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.
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.
We define a unitary phase operator for photons in a single momentum mode. The operator acts on a Hilbert space with basis consisting of all number states in both polarizations. The Susskind Glogower operator, E^=eiϕ^, thus defined is unitary and acts as a true ladder operator over all the states. It satisfies the commutation relation [E^,n^]=E^; consequently, the phase ϕ^ satisfies the canonical commutation relation [n^,ϕ^]=i. The eigenstates of the Susskind Glogower operator are found. Our model is able to directly account for phase measurements when polarization changes cannot be neglected and at the same time gives a well defined phase operator.
We have already shown that the coherent states are very close to the classical state of the well-defined amplitude and phase. A question not answered yet is: Does there exist an analogous quantum mechanical observable for the phase that is given by some Hermitian operator? We know from classical optics that phase of an EM field is an observable quantity. Hence, in quantum physics it should be associated with a Hermitian operator. In order to set up a quantitative description of the phase operator, we will introduce some of the ideas presented in the literature which led to the formulation of the Hermitian phase operator.
We will begin our journey through the background of quantum optics with an elementary, but quantitative, classical theory of radiative fields.Wewill first briefly outline the electromagnetic (EM) theory of radiation, and describe how the EM radiation may be understood as a wavewhich can be represented by a set of harmonic oscillators. We shall describe how the free or non-interacting EM field may be understood as a collection of harmonic oscillators which is quantized in the standard manner, and whose energy cannot be zero as a consequence of the basic non-commutability of the canonical field variables. This chapter discusses the properties of plane EM waves and normalization of the EM field in one dimension. This is followed by a description of the Hamiltonian and the amplitudes of the EM field in terms of the annihilation and creation operators. Based on this chapter, it is possible to considerably simplify the formulation of the physical basis for the mathematical description of the major problems of quantum optics.
In Chapter 4 we saw that the canonical commutation relations force us to use a space of states of infinite dimension, in which rigor would require the use of advanced mathematical tools. Fortunately, physicists generally need only to carry the results for finite dimension over to infinite dimension with some simple modifications which we shall indicate here, without embarking on sophisticated mathematics. Nevertheless, it is useful to be aware of the lapses in rigor which are customarily made in physics in order to avoid possible unpleasant surprises. The objective of this chapter is, on the one hand, to present some concrete examples illustrating the new features which arise in infinite dimension and, on the other, to give the rules for practical calculations, in particular to write down the spectral decomposition of Hermitian and unitary operators. The mathematics we use is a bit more detailed than commonly found in most quantum mechanics textbooks. The reader interested purely in the practical aspects can proceed directly to Section 7.3, where the results essential for later on are summarized. Hilbert spaces Definitions The space of states of quantum mechanics is a Hilbert space ℌ, which in general is of infinite dimension. The axiomatic definition of a Hilbert space is the following. It is a vector space which, for the needs of quantum mechanics, is defined on complex numbers. The vectors of this space are denoted |ϕ〉. […]
Methods are developed for discussing the photon statistics of arbitrary fields in fully quantum-mechanical terms. In order to keep the classical limit of quantum electrodynamics plainly in view, extensive use is made of the coherent states of the field. These states, which reduce the field correlation functions to factorized forms, are shown to offer a convenient basis for the description of fields of all types. Although they are not orthogonal to one another, the coherent states form a complete set. It is shown that any quantum state of the field may be expanded in terms of them in a unique way. Expansions are also developed for arbitrary operators in terms of products of the coherent state vectors. These expansions are discussed as a general method of representing the density operator for the field. A particular form is exhibited for the density operator which makes it possible to carry out many quantum-mechanical calculations by methods resembling those of classical theory. This representation permits clear insights into the essential distinction between the quantum and classical descriptions of the field. It leads, in addition, to a simple formulation of a superposition law for photon fields. Detailed discussions are given of the incoherent fields which are generated by superposing the outputs of many stationary sources. These fields are all shown to have intimately related properties, some of which have been known for the particular case of blackbody radiation.
In a series of papers, Barnett, Pegg (1986, 1988, 1989, 1992) and various co-authors have proposed a description of quantum phase by means of a collection of s-dimensional states and operators, s>or=1. We analyse the limiting procedure they employ for large s, which is known not to be compatible with quantum mechanics in the usual sense. Further, we supply a rigorous demonstration of the asymptotic limits of the `mean` and `variance` of their system of operators in coherent states. These values had previously been given but not justified mathematically. Our analysis, based on the asymptotic analysis of certain random variables, shows that the physical deductions that can be drawn from these limits are limited. We also prove that the (s+1)-dimensional `pure phase` LHW-states they consider form a sequence of approximate eigenvectors for the Weyl-quantized angle operator Delta ( phi ) and the Toeplitz phase operator X proposed by Garrison and Wong (1970), Popov and Yarunin (1992), and others. These states X s ( theta ), to our knowledge first introduced by Lerner, Huang and Waiters (1970), can be used to construct a system of measurement as in the usual quantum theory, sensitive to certain qualities of phase, but not all. Indeed, a feature of the Barnett-Pegg method, when it gives finite answers, is the construction of associated measurement systems for different observables. We give examples of sequences of (s+1)-dimensional devices which represent measurements significantly closer to ideal for X. This serves as a model for corresponding devices for Delta ( phi ), or indeed, any observable with a continuous spectrum, contingent on its spectral decomposition being obtained explicitly.