Andrei Rybin

Similarity solutions and collapse in the attractive gross-pitaevskii equation

We analyze a generalized Gross-Pitaevskii (GP) equation involving a paraboloidal trap potential in D space dimensions and generalized to a nonlinearity of order 2n+1. For attractive coupling constants collapse of the particle density occurs for Dn>/=2 and typically to a delta function centered at the origin of the trap. By introducing a special variable for the spherically symmetric solutions, we show that all such solutions are self-similar close to the center of the trap. Exact self-similar solutions occur if, and only if, Dn=2, and for this case of Dn=2 we exhibit an exact but rather special D=1 analytical self-similar solution collapsing to a delta function which, however, recovers and collapses periodically, while the ordinary GP equation in two space dimensions also has a special solution with periodic delta function collapses and revivals of the density. The relevance of these various results to attractive Bose-Einstein condensation in spherically symmetric traps is discussed.

The su(1,1) Tavis-Cummings model

A generic su(1,1) Tavis-Cummings model is solved both by the quantum inverse method and within a conventional quantum-mechanical approach. Examples of corresponding quantum dynamics including squeezing properties of the su(1,1) Perelomov coherent states for the multiatom case are given.

Theory of slow-light solitons.

In the framework of the nonlinear lambda model we investigate propagation of solitons in atomic vapors and Bose-Einstein condensates. We show how the complicated nonlinear interplay between fast solitons and slow-light solitons in the lambda-type media points to the possibility to create optical gates and, thus, to control the optical transparency of the lambda-type media. We provide an exact analytic description of decelerating, stopping and reaccelerating of slow-light solitons in atomic media in the nonadiabatic regime. Dynamical control over slow-light solitons is realized via a controlling field generated by an auxiliary laser. For a rather general time dependence of the field; we find the dynamics of the slow-light soliton inside the medium. We provide an analytical description for the nonlinear dependence of the velocity of the signal on the controlling field. If the background field is turned off at some moment of time, the signal stops. We find the location and shape of the spatially localized memory bit imprinted into the medium. We discuss physically interesting features of our solution, which are in a good agreement with recent experiments.

Darboux transformations and coherent interaction of the light pulse with two-level media

Applications of the Darboux transformation method to study the reduced Maxwell-Bloch (RMB) system and of the self-induced transparency (SIT) equations are considered. Both systems describe the propagation of ultrashort optical pulses in a two-level medium to a reasonable approximation. The main result of the present work is the construction of multisoliton solutions on an arbitrary background for RMB and SIT equations. Particular cases of these solutions are discussed in some detail.

Similarity solutions of the deformed Maxwell-Bloch system

The authors have found similarity reductions for the deformed Maxwell-Bloch system to the fifth and second Painleve equations. Asymptotics of the solutions of these equations, which are relevant to two-level atomic systems with pumping, have also been derived.

Algebraic approach to the Tavis-Cummings problem

An algebraic method is introduced for an analytical solution of the eigenvalue problem of the Tavis-Cummings Hamiltonian, based on polynomially deformed su(2), i.e., su{sub n}(2) algebras. In this method the eigenvalue problem is solved in terms of a specific perturbation theory, developed here up to third order. Generalization to the N-atom case of the Rabi frequency and dressed states is also provided. A remarkable enhancement of spontaneous emission of N atoms in a resonator is found to result from collective effects.

An integrable q-deformed model for bosons interacting with spin impurities

A composite quantum model on a lattice which describes the system of q-bosons interacting with Uq(su(2)) spin impurities is introduced and solved exactly under periodic boundary conditions. In one limit the model is shown to become a new exactly soluble quantum system on a lattice which can be interpreted as a q-deformed version of the quantum Dicke model. In the limit of infinitesimal spacing the model is further reduced to a multilevel version of the previously introduced continuum-limit Dicke model. For spin 1/2 the previous results for this particular case are confirmed.

q-deformed solitons and quantum solitons of the Maxwell-Bloch lattice

We report for the first time exact solutions of a completely integrable nonlinear lattice system for which the dynamical variables satisfy a q-deformed Lie algebra - the Lie-Poisson algebra suq(2). The system considered is a q-deformed lattice for which in the continuum limit the equations of motion become the envelope Maxwell-Bloch (or SIT) equations describing the resonant interaction of light with a nonlinear dielectric. Thus the N-soliton solutions we report here are the natural q-deformations, necessary for a lattice, of the well known multi-soliton and breather solutions of self-induced transparency (SIT). The method we use to find these solutions is a generalization of the Darboux-Backlund dressing method. The extension of these results to quantum solitons is sketched.

Stopping a slow-light soliton: an exact solution

We investigate propagation of a slow-light soliton in Λ-type media such as atomic vapours and Bose–Einstein condensates. We show that the group velocity of the soliton monotonically decreases with the intensity of the controlling laser field, which decays exponentially after the laser is switched off. The shock wave of the vanishing controlling field overtakes the slow soliton and stops it, while the optical information is recorded in the medium in the form of spatially localized polarization. In the strongly nonlinear regime we find an explicit exact solution describing the whole process.

Manipulation of optical solitons in Bose-Einstein condensates

We propose a method to control the optical transparency of a Bose–Einstein condensate with working energy levels of the Λ-type. The reported effects are essentially nonlinear and are considered in the framework of an exactly solvable model describing the interaction of light with a Λ-type medium. We show how the complicated nonlinear interplay between fast and slow solitons in the Λ-type medium points to a possibility to create optical gates as well as to a possibility to store optical information.