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Physical Review A

Displaced negative-binomial mixed states: Generalized thermo-field-dynamics

Apostol Vourdas, R. F. Bishop

Published Mar 1, 1995• 14 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.51.2353

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Acta Physica Sinica•Jan 1, 2015•1 citation
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.
We introduce coherent mixed states (or thermal coherent states) associated with the displaced harmonic oscillator at nonzero temperature (T ≠ 0), as a "random" (or "thermal" or "noisy") basis in Hilbert space. A resolution of the identity for these states is proven and is used to generalize the usual pure (T = 0) coherent state formalism to the mixed (T ≠ 0) case. This new formalism for thermal coherent states is then further generalized to a broader class of so-called negative-binomial mixed states. It is known that the negative-binomial distribution is itself intimately related to the discrete series of SU(1,1) representations. We consider the pure SU(1,1) coherent states in the two-mode harmonic oscillator space, and show how our negative-binomial mixed states arise from taking the partial trace with respect to one of these two modes. This observation is then used to show how the formalism of thermo-field dynamics may be generalized to a correspondingly much broader negative-binomial-field dynamics, which we expect to have many uses.
Coherent mixed states (or thermal coherent states) associated with the displaced harmonic oscillator at finite temperature, are introduced as a 'random' (or 'thermal' or 'noisy') basis in Hilbert space. A resolution of the identity for these states is proved and used to generalize the usual coherent state formalism for the finite temperature case. The Bargmann representation of an operator is introduced and its relation to the P and Q representations is studied. Generalized P and Q representations for the finite temperature case are also considered and several interesting relations among them are derived.
We study the non-classical properties and algebraic characteristics of the negative binomial states introduced by Barnett recently. The ladder operator formalism and displacement operator formalism of the negative binomial states are found and the algebra involved turns out to be the SU(1,1) Lie algebra via the generalized Holstein-Primarkoff realization. These states are essentially Peremolov's SU(1,1) coherent states. We reveal their connection with the geometric states and find that they are excited geometric states. As intermediate states, they interpolate between the number states and geometric states. We also point out that they can be recognized as the nonlinear coherent states. Their non-classical properties, such as sub-Poissonian distribution and squeezing effect are discussed. The quasiprobability distributions in phase space, namely the Q and Wigner functions, are studied in detail.
The thermal coherent states considered previously by the present authors represent an alternative mixed-state generalization of the usual pure-state coherent states. They describe displaced harmonic oscillators in thermodynamic equilibrium with a heat bath at nonzero temperature. We show how they provide a ``random'' (or ``thermal'' or ``noisy'') basis on a quantum-mechanical Hilbert space scrH. Their usefulness rests on the fact that the corresponding statistical density operator provides a probability operator measure on scrH. We thereby show how the thermal coherent states permit a generalization to nonzero temperatures of the well-known P and Q representations of operators in scrH. Particular emphasis here is placed on imbedding the formulation in the Bargmann or holomorphic representation of scrH. We examine the corresponding Bargmann representations of both state vectors and operators, and show how the former relate to the usual position and momentum representations and the latter to the usual P, Q, and W (or Weyl) representations. A particularly important and unexpected result is that the present temperature-dependent generalized P and Q representations are the analytic continuations to negative temperatures of each other. The usual Q and P representations thus represent the limits as the temperature approaches zero along the positive and negative real axes, respectively, of the enlarged generalized Q representation, suitably analytically continued to negative temperatures. We discuss the possible physical applications of the present thermal coherent states to both quantum optics situations involving coherent signals in the presence of thermal noise and to signal and image processing.
The pure Glauber (harmonic oscillator) coherent states provide a very useful basis for many purposes. They are complete in the sense that an arbitrary state in the Hilbert space may be expanded in terms of them. Furthermore, the well known P representation provides a diagonal expansion of an arbitrary operator in the Hilbert space in terms of the projection operators onto the coherent states. The authors study the extensions of these results to the analogous mixed states which describe comparable harmonic oscillator systems in thermodynamic equilibrium at non-zero temperatures. Their results are given for the general density operator which describes the mixed squeezed coherent states of the displaced and squeezed harmonic oscillator. They show how these squeezed coherent mixed states similarly provide a very convenient complete description of a Hilbert space. In particular they show how the usual P and Q representations of operators in terms of pure states may be extended to finite temperatures with the corresponding mixed states, and various relations between them are demonstrated. The question of the existence of the generalised P representation for an arbitrary operator is further examined and some pertinent theorems are proven. They also show how their results relate to the Glauber-Lachs formalism in quantum optics for mixtures of coherent and incoherent radiation. Particular attention is focused both on the interplay between the quantum mechanical and thermodynamical uncertainties and on the entropy associated with such mixed states.
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.