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],... 

It should be mentioned here that the quantum multipole radiation, defined in terms of the source, has also been considered [56]. 

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... 

In this chapter we have reviewed some results concerning the quantum multipole radiation. Although the representation of quantum electromagnetic radiation in terms of spherical waves of photons known since the first edition in 1936 of the famous book by Heitler on quantum theory of radiation [2], where this subject is discussed in the Appendix, this representation is not a widespread one. The spherical waves of photons are considered in very few advanced monographs on quantum optics [26]. The brilliant encyclopedic monographs [14,15] just touch on the subject. 

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