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Journal of Modern Optics

Spontaneous emission from a three-level atom placed in three-dimensional photonic crystals in the long-time limit

Hiroyuki Nihei, Atsushi Okamoto

Published Jul 1, 2002• 14 citations
Publisher

Abstract

The paper describes the spontaneous emission from a three-level atom placed in a periodic dielectric microstructure which exhibits a complete three-dimensional photonic band gap. By using the Euler approach, the upper level population of the atom is calculated for a wide range of relationships between the Rabi frequency and the detuning of the atomic transition frequency from the upper band edge. The results indicate that there are three cases of the relationship between Rabi frequency and detuning, which determine distinctive states of the atomic population in the long-time limit. When the detuning is greater than the Rabi frequency, the upper level has a zero steady-state atomic population, which leads to enhancement of spontaneous emission. When the magnitude of the detuning is less than the Rabi frequency, the upper level has a nonzero steady-state atomic population, which leads to suppression of spontaneous emission. When the negative detuning is greater than the Rabi frequency, the upper level has a nondecaying oscillatory-state atomic population due to long-time atomic splitting. These three properties of the spontaneous emission are relevant to several optical devices on an atomic scale, such as optical memories, switches and clocks.

DOI
10.1080/09500340110105920

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The paper describes the spontaneous emission from a three-level atom placed in a periodic dielectric microstructure which exhibits a complete three-dimensional photonic band gap. By using the Euler approach, the upper level population of the atom is calculated for a wide range of relationships between the Rabi frequency and the detuning of the atomic transition frequency from the upper band edge. The results indicate that there are three cases of the relationship between Rabi frequency and detuning, which determine distinctive states of the atomic population in the long-time limit. When the detuning is greater than the Rabi frequency, the upper level has a zero steady-state atomic population, which leads to enhancement of spontaneous emission. When the magnitude of the detuning is less than the Rabi frequency, the upper level has a nonzero steady-state atomic population, which leads to suppression of spontaneous emission. When the negative detuning is greater than the Rabi frequency, the upper level has a nondecaying oscillatory-state atomic population due to long-time atomic splitting. These three properties of the spontaneous emission are relevant to several optical devices on an atomic scale, such as optical memories, switches and clocks.
We demonstrate the coherent control of spontaneous emission for a three-level atom located within a photonic band gap (PBG) material, with one resonant frequency near the edge of the PBG. Spontaneous emission from the three-level atom can be totally suppressed or strongly enhanced depending on the relative phase between the steady-state control laser coupling the two upper levels and the pump laser pulse used to create an excited state of the atom in the form of a coherent superposition of the two upper levels. Unlike the free-space case, the steady-state inversion of the atomic system is strongly dependent on the externally prescribed initial conditions. This non-zero steady-state population is achieved by virtue of the localization of light in the vicinity of the emitting atom. It is robust to decoherence effects provided that the Rabi frequency of the control laser field exceeds the rate of dephasing interactions. As a result, such a system may be relevant for a single-atom, phase-sensitive optical memory device on the atomic scale. The protected electric dipole within the PBG provides a basis for a qubit to encode information for quantum computations. A detailed literature survey on the nature, fabrication and applications of PBG materials is presented to provide context for this research.
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.
We investigate spontaneous emissions from a three-level atom embedded in realistic anisotropic photonic band gap structures, where the band edge is assumed to be midway between the two upper levels of the atom. In the anisotropic model, the atom-field interaction is reduced to be smaller than that in the ideal isotropic model, which results in the reduction of suppression and oscillation effects in the spontaneous emission; however, we demonstrate that the suppression is strongly enhanced even in the anisotropic model by expanding the detuning between the upper level and the band edge, while the oscillation is greatly reduced. This demonstration is carried out by calculating the atomic state vector for a wide range of the detuning. Using the state vector, the atom-field interaction is studied for two field parts, a localized part and a propagating part, separately. For the localized part, the atom-field interaction is enhanced by expanding the detuning, in spite of the anisotropy, which leads to a strong suppression effect, while for the propagating part, the atom-field interaction is reduced, which leads to a weak oscillation effect. This result is relevant to strong memory effects with high-speed switching, which are attractive properties for a qubit.
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.
We demonstrate the coherent control of spontaneous emission for a three-level atom located within a photonic band-gap structure with one resonant frequency near the edge of the photonic band gap. Spontaneous emission from the three-level atom can be totally suppressed or strongly enhanced depending on the relative phase between the steady-state control laser coupling the two upper levels and the pump laser pulse used to create an excited state of the atom in the form of a coherent superposition of the two upper levels. Unlike the free-space case, the steady-state inversion of the atomic system is strongly dependent on the externally prescribed initial conditions. This nonzero steady-state population is achieved by virtue of the localization of light in the vicinity of the emitting atom. It is robust to decoherence effects provided that the Rabi frequency of the control laser field-atom interaction exceeds the rate of dephasing interactions. As a result, such a system may be relevant for a single-atom, phase-sensitive, optical memory device on the atomic scale. The protected electric dipole within the photonic band gap provides a basis for a qubit to encode information for quantum computations.
The spontaneous emission from a V-type three-level atom embedded in an isotropic photonic crystal with dynamic photonic band edge is studied. We consider the situation where the atom interacts with all possible radiation modes, and calculate numerically the evolution of atomic population without using Markov approximation. The calculation method can be used in related researches. In the present paper, we mainly discuss the effects of modulation parameters and the quantum interference on spontaneous emission when the band edge is modulated with step function or triangle function. We hope that the results can contribute to the applications in the dynamic photonic crystal environment in controlling the spontaneous emission via the quantum interference. The results show that in the step-modulated situation, the number of the photon-atom bound dressed states after the modulation has happened depends on atomic transition frequencies and the band edge frequency at that time, and is identical to the one in the unmodulated situation with the same parameters. The long-time evolution of the atomic population is affected by the time when the modulation happens. Depending on the system initial state, after the modulation has happened, the quantum interference can weaken the probability amplitude components corresponding to the photon-atom bound dressed states, and cause the upper-level population to decay quickly from a great value to a value near zero; or on the contrary, it can strengthen the bound dressed states, and make the upper levels retain a high population. In the modulated situation with trigonometric functions, after long enough time, the total upper-level population presents a decaying quasi-periodic oscillation behaviour. And the evolution of the total upper-level population tends to synchronize with the modulation, so the frequency of the quasi-periodic oscillation is approximately equal to the modulation frequency. But, the quantum interference can destroy the synchronization under some conditions. The decay rate of the total upper-level population is affected by the modulation frequency, and also by the initial state of the system and the angle between two dipole moment because of the quantum interference.