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Chinese Physics B

New formulas for normalizing photon-added (-subtracted) two-mode squeezed thermal states

Liyun Hu, Hong-Yi Fan, Zhi‐Ming Zhang

Published Mar 1, 2013• 14 citations
Publisher

Abstract

For the first time, we derive the compact forms of normalization factors for photon-added (-subtracted) two-mode squeezed thermal states by using the P -representation and the integration within an ordered product of operators (IWOP) technique. It is found that these two factors are related to the Jacobi polynomials. In addition, some new relationships for Jacobi polynomials are presented.

DOI
10.1088/1674-1056/22/3/034202

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For the first time, we derive the compact forms of normalization factors for photon-added (-subtracted) two-mode squeezed thermal states by using the P -representation and the integration within an ordered product of operators (IWOP) technique. It is found that these two factors are related to the Jacobi polynomials. In addition, some new relationships for Jacobi polynomials are presented.
We show that the two-mode photon-subtracted squeezed state (TPSSS) is a squeezed two-variable Hermite polynomial excitation state, and we can therefore determine its normalization as a Jacobi polynomial of the squeezing parameter. Some new relations about the Jacobi polynomials are obtained by this analysis. We also show that the TPSSS can be treated as a two-variable Hermite-polynomial excitation on squeezed vacuum state. The technique of integration within an ordered product of operators brings convenience in our derivation.
We introduce a kind of entangled state—a photon-addition two-mode squeezed thermal state (TMSTS)—by adding photons to each mode of the TMSTS. Using the P-representation of thermal state, the compact expression of the normalization factor is derived, a Jacobi polynomial. The nonclassicality is investigated by exploring especially the negativity of Wigner function. The entanglement is discussed by using Shchukin–Vogel criteria. It is shown that the photon addition to the TMSTS may be more effective for the entanglement enhancement than the photon subtraction from the TMSTS. In addition, the quantum teleportation is also examined, which shows that symmetrical photon-added TMSTS may be more useful for quantum teleportation than the nonsymmetric case.
For directly normalizing the photon non-Gaussian states (e.g., photon added and subtracted squeezed states), we use the method of integration within an ordered product (IWOP) of operators to derive some new bosonic operator-ordering identities. We also derive some new integration transformation formulas about one- and two-variable Hermite polynomials in complex function space. These operator identities and associative integration formulas provide much convenience for constructing non-Gaussian states in quantum engineering.