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Physical Review A

Non-Markovian decay of a three-level Λ -type atom in a photonic-band-gap reservoir

Xiangqian Jiang, Yongyuan Jiang, Yunliang Wang, Xiudong Sun

Published Mar 2, 2006• 17 citations
Publisher

Abstract

Considering a three-level $\ensuremath{\Lambda}$-type atom with both transitions coupled to a single photonic-band-gap reservoir, we investigate the upper-state population and spontaneous emission spectra by using the resolvent operator for the isotropic dispersion model, and give their analytical solutions. For the case of two transitions coupled to the same reservoir, the upper-state population always displays an oscillatory behavior and finally reaches a steady-state value. For atomic transition frequencies in the vicinity of the band-gap edge or inside the gap, the spontaneous emission spectra become strongly non-Lorentzian.

DOI
10.1103/physreva.73.033802

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Considering a three-level \ensuremathΛ\ensuremath{\Lambda}\ensuremathΛ-type atom with both transitions coupled to a single photonic-band-gap reservoir, we investigate the upper-state population and spontaneous emission spectra by using the resolvent operator for the isotropic dispersion model, and give their analytical solutions. For the case of two transitions coupled to the same reservoir, the upper-state population always displays an oscillatory behavior and finally reaches a steady-state value. For atomic transition frequencies in the vicinity of the band-gap edge or inside the gap, the spontaneous emission spectra become strongly non-Lorentzian.
For a three-level Λ-type atom embedded in three-dimensional photonic crystals (PCs), the upper state population and spontaneous emission spectra are investigated with both transitions coupled to the same modified reservoir. By using an anisotropic dispersion model the analytic expressions of the upper state population and spontaneous emission spectra were obtained. The upper state population reaches a steady state value with a slight oscillation, and the spectra show a non-Lorentzian for both transition frequencies inside the band gap.
The spontaneous emission spectrum of a three-level $\ensuremath{\Lambda}$-type atom driven by a microwave field was studied. For the two transitions coupled to the same modified reservoir, we discussed the influence of photonic band gap and Rabi frequency of the microwave field on the emission spectrum. The emission spectrum is given for different locations of the upper band-edge frequency. With the transition frequencies moving from outside the band gap to inside, the number of peaks decreases in the emission spectrum and the multipeak structure of spectral line is finally replaced by a strong non-Lorentzian shape. With increase of the Rabi frequency of the microwave field, we find the spectral line changes from a multipeak structure to a two-peak structure, originating from the inhibition of spontaneous emission for the corresponding decay channel.
The dynamics of a three-level atom in a cascade configuration with both transitions coupled to a single structured reservoir of quantized field modes is treated using Laplace transform methods applied to the coupled amplitude equations. Results are also obtained from master equations by two different approaches, that is, involving either pseudomodes or quasimodes. Two different types of reservoir are considered, namely a high-Q cavity and a photonic band-gap system, in which the respective reservoir structure functions involve Lorentzians. Non-resonant transitions are included in the model. In all cases non-Markovian behaviour for the atomic system can be found, such as oscillatory decay for the high-Q cavity case and population trapping for the photonic band-gap case. In the master equation approaches, the atomic system is augmented by a small number of pseudomodes or quasimodes, which in the quasimode approach themselves undergo Markovian relaxation into a flat reservoir of continuum quasimodes. Results from these methods are found to be identical to those from the Laplace transform method including two-photon excitation of the reservoir with both emitting sequences. This shows that complicated non-Markovian decays of an atomic system into structured EM field reservoirs can be described by Markovian models for the atomic system coupled to a small number of pseudomodes or quasimodes.
We study spontaneous emission near the edge of a photonic band gap. Instead of a simple exponential decay in the vacuum, spontaneous emission displays an oscillatory behavior. A single photon-atom bound dressed state exhibits a fractional steady-state atomic population on the excited state. For a three-level atom we evaluate the spectral splitting and subnatural linewidth of spontaneous emission. In the presence of N-1 unexcited atoms we show that the collective time scale factor is equal to ${\mathit{N}}^{\mathrm{\ensuremath{\varphi}}}$, where \ensuremath{\varphi}=2/3 for an isotropic band gap and \ensuremath{\varphi}=1 or 2 for anisotropic two-dimensional or three-dimensional band edges, respectively.
The dynamics of a three-level atom in a cascade configuration with both transitions coupled to a single structured reservoir of quantized field modes is treated using Laplace transform methods applied to the coupled amplitude equations. Results are also obtained from master equations by two different approaches, that is, involving either pseudomodes or quasimodes. Two different types of reservoir are considered, namely a high-Q cavity and a photonic band-gap system, in which the respective reservoir structure functions involve Lorentzians. Non-resonant transitions are included in the model. In all cases non-Markovian behaviour for the atomic system can be found, such as oscillatory decay for the high-Q cavity case and population trapping for the photonic band-gap case. In the master equation approaches, the atomic system is augmented by a small number of pseudomodes or quasimodes, which in the quasimode approach themselves undergo Markovian relaxation into a flat reservoir of continuum quasimodes. Results from these methods are found to be identical to those from the Laplace transform method including two-photon excitation of the reservoir with both emitting sequences. This shows that complicated non-Markovian decays of an atomic system into structured EM field reservoirs can be described by Markovian models for the atomic system coupled to a small number of pseudomodes or quasimodes.
The two models of three-level (one upper level and two lower levels, or two upper levels and one lower level) atom embedded in a double-band photonic crystal are adopted. The atomic transitions from the upper levels to the lower levels are assumed to be coupled by the same reservoir which are respectively the isotropic photonic band gap (PBG) modes, the anisotropic PBG modes and the free vacuum modes. The effects of the fine structure of the atomic ground state levels in the model with one upper level and two lower levels and the quantum interferences in the model with two upper levels and one lower level on the spontaneous emission spectrum of an atom are investigated in detail. Most interestingly, it is shown that new spontaneous emission lines are produced from the fine splitting of atomic ground state levels in the isotropic PBG case. The quantum interferences induce additional narrow spontaneous lines near the transition from the empty upper level to the lower level.