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Physical review. B, Condensed matter

Dark and bright excitons in systems with exchange and dipole-dipole interactions

V. V. Konotop, Sachio Takeno

Published May 1, 1997• 20 citations
Publisher

Abstract

Dark and bright soliton solutions (excitons) in one-dimensional chains of two-level atoms with exciton-exciton, dipole-dipole, and exchange interactions are obtained. It is shown that in some limiting cases evolution equations resulting from model Hamiltonians are reduced to the exactly integrable Ablowitz-Ladik model. Bloch oscillations of the excitons in inhomogeneous magnetic field and effects originated by the energy shift are discussed. An example of a multidimensional dark exciton is represented.

DOI
10.1103/physrevb.55.11342

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Dark and bright soliton solutions (excitons) in one-dimensional chains of two-level atoms with exciton-exciton, dipole-dipole, and exchange interactions are obtained. It is shown that in some limiting cases evolution equations resulting from model Hamiltonians are reduced to the exactly integrable Ablowitz-Ladik model. Bloch oscillations of the excitons in inhomogeneous magnetic field and effects originated by the energy shift are discussed. An example of a multidimensional dark exciton is represented.
We use a small-amplitude multiple scale expansion to investigate the existence of shock waves in a chain of two-level atoms with both exchange and dipole-dipole interactions. We show that the exchange interaction allows the formation of the system of both bright and dark shock waves. Conversely, the dipole-dipole interaction results in the instability of the background and, as a consequence, in the prevention of the formation of shock waves. The analytical results are found to be in good qualitative agreement with a direct numerical integration of the system.
We consider vector solitons of mixed bright-dark types in quasi-one-dimensional spinor (F=1)(F=1)(F=1) Bose-Einstein condensates. Using a multiscale expansion technique, we reduce the corresponding nonintegrable system of three coupled Gross-Pitaevskii equations (GPEs) to an integrable Yajima-Oikawa system. In this way, we obtain approximate solutions for small-amplitude vector solitons of dark-dark-bright and bright-bright-dark types, in terms of the ${m}_{F}=+1,\ensuremath{-}1,0$ spinor components, respectively. By means of numerical simulations of the full GPE system, we demonstrate that these states indeed feature soliton properties, i.e., they propagate undistorted and undergo quasielastic collisions. It is also shown that in the presence of a parabolic trap the bright component(s) is (are) guided by the dark one(s) and, as a result, the small-amplitude vector soliton as a whole performs quasiharmonic oscillations. The oscillation frequency is found as a function of the spin-dependent interaction strength for both small-amplitude and large-amplitude solitons.
Abstract We develop a direct perturbation theory for dark-bright solitons and derive evolution equations for the soliton parameters. In particular, first the linearization equation around the solitons is solved by expanding its solution into a set of complete eigenfunctions of the linearization operator. Then, suppression of secular growth in the linearized solution leads to the evolution equations of soliton parameters. The results are applied to a number of case examples motivated by the physics of atomic Bose-Einstein condensates, where dark-bright solitons have recently been studied both in theory and in experiments. We thus consider perturbations corresponding to (a) finite temperature-induced thermal losses, and (b) the presence of localized ( δ -function) impurities. In these cases, relevant equations of motion for the dark-bright soliton center are in agreement with ones previously obtained via alternative methods, including energy-based methods, as well as numerical linear stability analysis and direct simulations.
A formal, regorous theory of Frenkel excitons interacting with an external electromagnetic field is doveloped within the framework of a two-level-atom model in which the dipole approximation is employed in describing the atom-field (taken as a c-number) interaction. The model exciton Hamiltonian is shown to be identical in form with the Heisenberg model Hamiltonian for S=1/2 spins with anisotropic long-range exchange interactions. Instead of treating excitons as a non-ideal Bose gas, atomic polarization operators and level population defference are taken up as two fundamental quantities to describe Frenkel excitons. A pair of equations satisfied by these two quantities are derived. It is shown that Frenkel excitons are generally expressed as quantum-mechanical nonlinear polarization waves, the nonlinearity being characterized by the level population. A discussion is given on the interrelationship between such polarization waves and those derivable from the classical Drude-Lorentz model. With the aid of several approximation procedures similar to those employed in nonlinear optics, a set of equations describing nonlinear exciton-photon coupled modes are derived. Brief discussion is given on the eigenfrequencies of nonlinear Frenkel excitons and nonlinear polaritons. The existence of nonlinear exciton-photon coupled modes whose properties are different from those of polariton modes is suggested.
We analyze the dynamics of two-component vector solitons, namely, dark-bright solitons, via the variational approximation in Bose-Einstein condensates. The system is described by a vector nonlinear Schr\"odinger equation appropriate to multicomponent Bose-Einstein condensates. The variational approximation is based on a hyperbolic tangent (hyperbolic secant) for the dark (bright) component, which leads to a system of coupled ordinary differential equations for the evolution of the ansatz parameters. We obtain the oscillation dynamics of two-component dark-bright solitons. Analytical calculations are performed for same-width components in the vector soliton, and numerical calculations extend the results to arbitrary widths. We calculate the binding energy of the system and find it to be proportional to the intercomponent coupling interaction and numerically demonstrate the breakup or unbinding of a dark-bright soliton. Our calculations explore observable eigenmodes, namely, the internal oscillation eigenmode and the Goldstone eigenmode. We find analytically that the number of atoms in the bright component is required to be less than the number of atoms displaced by the dark soliton in the other component in order to find the internal oscillation eigenmode of the vector soliton and support the existence of the dark-bright soliton. This outcome is confirmed by numerical results. Numerically, we find that the oscillation frequency is amplitude independent. For dark-bright solitons in 87Rb^{87}\mathrm{Rb}87Rb we find that the oscillation frequency range is 90 to 405 Hz and therefore observable in multicomponent Bose-Einstein condensate experiments.
Dark and bright excitons are shown to appear naturally in a linear atomic super-lattice with two atoms per unit cell.In bringing the super-lattice into a strong coupling regime with a one-dimensional nanophotonic waveguide, bright excitons and photons are coherently mixed to form polaritons. Treating excitons as bosons implies a mechanism that forbids two excitations from being in the same atomic state, which is included here through a bosonization procedure with kinematic interactions.Interestingly these interactions couple dark and bright excitons, which we exploit as a new tool for exciting dark states in a controllable way.We suggest a pump-probe experiment where two polaritons scatter into two dark excitons that are found to be correlated and are represented as dark bogolon excitons.The results can be adapted for any super-lattice of active materials; for example, of organic molecules.