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Proceedings of the Physical Society

Non-linear processes to result from multipole interactions between molecules and electromagnetic fields

S. Kielich

Published Oct 1, 1965• 59 citations
Publisher

Abstract

A compact and at the same time general tensor formalism adapted to the theoretical prediction of a wide range of non-linear processes due in general to multipole interactions between atomic or molecular systems and electromagnetic fields is proposed. Recurring to a semi-classical method, the expectation values of the 2 n -pole induced electric moment P e ( n ) and 2 n -pole magnetic moment P m ( n ) for a transition k → l are computed explicitly to within the third order of quantum-mechanical perturbation theory approximation. The formalism is demonstrated with regard to its utility and simplicity on the example of electric and magnetic multipole light scattering with multiharmonic frequencies ω kl + ω 1 + ω 2 +... + ω r .

DOI
10.1088/0370-1328/86/4/305

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A compact and at the same time general tensor formalism adapted to the theoretical prediction of a wide range of non-linear processes due in general to multipole interactions between atomic or molecular systems and electromagnetic fields is proposed. Recurring to a semi-classical method, the expectation values of the 2 n -pole induced electric moment P e ( n ) and 2 n -pole magnetic moment P m ( n ) for a transition k → l are computed explicitly to within the third order of quantum-mechanical perturbation theory approximation. The formalism is demonstrated with regard to its utility and simplicity on the example of electric and magnetic multipole light scattering with multiharmonic frequencies ω kl + ω 1 + ω 2 +... + ω r .
A quantum theory of nonlinear phenomena is considered in which account is taken of all contributions from electric and magnetic multipoles, without any multipolar approximation. Exact closed analytical expressions for the multipoles are first obtained from quantum electrodynamics by applying a generalization of the Power-Zienau transformation. The standard procedure is invoked to calculate general electromagnetic susceptibility tensors ${\ensuremath{\chi}}_{\ensuremath{\mu}\ensuremath{\nu}{\ensuremath{\alpha}}_{1}{\ensuremath{\beta}}_{1}\ensuremath{\cdots}{\ensuremath{\alpha}}_{n}{\ensuremath{\beta}}_{n}}^{[n]}$ (${\ensuremath{\omega}}_{1}, \dots{}, {\ensuremath{\omega}}_{n}$). Such a tensor is a shorthand for the electric and magnetic susceptibilities as well as mixed susceptibilities. By considering a relativistic theory based on the Dirac equation, the spin susceptibilities are automatically included. The formalism in this way allows for the computation of a wide range of electro- and magneto-optical effects to any order of perturbation and to any order of multipolar approximation.
We investigate in this chapter the general problem of the interaction of an atom (or a molecule) and a classical electromagnetic field. Its importance derives firstly from the fact that a large part of our knowledge of atoms is obtained from a study of the radiation they absorb or emit (we shall consider equally visible radiation and radio-frequency and X-rays) and, secondly, that the interaction with matter modifies the propagation of the electromagnetic field itself, notably through absorption, refraction or scattering. The interactions of atoms with light therefore encompass a vast range of physical effects that we could not hope to cover adequately in a single chapter. In this chapter we shall therefore present the fundamental features of the interaction of an atom, which will be treated quantum mechanically, with a classical electromagnetic field, that is an electromagnetic field described by real electric and magnetic vectors obeying Maxwell's equations. A rigorous description of the atom–light interaction would have to take into account the quantum nature of light. This we shall leave to Chapter 6. It turns out, nevertheless, that many important results can be obtained from the semi-classical viewpoint adopted here (although it is perhaps more appropriate to refer to this treatment as ‘semi-quantized’).
Reduced density matrix descriptions are developed for linear and non-linear interactions in electromagnetically induced transparency and related pump-probe optical phenomena involving moving atomic systems. Applied magnetic fields as well as atomic collisions, together with other environmental decoherence and relaxation processes, are taken into account. Time-domain (equation-of-motion) and frequency-domain (resolvent-operator) formulations are developed in a unified manner. The standard Born (lowest-order perturbation-theory) and Markov (short-memory-time) approximations are systematically introduced within the framework of the general non-perturbative and non-Markovian formulations. A preliminary semiclassical treatment of the electromagnetic interaction is adopted. However, the need for a fully quantum mechanical approach is emphasized. Compact Liouville-space operator expressions are derived for the linear and the general (n'th order) non-linear macroscopic electromagnetic-response tensors within the framework of a perturbation-theory treatment of the semiclassical electromagnetic interaction. These expressions can be evaluated for coherent initial atomic excitations and for the full tetradic-matrix form of the Liouville-space self-energy operator representing the environmental interactions in the Markov approximation. Collisional interactions between atoms can be treated in various approximations for the self-energy operator, and the influence of Zeeman coherences on the electromagnetic-pulse propagation can be investigated.
A recent investigation of the interaction between the Maxwell radiation field and neutral atoms or molecules has led us to consider the relation between the straightforward interaction Hamiltonian between particle (k) and the radiation field, and the more physical, and explicitly gauge invariant, couplings of the form q. E and µ. H, etc., where q and µ are the electric and magnetic dipole moments of a system of particles. That these latter form of interaction energies follow from well-formulated approximations for atoms in external fields has been known for many years (Goeppert-Mayer(1); see also Rosenfeld(2)). When the Maxwell field is an internal and dynamical field further discussion is necessary, especially concerning the Coulomb interactions. Their advantage has recently been shown in dealing with dipole interactions in the long range Van der Waals force between two neutral systems where the complete coupling, in dipole approximation, is given by — q.ET for each neutral system, where ET is the transverse electric field. The short-range London limit is contained in this coupling despite the absence of longitudinal electric fields (Power and Zienau (3)). In this note we extend these arguments to higher electric multipoles and magnetic multipoles. We also demonstrate their use in the quantum theory of scattering of light, and in particular the derivations of Stephen (4) of the phase angle between emergent beams in optical double refraction phenomena are shown to follow immediately.
We present a formalism that enables the analytic calculation of the interaction of a spin-half particle with a polychromatic electromagnetic field. This powerful new approach provides a clear physical picture even for cases with highly degenerate energy levels, which are complicated to interpret in the standard dressed-atom picture. Typically semi-classical methods are used for such problems (leading to equations that are solved by Floquet theory). Our formalism is derived from quantum electrodynamics and thus is more widely applicable. In particular it makes accessible the intermediate regime between quantum and semi-classical dynamics. We give examples of the application to multi-frequency multi-photon processes in strong fields by deriving the Hamiltonians of such systems, and also to the dynamics of weak fields at long times for which semi-classical methods are insufficient.
Classical, semi-classical, and quantum-field descriptions for the interaction of light with matter are systematically discussed. Applications of interest include precise determinations of the linear and the non-linear electromagnetic response relevant to resonant pump-probe optical phenomena, such as electromagnetically induced transparency. In the quantum-mechanical description of matter systems, we introduce a general reduced-density-matrix framework. Time-domain (equation-of-motion) and frequency-domain (resolvent-operator) formulations are developed in a unified and self-consistent manner, using a Liouville-space operator representation. A preliminary semi-classical perturbation treatment of the electromagnetic interaction is adopted, in which the electromagnetic field is described as a classical field satisfying the Maxwell equations. Compact Liouville-space operator expressions are derived for the linear and the general (nth order) non-linear electromagnetic-response tensors describing moving many-electron systems. The tetradic matrix elements of the Liouville-space self-energy operators, which are introduced in the time-domain and frequency-domain formulations, are evaluated for environmental collisional and radiative interactions, in order to provide explicit forms for the quantum kinetic equations and the spectral-line shape formulas. It is emphasized that a quantized-field approach is essential for a fully self-consistent quantum-mechanical description of the interacting light-matter system.