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Physical Review A

Normal ordering of the SU(1,1) and SU(2) squeeze operators

Chi‐Fai Lo

Published Feb 1, 1995• 6 citations
Publisher

Abstract

We have derived the normal-ordered form of the SU(1,1) and SU(2) squeeze operators in the multimode bosonic representation and evaluated the coherent-state matrix elements of these operators. The analysis can be applied to any exponential functions of the generators of the Lie algebras su(1,1) and su(2) without difficulty.
DOI
10.1103/physreva.51.1706

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We have constructed the multimode bosonic realization of the su(1,1) Lie algebra, on the basis of which the su(1,1) generalized coherent state in the multimode Fock space is derived. It is shown that the multimode coherent state is, in fact, the generalized multimode squeezed vacuum state discussed by Lo and Sollie [Phys. Rev. A 47, 733 (1993)], and that the unitary displacement operator can be identified as the generalized multimode squeezing operator. Hence, similar to the single- and two-mode cases, our multimode bosonic realization of the su(1,1) Lie algebra has immediate relevance to the squeezing properties of boson fields.
In a previous paper Zhan et al. (2022) proposed the coherent state representation for linear operator, which shows a beautiful theoretical value. Inspired by the work, we further extend it to the nonlinear case, including single- and two-mode squeezing process. Based on squeezing operators’ transformation relation, we introduce their coherent state representations and derive their normal ordering forms by using the technique of integration within product of operator. As applications, we examine some operators’s normal ordering forms and show the entanglement property of prepared quantum states via the Schmidt decomposition. In particular, we further consider two cascaded two-mode squeezing operators, in which a generalized SU(1,1) interferometer is involved. In addition, the application of two-mode squeezing operator followed by photon-subtraction to generate non-Gaussian entangled states in quantum teleportation is discussed. Our results will be useful in preparation of quantum states and quantum metrology.
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.
Generalized normal- and antinormal-order decomposition formulas are derived for exponential functions of the generators of su(1, 1) and su(2) Lie algebras. The expectation values of analytic functions of these generators are calculated in terms of the generalized decomposition formulas, allowing su(1, 1) and su(2) fluctuations and their squeezing properties to be discussed. The second-order correlation functions are also considered. It is shown that the Mach–Zehnder interferometer, which is described by su(2) Lie algebra, is equivalent to a three-dimensional rotation and is characterized by the quantum counterparts of the classical Euler angles.