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Photonic band control in a quantum metamaterial

Didier Felbacq, Emmanuel Rousseau

Published Aug 24, 2017• 0 citations
Publisher

Abstract

In the present work, we present a metamaterial made of a periodic collection of dielectric resonators in which a quantum oscillator (denoted QO in the following) is inserted. The geometry at stake here is much more complicated than the textbook 1D cavity usually dealt with theoretically in quantum optics. We do provide a treatment essentially based on the scattering matrix non-perturbative approach, in order to investigate the various effects that could be expected to exist in such structures.The theoretical methods used are the Feshbach projection method associated with multiple scattering theory. First, the phenomenology for one scatterer with a QO inserted is presented, then the collective behavior of a finite periodic set of such scatterers is investigated and it is shown that it is possible to open and close a conduction band according to the state of the oscillators when the inserted quantum oscillators are put in the inversion regime by means of a pump field. They add gain to the system, allowing to reach the amplification regime in the vicinity of the Mie resonances of the dielectric resonators. When the transition frequency is situated at the photonic band gap edge, it creates switchable conducting modes within the bandgap.

DOI
10.1117/12.2272921

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In the present work, we present a metamaterial made of a periodic collection of dielectric resonators in which a quantum oscillator (denoted QO in the following) is inserted. The geometry at stake here is much more complicated than the textbook 1D cavity usually dealt with theoretically in quantum optics. We do provide a treatment essentially based on the scattering matrix non-perturbative approach, in order to investigate the various effects that could be expected to exist in such structures.The theoretical methods used are the Feshbach projection method associated with multiple scattering theory. First, the phenomenology for one scatterer with a QO inserted is presented, then the collective behavior of a finite periodic set of such scatterers is investigated and it is shown that it is possible to open and close a conduction band according to the state of the oscillators when the inserted quantum oscillators are put in the inversion regime by means of a pump field. They add gain to the system, allowing to reach the amplification regime in the vicinity of the Mie resonances of the dielectric resonators. When the transition frequency is situated at the photonic band gap edge, it creates switchable conducting modes within the bandgap.
Abstract Metamaterials made of periodic collections of dielectric nanorods are considered theoretically. When quantum resonators are embedded within the nanorods, one obtains a quantum metamaterial, whose electromagnetic properties depend upon the state of the quantum resonators. The theoretical model predicts that when the resonators are pumped and reach the inversion regime, the quantum metamaterial exhibits an all‐optical switchable conduction band. The phenomenon can be described by considering the pole stucture of the scattering matrix of the metamaterial.
Abstract Embedding tunable quantum emitters in a photonic bandgap structure enables control of dissipative and dispersive interactions between emitters and their photonic bath. Operation in the transmission band, outside the gap, allows for studying waveguide quantum electrodynamics in the slow-light regime. Alternatively, tuning the emitter into the bandgap results in finite-range emitter–emitter interactions via bound photonic states. Here, we couple a transmon qubit to a superconducting metamaterial with a deep sub-wavelength lattice constant (λ/60). The metamaterial is formed by periodically loading a transmission line with compact, low-loss, low-disorder lumped-element microwave resonators. Tuning the qubit frequency in the vicinity of a band-edge with a group index of ng = 450, we observe an anomalous Lamb shift of −28 MHz accompanied by a 24-fold enhancement in the qubit lifetime. In addition, we demonstrate selective enhancement and inhibition of spontaneous emission of different transmon transitions, which provide simultaneous access to short-lived radiatively damped and long-lived metastable qubit states.
We illustrate a mechanism based on coherent optical nonlinearities to realize optically tunable photonic crystals built from media supporting electromagnetically induced transparency. These exhibit specific periodic patterns where a light probe can experience a fully developed photonic band-gap with negligible absorption. An analytical method based on a two-mode approximation is developed to study the optical response of such a periodically modulated medium driven into a regime of standing-wave electromagnetically induced transparency. A comparison with a transfer matrix approach and with techniques based on the coupled Maxwell–Liouville equations shows that our method is very accurate to describe the optical properties of such photonic metamaterials in the frequency region of interest. Nearly perfect reflectivities may be attained for ultracold 87Rb atoms samples which are seen to reflect with little loss and deformation light pulses whose frequency components are contained within the gap. When the same mechanism is implemented for inhomogeneously broadened optical transitions of nitrogen-vacancy centers in diamond, which is more amenable to device applications, fully developed photonic band-gap structures with negligible absorption, reflectivities very close to unity and sufficiently large bandwidths may also be attained. We further examine more flexible schemes based on four level systems under double driving conditions showing more accurate and efficient coherent optical control of the photonic band-gap. The remarkable experimental simplicity of the schemes that we presented is set to ease quantum nonlinear optics applications.
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.
This paper deals with atomic systems coupled to a structured reservoir of quantum EM field modes, with particular relevance to atoms interacting with the field in photonic band gap materials. The case of high Q cavities has been treated elsewhere using Fano diagonalization based on a quasimode approach, showing that the cavity quasimodes are responsible for pseudomodes introduced to treat non-Markovian behaviour. The paper considers a simple model of a photonic band gap case, where the spatially dependent permittivity consists of a constant term plus a small spatially periodic term that leads to a narrow band gap in the spectrum of mode frequencies. Most treatments of photonic band gap materials are based on the true modes, obtained numerically by solving the Helmholtz equation for the actual spatially periodic permittivity. Here the field modes are first treated in terms of a simpler quasimode approach, in which the quasimodes are plane waves associated with the constant permittivity term. Couplings between the quasimodes occur owing to the small periodic term in the permittivity, with selection rules for the coupled modes being related to the reciprocal lattice vectors. This produces a field Hamiltonian in quasimode form. A matrix diagonalization method may be applied to relate true mode annihilation operators to those for quasimodes. The atomic transitions are coupled to all the quasimodes, and the true mode atom-EM field coupling constants (one-photon Rabi frequencies) are related to those for the quasimodes and also expressions are obtained for the true mode density. The results for the one-photon Rabi frequencies differ from those assumed in other work. Expressions for atomic decay rates are obtained using the Fermi Golden rule, although these are valid only well away from the band gaps.
We propose one-dimensional bright and dark optical lattices where Bragg scattering occurs, respectively, via periodic generation of gain without inversion and electromagnetically induced absorption in the absence of any refractive index perturbation. The absorption and gain processes are caused by quantum interference in the conduction intersubband transitions of a corrugated nnn-doped quantum well structure interacting with an intense infrared laser. The quantum well structure is embedded between two optical confinement layers, forming a periodic structure with a Bragg wavelength similar to that of the intersubband transition between the second and third conduction subbands of the quantum well. We show that in the absence of the intense infrared laser this transition is transparent, and the periodic structure forms a passive one-dimensional photonic gap generated by the background refractive index contrast of the corrugated region. In the presence of this laser, the intersubband transition is influenced by the quantum coherence effects, causing dramatic changes in the photonic band gap. We show that when one adjusts the intensity and wavelength of such a laser properly these effects can (i) destroy the one-dimensional photonic band gap, and (ii) form periodic regions of gain without inversion or electromagnetically induced loss, while the index perturbation is completely canceled out. Such bright and dark optical lattices can scatter a light at Bragg wavelength when its frequency is the same or close to that of the intersubband transition.