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Journal of Physics A Mathematical and General

Squeezing by tuning the oscillator frequency

Chi‐Fai Lo

Published Apr 7, 1990• 52 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/23/7/021

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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 Hilbert space of a harmonic oscillator with frequency ω can be spanned by the overcomplete set of coherent states of a second oscillator with frequency ω'. These states are squeezed states of the first oscillator. We examine the possibility of generating squeezed states out of coherent states by external changes of the oscillator frequency and explore the two limiting cases of an adiabatic and a sudden change. Only the latter results in squeezing, while in the former symmetrical uncertainities in coordinate and momentum turn out to be adiabatic invariants.
In this paper we examine the possibility of generating displaced and squeezed number states out of number states by the general driven time-dependent oscillator. Using the evolution-operator method developed by Cheng and Fung [J. Phys. A 21, 4115 (1988)], we investigate the evolution of a number state of a general driven time-dependent oscillator as well as its squeezing property. Our analyses indicate that the wave function of the time-dependent oscillator starts as a number state at t=0 and evolves as a displaced and squeezed number state at a later time.
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.
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.
Recent papers by Cheng and Fung (1988) and Lo (1990) have studied the generation of squeezed states out of coherent states in systems described by some time-dependent Hamiltonians. They have employed the evolution operator method for the Schrodinger equation with a Hamiltonian expressible as H(t)=a l (t)J + +a 2 (t)J 0 + +a 3 (t)J - , where J +- ,J 0 are the SU(2) group generators. Here, for the sake of comparison of methods and results, the present authors employ the normal ordered Hamiltonian H(t)= Sigma h l,m (t)(a) l a m and investigate squeezing through an alternative approach using the normal ordering technique to solve the evolution operator for the same system.
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.