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.
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.
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.
A critical analysis of the problem of causality in the energy transfer between a pair of atoms is carried out. One of the pair, the source, is initially excited and the other, the detector, is in its ground state. The probabilities are calculated for the detector to be excited at a later time t under different experimental conditions. In the first, a precise specification is made of the state of the source and that of the radiation field at the observation time t. In the second, the final state of the field is unspecified, whereas in the third, the state of neither the field nor the source is specified at time t. These three cases correspond to predicting the outcomes of different experimental measurements. It is shown that of the three cases, only the third exhibits exact causal behavior. The other two are noncausal. The noncausal contributions to the probability arise from the correlation of vacuum fluctuations at the two atoms at different times in the interval 0 to t. The probabilities for these cases can be significant for times slightly less than R/c, where R is the interatomic separation. Explicit expressions, correct to fourth order in the transition moments, for the probabilities in the three cases are given. A proof of strict causality to all orders is presented for the third case where the final measurement is solely made on the detector. In Appendix A the related problem of the time dependence of the electric energy density associated with the source is analyzed and shown to be causal.
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.
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.
When a time-dependent perturbation acts on a quantum system that is initially in the nondegenerate ground state ∣0> of an unperturbed Hamiltonian H(0), the wave function acquires excited-state components ∣k> with coefficients c(k)(t) exp(-iE(k)t/ℏ), where E(k) denotes the energy of the unperturbed state ∣k>. It is well known that each coefficient c(k)(t) separates into an adiabatic term a(k)(t) that reflects the adjustment of the ground state to the perturbation--without actual transitions--and a nonadiabatic term b(k)(t) that yields the probability amplitude for a transition to the excited state. In this work, we prove that the energy at any time t also separates completely into adiabatic and nonadiabatic components, after accounting for the secular and normalization terms that appear in the solution of the time-dependent Schrödinger equation via Dirac's method of variation of constants. This result is derived explicitly through third order in the perturbation. We prove that the cross-terms between the adiabatic and nonadiabatic parts of c(k)(t) vanish, when the energy at time t is determined as an expectation value. The adiabatic term in the energy is identical to the total energy obtained from static perturbation theory, for a system exposed to the instantaneous perturbation λH'(t). The nonadiabatic term is a sum over excited states ∣k> of the transition probability multiplied by the transition energy. By evaluating the probabilities of transition to the excited eigenstates ∣k'(t)> of the instantaneous Hamiltonian H(t), we provide a physically transparent explanation of the result for E(t). To lowest order in the perturbation parameter λ, the probability of finding the system in state ∣k'(t)> is given by λ(2) ∣b(k)(t)∣(2). At third order, the transition probability depends on a second-order transition coefficient, derived in this work. We indicate expected differences between the results for transition probabilities obtained from this work and from Fermi's golden rule.
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.