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Physics Letters A

Evolution of a coherent state of an harmonic oscillator with time-dependent frequency

B. Crosignani, Paolo Di Porto, S. Solimeno

Published Dec 1, 1968• 12 citations
Publisher
DOI
10.1016/0375-9601(68)90278-8

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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.
The authors study the evolution of SU(1,1) coherent states, which may represent squeezed vacuum states, in harmonic oscillators with time-dependent masses. Such oscillators may model a quantised electromagnetic field in a cavity where the field changes with time under the action of some reservoir. They consider three types of variable mass: a decaying mass, a pulsating mass, and a mass with damping and pulsation. They have found that in the most interesting cases involving pulsation, a high degree of squeezing (enhanced squeezing) may be obtained in the vicinity of the resonance where the pulsation frequency is twice the natural oscillator frequency.
The temporal evolution of the state vector relative to a harmonic oscillator with time-dependent frequency is examined. The Schrödinger equation is solved by choosing the instantaneous eigenstates of the Hamiltonian as the basis, thus getting an infinite set of coupled linear differential equations. This formulation is particularly suitable for studying the cases in which the Hamiltonian undergoes a very slow or a sudden variation from an initial constant value into a final one. A rigorous proof of adiabatic invariance to all orders in the slowness parameter is given for the transition probabilities. An application to the evolution of an initially coherent state is made.