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Acta Physica Sinica

Deriving operator identities by two-mode coherent-entangled state representation

Nanrun Zhou, Gong Li-Hua, Jia Fang, (1)江西师范大学物理与通信电子学院，南昌 330022; (2)南昌大学电子信息工程系，南昌 330031

Published Jan 1, 2009• 2 citations
Publisher

Abstract

On the basis of the new introduced two-mode coherent entangled state representation and the technique of integration within an ordered product of operators, we derived some operator identities of the operator functions about the quadrature of two-mode optical field, such as （X1+X2）/21/2 and （μX1+νX2）/λ, which should be helpful to further studying the high order squeezing properties of optical field and to constructing some generalized-squeezed states.

DOI
10.7498/aps.58.2179

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On the basis of the new introduced two-mode coherent entangled state representation and the technique of integration within an ordered product of operators, we derived some operator identities of the operator functions about the quadrature of two-mode optical field, such as （X1+X2）/21/2 and （μX1+νX2）/λ, which should be helpful to further studying the high order squeezing properties of optical field and to constructing some generalized-squeezed states.
The coherent-intermediate-entangled state |α, x ⟩ λ,ν is proposed in the two-mode Fock Space, which exhibits both the properties of the coherent and entangled states. The |α, x ⟩ λ,ν makes up a new quantum mechanical representation, and the completeness relation of |α, x ⟩ λ,ν is proved by virtue of the technique of integral within an ordered product of operators. The corresponding squeezing operators are derived. Furthermore, Generalized P-representation is constructed in the coherent-intermediate-entangled state |α, x ⟩ λ,ν and the Schmidt decomposition of |α, x ⟩ λ,ν is investigated.
We introduce the three-mode entangled state and set up an experiment to generate it. Then we discuss the three-mode squeezing operator squeezed | p, x2, x3> →μ-3/2|p/μ, x2/μ, x3/μ> and the optical implement to realize such a squeezed state. We also reveal that c-number asymmetric shrink transform in the three-mode entangled state, i.e. |p, x2,x3> →μ-1/2|p/μ,x2,x3>, maps onto a kind of one-sided three-mode squeezing operator exp{iλ/3(∑3i=1 Pi) (∑3i=1 Qi) - λ/2}. Using the technique of integration within an ordered product (IWOP) of operators, we derive their normally ordered forms and construct the corresponding squeezed states.
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.
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.