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Physical Review A

Distance dependence of two-atom dipole interactions with one atom in an excited state

Peter W. Milonni, Seyed Mohammad Hashemi Rafsanjani

Published Dec 17, 2015• 45 citations
Publisher

Abstract

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 $r$ is purely monotonic or is sinusoidally modulated as it falls off with $r$. 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.

DOI
10.1103/physreva.92.062711

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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.
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.
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.
I revisit the problem of the interaction between two dissimilar atoms with one atom in an excited state, recently addressed by Berman [Phys. Rev. A 91, 042127 (2015)], Donaire et al. [Phys. Rev. Lett. 115, 033201 (2015)], and Milonni and Rafsanjani [Phys. Rev. A 92, 062711 (2015)], for which precedent approaches have given conflicting results. In the first place, I discuss to what extent these works provide equivalent results. I show that the phase-shift rate of the two-atom wave function computed by Berman, the van der Waals potential of the excited atom by Donaire et al., and the level shift of the excited atom by Milonni and Rafsanjani possess equivalent expressions in the quasistationary approximation. In addition, I show that the level shift of the ground-state atom computed by Milonni and Rafsanjani is equivalent to its van der Waals potential. A diagrammatic representation of all those quantities is provided. The equivalences among them are, however, not generic. In particular, it is found that for the case of the interaction between two identical atoms excited, the phase-shift rate and the van der Waals potentials differ. Concerning the conflicting results of previous approaches in regards to the spatial oscillation of the interactions, I conclude, in agreement with Berman and with Milonni and Rafsanjani, that they refer to different physical quantities. The impacts of free-space dissipation and finite excitation rates on the dynamics of the potentials are analyzed. In contrast with Milonni and Rafsanjani, the oscillatory versus monotonic spatial forms of the potentials of each atom are found not to be related to the reversible versus irreversible nature of the excitation transfer involved.
Energy transfer and coherence transfer between two atoms of different species are studied theoretically. The electric dipole-dipole interaction is the mechanism considered for the excitation transfer. A generalized perturbation method for a damping system is used. A general formula is derived for the polarization of the sensitized fluorescence resulting from the excitation transfer. This polarization is shown to be dependent on the angular momenta of the two atoms, and to be independent of their relative velocity as well as of their difference in the energy of excitation. The polarizations are found to be nonzero in many cases which demonstrate the transfer of coherence through collision processes. A general and compact formula is also derived for the polarization of pure fluorescence where excitation exchange has not taken place. The polarization of the sensitized fluorescence is in general smaller than that of the pure fluorescence. This means that the coherence is transferred only partially. Finally, the magnetic field effect has been included. This field depolarization depends on the gj{g}_{j}gj​ values and the lifetimes of the excited states of both atoms. The general feature of the field dependence also differs considerably from that of the Hanle effect for resonance fluorescence.
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.
The dipole-dipole interaction among ultracold Rydberg atoms is simulated. We examine a general interaction scheme in which two atoms excited to the xxx and ${x}^{\ensuremath{'}}$ states are converted to yyy and ${y}^{\ensuremath{'}}$ states via a F\"orster resonance. The atoms are arranged in two spatially separated groups, each consisting of only one species of atom. We monitor the state mixing by recording the fraction of atoms excited to the ${y}^{\ensuremath{'}}$ state as the distance between the two groups is varied. With zero detuning a many-body effect that relies on always resonant interactions causes the state mixing to have a finite range. When the detuning is greater than zero, another many-body effect causes a peak in the state mixing when the two groups of atoms are some distance away from each other. To obtain these results it is necessary to include multiple atoms and solve the full many-body wave function. These simulation results are supported by recent experimental evidence. These many-body effects, combined with appropriate spatial arrangement of the atoms, could be useful in controlling the energy exchange among the atoms.