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Communications in Theoretical Physics

Two-Variable Hermite Polynomial Excitation of Two-Mode Squeezed Vacuum State as Squeezed Two-Mode Number State

Liyun Hu, Fan Hong‐Yi

Published Oct 1, 2008• 12 citations
Publisher

Abstract

We find that the squeezed two-mode number state is just a two-variable Hermite polynomial excitation of the two-mode squeezed vacuum state (THPES). We find that the Wigner function of THPES and its marginal distributions are just related to two-variable Hermite polynomials (or Laguerre polynomials) and that the tomogram of THPES can be expressed by one-mode Hermite polynomial.

DOI
10.1088/0253-6102/50/4/35

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We find that the squeezed two-mode number state is just a two-variable Hermite polynomial excitation of the two-mode squeezed vacuum state (THPES). We find that the Wigner function of THPES and its marginal distributions are just related to two-variable Hermite polynomials (or Laguerre polynomials) and that the tomogram of THPES can be expressed by one-mode Hermite polynomial.
Based on the two-mode squeezing operator which is the quantum version of the symplectic transformation, we find that the corresponding squeezed two-mode number state is just a two-variable Hermite polynomial excited state, which possesses well-behaved features, e.g. its Wigner function is a direct product of two Laguerre polynomials; its Husimi function, a Gaussian broaden version of the Wigner function, is related to two-variable Hermite polynomials. Moreover, its quantum statistical features, such as squeezing properties and the inter-mode photon bunching, are discussed.
The nonclassicality of the two-variable Hermite polynomial state is investigated. It is found that the two-variable Hermite polynomial state can be considered as a two-mode photon subtracted squeezed vacuum state. A compact expression for the Wigner function is also derived analytically by using the Weyl-ordered operator invariance under similar transformations. Especially, the nonclassicality is discussed in terms of the negativity of the Wigner function. Then violations of Bell’s inequality for the two-variable Hermite polynomial state are studied.
The nonclassicality of the two-variable Hermite polynomial state is investigated. It is found that the two-variable Hermite polynomial state can be considered as a two-mode photon subtracted squeezed vacuum state. A compact expression for the Wigner function is also derived analytically by using the Weyl-ordered operator invariance under similar transformations. Especially, the nonclassicality is discussed in terms of the negativity of the Wigner function. Then violations of Bell's inequality for the two-variable Hermite polynomial state are studied.
We investigate the statistical properties of the photon-subtractions from the two-mode squeezed vacuum state and its decoherence in a thermal environment. It is found that the state can be considered as a squeezed two-variable Hermite polynomial excitation vacuum and the normalization of this state is the Jacobi polynomial of the squeezing parameter. The compact expression for Wigner function (WF) is also derived analytically by using the Weyl ordered operators' invariance under similar transformations. Especially, the nonclassicality is discussed in terms of the negativity of WF. The effect of decoherence on this state is then discussed by deriving the analytical time evolution results of WF. It is shown that the WF is always positive for any squeezing parameter and photon-subtraction number if the decay time exceeds an upper bound (}$κt>{1/2}\ln \frac{2\bar{n}+2}{2\bar{n}+1}).
Based on the fact that a two-mode squeezed number state is a two-variable Hermite polynomial excitation of the two-mode squeezed vacuum state, the result of one-mode l -photon measurement for the two-mode squeezed number state S 2 | m , n ⟩ is discussed. It is found that a remaining field-mode simultaneously collapses into a number state | n – m + l ⟩ with the coefficient being a Jacobi polynomial of n , m and l , which manifestly exhibits the entanglement between the two modes, i.e. it depends on the number-difference between the two modes. The second mode collapses into an excited coherent state when the first mode is measured as a coherent state.
We emphasize the importance of two-mode Hermite poly-nomials in quantum optics theory,we consider it as the prominent function in studying the entangled state of continuum variables and two-mode squeezed state,and that possesses a clear physical meaning.By virtue of the two-mode Hermite polynomials and combining operators' integration technique within normal and antinormal orderings,we derive some new operator identities and mutual invertible integration transformation,and prove that squeezed two-mode number state is just the two-mode Hermite polynomial excited squeezed two-mode vacuum state.