One-dimensional Gross-Pitaevskii Equation Research Articles

Bright solitons and interaction in the higher-order Gross-Pitaevskii equation investigated with Hirota's bilinear method

In this paper, we investigate bright solitons and their interactions in trapless Bose-Einstein condensate described by a one-dimensional Gross-Pitaevskii equation in the presence of higher-order nonlinearity. By constructing appropriate Hirota bilinear form of the problem, we derive the one- and two-bright soliton solutions where the effects of the higher-order interaction term are discussed in details. The profile of different solutions obtained clearly presents the phenomenon of soliton interaction that embodies the properties of soliton. It also highlights the impact of higher-order interactions that plays the role of wave amplification/compression resulting from the condensate. The theoretical results confirm that Hirota bilinear method is an efficient and interesting tool for the analysis of nonlinearity in physical science. In addition, the higher-order interactions have significant effects on the stabilization process of Bose-Einstein condensates.

The Study of Formation of Sonic Black Hole in One-Dimensional Bose-Einstein Condensate

Based on the one-dimensional Gross-Pitaevskii equation (GPE) model, this work studies the formation of acoustic black holes in one-dimensional Bose Einstein condensate (BEC) in harmonic oscillator external potential, and extending the applicable scenarios of the GPE model for its specific application of simulating the formation conditions of acoustic black holes in one-dimensional systems. At the same time, We deepen the understanding of the analytical formula for determining the production conditions of acoustic black holes and the evolution formula of the radius of acoustic black holes.

Training and projecting: A reduced basis method emulator for many-body physics

We present the reduced basis method as a tool for developing emulators for equations with tunable parameters within the context of the nuclear many-body problem. The method uses a basis expansion informed by a set of solutions for a few values of the model parameters and then projects the equations over a well-chosen low-dimensional subspace. We connect some of the results in the eigenvector continuation literature to the formalism of reduced basis methods and show how these methods can be applied to a broad set of problems. As we illustrate, the possible success of the formalism on such problems can be diagnosed beforehand by a principal component analysis. We apply the reduced basis method to the one-dimensional Gross-Pitaevskii equation with a harmonic trapping potential and to nuclear density functional theory for $^{48}$Ca, achieving speed-ups of more than x150 in both cases when compared to traditional solvers. The outstanding performance of the approach, together with its straightforward implementation, show promise for its application to the emulation of computationally demanding calculations, including uncertainty quantification.

Bose-Einstein condensates in quasiperiodic lattices: Bosonic Josephson junction, self-trapping, and multimode dynamics

A Bose-Einstein condensate loaded in a one-dimensional bichromatic optical lattice with constituent sublattices having incommensurate periods is considered. Using the rational approximations for the incommensurate periods, we show that below the mobility edge the localized states are distributed nearly homogeneously in the space and explore the versatility of such potentials. We show that superposition of symmetric and antisymmetric localized states can be used to simulate various physical dynamical regimes, known to occur in double-well and multiwell traps. As examples, we obtain an alternative realization of a bosonic Josephson junction, whose coherent oscillations display beatings or switching in the weakly nonlinear regime, and describe self-trapping and four-mode dynamics, mimicking coherent oscillations and self-trapping in four-well potentials. These phenomena can be observed for different pairs of modes, which are localized due to the interference rather than due to a confining trap. The results obtained using few-mode approximations are compared with the direct numerical simulations of the one-dimensional Gross-Pitaevskii equation. The localized states and the related dynamics are found to persist for long times even in the repulsive condensates. We also described bifurcations of the families of nonlinear modes, the symmetry breaking, and stable minigap solitons.

Enhanced transport of spin-orbit-coupled Bose gases in disordered potentials.

Anderson localization is a single-particle localization phenomena in disordered media that is accompanied by an absence of diffusion. Spin-orbit coupling (SOC) describes an interaction between a particle's spin and its momentum that directly affects its energy dispersion, for example, creating dispersion relations with gaps and multiple local minima. We show theoretically that combining one-dimensional spin-orbit coupling with a transverse Zeeman field suppresses the effects of disorder, thereby increasing the localization length and conductivity. This increase results from a suppression of backscattering between states in the gap of the SOC dispersion relation. Here, we focus specifically on the interplay of disorder from an optical speckle potential and SOC generated by two-photon Raman processes in quasi-one-dimensional Bose-Einstein condensates. We first describe backscattering by using a Fermi golden rule approach, and then numerically confirm this picture by solving the time-dependent one-dimensional Gross-Pitaevskii equation for a weakly interacting Bose-Einstein condensate with SOC and disorder. We find that on the tens of millisecond timescale of typical cold atom experiments moving in harmonic traps, initial states with momentum in the zero-momentum SOC gap evolve with negligible backscattering, while without SOC these same states rapidly localize.

Emulation of spin-orbit coupling for solitons in nonlinear optical media

We design a framework based on the spatial-domain copropagation of two light beams with mutually orthogonal polarizations and opposite transverse components of carrier wave vectors in a nonlinear waveguide with randomly varying birefringence, the averaging with respect to which introduces an effective Manakov nonlinearity in the system. The corresponding two-component system of nonlinear Schr\odinger equations is derived, being similar to the system of coupled one-dimensional Gross-Pitaevskii equations for a binary spin-orbit-coupled Bose-Einstein condensate. The system may also include an effective Rabi coupling (direct linear mixing of the components) and a periodic potential, representing a photonic-crystal structure in the underlying waveguide. For self-focusing and self-defocusing signs of nonlinearity, soliton solutions of several symmetry types are obtained by means of numerical methods, and their stability is investigated, including gap solitons in the case when the periodic potential is present.

Observation of nonequilibrium motion and equilibration in polariton rings

We present a study of the macroscopic dynamics of a polariton condensate formed by non-resonant optical excitation in a quasi-one-dimensional ring-shaped microcavity. The presence of a gradient in the cavity photon energy creates a macroscopic trap for the polaritons in which a single mode condensate is formed. With time- and energy-resolved imaging we show the role of interactions in the motion of the condensate as it undergoes equilibration in the ring. These experiments also give a direct measurement of the polariton-polariton interaction strength above the condensation threshold. Our observations are compared to the open-dissipative one-dimensional Gross-Pitaevskii equation which shows excellent qualitative agreement.

Ideal gas model of Bose-Einstein condensates confined in the parabolic trap

By using the one-dimensional canonical partition function, we modelled an ideal gas-like form of a set of Bose-Einstein condensates confined by a three-dimensional anisotropic parabolic trap. The model itself was constructed by taking the eigenenergies of the one-dimensional Gross-Pitaevskii equation in the longitudinal direction and enabling the harmonic volume as the inverse cube of average geometric trapping frequency to substitute the real volume. In this paper, we showed that the condensates form an ideal gas represented by its equation of state and have similar mature to Einstein’s solid-like model with the corrections in both the low and high temperatures in the internal energy formulations.

Energies of the static solitary wave solutions of the one-dimensional Gross-Pitaevskii equation

We calculated the energies of the static solitary wave solutions of the one-dimensional Gross-Pitaevskii equation with the time-dependent parabolic trap, the time-dependent scattering wave length of s-wave, and the time-dependent external potential describing a gain or loss term. Some written solutions of the equation were used, two of which are based on the experimental results. The solutions satisfy the condition of solitary wave solution since they are localized over the space. By this argument, the energies were obtained by integrating the Hamiltonian density over the space formulated in the classical field theory. To do that, we constructed the appropriate Lagrangian density representing the equation by initially writing the ansatz Lagrangian density and then substituting into the Euler-Lagrange equation. We found that two of the solutions have the same energies and the other one should mathematically have the pure imaginary function describing the gain-loss term to achieve the real energy.

Excitation Modes of Bright Matter-Wave Solitons.

We experimentally study the excitation modes of bright matter-wave solitons in a quasi-one-dimensional geometry. The solitons are created by quenching the interactions of a Bose-Einstein condensate of cesium atoms from repulsive to attractive in combination with a rapid reduction of the longitudinal confinement. Adeliberate mismatch of quench parameters allows for the excitation of breathing modes of the emerging soliton and for the determination of its breathing frequency as a function of atom number and confinement. In addition, we observe signatures of higher-order solitons and the splitting of the wave packet after the quench. Our experimental results are compared to analytical predictions and to numerical simulations of the one-dimensional Gross-Pitaevskii equation.

Symmetry breaking of quantum droplets in a dual-core trap

We consider the dynamical model of a binary bosonic gas trapped in a symmetric dual-core cigar-shaped potential. The setting is modeled by a system of linearly-coupled one-dimensional Gross-Pitaevskii equations with the cubic self-repulsive terms and quadratic attractive ones,which represent the Lee-Huang-Yang corrections to the mean-field theory in this geometry. The main subject is spontaneous symmetry breaking (SSB) of quantum droplets (QDs), followed by restoration of the symmetry, with respect to the symmetric parallel-coupled trapping cores, following the increase of the QD's total norm. The SSB transition and inverse symmetry-restoring one form a bifurcation loop, whose shape in concave at small values of the inter-core coupling constant, $\kappa$, and convex at larger $\kappa $. The loop does not exist above a critical value of $\kappa $. At very large values of the norm, QDs do not break their symmetry, featuring a flat-top shape. Some results are obtained in an analytical form, including an exact front solution connecting constant zero and finite values of the wave function. Collisions between moving QDs are considered too, demonstrating a trend to merger into breathers.

Dipolar-Induced Formation of Domain in Spin-2 Bose-Einstein Condensates

In this paper, we investigate the formation of domain in a spin-2 Bose-Einstein condensates by numerically solving the one-dimensional Gross-Pitaevskii equation. We focus the effect of dipolar interaction on the formation of domain. We reveal that the density distribution obeys the single-mode approximation at small dipole strength $$ {c}_d^{\prime } $$ . In this case, the domain isn’t observed. Nevertheless, the formation of domain can be realized by further increasing the dipole strength to a critical value. In addition, we also calculate the domain structure by changing the trap geometric, the rich domain structures are observed. Fianlly, we show that the number of position Nd increases with increasing dipole strength $$ {c}_d^{\prime } $$ and it keeps even in current considered parameters.

Modeling of matter-wave solitons in a nonlinear inductor-capacitor network through a Gross-Pitaevskii equation with time-dependent linear potential.

A lossless nonlinear LC transmission network is considered. With the use of the reductive perturbation method in the semidiscrete limit, we show that the dynamics of matter-wave solitons in the network can be modeled by a one-dimensional Gross-Pitaevskii (GP) equation with a time-dependent linear potential in the presence of a chemical potential. An explicit expression for the growth rate of a purely growing modulational instability (MI) is presented and analyzed. We find that the potential parameter of the GP equation of the system does not affect the different regions of the MI. Neglecting the chemical potential in the GP equation, we derive exact analytical solutions which describe the propagation of both bright and dark solitary waves on continuous-wave (cw) backgrounds. Using the found exact analytical solutions of the GP equation, we investigate numerically the transmission of both bright and dark solitary voltage signals in the network. Our numerical studies show that the amplitude of a bright solitary voltage signal and the depth of a dark solitary voltage signal as well as their width, their motion, and their behavior depend on (i) the propagation frequencies, (ii) the potential parameter, and (iii) the amplitude of the cw background. The GP equation derived in this paper with a time-dependent linear potential opens up different ideas that may be of considerable theoretical interest for the management of matter-wave solitons in nonlinear LC transmission networks.

Gap solitons for the repulsive Gross-Pitaevskii equation with periodic potential: Coding and method for computation

The paper is devoted to nonlinear localized modes ("gap solitons") for the spatially one-dimensional Gross-Pitaevskii equation (1D GPE) with a periodic potential and repulsive interparticle interactions. It has been recently shown (G. L. Alfimov, A. I. Avramenko, Physica D, 254, 29 (2013)) that under certain conditions all the stationary modes for the 1D GPE can be coded by bi-infinite sequences of symbols of some finite alphabet (called "codes" of the solutions). We present and justify a numerical method which allows to reconstruct the profile of a localized mode by its code. As an example, the method is applied to compute the profiles of gap solitons for 1D GPE with a cosine potential.

Dynamics of matter-wave solitons in Bose-Einstein condensates with time-dependent scattering length and complex potentials

We investigate the dynamics of matter-wave solitons in the one-dimensional (1-D) Gross-Pitaevskii (GP) equation describing Bose-Einstein condensates (BECs) with time-dependent scattering length in varying trapping potentials with feeding/loss term. By performing a modified lens-type transformation, we reduce the GP equation into a classical nonlinear Schrodinger (NLS) equation with distributed coefficients and find its integrable condition. Under the integrable condition, we apply the generalized Jacobian elliptic function method (GJEFM) and present exact analytical solutions which describe the propagation of a bright and dark solitons in BECs. Their stability is examined using analytic method. The obtained exact solutions show that the amplitude of bright and dark solitons depends on the scattering length, while their motion and the total number of BEC atoms depend on the external trapping potential. Our results also shown that the loss of atoms can dominate the aggregation of atoms by the attractive interaction, and thus the peak density can decrease in time despite that the strength of the attractive interaction is increased.

Dynamics of solitary-wave structures in one-dimensional Gross-Pitaevskii equation with distributed coefficients

Motivated by recent experimental investigations in the context of matter wave solitons in Bose-Einstein condensates (BECs), we consider the 1+1 Gross-Pitaevskii equation with complex time-varying harmonic potential, and time-varying cubic and quintic nonlinearities. By performing a modified lens-type transformation for the one-dimensional GP equation, we present one and/or two parameter exact analytical solutions which describe the propagation of bright, kink, and dark solitary waves on the vanishing continuous wave (cw) background. Based on exact analytical solutions of the GP equation, we investigate analytically the dynamics of matter-wave solitons in the BEC systems. Our studies show that the solitons’ amplitude depends on both the scattering length and the feeding/loss term of the potential while their motion depends on the external trapping potential and solution parameters.

Asymptotic stability of the black soliton for the Gross–Pitaevskii equation

We introduce a new framework for the analysis of the stability of solitons for the one-dimensional Gross-Pitaevskii equation. In particular, we establish the asymptotic stability of the black soliton with zero speed.

On the Cauchy Problem and the Black Solitons of a Singularly Perturbed Gross--Pitaevskii Equation

We consider the one-dimensional Gross-Pitaevskii equation perturbed by a Dirac potential. Using a fine analysis of the properties of the linear propagator, we study the well-posedness of the Cauchy Problem in the energy space of functions with modulus 1 at infinity. Then we show the persistence of the stationary black soliton of the unperturbed problem as a solution. We also prove the existence of another branch of non-trivial stationary waves. Depending on the attractive or repulsive nature of the Dirac perturbation and of the type of stationary solutions, we prove orbital stability via a variational approach, or linear instability via a bifurcation argument.

On the Marchenko System and the Long-time Behavior of Multi-soliton Solutions of the One-dimensional Gross-Pitaevskii Equation

We establish a rigorous well-posedness results for the Marchenko system associated to the scattering theory of the one dimensional Gross-Pitaevskii equation (GP). Under some assumptions on the scattering data, these well-posedness results provide regular solutions for (GP). We also construct particular solutions, called $N-$soliton solutions as an approximate superposition of traveling waves. A study for the asymptotic behaviors of such solutions when $t\rightarrow\pm \infty$ is also made.

Nonlinear effects in time-dependent transonic flows: An analysis of analog black hole stability

We study solutions of the one-dimensional Gross-Pitaevskii equation to better understand dynamical instabilities occurring in flowing atomic condensates. Whereas transonic stationary flows can be fully described in simple terms, time-dependent flows exhibit a wide variety of behaviors. When the sound speed is crossed once, we observe that flows analogous to black holes obey something similar to the so-called no hair theorem since their late time profile is stationary and uniquely fixed by parameters entering the Hamiltonian and conserved quantities. For flows analogous to white holes, at late time one finds a macroscopic undulation in the supersonic side which has either a fixed amplitude, or a widely varying one signaling a quasi periodic emission of solitons on the subsonic side. When considering flows which cross the sound speed twice, we observe various scenarios which can be understood from the above behaviors, and from the hierarchy of the growth rates of the dynamical instabilities characterizing such flows.