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Acta Physica Sinica

Thermo-vacuum state in a negative binomial optical field and its application

Zhi-Long Wan, Fan Hong‐Yi

Published Jan 1, 2015• 1 citation
Publisher

Abstract

The core of optical field theory at finite temperature is how to introduce the thermo-vacuum state which is the basis of comprehensive investigation of electromagnetic field by virtue of quantum statistic method. Based on the spirit of thermo-field dynamics initiated by Takahashi and Umezawa, we first employ the integration method within the ordered product of operators to search for thermo-vacuum state for the optical negative binomial state (NBS)s=s+1(1-)n|n,+,+,which takes the form of pure negative binomial state. The newly found thermo-vacuum state brings convenience for evaluating the Wigner function of NBS and the fluctuation of photon numbers in NBS.

DOI
10.7498/aps.64.190301

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The core of optical field theory at finite temperature is how to introduce the thermo-vacuum state which is the basis of comprehensive investigation of electromagnetic field by virtue of quantum statistic method. Based on the spirit of thermo-field dynamics initiated by Takahashi and Umezawa, we first employ the integration method within the ordered product of operators to search for thermo-vacuum state for the optical negative binomial state (NBS)s=s+1(1-)n|n,+,+,which takes the form of pure negative binomial state. The newly found thermo-vacuum state brings convenience for evaluating the Wigner function of NBS and the fluctuation of photon numbers in NBS.
We propose a new two-mode thermo- and squeezing-mixed optical field, described by the new density operator ρ = ( 1 − e f − | g | 2 ) e g a † b † e f a † a | 0 ⟩ b b ⟨ 0 | e g ∗ a b , where | 0 ⟩ b b ⟨ 0 | is the b -mode vacuum, e f a † a represents the thermo-field, and e g a † b † indicates squeezing. The photon statistics for ρ is studied by virtue of the method of integration within ordered product (IWOP) of operators. Such a field can be generated when a two-mode squeezed state passes through a one-mode dissipation channel.
Following the spirit of thermo field dynamics initiated by Takahashi and Umezawa, we employ the technique of integration within an ordered product of operators to derive the thermal vacuum state (TVS) for the Hamiltonian H of the two-coupled-oscillator model. The ensemble averages of the system are derived conveniently by using the TVS. In addition, the entropy for this system is discussed based on the relation between the generalized Hellmann—Feynman theorem and the entroy variation in the context of the TVS.
Displaced negative-binomial mixed states are introduced and their properties are studied. It is shown that they may be used to construct a resolution of the identity operator and, therefore, that they can be used as a ``basis'' in the Hilbert space. Generalized P and Q representations with respect to these states are introduced and their properties are discussed. The pure SU(1,1) coherent states discussed by Perelomov are considered in the two-mode harmonic oscillator Hilbert space and it is shown that the partial trace with respect to one of the modes leads to negative-binomial mixed states. Using this observation, the formalism of thermo-field-dynamics is generalized in a corresponding negative-binomial field dynamics.