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Communications in Theoretical Physics

One- and Two-Mode Combination Squeezing Operator for Two Harmonic Oscillators with Coordinate-Momentum Coupling

Fan Hong‐Yi, Peng Yan

Published Sep 1, 2007• 10 citations
Publisher

Abstract

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.

DOI
10.1088/0253-6102/48/3/008

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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.
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.
We find that the exponential operators U=exp[-i(${\ensuremath{\lambda}}_{1}$Q1{\mathit{Q}}_{1}Q1​P2{\mathit{P}}_{2}P2​+${\ensuremath{\lambda}}^{2}$Q2{\mathit{Q}}_{2}Q2​P1{\mathit{P}}_{1}P1​] and W=exp[-i(${\ensuremath{\lambda}}_{1}$Q1{\mathit{Q}}_{1}Q1​Q2{\mathit{Q}}_{2}Q2​-${\ensuremath{\lambda}}_{2}$pP1{\mathrm{P}}_{1}P1​P2{\mathit{P}}_{2}P2​)] (where ${\ensuremath{\lambda}}_{1}$ and ${\ensuremath{\lambda}}_{2}$ are real; Pi{P}_{i}Pi​ and Qi{Q}_{i}Qi​ are coordinate and momentum operators, respectively; and i=1,2) are two types of generalized squeezing operators responsible for generating two types of one- and two-mode combination squeezed states, respectively. The coordinate and/or momentum representations of U and W are presented, and their normal product forms are first derived in terms of the newly developed technique of integration within an ordered product, which provides us with a direct approach to constructing these new squeezed states. The fluctuations in quadrature phases for these states are analyzed.
We find that the exponential operator V ≡exp[ i λ( Q 1 P 2 + Q 2 P 3 +⋅⋅⋅+ Q n −1 P n + Q n P 1 )], Q i , P i are, respectively, the coordinate and momentum operators, is an n -mode squeezing operator which engenders standard squeezing. By virtue of the technique of integration within an ordered product of operators we derive V 's normally ordered expansion and obtain the n -mode squeezed vacuum states, its Wigner function is calculated by using the Weyl ordering invariance under similar transformations.
In a squeezed state, the variance in one canonical variable may be suppressed below that normally associated with either the ground state or a coherent state, at the expense of an expansion in the variance of the conjugate variable. Squeezed states are usually discussed in the context of quantized light fields. The principal properties of squeezed states are demonstrated using the motion of quantum mechanical simple harmonic oscillators. The motion of a single oscillator is discussed to introduce the key concepts of squeezing, including quadrature operators, and the error contours of the Wigner function describing the quantum quasiprobability distributions in phase space of the oscillator motion. The main topic of two-mode squeezing is then addressed, in which the fluctuations in a system of two oscillators are tightly correlated. The existence of squeezing is demonstrated in normal mode coordinates representing motion of superpositions of the motion of the two oscillators. The fluctuations within a single oscillator are shown to increase when squeezing increases in the normal modes, generating thermal noise in individual mode subsystems.