David T. Pegg

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.

The difficulties in formulating a natural and simple operator description of the phase of a quantum oscillator or single-mode electromagnetic field have been known for some time. We present a unitary phase operator whose eigenstates are well-defined phase states and whose properties coincide with those normally associated with a phase. The corresponding phase eigenvalues form only a dense subset of the real numbers. A natural extension to the definition of a time-measurement operator yields a corresponding countable infinity of eigenvalues.

Abstract It has long been believed that no Hermitian optical phase operator exists. However, such an operator can be constructed from the phase states. We demonstrate that its properties are precisely in accord with the results of semiclassical and phenomenological approaches when such approximate methods are valid. We find that the number-phase commutator differs from that originally postulated by Dirac. This difference allows the consistent use of the commutator for inherently quantum states. It also leads to the correct periodic phase behaviour of the Poisson bracket in the classical regime.

The DEPT pulse sequence (π/2)(H,y)−(2J)−1−π(H), (π/2)(C,x)−(2J)−1 −ϑ(H,x)π(C)−(2J)−1−(acquire 13C) is analyzed theoretically for a variable ϑ pulse for three spin systems: CH, CH2, and CH3. It is shown that the pulse train produces an enhanced distortion-free 13C signal which has the following characteristics: (a) there is phase coherency within and between the components of the 13C multiplets; (b) the enhancements vary with ϑ as (γH/γC)sin ϑ for CH, (γH/γC)sin 2ϑ for CH2, and (3γH/4γC) (sin ϑ+sin 3ϑ) for CH3. Experimental evidence is provided for these predictions. An important application of the DEPT pulse train is for the generation of both individual proton-coupled and proton-decoupled 13C methine (CH), methylene (CH2), and methyl (CH3) subspectra. This can be readily achieved by forming suitable combinations of DEPT spectra determined at ϑ = (π/4), (π/2), and (3π/4). Such spectral editing is less sensitive to variations in J values than the INEPT pulse sequence. Signal enhancement for 195Pt and 29Si NMR signals are also demonstrated using the DEPT sequence. The only disadvantage of this pulse train compared with the INEPT sequence appears to be its greater sensitivity to spin relaxation, a consequence of its time span.