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Physica Scripta

Some formal properties of operator polar decomposition

Stig Stenholm

Published Jan 1, 1993• 7 citations
Publisher

Abstract

In this paper I consider the general mathematical problem of the polar decomposition of an operator in a linear space. Extending the space makes it possible to define a unitary operator related to the original nonhermitean one. By Stone's theorem this guarantees the existence of a phase operator in the extended space. The connection with supersymmetry is pointed out. Applying the general results to harmonic oscillator creation and annihilation operators we regain a phase description originally introduced by Newton. Projecting the phase operator from the extended space to the original one, we find a phase representation for the Boson operators. Introducing the conjugate rotation operator, one can describe the oscillator dynamics in the phase representation. The connection with the Barnett–Pegg phase operator is pointed out.

DOI
10.1088/0031-8949/1993/t48/011

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In this paper I consider the general mathematical problem of the polar decomposition of an operator in a linear space. Extending the space makes it possible to define a unitary operator related to the original nonhermitean one. By Stone's theorem this guarantees the existence of a phase operator in the extended space. The connection with supersymmetry is pointed out. Applying the general results to harmonic oscillator creation and annihilation operators we regain a phase description originally introduced by Newton. Projecting the phase operator from the extended space to the original one, we find a phase representation for the Boson operators. Introducing the conjugate rotation operator, one can describe the oscillator dynamics in the phase representation. The connection with the Barnett–Pegg phase operator is pointed out.
Abstract. The kth root taken of the bosonic phase operator is considered. This leads to the extension of the Hilbert space. In the case k = 2 an ordinary fermionic extension arises. Particles whose statistics depends on k are introduced for other values of k. 1. Phase operators The problem of polar decomposition of the creation and annihilation operators (a ∗ , a) of a harmonic oscillator goes back to Dirac [1]. In this section some aspects of this problem will be reviewed [2]. Let h be the oscillator hamiltonian h = (p 2 + q 2)/2 (1.1) on the phase plane Γ with coordinates p, q and Poisson brackets (PB) If the polar angle ϕ is introduced by then for the complex variables a and a ∗ {q, p} = 1. (1.2) p = √ 2h sin ϕ, q = √ 2h cos ϕ, (1.3) a = q + ip
A simple formalism is introduced for describing the quantum phase of a single-mode field (or, equivalently, a harmonic oscillator) using a particular infinite-dimensional Hilbert space Hsym{\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}Hsym​. This space is spanned by vectors which represent states of the single field mode containing non-negative numbers of photons. It has a symmetry in the sense that it contains vectors in the neighborhood of both the vacuum state and the infinite photon number state. A striking property of Hsym{\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}Hsym​ is that it supports Hermitian and unitary phase operators. Indeed the Pegg-Barnett unitary phase operators are shown to converge strongly on the Hilbert space Hsym{\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}Hsym​ and so the correponding limit points are ordinary operators on Hsym{\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}Hsym​. This can be compared with the situation for the infinite-dimensional Hilbert space H conventionally used in quantum optics on which the Pegg-Barnett unitary phase operators converge weakly, in general.
The well-known difficulties of defining a phase operator of an oscillator are considered from the point of view of the canonical transformation to action and phase-angle variables. This transformation turns out to be nonbijective, i.e., it is not a one-to-one onto mapping. In order to make possible the unitarity of its representations in quantum optics we should enlarge the Hilbert space of the problem. In this enlarged space we find a phase operator that, after projection, reproduces previous candidates to represent a well-behaved phase operator in the quantum domain.
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.
Pegg and Barnett [Phys. Rev. A 39, 1665 (1989)] have proposed a formalism for a Hermitian optical phase operator, constructed from the so-called phase states. In this paper we give an alternative derivation. We begin by defining a unitary phase operator ${e}^{i\stackrel{^}{\ensuremath{\varphi}}}$ from the relationship $\stackrel{^}{a}={e}^{i\stackrel{^}{\ensuremath{\varphi}}}R(\stackrel{^}{N})$ ($\stackrel{^}{a}$ is an annihilation operator for a single-mode boson field), but borrow the idea of a finite- (but arbitrarily large) dimensional space from Pegg and Barnett. Finally, we comment on an experiment to determine quantum phase uncertainty. Our analysis uses the operator ${e}^{i\stackrel{^}{\ensuremath{\varphi}}}$, which we feel is a more natural choice than $\stackrel{^}{\ensuremath{\varphi}}$. The use of ${e}^{i\stackrel{^}{\ensuremath{\varphi}}}$ also avoids the multivalued problems associated with $\stackrel{^}{\ensuremath{\varphi}}$.
The algebra of generalized linear quantum canonical transformations is examined in the prespective of Schwinger's unitary-canonical basis. Formulation of the quantum phase problem within the theory of quantum canonical transformations and in particular with the generalized quantum action-angle phase space formalism is established and it is shown that the conceptual foundation of the quantum phase problem lies within the algebraic properties of the quantum canonical transformations in the quantum phase space. The representations of the Wigner function in the generalized action-angle unitary operator pair for certain Hamiltonian systems with the dynamical symmetry are examined. This generalized canonical formalism is applied to the quantum harmonic oscillator to examine the properties of the unitary quantum phase operator as well as the action-angle Wigner function.