Vats, John, and Busch [Phys. Rev. A 65, 043808 (2002)] developed a formalism for fluorescence from emitters embedded in photonic crystals, expressing emission spectra, non-Markovian emission dynamics, and the Lamb shift in terms of the orientationally averaged local density of photon modes (LDOS) at the emitter position, and illustrating it with model densities of states. The paper itself identifies four unresolved steps needed to connect this formalism to real materials and experiments: (i) material- and geometry-specific determination of the band-edge dispersion curvature (the tensor generalization of the curvature parameter AA, fixing the characteristic scale βA\beta_A) and hence of the true LDOS near band edges; (ii) a fully controlled, relativistic-field-theoretic treatment of the high-frequency part of the Lamb shift in an inhomogeneous dielectric, replacing the Bethe-style cutoff and mass renormalization used; (iii) quantitative incorporation of realistic-sample effects—defects, emitter broadening, local-field corrections, and defect-assisted emission channels; and (iv) extension to lossy, dispersive, or nonlinear host dielectrics and to many-emitter coherent processes. Later literature has advanced several of these points—accurate LDOS computations for inverse opals and finite three-dimensional photonic crystals, experimental observation of Lamb shifts in inverse opals, and the macroscopic-QED framework for absorbing and dispersive inhomogeneous media—but no later work is known to close the composite objective. The surviving open core is a fully quantitative, experiment-calibrated theory of band-edge fluorescence in realistic, disordered, lossy photonic crystals with material-specific dispersion.

Background

A photonic crystal is a dielectric material whose refractive index varies periodically on the scale of the optical wavelength, so that light propagates as Bloch modes—electromagnetic eigenmodes whose field amplitudes repeat with the lattice periodicity and can be strongly nonuniform within a unit cell. For a sufficiently strong index contrast the crystal can possess a photonic band gap, a frequency interval in which no propagating Bloch modes exist, or a pseudogap, a frequency range in which the density of modes is suppressed but not zero. Because spontaneous emission of an excited atom or other quantum emitter requires an available photon mode, its rate is proportional not to the total density of photon states but to the local density of states (LDOS)—the density of modes weighted by the field intensity and dipole orientation at the emitter's position. Emitters in the high- versus low-dielectric regions of the unit cell can therefore have very different decay rates and spectra, and near a band edge the LDOS varies rapidly with frequency, invalidating the Markov (memory-less) approximation that underlies standard exponential-decay theory.

Vats, John, and Busch (2002) made this physics explicit. Writing the atom–field coupling in the Bloch-mode basis, they showed that the ensemble-averaged fluorescence dynamics obey a non-Markovian integro-differential equation whose memory kernel Gf(τ)G_f(\tau) is determined by the orientationally averaged LDOS Nl(r,ω)N_l(\mathbf{r},\omega) at the emitter positions, that the stationary emission spectrum is a closed-form function of NlN_l and of the Lamb-shifted transition frequency, and that the Lamb shift (the small shift of an atomic transition frequency caused by coupling to vacuum field fluctuations) satisfies a renormalized principal-value integral over the LDOS, cut off at the electron Compton frequency (the frequency scale set by the electron rest-mass energy) in the style of Bethe's original nonrelativistic Lamb-shift calculation, together with an explicit mass renormalization. To illustrate the formalism they used model LDOS shapes: near a band edge the dispersion was approximated by a quadratic (effective-mass) expansion ωk=ωu+A∣k−k0∣2\omega_{\mathbf{k}}=\omega_u+A|\mathbf{k}-\mathbf{k}_0|^2 with curvature parameter AA (a scalar stand-in for the full curvature tensor), giving a square-root band-edge LDOS, a characteristic coupling frequency βA\beta_A that sets the scale of non-Markovian band-edge dynamics, strongly non-Lorentzian spectra, fractional localization (a partial atom–photon bound state) inside the gap, and band-edge Lamb shifts potentially an order of magnitude above the free-space value.

The formalism, however, was demonstrated on these idealized model densities of states, not on band structures of actual materials. The quantitative accuracy of band-edge predictions therefore hinges on knowing the true LDOS—equivalently the curvature tensor of the band-edge dispersion—for the specific crystal geometry, on a controlled Lamb-shift theory in a spatially varying dielectric, and on corrections from disorder, finite crystal size, emitter linewidth broadening, local-field effects, and dielectric absorption and dispersion. Later work provides the relevant toolbox: macroscopic quantum electrodynamics (macroscopic QED) quantizes the electromagnetic field in arbitrary dispersive and absorbing inhomogeneous media by coupling it to a continuum of fluctuating material polarization modes, yielding Green-function expressions for decay rates and level shifts that remain valid in the presence of loss; and accurate numerical methods (plane-wave expansion, finite-difference time-domain, transfer-matrix, and multiple-scattering calculations) can now compute LDOS in real photonic-crystal geometries.

Problem Statement

Extend the LDOS-based fluorescence formalism of Vats, John, and Busch (2002) from idealized model densities of photon states to realistic photonic crystals, to the level of quantitative, experiment-calibrated prediction. The common objective is a predictive theory in which every input is material- and geometry-specific, and the requested, scientifically linked results are: (i) determination of the true local density of photon modes near photonic band edges for actual crystals—equivalently the dispersion curvature tensor (the multicomponent generalization of the scalar curvature parameter AA in the quadratic band-edge expansion), which fixes the characteristic frequency scale βA\beta_A and hence the quantitative band-edge fluorescence spectra, non-Markovian emission dynamics, and fractional localization; (ii) a fully controlled theoretical treatment of the Lamb shift in an inhomogeneous dielectric—replacing the nonrelativistic Bethe-style cutoff and mass renormalization with a controlled (relativistic-field-theoretic or equivalent macroscopic-QED) account of the high-frequency contributions—so that band-edge and pseudogap level shifts can be predicted rather than estimated; (iii) quantitative incorporation, in concrete experimental geometries, of all of the following realistic-sample effects: point and extended defects, inhomogeneous and homogeneous emitter broadening, local-field corrections for emitters in or near the dielectric host, and scattering-induced defect-assisted emission channels, with their impact on the LDOS, spectra, and dynamics computed rather than discussed qualitatively; and (iv) extension of the formalism to hosts that are lossy, dispersive, or nonlinear (whichever the specified material presents) and, separately, to multiphoton and many-emitter coherent processes. All four results (i)–(iv) are required. Results (i)–(iii) must be instantiated together in a quantitative comparison with experiment for a specified real crystal: the computed fluorescence spectra, emission dynamics, and level shifts for specified emitter distributions must agree with the corresponding measured quantities within their stated experimental uncertainties, with no free parameter adjusted to fit the data being predicted. Result (iv) must also be delivered, but may be a separate, non-experiment-calibrated extension to lossy, dispersive, or nonlinear hosts and to multiphoton and many-emitter coherent processes.

References

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