arXiv subjects
Publications and source records attributed to V. V. Konotop.
A theory of the non-symmetric Landau-Zener tunneling of Bose-Einstein condensates in deep optical lattices is presented. It is shown that periodic exchange of matter between the bands is described by a set of linearly coupled nonlinear Schrödinger equations. The key role of the modulational instability in rendering the inter-band transitions irreversible is highlighted.
The modulational stability of the nonlinear Schr{ö}dinger (NLS) equation is examined in the cases with linear and quadratic external potential. This study is motivated by recent experimental studies in the context of matter waves in Bose-Einstein condensates. The linear case can be examined by means of the Tappert transformation and can be mapped to the NLS in the appropriate (constant acceleration) frame. The quadratic case can be examined by using a lens-type transformation that converts it into a regular NLS with an additional linear growth term.
We consider localized modes (discrete breathers) of the discrete nonlinear Schrödinger equation $i\frac{dψ_n}{dt}=ψ_{n+1}+ψ_{n-1}-2ψ_n+σ|ψ_n|^2ψ_n$, $σ=\pm1$, $n\in \mathbb{Z}$. We study the diversity of the steady-state solutions of the form $ψ_n(t)=e^{iωt}v_n$ and the intervals of the frequency, $ω$, of their existence. The base for the analysis is provided by the anticontinuous limit ($ω$ negative and large enough) where all the solutions can be coded by the sequences of three symbols "-", "0" and "+". Using dynamical systems approach we show that this coding is valid for $ω<ω^*\approx -3.4533$ and the point $ω^*$ is a point of accumulation of saddle-node bifurcations. Also we study other bifurcations of intrinsic localized modes which take place for $ω>ω^*$ and give the complete table of them for the solutions with codes consisting of less than four symbols.
We study expansion of quasi-one-dimensional Bose-Einstein condensate (BEC) after switching off the confining harmonic potential. Exact solution of dynamical equations is obtained in framework of the hydrodynamic approximation and it is compared with the direct numerical simulation of the full problem showing excellent agreement at realistic values of physical parameters. We analyze the maximum of the current density and estimate the velocity of expansion. The results of the 1D analysis provides also qualitative understanding of some properties of BEC expansion observed in experiments.
Evolution of periodic matter waves in one-dimensional Bose-Einstein condensates with time dependent scattering length is described. It is shown that variation of the effective nonlinearity is a powerful tool for controlled generation of bright and dark solitons starting with periodic waves.
Coupled nonlinear Schrodinger equations (CNLS) with an external elliptic function potential model a quasi one--dimensional interacting two-component Bose-Einstein condensate trapped in a standing light wave. New families of stationary solutions of the CNLS with a periodic potential are presented and their stability studied by direct numerical simulations. Some of these solutions allow reduction to Manakov system. From a physical point of view these solutions can be interpreted as exact Bloch states at the edge of the Brillouin zone. Some of them are stable while others are found to be unstable against modulations of long wavelength. The solutions which are modulationally unstable are shown to lead to the formation of localized ground states of the coupled BEC system.
It is shown that the generalized discrete nonlinear Schrödinger equation can be reduced in a small amplitude approximation to the KdV, mKdV, KdV(2) or the fifth-order KdV equations, depending on values of the parameters. In dispersionless limit these equations lead to wave breaking phenomenon for general enough initial conditions, and, after taking into account small dispersion effects, result in formation of dissipationless shock waves. The Whitham theory of modulations of nonlinear waves is used for analytical description of such waves.
We show that the phenomenon of modulational instability in arrays of Bose-Einstein condensates confined to optical lattices gives rise to coherent spatial structures of localized excitations. These excitations represent thin disks in 1D, narrow tubes in 2D, and small hollows in 3D arrays, filled in with condensed atoms of much greater density compared to surrounding array sites. Aspects of the developed pattern depend on the initial distribution function of the condensate over the optical lattice, corresponding to particular points of the Brillouin zone. The long-time behavior of the spatial structures emerging due to modulational instability is characterized by the periodic recurrence to the initial low-density state in a finite optical lattice. We propose a simple way to retain the localized spatial structures with high atomic concentration, which may be of interest for applications. Theoretical model, based on the multiple scale expansion, describes the basic features of the phenomenon. Results of numerical simulations confirm the analytical predictions.
Generation of dark solitons from large initial excitations and their evolution in a quasi-one-dimensional Bose-Einstein condensate trapped by a harmonic potential is studied analytically and numerically. In the case of a single deep soliton main characteristics of its motion such as a frequency and amplitude of oscillations are calculated by means of the perturbation theory which in the leading order results in a Newtonian dynamics, corrections to which are computed as well. It is shown that long-time dynamics of a dark soliton in a generic situation deviates substantially from outcomes of the naive application of the Ehrenfest theorem. We also consider three different techniques of controllable creation of multi-soliton structures (soliton trains) from large initial excitations and calculate their initial parameters (depths and velocities) with the use of a generalized Bohr-Sommerfeld quantization rule. Multi-soliton effects are discussed.
In the present Letter we use the Wannier function basis to construct lattice approximations of the nonlinear Schrödinger equation with a periodic potential. We show that the nonlinear Schrödinger equation with a periodic potential is equivalent to a vector lattice with long-range interactions. For the case-example of the cosine potential we study the validity of the so-called tight-binding approximation i.e., the approximation when nearest neighbor interactions are dominant. The results are relevant to Bose-Einstein condensate theory as well as to other physical systems like, for example, electromagnetic wave propagation in nonlinear photonic crystals.
A self consistent theory of a cigar shaped Bose-Einstein condensate (BEC) periodically modulated by a laser beam is presented. We show, both theoretically and numerically, that modulational instability/stability is the mechanism by which wavefunctions of soliton type can be generated in cigar shaped BEC subject to a 1D optical lattice. The theory explains why bright solitons can exist in BEC with positive scattering length and why condensate with negative scattering length can be stable and give rise to dark solitary pulses.
The dynamics of localized excitations in array of Bose-Einstein condensates is investigated in the framework of the nonlinear lattice theory. The existence of temporarily stable ground states displaying an atomic population distributions localized on very few lattice sites (intrinsic localized modes), as well as, of atomic population distributions involving many lattice sites (envelope solitons), is studied both numerically and analytically. The origin and properties of these modes are shown to be inherently connected with the interplay between macroscopic quantum tunnelling and nonlinearity induced self-trapping of atoms in coupled BECs. The phenomenon of Bloch oscillations of these excitations is studied both for zero and non zero backgrounds. We find that in a definite range of parameters, homogeneous distributions can become modulationally unstable. We also show that bright solitons and excitations of shock wave type can exist in BEC arrays even in the case of positive scattering length. Finally, we argue that BEC array with negative scattering length in presence of linear potentials can display collapse.
The dynamics of a 2D Bose-Einstein condensate in optical trap is studied taking into consideration fluctuations of the far-off-resonance laser field intensity. The problem is described in the frame of the mean field Gross-Pitaevskii equation with randomly varying trap potential. An analytic approach based on the moments method has been employed to describe the noise induced evolution of the condensate properties. Stochastic parametric resonance in oscillations of the condensate width is proved to exist. For the condensate with negative scattering length of atoms, it is shown that the noise can delay or even arrest the collapse. Analytical predictions are confirmed by numerical simulations of the underlying PDE and ODE models.
A way of quantizing weakly nonlinear lattices is proposed. It is based on introducing "pseudo-field" operators. In the new formalism quantum envelope solitons together with phonons are regarded as elementary quasi-particles making up boson gas. In the classical limit the excitations corresponding to frequencies above linear cut-off frequency are reduced to conventional envelope solitons. The approach allows one to identify the quantum soliton which is localized in space and understand existence of a narrow soliton frequency band.
In a one-dimensional periodic nonlinear $χ^{(2)}$ medium, by choosing a proper material and geometrical parameters of the structure, it is possible to obtain two matching conditions for simultaneous generation of second and third harmonics. This leads to new diversity of the processes of the resonant three-wave interactions, which are discussed within the framework of slowly varying envelope approach. In particular, we concentrate on the fractional conversion of the frequency $ω\to (2/3) ω$. This phenomenon occurs by means of intermediate energy transfer to the first harmonic at the frequency $ω/3$ and can be controlled by this mode. By analogy the same medium allows "nondirect" second harmonic generation, controlled by the cubic harmonic. Propagation of localized pulses in the form two coupled bright solitons on first and third harmonics and a dark soliton on the second harmonic is possible.
We prove global existence and uniqueness of solutions of some important nonlinear lattices which include the Fermi-Pasta-Ulam (FPU) lattice. Our result shows (on a particular example) that the FPU lattice with high nonlinearity and its continuum limit display drastically different behaviour with respect to blow up phenomenon.
We study the effect of nonlocality on the collapse properties of a self-focusing Nonlinear Schrödinger system related to Bose-Einstein condensation problems. Using a combination of moment techniques, time dependent variational methods and numerical simulations we present evidences in support of the hypothesis that nonlocal attractively interacting condensates cannot collapse when the dominant interaction term is due to finite range interactions. Instead there apppear oscillations of the wave packet with a localized component whose size is of the order of the range of interactions. We discuss the implications of the results to collapse phenomena in negative scattering length Bose-Einstein condensates.
An integrable discretization of the inhomogeneous Ablowitz-Ladik model with a linear force is introduced. Conditions on parameters of the discretization which are necessary for reproducing Bloch oscillations are obtained. In particular, it is shown that the step of the discretization must be comensurable with the period of oscillations imposed by the inhomogeneous force. By proper choice of the step of the discretization the period of oscillations of a soliton in the discrete model can be made equal to an integer number of periods of oscillations in the underline continuous-time lattice.