as its squeezing and coherence property. Two specific models of frequency variation are explored; namely, one whose frequency is exponentially decreasing in time and the other with a periodic frequency. His analyses indicates that the wavefunction of the time-dependent oscillator starts as a coherent state at t=O and evolves as a squeezed state at a later time. It is also shown that squeezing cannot be generated by an adiabatic change of the oscillator frequency because the variances of q and p turn out to be adiabatic invariants, whereas a sudden change can produce squeezing.