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Journal of Physics A Mathematical and General

A new coherent paired state with possible applications to fluctuation-dissipation phenomena

R. F. Bishop, Apostol Vourdas

Published Aug 21, 1987• 22 citations
Publisher

Abstract

The authors extend their previous work on coherent paired states associated with the Lie group SU(1,1). Whereas the earlier states were defined with respect to a single type of canonical boson or (linear) quantum harmonic oscillator, the new states are defined in terms of two distinct types of bosons or oscillators. The new coherent states may again, on the one hand, be viewed as ordinary (Glauber) coherent states in the two-boson Hilbert space spanned by arbitrary numbers of two distinct Bogoliubov quasiparticles associated with the original bosons via a generalised Bogoliubov transformation. Alternatively, expressed wholly in terms of the original bosons these new coherent states are reached from the ordinary coherent states via a unitary (pairing) transformation which is shown to be associated with the entire so-called discrete series of representations of the group SU(1,1). As an important illustration of the use of these states and transformations, they study in detail a rather general class of quantum Lagrangians which includes the damped linear harmonic oscillator. They thereby illustrate their possible usefulness in applications to quantum many-body or field-theoretic processes involving fluctuation-dissipation phenomena in general.

DOI
10.1088/0305-4470/20/12/024

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The authors extend their previous work on coherent paired states associated with the Lie group SU(1,1). Whereas the earlier states were defined with respect to a single type of canonical boson or (linear) quantum harmonic oscillator, the new states are defined in terms of two distinct types of bosons or oscillators. The new coherent states may again, on the one hand, be viewed as ordinary (Glauber) coherent states in the two-boson Hilbert space spanned by arbitrary numbers of two distinct Bogoliubov quasiparticles associated with the original bosons via a generalised Bogoliubov transformation. Alternatively, expressed wholly in terms of the original bosons these new coherent states are reached from the ordinary coherent states via a unitary (pairing) transformation which is shown to be associated with the entire so-called discrete series of representations of the group SU(1,1). As an important illustration of the use of these states and transformations, they study in detail a rather general class of quantum Lagrangians which includes the damped linear harmonic oscillator. They thereby illustrate their possible usefulness in applications to quantum many-body or field-theoretic processes involving fluctuation-dissipation phenomena in general.
The authors study the properties of the states U 2 ( rho , theta , lambda ) mod A) where U 2 is an operator associated with the group SU(1,1), and mod A) is a standard (atomic or Glauber) coherent state defined in terms of the usual boson creation and destruction operators a Dagger and a. They show how these states may be viewed as ordinary coherent states in terms of the Bogoliubov quasiparticles whose creation and destruction operators b Dagger and b are associated with the operators a Dagger and a by a Bogoliubov transformation. As an important example of the use of these states, they show that they are the coherent states associated with a uniformly accelerated (Rindler) observer moving through Minkowski space. The previous results then simply show how the Minkowski (inertial) vacuum appears to the Rindler observer as a black-body radiator with a Planckian distribution corresponding to a temperature proportional to the proper acceleration.
Unitary operator coherent states, as defined by Klauder (1963), Perelomov (1972) and Gilmore (1974), are considered for the harmonic series irreducible representations of U(p,q). Various properties of these states are reviewed, including their reproducing kernel and measure explicit forms, as well as the representation they lead to for the U(p,q) generators. Starting from a contraction of the su(p,q) Lie algebra, both Dyson and Holstein-Primakoff boson realisations of u(p,q) are obtained in terms of pq pairs of boson creation, annihilation operators and of the generators of a U(p)*U(q) intrinsic group. Such boson realisations are applied to determine the matrix elements of the U(p,q) generators between discrete bases classified according to the chain U(p,q) contains/implies U(p)*U(q). Finally, the isomorphism between so(4,2) and su(2,2) is employed to derive the corresponding properties of the SO(4,2) unitary operator coherent states for application in atomic physics.
We introduce and study the properties of a class of coherent states for the group SU(1,1) X SU(1,1) and derive explicit expressions for these using the Clebsch-Gordan algebra for the SU(1,1) group. We restrict ourselves to the discrete series representations of SU(1,1). These are the generalization of the `Barut Girardello' coherent states to the Kronecker Product of two non-compact groups.The resolution of the identity and the analytic phase space representation of these states is presented. This phase space representation is based on the basis of products of `pair coherent states' rather than the standard number state canonical basis. We discuss the utility of the resulting `bi-pair coherent states' in the context of four-mode interactions in quantum optics.
A new method for describing the two-level pairing model in a many-fermion system is presented. The basic idea comes from the Schwinger boson representation for the su(2) ⊗ su(2)-algebra, in which four kinds of boson operators govern the dynamics. It is well known that in the original fermion space, it is almost impossible to introduce any trial state for the variation except the BCS state in the framework of the mean field approximation. However, in the Schwinger boson representation, various trial states can be defined on the condition that the mean field approximation is useful. In these states, the su(2) ⊗ su(1,1)-coherent state is mainly discussed in this paper.
Using Lie algebra methods, we find a new class of Bogoliubov transformations which generalize the notion of squeezed states. The Hamiltonians for the simple harmonic and anharmonic oscillators, turn out to be the generators of a Lie group, whose other generators may be found exactly, or up to any desired order of the perturbation parameter. An element of this Lie group, which is realized as the multi-photon operator, transforms the anharmonic Hamiltonian to the harmonic one. The transformation of the ordinary annihilation and creation operators under this unitary transformation leads to the introduction of multi-photon coherent states. We specifically consider four-photon coherent states in detail and study the time dependent position and momentum uncertainties in these states.
Abstract From the definition of the standard Perelomov coherent states we introduce the Perelomov number coherent states for any su(2) Lie algebra. With the displacement operator we apply a similarity transformation to the su(2) generators and construct a new set of operators which also close the su(2) Lie algebra, being the Perelomov number coherent states the new basis for its unitary irreducible representation. We apply our results to obtain the energy spectrum, the eigenstates and the partition function of two coupled oscillators. We show that the eigenstates of two coupled oscillators are the SU(2) Perelomov number coherent states of the two-dimensional harmonic oscillator with an appropriate choice of the coherent state parameters.