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arXiv:0711.1499 · Physical Review A

Emission spectra of atoms with non-Markovian interaction: Fluorescence in a photonic crystal

Inés de Vega, Daniel Alonso

Published Apr 29, 2008• 27 citations
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Abstract

We present a formula to evaluate the spontaneous emission spectra of an atom in contact with a radiation field with non-Markovian effects. This formula is written in terms of a two-time correlation of system observables and the environmental correlation function, and depends on the distance between the emitting atom and the detector. As an example, we use it to analyze the fluorescence spectra of a two level atom placed as an impurity in a photonic crystal. The radiation field within those materials has a gap or discontinuity where electromagnetic modes cannot propagate in the stationary limit. In that situation, the atomic emission occurs in the form of evanescent waves which are detected with less efficiency the farther we place the detector. The methodology presented in this paper may be useful to study the non-Markovian dynamics of any quantum open system in linear interaction with a harmonic oscillator reservoir and within the weak-coupling approximation.

DOI
10.1103/physreva.77.043836

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arXiv:quant-ph/0008093•Physical Review A•Mar 7, 2001•25 citations
We present a formula to evaluate the spontaneous emission spectra of an atom in contact with a radiation field with non-Markovian effects. This formula is written in terms of a two-time correlation of system observables and the environmental correlation function, and depends on the distance between the emitting atom and the detector. As an example, we use it to analyze the fluorescence spectra of a two level atom placed as an impurity in a photonic crystal. The radiation field within those materials has a gap or discontinuity where electromagnetic modes cannot propagate in the stationary limit. In that situation, the atomic emission occurs in the form of evanescent waves which are detected with less efficiency the farther we place the detector. The methodology presented in this paper may be useful to study the non-Markovian dynamics of any quantum open system in linear interaction with a harmonic oscillator reservoir and within the weak-coupling approximation.
The non-Markovian correlation spectra of spontaneous emission of an excited two-level atom are derived including the effect of finite size of the atom and all the possible contribution of allowed multipole radiations. The emission process is then analyzed by the quantum stochastic trajectory approach. The non-Markovian effect is counted in by expanding the original system to an enlarged system with Markovian reservoirs. In the case of a hydrogenlike atom with large atomic number Z, the deviation from the Weisskopf-Wigner result is quite evident.
In contrast to the situation in free space where the Markovian approximation and the quantum regression theorem are valid, the atom-photon interaction in photonic bandgap materials is in general nonMarkovian. In the present paper we treat the photonic crystals as background reservoirs and introduce the nonuniform noise operators caused by such nonuniform DOS distributions near the band edge. The steady state fluorescence spectrum of a two-level atom embedded in a three-dimensional photonic bandgap crystal and driven by a monochromatic classical electrical field is calculated theoretically for the first time as we known. With the help of the Liouville operator expansion, we can calculate the stationary two-time correlation functions of the atomic operators as well as the spectral distribution of the resonance fluorescence. We also use various dispersion relations to model this nonuniform DOS distribution near the bandgap. We find that if the atomic energy level is far from the bandgap, fluorescence spectra with Mollow triplets are observed.
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.
We present a formalism for the description of fluorescence from optically active materials embedded in a photonic crystal structure possessing a photonic band gap or pseudogap. An electromagnetic field expansion in terms of Bloch modes of the crystal is used to develop the equations for fluorescence in terms of the local density of photon modes available to the emitting atoms in either the high or low dielectric regions of the crystal. We then obtain expressions for fluorescence spectra and emission dynamics for luminescent materials in photonic crystals. The validity of our formalism is demonstrated through the calculation of relevant quantities for model photon densities of states. The connection of our calculations to the description of realistic systems is discussed. We also describe the consequences of these analyses on the accurate description of the interaction between radiative systems and the electromagnetic reservoir within photonic crystals.
The spontaneous emission field and spectrum of a two-level atom, located in an isotropic photonic crystal with dynamic band edges, are investigated by means of numeric calculation. The investigation is expected to help comprehend the characteristics of the atomic spontaneous emission in the dynamic photonic crystal, and provide a possible way to control dynamically the spontaneous emission in photonic crystal. The expression of the spontaneous radiation field is obtained without using the far-zone approximation and the Weisskopf-Wigner approximation, and expected to be applicable in other relevant researches. In the investigation, the spontaneous radiation field and spectrum are calculated when the band edge frequency is unmodulated, or modulated by a step function or triangle function. In the unmodulated situation, the radiation field intensity tends to a constant which is equal to the intensity of the localized field component. The radiation field pulse presents a wave packet behavior as propagation distance increases. The components of the radiation field correspond one-to-one to the peaks in the spontaneous radiation spectrum. When the band edge frequency is modulated by step function, the radiation field intensity tends to a steady-state value after the modulation has happened. And the steady-state intensity is affected by the time when the modulation happens. The components of the non-localized field and the frequency of the localized field after modulation depend on the atomic transition frequency and the band edge frequency, and are identical to those in the unmodulated situation with the same parameters. When the band edge frequency is modulated by a triangle function, the field intensity presents a decaying quasi-periodic oscillation after a long enough time. The modulation frequency determines the frequency of the oscillation, and influences the decay rate. The radiation energy becomes sharp peaks around a set of the discrete frequencies which are evenly spaced with the modulation frequency. The central frequency of these frequencies depends on the atomic transition frequency and the value range of the band edge frequency. The modulation initial phase affects the intensity of the radiation field emitted in an initial period of time.
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.