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arXiv:1504.06803

Cavity quantum electrodynamics with three-dimensional photonic bandgap crystals

Willem L. Vos, Léon A. Woldering

Published Apr 26, 2015• 7 citations
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Abstract

This paper gives an overview of recent work on three-dimensional (3D) photonic crystals with a "full and complete" 3D photonic band gap. We review five main aspects: 1) spontaneous emission inhibition, 2) spatial localization of light within a tiny nanoscale volume (aka "a nanobox for light"), 3) the introduction of a gain medium leading to thresholdless lasers, 4) breaking of the weak-coupling approximation of cavity QED, both in the frequency and in the time-domain, 5) decoherence, in particular the shielding of vacuum fluctuations by a 3D photonic bandgap. In addition, we list and evaluate all known photonic crystal structures with a demonstrated 3D band gap.

DOI
10.48550/arxiv.1504.06803

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This paper gives an overview of recent work on three-dimensional (3D) photonic crystals with a "full and complete" 3D photonic band gap. We review five main aspects: 1) spontaneous emission inhibition, 2) spatial localization of light within a tiny nanoscale volume (aka "a nanobox for light"), 3) the introduction of a gain medium leading to thresholdless lasers, 4) breaking of the weak-coupling approximation of cavity QED, both in the frequency and in the time-domain, 5) decoherence, in particular the shielding of vacuum fluctuations by a 3D photonic bandgap. In addition, we list and evaluate all known photonic crystal structures with a demonstrated 3D band gap.
Introduction The propagation of light in periodically ordered photonic crystals bears a strong analogy to the wave propagation of conduction electrons in atomic crystals. The development of Bloch modes and dispersion are determined by the interference of waves that are diffracted by lattice planes [19, 52]. The periodicity commensurate with the wavelength (a ≈ λ/2) gives rise to Bragg diffraction that is associated with frequency windows for which waves are forbidden to propagate in a particular direction. Such stop gaps have long been known to arise for electromagnetic waves, notably for X-rays in atomic crystals [214]. At optical frequencies, the stop gaps that occur in one-dimensional periodic structures, known as Bragg stacks, are widely used in optical laboratories as broadband highly reflecting dielectric mirrors [556]. The distinguishing feature of three-dimensional (3D) photonic crystals is that a common stop gap can be achieved for all directions and for all polarizations simultaneously: the widely pursued 3D photonic band gap . Historically a 3D band gap has never been considered for the propagation of X-rays in periodic media, since a gap requires a high refractive index contrast of order unity, whereas the refractive index varies by less then 10 −4 in the X-ray range. At frequencies inside the 3D photonic band gap, the density of optical states (DOS) completely vanishes. As the density of states can also be interpreted as the density of vacuum fluctuations [334], a 3D band gap thus serves as an effective shield to these fluctuations. The total absence of optical modes in a photonic band gap has implications beyond classical optics that break the analogy between the behavior of light in photonic crystals and the behavior of electrons in atomic crystals. 3D photonic band gap crystals play an important role in cavity quantum electrodynamics (cQED) [64, 549], where they offer at least five prospects for new physics. Firstly, probably the most eagerly pursued phenomenon is the complete inhibition of spontaneous emission: an excited quantum emitter – such as an atom, molecule, or quantum dot – embedded in a crystal with its transition frequency tuned to within the 3D band gap remains forever excited since it cannot decay to the ground state by emitting a photon.
In certain three-dimensional crystals, a frequency range exist for all polarizations for which light is not allowed to propagate in any direction, called the 3D photonic band gap: a frequency range where the density of vacuum fluctuations vanishes in an ideal infinitely large and perfect system. The complete absence of vacuum fluctuations can be interpreted as zero density of states and vice versa. It is well known that the characteristics of spontaneous emission of light depend strongly on the environment of the light source. According to quantum electrodynamics, the emission rate of a two-level quantum emitter as described by Fermi’s “golden rule”, is factorized into one part describing the emitter’s properties and a second part describing the effect of the environment on the light field via the local density of optical state (LDOS). The radiative decay rate of the emitters is proportional to the LDOS. The absence of vacuum fluctuations or the zero LDOS leads to complete inhibition of the spontaneous emission. While many great advances towards a band gap have been made, no complete inhibition has ever been observed to date. In this thesis, for the first time, we have studied the spontaneous emission of PbS quantum dots inside real 3D photonic band gap crystal as a function of frequency throughout the band gap. In our experiments we have observed an inhibited emission of 18x in the band gap. The time-resolved decay curves show evidence for intriguing finite-size effects whereby vacuum fluctuations tunnel into the band gap. Unfortunately, most theories assume crystals of infinite extent, where the inhibition in the band gap is also infinite, as vacuum fluctuations don’t exist inside the crystal. Here we propose an original model to calculate the LDOS in the band gap of a finite photonic crystal based on tunneling of the vacuum fluctuations into the crystal. We validate our model for the one-dimensional situation with a direct comparison to the analytic calculations which exist for one-dimensional photonic crystals. For three dimensional crystals, the theory shows an excellent match to the experiment.
Photonic crystals are optical materials that have an intricate structure with length scales of the order of the wavelength of light. The flow of photons is controlled in a manner analogous to how electrons propagate through semiconductor crystals, i.e., by Bragg diffraction and the formation of band structures. If the interaction between light and matter is made strong, multiple diffraction and multiple scattering effects dominate. A main research goal is the realization of a "photonic band gap", that is, a frequency range for which no light can propagate in a crystal in any direction, which causes radical modifications of the density of radiative states. Important consequences of photonic band gaps are the complete control over spontaneous and stimulated emission of light, as well as over the propagation of light, in particular photon localization. This opens up the possibility to achieve a "cage for light": trap photons and do with them whatever one chooses. In this talk we will also review means for making such structures, and recent experimental advances in optical experiments.
In the first part of this introductory review we outline the developments in photonic band gap materials from the physics of photonic band gap formation to the fabrication and potential applications of photonic crystals. We briefly describe the analogies between electron and photon localization, present a simple model of a band structure calculation and describe some of the techniques used for fabricating photonic crystals. Also some applications in the field of photonics and optical circuitry are briefly presented. In the second part, we discuss the consequences for the interaction between an atom and the light field when the former is embedded in photonic crystals of a specific type, exhibiting a specific form of a gap in the density of states. We first briefly review the standard treatment (Weisskopf – Wigner theory) in describing the dynamics of spontaneous emission in free space from first principles, and then proceed by explaining the alterations needed to properly treat the case of a two-level atom embedded in a photonic band gap material.
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.
A theoretical model that takes into full account the vectorial nature of electromagnetic (EM) fields is developed to investigate various types of quantum electrodynamics (QED) in a three-dimensional (3D) photonic crystal. The EM fields are quantized via solving the eigenproblem of photonic crystals with the use of a plane-wave expansion method. It is found that the light-atom coupling coefficients strongly depend on the Bloch states, and the key physical function concerning the atomic QED is the photon local density of states (LDOS) instead of the DOS. Both the DOS and the LDOS vary slowly near the band edge and no singularity takes place. This vectorial model show that the spontaneous emission from a two-level atom can be solved via the conventional Weisskopf-Wigner approximation theory, which exhibits a pure exponential decay behavior with a rate proportional to the LDOS. The quantum interference effect from a three-level atom is greatly weakened at the band edge, instead of enhanced. Neglecting this vectorial nature will lead to many discrepancies in understanding quantum optics in 3D photonic crystals and other inhomogeneous media.