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Physical Review A

Significance of Röntgen current in quantum optics: Spontaneous emission of moving atoms

Martin Wilkens

Published Jan 1, 1994• 52 citations
Publisher

Abstract

We argue that for moving atoms, the conventional dipole interaction must be supplemented by a so-called R\"ontgen term in order to obtain results which are trustworthy to order V/c in the atomic velocity. As an illustration, we include the R\"ontgen interaction in a Hamiltonian model to derive the angular distribution of photons spontaneously emitted by a freely moving atom. We find agreement with an independent calculation which is based on the transformation properties of such distribution under Lorentz boosts. This resolves the previously reported inconsistency of the conventional dipole interaction for moving atoms [Phys. Rev. A 47, 671 (1993)].

DOI
10.1103/physreva.49.570

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We argue that for moving atoms, the conventional dipole interaction must be supplemented by a so-called R"ontgen term in order to obtain results which are trustworthy to order V/c in the atomic velocity. As an illustration, we include the R"ontgen interaction in a Hamiltonian model to derive the angular distribution of photons spontaneously emitted by a freely moving atom. We find agreement with an independent calculation which is based on the transformation properties of such distribution under Lorentz boosts. This resolves the previously reported inconsistency of the conventional dipole interaction for moving atoms [Phys. Rev. A 47, 671 (1993)].
We present a time-dependent calculation of the velocity-dependent forces which act on an excited atomic dipole in relative motion with respect to ground state atoms of a different kind. Both its interaction with a single atom and with a dilute atomic plate are evaluated. In either case, the total force consists of a conservative van der Waals component and a nonconservative R\"ontgen component. On physical grounds, the former corresponds to the velocity-dependent recoil experienced by the excited atom in the processes of absorption and emission of the photons that it exchanges with the ground-state atoms on a periodic basis. The latter corresponds to the time-variation of the R\"ontgen momentum, which is also mediated by the periodic exchange of quasiresonant photons. We find that, at leading order, all these interactions are linear in velocity. In the nonretarded regime the van der Waals force dominates, being antiparallel to the velocity. On the contrary, in the retarded regime the velocity-dependent forces oscillate in space, van der Waals and R\"ontgen forces are of the same order in the atom-atom interaction, and the R\"ontgen component dominates in the atom-surface interaction.
The spectrum of spontaneously emitted photons from an atom interacting with a single cavity mode depends on the relative sizes of the mode coupling constant and the cavity loss rate. If the coupling constant is not too large, perturbation theory can he used to show that the atomic linewidth is either increased (enhanced spontaneous emission) or decreased (inhibited spontaneous emission) by the presence of the cavity. When the coupling constant is so large that perturbation theory cannot be used, the spontaneous- emission spectrum forms a doublet, with peaks located at the normal-mode frequencies for the coupled-dipole and cavity-mode oscillators (the so-called "vacuum" Rabi splitting).
A theory of the changes in momenta suffered by the translational motion of a two-level atom due to interaction near resonance within uniform light, is presented. The leading changes in the translational energy are first derived and the changes in the momenta due to the irradiation are then deduced from these in a simple and transparent manner. The derivation depends on the inclusion of a quantum interaction term whose classical origin is the Rontgen current in the electric dipole approximation. Thus the theory presented differs considerably from conventional approaches which are normally based on considerations of the transition rates and do not include the effects of the Rontgen term. The momentum changes obtained can be expressed in terms of the susceptibility and its first derivative, and are shown to be independent of the state of the atomic translation. They depend, mainly in sign, on whether the atom is in its ground or excited state. These features lead to a simple derivation of the average momentum changes induced by the irradiation of an atomic system.
The Röntgen term is an often neglected contribution to the interaction between an atom and an electromagnetic field in the electric dipole approximation. In this work we discuss how this interaction term leads to a difference between the kinetic and canonical momentum of an atom which, in turn, leads to surprising radiation forces acting on the atom. We use a number of examples to explore the main features of this interaction, namely forces acting against the expected dipole force or accelerations perpendicular to the beam propagation axis.
We consider the relationship between the spontaneous radiative decay of an atom at rest and of an atom moving with a constant velocity. Based on the relativistic form, Aμjμ, of the field–current interaction we analyse two situations. In the first of them the radiative decay of a single atom is considered by using two different reference frames. In the second, a single reference frame is used to describe the radiative decay of two identical atoms, one of which is at rest in this frame and the other one moves with a constant velocity. Although in special relativity only the relative velocity between a source and an observer matters, there are subtle differences in the analysis of these situations: within the framework of the first situation one deals only with the Lorentz transformation, which is insensitive to whether exact or approximate atomic wavefunctions are used in the analysis, whereas the analysis of the second situation involves a gauge transformation which in general is not compatible with using approximate atomic wavefunctions. We have also shown that this gauge transformation can be used as a test for the accuracy of approximate atomic states obtained in atomic structure calculations.
It is shown that the distribution of the momentum transfer to an atom in resonance fluorescence may be easily derived from the angular distribution of spontaneously emitted photons. Some general properties of the momentum distribution are found, and explicit forms of the distribution are derived for several special cases. It is shown that the distribution tends to become Gaussian after a large number n of spontaneous photon emissions, with variance proportional to n, but it is already close to the asymptotic form when n=3. When the appropriate averaging is performed to allow for the velocity spread of the atoms, the derived distributions account for the results of atomic beam deflection measurements, but much of the information about the original distributions is thereby lost. In a standing wave field both stimulated and spontaneous emissions contribute to the atomic momentum, in such a way that the variance of the momentum spread is again proportional to the number n of photon emissions.