In 1998 Vats and John analyzed the collective spontaneous emission of N two-level atoms near a photonic band edge using effective-mass (quadratic) dispersion models, a mean-field factorization for high excitation, a stochastic Stark-shift model of dephasing, and a classical stochastic reservoir simulation. They left open the program needed to turn this into a quantitative theory of a band-edge laser: memory kernels derived from full realistic (double-valued, both-band-edge, crystal-symmetry-dependent) dispersions of three-dimensional photonic crystals; a microscopic, non-Markovian treatment of homogeneous broadening (collisions, phonons) consistent with the reservoir memory kernel; a quantitative theory of pumping energy into and out of the localized atom-photon gap-dressed states; and a non-perturbative master equation (or equivalent) that can handle the van Hove singularity of the photonic density of states together with externally driven and pumped atoms. Later literature has closed parts of each component - realistic Bloch-mode densities of states for single emitters, a semiclassical pumped lasing theory, a microscopic single-emitter phonon-dephasing theory, and exact single-emitter band-edge decay laws - but no known work combines pumping, microscopic dephasing, realistic three-dimensional dispersion, and non-perturbative non-Markovian multi-emitter dynamics into the requested complete theory, whose predictions (pumping thresholds, steady-state inversion and polarization, spectra, and photon statistics of a band-edge laser) remain unestablished.

Background

A photonic band gap (PBG) material is a periodic dielectric whose Bragg scattering of light creates frequency intervals in which electromagnetic propagation is forbidden. The band edge - the cutoff frequency ωc\omega_c of such a gap - is the physically singular place of this problem: the photonic density of states there has a van Hove singularity, a non-analytic behavior inherited from the extremum of the dispersion relation ωk\omega_k. Vats and John worked with two simplified models. In the isotropic effective-mass model, the dispersion near the edge is expanded quadratically, ωk=ωc+A(∣k∣−k0)2\omega_k=\omega_c+A(|k|-k_0)^2, giving a density of states ρ(ω)∝(ω−ωc)1/2\rho(\omega)\propto(\omega-\omega_c)^{1/2}; in the anisotropic model, which associates the gap with one Bragg point in k-space, the density of states diverges as ρ(ω)∝(ω−ωc)−1/2\rho(\omega)\propto(\omega-\omega_c)^{-1/2}. An atom with resonance near ωc\omega_c then couples to a reservoir whose correlation (memory) function - the kernel G(t−t′)G(t-t') of the atom-field interaction - decays only as a power law, (t−t′)−3/2(t-t')^{-3/2} or (t−t′)−1/2(t-t')^{-1/2}, instead of being short-lived. The usual Born-Markov approximation of quantum optics, which assumes the reservoir loses memory much faster than the system evolves and yields a Markovian master equation, therefore fails. The resulting non-Markovian dynamics produce the phenomena that motivate this problem: atoms whose resonance lies inside the gap form localized atom-photon bound states (photon localization), the excited-state population exhibits fractional localization with power-law decay tails, and the cooperative emission of N atoms (superradiance, the collective enhancement of radiative decay identified by Dicke) is drastically altered, as shown earlier by John and Quang. Vats and John solved the low-excitation dynamics exactly in a point model, treated high excitation by a mean-field factorization of the atom-atom correlations (the analogue of the semiclassical Maxwell-Bloch equations of standard laser theory, which couple the medium polarization and inversion to a classical field), and found a laser-like steady state - fractional inversion with a macroscopic polarization and a nonzero local electric field - sustained by photon localization rather than by a cavity mode. Dephasing of the atomic dipoles by collisions or phonons was represented only by random Stark shifts of the atomic frequency (which detune each atom from the band edge, where the dynamics is extremely detuning-sensitive), simulated stochastically; and the steady-state statistics of a pumped band-edge laser were left beyond reach, because the non-perturbative techniques then available for going beyond the Born-Markov approximations could not treat the van Hove singularity nor externally driven atoms. For pumping energy into and extracting it from the localized gap-dressed states, the paper raised, without establishing a kinetic framework, two possible mechanisms: an auxiliary atomic transition lying outside the gap, and intense ultrashort pulses that transiently modify the nonlinear dielectric so that light can propagate into the gap. A homogeneous-broadening process treated microscopically means one derived from a Hamiltonian interaction (e.g., phonon coupling) rather than inserted as a phenomenological exponential decay constant; a van Hove singularity is the divergence or non-analyticity of the density of states at the band edge; a memory kernel is the reservoir autocorrelation function appearing in the integro-differential (non-Markovian) equations of motion for the atoms.

Problem Statement

Construct the complete quantitative theory of collectively radiating atoms near a photonic band edge in a realistic photonic band-gap material, including pumping and microscopic dephasing - the quantum theory of the band-edge laser that Vats and John (Phys. Rev. A 58, 4168 (1998)) formulated only within effective-mass dispersion models, a mean-field high-excitation treatment, a stochastic Stark-shift model of dephasing, and a stochastic surrogate for the quantum reservoir. The problem comprises four scientifically linked requests made by that paper, all of which are jointly required for a complete solution: (1) replace the effective-mass memory kernels by kernels derived from the full photonic dispersion - the double-valued model of Eq. (A8) capturing both band edges for arbitrary gap width and atomic frequency, and, beyond that, the crystal-symmetry-dependent band-edge wavevectors and curvature of realistic three-dimensional photonic crystals (the Bloch-mode dispersions of refs. 4-5) - and work out their quantitative consequences for photon localization, decay, and collective dynamics as functions of detuning from the band edge; (2) include homogeneous broadening (atom-atom collisions and phonon interactions) microscopically, in a form consistent with the non-Markovian reservoir memory kernel rather than as a phenomenological decay constant or stochastic Stark-shift ansatz, sufficient to determine pumping thresholds and steady-state behavior of a band-edge laser; (3) give a quantitative theory of how energy is pumped into and extracted from the localized atom-photon gap-dressed states that carry the fractional steady-state inversion, including the relevant kinetics and rates; and (4) formulate a non-perturbative master equation (or equivalent) for driven, externally pumped atoms that can treat the van Hove singularity of the band-edge density of states, going beyond the Born and Markov approximations, and derive from it the steady state, emission spectra, and photon statistics of a band-edge laser. The theory should reduce correctly to the known limits (free-space Markovian dynamics, the exact single-emitter band-edge decay laws, the semiclassical Maxwell-Bloch lasing theory) and yield quantitative predictions for the pumping threshold, fractional steady-state inversion and polarization, linewidth, and photon statistics of band-edge atom-field systems in realistic three-dimensional photonic crystals.

References

  1. Nipun Vats and Sajeev John, 'Non-Markovian quantum fluctuations and superradiance near a photonic band edge', Physical Review A 58, 4168 (1998). DOI: 10.1103/PhysRevA.58.4168; arXiv:quant-ph/9806038; https://arxiv.org/abs/quant-ph/9806038
  2. Sajeev John and Tran Quang, 'Localization of superradiance near a photonic band gap', Physical Review Letters 74, 3419 (1995). DOI: 10.1103/PhysRevLett.74.3419; https://doi.org/10.1103/PhysRevLett.74.3419
  3. Mesfin Woldeyohannes and Sajeev John, 'Coherent control of spontaneous emission near a photonic band edge: A qubit for quantum computation', Physical Review A 60, 5046 (1999). DOI: 10.1103/PhysRevA.60.5046; https://doi.org/10.1103/PhysRevA.60.5046
  4. Kurt Busch, Nipun Vats, Sajeev John, and Barry C. Sanders, 'Radiating dipoles in photonic crystals', Physical Review E 62, 4251 (2000). DOI: 10.1103/PhysRevE.62.4251; arXiv:physics/0006002; https://arxiv.org/abs/physics/0006002
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  13. Devashish Pandey and Martijn Wubs, 'Collective photon emission in solid state environments: Concatenating non-Markovian and Markovian dynamics', Physical Review Research 6, 033044 (2024). DOI: 10.1103/PhysRevResearch.6.033044; https://doi.org/10.1103/PhysRevResearch.6.033044
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