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Journal of Physics A Mathematical and Theoretical

Generalized multi-mode bosonic realization of theSU(1, 1) algebra and its corresponding squeezing operator

Hong‐Chun Yuan, Heng-Mei Li, Hong-Yi Fan

Published Jan 29, 2010• 3 citations
Publisher

Abstract

By constructing a generalized multi-partite entangled state representation and introducing the ket-bra integral in this representation, we find a new set of generalized bosonic realization of the generators of the SU (1, 1) algebra, which can compose a generalized multi-mode squeezing operator. This operator squeezes the multi-partite entangled state in a natural way. Then the corresponding multi-mode squeezed vacuum states | r ⟩ is obtained. Based on this, the variances of the n -mode quadratures and the higher-order squeezing in | r ⟩ are evaluated. In addition, we examine the violation of the Bell inequality for | r ⟩ by using the formalism of Wigner representation.

DOI
10.1088/1751-8113/43/7/075304

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By constructing a generalized multi-partite entangled state representation and introducing the ket-bra integral in this representation, we find a new set of generalized bosonic realization of the generators of the SU (1, 1) algebra, which can compose a generalized multi-mode squeezing operator. This operator squeezes the multi-partite entangled state in a natural way. Then the corresponding multi-mode squeezed vacuum states | r ⟩ is obtained. Based on this, the variances of the n -mode quadratures and the higher-order squeezing in | r ⟩ are evaluated. In addition, we examine the violation of the Bell inequality for | r ⟩ by using the formalism of Wigner representation.
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.
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.