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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Emission decay and lamb shift in photonic crystals

Xuehua Wang, Ben–Yuan Gu, Yuri S. Kivshar

Published Feb 9, 2005• 1 citation
Publisher

Abstract

Motivated by the significant controversy between the two dispersion models and Weisskopf-Wigner approximation (WWA), for the first time to our knowledge, we introduce the position-dependent photon-atom interaction into the Green function method of the evolution operator and develop a universal theoretical treatment on spontaneous emission of atoms in photonic crystals (PCs). A position-sensitive generalized Lorentzian formulism (non-Lorentzian shape) for the decay of an excited atom in PCs is derived, an exact numerical method for calculating the local coupling strength, proportional to the photonic local density of state (LDOS), is presented. For weak interaction PCs with pseudo gaps, the generalized Lorentzian formulism may be reduced to the famous Lorentzian spectrum. In this case, we introduced a lifetime distribution function for an assembly of atoms and found that the lifetime distribution strongly depend on the spread configuration of these atoms in space, which clarifies successfully the tremendous discrepancy between different experiments. For the PCs with large full gaps, we found that the atomic position can fundamentally change the decay behavior of an excited atom: in strong interaction positions, the atomic decay is non-classical or exhibits an envelope-damped Rabi oscillation, while in weak interaction positions the WWA is valid. Recently, we also predicted giant Lamb shifts for hydrogen atoms in PCs, and revealed that in inhomogeneous electromagnetic environment, the dominant contribution to the Lamb shift comes from real photon emission, while the contribution from emission and reabsorption of virtual photon is negligible, in vast contrast with the case of free space where the virtual photon processes play a key role.

DOI
10.1117/12.574320

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Motivated by the significant controversy between the two dispersion models and Weisskopf-Wigner approximation (WWA), for the first time to our knowledge, we introduce the position-dependent photon-atom interaction into the Green function method of the evolution operator and develop a universal theoretical treatment on spontaneous emission of atoms in photonic crystals (PCs). A position-sensitive generalized Lorentzian formulism (non-Lorentzian shape) for the decay of an excited atom in PCs is derived, an exact numerical method for calculating the local coupling strength, proportional to the photonic local density of state (LDOS), is presented. For weak interaction PCs with pseudo gaps, the generalized Lorentzian formulism may be reduced to the famous Lorentzian spectrum. In this case, we introduced a lifetime distribution function for an assembly of atoms and found that the lifetime distribution strongly depend on the spread configuration of these atoms in space, which clarifies successfully the tremendous discrepancy between different experiments. For the PCs with large full gaps, we found that the atomic position can fundamentally change the decay behavior of an excited atom: in strong interaction positions, the atomic decay is non-classical or exhibits an envelope-damped Rabi oscillation, while in weak interaction positions the WWA is valid. Recently, we also predicted giant Lamb shifts for hydrogen atoms in PCs, and revealed that in inhomogeneous electromagnetic environment, the dominant contribution to the Lamb shift comes from real photon emission, while the contribution from emission and reabsorption of virtual photon is negligible, in vast contrast with the case of free space where the virtual photon processes play a key role.
Being motivated by the controversial results based on two dispersion models and Weisskopf–Wigner approximation (WWA), we introduce, for the first time to our knowledge, the position-dependent photon-atom interaction into the Green function method of the evolution operator and develop a universal theoretical approach to study spontaneous emission of atoms in photonic crystals (PCs). A position-sensitive generalized Lorentzian formalism (non-Lorentzian shape) for the decay of an excited atom in PCs is derived, and an exact numerical method for calculating the local coupling strength, proportional to the photonic local density of state (LDOS), is presented. For weak interaction PCs with pseudo gaps, the generalized Lorentzian formalism may be reduced to the famous Lorentzian spectrum. In this case, we introduce a lifetime distribution function for an assembly of atoms and find that it depends strongly on the atomic configuration in space, which clarifies successfully the tremendous discrepancy between different experiments. For the PCs with large full gaps, we find that the atomic position can fundamentally change the decay behavior of an excited atom: in strong interaction positions, the atomic decay is non-classical or exhibits an envelope-damped Rabi oscillation, while in weak interaction positions the WWA is valid. Recently, we also predicted giant Lamb shifts for hydrogen atoms in PCs, and revealed that in inhomogeneous electromagnetic environment, the dominant contribution to the Lamb shift comes from real photon emission, while the contribution from emission and reabsorption of virtual photon is negligible, in vast contrast with the case of free space where the virtual photon processes play a key role.
Quantum electrodynamics of atom spontaneous emission from a three-dimensional photonic crystal is studied in a full vectorial framework. The electromagnetic fields are quantized via solving the eigenproblem of photonic crystals with use of a plane-wave expansion method. It is found that the photon density of states and local density of states (LDOS) with a full band gap vary slowly near the edge of band gap, in significant contrast to the singular character predicted by the previous isotropic model. Therefore, the spontaneous emission can be solved by conventional Weisskopf-Wigner approximate theory, which yields a pure exponentially decaying behavior with a rate proportional to the LDOS.
Decay kinetic properties of a two-level atom near the band edges of photonic crystals (PCs) with absolute gaps are studied based on the Green's function expression for the evolution operator. The local coupling strength between the photons and an atom is evaluated by an exact numerical method. It is found that the decay behavior of an excited atom can be fundamentally changed by the variation of the atomic position: Weisskopf-Wigner and non-Weisskopf-Wigner decay phenomena occur at different atomic positions in the PCs as a result of a significant difference in the local coupling strength. Our finding implies that it is possible to engineer the luminescence spectrum by controlling the atomic position.
We obtain a general result for the Lamb shift of excited states of multilevel atoms in inhomogeneous electromagnetic structures and apply it to study atomic hydrogen in inverse-opal photonic crystals. We find that the photonic-crystal environment can lead to very large values of the Lamb shift, as compared to the case of vacuum. We also suggest that the position-dependent Lamb shift should extend from a single level to a miniband for an assembly of atoms with random distribution in space, similar to the velocity-dependent Doppler effect in atomic/molecular gases.
We investigate spontaneous emission from a two-level atom embedded in a photonic crystals with three-dimensional dispersion relation, where the dependence of wave vectors on direction is taken into account. Due to the difference of the density of states of electromagnetic modes between a three-dimensional and a one-dimensional photonic crystal (lack of singularity), the dynamics of the population in the excited state and the emitted field are quite different from those of a one-dimensional photonic crystal. There is no quasi-oscillation for the population evolution. The localized field and the propagating field cannot coexist. When the upper level moves from the gap to the band, the emitted field changes from a localized field to a diffusion field and then to a propagating field. The energy of the diffusion and propagating fields propagates out incoherently and coherently, respectively. An analytical expression for the Lamb shift is obtained. The Lamb shift for the current case is smaller than that in an ordinary vacuum or in one-dimensional photonic crystals.
Spontaneous decay behaviors from an assembly of atoms (or molecules) in 3D photonic crystals (PC's) with pseudogaps are investigated. Theoretically, a lifetime distribution function for these atoms or molecules is defined to reveal decay kinetics. Our calculations show that quite wide or narrow lifetime distributions can occur for different spread configurations of the atoms (or molecules). The pure PC effect may lead to coexistence of both accelerated and inhibited decay processes. These results provide theoretical clarification for substantial discrepancies in the recent reported experiments.