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Physical Review A

Theory of one-dimensional quantum gap solitons

Ze Cheng, Gershon Kurizki

Published Oct 1, 1996• 17 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.54.3576

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We theoretically show that a self-induced transparency (SIT) soliton and a Bragg soliton can coexist in a nonlinear photonic band gap (PBG) medium doped uniformly with inhomogeneous-broadening two-level atoms. The Maxwell-Bloch equations for the pulse propagating through such a uniformly doped PBG structure are derived first and further reduced to an effective nonlinear Schrödinger equation. This model describes an equivalent physical mechanism for a Bragg-soliton propagation resulting from the effective quadratic dispersion balancing with the effective third-order nonlinearity. Because the resonant atoms are taken into account, the original band gap can be shifted both by the dopants and the instantaneous nonlinearity response originating from an intense optical pulse. As a result, even if a SIT soliton with its central frequency deep inside the original forbidden band, it still can propagate through the resonant PBG medium as long as this SIT soliton satisfies the effective Bragg-soliton propagation. An approximate soliton solution describing such coexistence is found. We also show that the pulse width and group velocity of this soliton solution can be uniquely determined for given material parameters, atomic transition frequency, and input central frequency of the soliton. The numerical examples of the SIT soliton in a one-dimensional As2S3-based PBG structure doped uniformly with Lorentzian line-shape resonant atoms are shown. It is found that a SIT soliton with approximately 100-ps width in such a resonant PBG structure can travel with the velocity being two orders of magnitude slower than the light speed in an unprocessed host medium.

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Nov 20, 2002•8 citations
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.
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.
Quantum electrodynamics of a single two-level atom placed within a frequency dispersive medium whose polariton spectrum has a gap due to photon coupling to a medium excitation (exciton, optical phonon, etc.) is studied. Despite a nonlocal effective polariton-polariton coupling, the system is proven to be integrable and is diagonalized exactly by means of the Bethe-ansatz technique. Its spectrum consists of bound many-polariton complexes (quantum solitons) and exhibits some unusual features due to the existence of the polaritonic gap. Only solitons containing an even number of polaritons propagate within the gap, while a soliton with an odd number of polaritons is pinned to the atom and forms a many-polariton-atom bound state, in which the radiation and the medium polarization are localized in the vicinity of the atom. The model under consideration has a quite general structure and can also be applied to the case of photonic-band-gap materials. \textcopyright{} 1996 The American Physical Society.
A system of polaritons interacting with a two-level atom placed within a frequency dispersive medium is proven to be integrable and diagonalized exactly by the Bethe ansatz method, despite a nonlocal effective polariton-polariton coupling. Its spectrum consists of bound many-polariton complexes (quantum solitons) and exhibits unusual features due to the existence of the polaritonic gap. Only solitons containing an even number of polaritons (``even'' solitons) propagate within the gap, while an ``odd'' soliton is pinned to the atom and forms a many-polariton--atom bound state.
We derive the properties of self-induced transparency (SIT) solitary waves in a one-dimensional periodic structure doped uniformly with resonance two-level atoms. In our model, the electromagnetic field is treated classically and the dopant atoms are described quantum mechanically. The resulting solitary waves take the form of ultrashort (picosecond) laser pulses which propagate near the band edge of the nonlinear photonic band gap (PBG) material doped with rare-earth atoms such as erbium. Solitary wave formation involves the combined effects of group velocity dispersion (GVD), nonresonant Kerr nonlinearity, and resonant interaction with dopant atoms. We derive the general Maxwell-Bloch equations for a nonlinear PBG system and then demonstrate the existence of elementary solitary wave solutions for frequencies far outside the gap where GVD effects are negligible and for frequencies near the photonic band edge where GVD effects are crucial. We find two distinct new types of propagating SIT solitary wave pulses. Far from Bragg resonance, we recapture the usual McCall-Hahn soliton with hyperbolic secant profile when the nonlinear Kerr coefficient ${\ensuremath{\chi}}^{(3)}=0.$ However, when the host nonresonant Kerr coefficient is nonzero, we obtain the first new type of soliton. In this case, the optical soliton envelope function deviates from the hyperbolic secant profile and pulse propagation requires nontrivial phase modulation (chirping). We derive the dependence of the solitary wave structure on the Kerr coefficient ${\ensuremath{\chi}}^{(3)},$ the resonance impurity atom density, and the detuning of the average laser frequency from the atomic transition. When the laser frequency and the atomic transition frequencies are near the photonic band edge we obtain the second type of soliton. To illustrate the second type of soliton we consider two special cases. In the first case, GVD facilitates the propagation of an unchirped SIT-gap soliton moving at a velocity fixed by the material's parameters. The soliton structure changes dramatically as the laser frequency is tuned through the atomic resonance. In the second illustrative case we set the Kerr coefficient ${\ensuremath{\chi}}^{(3)}=0.$ In this case, the solution is a chirped pulse which arises from the balance between GVD and the resonance interaction with the dopant atoms. Finally, we show that under certain circumstances, these solitary wave solutions may persist even in the presence of (subpicosecond) dipolar dephasing of the dopant atoms and absorption losses of the host PBG material, provided that the system is incoherently pumped. These results may be relevant to the application of PBG materials as optical devices in fiber-optic networks.
We theoretically show that a self-induced transparency (SIT) soliton and a Bragg soliton can coexist in a nonlinear photonic band gap (PBG) medium doped uniformly with inhomogeneous-broadening two-level atoms. The Maxwell-Bloch equations for the pulse propagating through such a uniformly doped PBG structure are derived first and further reduced to an effective nonlinear Schrödinger equation. This model describes an equivalent physical mechanism for a Bragg-soliton propagation resulting from the effective quadratic dispersion balancing with the effective third-order nonlinearity. Because the resonant atoms are taken into account, the original band gap can be shifted both by the dopants and the instantaneous nonlinearity response originating from an intense optical pulse. As a result, even if a SIT soliton with its central frequency deep inside the original forbidden band, it still can propagate through the resonant PBG medium as long as this SIT soliton satisfies the effective Bragg-soliton propagation. An approximate soliton solution describing such coexistence is found. We also show that the pulse width and group velocity of this soliton solution can be uniquely determined for given material parameters, atomic transition frequency, and input central frequency of the soliton. The numerical examples of the SIT soliton in a one-dimensional As2S3-based PBG structure doped uniformly with Lorentzian line-shape resonant atoms are shown. It is found that a SIT soliton with approximately 100-ps width in such a resonant PBG structure can travel with the velocity being two orders of magnitude slower than the light speed in an unprocessed host medium.