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Physical Review A

Method of collective operators for resonance fluorescence near a photonic band edge

Dmitri S. Mogilevtsev, S. Ya. Kilin

Published Feb 28, 2003• 8 citations
Publisher

Abstract

In this work a formalism of collective atom-field operators is described and applied to study the dynamics of a classically driven two-level atom inside a band-gap material. We show that if a photon-atom bound state can exist when driving is absent, then it is possible to assume mutual independence of collective operators and develop an effective approximation for the case when driving is present.

DOI
10.1103/physreva.67.023815

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Summary form only given. In this work we suggest and discuss the method of collective atomic-field operators for the resonance fluorescence of two-level atom with the transition frequency near the band gap edge. The method is based on the formal diagonalization of the interaction Hamiltonian in the absence of driving. In the absence of driving the method also allows to obtain the exact solution for the case when there are two photons in the field. We show that due to closeness to the gap edge the collective operators are weakly coupled. This allows to develop an effective approximation applicable when the problem is strongly non-Markovian, i.e. when in the absence of driving the effect of "freezing" of upper-level atomic population takes place. Then the dynamics of all but one collective operators becomes similar to the bosonic one. Only the dynamics of this one collective operators is nonlinear. We show that in this case a perturbation theory could be developed using as a small parameter a respective coupling strength between this one collective operator and others. In fact, the method of collective operators gives possibility to develop a perturbation theory "tailored" especially for the essentially non-Markovian problem. We derive a condition of applicability for this theory. Essentially, it is the condition of weak excitation of all collective atom-field "modes" corresponding to the collective operators with dynamics similar to the bosonic one. For the moderate intensities of the driving field even an approximation of independent collective operators gives good results. In the limit of weak driving field a simple analytic approximation is developed. As comparison shows, this approximation allows to describe a basic features of atomic population dynamics even for the cases when the driving field could not any more be considered weak.

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In this work a formalism of collective atom-field operators is described and applied to study the dynamics of a classically driven two-level atom inside a band-gap material. We show that if a photon-atom bound state can exist when driving is absent, then it is possible to assume mutual independence of collective operators and develop an effective approximation for the case when driving is present.
Summary form only given. In this work we suggest and discuss the method of collective atomic-field operators for the resonance fluorescence of two-level atom with the transition frequency near the band gap edge. The method is based on the formal diagonalization of the interaction Hamiltonian in the absence of driving. In the absence of driving the method also allows to obtain the exact solution for the case when there are two photons in the field. We show that due to closeness to the gap edge the collective operators are weakly coupled. This allows to develop an effective approximation applicable when the problem is strongly non-Markovian, i.e. when in the absence of driving the effect of "freezing" of upper-level atomic population takes place. Then the dynamics of all but one collective operators becomes similar to the bosonic one. Only the dynamics of this one collective operators is nonlinear. We show that in this case a perturbation theory could be developed using as a small parameter a respective coupling strength between this one collective operator and others. In fact, the method of collective operators gives possibility to develop a perturbation theory "tailored" especially for the essentially non-Markovian problem. We derive a condition of applicability for this theory. Essentially, it is the condition of weak excitation of all collective atom-field "modes" corresponding to the collective operators with dynamics similar to the bosonic one. For the moderate intensities of the driving field even an approximation of independent collective operators gives good results. In the limit of weak driving field a simple analytic approximation is developed. As comparison shows, this approximation allows to describe a basic features of atomic population dynamics even for the cases when the driving field could not any more be considered weak.
We studied the effects of a coherent monochromatic resonant laser field driving the transition of a two-level atom embedded in a photonic band gap (PBG) material on the emission dynamics of the atom. When the transition frequency of the atom lies outside a PBG and sufficiently far from the band edge that the emission dynamics of the atom is not normally affected by the gap, the dynamic stark splitting of the upper level of the atom into two dressed states by a sufficiently strong resonant driving field can bring the lower dressed state close enough to the band edge or even inside the gap so as to be affected by the gap, providing a switchable means of extending the novel effects of atom–photon interaction near the edge of a PBG for atomic transitions outside the gap and far from the band edge. The net effect is as if a sufficiently strong resonant field driving a transition of an atom embedded in a PBG actively shifts the PBG close enough to the transition so that the gap affects the emission dynamics associated with the transition.
We introduce a dressed-state Monte Carlo wave-function technique to describe resonance fluorescence in a broad class of non-Markovian reservoirs with strong atom-reservoir interaction. The method recaptures photon localization effects which are beyond the Born and Markovian approximations, and describes the influence of the driving field on the atom-reservoir interaction. Using this approach, we predict a number of fundamentally new features in resonance fluorescence near the edge of a photonic band gap. In particular, the atomic population exhibits inversion for moderate applied field intensity. For a low external field intensity, the atomic system retains a long-time memory of its initial state.
A numerical method of calculating the non-Markovian evolution of a driven atom radiating into a structured continuum is developed. The formal solution for the atomic reduced density matrix is written as a Markovian algorithm by introducing a set of additional, virtual density matrices which follow, to the level of approximation of the algorithm, all the possible trajectories of the photons in the electromagnetic field. The technique is perturbative in the sense that more virtual density matrices are required as the product of the effective memory time and the effective coupling strength becomes larger. The number of density matrices required is given by 3M{3}^{M}3M where M is the number of time steps per memory time. The technique is applied to the problem of a driven two-level atom radiating close to a photonic band gap and the steady-state correlation function of the atom is calculated.
We discuss a point model for the collective emission of light from NNN two-level atoms in a photonic band-gap material, each with an atomic resonant frequency near the edge of the gap. In the limit of a low initial occupation of the excited atomic state, our system is shown to possess atomic spectra and population statistics that are radically different from free space. For a high initial excited-state population, mean-field theory suggests a fractionalized inversion and a macroscopic polarization for the atoms in the steady state, both of which can be controlled by an external dc field. This atomic steady state is accompanied by a nonzero expectation value of the electric field operators for field modes located in the vicinity of the atoms. The nature of homogeneous broadening near the band edge is shown to differ markedly from that in free space due to non-Markovian memory effects in the radiation dynamics. Non-Markovian vacuum fluctuations are shown to yield a partially coherent steady-state polarization with a random phase. In contrast with the steady state of a conventional laser, near a photonic band edge this coherence occurs as a consequence of photon localization in the absence of a conventional cavity mode. We also introduce a classical stochastic function with the same temporal correlations as the electromagnetic reservoir, in order to stochastically simulate the effects of vacuum fluctuations near a photonic band edge.
The forbidden band in a photonic crystal leads to the existence of trapped atom-field state whose properties depend on the followings: the number of atoms embedded in a photonic crystal,the energy level structure of atom,the relative position between the level and the edge of band gap and the existence of driving field.By using the method of searching for the eigenstate of the trapped state,that the trapped atom-field state for a three-level atom with driving field in a photonic crystal exists due to the interference between atom and field, was discussed. That the trapped states and the population in various levels can be influenced by the strength and the Rabi frequency of the driving field was proved. That more than one trapped state exist and they depend on the position of the upper level also were proved. Analysis show that there is only one eigenstate when the upper energy level locates outside the band gap, and atom hardly stays in upper energy level; there are two eigenstates when the upper energy level locates in the band gap, but the properties of the two are quite different each to other. The distribution of particle number on every energy level varies obviously as the driving field parameter varies.