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Physical Review A

Nonzero-temperature coherent and squeezed states for the harmonic oscillator: The time-dependent frequency case

J. Aliaga, G. Crespo, Araceli N. Proto

Published Jul 1, 1990• 35 citations
Publisher

Abstract

We present a density-matrix method, based on information theory, which makes it possible to find nonzero-temperature coherent and squeezed states for the harmonic oscillator with time-dependent frequency. We establish a connection between the appearance of squeezed states and the relevant operators included in the density matrix, compare our results with previous ones that were obtained using wave functions, and analyze the particular case of a sudden jump in the oscillator's frequency.

DOI
10.1103/physreva.42.618

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We present a density-matrix method, based on information theory, which makes it possible to find nonzero-temperature coherent and squeezed states for the harmonic oscillator with time-dependent frequency. We establish a connection between the appearance of squeezed states and the relevant operators included in the density matrix, compare our results with previous ones that were obtained using wave functions, and analyze the particular case of a sudden jump in the oscillator's frequency.
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.
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.