Fatkhulla Kh. Abdullaev

Controlling collapse in Bose-Einstein condensates by temporal modulation of the scattering length

We consider, by means of the variational approximation (VA) and direct numerical simulations of the Gross-Pitaevskii (GP) equation, the dynamics of two-dimensional (2D) and 3D condensates with a scattering length containing constant and harmonically varying parts, which can be achieved with an ac magnetic field tuned to the Feshbach resonance. For a rapid time modulation, we develop an approach based on the direct averaging of the GP equation, without using the VA. In the 2D case, both VA and direct simulations, as well as the averaging method, reveal the existence of stable self-confined condensates without an external trap, in agreement with qualitatively similar results recently reported for spatial solitons in nonlinear optics. In the 3D case, the VA again predicts the existence of a stable self-confined condensate without a trap. In this case, direct simulations demonstrate that the stability is limited in time, eventually switching into collapse, even though the constant part of the scattering length is positive (but not too large). Thus a spatially uniform ac magnetic field, resonantly tuned to control the scattering length, may play the role of an effective trap confining the condensate, and sometimes causing its collapse.

DYNAMICS OF BRIGHT MATTER WAVE SOLITONS IN A BOSE EINSTEIN CONDENSATE

Recent experimental and theoretical advances in the creation and description of bright matter wave solitons are reviewed. Several aspects are taken into account, including the physics of soliton train formation as the nonlinear Fresnel diffraction, soliton-soliton interactions, and propagation in the presence of inhomogeneities. The generation of stable bright solitons by means of Feshbach resonance techniques is also discussed.

Gap solitons in Bose-Einstein condensates in linear and nonlinear optical lattices

Abstract Properties of localized states on array of BEC confined to a potential, representing superposition of linear and nonlinear optical lattices are investigated. For a shallow lattice case the coupled mode system has been derived. We revealed new types of gap solitons and studied their stability. For the first time a moving soliton solution has been found. Analytical predictions are confirmed by numerical simulations of the Gross–Pitaevskii equation with jointly acting linear and nonlinear periodic potentials.

Gap solitons in a spin-orbit-coupled bose-einstein condensate

We report a diversity of stable gap solitons in a spin-orbit-coupled Bose-Einstein condensate subject to a spatially periodic Zeeman field. It is shown that the solitons can be classified by the main physical symmetries they obey, i.e., symmetries with respect to parity (P), time (T), and internal degree of freedom, i.e., spin (C), inversions. The conventional gap and gap-stripe solitons are obtained in lattices with different parameters. It is shown that solitons of the same type but obeying different symmetries can exist in the same lattice at different spatial locations. PT and CPT symmetric solitons have antiferromagnetic structure and are characterized, respectively, by nonzero and zero total magnetizations.

Transmission of matter-wave solitons through nonlinear traps and barriers

The transmission of matter wave packets through inhomogeneities of different types of Bose-Einstein condensates BECs has recently attracted a lot of attention, because this phenomenon is important for the design of control methods of the soliton parameters and atomic soliton lasers 1. The transmission and reflection of bright and dark matter wave solitons has been studied in the case of linear inhomogeneities, induced by the variations in space of the potential field 2‐8. In particular, the effect of a potential step or impurity, including the soliton train evolution, has been analyzed in Ref. 6. The adiabatic dynamics of a dark soliton, as well as the radiative wave emission leading to the dark soliton degradation, has been studied in Ref. 7. Finally, the continuous wave emission by a bright soliton in an optical lattice has been addressed in Ref. 8. The case of inhomogeneities produced by spatial variations of the scattering length has been less investigated. In Refs. 9‐11 the variational approach has been applied and numerical simulations have been performed. When the linear and nonlinear inhomogeneities compete with each other, direct numerical simulations of the soliton propagation have been carried out. The enhanced soliton transmission through a linear barrier is observed for well-chosen parameters of the nonlinear potential 12. The explanation of this phenomenon, as shown in this paper, requires to take into account the radiative effects when the soliton interacts with the nonlinear potential. The purpose of this work is to develop the theory describing the transmission of matter wave solitons through nonlinear barriers and traps. Such barriers can be produced by using the Feshbach resonance method, namely by the local variation of the external magnetic field Bz in space near the resonant value Bc 13. By the small variation of the field near the resonant value we can induce the large variations of the scattering length in space according to the formula

Modulational instability of electromagnetic waves in inhomogeneous and in discrete media

Publisher Summary This chapter discusses the modulational instability (MI) of electromagnetic waves in inhomogeneous and in discrete media. MI exists because of the interplay between the nonlinearity and dispersion/diffraction effects. Important models for investigating MI of electromagnetic waves in nonlinear media represent the scalar and vectorial nonlinear Schrodinger (NLS) equations, the system describing evolution of the envelopes of fundamental and second harmonics waves in quadratically nonlinear media, and sine-Gordon equation. The methods such as periodic solutions of the NLS equation and the coupled-mode theory with three modes are discussed. The chapter discusses the MI of electromagnetic waves in optical media with periodic inhomogeneities. The origin of the random fluctuations of parameters in optical fibers and other nonlinear optical media is described. MI in fibers with random amplification and dispersion and MI in randomly birefringent fibers are discussed. The chapter discusses the MI of electromagnetic waves in nonlinear discrete optical systems such as an array of planar waveguides and fibers. Particular cases of MI in discrete media with cubic nonlinearity and quadratic nonlinearity are investigated.

Stable localized modes in asymmetric waveguides with gain and loss

It is shown that asymmetric waveguides with gain and loss can support a stable propagation of optical beams. This means that the propagation constants of modes of the corresponding complex optical potential are real. A class of such waveguides is found from a relation between two spectral problems. A particular example of an asymmetric waveguide, described by the hyperbolic functions, is analyzed. The existence and stability of linear modes and of continuous families of nonlinear modes are demonstrated.

Stable two-dimensional dispersion-managed soliton

The existence of a dispersion-managed soliton in two-dimensional nonlinear Schrödinger equation with periodically varying dispersion has been explored. The averaged equations for the soliton width and chirp are obtained which successfully describe the long time evolution of the soliton. The slow dynamics of the soliton around the fixed points for the width and chirp are investigated and the corresponding frequencies are calculated. Analytical predictions are confirmed by direct partial differential equation (PDE) and ordinary differential equation (ODE) simulations. Application to a Bose-Einstein condensate in optical lattice is discussed. The existence of a dispersion-managed matter-wave soliton in such system is shown.

Resonances in a trapped 3D Bose?Einstein condensate under periodically varying atomic scattering length

Nonlinear oscillations of a 3D radial symmetric Bose–Einstein condensate under periodic variation in time of the atomic scattering length have been studied. The time-dependent variational approach is used for the analysis of the characteristics of nonlinear resonances in the oscillations of the condensate. The bistability in oscillations of the BEC width is investigated. The dependence of the BEC collapse threshold on the drive amplitude and parameters of the condensate and trap is found. Predictions of the theory are confirmed by numerical simulations of the full Gross–Pitaevskii equation.

Tunable spin-orbit-coupled Bose-Einstein condensates in deep optical lattices

Binary mixtures of Bose-Einstein condensates (BECs) trapped in deep optical lattices and subjected to equal contributions of Rashba and Dresselhaus spin-orbit coupling (SOC) are investigated in the presence of a periodic time modulation of the Zeeman field. SOC tunability is explicitly demonstrated by adopting a mean-field tight-binding model for the BEC mixture and by performing an averaging approach in the strong modulation limit. In this case, the system can be reduced to an unmodulated vector discrete nonlinear Schr¨odinger equation with a rescaled SOC tuning parameter α, which depends only on the ratio between amplitude and frequency of the applied Zeeman field. We consider the attractive interaction case and focus on the effect of the SOC tuning on the localized ground states. The dependence of the spectrum of the linear system on α has been analytically characterized. In particular, we show that extremal curves (ground and highest excited states) of the linear spectrum are continuous piecewise functions (together with their derivatives) of α, which consist of a finite number of decreasing band lobes joined by constant lines. This structure also remains in the presence of inter- and intra-species interactions, the nonlinearity mainly introducing a number of localized states in the band gaps. The stability of ground states in the presence of the modulating field has been demonstrated by real-time evolutions of the original (unaveraged) system. Localization properties of the ground state induced by the SOC tuning, and a parameter design for possible experimental observation, have also been discussed.