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arXiv:1412.3218 · Physica Scripta

Regular phase operator and SU(1,1) coherent states of the harmonic oscillator

Sándor Varró

Published Jun 1, 2015• 8 citations
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Abstract

A new solution is proposed to the longstanding problem of describing the quantum phase of a harmonic oscillator. In terms of an 'exponential phase operator', defined by a new 'polar decomposition' of the quantized amplitude of the oscillator, a regular phase operator is constructed in the Hilbert–Fock space as a strongly convergent power series. It is shown that the eigenstates of the new 'exponential phase operator' are SU(1,1) coherent states associated to the Holstein–Primakoff realization. In terms of these eigenstates the diagonal representation of phase densities and a generalized spectral resolution of the regular phase operator are derived, which are very well suited to our intuitive pictures of classical phase-related quantities.

DOI
10.1088/0031-8949/90/7/074053

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A new solution is proposed to the longstanding problem of describing the quantum phase of a harmonic oscillator. In terms of an 'exponential phase operator', defined by a new 'polar decomposition' of the quantized amplitude of the oscillator, a regular phase operator is constructed in the Hilbert–Fock space as a strongly convergent power series. It is shown that the eigenstates of the new 'exponential phase operator' are SU(1,1) coherent states associated to the Holstein–Primakoff realization. In terms of these eigenstates the diagonal representation of phase densities and a generalized spectral resolution of the regular phase operator are derived, which are very well suited to our intuitive pictures of classical phase-related quantities.
For a quantized mode of the radiation field, the operator whose classical analog is the ordinary phase factor of the mode amplitudes has been shown to be nonunitary. A mathematically rigorous formulation of the phase P is given on the basis of the canonical factorization theorem. All other phase quantities are defined in terms of P and P†. Many of the seemingly complex features of phase operators are found to be simple direct consequences of the general mathematical theory. It is easily seen that P is a partial isometry but not a unitary operator. In contrast to the amplitude operator, it is found that P is not a spectral operator and the set of phase eigenstates is not complete. Mathematically precise operator relations, including the rigorous statement of commutation rules, are developed. For each of the phase operators, a complete spectral analysis is given, with the nature of the spectrum, spectral decomposition (if any), and eigenstates shown explicitly. The values of the various phase operators are compared between phase states and coherent states.
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.
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.
The problem of quantization of the classical phase of a harmonic oscillator (HO) is solved here in two steps. First, polar decomposition of the step operators of the u(2) algebra is performed. Second, the method of group contraction is used through which, in the limit j→∞, â,â° is passed to for the quantized HO and its Hermitian phase operators. Also, phase states, i.e., states with sharply defined phase, are constructed and the dynamical aspects of the contraction limit between the Jaynes–Cummings model (JCM) and a finite-dimensional counterpart with increasing j parameter are studied. Finally, the old problem of the phase operators is discussed in the wider frame of rank-1 algebras and classify the previous works in this frame.