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arXiv:0906.0743 · Physical Review A

Spontaneous emission from a two-level atom in anisotropic one-band photonic crystals: A fractional calculus approach

Jing-Nuo Wu, Chih‐Hsien Huang, Szu-Cheng Cheng, Wen–Feng Hsieh

Published Feb 24, 2010• 38 citations
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Abstract

Spontaneous emission (SE) from a two-level atom in an anisotropic photonic crystal (PC) is investigated by the fractional calculus. Physical phenomena of the SE are studied analytically by solving the fractional kinetic equations of the SE. There is a dynamical discrepancy between the SE of anisotropic and isotropic PCs. We find that, contrary to the SE phenomenon of the isotropic PC, the SE near the band edge of an anisotropic PC shows no photon-atom bound state. It is consistent with the experimental results of Barth, Schuster, Gruber, and Cichos [Phys. Rev. Lett. 96, 243902 (2006)] that the anisotropic property of the system enhances the SE. We also study effects of dispersion curvatures on the changes of the photonic density of states and the appearance of the diffusion fields in the SE.

DOI
10.1103/physreva.81.023827

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Spontaneous emission (SE) from a two-level atom in an anisotropic photonic crystal (PC) is investigated by the fractional calculus. Physical phenomena of the SE are studied analytically by solving the fractional kinetic equations of the SE. There is a dynamical discrepancy between the SE of anisotropic and isotropic PCs. We find that, contrary to the SE phenomenon of the isotropic PC, the SE near the band edge of an anisotropic PC shows no photon-atom bound state. It is consistent with the experimental results of Barth, Schuster, Gruber, and Cichos [Phys. Rev. Lett. 96, 243902 (2006)] that the anisotropic property of the system enhances the SE. We also study effects of dispersion curvatures on the changes of the photonic density of states and the appearance of the diffusion fields in the SE.
We suggest a better mathematical method, fractional calculus, for studying the behavior of the atom-field interaction in photonic crystals. By studying the spontaneous emission of an atom in a photonic crystal with a one-band isotropic model, we found that the long-time inducing memory of the spontaneous emission is a fractional phenomenon. This behavior could be well described by fractional calculus. The results show no steady photon-atom bound state for the atomic resonant transition frequency lying in the proximity of the allowed band edge which was encountered in a previous study (Woldeyohannes and John 2003 J. Opt. B: Quantum Semiclass. Opt. 5 R43). The correctness of this result is validated by the 'cut-off smoothing' density of photon states (DOS) with fractional calculus. By obtaining a rigorous solution without the multiple-valued problem for the system, we show that the method of fractional calculus has a logically concise property.
We suggest a better mathematical method, fractional calculus, for studying the behavior of the atom-field interaction in photonic crystals. By studying the spontaneous emission of an atom in a photonic crystal with one-band isotropic model, we found that the long-time inducing memory of the spontaneous emission is a fractional phenomenon. This behavior could be well described by the fractional calculus. And the results show no steady photon-atom bound state for the atomic resonant transition frequency lying in the proximity of allowed band edge which is encountered in the previous study [J. Opt. B: Quantum Semiclass. Opt. {\bf 5}, R43 (2003)]. The correctness of this result is validated by the ``cut-off smoothing'' density of photon states (DOS) with fractional calculus. By obtaining a rigorous solution without the multiple-valued problem for the system, we show the method of fractional calculus has logically concise property.
We have studied the properties of the spontaneous emission from a two-level atom embedded in two-band photonic crystals. Due to the anisotropic two-band dispersion relation, the localized and propagating fields cannot coexist, this leads to the non-quasi-oscillation behaviour of the upper level population. Thus the upper level population will tend to a constant as time goes to infinity, or decay with time in a power law, or decay with time exponentially, different from that in the isotropic one-band dispersion relation case. These properties are directly related to the relative position of the upper level from the band edge and the width of the gap.
Studying the spontaneous emission of a V-type three-level atom embedded in a photonic crystal (PC) by fractional calculus, we found that the atomic excited states in the anisotropic PC can be expressed as a superposition of four dressed states analytically. Through detuning two allowed atomic transition energies with respect to the photonic band edge, the coupling between these two transitions leads to three dynamic regimes, namely non-Markovian decay, damped quantum interference and quantum interference, classified by the numbers of contributed bounded dressed states. From the degree of quantum interference of two atomic transitions, we found the energy exchange between the atom and PC reservoir is the lowest as the excited states become degenerate but with maximum quantum interference when the atom is prepared at one of the excited states. The results also show that excited states prefer to stay out of phase at all detuning energy except for near degenerate. Therefore, we can control the spontaneous emission rate not only by the amount of detuning frequencies but also the relative phase of initial states.
The spontaneous emission from a microwave-driven four-level atom embedded in an anisotropic photonic crystal is studied. Due to the modified density of state (DOS) in the anisotropic photonic band gap (PBG) and the coherent control induced by the coupling fields, spontaneous emission can be significantly enhanced when the position of the spontaneous emission peak gets close to the band gap edge. As a result of the closed-loop interaction between the fields and the atom, the spontaneous emission depends on the dynamically induced Autler–Townes splitting and its position relative to the PBG. Interesting phenomena, such as spectral-line suppression, enhancement and narrowing, and fluorescence quenching, appear in the spontaneous emission spectra, which are modulated by amplitudes and phases of the coherently driven fields and the effect of PBG. This theoretical study can provide us with more efficient methods to manipulate the atomic spontaneous emission.