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Acta Sinica Quantum Optica

New Applications of Two-mode Hermite Polynomials in Quantum Optics Theory

Fan Hong‐Yi

Published Jan 1, 2012• 0 citations
OpenAlex

Abstract

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.

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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.
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.
Special polynomials (e.g. Hermite polynomials) are very important for the development of physics and mathematics. As a further extension of ordinary Hermite polynomials, we introduce new generalized Hermite polynomials with three variables and find their generating functions using the operator ordering method in quantum optics. Also, some new operator identities and integral formulas are obtained. As applications, the normalization, Wigner functions and evolutions for certain quantum states are analytically presented. These analytical results can provide conveniences for numerically studying the properties and applications of such quantum states.
By virtue of the entangled state representation we concisely derive some new operator identities with regard to the two-variable Hermite polynomial (TVHP). By them and the technique of integration within an ordered product (IWOP) of operators we further derive new generating function formulas of the TVHP. They are useful in quantum optical theoretical calculations. It is seen from this work that by combining the IWOP technique and quantum mechanical representations one can derive some new integration formulas even without really performing the integration.
In this paper, by virtue of the entangled state representation we concisely derive some new operator identities regarding to two-variable Hermite polynomial (TVHP). By them and the technique of integration within an ordered product (IWOP) of operators we further derive new generating function formulas of TVHP. They are useful in quantum optical theoretical calculations. It is seen from this work that by combining the IWOP technique and quantum mechanical representations one can derive some new integration formulas even without really performing the integration.