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Physical Review A

Calculation of spontaneous emission from a V-type three-level atom in photonic crystals using fractional calculus

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

Published Jul 5, 2011• 9 citations
Publisher

Abstract

Fractional time derivative, an abstract mathematical operator of fractional calculus, is used to describe the real optical system of a V-type three-level atom embedded in a photonic crystal. A fractional kinetic equation governing the dynamics of the spontaneous emission from this optical system is obtained as a fractional Langevin equation. Solving this fractional kinetic equation by fractional calculus leads to the analytical solutions expressed in terms of fractional exponential functions. The accuracy of the obtained solutions is verified through reducing the system into the special cases whose results are consistent with the experimental observation. With accurate physical results and avoiding the complex integration for solving this optical system, we propose fractional calculus with fractional time derivative as a better mathematical method to study spontaneous emission dynamics from the optical system with non-Markovian dynamics.

DOI
10.1103/physreva.84.013802

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Fractional time derivative, an abstract mathematical operator of fractional calculus, is used to describe the real optical system of a V-type three-level atom embedded in a photonic crystal. A fractional kinetic equation governing the dynamics of the spontaneous emission from this optical system is obtained as a fractional Langevin equation. Solving this fractional kinetic equation by fractional calculus leads to the analytical solutions expressed in terms of fractional exponential functions. The accuracy of the obtained solutions is verified through reducing the system into the special cases whose results are consistent with the experimental observation. With accurate physical results and avoiding the complex integration for solving this optical system, we propose fractional calculus with fractional time derivative as a better mathematical method to study spontaneous emission dynamics from the optical system with non-Markovian dynamics.
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.
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.
The spontaneous emission dynamics of atoms embedded in an omnidirectional waveguide (ODWG), a novel optical waveguide, is studied on the basis of the complete reflection of one-dimensional photonic crystals. With the dispersion curve of the single waveguide mode within the photonic band gap and various extents of background dissipation, we characterize the photon-atom interaction in the ODWG. The photon emitter of the system is a two-level atom embedded in the low-index medium of the multilayer-film ODWG or the atom-ODWG system. Fractional calculus, an innovative mathematical method in optical systems, is applied to solve the equation of motion for this atom-ODWG system. Two kinds of states with different group velocities exhibit totally distinctive dynamical behavior. The high frequency waveguide mode with a fast group velocity shows fast exponential decay in propagation while the band-edge mode with a slow group velocity displays non-Markovian dynamics with non-exponential oscillating time evolution. We therefore suggest different functions of this atom-ODWG system for these two kinds of states. The richness of the physical content of the system is also revealed through investigating the dynamical behavior of the band-edge mode. These results aid in further application and fundamental understanding of the atom-ODWG system.
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 dynamic evolution of atomic spontaneous emission in one-dimensional photonic crystals (1DPC) is investigated by using mode functions for the 1DPC. Our attention is focused on the non-Markovian processes in the first several optical cycles of the spontaneous emission decay. The integral-differential dynamic equation is directly solved by numerical calculation without using the Markovian approximation. The interference of the multireflected fields plays an important role in the non-Markovian processes. Due to the interference of the fields successive arrived at the atom due to the reflection at the interfaces, the decay rate shows a series of pulselike peaks. If the interference is constructive, the spontaneous decay is enhanced, while if the interference is destructive, the spontaneous decay is suppressed. The steady decay rate is approximately equal to the vacuum decay rate, when the atomic transition frequency corresponds to an allowed normal mode of the 1DPC. The spontaneous emission decay is suppressed if the atomic transition frequency is within the forbidden gap of the 1DPC.