Describing the phase of an electromagnetic field mode or harmonic oscillator has been an obstacle since the early days of modern quantum theory. The quantum phase operator was even more problematic with the invention of the maser and laser in the 1950s and 1960s. This problem was not solved until the Pegg-Barnett formalism was developed in the 1980
Describing the phase of an electromagnetic field mode or harmonic oscillator has been an obstacle since the early days of modern quantum theory. The quantum phase operator was even more problematic with the invention of the maser and laser in the 1950s and 1960s. This problem was not solved until the Pegg-Barnett formalism was developed in the 1980
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.
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 development of the theory of quantum phase had been forestalled by the lack of a consistent and physically meaningful definition for the phase angle. However, once the Pegg-Barnett phase formalism was introduced this impasse was removed and the theoretical and experimental research into quantum phase began with vigour. From the outset there was a need to further develop the more mathematical aspects of the formalism. These included exploring a new Wigner representation of the state of a harmonic oscillator, the nature of the canonical conjugate relationship between Nˆ and φˆθ , the implications of the special limiting procedure s → ∞ and the consistency of the formalism with the traditional description of a harmonic oscillator. In this chapter we review the main results of the investigations into these issues.
The phase of a simple-harmonic vibration or wave is a useful and easily-understood concept in classical mechanics. It is the angle that determines the sinusoidal development of the motion, and it varies linearly with time, increasing by 360°(or 2π radians) over a complete cycle of oscillation. By contrast, the concept of phase in the quantum mechanics of harmonic motion has been fraught with difficulties, which have hampered its use. Recently, D T pegg and S M Barnett (Europhys. Lett. 6 483) have proposed a new definition of the quantum-mechanical phase operator, which they suggest will lead to important and far-reaching consequences.
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.