Pegg-Barnett Hermitian phase operator

The Pegg-Barnett Hermitian phase operator is defined as... 

The Pegg-Barnett Hermitian phase formalism allows for direct calculations of quantum phase properties of optical fields. As the Hermitian phase operator is defined, one can calculate the expectation value and variance of this operator for a given state /). Moreover, the Pegg-Barnett phase formalism allows for the introduction of the continuous phase probability distribution, which is a representation of the quantum state of the field and describes the phase properties of the field in a very spectacular fashion. For so-called physical states, that is, states of finite energy, the Pegg-Barnett formalism simplifies considerably. In the limit as a —> oo one can introduce the continuous phase distribution... 

Polar decomposition of the field amplitude, as in (36), which is trivial for classical fields becomes far from being trivial for quantum fields because of the problems with proper definition of the Hermitian phase operator. It was quite natural to associate the photon number operator with the intensity of the field and somehow construct the phase operator conjugate to the number operator. The latter task, however, turned out not to be easy. Pegg and Barnett [11-13] introduced the Hermitian phase formalism, which is based on the observation that in a finite-dimensional state space, the states with well-defined phase exist [14]. Thus, they restrict the state space to a finite (cr + l)-dimensional Hilbert space H-+ spanned by the number states 0), 1),. .., a). In this space they define a complete orthonormal set of phase states by... 

It is readily apparent that the Hermitian phase operator 4>t) has well-defined matrix elements in the number-state basis and does not suffer from the problems as those the original Dirac phase operator suffered. Indeed, using the Pegg-Barnett phase operator (43) one can readily calculate the phase-number commutator [13]... 

A way to overcome the difficulties in the definition of the Hermitian phase operator has been proposed by Pegg and Barnett [40,45]. Their method is based on a contraction of the infinite-dimensional Hilbert-Fock space of photon states Within this method, the quantum phase variable is determined first in a finite 5-dimensional subspace of //, where the polar decomposition is allowed. The formal limit, v oc is taken only after the averages of the operators, describing the physical quantities, have been calculated. Let us stress that any restriction of dimension of the Hilbert-Fock space of photons is equivalent to an effective violation of the algebraic properties of the photon operators and therefore can lead to an inadequate picture of quantum fluctuations [46]. 

The phase states were applied by Pegg and Barnett in their definition of the Hermitian quantum-optical phase operator [26] ... 

The Hermitian operator phase formalism of Pegg and Barnett [11-13] allows for quantum calculations of phase distribution for the fields produced in the... 

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