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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Non-Markovian stochastic dynamics of a two level system inmersed in a PBG material

Inés de Vega, Daniel Alonso, Pierre Gaspard

Published May 25, 2004• 0 citations
Publisher

Abstract

Spontaneous emission of an atom in contact with the modified radiation field of a Photonic band gap (PBG) material is known to present a typically non-Markovian behavior. In addition, the presence of a gap of forbidden frequencies in the photonic density of states (DOS), produces residual population of the excited level, and population inversion when the atomic system is driven by an external laser field. It's our aim to study those effects using non-Markovian Stochastic Schroedinger equations (NMSSE). This formalism permit us to describe the evolution of the atom in terms of a wave function, |ψt(z)], which due to the presence of the radiation field depends on a stochastic variable zt. Having a singularity at the origin, the models of field correlation function usually found in literature, are not suitable to generate the corresponding zt. We have proposed a new correlation function that do not have this problem, and reproduces all the important physical phenomena of the system under study. The results obtained with NMSSE are compared with those of the master equation, since the ensemble average of stochastic trajectories gives the reduced density matrix of the system, ρs. In addition, we propose a new non-Markovian master equation derived from the stochastic formalism, which in contrast to the current models of master equation preserves positivity for the problem under study.

DOI
10.1117/12.548641

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Spontaneous emission of an atom in contact with the modified radiation field of a Photonic band gap (PBG) material is known to present a typically non-Markovian behavior. In addition, the presence of a gap of forbidden frequencies in the photonic density of states (DOS), produces residual population of the excited level, and population inversion when the atomic system is driven by an external laser field. It's our aim to study those effects using non-Markovian Stochastic Schroedinger equations (NMSSE). This formalism permit us to describe the evolution of the atom in terms of a wave function, |ψt(z)], which due to the presence of the radiation field depends on a stochastic variable zt. Having a singularity at the origin, the models of field correlation function usually found in literature, are not suitable to generate the corresponding zt. We have proposed a new correlation function that do not have this problem, and reproduces all the important physical phenomena of the system under study. The results obtained with NMSSE are compared with those of the master equation, since the ensemble average of stochastic trajectories gives the reduced density matrix of the system, ρs. In addition, we propose a new non-Markovian master equation derived from the stochastic formalism, which in contrast to the current models of master equation preserves positivity for the problem under study.
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.
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 coherent control of cooperative spontaneous emission from two identical non-overlapping three-level atoms in the V-configuration located within a photonic band gap (PBG) material with two resonant frequencies near the upper band edge of the PBG and confined to a region small in comparison to their radiation wavelengths but still greater than their atomic sizes is investigated. The dependencies of cooperative effects in which a photon emitted by one atom is reabsorbed by the other atom on the inter-atomic separation, on the initial state of the two-atom system, on the strength of the driving control laser field, and on the detuning of the atomic resonant frequencies from the upper band edge frequency is analyzed so as to identify the conditions for which these cooperative effects are enhanced or inhibited. Cooperative effects between atoms are shown to be influenced more by the PBG than by the nature of the atomic transitions involved. Excited state populations as well as coherences between excited levels are expressed in terms of time-dependent amplitudes which are shown to satisfy coupled integro-differential equations for which analytic solutions are derived under special conditions. Unlike for the case of one atom in a PBG where the fractional non-zero steady state populations on the excited levels as well as the coherence between the excited levels are constants independent of time, in the case of two atoms in PBG these quantities continuously oscillate as a manifestation of beating due to the continuous exchange between the two atoms of the photon trapped by the PBG. The values of these quantities as well as the amplitudes and frequencies of their oscillations depend of the parameters of the system, providing different ways of manipulating the system. The general formalism presented here is shown to recapture the special results of investigations of similar systems in free space when the non-Markovian memory kernels of the PBG are replaced by delta function dependent Markovian memory kernels.
We present a comprehensive quantum electrodynamical analysis of the interaction between a continuum with photonic band gaps (PBGs) or frequency cut-off and an excited two-level atom, which can be either ‘bare’ or ‘dressed’ by coupling to a near-resonant field mode. A diversity of novel features in the atom and field dynamics is shown to arise from the non-Markovian character of radiative decay into such a continuum of modes. Firstly the excited atom is shown to evolve, by spontaneous decay, into a superposition of non-decaying single-photon dressed states, each having an energy in a different PBG, and a decaying component. This superposition is determined by the atomic resonance shift, induced by the spontaneously emitted photon, into or out of a PBG. The main novel feature exhibited by the decaying excited-state component is the occurrence of beats between the shifted atomic resonance frequency and the PBG cut-off frequencies, corresponding to a non-Lorentzian emission spectrum. Secondly the induced decay of a resonantly driven atom into such a continuum exhibits a cascade of transitions down the ladder of dressed states, which are labelled by decreasing photon numbers of the driving mode. Remarkably, this cascade is terminated at the dressed-state doublet, from which all subsequent transitions to lower doublets are forbidden because they fall within the PBG. This doublet then becomes an attractor state for the populations of higher-lying doublets. As a result, the photon-number distribution of the driving mode becomes strongly sub-Poissonian.
Summary form only given. Unlike semiconductors which facilitate the coherent propagation of electrons, photonic band gap (PBG) materials execute their novel functions through the coherent localization of photons. When a PBG material is doped with impurity atoms which have an electronic transition that lies within the gap, spontaneous emission of light from the atom is inhibited. Inside the gap, the photon forms a bound state to the atom. Outside the gap, radiative dynamics in the colored vacuum is highly nonMarkovian. I discuss the influence of these memory effects on laser action. When spontaneous emission is absent, the next order radiative effect (resonance dipole-dipole interaction between atoms) must be incorporated, leading to anomalous nonlinear optical effects which occur at a much lower threshold than in ordinary vacuum. I describe the collective switching of two-level atoms near a photonic band edge, by external laser field, from a passive stare to one exhibiting population inversion. This effect is forbidden in ordinary vacuum. Finally, I discuss the prospects for a phase sensitive, single (3-level) atom quantum memory device, onto which information may be written by an external laser pulse. By scattering polarized photons (flying qubit) this system may execute certain quantum logic operations.