Quantum phase via the angular momentum

Abstract

An approach to the quantum phase, taking into account the process of light generation and extending in this way the operational approach, is proposed. The cosine and sine operators of the phase difference of the two circularly polarized modes are determined with the aid of a polar decomposition of the angular momentum of radiative transition and conservation of the total angular momentum. Application of the approach to the Jaynes-Cummings model for an electric dipole transition shows consistency with the classical definition of the phase difference and with the results of standard operational approach as well. The main idea of the operational approach to the quantum phase of light [l-3] is to define the phase operator in terms of quantities that can be measured in real experiments. It should be stressed that the polarization measurements represent an important method of operational definition of the phase difference between two circularly orthogonal polarized modes [2-51. The analysis of various schemes of measurement has shown that there is no unique phase operator at all [2]. Then, consideration of the so-called intrinsic and operational observables [6] has lead to the conclusion that there should be a unique intrinsic phase operator corresponding to the set of operational quantities (see also discussion in Refs. . Further analysis of the relation between the operational approach and the method, based on the phase distributions [IO,1 I] has shown that the moments of periodic functions of the measured phase difference are those calculated from the appropriate integrated Wigner or Q functions [ l2-141.All these results are based on the consideration of a '

Dual representation of dipole photons

“…An approach focused on overcoming this difficulty has been proposed recently [4]. The main idea, which seems to be a very natural one, is to consider the radiation of a given quantum source rather than a free electromagnetic field.…”

“…Since the total angular momentum JE + ME is conserved in the process of generation, we can first construct the polar decomposition of JE in the (2j + 1)-dimensional atomic Hilbert space HA, following the method by Vourdas [2]. Then, for the 'phase-dependent' dual representation of theatomicSU(2)algebra, wehavetodeterminetheradiationcounterpart, whichconsistsofthe 0305-4470/99/376589+16$30.00 © 1999 IOP Publishing Ltd 6589 operators complementary to the atomic operators with respect to the integrals of motion [4,6]. In particular, the cosine and sine of the so-called radiation phase have been determined in this way for the electric dipole radiation by a two-level atom [4,6].…”

“…Then, for the 'phase-dependent' dual representation of theatomicSU(2)algebra, wehavetodeterminetheradiationcounterpart, whichconsistsofthe 0305-4470/99/376589+16$30.00 © 1999 IOP Publishing Ltd 6589 operators complementary to the atomic operators with respect to the integrals of motion [4,6]. In particular, the cosine and sine of the so-called radiation phase have been determined in this way for the electric dipole radiation by a two-level atom [4,6]. The radiation phase is, by construction, the quantum phase of angular momentum of radiation.…”

“…momentum (e.g., see [7]). The new element of our approach [4] is that we determine the quantum phase of radiation via the quantum phase of angular momentum of its source. We also note that one of the most popular and important methods in the field of quantum phase, proposed by Pegg and Barnett [8] (for a recent review, see [9]), is based on a contraction of the infinite-dimensional Hilbert space of photon states HR.…”

Polarization of Dipole Radiation in Quantum Domain

“…Moreover, the operators S 1 , S 2 are directly connected with the operators of cosine and sine of the azimuthal phase of angular momentum of the dipole radiation [7,9] and can be obtained from the conservation of the angular momentum in the process of radiation. The difference in the commutation properties can be traced in the most clear way in the quantum fluctuations of the Stokes parameters.…”