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Ukrainian Journal of Physics

One-Photon Scattering by an Atomic Chain in One- and Two-Mode Resonators

Андрій Сіжук, S. M. Yezhov

Published Dec 14, 2015• 4 citations
Publisher

Abstract

A chain of N identical two-level atoms coupled with the electromagnetic field, prepared via a single-photon Fock state, is investigated. It is found that, if the interaction between atoms is negligible, than the obtained dynamic equations for the probability amplitudes allow, in a certain sense, an interpretation of the dynamics of states in the classical fashion in terms of a superposition of oscillatory modes of the system under study. The derived equations reveal how a space configuration of the system of atoms affects the dynamics of the atomic states, particularly the “decay” rates of separate atoms and the system as the whole.

DOI
10.15407/ujpe58.10.1009

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A chain of N identical two-level atoms coupled with the electromagnetic field, prepared via a single-photon Fock state, is investigated. It is found that, if the interaction between atoms is negligible, than the obtained dynamic equations for the probability amplitudes allow, in a certain sense, an interpretation of the dynamics of states in the classical fashion in terms of a superposition of oscillatory modes of the system under study. The derived equations reveal how a space configuration of the system of atoms affects the dynamics of the atomic states, particularly the “decay” rates of separate atoms and the system as the whole.
In this work, a chain of N identical two-level atoms coupled with a quantized electromagnetic field, initially prepared via a single-photon Fock state, is investigated. The N-particle state amplitude of the system is calculated for several space configurations of the atoms in the Weisskopf-Wigner approximation. It was shown that the space configuration of an atomic chain, the total number of atoms, and even the available volume for the field modes define the behavior of the system state amplitude with time. Applying the condition of 'cyclic bonds', presented in this work, to the elaborated theory allows to describe the system time evolution, practically, for any space configuration.
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.
We describe the interaction of two two-level atoms in free space with propagating modes of the quantized electromagnetic field, using the time-dependent Heisenberg-Langevin method. For single- photon pulses, we consider the effect of the pulse's spatial and temporal profiles on the atomic excitation. In particular, we find the ideal shape for a pulse to put exactly one excitation in any desired state of the bi-atomic system. Furthermore, we analyze the differences in the atomic dynamics between the cases of Fock state pulses and coherent state pulses.
The dynamics of a degenerate two-mode electromagnetic field coupled to a single two-level atom is investigated both analytically and numerically. New quantum effects are discussed concerning the time dependence of the photon number and of its fluctuations, assuming that at t = 0 one of the modes is coherent and the other is empty. The field dynamics are dominated by oscillatory net exchanges of a large number of photons between the two modes, displaying amplitude decay. Over a longer time scale, revivals and collapses in the field populations take place. The time scales of these phenomena are much larger than those of the atomic Rabi oscillations decay. Moreover, the system attains a highly non-classical regime in which the field intensity in each mode is subjected to strong fluctuations even for a large initial number of photons in the coherent mode. Various correlations between atomic and field dynamics are pointed out, and the physical origin of the new quantum effects is briefly discussed. Finally, using our fully quantized calculations, we comment on the validity of the atomic homodyne detection proposed by Wilkens and Meystre.