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Physical Review A

Canonical transformations to action and phase-angle variables and phase operators

Alfredo Luis, Luis Lorenzo Sánchez-Soto

Published Jul 1, 1993• 18 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.48.752

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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.
In the present paper we analyze the representations in quantum mechanics of classical canonical transformations that are nonbijective, i.e., not one-to-one onto. We take as the central example the canonical transformation that changes the Hamiltonian of a one-dimensional oscillator of frequency κ−1 into one of frequency k−1 where κ, k are relatively prime integers. For the particular case k=1, the mapping of the original phase space (x,p) onto the new one (x̄, p̄) is κ to 1 and the equivalent points in (x,p) are related by a cyclic group 𝒞κ of linear canonical transformations. When formulating this problem in Bargmann Hilbert space, the canonical transformation can be related to the conformal transformation w=zκ, which again is κ to 1 and where a group 𝒞κ also appears. This cyclic group proves fundamental for the determination of representations of the conformal transformation in Bargmann Hilbert space. To begin with, it suggests that while we can take in the original Bargmann Hilbert space a single component function, in the new Bargmann Hilbert space we must take a κ component one. In this way we can map in a one-to-one fashion the states and operators in the old and new Bargmann Hilbert spaces. When translating these results to ordinary Hilbert space, we get in an unambiguous way the quantization of the observables appearing in the equations that determine the representation of the classical canonical transformation relating oscillators of frequencies κ−1 and k−1. Furthermore, we also get the solutions of these equations, and the resulting representation is unitary. While our discussion is restricted to the problem mentioned above, in the concluding section we indicate our surmise for deriving systematically the unitary representation in quantum mechanics of arbitrary canonical transformations.
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.
The algebra of generalized linear quantum canonical transformations is examined in the perspective of Schwinger's unitary-canonical operator 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 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 dynamical symmetry is 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.
For a time-dependent classical quadratic oscillator we introduce pairs of real and complex invariants that are linear in position and momentum. Each pair of invariants realize explicitly a canonical transformation from the phase space to the invariant space, in which the action-phase variables are defined. We find the action operator for the time-dependent quantum oscillator via the classical-quantum correspondence. Candidate phase operators conjugate to the action operator are discussed, but no satisfactory ones are found.
The correct description of the phase variable in quantum mechanics is a question rooted in its earliest formulations. The atom mechanics [1] of Bohr and Sommerfeld—the precursor of modern quantum mechanics—ascribes a central role to action-angle variables. However, Heisenberg's matrix mechanics and Schrödinger's wave mechanics are formulated [2] in terms of the canonical variables representing cartesian coordinates and momenta. Stimulated by this work London [3] attempted a reformulation of these theories in the previously favored action-angle variables. However, this attempt failed due to the difficulty in ascribing quantum operators to the angle variables of classical theory. Despite these difficulties London found [4] an operator representation of the complex exponential of these angle variables. The quantum phase of light made its first appearance in Dirac's classic paper [5] on the quantization of the radiation field. In contrast to modern methods, he constructed the annihilation and creation operators for each field mode from the corresponding amplitude and phase operators. Phase and its quantum nature acquired new significance with the development of lasers in the early sixties: theoretical investigations highlighted significant problems with Dirac's original proposal for the phase operator. A particularly elegant illustration of the difficulty was given by Louisell [6] in 1963. The advent of phase sensitive quantum noise as demonstrated experimentally in the production and detection of squeezed light [7] has created a new wave of interest in the nature of quantum optical phase leading to the discovery of the hermitian optical phase operator [8]. There has followed an explosion of theoretical activity in this area stimulating fresh experimental investigations. In preparing this special issue we have attempted to present the current state of this active and rapidly moving field. We have arranged the papers into what we hope is a coherent representation of the field. The first papers give a historical perspective and overview of current thinking. The two recent experimental investigations which follow are intimately connected to the phase space description of quantum mechanics based on quasi-probability distributions. The representa tion of phase via phase space and its connection with phase-dependent measurements and the phase operator are addressed in the next section. Some more formal considerations pertinent to phase are presented in the following section. Gravitational wave detection and optical communication have motivated the study of the limits of phase noise. Some recent investigations on such optimal phase states are presented. The issue concludes with two papers discussing the significance of phase in light-matter interactions. In concluding we express our gratitude to the authors of the papers in this volume for their efforts in preparing their high quality presentations.
It is generally thought that, in the same way that position and momentum are canonical variables in quantum mechanics, the number operator should have a canonical operator, i.e., there should be a possibility of defining a well-defined phase operator associated with it. From the point of view of the phase space, it becomes clearer as position and momentum uncertainties are well reflected. Indeed, the same occurs in the phase space for phase and intensity. In this contribution, we do a small survey on the possibilities of defining a phase operator via discrete Fourier operators and introduce some possible applications.