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Journal of Physics B Atomic and Molecular Physics

On the matrix element for two-photon absorption

Edwin Albert Power, T. Thirunamachandran

Published Jul 11, 1975• 14 citations
Publisher

Abstract

The matrix element for two-photon absorption by atoms and molecules is obtained from different forms of the interaction Hamiltonian. The use of the minimal coupling, multipolar and space-translated interaction energies are compared and terms arising from higher multipoles are discussed.
DOI
10.1088/0022-3700/8/10/002

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Shahram Mohammad-Mehdi Golshan

A pair of nonidentical two-level atoms, separated by a fixed distance R, interact through photon exchange. The system is described by a state vector which is assumed to be a superposition of four "essential states": (1) the first atom is excited, the second one is in the ground state, and no photon is present, (2) the first atom is in its ground state, the second one is excited, and no photon is present, (3) both atoms are in their ground states and a photon is present, and (4) both atoms are excited and a photon is also present. The system is initially in state (1). The probabilities of each atom being excited are calculated for both the minimally-coupled interaction and the multipolar interaction in the electric dipole approximation. For the minimally-coupled interaction Hamiltonian, the second atom has a probability of being instantaneously excited, so the interaction is not retarded. For the multipolar interaction Hamiltonian, the second atom is not excited before the retardation time, which agrees with special relativity. For the minimally-coupled interaction the nonphysical result occurs because the unperturbed Hamiltonian is not the energy operator in the Coulomb gauge. For the multipolar Hamiltonian in the electric dipole approximation the unperturbed Hamiltonian is the energy operator. An active view of unitary transformations in nonrelativistic quantum electrodynamics is used to derive transformation laws for the potentials of the electromagnetic field and the static Coulomb potential. For a specific choice of unitary transformation the transformation laws for the potentials are used in the minimally-coupled second-quantized Hamiltonian to obtain the multipolar Hamiltonian, which is expressed in terms of the quantized electric and magnetic fields.

•May 1, 1987•0 citations
In nonrelativistic quantum electrodynamics there is a class of equivalent Hamiltonians that describe a system of radiation and atoms or molecules in interaction. Three specific examples, namely, the minimal coupling, the multipolar, and the space-translated forms are analyzed here in detail. Their equivalence is demonstrated. The Lagrangian approach has also been examined to give further insight into the dynamics. The equations of motion are compared. Finally, the process of two-photon absorption by an atom is used as an application of the various interaction Hamiltonians in a discussion of their computational differences.
The two-photon absorption process is discussed using a density matrix formalism. The electromagnetic field is described by means of coherent states, a technique that permits a simple evaluation of the effect of coherence on the transition probability. It is concluded that the latter is the same for a coherent and an incoherent pulse of radiation. In the case of a stationary field containing many modes, the transition probability is nearly the same in the coherent and incoherent cases, whereas for a single mode the transition probability is twice as high for the incoherent field. These conclusions are independent of the multipole expansion of the interaction Hamiltonian and are valid for electric dipole and higher multipole transitions.
The off-diagonal matrix elements of the polarizability operator when operating within an lN{l}^{N}lN configuration of a complex atom can be approximately written in simple tensor-operator form. The resulting expressions are discussed in terms of two-photon absorption and the recently observed electronic Raman scattering in the trivalent rare-earth ions.
Applying a space translation operation, the Schrodinger equation for an atom in an electromagnetic field is solved with sufficient accuracy to obtain probabilities for multiple absorption of photons from a monochromatic laser beam of arbitrary intensity or frequency. It is shown that the derived expression for the N-photon T-matrix contains the usual single photon matrix elements given by the perturbation theory and that the perturbative result is obtained in the limit of low intensity. Other explicit examples are considered. The conditions of applicability of the method are specified.
A pair of nonidentical two-level atoms, separated by a fixed distance R, interact through photon exchange. The system is described by a state vector which is assumed to be a superposition of four "essential states": (1) the first atom is excited, the second one is in the ground state, and no photon is present, (2) the first atom is in its ground state, the second one is excited, and no photon is present, (3) both atoms are in their ground states and a photon is present, and (4) both atoms are excited and a photon is also present. The system is initially in state (1). The probabilities of each atom being excited are calculated for both the minimally-coupled interaction and the multipolar interaction in the electric dipole approximation. For the minimally-coupled interaction Hamiltonian, the second atom has a probability of being instantaneously excited, so the interaction is not retarded. For the multipolar interaction Hamiltonian, the second atom is not excited before the retardation time, which agrees with special relativity. For the minimally-coupled interaction the nonphysical result occurs because the unperturbed Hamiltonian is not the energy operator in the Coulomb gauge. For the multipolar Hamiltonian in the electric dipole approximation the unperturbed Hamiltonian is the energy operator. An active view of unitary transformations in nonrelativistic quantum electrodynamics is used to derive transformation laws for the potentials of the electromagnetic field and the static Coulomb potential. For a specific choice of unitary transformation the transformation laws for the potentials are used in the minimally-coupled second-quantized Hamiltonian to obtain the multipolar Hamiltonian, which is expressed in terms of the quantized electric and magnetic fields.