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Physical Review A

Unitary transformation for four harmonically coupled identical oscillators

Fan Hong-Yi

Published Oct 1, 1990• 20 citations
Publisher

Abstract

A unitary operator U, which can transform the Fock space of a four-dimensional oscillator into the space in which the Hamiltonian of four harmonically coupled identical one-dimensional harmonic oscillators is diagonalized, is found. The coordinate representation of U is presented and is used to directly derive the wave function of the energy eigenstate of the coupled oscillators. The normally ordered form of U is also deduced with the use of integration within an ordered product.

DOI
10.1103/physreva.42.4377

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A unitary operator U, which can transform the Fock space of a four-dimensional oscillator into the space in which the Hamiltonian of four harmonically coupled identical one-dimensional harmonic oscillators is diagonalized, is found. The coordinate representation of U is presented and is used to directly derive the wave function of the energy eigenstate of the coupled oscillators. The normally ordered form of U is also deduced with the use of integration within an ordered product.
In view of a recent paper [F. Michelot, Phys. Rev. A 45, 4271 (1992)] tackling the Hamiltonian of an arbitrary number of harmonically coupled oscillators, we present a coordinate representation of the unitary operator that can diagonalize the Hamiltonian. The normally ordered form of the unitary operator, which manifestly connects two Fock spaces associated with the uncoupled and coupled oscillators, is also derived by virtue of the technique of integration within an ordered product of operators.
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We derive the density matrix ρ(x1, x2; x1', x2', β) for two coupled harmonic oscillators by taking a unitary transformation approach. The corresponding unitary operator U is characteristic of including frequency-jump squeezing transformation. By virtue of the technique of integration within ordered product and the coordinate representation of U we manifestly show that the ground state of the coupled oscillators is a squeezed state. Based on the derived ρ(x1, x2; x1', x2', β) we calculate the average coupling energy.