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arXiv:1011.3014 · Revista Mexicana de Física

Anomalous decay of an atom in structured band gap reservoirs

Filippo Giraldi, Francesco Petruccione

Published Nov 12, 2010• 0 citations
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Abstract

We analyze the spontaneous emission of a two-level atom interacting with a special class of structured reservoirs of field modes with band gap edge coinciding with the atomic transition frequency. The exact time evolution of the population of the excited level is evaluated analytically through series of Fox-$H$ functions. Over estimated long time scales, inverse power law relaxations emerge, with powers decreasing continuously to 2 according to the choice of the special reservoir. No trapping of the population of the excited level emerges. The same results are recovered in presence of $N-1$ atoms, each one in the ground state, described by the Dicke model. The power of the inverse power law decay results to be independent of $N$. A critical number $N_α^{(\star)}$ is evaluated, such that, for $N \gg N_α^{(\star)}$, the inverse power law decay vanishes.

DOI
10.48550/arxiv.1011.3014

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We analyze the spontaneous emission of a two-level atom interacting with a special class of structured reservoirs of field modes with band gap edge coinciding with the atomic transition frequency. The exact time evolution of the population of the excited level is evaluated analytically through series of Fox-HHH functions. Over estimated long time scales, inverse power law relaxations emerge, with powers decreasing continuously to 2 according to the choice of the special reservoir. No trapping of the population of the excited level emerges. The same results are recovered in presence of N−1N-1N−1 atoms, each one in the ground state, described by the Dicke model. The power of the inverse power law decay results to be independent of NNN. A critical number Nα(⋆)N_α^{(\star)}Nα(⋆)​ is evaluated, such that, for N≫Nα(⋆)N \gg N_α^{(\star)}N≫Nα(⋆)​, the inverse power law decay vanishes.
A formalism for studying spontaneous decay of an excited two-level atom in the presence of dispersing and absorbing dielectric bodies is developed. An integral equation, which is suitable for numerical solution, is derived for the atomic upper-state-probability amplitude. The emission pattern and the power spectrum of the emitted light are expressed in terms of the Green tensor of the dielectric-matter formation, including absorption and dispersion. The theory is applied to the spontaneous decay of an excited atom at the center of a three-layered spherical cavity, with the cavity wall being modeled by a band-gap dielectric of Lorentz type. Both weak and strong coupling are studied, the latter with a special emphasis on cases where the atomic transition is (i) in the normal-dispersion zone near the medium resonance, and (ii) in the anomalous-dispersion zone associated with the band gap. In a single-resonance approximation, conditions of the appearance of Rabi oscillations and closed solutions to the evolution of the atomic state population are derived, which are in good agreement with the exact numerical results.
The exact dynamics of a Jaynes-Cummings model for a qubit interacting with a continuous distribution of bosons, characterized by a special form of the spectral density, is evaluated analytically. The special reservoir is designed to induce anomalous decoherence, resulting in an inverse power-law relaxation, of power 3/23/23/2, over an evaluated long time scale. If compared to the exponential-like relaxation obtained from the original Jaynes-Cummings model for Lorentzian-type spectral density functions, decoherence is strongly suppressed. The special reservoir exhibits a band-edge frequency coinciding with the qubit transition frequency. Known theoretical models of photonic band-gap media suitable for the realization of the designed reservoir are proposed.
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.
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.
Anomalous decoherence in the Jaynes-Cummings model emerges for a certain class of bosonic reservoirs, described by spectral densities with a band gap edge frequency coinciding with the qubit transition frequency. The special reservoirs are piecewise similar to those usually adopted in quantum optics, i.e., sub-ohmic at low frequencies and inverse power laws at high frequencies. The exact dynamics of the qubit is described analytically through Fox H-functions. Over estimated long time scales, decoherence results in inverse power laws with powers decreasing continuously to unity, according to the particular choice of the special reservoir. The engineering reservoir approach is a new way of strongly delaying the decoherence process with possible applications to quantum technologies, due to the simple form of the designed reservoirs.
We extend recent work on the discretization of photonic continua. A two-level atom is coupled to a photonic bandgap reservoir, which we model by a large number of discrete modes. The atom is also coupled to a defect mode inside the gap, which acts as a photon source that can pump the atom. Propagating the wavefunction of the system, we obtain results pertaining to the problem of two and three photons in the reservoir, for various states of the defect field. For a wide range of parameters we find that part of the total initial excitation remains trapped to the atom and the defect mode in the long-time limit, while the atomic population may exhibit inversion. We further investigate the dynamics of an atomic Dicke system, with transition frequencies at the edge of a photonic bandgap.