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arXiv:quant-ph/9708030 · Physical Review Letters

Fluorescence into Flat and Structured Radiation Continua: An Atomic Density Matrix without a Master Equation

Søren Bay, P. Lambropoulos, Klaus Mølmer

Published Oct 6, 1997• 96 citations
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Abstract

We investigate an atomic $\ensuremath{\Lambda}$ system with one transition coupled to a laser field and a flat continuum of vacuum modes and the other transition coupled to field modes near the edge of a photonic band gap. The system requires simultaneous treatment of Markovian and non-Markovian dissipation processes, but the photonic-band-gap continuum cannot be eliminated within a density matrix treatment. Instead we propose a formalism based on Monte Carlo wave functions, and we present results relevant to the experimental characterization of a structured continuum.

DOI
10.1103/physrevlett.79.2654

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We investigate an atomic \ensuremathΛ\ensuremath{\Lambda}\ensuremathΛ system with one transition coupled to a laser field and a flat continuum of vacuum modes and the other transition coupled to field modes near the edge of a photonic band gap. The system requires simultaneous treatment of Markovian and non-Markovian dissipation processes, but the photonic-band-gap continuum cannot be eliminated within a density matrix treatment. Instead we propose a formalism based on Monte Carlo wave functions, and we present results relevant to the experimental characterization of a structured continuum.
An atomic Dicke system is coupled to a photonic band-gap continuum, which we model by a Fano-profile density of states. By introduction of two pseudomodes, a Markovian master equation can be derived governing only the degrees of freedom of the atoms plus the two pseudomodes. One of the modes can be adiabatically eliminated, and effectively we then have an atomic Dicke system coupled to a harmonic oscillator and both systems coupled to the same flat continuum. We find that following the superradiant regime, a metastable state is reached for the atomic system. The decay of the metastable state is nonexponential, and we derive an analytical expression for the decay based on perturbation theory and trapping states identified by the Monte Carlo wave-function method. Further, we investigate mean-value equations of motion for the operators of the system and discuss different decorrelation approximations of the operator expectation values.
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.
This paper deals with non-Markovian behavior in atomic systems coupled to a structured reservoir of quantum electromagnetic field modes, with particular relevance to atoms interacting with the field in high-Q cavities or photonic band-gap materials. In cases such as the former, we show that the pseudomode theory for single-quantum reservoir excitations can be obtained by applying the Fano diagonalization method to a system in which the atomic transitions are coupled to a discrete set of (cavity) quasimodes, which in turn are coupled to a continuum set of (external) quasimodes with slowly varying coupling constants and continuum mode density. Each pseudomode can be identified with a discrete quasimode, which gives structure to the actual reservoir of true modes via the expressions for the equivalent atom-true mode coupling constants. The quasimode theory enables cases of multiple excitation of the reservoir to now be treated via Markovian master equations for the atom-discrete quasimode system. Applications of the theory to one, two, and many discrete quasimodes are made. For a simple photonic band-gap model, where the reservoir structure is associated with the true mode density rather than the coupling constants, the single quantum excitation case appears to be equivalent to a case with two discrete quasimodes.
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.
It is our aim to study the dynamics of a two-level atom immersed in the modified radiation field of a photonic band-gap material using non-Markovian stochastic Schr\"odinger equations. Up to now, such methodology has only been applied to toy models and not to physically realistic systems as the one presented here. In order to check its validity, we shall study several of the physical phenomena already described in the literature within non-Markovian master equations, such as the long-time-limit residual population in the excited level of the atom and the population inversion which occurs in the atomic system when applying an external laser field. In addition to the stochastic equation, we propose a non-Markovian master equation derived from the stochastic formalism, which in contrast to the current models of master equation preserves positivity. We propose a correlation function for the radiation field (environment) that captures many of the physically relevant aspects of the problem and describes the short-time behavior in a more accurate way than previously proposed ones. This characteristic permits a correct description of the fluctuations of the electromagnetic field, which in the stochastic formalism are represented by the noise, and a better description of the non-Markovian effects in the atomic dynamics. The methodology presented in this paper to apply stochastic Schr\"odinger equations can be followed to study more complex systems, like many-level atoms embedded in more complicated photonic band-gap structures.