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Physical Review A

Generalized multimode squeezed states

Chi‐Fai Lo, Roger Sollie

Published Jan 1, 1993• 49 citations
Publisher

Abstract

The notion of squeezing is generalized to the case of multimodes. The multimode generalized squeezing operator has similar algebraic properties to those of the single-mode case and reduces to the usual two-mode squeezing operator in the case of two modes. It is also shown that the generalized multimode squeezed state is a multimode minimum-uncertainty state if the squeezing parameters are real.

DOI
10.1103/physreva.47.733

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The notion of squeezing is generalized to the case of multimodes. The multimode generalized squeezing operator has similar algebraic properties to those of the single-mode case and reduces to the usual two-mode squeezing operator in the case of two modes. It is also shown that the generalized multimode squeezed state is a multimode minimum-uncertainty state if the squeezing parameters are real.
Multimode squeeze and rotation operators are defined such that they have extremely similar algebraic properties as those of their single-mode counterparts. It is shown that the introduction of N-mode squeeze operators provides a convenient set of parameters to describe the variances of the quadrature amplitudes in multimode Gaussian squeezed states. Some important properties of these N-mode unitary operators are investigated. It is also shown that the time-evolution operator for a general N-mode quadratic Hamiltonian can be conveniently expressed as an operator product containing an N-mode squeeze operator, an N-mode rotation operator, and an N-mode displacement operator.
By the complex multimode Bogoliubov transformation, we obtain the general forms of squeeze operators and squeezed states including squeezed vacuum states, squeezed coherent states, squeezed Fock states and squeezed coherent Fock states, for a general multimode boson system. We decompose the squeezed operator into disentangling form in normal ordering to simplify the expressions of the squeezed states. We also calculate the statistical properties of the SS. Furthermore we prove that if it is non-degenerate, a Bogoliubov transformation matrix can be decomposed into three basic matrix, by which we can not only check the criterion of the minimum uncertainty state (MUS) found by Milburn, but also prove that except the special cases, any multimode squeezed state is MUS after the original creation and annihilation operators rotate properly. We also discuss some special cases of the three basic matrices. Finally we give an analytical results of a two-mode system as an example.
We show that the best way to study the relationship between quantum entanglement and quantum squeezing for the multimode case is through constructing the new n -pair entangled state representation . The explicit form of the 2 n -mode squeezing operator which squeezes the n -pair entangled state can be directly derived via the transition from to where M is an n × n complex matrix, and the technique of integration within an ordered product of operators. We also analyze the squeezing properties of the 2 n -mode squeezed state for M being Hermitian, and obtain the variances of the 2 n -mode quadrature operators in the 2 n -mode squeezed vacuum state with a concise result; the condition for reaching the minimum of the uncertainty relationship is also investigated.