Hermitian phase operator

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... 

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

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]... 

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... 

The generalized phase CS, p,0o)(s), and truncated phase CS, j3,0o)(s), are associated with the Pegg-Bamett formalism of the Hermitian phase operator S. The operators 4>s, Hilbert space Thus the generalized and truncated phase CS are properly defined only in of finite dimension. States p,0o)( and p, 0o)(s), similar to a)(s) and a)(s), approach each other for p 2 = p 2 < C s/n [20]. This can be shown explicitly by calculating the scalar product between generalized and truncated phase CS. We find (p = p)... 

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]. 

Phase measurements do not belong to the category of conventional measurements since a Hermitian phase operator does not exist. What is usually measured in practice is energy, and various phase sensitive devices (interferometers, etc.) are used to transform phase shifts into variations of output energies. Because of the statistical nature of quantum theory the resulting relationship between the measured quantities and the parameters of interest is not deterministic. This sort of indirect inferences is usually called estimation. The scheme of an estimation procedure is the following... 

A theory for nonequilibrium quantum statistical mechanics can be developed using a time-dependent, Hermitian, Hamiltonian operator Hit). In the quantum case it is the wave functions [/ that are the microstates analogous to a point in phase space. The complex conjugate / plays the role of the conjugate point in phase space, since, according to Schrodinger, it has equal and opposite time derivative to v /. 

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

Thus the current operator indeed transforms like a vector. This must be the case in order that the equation Qdu(x) = ju(x) transform properly, assuming the transformation property (11-267) for Au(x). We now inquire briefly into tike question of the uniqueness of the U(ia) operator, in particular into the question of the phase associated with the fermion field operator. Note that the phase of the photon field operator is uniquely determined (Eq. (11-267)) by the fact that An is a hermitian field which commutes with the total charge operator Q. The negaton-positon field operator on the other hand does not commute with the total charge operator, in fact... 

It is also useful to define the transformed operator L whose operation on a function f is L f = L[Peqf). This operator coincides with the time reversed backward operator, further details on these relationships may be found in Refs. 43,44. L operates in the Hilbert space of phase space functions which have finite second moments with respect to the equilibrium distribution. The scalar product of two functions in this space is defined as (f, g) = (fgi q. It is the phase space integrated product of the two functions, weighted by the equilibrium distribution P The operator L is not Hermitian, its spectrum is in principle complex, contained in the left half of the complex plane. 

Also, since S + is a raising operator, we must have / — ca, where c is some constant. We can evaluate c by use of the normalization of the spin functions and the Hermitian property of Sx and Sy one finds (Problem 1.8) c = h. Choosing the phase of c as zero, we have... 

Thus, a generic hermitian operator can be expressed in terms of the representation of the group Hn. The existence of this expansion is demonstrated in Appendix 1. There, we also prove that the functions B(g) have to be obtained as the inverse Fourier transform of the functions Bw (q,p) of the phase-space variables (q,p) = qi,. . ., qn,Pi, , Pn), which are associated to the quantum operators, B, via the Wigner transform [10]. Since we are considering hermitian operators of the form B X, IiD J, the coefficients in eq.(22) are... 

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

It is well known that the angular momentum of a quantum mechanical system is specified by a representation of the SU(2) algebra. If the corresponding enveloping algebra contains a uniquely defined scalar (the Casimir operator), the polar decomposition of the angular momentum can be obtained [51]. This polar decomposition determines a dual representation of the SU(2) algebra expressed in terms of so-called phase states [51], In particular, the Hermitian operator of the SU(2) quantum phase can be constructed [51],... 

We now note that the operators (69) and (84) introducing the radiation phase are defined in terms of bilinear forms in the photon operators. At first glance, such a definition runs counter to the original idea by Dirac to determine the Hermitian quantum phase via linear forms in the photon operators [1] (see also Refs. 38, 42, and 44). Leaving aside Dirac s problem of existence of a Hermitian quantum phase variable of a harmonic oscillator, we should emphasize that the use of bilinear forms seems to be quite reasonable from the physical point of view. It can be argued in the following way ... 

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