Polarization of Dipole Radiation in Quantum Domain

Abstract

For a dipole radiation, the set of generalized Stokes parameters and corresponding Stokes operators are discussed. A qualitatively new behavior of quantum fluctuations of the Stokes parameters is predicted. The possibility to check this behavior in the eight-port operational measurement is shown.[ S0031-9007(98) PACS numbers: 42.50. Dv, 42.25.Ja, 42.50.Lc The polarization properties of a classical radiation are usually specified by the set of Stokes parameters [1] determined for a transverse field either in the linear polarization basis or in the circular polarization basis. The quantum counterpart is provided by the Stokes operators which can be obtained from the Stokes parameters by standard quantization of the field amplitudes [2,3]. Let us stress here that within the quantum optics, describing the radiation as a beam of photons, the polarization should be determined as a given spin state of photons, forming the beam. Spin of a photon is defined as the minimum value of the angular momentum and is equal to 1 (e.g., see [4]). Thus, it has three projections and therefore just three spin states and corresponding polarizations should be taken into consideration.An example is provided by a dipole radiation when, due to the selection rules, the photons with the angular momentum 1 are emitted. It is well known that even in the classical picture, the dipole radiation always has a longitudinal component in addition to the transversal components [5]. Since this component decays with the distance quite rapidly, it is neglected in the far zone where the standard Stokes parameters for a completely transverse field are determined. Thus, the conventional definition of the Stokes parameters of the dipole radiation should be considered as an approximation which is known to be valid in the far zone. However, it is not a case in the quantum domain where one cannot neglect the longitudinal component a priori. Actually, even in the far zone, where the longitudinal component with the projection of spin m 0 contains very few photons and could be approximated by the vacuum state, it may contribute into the quantum fluctuations of different physical parameters. Therefore, it seems to be important to estimate the contribution of the longitudinal component with no resort to the transverse field approximation.In view of this aim, let us consider the classical tensor of polarization [6] with the components which are slowly varying bilinear forms with respect to the complex electric field amplitudes E ljm where the index l shows the type of radiation (either electric or magnetic) and j, m ͑jmj # j͒ are the indexes of the multipole expansion [5]. In the case of a dipole radiation j 1, m 0, 61. The modes with m 61 correspond to the circularly polarized components with the opposite helicities while m 0 specifies the linearly polarized longitudinal component. Hence, the tensor of polarization has nine components and as usual it is represented as a superposition of a diagonal and Hermitian parts. Therefore, the polarization is specified by ...

Dual representation of dipole photons

“…Here the coefficients bm are determined by the source, f(kr) is a certain combination of the Hankel functions describing the radial dependence, and YE is the vector spherical harmonics describing the angular dependence of radiation. The component EE0 of field (13) Following the standard method [15], one can define the 3 × 3 Hermitian matrix of polarization whose elements are [12] . (14) Clearly, this is a generalization of the conventional polarization matrix determined in the circular polarization basis [15], which can be obtained from (14) at E0 = 0.…”

“…Then, the generalized Stokes parameters of the electric dipole radiation can be chosen as follows [12]:…”

“…Inrecentpublications [11]wehaveshownthattheradiationphaseasdefinedin [4]hasclose connectionwiththegeneralizedStokesoperatorsintroducedin [12]todescribethepolarization propertiesoftheelectricdipoleradiationinthequantumdomain. Thisisnotsurprising,because the radiation phase describes the quantum phase of angular momentum which, generally speaking, consists of the spin and 'orbital' contributions, while the polarization is defined to be a given spin state of photons (e.g., see [13]).…”

“…Inthecaseofmultipoleradiation, inadditiontothetransversalcomponents thepolarizationalwayshastheradialcomponent, atleastinthenearandintermediatezones [5], and the spatial anisotropy should be considered rather than the transversal one. In spite of the factthattheradialcomponentusuallyhaslowintensityatfardistances,itmaystronglyinfluence the quantum fluctuations of the polarization of the two other components [12]. Therefore it cannot be neglected, even in the vacuum state.…”

“…We examine the spectrum of the cosine and sine of radiation phase operators and show that the eigenstates of these operators are certain new Fock number states, forming a dual representation with respect to the conventional number states of dipole photons. In section 2, based on our previous results [4,6,11,12], we briefly discuss the 'transmission' of quantum phase information from the atomic angular momentum to photons. In section 3 we consider the quantum polarization properties of the electric dipole radiation in the quantum domain.…”