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Physical Review A

Multimode squeeze operators and squeezed states

Xin Ma, William Rhodes

Published May 1, 1990• 149 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.41.4625

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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.
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.
In the field of continuous variables and particularly of Gaussian states an important role is played by the statistics of photon numbers. This issue has been considered in pioneering works for the one- and two-mode cases, but a systematic approach seems not to be available. In this paper, the most general multimode mixed Gaussian state, which is generated starting from a multimode thermal state and processed by the most general Gaussian unitary, is considered. In the NNN-mode the photon numbers are represented by NNN random variables, one for each mode, and the target is the evaluation of the means and of the covariances. The means describe statistically the amount of photon numbers in each mode, while the covariances give the correlations between the photon numbers in the different modes. For both means and covariances a closed-form result is obtained, expressed in simple and compact formulas. The main tools are provided by the representation of Gaussian unitaries (given by a cascade of an NNN-mode squeeze operator, an NNN-mode rotation operator, and a parallel set of NNN single-mode displacement operators) and by the corresponding Bogoliubov transformations.
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 consider the multimode generalization of the normally ordered factorization formula of squeezings. This formula allows us to establish relationships between various representations of squeezed states, to calculate partial traces, mean values, and variations. The main results are expressed in terms of the matrix representation of canonical transformations, which is a convenient and numerically stable mathematical tool. Explicit representations are given for the inner product and the composition of generalized multimode squeezings. Explicitly solvable evolution problems are considered.
Gaussian quantum states of light have many applications in quantum technologies. Two of the most widely used Gaussian quantum states of light are the single- and two-mode squeezed states. In this thesis, we propose the generation of a hybrid of these two states, which has both properties of single- and two-mode squeezed states. We then extend our method to the generation of multimode squeezed states which posses N-partite entanglement. Our states are defined using Gaussian frequency modes, and our method relies on a nonlinear-optical process called spontaneous parametric downconversion. We shape the joint spectral amplitude of the generated light in two ways, by spectrally engineering the light incident on the crystal and by the engineering of the nonlinear crystal to have desired properties. We first use our method to generate tuneable hybrid squeezed states and then extend it to multimode squeezed states. We then investigate design considerations for the nonlinear crystal and study the effect of fabrication errors. The tuneable hybrid and multimode squeezed states are localized in both frequency and time, making them ideal for a variety of quantum information protocols. We expect that the states will be used to generalize many quantum information protocols to their multimode counterparts, such as, multi-parameter quantum metrology, multi-channel quantum imaging and multi-partite teleportation. Our work will therefore broaden the applicability of optics for the development of quantum technologies.