Gross-Pitaevskii Model Research Articles

Localized Analytical Solutions and Parameters Analysis in the Nonlinear Dispersive Gross–Pitaevskii Mean‐Field GP (m,n) Model with Space‐Modulated Nonlinearity and Potential

The novel nonlinear dispersive Gross–Pitaevskii (GP) mean‐field model with the space‐modulated nonlinearity and potential (called GP equation) is investigated in this paper. By using self‐similar transformations and some powerful methods, we obtain some families of novel envelope compacton‐like solutions spikon‐like solutions to the GP equation. These solutions possess abundant localized structures because of infinite choices of the self‐similar function . In particular, we choose as the Jacobi amplitude function and the combination of linear and trigonometric functions of space x so that the novel localized structures of the GP(2, 2) equation are illustrated, which are much different from the usual compacton and spikon solutions reported. Moreover, it is shown that GP(m, 1) equation with linear dispersion also admits the compacton‐like solutions for the case and spikon‐like solutions for the case .

Quantum vortex reconnections

We study reconnections of quantum vortices by numerically solving the governing Gross-Pitaevskii equation. We find that the minimum distance between vortices scales differently with time before and after the vortex reconnection. We also compute vortex reconnections using the Biot-Savart law for vortex filaments of infinitesimal thickness, and find that, in this model, reconnections are time symmetric. We argue that the likely cause of the difference between the Gross-Pitaevskii model and the Biot-Savart model is the intense rarefaction wave which is radiated away from a Gross-Pitaeveskii reconnection. Finally we compare our results to experimental observations in superfluid helium and discuss the different length scales probed by the two models and by experiments.

Dissipative solitons and vortices in polariton Bose-Einstein condensates

We examine spatial localisation and dynamical stability of Bose-Einstein condensates of exciton-polaritons in microcavities under the condition of off-resonant spatially inhomogeneous optical pumping both with and without a harmonic trapping potential. We employ the open-dissipative Gross-Pitaevskii model for describing an incoherently pumped polariton condensate coupled to an exciton reservoir, and reveal that spatial localisation of the steady-state condensate occurs due to the effective self-trapping created by the polariton flows supported by the spatially inhomogeneous pump, regardless of the presence of the external potential. A ground state of the polariton condensate with repulsive interactions between the quasiparticles represents a dynamically stable bright dissipative soliton. We also investigate the conditions for sustaining spatially localised structures with non-zero angular momentum in the form of single-charge vortices.

Quantized vortex reconnection: Fixed points and initial conditions

Quantized vortices are phase singularities in complex fields. In superfluids, they appear as mobile interacting defects that may cross and reconnect by exchanging tails. Reconnection is a topology-changing event that allows vortex tangles to decay; it is a defining signature of quantum turbulence. We report a family of fixed points (i.e., stationary solutions), including planar forms, that capture reconnection in the Gross-Pitaevskii model in contrast to previous suggestions of pyramidal structures. These are obtained using a well known, systematic method for generating low-energy relaxed initial conditions for Gross-Pitaevskii simulations.

Fluctuating and dissipative dynamics of dark solitons in quasicondensates

The fluctuating and dissipative dynamics of matter-wave dark solitons within harmonically trapped, partially condensed Bose gases is studied both numerically and analytically. A study of the stochastic Gross-Pitaevskii equation, which correctly accounts for density and phase fluctuations at finite temperatures, reveals dark-soliton decay times to be lognormally distributed at each temperature, thereby characterizing the previously predicted long-lived soliton trajectories within each ensemble of numerical realizations [S. P. Cockburn et al., Phys. Rev. Lett. 104, 174101 (2010)]. Expectation values for the average soliton lifetimes extracted from these distributions are found to agree well with both numerical and analytic predictions based upon the dissipative Gross-Pitaevskii model (with the same ab initio damping). Probing the regime for which 0.8 ${k}_{B}T<\ensuremath{\mu}<1.6$ ${k}_{B}T$, we find average soliton lifetimes to scale with temperature as $\ensuremath{\tau}\ensuremath{\sim}{T}^{\ensuremath{-}4}$, in agreement with predictions previously made for the low-temperature regime ${k}_{B}T\ensuremath{\ll}\ensuremath{\mu}$. The model is also shown to capture the experimentally relevant decrease in the visibility of an oscillating soliton due to the presence of background fluctuations.

Interaction of half-quantized vortices in two-component Bose-Einstein condensates

We study the asymptotic interaction between two half-quantized vortices in two-component Bose-Einstein condensates. When two vortices in different components are placed at distance $2R$, the leading order of the force between them is found to be $(\mathrm{ln}R/\ensuremath{\xi}\ensuremath{-}1/2)/{R}^{3}$, in contrast to $1/R$ between vortices placed in the same component. We derive it analytically using the Abrikosov ansatz and the profile functions of the vortices, confirmed numerically with the Gross-Pitaevskii model. We also find that the short-range cutoff of the intervortex potential linearly depends on the healing length.

Gross-Pitaevskii model of pulsar glitches

The first large-scale quantum mechanical simulations of pulsar glitches are presented, using a Gross-Pitaevskii model of the crust-superfluid system in the presence of pinning. Power-law distributions of simulated glitch sizes are obtained, in accord with astronomical observations, with exponents ranging from -0.55 to -1.26. Examples of large-scale simulations, containing $\sim 200$ vortices, reveal that these statistics persist in the many-vortex limit. Waiting-time distributions are also constructed. These and other statistics support the hypothesis that catastrophic unpinning mediated by collective vortex motion produces glitches; indeed, such collective events are seen in time-lapse movies of superfluid density. Three principal trends are observed. (1) The glitch rate scales proportional to the electromagnetic spin-down torque. (2) A strong positive correlation is found between the strength of vortex pinning and mean glitch size. (3) The spin-down dynamics depend less on the pinning site abundance once the latter exceeds one site per vortex, suggesting that unpinned vortices travel a distance comparable to the inter-vortex spacing before repinning.

Short-wave excitations in non-local Gross-Pitaevskii model

Abstract We show that a non-local form of the Gross-Pitaevskii equation describes not only long-wave excitations, but also the short-wave ones in Bose-condensate systems. At certain parameter values, the excitation spectrum mimics the Landau spectrum of quasi-particle excitations in superfluid helium with roton minimum. The excitation wavelength, at which the roton minimum exists, is close to the inter-particle interaction range. We determine how the roton gap and the effective roton mass depend on the interaction potential parameters, and show that the existence domain of the spectrum with a roton minimum is reduced if one accounts for an inter-particle attraction.

Annular Bose-Einstein Condensates in the Lowest Landau Level

A rotating superuid such as a Bose-Einstein condensate is usually described by the GrossPitaevskii (GP) model. An important issue is to determine from this model the properties of the quantized vortices that a superuid nucleates when set into rotation. In this paper we address the minimization of a two dimensional GP energy functional describing a rotating annular Bose-Einstein condensate. In a certain limit it is physically relevant to restrict the minimization to the LowestLandau-Level, that is the rst eigenspace of the Ginzburg-Landau operator. Taking the particular structure of this space into account we obtain theoretical results concerning the vortices of the condensate. We also compute the vortices’ locations by a numerical minimization procedure. We nd that they lie on a distorted lattice and that multiple quantized vortices appear in the central hole of low matter density.

Static properties of coupled atomic-molecular Bose-Einstein condensates in modified Gross-Pitaevskii approach

We analyze the stationary state properties of an atomic Bose-Einstein condensate coupled to a molecular condensate via a Raman photoassociation process using Gross-Pitaevskii and modified Gross-Pitaevskii model and compare the results. We find that the static properties depend crucially on the Raman detuning parameter, which can be experimentally varied. We also show how the analytical expressions for densities of atoms and molecules of the hybrid system can be found in the Thomas-Fermi approximation where kinetic energies are not included. We have explored the feasibility of maximum molecular BEC formation by varying the Raman detuning.

Direct energy cascade in two-dimensional compressible quantum turbulence

We numerically study two-dimensional quantum turbulence with a Gross-Pitaevskii model. With the energy initially accumulated at large scale, quantum turbulence with many quantized vortex points is generated. Due to the lack of enstrophy conservation in this model, direct energy cascade with a Kolmogorov-Obukhov energy spectrum $E(k)\ensuremath{\propto}{k}^{\ensuremath{-}5/3}$ is observed, which is quite different from two-dimensional incompressible classical turbulence in the decaying case. A positive value for the energy flux guarantees a direct energy cascade in the inertial range (from large to small scales). After almost all the energy at the large scale cascades to the small scale, the compressible kinetic energy realizes the thermodynamic equilibrium state without quantized vortices.

Nonautonomous matter-wave solitons near the Feshbach resonance

By means of analytical and numerical methods, we reveal the main features of nonautonomous matter-wave solitons near the Feshbach resonance in a one-dimensional Bose-Einstein condensate confined by a harmonic potential with a varying-in-time longitudinal trapping frequency. Based on the generalized nonautonomous Gross-Pitaevskii model, we show that solitons in nonautonomous physical systems exist only under certain conditions so that varying-in-time nonlinearity and confining harmonic potential cannot be chosen independently; they satisfy the exact integrability scenarios and complement each other. We focus on the most physically important examples where the applied magnetic field is either a linearly or a periodically varying-in-time function. In the case of periodically varying scattering length, variations of confining harmonic potential are found to be sign-reversible (periodic attractive and repulsive) to support the soliton-management regime. We investigate the losses of validity of one-dimensional (1D) approximation in the cases when, by the joint action of varying-in-time nonlinearity and confining potential, the atom cloud can be compressed from an initially elongated quasi-1D cigar-shaped geometry to a final ball-shaped three-dimensional geometry and the induced soliton collapse may occur.

Possibility of Inverse Energy Cascade in Two-Dimensional Quantum Turbulence

We numerically study two-dimensional quantum turbulence by using the Gross-Pitaevskii model and the Vortex-Point model. Two-dimensional classical turbulence has long been investigated as an ideal system of geophysical phenomena. The amazing character of this turbulence is inverse energy cascade which carries energy toward low wavenumbers and excites large-scale motion. We expect these phenomena in two-dimensional quantum turbulence because in three-dimensional turbulence we know classical and quantum analogue. However, we have not yet confirmed inverse cascade in two-dimensional quantum turbulence. In this work, we show numerical results and discuss why inverse cascade does not occur in two-dimensional quantum turbulence by referring to the mechanism of two-dimensional classical turbulence.

Role of the relative phase in the merging of two independent Bose-Einstein condensates

We study the merging of two independent Bose-Einstein condensates with arbitrary initial phase difference, in the framework of a one-dimensional time-dependent Gross-Pitaevskii model. The role of the initial phase difference in the process is discussed, and various types of phase-sensitive excitations are identified.

Quantum Turbulence in Trapped Bose-Einstein Condensates

Quantum turbulence is recently one of the most important topics in low temperature physics. Physics of quantum turbulence and quantized vortices have been studied chiefly in superfluid helium. The realization of atomic Bose-Einstein condensation has allowed us to study quantized vortives in the fascinating systems too. Generally there are two kinds of cooperative phenomena comprised of quantized vortices, from the research history of superfluid helium. Lots of studies in atomic Bose-Einstein condensates (BECs) are devoted mainly to the issues of vortex lattices under rotation. However, another important state on quantized vortices, namely quantum turbulence, has been never studied in atomic BECs. In this work, we address for the first time quantum turbulence in atomic BECs theoretically and numerically. After reviewing shortly the recent developments on quantum turbulecne, we propose how to make quantum turbulence in a trapped BEC by combining rotation around two axes, and confirm the Kolmogorov spectra by the Gross-Pitaevskii model.

Quantum tunneling of vortices in two-dimensional superfluids

We examine the problem of quantum vortex tunneling in the Gross-Pitaevskii model. The effect of gapless phonons is to produce a super-Ohmic quantum dissipation that renormalizes the effective tunneling parameters but does not destroy quantum coherence. The renormalization of the instanton frequency is computed within a specific model tunneling potential.

Quantum Turbulence in Trapped Atomic Bose-Einstein Condensates

Generally there are two kinds of cooperative phenomena comprised of quantized vortices. One is a vortex lattice under rotation, and the other is a vortex tangle (quantum turbulence) made by some flow. Both have been studied in the field of superfluid helium through the long research history. On the other hand, the research of atomic Bose-Einstein condensates (BECs) has been limited to the former case, namely a vortex lattice. In this work, we address for the first time quantum turbulence in atomic BECs theoretically and numerically. We propose how to make quantum turbulence in a trapped BEC by combining rotation around two axes, and confirm the Kolmogorov spectra by the Gross-Pitaevskii model.

Dark matter-wave solitons in the dimensionality crossover

We consider the statics and dynamics of dark matter-wave solitons in the dimensionality crossover regime from three dimensions (3D) to one dimension (1D). There, using the nonpolynomial Schr\odinger mean-field model, we find that the anomalous mode of the Bogoliubov spectrum has an eigenfrequency which coincides with the soliton oscillation frequency obtained by the 3D Gross-Pitaevskii model. We show that substantial deviations (of the order of 10% or more) from the characteristic frequency ${\ensuremath{\omega}}_{z}∕\sqrt{2}$ (${\ensuremath{\omega}}_{z}$ being the longitudinal trap frequency) are possible even in the purely 1D regime.

Vortex Mass in a Superfluid at Low Frequencies

An inertial mass of a vortex can be calculated by driving it around in a circle with a steadily revolving pinning potential. We show that in the low-frequency limit this gives precisely the same formula that was used by Baym and Chandler, but find that the result is not unique and depends on the force field used to cause the acceleration. We apply this method to the Gross-Pitaevskii model, and derive a simple formula for the vortex mass. We study both the long-range and short-range properties of the solution. We agree with earlier results that the nonzero compressibility leads to a divergent mass. From the short-range behavior of the solution we find that the mass is sensitive to the form of the pinning potential, and diverges logarithmically when the radius of this potential tends to zero.

Limits to the flux of a continuous atom laser

We present experimental results demonstrating that the flux of a continuous atom laser beam cannot be arbitrarily increased by increasing the strength of output coupling. We find that the continuous atom laser has large density fluctuations when the Rabi frequency becomes greater than the radial trapping frequency, due to the output coupling populating all accessible Zeeman states. In addition, we find that a state-changing output coupler for a continuous atom laser switches off as the output-coupling strength, parametrized in general by the Rabi frequency, is increased above a critical value. The numerical solution of a one-dimensional five-state Gross-Pitaevskii model for the atom laser is in good qualitative agreement with the experiment.