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Physica Scripta

Phase states: an analytic approach in the unit disc

Apostol Vourdas

Published Jan 1, 1993• 38 citations
Publisher

Abstract

Phase states are defined as eigenkets of the "exponential of the phase" operator E − . It is shown that they are SU(1, 1) Perelomov coherent states corresponding to the k = ½ representation. The properties of these states are discussed and used for the calculation of the phase distribution of coherent, squeezed and phase states.

DOI
10.1088/0031-8949/1993/t48/012

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Phase states are defined as eigenkets of the "exponential of the phase" operator E − . It is shown that they are SU(1, 1) Perelomov coherent states corresponding to the k = ½ representation. The properties of these states are discussed and used for the calculation of the phase distribution of coherent, squeezed and phase states.
We cast the phase state as a SU(1,1) nonlinear coherent state to support the idea that the SU(1,1) representation of the electromagnetic field may be helpful in some instances and to bring forward that it may relate to the phase state problem. We also construct nonlinear coherent states related to the exponential phase operator and provide their corresponding nonlinear annihilation operators. Finally, we discuss the propagation of classical fields through arrays of coupled waveguides that are solved through the use of nonlinear coherent states of SU(1,1) or the exponential phase operator.
We introduce the concept of algebra eigenstates which are defined for an arbitrary Lie group as eigenstates of elements of the corresponding complex Lie algebra. We show that this concept unifies different definitions of coherent states associated with a dynamical symmetry group. On the one hand, algebra eigenstates include different sets of Perelomov's generalized coherent states. On the other hand, intelligent states (which are squeezed states for a system of general symmetry) also form a subset of algebra eigenstates. We develop the general formalism and apply it to the SU(2) and SU(1,1) simple Lie groups. Complete solutions to the general eigenvalue problem are found in the both cases, by a method that employs analytic representations of the algebra eigenstates. This analytic method also enables us to obtain exact closed expressions for quantum statistical properties of an arbitrary algebra eigenstate. Important special cases such as standard coherent states and intelligent states are examined and relations between them are studied by using their analytic representations.
We consider two analytic representations of the SU(1,1) Lie group: the representation in the unit disc based on the SU(1,1) Perelomov coherent states and the Barut - Girardello representation based on the eigenstates of the SU(1,1) lowering generator. We show that these representations are related through a Laplace transform. A `weak' resolution of the identity in terms of the Perelomov SU(1,1) coherent states is presented which is valid even when the Bargmann index k is smaller than . Various applications of these results in the context of the two-photon realization of SU(1,1) in quantum optics are also discussed.
The Wigner quasiprobability, along with some of its essentialproperties, is introduced and discussed in two versions, first covering real canonical variables such as W(q,p) and second a pair of complex conjugate coordinates such as W(α,α*). The reconstruction of the density operator ϱ of states is also given. Building upon the Susskind–Glogower concept of quantum phase operators, further aspects of phase operator algebras in the quantum optics of a harmonic oscillator are discussed in relation to the realization of the su(1,1) Lie algebra. Coherent phase states |ε⟩ are introduced in analogy to the common coherent states |α⟩ in two ways, as both eigenstates of certain operators and as states generated from a ground state |0⟩ by operators of the Lie group SU(1,1). The limiting transition to the non-normalizable Fritz London phase states |eiφ⟩ on the unit circle and an (over)-completeness relation for the coherent phase states are derived. The Wigner quasiprobability W(q,p) for the coherent phase states is calculated and graphically represented. From the Wigner quasiprobability, a phase distribution W(φ) is calculated by integrating over the radius, and its uncertainty is defined and presented. The Hilbert–Schmidt distance is discussed as a measure of the non-classicality of states, where most of our with Viktor Dodonov work was carried out.