Photon operators quantum multipole radiation

In turn, the monochromatic multipole photons are described by the scalar wavenumber k (energy), parity (type of radiation either electric or magnetic), angular momentum j 1,2,..., and projection m = —j,..., / [2,26,27]. This means that even in the simplest case of monochromatic dipole (j = 1) photons of either type, there are three independent creation or annihilation operators labeled by the index m = 0, 1. Thus, the representation of multipole photons has much physical properties in comparison with the plane waves of photons. For example, the third spin state is allowed in this case and therefore the quantum multipole radiation is specified by three different polarizations, two transversal and one longitudinal (with respect to the radial direction from the source) [27,28], In contrast to the plane waves of photons, the projection of spin is not a quantum number in the case of multipole photons. Therefore, the polarization is not a global characteristic of the multipole radiation but changes with distance from the source [22],... 

Due to the form of the operator polarization matrix (142) and corresponding Stokes operators, the polarization, defined to be the spin state of photons [4,27], is not a global property of the quantum multipole radiation. Any atomic transition emits photons with given quantum number m, which yields, in view of (18), (24), and (142), the polarization of all three types depending on the distance from the atom. The structure of (152) and (154) just shows us how the photons with different m contribute into the polarization at an arbitrary point r. Using the operators (154), we can construct, for example, the local bare operator representation of the polarization matrix (142) as follows... 

Let us stress a very important difference between the representations of Stokes operators (137) and (157). If the former is valid only for the electric dipole photons, the latter describes an arbitrary multipole radiation with any X and j. The similarity in the operator structure and quantum phase properties is caused by the same number of degrees of freedom defining the representation of the SU(2) subalgebra in the Weyl-Heisenberg algebra. 

There is a principal difference that complicates the direct use of the operation approach to the problem of localizing photons in the case of multipole radiation. The point is that the multipole photons are in a state with given angular momentum and therefore they have no well-defined direction of propagation. In view of the wave-particle dualism, one can say that the multipole photons emitted by a point-like quantum source propagate as outgoing spherical waves. Definitely, these photons are localized initially inside the source. 

The simplest way to show the principal difference between the representations of plane and multipole photons is to compare the number of independent quantum operators (degrees of freedom), describing the monochromatic radiation field. In the case of plane waves of photons with given wavevector k (energy and linear momentum), there are only two independent creation or annihilation operators of photons with different polarization [2,14,15]. It is well known that QED (quantum electrodynamics) interprets the polarization as given spin state of photons [4]. The spin of photon is known to be 1, so that there are three possible spin states. In the case of plane waves, projection of spin on the... 

We also note that, in contrast to the Pegg-Bamett formalism [45], we consider an extended space of states, including the Hilbert-Fock state of photons as well as the space of atomic states [36,46,53,54]. The quantum phase of radiation is defined, in this case, by mapping of corresponding operators from the atomic space of states to the whole Hilbert-Fock space of photons. This procedure does not lead to any violation of the algebraic properties of multipole photons and therefore gives an adequate picture of quantum phase fluctuations [46],... 

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