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Communications in Theoretical Physics

Optical Exciton Molecules in Three-Dimensional Kerr-Nonlinear Photonic Crystals

Ze Cheng

Published Dec 15, 1998• 1 citation
Publisher

Abstract

Three-dimensional Kerr-nonlinear photonic crystals support a new kind of quantum gap solitons. Such 3D-quantum gap solitons are described by an approximate Bethe-ansatz solution of the quantized nonlinear-field Hamiltonian under the effective mass approximation. Their main manifestations are predicted: (i) a series of discrete transmission peaks in the band gap, which progresses linearly with the diphoton number; (ii) Bessel-function dependence of the intensity correlation function on the detector separation.

DOI
10.1088/0253-6102/30/4/517

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Three-dimensional Kerr-nonlinear photonic crystals support a new kind of quantum gap solitons. Such 3D-quantum gap solitons are described by an approximate Bethe-ansatz solution of the quantized nonlinear-field Hamiltonian under the effective mass approximation. Their main manifestations are predicted: (i) a series of discrete transmission peaks in the band gap, which progresses linearly with the diphoton number; (ii) Bessel-function dependence of the intensity correlation function on the detector separation.
We establish a quantum theory of gap solitons in a one-dimensional periodic dielectric structure [distributed Bragg reflector (DBR)] of a Kerr nonlinearity. The electromagnetic field in the structure is quantized within the two-band approximation. The basic idea for this quantization is that incident photons with frequencies in the band gap of the DBR are scattered into the upper ``conduction'' and lower ``valence'' bands by the nonlinearity. The effective Hamiltonian of quantum gap solitons is derived within the effective-mass regime. The eigenstates of the Hamiltonian are constructed exactly by Bethe's ansatz method. We find that in certain band gaps of the DBR these eigenstates can be bound, consisting of one or more photon pairs from the conduction and valence bands. These bound multiphoton states are optical analogs of exciton molecules [optical multiexcitons (OMEs)]. Their existence should be manifested by the discrete spectrum of band-gap transmission as a function of the transmitted photon number and by the multiexponential falloff of intensity-intensity correlations on a 0.1-mm scale. Quantum gap solitons are shown to arise as superpositions of OMEs whenever the nonlinear binding exceeds phonon scattering. \textcopyright{} 1996 The American Physical Society.
We find a Bethe-ansatz solution for pairwise interacting quanta within the effective-mass regime of band-gap propagation in nonlinear Bragg reflectors. Our theory predicts a new kind of collective excitation of the electromagnetic field dressed by such media, namely, optical multiexciton (OME) complexes (or condensates), which are quantum states associated with gap solitary waves. Their existence should be manifested by the discrete spectrum of band-gap transmission as a function of the transmitted photon number and by the multiexponential falloff of intensity-intensity correlations on a 0.1 mm scale. OMEs should have advantageous stability properties.
Nonlinear photonic time crystals (PTCs) can support solitons inside momentum k gaps, where the amplification of k gap modes is saturated by Kerr nonlinearity, forming spatially homogeneous but temporally localized excitations. Yet their quantum nature remains unclear. Here we quantize nonlinear k gap dynamics of PTCs and show that k gap solitons are represented by biphoton Fock ladder states. K gap amplification drives two-mode squeezing of the biphoton, while Kerr nonlinearity generates an anharmonic potential along the biphoton Fock ladder that balances this squeezing process, creating a finite biphoton number turning point and giving rise to quantum collapse and revival dynamics and nonclassical phase space interference. We further analyze how photon loss and dephasing reshape the biphoton statistics of quantized k gap solitons. Our results establish a biphoton Fock space description of k gap soliton quantization and provide a framework for studying quantum nonlinear excitations and entangled light generation in photonic time crystals.
We survey basic quantum optical processes that undergo modifications in photonic crystals doped with resonant atoms: (a) Solitons and multi-dimensional localized 'bullets' propagating at photonic band gap frequencies. These novel entities differ substantially from solitons in Kerr-nonlinear photonic crystals. (b) Giant photon-photon cross-coupling that can give rise to fully entangled two-photon states. We conclude that doped photonic crystals have the capacity to form efficient networks for high-fidelity classical and quantum optical communications.
Optical phenomena in materials with a space modulation of dielectric constant at distances close to the light wavelengths (so called photonic band-gap structures or photonic crystals) are of a great interest now because of the existence of band gaps in their photonic band structure (Bykov, 1972; Yablonovitch, 1987; John, 1987). These band gaps represent frequency regions where electromagnetic waves are forbidden, irrespective of the spatial propagation directions. Inside the band gaps, the photon density of states is equal to zero and so the emission of light sources embedded in these crystals should be inhibited in these spectral regions (Vats et al., 2002). Since the time the effect is predicted, many experiments have been devoted to studies of spontaneous emission of molecules embedded in photonic crystals (Gaponenko et al., 1999; Gorelik, 2007). Typical structures of photonic crystals and calculations of corresponding photonic band structures are presented in a book by Prof. Joannopoulos (Joannopoulos et al., 2008). Besides the emission inhibition effect, a number of new optical phenomena in 3D photonic crystals, interesting from the applied point of view, are under intensive study now. The main research directions are the following:  Effects of light localisation (John, 1987; Kaliteevskii et al., 2005; Vignolini et al., 2008).  Radiation of photonic crystals filled with organic (rhodamine 6G, 1,8-naphthoylene1’,2’-benzimidazole, stilbene) and inorganic (ZnO, ZnS, rare-earth ions Eu3+, Tb3+, Er3+) luminophores near by the edges of photonic band-gap (Gaponenko et al., 1999; Aliev et al., 2002; Emel’chenko et al., 2005; Li et al., 2007).  Radiation of CdTe, CdSe/ZnS quantum dots in photonic crystal volume (Gruzintsev et al., 2009; Ambrozevich et al., 2009).  Quantum optics phenomena in nano-structured materials based on photonic crystals and nonlinear optical substances (Lin & Vuckovic, 2010).  Effects of the radiation field amplification in photonic crystals (Lin & Vuckovic, 2010).  Increase of solar cells efficiency with the use of photonic crystals (Florescu et al., 2007).
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.