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Communications in Theoretical Physics

Two-Variable Hermite Function as Quantum Entanglement of Harmonic Oscillator's Wave Functions

Lu Hai-Liang, Fan Hong-Yi

Published Jun 1, 2007• 8 citations
Publisher

Abstract

We reveal that the two-variable Hermite function h m , n , which is the generalized Bargmann representation of the two-mode Fock state, involves quantum entanglement of harmonic oscillator's wave functions. The Schmidt decomposition of h m , n is derived. It also turns out that h m , n can be generated by windowed Fourier transform of the single-variable Hermite functions. As an application, the wave function of the two-variable Hermite polynomial state S ( r ) H m , n (μ a 1 † ,μ a 2 † )|00⟩, which is the minimum uncertainty state for sum squeezing, in ⟨η| representation is calculated.

DOI
10.1088/0253-6102/47/6/011

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We reveal that the two-variable Hermite function h m , n , which is the generalized Bargmann representation of the two-mode Fock state, involves quantum entanglement of harmonic oscillator's wave functions. The Schmidt decomposition of h m , n is derived. It also turns out that h m , n can be generated by windowed Fourier transform of the single-variable Hermite functions. As an application, the wave function of the two-variable Hermite polynomial state S ( r ) H m , n (μ a 1 † ,μ a 2 † )|00⟩, which is the minimum uncertainty state for sum squeezing, in ⟨η| representation is calculated.
We deduce entangled fractional Fourier transformation (EFFT) for the multipartite entangled state representation, which was newly constructed with two mutually conjugate n -mode entangled states of continuum variables in n -mode Fock space. We establish a formalism of EFFT for quantum mechanical wave functions, which provides us a convenient way to derive some wave functions. We find that the eigenmode of EFFT is different from the usual Hermite Polynomials. We also derive the EFFT of the n -mode squeezed 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.