Table of contents

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.

After reviewing the role of phase in quantum mechanics, I discuss, with the aid of a number of unpublished documents, the development of quantum phase operators in the 1960's. Interwoven in the discussion are the critical physics questions of the field: are there (unique) quantum phase operators and are there quantum systems which can determine their nature? I conclude with a critique of recent proposals which have shed new light on the problem.

Much confusion has resulted from the failure to recognise the different concepts embodied in the single word "phase" as applied to a quantized electromagnetic field mode. In this paper we describe three conceptions of phase and the connections between them.

In the Ψ-space limiting procedure limits of expectation values, rather than operator or states, are found as the state space dimensionality tends to infinity. This approach has been applied successfully to the calculation of the phase properties of various states of light, but its status as a valid quantum mechanical thory equivalent to the usual infinite Hilbert space (H-space) approach has not yet been fully accepted. Here we address this issue by investigating the formal relationship between the two approaches. We establish the consistency between the Ψ-space and H-space approaches for observables which are amenable to an H-space treatment. Such observables are represented in H by operators which are strong limits of Ψ-space operators and which obey the same algebra as the corresponding Ψ-space operators. The phase operator, however, exists in H only as a weak limit of a Ψ-space operator. For such limits the Ψ-space operator algebra is not preserved, which is the fundamental reason for the difficulties in constructing a consistent quantum description of phase in H. We show that for the phase observable the Ψ-space approach is consistent with the probability-operator measure (POM) method with the important distinction that, whereas the relation between non-orthogonal POMs and probability has to be accepted in the latter method as a postulate, the corresponding relation is derived in the Ψ-space approach. We conclude that the Ψ-space approach is not only equivalent to, but is also more fundamental than both the H-space and POM approaches.

Our operational approach to the problem of identifying dynamical variables to represent the phase, or the sine and cosine of the phase, of a quantum field is reviewed. We start from classical optics and examine what is usually measured to determine the phase difference between two fields. We point out that difficulties in phase measurements exist even in the classical domain, and these difficulties persist when the classical relations are used as guides in the construction of phase operators. We find that different measurement procedures lead naturally to different phase operators, so that there appears to be no one "correct" phase operator. We carry out a series of experiments to measure the phase difference between two fields in various quantum states, and then compare the results with the predictions of our formalism and some other formalisms. We obtain good agreement between experiment and our theoretical approach in every case, so far without exception.

We have used the recently demonstrated method of optical homodyne tomography (OHT) to measure the Wigner quasiprobability distribution (Wigner function) and the density matrix for both a squeezed-vacuum and a vacuum state of a single spatial-temporal mode of the electromagnetic field. This method consists of measuring a set of probability distributions for many different Hilbert-space representations of the field-quadrature amplitude, using balanced homodyne detection, and then using tomography to obtain the Wigner function. Once the Wigner function is obtained, one can acquire the density matrix, including its complex phase. In the case of a pure state, this technique yields an experimentally determined complex wavefunction, as demonstrated here for the vacuum. The density matrix represents a complete quantum mechanical characterization of the state. From the measured density matrix we have obtained the Pegg–Barnett optical phase distribution, and from the Wigner function, the Wigner optical phase distribution.

An overview is given on experimental schemes for measuring phase properties (in particular, phase distributions) discussed and partly even realized thus far. It is pointed out that all of them amount to measuring the Q function for the field. Moreover, experimental procedures for determining the Wigner function for the field more or less directly are described which thus allow to assign experimental significance to the concept of area of overlap in phase space when applied to the quantum mechanical description of phase.

We address the problem of strong local oscillator limit of the recently introduced operational approach to phase uncertainty measurements. In this limit the operator character of the local oscillator mode is neglected and its coherent amplitude tends to infinity. Thereby reasonable new operators for the sine and the cosine of the phase difference between an arbitrary signal field and the coherent local oscillator field are found which describe an already well-known physical situation connected with strong amplification of the signal field.

The phase distribution obtained within the Pegg–Barnett Hermitian phase formalism is compared to the phase distributions obtained from the s-parametrized quasiprobability distributions integrated over the "radial" variable for some real states of the field. Exact analytical formulas for the s-parametrized phase distributions of coherent states, squeezed states, and displaced number states are obtained. A general formula relating the s-parametrized phase distributions to the Pegg–Barnett distribution is derived. Numerical examples illustrating the similarities and differences are presented in a graphical form.

The Wigner function for a phase state |ϕ⟩ is expanded, as a function of polar coordinates ϱ' and ϕ', in terms of cos [k(ϕ' − ϕ)] (k = 0, 1, 2,...). The ϱ'-dependent expansion coefficients are evaluated numerically. The Wigner function thus obtained fits well to an approximate analytic solution valid for ϱ' ≫ 1.

The probability distribution for finding a quantum state of the light field with a particular phase can be described by a number of theoretical methods. One of the most frequently used in quantum optics is the Pegg–Barnett phase distribution which identifies the phase distribution as the (necessarily positive) modulus square of the overlap of the quantum state with the relevant phase state. Other phase distributions have been proposed which employ the phase-sensitivity of various quantum quasi-probabilities such as the Wigner function. In this paper we discuss the connections between these approaches and illustrate their similarities and important differences for a number of pertinent quantum states of light. For some of these states, quantum interference can generate significant regions of phase space for which the Wigner quasi-probability is negative, and as we show, this may render the quasi-probability phase distribution negative. We discuss the general significance of this phenomenon.

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.

Phase states are defined as eigenkets of the "exponential of the phase" operator E₋. It is shown that they are SU(1, 1) Perelomov coherent states corresponding to the k = ½ representation. The properties of these states are discussed and used for the calculation of the phase distribution of coherent, squeezed and phase states.

We discuss the operators for the spherical angles of a quantum-mechanical angular momentum vector. Since these operators are restricted to the 2j + 1 dimensional state space, both the polar angle and the azimuthal angle have discrete eigenstates. These two operators do not commute, and thus they have a different eigenvector basis. Their commutation rule reflects the commutation rule for the Cartesian components of angular momentum. The relation with coherent spin states is analyzed.

A suitable ordering of phase exponential operators has been compared with the antinormal ordering of the annihilation and creation operators of a single mode optical field. The extended Wigner function for number and phase in the enlarged Hilbert space has been used for the derivation of the Wigner function for number and phase in the original Hilbert space.

Using a single mode analysis, we assess the utility of Shannon information as a robust performance criterion for phase detection. To provide a sharp test, we apply it to the Shapiro, Shepard and Wong optimal phase state. This has well-known pathologies, and is thus a good hurdle. Numerical studies show that the SSW-state information gain converges rapidly to the finite value ⟨ΔI⟩ ∼ 0.966 bits at infinite photon number. For large photon numbers N the truncated phase state scheme yields ⟨ΔI⟩ ∼ log₂ (2N + 1) − 1.220 bits whereas the coherent state scheme returns ⟨ΔI⟩ ∼ log₂N^1/2 − 1.604 bits. Numerical studies confirm these asymptotic formulae and reveal an interesting crossover regime where the coherent state scheme proves superior to the truncated phase state scheme at low photon numbers N ⩽ 10. The anomalous behaviour of the SSW-state is studied with the aid of a simple scaling argument.

The Paley–Wiener restriction on the statistics of single-mode quantum phase measurements implies that single-mode, phase-modulated quantum communication always has nonzero error probability at finite average photon number. A two-mode formulation is demonstrated which circumvents the Paley–Wiener constraint, leading to a scheme for zero-error probability phase-conjugate quantum communication at finite average photon number. The minimum root-mean-square (RMS) total photon number for error-free K-ary phase-conjugate communication turns out to be K/2, and the state achieving this optimum performance is exhibited. Application of the construct to precision measurements is briefly discussed. Here, the optimum state with RMS photon number K/2 can be used to guarantee that the phase estimate is within ±π/K radians of the true value.

We compare and contrast five measures of phase uncertainty of a quantum state corresponding to a single mode of the electromagnetic field. The basis of this study are the states which minimize a particular measure for a fixed number of Fock states and normalization. We find these optimal states and study their characteristic properties. These optimal states allow us to establish an ordering of the different definitions for phase uncertainty.

We investigate those quantum states, which lead to minimum phase uncertainty for a fixed mean intensity. In contrast to the work of Bandilla, Paul, and Ritze (BPR) we employ instead of the usual variance measure for the phase the Süssmann measure. It turns out that our minimal states are of the form |ψ⟩ ∝ ∑ γⁿ | n⟩. Because of this simple expression the phase distribution can be given analytically. It is shown that this distribution is about twice as peaked than the BPR phase minimum states. The Q-function is also obtained. It shows an asymmetry whereas the BPR states lead to a Q-function with a nearly symmetric ellipsoidal shape.

An explicit expression is given for the phase noise in a high intensity squeezed state. For fixed intensity this is found to have a minimum value of ln /4². The principal limit on the noise reduction is given by the non-zero probability for the field to have the opposite phase from that intended. A weaker limit is set by the occupation of states close to the origin, in particular of the vacuum state.

A scheme is proposed whereby the phase distribution of a field in a coherent state can be narrowed by the process of continuous measurements. The amplitude of a small signal is built up by the measurement process thereby resulting in phase narrowing.

We study phase properties of quantum mechanical superposition states decaying into phase-sensitive reservoirs. We show that the Wigner phase quasiprobability distribution reflects the quantum interference effects very well while the Pegg–Barnett and the Husimi phase probability distributions are quite insensitive with respect to the character of the quantum interference between component states. We study in detail phase properties of the even coherent state which decays into the thermal and the squeezed reservoirs. In thermal reservoirs the phase of the superposition state becomes randomized under the influence of thermal fluctuations. The higher the number of thermal photons the faster the randomization process is. On the other hand we show that depending on the relative phase between the superposition state and the phase-sensitive reservoir the process of phase randomization can be either enhanced or completely suppressed.