The problem of phase in quantum mechanics is an old one but one that has received increasing attention of late. For a single-mode radiation field with annihilation operator â there have been two recent formulations for its quantum phase, Pegg and Bamett1 have created a Hermitian phase operator ϕ ^ s by truncating the state space associated with â to the subspace spanned by the number kets {|n〉:0 ≤ n ≤ S}.
The problem of phase in quantum mechanics is an old one but one that has received increasing attention of late. For a single-mode radiation field with annihilation operator â there have been two recent formulations for its quantum phase, Pegg and Bamett1 have created a Hermitian phase operator ϕ ^ s by truncating the state space associated with â to the subspace spanned by the number kets {|n〉:0 ≤ n ≤ S}.
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 $\stackrel{^}{\ensuremath{\varphi}}$.
For a long time, about 60 years, the phase as an observable of the quantized light fields has r emained a fundamental problem with considerable difficulties on quantum mechanics and also q uantum optics.Recently, however, the hermitian phase operator and its eigenstates in a finite H ilbert space are found by Pegg and Barnett.In this paper, we provide a historical review and t he background of the problem on the phase operator, explore the Pegg-Barnett formalism, and s how the recent developements of this fields and its application to various light fields.Further problems on this formalism and other approaches are also discussed.
More particularly, the sets {|n〉 : n = 0, 1, · · · s} and {|θm〉 : m = 0, 1, · · · s} are two mutually unbiased bases for s as |〈n|θm〉|2 = 1/(s + 1) for n, m = 0, 1, · · · , s. The exponential phase operator is defined in terms of the phase state basis as ei φˆθ = s∑ m=0 eiθm |θm〉〈θm|, (2.4) or equivalently, in terms of the energy eigenstates as |n〉〈n + 1| + ei(s+1)θ0 |s〉〈0|. (2.5) The subscript θ serves as a reminder that the values of phase variable θm lie in the range from θ0 to θ0 + 2π . This construction ensures that the phase states are the eigenstates of ei φˆθ , i.e., ei φˆθ |θm〉 = eiθm |θm〉. (2.6) The exponential phase operator is unitary, because n=0 |n〉〈n| = 1ˆ s (2.7) where 1ˆ s is the identity operator on s . The Hermitian phase operator representing the phase angle itself was introduced in (Pegg and Barnett 1988b Paper 2.2) and (Barnett and Pegg 1989 Paper 2.3) as simply φˆθ = s∑ m=0 θm|θm〉〈θm|. (2.8) The operator on the left hand sides of Eqs. (2.4) and (2.5) is the exponential of i φˆθ .Moreover, the operator corresponding to an arbitrary function of the phase angle f (θ) was introduced in Barnett and Pegg (1992 Paper 3.3) and also discussed in Pegg and Barnett (1997a Paper 2.6) as f (φˆθ ) = s∑ m=0 f (θm)|θm〉〈θm|. (2.9) It follows that the Hermitian conjugate of ei φˆθ is the operator (ei φˆθ )† = s∑ m=0 e−iθm |θm〉〈θm| = e−i φˆθ . (2.10) This construction on the finite dimensional spaces elegantly sidesteps the problems faced with the infinite dimensional Hilbert space discussed in the previous chapter. Its simplicity is particularly appealing. The question remains, however, of how to deal with states such as | f 〉 = ∑∞n=0 cn|n〉 which reside in the infinite dimensional Hilbert space. For each value of s, there exists a phase operator defined on s ; this leads naturally to sequences of operators, such as e.g. φˆθ |s=S, φˆθ |s=S+1, φˆθ |s=S+2 · · · which are phase angle operators defined on the sequence of spaces S, S+1, S+2, · · · for arbitrarily large integer S. The space s contains a state | f 〉s which approximates | f 〉 to arbitrary precision for a sufficiently large value of s, such that the modulus of the overlap 〈 f | f 〉s is as close to unity as desired. Thus the issue of including states such as | f 〉 entails dealing with such sequences as s tends to infinity.
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.
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. 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 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 and so the correponding limit points are ordinary operators on 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 phase associated with a single mode of the electromagnetic field is complementary to the photon number. This simple idea leads us to construct a probability density for the phase. The phase operator cannot be represented exactly in the usual-infinite Hilbert space but can be constructed in a subspace of it and provides a valid representation of phase if used together with a suitable limiting procedure. We introduce the Hermitian optical phase operator and describe some of its most important properties. We also provide brief discussions of some alternative approaches to the phase problem and to the question of phase measurements. The paper concludes with an extensive bibliography.