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Physical Review A

Dynamics of an excited two-level atom in the presence ofN-1unexcited atoms in the free space

Vladimír Bužek

Published Feb 1, 1989• 18 citations
Publisher

Abstract

The dynamics of the system of N two-level atoms in the free space is studied, when at the initial moment just one of the atoms is excited. It is shown that though the number of the field modes is sufficiently larger than the number of the atoms and all energy is transferred into radiation field for t\ensuremath{\rightarrow}\ensuremath{\infty}, the radiation suppression (in a broader sense) can be still observed.

DOI
10.1103/physreva.39.2232

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The dynamics of the system of N two-level atoms in the free space is studied, when at the initial moment just one of the atoms is excited. It is shown that though the number of the field modes is sufficiently larger than the number of the atoms and all energy is transferred into radiation field for t\ensuremath{\rightarrow}\ensuremath{\infty}, the radiation suppression (in a broader sense) can be still observed.
The quantum-electrodynamical problem of N identical two-level atoms interacting with M field modes is considered, and an exact solution for the case of spontaneous emission from a single atom is obtained. The behavior of this system is shown to be quite different from the exponential decay of a single excited atom radiating into free space. A ``ringing'' behavior at the enigmatic frequency \ensuremath{\sim} \ensuremath{\surd}N which occurs when only one mode is accessible to the field persists when many modes are present. When a continuum of modes is accessible, the long-time limit shows that the energy of the originally excited atom is equally likely to be shared by the N-1 originally unexcited atoms and the electromagnetic field when M/N\ensuremath{\rightarrow}1. ``Radiation suppression'' occurs whenever N\ensuremath{\gg}1 and M\ensuremath{\ll}N, when the originally excited atom does not emit its energy. When M=N\ensuremath{\gg}1, the single atom decays monotonically to zero (and approximately exponentially) but with a decay rate that is \ensuremath{\surd}N times the Rabi frequency. Such unexpected effects are presently within experimental range.
A method utilising the coherent state variables for the electromagnetic field and appropriate spin variables for two-level atomsis developed for obtaining the dynamical evolution of the field and the atoms separately. The case of a single radiation mode interacting with two two-level atoms is studied. The time evolution of the photon number, starting from n photons and both atoms at their higher levels, is given. Furthermore, when the initial state consists of n photons and one of the atoms in its excited state with the other in its ground state, the temporal development of the photon number, as well as the motion of each atom, are obtained. In the latter case it becomes apparent that energy exchange takes place among the atomic systems, while the field acts as a transfer agent.
The long-time behavior of the number of photons in a mode of the electromagnetic field is given in closed form in terms of elementary functions when there is initially one (two-level) atom excited in the presence of NNN unexcited identical atoms and MMM field modes. The radiation trapping which occurs with only one mode disappears when the number of modes MMM approaches the total number of atoms in the continuum mode approximation.
An exact solution for the problem of two two-level atoms interacting with a single-mode detuned quantized radiation field is presented. On resonance, an asymptotic result is derived for large coherent initial fields. It is shown that when the atoms are initially prepared in one of the semiclassical eigenstates, the atom-plus-field wavefunction remains almost factored into an atomic and a field part throughout the interaction. Under certain conditions, the atomic part evolves into a unique pure state at half-revival time. The effects of the cavity detuning on the dynamics of the atomic level occupation probabilities and emission spectra are studied. It is found, for non-symmetrical excitation, that far off-resonance the field acts like a virtual level by means of which the energy is transferred between the atoms. Due to non-zero detuning, nine-peaked spectra are observed and are given a clear explanation from the dressed-state viewpoint. It is established that in the case of symmetrical excitation, the intensity of the emission spectrum is two times larger than that in the case of non-symmetrical excitation.
The theoretical model presented here is motivated by experimental works demonstrating the suppression of spontaneous emission by atoms placed inside a cavity with a fundamental frequency above that for the atomic transition. In my model, two distinguishable two-level atoms, one of them initially excited, the other in the ground state, are coupled to a nonresonant radiation mode, with no direct interaction between the atoms. The exchange of energy between the atoms is investigated, and it is found that it oscillates from one atom to the other with angular frequency \ensuremath{\Omega} related to the radiative lifetime \ensuremath{\tau} by \ensuremath{\Omega}\ensuremath{\tau}=(3/2${\mathrm{\ensuremath{\pi}}}^{2}$)(${\mathrm{\ensuremath{\omega}}}_{0}$/${\mathrm{\ensuremath{\omega}}}_{0}$-\ensuremath{\omega})(${\mathrm{\ensuremath{\omega}}}_{0}$/\ensuremath{\omega})3{)}^{3})3, with ${\mathrm{\ensuremath{\omega}}}_{0}$ and \ensuremath{\omega}, the frequencies for the radiation mode and the atomic transition, respectively.
The dynamics of few two-level atoms interacting with one cavity mode is described in master equation formalism. Two different configurations are considered: a transient one with all atoms initially in the excited state and a stationary one where a coherent pump acts on either the atoms or the cavity mode. Superradiance, resonant scattering and switching between a low and a high emission state are considered. Cooperation and decoherence as well as the statistics of the emitted radiation field in both cases are discussed.