Self-trapping and Josephson tunneling solutions to the nonlinear Schr" odinger/Gross-Pitaevskii Equation (original) (raw)
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Physica D: Nonlinear Phenomena, 2003
We introduce a model of a two-core system, based on an equation of the Ginzburg-Landau (GL) type, coupled to another GL equation, which may be linear or nonlinear. One core is active, featuring intrinsic linear gain, while the other one is lossy. The difference from previously studied models involving a pair of linearly coupled active and passive cores is that the stabilization of the system is provided not by a linear diffusion-like term, but rather by a cubic or quintic dissipative term in the active core. Physical realizations of the models include systems from nonlinear optics (semiconductor waveguides or optical cavities), and a double-cigar-shaped Bose-Einstein condensate with a negative scattering length, in which the active "cigar" is an atom laser. The replacement of the diffusion term by the nonlinear loss is principally important, as diffusion does not occur in these physical media, while nonlinear loss is possible. A stability region for solitary pulses is found in the system's parameter space by means of direct simulations. One border of the region is also found in an analytical form by means of a perturbation theory. Moving pulses are studied too. It is concluded that collisions between them are completely elastic, provided that the relative velocity is not too small. The pulses withstand multiple tunneling through potential barriers. Robust quantum-rachet regimes of motion of the pulse in a time-periodic asymmetric potential are found as well.
Physical Review Letters, 2005
It is proven that periodically varying and sign definite nonlinearity in a general case does not prevent collapse in two-and three-dimensional nonlinear Schrödinger equations: at any oscillation frequency of the nonlinearity blowing up solutions exist. Contrary to the results known for a sign alternating nonlinearity, increase of the frequency of oscillations accelerates collapse. The effect is discussed from the viewpoint of scaling arguments. For the three-dimensional case a sufficient condition for existence of collapse is rigorously established. The results are discussed in the context of the meanfield theory of Bose-Einstein condensates with time dependent scattering length.
Physical Review A, 2002
The criteria for validity of adiabaticity for nonlinear wave equations are considered within the context of atomic matter-waves tunneling from macroscopically populated optical standing-wave traps loaded from a Bose-Einstein condensate. We show that even when the optical standing wave is slowly turned on and the condensate behaves adiabatically during this turn-on, once the tunnelingtime between wells in the optical lattice becomes longer than the nonlinear timescale , adiabaticity breaks down and a significant spatially varying phase develops across the condensate wave function from well to well. This phase drastically affects the contrast of the fringe pattern in Josephson-effect interference experiments, and the condensate coherence properties in general.
Solitons in Bose-Einstein condensates trapped in a double-well potential
Physica D-Nonlinear Phenomena, 2004
We investigate, analytically and numerically, families of bright solitons in a system of two linearly coupled nonlinear Schrödinger/Gross-Pitaevskii equations, describing two Bose-Einstein condensates trapped in an asymmetric double-well potential, in particular, when the scattering lengths in the condensates have arbitrary magnitudes and opposite signs. The solitons are found to exist everywhere where they are permitted by the dispersion law. Using the Vakhitov-Kolokolov criterion and numerical methods, we show that, except for small regions in the parameter space, the solitons are stable to small perturbations. Some of them feature self-trapping of almost all the atoms in the condensate with no atomic interaction or weak repulsion coupled to the self-attractive condensate. An unusual bifurcation is found, when the soliton bifurcates from the zero solution without a visible jump in the shape, but with a jump in the number of trapped atoms. By means of numerical simulations, it is found that, depending on values of the parameters and the initial perturbation, unstable solitons either give rise to breathers or completely break down into incoherent waves ("radiation"). A version of the model with the self-attraction in both components, which applies to the description of dual-core fibers in nonlinear optics, is considered too, and new results are obtained for this much studied system.
Macroscopic quantum tunnelling of Bose–Einstein condensates in a finite potential well
Journal of Physics B-atomic Molecular and Optical Physics, 2005
Bose-Einstein condensates are studied in a potential of finite depth which supports both bound and quasi-bound states. This potential, which is harmonic for small radii and decays as a Gaussian for large radii, models experimentally relevant optical traps. The nonlinearity, which is proportional to both the number of atoms and the interaction strength, can transform bound states into quasi-bound ones. The latter have a finite lifetime due to tunnelling through the barriers at the borders of the well. We predict the lifetime and stability properties for repulsive and attractive condensates in one, two, and three dimensions, for both the ground state and excited soliton and vortex states. We show, via a combination of the variational and WKB approximations, that macroscopic quantum tunnelling in such systems can be observed on time scales of 10 milliseconds to 10 seconds.
Physical Review A, 2002
We present a representative set of analytic stationary state solutions of the Nonlinear Schrödinger equation for a symmetric double square well potential for both attractive and repulsive nonlinearity. In addition to the usual symmetry preserving even and odd states, nonlinearity introduces quite exotic symmetry breaking solutions-among them are trains of solitons with different number and sizes of density lumps in the two wells. We use the symmetry breaking localized solutions to form macroscopic quantum superposition states and explore a simple model for the exponentially small tunneling splitting.
Quenched dynamics of two-dimensional solitons and vortices in the Gross-Pitaevskii equation
2012
We consider a two-dimensional (2D) counterpart of the experiment that led to the creation of quasi-1D bright solitons in Bose-Einstein condensates (BECs) (2002 Nature 417 150-3). We start by identifying the fundamental state of the 2D Gross-Pitaevskii equation for repulsive interactions, with a harmonic-oscillator (HO) trap, and with or without an optical lattice (OL). Subsequently, we switch the sign of the interaction to induce interatomic attraction and monitor the ensuing dynamics. Regions of stable self-trapping and a catastrophic collapse of 2D fundamental states are identified in the parameter plane of the OL strength and BEC norm. The increase of the OL strength expands the persistence domain for the solitary waves to larger norms. For single-charged solitary vortices, in addition to the survival and collapse regimes, an intermediate one is identified, where the vortex resists the collapse but loses its structure, transforming into a single-hump state. The same setting may also be implemented in the context of optical solitons and vortices, using photonic-crystal fibers.
Stability of matter–wave soliton in a time-dependent complicated trap
Chaos, Solitons & Fractals, 2012
We examine the possibility to generate localized structures in effectively one-dimensional Gross-Pitaevskii with a time-dependent scattering length and a complicated potential. Through analytical methods invoking a generalized lens-type transformation and the Darboux transformation, we present the integrable condition for the Gross-Pitaevskii equation and obtain the exact analytical solution which describes the modulational instability and the propagation of bright solitary waves on a continuous wave background. The dynamics and stability of this solution are analyzed. Moreover, by employing the extended tanhfunction method we obtain the exact analytical solutions which describes the propagation of dark and other families of solitary waves.
Physical Review E, 2013
Motivated by recent proposals of "collisionally inhomogeneous" Bose-Einstein condensates (BECs), which have a spatially modulated scattering length, we introduce a phase imprint into the macroscopic order parameter governing the dynamics of BECs with spatiotemporal varying scattering length described by a cubic Gross-Pitaevskii (GP) equation and then suitably engineer the imprinted phase to generate the modified GP equation, also called the cubic derivative nonlinear Schrödinger (NLS) equation. This equation describes the dynamics of condensates with two-body (attractive and repulsive) interactions in a time-varying quadratic external potential. We then carry out a theoretical analysis which invokes a lens-type transformation that converts the cubic derivative NLS equation into a modified NLS equation with only explicit temporal dependence. Our analysis suggests a particular interest in a specific time-varying potential with the strength of the magnetic trap ∼1/(t + t *) 2. For a time-varying quadratic external potential of this kind, an explicit expression for the growth rate of a purely growing modulational instability is presented and analyzed. We point out the effect of the imprint parameter and the parameter t * on the instability growth rate, as well as on the solitary waves of the BECs.
arXiv: Other Condensed Matter, 2004
We study, analytically and numerically, the stationary states in the system of two linearly coupled nonlinear Schr{\"o}dinger equations in two spatial dimensions, with the nonlinear interaction coefficients of opposite signs. This system is the two-dimensional analog of the coupled-mode equations for a condensate in the double-well trap [\textit{Physical Review A} \textbf{69}, 033609 (2004)]. In contrast to the one-dimensional case, where the bifurcation from zero leads to stable bright solitons, in two spatial dimensions this bifurcation results in the appearance of unstable soliton solutions (the Townes-type solitons). With the use of a parabolic potential the ground state of the system is stabilized. It corresponds to strongly coupled condensates and is stable with respect to collapse. This is in sharp contrast to the one-dimensional case, where the ground state corresponds to weakly coupled condensates and is unstable. Moreover, the total number of atoms of the stable groun...