Nonlinear Matter Waves Research Articles

Quasi-two-dimensional Bose–Einstein condensates with spatially modulated cubic–quintic nonlinearities

We investigate exact nonlinear matter wave functions with odd and even parities in the framework of quasi-two-dimensional Bose–Einstein condensates (BECs) with spatially modulated cubic–quintic nonlinearities and harmonic potential. The existence condition for these exact solutions requires that the minimum energy eigenvalue of the corresponding linear Schrödinger equation with harmonic potential is the cutoff value of the chemical potential λ . The competition between two-body and three-body interactions influences the energy of the localized state. For attractive two-body and three-body interactions, the larger the matter wave order number n , the larger the energy of the corresponding localized state. A linear stability analysis and direct simulations with initial white noise demonstrate that, for the same state (fixed n ), increasing the number of atoms can add stability. A quasi-stable ground-state matter wave is also found for repulsive two-body and three-body interactions. We also discuss the experimental realization of these results in future experiments. These results are of particular significance to matter wave management in higher-dimensional BECs.

Weak Nonlinear Matter Waves in a Trapped Spin-1 Condensates

The dynamics of the weak nonlinear matter solitary waves in a spin-1 condensates with harmonic external potential are investigated analytically by a perturbation method. It is shown that, in the small amplitude limit, the dynamics of the solitary waves are governed by a variable-coefficient Korteweg-de Vries (KdV) equation. The reduction to the (KdV) equation may be useful to understand the dynamics of nonlinear matter waves in spinor BECs. The analytical expressions for the evolution of soliton show that the small-amplitude vector solitons of the mixed types perform harmonic oscillations in the presence of the trap. Furthermore, the emitted radiation profiles and the soliton oscillation frequency are also obtained.

Effects of long-range nonlinear interactions in double-well potentials

We consider the interplay of linear double-well-potential (DWP) structures and nonlinear long-range interactions of different types, motivated by applications to nonlinear optics and matter waves. We find that, while the basic spontaneous-symmetry-breaking (SSB) bifurcation structure in the DWP persists in the presence of the long-range interactions, the critical points at which the SSB emerges are sensitive to the range of the nonlocal interaction. We quantify the dynamics by developing a few-mode approximation corresponding to the DWP structure, and analyze the resulting system of ordinary differential equations and its bifurcations in detail. We compare results of this analysis with those produced by the full partial differential equation, finding good agreement between the two approaches. Effects of the competition between the local self-attraction and nonlocal repulsion on the SSB are studied too. A far more complex bifurcation structure involving the possibility for not only supercritical but also subcritical bifurcations and even bifurcation loops is identified in that case.

Localized nonlinear matter waves in two-component Bose-Einstein condensates with time- and space-modulated nonlinearities

We investigate the localized nonlinear matter waves in the two-component Bose-Einstein condensates with time- and space-modulated nonlinearities analytically and numerically. The similarity transformations are developed to solve the coupled Gross-Pitaevskii equations and two families of explicitly exact solutions are derived. Our results show that not only the attractive spatiotemporal inhomogeneous interactions but the repulsive ones support novel localized nonlinear matter waves in two-component Bose-Einstein condensates. The dynamics of these matter waves, including the breathing solitons, quasibreathing solitons, resonant solitons, and moving solitons, is discussed. We confirm the stability of the exact solutions by adding various initial stochastic noise and study the general cases of the interaction parameters numerically. We also provide the experimental parameters to produce these phenomena in future experiments.

Adiabatic splitting, transport, and self-trapping of a Bose-Einstein condensate in a double-well potential

We show that the adiabatic dynamics of a Bose-Einstein condensate (BEC) in a double well potential can be described in terms of a dark variable resulting from the combination of the population imbalance and the spatial atomic coherence between the two wells. By means of this dark variable, we extend, to the non-linear matter wave case, the recent proposal by Vitanov and Shore [Phys. Rev. A 73, 053402 (2006)] on adiabatic passage techniques to coherently control the population of two internal levels of an atom/molecule. We investigate the conditions to adiabatically split or transport a BEC as well as to prepare an adiabatic self trapping state by the optimal delayed temporal variation of the tunneling rate via either the energy bias between the two wells or the BEC non-linearity. The emergence of non-linear eigenstates and unstable stationary solutions of the system as well as their role in the breaking down of the adiabatic dynamics is investigated in detail.

Solitary waves in nonautonomous nonlinear and dispersive systems: nonautonomous solitons

The fundamental concept of colored nonautonomous solitons in nonlinear and dispersive nonautonomous physical systems is introduced. Novel soliton solutions for the nonautonomous nonlinear Schrödinger equation models with linear and harmonic oscillator potentials substantially extend the concept of classical solitons and generalize it to the plethora of nonautonomous solitons that interact elastically and generally move with varying amplitudes, speeds and spectra adapted both to the external potentials and to the dispersion and nonlinearity variations. The parallels between nonlinear guided wave phenomena in optics and nonlinear guided wave phenomena in Bose condensates are clearly demonstrated by considering optical and matter wave soliton dynamics in the framework of nonautonomous evolution equations. The exact analytical solutions and numerical experiments reveal many specific features of nonautonomous solitons. Fundamental laws of the soliton adaptation to the external potentials are derived. Bound states of colored nonautonomous solitons are studied in detail and a comparison of the canonical Satsuma–Yajima breather dynamics with a nonautonomous ‘agitated’ breather is presented. The nonautonomous soliton concept can be applied to different physical systems, from hydrodynamics and plasma physics to nonlinear optics and matter waves, and offer many opportunities for further scientific studies

Quantized quasi-two-dimensional Bose-Einstein condensates with spatially modulated nonlinearity

We investigate the localized nonlinear matter waves of the quasi-two dimensional Bose-Einstein condensates with spatially modulated nonlinearity in harmonic potential. It is shown that the whole Bose-Einstein condensates, similar to the linear harmonic oscillator, can have an arbitrary number of localized nonlinear matter waves with discrete energies, which are mathematically exact orthogonal solutions of the Gross-Pitaevskii equation. Their novel properties are determined by the principle quantum number n and secondary quantum number l: the parity of the matter wave functions and the corresponding energy levels depend only on n, and the numbers of density packets for each quantum state depend on both n and l which describe the topological properties of the atom packets. We also give an experimental protocol to observe these novel phenomena in future experiments.

Analyzing Dynamics of a Bright Soliton in Bose–Einstein Condensates with Time-Dependent Atomic Scattering Length

We analyze the dynamics of a bright soliton in Bose–Einstein condensates (BECs) with time-dependent atomic scattering length in an expulsive parabolic potential. Under a safe range of parameters in which the Gross–Pitaevskii (GP) equation is effective in one dimension, our results show that, the dynamics of the bright soliton can be classed into two phases, depending on the value of the scattering length. Meanwhile, there exists a critical value of the absolute value of the atomic scattering length, below which, the dynamics of the bright soliton is very regular. Those phenomena can be useful for developing concrete applications of the nonlinear matter waves. We also obtain the orbital equation of the bright soliton and get some interesting data which may be useful for the experimental observation of the bright soliton and the application of the atom laser with manipulated intensity.

Dissipation-Induced Coherent Structures in Bose-Einstein Condensates

We discuss how to engineer the phase and amplitude of a complex order parameter using localized dissipative perturbations. Our results are applied to generate and control various types of atomic nonlinear matter waves (solitons) by means of localized dissipative defects.

Solitons and vortices in an evolving Bose-Einstein condensate

Spatiotemporal evolution of a confined Bose-Einstein condensate is studied by numerically integrating the time-dependent Gross-Pitaevskii equation. Self-interference between the successively expanding and reflecting nonlinear matter waves results in spiral atomic density profile, which subsequently degenerates into an embedding structure: The inner part preserves memory of the initial states while the outer part forms a sequence of necklacelike rings. The phase plot reveals a series of discrete concentric belts. The large gradients between adjacent belts indicate that the ring density notches are dark solitons. In the dynamical process, a scenario of vortex-antivortex pairs are spontaneously created and annihilated, whereas the total vorticity keeps invariant.

Weak nonlinear matter waves in a trapped two-component Bose–Einstein condensates

The dynamics of the weak nonlinear matter solitary waves in two-component Bose–Einstein condensates (BEC) with cigar-shaped external potential are investigated analytically by a perturbation method. In the small amplitude limit, the two-components can be decoupled and the dynamics of solitary waves are governed by a variable-coefficient Korteweg–de Vries (KdV) equation. The reduction to the KdV equation may be useful to understand the dynamics of nonlinear matter waves in two-component BEC. The analytical expressions for the evolution of soliton, emitted radiation profiles and soliton oscillation frequency are also obtained.

Generating ring dark solitons in an evolving Bose-Einstein condensate

The successive dynamical evolution of a Bose-Einstein condensate confined in a cylindrical well is numerically studied in the framework of the time-dependent Gross-Pitaevskii equation. Interference in the nonlinear matter wave leads to concentric density rings. The phase distribution exhibits a discontinuous sequence of plateaulike belts. Abrupt jumps in the phase between adjacent belts imply large radial superfluid velocity at the borderline. This, however, does not mean large particle current because the corresponding superfluid density is nearly zero. The density zeros along with the large gradient are identified as ring dark solitons, which have a brief lifetime before evolving into other soliton states.

Nonautonomous Solitons in External Potentials

Novel soliton solutions for the nonautonomous nonlinear Schrödinger equation models with linear and harmonic oscillator potentials substantially extend the concept of classical solitons and generalize it to the plethora of nonautonomous solitons that interact elastically and generally move with varying amplitudes, speeds, and spectra adapted both to the external potentials and to the dispersion and nonlinearity variations. The nonautonomous soliton concept can be applied to different physical systems, from hydrodynamics and plasma physics to nonlinear optics and matter waves, and offer many opportunities for further scientific studies.

Lie Symmetries and Solitons in Nonlinear Systems with Spatially Inhomogeneous Nonlinearities

Using Lie group theory and canonical transformations, we construct explicit solutions of nonlinear Schrödinger equations with spatially inhomogeneous nonlinearities. We present the general theory, use it to show that localized nonlinearities can support bound states with an arbitrary number solitons, and discuss other applications of interest to the field of nonlinear matter waves.

Self-Trapped Nonlinear Matter Waves in Periodic Potentials

We demonstrate that the recent observation of nonlinear self-trapping of matter waves in one-dimensional optical lattices [Th. Anker, Phys. Rev. Lett. 94, 020403 (2005)10.1103/PhysRevLett.94.020403] can be associated with a novel type of broad nonlinear state existing in the gaps of the matter-wave band-gap spectrum. We find these self-trapped localized modes in one-, two-, and three-dimensional periodic potentials, and demonstrate that such novel gap states can be generated experimentally in any dimension.

Modeling and simulation of multicellular tumor growth using a nonlinear matter wave equation

We show that the observed oscillatory pattern in the growth dynamics of a multicellular tumor may be successfully simulated with a classical field limit nonlinear matter wave equation given by the mean-field Gross-Pitaevskii model. Constructing and scaling the many-body Hamiltonian we account for the overall physics of the cell aggregate (e.g., the kinetic pressure, the cell-cell interaction, and the external trapping of the cells) together with the nonconservative effect of mitosis and necrosis, and find a compelling agreement with the tumor culture data for the V79 cells both in qualitative and quantitative sense. Results, obtained through an scaling of the characteristic length √ħ/ mw (ħ = h/2π, h = Planck's constant; m = mass; ω ≡ frequency) associated with the wave equation by the ratio of an effective cell size in tumor to the effective atom size in a condensate, demonstrate a scaling behavior for the tumor and reveal the role of underlying physical interactions, in particular the presence and influence of nonlinear cell-cell interactions, in the growth dynamics.

THEORY OF NONLINEAR MATTER WAVES IN OPTICAL LATTICES

We consider several effects of the matter wave dynamics which can be observed in Bose–Einstein condensates embedded into optical lattices. For low-density condensates, we derive approximate evolution equations, the form of which depends on relation among the main spatial scales of the system. Reduction of the Gross–Pitaevskii equation to a lattice model (the tight-binding approximation) is also presented. Within the framework of the obtained models, we consider modulational instability of the condensate, solitary and periodic matter waves, paying special attention to different limits of the solutions, i.e. to smooth movable gap solitons and to strongly localized discrete modes. We also discuss how the Feshbach resonance, a linear force and lattice defects affect the nonlinear matter waves.

Dynamics of bright matter-wave solitons in a Bose–Einstein condensate with inhomogeneous scattering length

We investigate dynamical effects of a bright soliton in Bose–Einstein condensed (BEC) systems with local and smooth space variations of the two-body atomic scattering length. It includes a discussion about the possible observation of a new type of standing nonlinear atomic matter wave in cigar-type traps. A rich dynamics is observed in the interaction between the soliton and an inhomogeneity. By considering an analytical time-dependent variational approach and also full numerical simulation of one-dimensional and three-dimensional Gross–Pitaevskii equations, we study processes such as trapping, reflection and transmission of the bright matter soliton due to the impurity. We also derive conditions for the collapse of the bright solitary wave, considering a quasi-one-dimensional BEC with attractive local inhomogeneity.

Simulation of multicellular tumor growth using a nonlinear matter wave equation

Simulation of multicellular tumor growth using a nonlinear matter wave equation

Cold Quantum Gases

This Special Issue of Journal of Optics B: Quantum and Semiclassical Optics brings together the contributions of various researchers working on theoretical and experimental aspects of cold quantum gases. Different aspects of atom optics, matter wave interferometry, laser manipulation of atoms and molecules, and production of very cold and degenerate gases are presented. The variety of subjects demonstrates the steadily expanding role associated with this research area. The topics discussed in this issue, extending from basic physics to applications of atom optics and of cold atomic samples, include: •Bose--Einstein condensation •Fermi degenerate gases •Characterization and manipulation of quantum gases •Coherent and nonlinear cold matter wave optics •New schemes for laser cooling •Coherent cold molecular gases •Ultra-precise atomic clocks •Applications of cold quantum gases to metrology and spectroscopy •Applications of cold quantum gases to quantum computing •Nanoprobes and nanolithography. This special issue is published in connection with the 7th International Workshop on Atom Optics and Interferometry, held in Lunteren, The Netherlands, from 28 September to 2 October 2002. This was the last in a series of Workshops organized with the support of the European Community that have greatly contributed to progress in this area. The scientific part of the Workshop was managed by A Hemmerich, W Hogervorst, W Vassen and J T M Walraven, with input from members of the International Programme Committee who are listed below. The practical aspects of the organization were ably handled by Petra de Gijsel from the Vrije Universiteit in Amsterdam. The Workshop was funded by the European Science Foundation (programme BEC2000+), the European Networks 'Cold Quantum Gases (CQG)', coordinated by E Arimondo, and 'Cold Atoms and Ultraprecise Atomic Clocks (CAUAC)', coordinated by J Henningsen, by the German Physical Society (DFG), by the Dutch Foundation for Fundamental Research on Matter (FOM) and by the Dutch Gelderland province. We thank all these sponsors and the members of the International Programme Committee for making the Workshop such a success. At this point we take the opportunity to express our gratitude to both authors and reviewers, for their efforts in preparing and ensuring the high quality of the papers in this special issue. Wim Vassen Vrije Universiteit, Amsterdam Andreas Hemmerich Universitat Hamburg Ennio Arimondo Universita di Pisa Guest Editors International Programme Committee A Aspect Orsay, France E Cornell Boulder, USA W Ertmer Hannover, Germany T W Haensch Munich, Germany A Hemmerich Hamburg, Germany W Hogervorst Amsterdam, The Netherlands D Kleppner Cambridge, USA C Salomon Paris, France G V Shlyapnikov Amsterdam, Paris, Moscow S Stringari Trento, Italy W Vassen Amsterdam, The Netherlands J T M Walraven Amsterdam, The Netherlands