Multipole radiation field

The transformation of the relativistic expression for the operator of magnetic multipole radiation (4.8) may be done similarly to the case of electric transitions. As has already been mentioned, in this case the corresponding potential of electromagnetic field does not depend on the gauge condition, therefore, there is only the following expression for the non-relativistic operator of Mk-transitions (in a.u.) ... 

Relativistic corrections of order v2/c2 to the non-relativistic transition operators may be found either by expanding the relativistic expression of the electron multipole radiation probability in powers of v/c, or semiclas-sically, by replacing p in the Dirac-Breit Hamiltonian by p — (l/c)A (here A is the vector-potential of the radiation field) and retaining the terms linear in A. Calculations show that in the general case the corresponding corrections have very complicated expressions, therefore we shall restrict ourselves to the particular case of electric dipole radiation and to the main corrections to the length and velocity forms of this operator. 

As an example of these methods, consider the B cyclic theorem for multipole radiation, which can be developed for the multipole expansion of plane-wave radiation to show that the B<3) field is irrotational, divergentless, and fundamental for each multipole component. The magnetic components of the plane wave are defined, using Silver s notation [112] as... 

This is the phaseless magnetic field of multipole radiation on the 0(3) level. The solution (777) reduces to the simple... 

In a multipole expansion of the interaction of a molecule with a radiation field, the contribution of the magnetic dipole is in general much smaller than that of the electric dipole. The prefactor for a magnetic dipole transition probability differs from the one for an electric dipole by a2/4 1.3 x 1 () 5. Magnetic dipoles may play an important role, however, when electric dipole transitions are symmetry-forbidden as, e.g., in homonuclear diatomics. 

Equations similar to (8.32) can be derived in terms of state multipoles or other parameters. They completely specify the radiation field and show how the properties of this field are related to the dynamical observables of the collision. In practice the time evolution of the excited state is not measured, since it is of no interest in determining X and x, and the total integrated intensity is usually determined experimentally. This means that the factor in C is replaced by 1/y. 

An alternative coupling scheme, that proves to be advantageous in many regards, and is now the preferred theoretical method, is the multipolar framework [32, 33] of molecular QED. In this formulation, molecules couple directly to the radiation fields through their molecular multipole moments. Because the Maxwell fields obey Einstein causality, interactions between species is properly retarded, which is mediated by the emission and absorption of transverse photons. It is convenient to partition the total many-body QED matter-field multipolar Hamiltonian into a sum of parficle, radiation field, and interaction terms as follows ... 

The inclusion of coherent states of the radiation field in the formalism describing opfically induced forces is most conveniently carried out within the induced multipole moment method delineated in Section 5. Instead of number sfafes n(fc, 2.)) = (k)), coherent states a= =) are defined... 

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... 

We now turn to the problem of the SU(2) quantum phase of multipole radiation. As a particular example of some considerable interest, we investigate the electric dipole field. All other types of the multipole radiation can be considered in the same way. 

In contrast to the plane waves, the field strengths of the multipole radiation can have any direction. In fact, the electric multipole radiation obey the condition -r = 0, while it can have nonzero longitudinal component (S r 0) of the electric field strength [25], In other words, this is the transversal magnetic radiation. In turn, the magnetic multipole field is characterized by the relations... 

Hence, the polarization of either multipole radiation should be specified by the spatial anisotrophy of the field strengths rather than the transversal anisotrophy as in the case of plane waves [28,46,54,88]. Thus, the polarization of the classical multipole field should be described by bilinear forms in all three components of the field strengths which leads to the Hermitian (3 x 3) matrix with the elements [28,46]... 

Here we again consider a monochromatic radiation field. Let us stress that this expression describes the spatial anisotrophy of the electric field and therefore specifies the polarization of the electric multipole radiation. In the case of magnetic multipole radiation, the spatial anisotrophy of magnetic induction can be described by the following polarization matrix [89] ... 

The quantum counterpart of the polarization matrices can be constructed in direct analogy to the field quantization [90]. We have to subject the field amplitudes in (124), (125), and (129) to the Weyl-Heisenberg commutation relations (22) and (23) respectively. Thus, we get the operator matrices of polarization of the multipole radiation of the form... 

To illustrate the spatial properties of the polarization of multipole radiation, consider the normal-ordered operator polarization matrix (133) in the case of monochromatic electric-type pure y-pole radiation. Assume that the radiation field is in a single-photon state lm) with given m. Then, the average of (133) takes the form... 

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. 

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