Multipole photons

We now stress that, in the usual treatment of photon localization, the radiation field is considered as though it consist of the plane waves of photons [14-20]. In reality, the radiation emitted by the atomic transitions corresponds to the multipole photons [23] represented by the quantized spherical waves [2]. 

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... 

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],... 

While the simplified picture based on the model of plane waves of photons, neglecting the presence of sources and absorbers, is incapable of describing the photon localization, we show here that the use of the rich physical properties of multipole photons leads to an adequate description of localization in the atom-... 

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],... 

As can be seen from the equations (21)-(22) and (23)-(24), there is an essential difference between the representations of plane and multipole waves of photons. In particular, a monochromatic plane wave of photons is specihed by only two different quantum numbers a = x, y, describing the linear polarization in Cartesian coordinates. In turn, the monochromatic multipole photons are described by much more quantum numbers. Even in the simplest case of the electric dipole radiation when X = E and j = 1, we have three different states of multipole photons in (23) with m = 0, 1. Besides that, the plane waves of photons have the same polarization a everywhere, while the states of multipole photons have given m. It is seen from (24) that, in this case, the polarization described by the spin index p can have different values at different distances from the singular point. In Section V we discuss the polarization properties of the multipole radiation in greater detail. 

In turn, employing the representation (23)-(24) then gives for (27) in the case of multipole photons the following equation ... 

It is seen from the definition of the mode functions (18) and (19) that, in contrast to (28), the zero-point oscillations of the electric field strength of multipole photons manifest the spatial inhomogeneity. 

The simplest quantum source of photons is the atomic transition, creating, according to the selection mles, multipole photons. The simplest model of the interaction of an atom with the electromagnetic radiation is associated with the notion of so-called two-level atom [64]. In fact, this model originates from the famous study of radiation kinetics by Einstein [65]. With the development of laser, the notion of two-level atom entered firmly into the practice of quantum optics. The fact is that, using lasers as sources of electromagnetic radiation, one... 

The first term in (34) describes the energy of the free cavity field and atom, while the second term gives the energy either of the transition 1, m) —> 0,0) with generation of the multipole photon or of the transition [0,0) —> l,m) accompanied by the absorption of corresponding electric dipole photon. 

The Weyl-Heisenberg algebra of multipole photons allows the dual representation in which we deal with the photons with given radiation phase (the 517(2) phase of angular momentum) instead of standard photons with given projection of the angular momentum. 

The radiation phase of multipole photons has discrete spectrum in the interval (0,2n). In the classical limit of high-intensity coherent field, the eigenvalues of the radiation phase are distributed uniformly over (0, 2ji). 

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. 

Considering into account that (159) is simply the positive-frequency part of the vector potential (21), we can introduce the multipole photon absorption operator as follows... 

Thus, the picture of measurement in the atom-detector system of two identical atoms is compatible with Mandel s operational approach to the photon localization. For example, the multipole photon statistics in finite volume can be examined in the same way as in Refs. 14 and 20. The commutators for different t can also be constructed in analogy to Ref. 20. 

In the previous subsection, we considered the held emitted by one atom and then absorbed by another atom as a superposition of outgoing and incoming spherical waves of multipole photons. This wave picture completely eliminates an inquiry concerning the trajectory of photons between the atoms. In fact, the path of a particle in quantum mechanics is not a well-defined notion. The most that we can state about the path of a quantum particle in many cases is that it is represented by a nondifferentiable, statistically self-similar curve [93]. For example, the path of a tunneling electron and time spending in the barrier are not still defined unambiguously [94]. Moreover, some experiments on photonic tunneling and transmission of information show the possibility of superluminal motion of photons inside an opaque barrier [95]. 

Localization of multipole photons at generation and absorption by an atom, described in terms of outgoing or incoming waves of photons, is compatible with Mandel s conception of localization. 

The two-atom scheme of the Hertz experiment with multipole photons, in which the radiation field is described by a superposition of outgoing and incoming waves focused on the emitting and detecting atoms respectively, obeys the causality principle even though the path of detecting photons is indefinite. 

One of the major trends of current research is the study of transmission of information between the atom and photons in the process of emission and absorption. In particular, the conservation of angular momentum provides the transmission of the quantum phase information in the atom-held system. The atomic quantum phase can be constructed as the 57/(2) phase of the angular momentum of the excited atomic state (Section III). It is shown that this phase has very close connection with the EPR paradox and entangled states in general. Via the integrals of motion, it is mapped into the Hilbert space of multipole photons (Section IV.A). This mapping is adequately described by the dual representation of multipole photons, constructed in another study [46] (see also Section IV.B, below). Instead of the quantum number m, corresponding to the... 

The polarization and quantum phase properties of multipole photons change with the distance from the source. This dependence can be adequately described with the aid of the local representation of the photon operators proposed in Ref. 91 and discussed in Section V.D. In this representation, the photon operators of creation and annihilation correspond to the states with given spin (polarization) at any point. This representation may be useful in the quantum near-field optics. As we know, so far near-field optics is based mainly on the classical picture of the field [106]. 

The local representation of multipole photons is compatible with the Mandel operational definition of photon localization [20]. In addition to the localization at photodetection, it permits us to describe a complete Hertz-type experiment with two identical atoms used as the emitter and detector (Section VI.A). Although the photon path is undefined from the quantum-mechanical point of view, the measurement process in such a system obeys the causality principle (Section IV.B). The two-atom Hertz experiment can be realized for the trapped... 

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