Pitaevskii Equation Research Articles

Effective spin dynamics of spin-orbit coupled matter-wave solitons in optical lattices

We consider matter–wave solitons in spin–orbit coupled Bose–Einstein condensates embedded in an optical lattice and study the dynamics of the soliton within the framework of Gross–Pitaevskii equations. We express spin components of the soliton pair in terms of nonlinear Bloch equations and investigate the effective spin dynamics. It is seen that the effective magnetic field that appears in the Bloch equation is affected by optical lattices, and thus the optical lattice influences the precessional frequency of the spin components. We make use of numerical approaches to investigate the dynamical behavior of density profiles and center-of-mass of the soliton pair in the presence of the optical lattice. It is shown that the spin density is periodically varying due to flipping of spinors between the two states. The amplitude of spin-flipping oscillation increases with lattice strength. We find that the system can also exhibit interesting nonlinear behavior for chosen values of parameters. We present a fixed point analysis to study the effects of optical lattices on the nonlinear dynamics of the spin components. It is seen that the optical lattice can act as a control parameter to change the dynamical behavior of the spin components from periodic to chaotic.

Inverse design of polaritonic devices

Polaritons, arising from the strong coupling between excitons and photons within microcavities, hold promise for optoelectronic and all-optical devices. They have found applications in various domains, including low-threshold lasers and quantum information processing. To realize complex functionalities, non-intuitive designs for polaritonic devices are required. In this contribution, we use finite-difference time-domain simulations of the dissipative Gross–Pitaevskii equation, written in a differentiable manner, and combine it with an adjoint formulation. Such a method allows us to use topology optimization to engineer the potential landscape experienced by polariton condensates to tailor its characteristics on demand. The potential directly translates to a blueprint for a functional device, and various fabrication and optical control techniques can experimentally realize it. We inverse-design a selection of polaritonic devices, i.e., a structure that spatially shapes the polaritons into a flat-top distribution, a metalens that focuses a polariton, and a nonlinearly activated isolator. The functionalities are preserved when employing realistic fabrication constraints such as minimum feature size and discretization of the potential. Our results demonstrate the utility of inverse design techniques for polaritonic devices, providing a stepping stone toward future research in optimizing systems with complex light–matter interactions.

New dynamics performances to the established dark solitons in Polariton Condensate

Abstract New diverse enormous soliton solutions to the Gross–Pitaevskii Equation, which describes the dynamics of two dark solitons in a polarization condensate under non-resonant pumping, have been constructed for the first time by using two various Schemes. The two utilized Schemes are the Generalized Kudryashov Scheme, the (G'/G)-Expansion Scheme. Throughout these two suggested schemata we construct new diverse forms solutions that include dark, bright-shaped soliton solutions, combined bright-shaped, dark-shaped soliton solutions, hyperbolic function soliton solution, singular shaped soliton solution and other rational soliton solutions. The two 2D and 3D figures designs have been configured by using the Mathematica-program. In addition, the Haar Wavelet Numerical Scheme has been applied to construct the identical numerical behavior for all soliton solutions achieved by the two suggested Schemes to show existing similarity between the soliton solutions and numerical solutions.

Using Artificial Neural Networks to Solve the Gross–Pitaevskii Equation

The current work proposes the incorporation of an artificial neural network to solve the Gross–Pitaevskii equation (GPE) efficiently, using a few realistic external potentials. With the assistance of neural networks, a model is formed that is capable of solving this equation. The adaptation of the parameters for the constructed model is performed using some evolutionary techniques, such as genetic algorithms and particle swarm optimization. The proposed model is used to solve the GPE for the linear case (γ=0) and the nonlinear case (γ≠0), where γ is the nonlinearity parameter in GPE. The results are close to the reported results regarding the behavior and the amplitudes of the wavefunctions.

Generation of patterns and higher harmonics in 1D quantum droplets in tilted and driven quasi-periodic confinements

The generation of patterns by breaking the spatial symmetry in external confinement is a captivating area of physics. The emergence of patterns is a fundamental inquiry spanning various disciplines such as nonlinear optics, condensed matter physics, and fluid dynamics. The article investigates the generation of a variety of patterns in a one-dimensional binary mixture of Bose–Einstein condensate forming quantum droplets. By solving the extended Gross–Pitaevskii equation in the presence of tilted and driven engineered bi-chromatic optical lattices (BOL), the out-of-equilibrium dynamics of droplets under strong dc and ac fields are illustrated. Under the influence of a dc field, a stripe-like pattern emerges in the temporal domain, while the scenario with ac fields demonstrates temporal periodic and bi-periodic oscillations of density waves. The width and period of formed patterns are directly correlated with the strength of ac and dc fields. Moreover, temporal modulation of the BOL potential depth yields various harmonics in the oscillations of the condensate density pattern. Through Fast Fourier Transform (FFT) analysis, it is confirmed that these harmonics encompass multiple and combinational frequencies, suggesting potential applications in generating desired frequency combs within quantum droplets. We have also carried out a thorough numerical stability check of the obtained solutions and found them sufficiently stable.

Rogue waves and nonzero background solutions for the Gross–Pitaevskii equation with a parabolic potential

In this paper an integrable Gross–Pitaevskii equation with a parabolic potential and a gain term is investigated. Its solutions with a nonzero background are derived. These solutions are constructed by using biliearization reduction approach and connections between the nonlinear Schrödinger equation and the Gross–Pitaevskii equation. The solutions are presented in double-Wronskian form and are classified in terms of canonical forms of a certain matrix. Various breathers and rogue waves are analyzed and illustrated.

Information theoretic measures on quantum droplets in ultracold atomic systems

We consider Shannon entropy, Fisher information, Rényi entropy, and Tsallis entropy to study the quantum droplet phase in Bose–Einstein condensates. In the beyond mean-field description, the Gross–Pitaevskii equation with Lee-Huang-Yang correction gives a family of quantum droplets with different chemical potentials. At a larger value of chemical potential, quantum droplet with sharp-top probability density distribution starts to form while it becomes flat top for a smaller value of chemical potential. We show that entropic measures can distinguish the shape change of the probability density distributions and thus can identify the onset of the droplet phase. During the onset of droplet phase, the Shannon entropy decreases gradually with the decrease of chemical potential and attains a minimum in the vicinity where a smooth transition from flat-top to sharp-top QDs occurs. At this stage, the Shannon entropy increases abruptly with the lowering of chemical potential. We observe an opposite trend in the case of Fisher information. These results are found to be consistent with the Rényi and Tsallis entropic measures.

Data-driven two-dimensional stationary quantum droplets and wave propagations in the amended Gross–Pitaevskii equation with two potentials via deep neural networks learning

In this paper, we develop a systematic deep learning approach to solve two-dimensional stationary quantum droplets (QDs) and investigate their wave propagation in the two-dimensional amended Gross–Pitaevskii equation with Lee–Huang–Yang correction and two kinds of potentials. Firstly, we use the initial-value iterative neural network algorithm for two-dimensional stationary QDs of stationary equations. Then, the learned stationary QDs are used as the initial value conditions for physics-informed neural networks to explore their evolutions in the space-time region. Especially, we consider two types of potentials, one is the two-dimensional quadruple-well Gaussian potential and the other is the P T -symmetric harmonic-oscillator (HO)-Gaussian potential, which lead to spontaneous symmetry breaking and the generation of multi-component QDs. The used deep learning method can also be applied to study wave propagations of other nonlinear physical models.

Spontaneous symmetry breaking and vortices in a tri-core nonlinear fractional waveguide

We introduce a waveguiding system composed of three linearly-coupled fractional waveguides, with a triangular (prismatic) transverse structure. It may be realized as a tri-core nonlinear optical fiber with fractional group-velocity dispersion (GVD), or, possibly, as a system of coupled Gross–Pitaevskii equations for a set of three tunnel-coupled cigar-shaped traps filled by a Bose–Einstein condensate of particles moving by Lévy flights. The analysis is focused on the phenomenon of spontaneous symmetry breaking (SSB) between components of triple solitons, and the formation and stability of vortex modes. In the self-focusing regime, we identify symmetric and asymmetric soliton states, whose structure and stability are determined by the Lévy index of the fractional GVD, the inter-core coupling strength, and the total energy, which determines the system’s nonlinearity. Bifurcation diagrams (of the supercritical type) reveal regions where SSB occurs, identifying the respective symmetric and asymmetric ground-state soliton modes. In agreement with the general principle of the SSB theory, the solitons with broken inter-component symmetry prevail with the increase of the energy in the weakly-coupled system. Three-components vortex solitons (which do not feature SSB) are studied too. Because the fractional GVD breaks the system’s Galilean invariance, we also address mobility of the vortex solitons, by applying a boost to them.

The Gross–Pitaevskii Equation for an Infinite Square Well with a Delta-Function Barrier

The Gross–Pitaevskii Equation for an Infinite Square Well with a Delta-Function Barrier

Approximation of a two‐dimensional Gross–Pitaevskii equation with a periodic potential in the tight‐binding limit

AbstractThe Gross–Pitaevskii (GP) equation is a model for the description of the dynamics of Bose–Einstein condensates. Here, we consider the GP equation in a two‐dimensional setting with an external periodic potential in the ‐direction and a harmonic oscillator potential in the ‐direction in the so‐called tight‐binding limit. We prove error estimates which show that in this limit the original system can be approximated by a discrete nonlinear Schrödinger equation. The paper is a first attempt to generalize the results from [19] obtained in the one‐dimensional setting to higher space dimensions and more general interaction potentials. Such a generalization is a non‐trivial task due to the oscillations in the external periodic potential which become singular in the tight‐binding limit and cause some irregularity of the solutions which are harder to handle in higher space dimensions. To overcome these difficulties, we work in anisotropic Sobolev spaces. Moreover, additional non‐resonance conditions have to be satisfied in the two‐dimensional case.

Coupling Dynamics and Linear Polarization Phenomena in Codirectional Polariton Waveguide Couplers

AbstractIn this work, the previous studies investigating the potential of integrated devices using polariton waveguides that form codirectional couplers is extended. For suitable coupler parameters, a transfer of condensates between the arms of the coupler occurs leading to the observation of previously predicted Josephson‐like oscillations. The ability to tune the periodicity of these oscillations opens the way to the design of polaritonic circuits in which the directionality of the signal towards the output terminals can be controlled, for a fixed separation between the arms, varying the coupling length. The response of the devices to linearly polarized excitation is also investigated, delving into the dynamics of linear polarization at the output terminals of long couplers, providing valuable insights into their potential applications, including polariton switches and logic gates with efficient operation. The results are supported by numerical simulations based on the generalized Gross–Pitaevskii equation describing the dynamics of coherent polaritons in spatially non‐uniform systems, reproducing the polariton distribution on the output terminals and an oscillatory dependence of the corresponding emission as a function of the polarization azimuth. How the coupling and the controllable polarization degree of freedom in polariton couplers opens avenues for innovative optical architectures and functionalities is shown here.

Gap solitons of spin–orbit-coupled Bose–Einstein condensates in a Jacobian elliptic sine potential

Gap solitons of quasi-one-dimensional Bose–Einstein condensate loaded in a Jacobian elliptic sine potential with spin–orbit-coupled are investigated theoretically. Under the mean-field approximation, the dynamical behaviors of such a system are described using the Gross–Pitaevskii equation (GPE). Firstly, we linearize the GPE to obtain the band-gap structures. Secondly, the Newton-Conjugate-Gradient (NCG) method is used to search for gap solitons. Thirdly, the dynamic stability of the gap solitons obtained by the NCG method was investigated using linear stability analysis and nonlinear dynamic evolution methods. It is found that in this nonlinear system, there exist both stable gap solitons and structurally rich unstable gap solitons. The external potential modulus and the strength of the spin–orbit coupling are important parameters that influence the stability of the gap solitons.

Accuracy of the Gross–Pitaevskii Equation in a Double-Well Potential

Accuracy of the Gross–Pitaevskii Equation in a Double-Well Potential

Bohmian analysis of dark solutions in interfering Bose–Einstein condensates: The dynamical role of underlying velocity fields

In the last decades, the experimental research on Bose–Einstein interferometry has received much attention due to promising technological implications. This has thus motivated the development of numerical simulations aimed at solving the time-dependent Gross–Pitaevskii equation and its reduced one-dimensional version to better understand the development of interference-type features and the subsequent soliton dynamics. In this work, Bohmian mechanics is considered as an additional tool to further explore and analyze the formation and evolution in real time of the soliton arrays that follow the merging of two condensates. An alternative explanation is thus provided in terms of an underlying dynamical velocity field, directly linked to the local phase variations undergone by the condensate along its evolution. Although the reduced one-dimensional model is considered here, it still captures the essence of the phenomenon, rendering a neat picture of the full evolution without diminishing the generality of the description. To better appreciate the subtleties of free versus bound dynamics, two cases are discussed. First, the soliton dynamics exhibited by a coherent superposition of two freely released condensates is studied, discussing the peculiarities of the underlying velocity field and the corresponding flux trajectories in terms of both the peak-to-peak distance between the two initial clouds and the addition of a phase difference between them. In the latter case, an interesting correspondence with the well-known Aharonov-Bohm effect is found. Then, the recurrence dynamics displayed by the more general case of two condensates released from the two opposite turning points of a harmonic trap is considered in terms of the distance between such turning points. In both cases, it is presumed that the initial superposition state is generated by splitting adiabatically a single condensate with the aid of an optical lattice, which is then turned off. Nonetheless, although the lattice does not play any active role in the simulations, the parameters defining the initial states are in compliance with it, which helps in the interpretation and understanding of the results observed.

Collective excitation of quantum droplet with different ranges of the interaction of Pöschl-Teller potential

In this article, we studied quantum droplet with the Pöschl-Teller (PT) interaction potential between the Bose atoms. The Gross–Pitaevskii (GP) equation governs the system. The range and strength of the PT interaction can be adjusted. First, we studied the quantum droplet’s density variation for various PT interaction parameters by the imaginary-time split-step Crank-Nicolson (CN) method. We then used the Bogoliubov theory to examine the collective excitation spectra. We observed that sharp roton forms and phonon modes are missing during long-range interactions. There is a gap at the zero momentum zone due to the long-range PT interaction, which increases with the range and strength of the interaction.

The localized excitation on the Jacobi elliptic function periodic background for the Gross–Pitaevskii equation

In this paper, the nonlinear wave solutions for Gross–Pitaevskii equation on the periodic wave background are investigated by Darboux-Bäcklund transformation, from which the soliton and breather wave solutions on the Jacobi elliptic cn and dn functions backgrounds are derived. The corresponding evolutions and dynamical properties of nonlinear wave solutions under different parameters are discussed. These results reported in this paper may raise the possibility of related experiments and potential applications in nonlinear science fields, such as nonlinear optics, oceanography and so on.

Ultrafast dynamics of exciton-polariton fluids at non-zero momenta

In this study, we have explored the ultrafast formation and decay dynamics of exciton-polariton fluids at non-zero momenta, non-resonantly excited by a small-spot femtosecond pump pulse in a ZnO microcavity. Using the femtosecond angle-resolved spectroscopic imaging technique, multidimensional dynamics in both the energy and momentum degrees of freedom have been obtained. Two distinct regions with different decay rate in the energy dimension and various decay-channels in the momentum dimension can be well-resolved. Theoretical simulations based on the generalized Gross–Pitaevskii equation can reach a qualitative agreement with the experimental observations, demonstrating the significance of the initial potential barrier induced by the pump pulse during the decay process. The finding of our study can provide additional insights into the fundamental understanding of exciton-polariton condensates, enabling further advancements for controlling the fluids and practical applications.

Quantum fluctuations of many-body dynamics around the Gross–Pitaevskii equation

We consider the evolution of a gas of N bosons in the three-dimensional Gross–Pitaevskii regime (in which particles are initially trapped in a volume of order one and interact through a repulsive potential with scattering length of order 1/N ). We construct a quasi-free approximation of the many-body dynamics, whose distance to the solution of the Schrödinger equation converges to zero, as N \to \infty , in the L^{2} (\mathbb{R}^{3N}) -norm. To achieve this goal, we let the Bose–Einstein condensate evolve according to a time-dependent Gross–Pitaevskii equation. After factoring out the microscopic correlation structure, the evolution of the orthogonal excitations of the condensate is governed instead by a Bogoliubov dynamics, with a time-dependent generator quadratic in creation and annihilation operators. As an application, we show a central limit theorem for fluctuations of bounded observables around their expectation with respect to the Gross–Pitaevskii dynamics.

Study of Bose–Einstein condensate in the presence of the extended uncertainty principle: infinite potential well

This paper investigates the influence of the extended uncertainty principle (EUP) and non-linearity on Bose–Einstein condensate (BEC) confined within an infinite potential well, described by a deformed one-dimensional Gross–Pitaevskii equation (GPE). Exact solutions are derived, and the impact of the EUP and the parameter of interaction g is explored through solution, position, and momentum uncertainties plots. The study reveals significant changes in the probability density and energy spectra, depending on the deformation and non-linearity parameters.