Applications of the Optical Phase Operator
Quantum optics has progressed a long way in the last sixty years and much is now understood about coherence. Nevertheless, the description of the quantum nature of optical phase has been a longstanding problem. Classically, phase is a useful and easily-understood concept, and it might be expected that the classical phase observable should, according to the usual quantization procedures, correspond to an Hermitian phase operator ϕ. Indeed, the existence of this operator was indeed originally postulated by Dirac,1 but problems associated with finding such an operator have led to the present wide-spread belief that no such operator exists.2 Attempts to construct a phase operator have involved the use of an infinite dimensional state space and a polar decomposition of the annihilation operator. This procedure does not provide a unique operator. Specifically, the action of exp (iϕ) on the vacuum is not determined. The usual ad hoc assumption is to set exp (iϕ) ∣ 0 > = 0. This removes the indeterminacy but destroys the unitarity of exp (iϕ), along with the possibility of extracting an Hermitian operator from the exponential.
KeywordsCoherent State Wigner Function Photon Number Phase Operator Hermitian Operator
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