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Acta Physica Sinica

Two-mode integral form projection operator and its application in quantum optics

Shimin Xu, Hongqi Li, Ji‐Suo Wang, Xing-Lei Xu, (1)菏泽学院物理系,菏泽 274015; (2)聊城大学物理系,聊城 252059

Published Jan 1, 2009• 7 citations
Publisher

Abstract

A two-mode integral form projection unitary operator is introduced. Using the technique of integration within an ordered product （IWOP） of operators, the transformation property of the integral form projection operator is studied, and its normal ordered expression is derived. Then the Hamiltonian of two-mode quantum harmonic oscillator system with coordinate-momentum coupling is diagonalized by virtue of the integral form projection operator, there by the eigenenergy and eigenfunction of the system are exactly solved. Lastly, a special case is discussed.

DOI
10.7498/aps.58.2174

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A two-mode integral form projection unitary operator is introduced. Using the technique of integration within an ordered product （IWOP） of operators, the transformation property of the integral form projection operator is studied, and its normal ordered expression is derived. Then the Hamiltonian of two-mode quantum harmonic oscillator system with coordinate-momentum coupling is diagonalized by virtue of the integral form projection operator, there by the eigenenergy and eigenfunction of the system are exactly solved. Lastly, a special case is discussed.
A double-mode integral form projection unitary operator is introduced. Using the technique of integration within an ordered product (IWOP) of operators, the transformation character of the integral form projection operator is studied. The Hamiltonian of double-mode quantum harmonic oscillator system with kinetic coupling is diagonalized by virtue of the integral form projection operator, and then the eigenenergy and eigenfunction of the system are accurately solved. Finally the particular case is discussed.
By virtue of the technique of integration within an ordered product of operators, we derive the normal ordering expansion of a one- and two-mode combination squeezing operator for two harmonic oscillators with coordinate-momentum coupling. It turns out that this squeezing operator just diagonalizes the Hamiltonian H = p 1 2 /2 m 1 + m 1 ω 1 2 x 1 2 /2+ p 2 2 /2 m 2 + m 2 ω 2 2 x 2 2 /2−λ x 2 p 1 , so its ground state is a one- and two-mode combination squeezed state. Quantum fluctuation in the ground state is calculated.
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.