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Physical Review A

Retarded and nonretarded interactions between a pair of nonidentical two-level atoms

Shahram Mohammad-Mehdi Golshan, Donald H. Kobe

Published Nov 1, 1986• 8 citations
Publisher

Abstract

Two nonidentical two-level atoms are separated by a fixed distance R. The probabilities of the atoms being excited are calculated for both the A\ensuremath{\cdot}p interaction and the E\ensuremath{\cdot}r interaction in the electric dipole approximation. For the A\ensuremath{\cdot}p interaction the second atom has a nonvanishing probability of being excited for times t<R/c, and so the interaction is not retarded. For the E\ensuremath{\cdot}r interaction the second atom has zero probability of being excited for times t<R/c, and so the interaction is completely retarded. Reasons for the difference are discussed.

DOI
10.1103/physreva.34.4449

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Two nonidentical two-level atoms are separated by a fixed distance R. The probabilities of the atoms being excited are calculated for both the A\ensuremath{\cdot}p interaction and the E\ensuremath{\cdot}r interaction in the electric dipole approximation. For the A\ensuremath{\cdot}p interaction the second atom has a nonvanishing probability of being excited for times t<R/c, and so the interaction is not retarded. For the E\ensuremath{\cdot}r interaction the second atom has zero probability of being excited for times t<R/c, and so the interaction is completely retarded. Reasons for the difference are discussed.
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.
The problem considered is that of two identical two-level atoms a fixed distance rrr apart, one of which is excited at t=0t=0t=0. A simple improvement on the usual on-shell approximation (which in the single-atom case is part of the Weisskopf-Wigner approximation) yields new solutions for the various probability amplitudes in the form of infinite series involving all the retardation times nrc\frac{\mathrm{nr}}{c}cnr​. The truncated solutions involving only the single retardation time rc\frac{r}{c}cr​ are compared with previously published results both when all photon modes are allowed and when only photons propagating along the interatomic axis are allowed. When the retardation times are neglected, the series are summed to give the well-known results of Stephen and others.
There are different results in the literature for the interaction energy of two separated nonidentical atoms in the case when one of the atoms is prepared in an excited state. Moreover, there are different ways to define this interaction energy. If the interaction energy is defined as a shift in the energy of the initial state of the combined atomic system, it is possible to carry out a time-dependent calculation that provides an unambiguous method for obtaining this shift. The time-dependent calculations lead to an interaction energy that is an oscillatory function of the interatomic separation, in contrast to the nonoscillatory behavior that is predicted using an alternative theory based on time-independent perturbation theory.
The calculation of the retarded fourth-order resonance interaction between identical molecules with one shared quantum of excitation energy is described. The interaction is calculated in the electric dipole approximation for molecular transitions with the help of time-ordered graphs and a reduced Hamiltonian which treats molecule-photon and molecule-molecule interactions on the same footing. Interactions involving spins, valence forces and permanent electric or magnetic multipoles are not considered. In the near and wave zones the leading terms of the retarded interaction are proportional to R -6 and R -2 respectively, where R is the intermolecular separation.
We present a Heisenberg-picture approach to the electric dipole interaction of two generally nonidentical atoms, one of which is initially excited, and address the question of whether the dependence of the interaction energy on the interatomic separation rrr is purely monotonic or is sinusoidally modulated as it falls off with rrr. We derive energies of both types and associate them with different model assumptions and physical effects. The sinusoidally modulated form is the interaction energy involved in reversible exchange of excitation (``pendulation''). The monotonic form characterizes an energy shift associated with effectively irreversible (F\"orster) excitation transfer.