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Physical Review A

Retardation in the resonant interaction of two identical atoms

Peter W. Milonni, P. L. Knight

Published Oct 1, 1974• 342 citations
Publisher

Abstract

The problem considered is that of two identical two-level atoms a fixed distance $r$ apart, one of which is excited at $t=0$. 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 $\frac{\mathrm{nr}}{c}$. The truncated solutions involving only the single retardation time $\frac{r}{c}$ 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.

DOI
10.1103/physreva.10.1096

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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.
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 system of N identical two-level atoms coupled with a quantized electromagnetic field prepared via a one-photon Fock state is investigated. The corresponding N-particle state amplitudes in one- and two-mode resonators are calculated for several space configurations in the cases of closed conservative and open dissipative systems. The nature and the structure of the Weisskopf–Wigner approximation is revealed in the many-body problem. It is shown that the space distribution of atoms, the total number of atoms, and even the available volume for the field modes define the behavior of system’s state amplitudes in time. The elaborated theory allows one to analytically describe the time evolution of the system for a quite wide range of the space configurations, if the specific “cyclic” restrictions are applied.
In this paper we analyze the dynamics of single-excitation states, which model the scattering of a single photon from multiple two level atoms. For short times and weak atom-field couplings we show that the atomic amplitudes are given by a sum of decaying exponentials, where the decay rates and Lamb shifts are given by the poles of a certain analytic function. This result is a refinement of the "pole approximation" appearing in the standard Wigner-Weisskopf analysis of spontaneous emission. On the other hand, at large times, the atomic field decays like O(1/t3)O(1/t^3)O(1/t3) with a known constant expressed in terms of the coupling parameter and the resonant frequency of the atoms. Moreover, we show that for stronger coupling, the solutions also feature a collection of oscillatory exponentials which dominate the behavior at long times. Finally, we extend the analysis to the continuum limit in which atoms are distributed according to a given density.
The interaction between two identical atoms, separated by a distance r that is large compared to an optical wavelength, is considered using an amplitude approach in the Schr\"odinger picture. When atom 1 is excited by a radiation pulse whose duration is long compared to an optical period but short compared to the time it takes for light to travel between the atoms, the probability for atom 2 to be excited is found to be identically zero for times t/c, provided only r-dependent terms in the probability are retained. The r-dependent probability amplitude for atom 2 to be excited and atom 1 to be in its ground state is not identically equal to zero for t/c. Rather it oscillates at the natural frequency of the optical transition. The Schr\"odinger picture approach allows one to clearly identify the various terms that give rise to the excitation probabilities. A diagrammatic technique is developed that facilitates the calculation.