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Quantum Optics Journal of the European Optical Society Part B

Squeezing in the Schrodinger picture: normal ordering technique to solve the mass-varying oscillator

A. L. De Brito, B. Baseia

Published Aug 1, 1992• 3 citations
Publisher

Abstract

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.

DOI
10.1088/0954-8998/4/4/001

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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.
In a previous paper the authors have employed a group theoretic method to find the evolution operator for a quantum system having a SU(2) Hamiltonian. In this paper, they consider a more complicated system whose Hamiltonian consists of SU(2) and h(4) group generators. A transformation method will be introduced, in conjunction with the above group theoretic method, to tackle this quantum problem. The result thus obtained will be applied to the problem of a mass-varying harmonic oscillator under an external force. Their result shows that the equation of motion of this oscillator is identical to a damped harmonic oscillator under an external force. In addition, it is shown that an initial coherent state will evolve as a squeezed state under the above Hamiltonian.
For investigating dynamic evolution of a mass-varying harmonic oscillator we constitute a ket-bra integration operator in coherent state representation and then perform this integral by virtue of the technique of integration within an ordered product of operators. The normally ordered time evolution operator is thus obtained. We then derive the Wigner function of u(t)|n⟩, where |n⟩ is a Fock state, which exhibits a generalized squeezing, the squeezing effect is related to the varying mass with time.
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 design of squeezing by a time-dependent harmonic oscillator is considered. The predetermined general time-dependent squeezing transformations are taken as the starting point from which the time evolution operator U(t,0) and then the corresponding time-dependent Hamiltonian are derived. The technique of integration within an ordered product of operators and the coherent-state representation are used in the intermediate derivations, where U(t,0) is obtained as an operator image of the classical symplectic transformation in phase space by a phase-space integral over coherent states.