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arXiv:quant-ph/9901015

Phase operator for one-photon quantized field

Hong-Yi Fan, Q. Y. Liu

Published Jan 8, 1999• 0 citations
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Abstract

Based on the quantization of one-photon electromagnetic field described in terms of the Riemann-Silberstein complex vector {\\bf F}, we construct the eigenvector set of {\\bf F}, this set is orthonormal and complete, with which we introduce the phase operator for one-photon state. It is shown that this phase operator manifestly exhibits phase behaviour.

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Based on the quantization of one-photon electromagnetic field described in terms of the Riemann-Silberstein complex vector {\bf F}, we construct the eigenvector set of {\bf F}, this set is orthonormal and complete, with which we introduce the phase operator for one-photon state. It is shown that this phase operator manifestly exhibits phase behaviour.
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.
A first-quantized free photon is a complex massless vector field A=(Aμ) whose field strength satisfies Maxwell’s equations in vacuum. We construct the Hilbert space H of the photon by endowing the vector space of the fields A in the temporal-Coulomb gauge with a positive-definite and relativistically invariant inner product. We give an explicit expression for this inner product, identify the Hamiltonian for the photon with the generator of time translations in H, determine the operators representing the momentum and the helicity of the photon, and introduce a chirality operator whose eigenfunctions correspond to fields having a definite sign of energy. We also construct a position operator for the photon whose components commute with each other and with the chirality and helicity operators. This allows for the construction of the localized states of the photon with a definite sign of energy and helicity. We derive an explicit formula for the latter and compute the corresponding electric and magnetic fields. These turn out to diverge not just at the point where the photon is localized but on a plane containing this point. We identify the axis normal to this plane with an associated symmetry axis and show that each choice of this axis specifies a particular position operator, a corresponding position basis, and a position representation of the quantum mechanics of a photon. In particular, we examine the position wave functions determined by such a position basis, elucidate their relationship with the Riemann-Silberstein and Landau-Peierls wave functions, and give an explicit formula for the probability density of the spatial localization of the photon.
Recently a photon position operator with commuting components and a particle density that is positive definite for both positive and negative frequency states has led to a first quantized theory of the photon. Here we extend this theory to a manifestly covariant form and show for the first time that, with invariant plane wave normalization, the position operator is Hermitian with real instantaneously localized eigenvectors. Reality of the field and wave function ensure causal propagation and zero net absorption of energy in the absence of charged matter. The position space wave function is the real part of the projection of the photon's state vector onto a basis of position eigenvectors. Manifest covariance and consistency with quantum field theory is maintained through use of the electromagnetic four-potential and the Gupta-Bleuler gauge condition.
In Chapt. 10, photon wave mechanics was based on the transverse part of the vector potential. As we shall realize soon, it is quite possible to claim rightfully that other quantities constructed from electromagnetic field vectors may be considered as the photon wave function in free space. The crucial point is that whatever choice one makes for the photon wave function the subsequent photon-matter interactions must all result in the same observable consequences. In this chapter an important and interesting choice based on the Riemann-Silberstein vectors, which were introduced in Sec. 2.8, will be described.
This paper discusses how the two ways in which the wavefunction of a free-propagating photon can be introduced-starting from the relativistic energy-momentum relationship or based on the electromagnetic field, in particular on Riemann-Silberstein vectors-are not entirely equivalent since they can lead to different consequences regarding photon localization. In the first case, a phase space localization in regions of the order of Planck's constant, in agreement with the quantum uncertainty principle, could be unambiguously obtained. In the second case, the choice of canonically conjugate variables and Fourier transforms determines if the state is treated quantumly or classically. Both formalisms are, however, compatible.
The usual mathematical model of the single-mode electromagnetic field is the harmonic oscillator with an infinite-dimensional state space, which unfortunately cannot accommodate the existence of a Hermitian phase operator. Recently we indicated that this difficulty may be circumvented by using an alternative, and physically indistinguishable, mathematical model of the single-mode field involving a finite but arbitrarily large state space, the dimension of which is allowed to tend to infinity after physically measurable results, such as expectation values, are calculated. In this paper we investigate the properties of a Hermitian phase operator which follows directly and uniquely from the form of the phase states in this space and find them to be well behaved. The phase-number commutator is not subject to the difficulties inherent in Dirac's original commutator, but still preserves the commutator--Poisson-bracket correspondence for physical field states. In the quantum regime of small field strengths, the phase operator predicts phase properties substantially different from those obtained using the conventional Susskind-Glogower operators. In particular, our results are consistent with the vacuum being a state of random phase and the phases of two vacuum fields being uncorrelated. For higher-intensity fields, the quantum phase properties agree with those previously obtained by phenomenological and semiclassical approaches, where such approximations are valid. We illustrate the properties of the phase with a discussion of partial phase states. The Hermitian phase operator also allows us to construct a unitary number-shift operator and phase-moment generating functions. We conclude that the alternative mathematical description of the single-mode field presented here provides a valid, and potentially useful, quantum-mechanical approach for calculating the phase properties of the electromagnetic field.