Bose-Einstein condensates are formed when a bosonic gas is cooled to a temperature near absolute zero. When this occurs, quantum properties that are microscopic become macroscopic, facilitating their study. We have studied matter waves in the Gross-Pitaevskii equation subject to a trap Kapitza potential, which is a quantum analog of the classical inverse Kapitza pendulum. Since this specific system was recently experimentally realized with ultracold atoms for the first time, exploring the theoretical models of this system is, therefore, significant and up-to-date. To find the analytical solutions of the corresponding Gross-Pitaevskii equation with such potential, the extended hyperbolic tangent function method was chosen, which leads to soliton solutions. The new class of solutions found in this work, written in terms of Mathieu functions, was used to analyze the influence of the Kapitza potential free model parameters on the soliton dynamics within the condensate. These mainly include dark and dark-bright solitons. Our findings suggest that there are specific classes of parameters of the system for which bright solitons do not exist.

Exciton-polariton condensates are explored using coupled driven-dissipative Gross-Pitaevskii equations under the Keeling-Berloff approximation in the presence of photonic spin-orbit coupling (SOC). An extended dissipative variational approximation method uses analytical expressions for coupled vortex ring solitons as minimizing functions of a generalized Lagrangian framework. The results reveal that the amplitudes of the vortices are linked to losses and gain, while the topological charges of the vortices are tied to the width of the vortex rings. Numerically, using the proposed solution as initial conditions, the emergence of even-numbered multilobed rings takes place as the topological charges of the seeded vortices are increased. The analytical results are then confronted by a detailed numerical analysis that further elucidates the rich vortex dynamics inferred by SOC and other combinations of various system parameters.

We present a theoretical framework for one-dimensional open Bose-Bose mixtures by coupling a driven-dissipative Gross-Pitaevskii equation to a dynamically evolving inhomogeneous reservoir while incorporating the beyond-mean-field Lee-Huang-Yang correction. An adiabatic elimination procedure yields a complex Ginzburg-Landau equation with quadratic-cubic nonlinearity in the regime of rapid reservoir relaxation. Through linear stability analysis, we identify regimes of modulational instability via a modified Lange-Newell criterion and numerically confirm the ensuing nucleation of dissipative quantum droplets. Our findings highlight the critical role of balancing beyond-mean-field effects and dissipation in shaping nontrivial, far-from-equilibrium states of quantum fluids.

We investigate the dynamical instabilities of a one-dimensional ultracold Bose-Bose mixture with long-range dipole-dipole interactions, trapped in deep optical lattices and subject to periodically varying contact interaction. The effect of beyond-mean-field corrections due to quantum fluctuations is considered. In the tight-binding regime, we employ Wannier functions to derive a discrete nonlinear Schr & ouml;dinger equation that captures the dynamics of the system. Using linear stability analysis coupled to multiple scale expansion, we systematically study the modulational and parametric instabilities of the system in the miscible regime, identifying the conditions under which these instabilities emerge. The corresponding instability domains are found and examined. The roles of long-range interactions and quantum fluctuations are highlighted, demonstrating their significant impact on the stability and dynamics of the lattice system. Our analytical predictions are validated through direct numerical simulations, which confirm the instability diagrams through the effective onset of instabilities and reveal the intricate interplay between interaction strength, quantum fluctuation, and the long-range nature of the dipole-dipole forces. This work provides insights into the control and manipulation of ultracold quantum gases in optical lattices, which have potential applications in quantum simulation and condensed-matter physics.

The letter explores modulational instability (MI) in chiral Bose-Einstein condensates (BECs) using a modified Gross-Pitaevskii (GP) equation that includes a density-dependent gauge potential and residual higher-order interactions from shape-dependent confinement corrections. It shows that the interaction between current density and residual nonlinearity can cause instability when balanced with attractive two-body interactions. The growth rate of MI is determined, and the combined effects extend instability windows based on BEC density and wavenumber. Numerical simulations confirm analytical predictions, including the growth rate for single-wavelength perturbations and areas of instability with noisy initial conditions. The evolution and directional response of chiral solitons to residual nonlinearity are discussed, revealing shape-changing structures, breathing soliton trains, and rogue waves that are stabilized by adjusting the residual nonlinearity and MI onset. Findings suggest a way to manipulate MI in BECs using residual higher-order and current nonlinearities.

We propose a comprehensive analysis of the modulational instability (MI) phenomenon in one-dimensional Bose-Einstein condensate (BEC) in the context of mutually symmetric components in the binary mixture. The proposed model considers weak nonlocal cubic mean-field and quadratic beyond mean-field interactions arising from quantum fluctuations. The instability regions are analyzed using linear stability analysis of a continuous wave, considering maximum perturbation wavenumber and growth rate. The regions for quantum droplets (QDs) and solitons are identified by varying the nonlocality parameter. Numerical simulations confirm analytical predictions, revealing that nonlocality influences the formation of various QD profiles and instability time. Furthermore, solitonic patterns exhibit nonlocality-dependent behavior. Our study opens new doors to a better characterization of QDs in binary mixtures in the presence of higher-order beyond mean-field effects.

The nonlinear dynamics induced by the modulation instability (MI) of a binary mixture in an atomic Bose-Einstein condensate (BEC) is investigated theoretically under the joint effects of higher-order residual nonlinearities and helicoidal spin-orbit (SO) coupling in a regime of unbalanced chemical potential. The analysis relies on a system of modified coupled Gross-Pitaevskii equations on which the linear stability analysis of plane-wave solutions is performed, from which an expression of the MI gain is obtained. A parametric analysis of regions of instability is carried out, where effects originating from the higher-order interactions and the helicoidal spin-orbit coupling are confronted under different combinations of the signs of the intra- and intercomponent interaction strengths. Direct numerical calculations on the generic model support our analytical predictions and show that the higher-order interspecies interaction and the SO coupling can balance each other suitably for stability to take place. Mainly, it is found that the residual nonlinearity preserves and reinforces the stability of miscible pairs of condensates with SO coupling. Additionally, when a miscible binary mixture of condensates with SO coupling is modulationally unstable, the presence of residual nonlinearity may help soften such instability. Our results finally suggest that MI-induced formation of stable solitons in mixtures of BECs with two-body attraction may be preserved by the residual nonlinearity even though the latter enhances the instability.

This letter studies the excitation of nonlinear patterns on the one-dimensional (1D) quantum droplet in the presence of polaron-like impurity under modulational instability (MI) activation. Analytically, it is shown that strong coupling between the impurity and Bose species expands the instability domain, especially in the quantum droplet regime, while pronounced self-repulsive nonlinearity tends to restrict the MI to regions of strong interspecies interactions. In the process, the wavenumber spectrum, capable of supporting MI, also shrinks. Direct numerical simulations confirm the existence of coupled boson-impurity wave patterns, mainly governed by trains of solitary waves of different profiles for the impurity and quasi-droplet waveforms for the boson. Additionally, the two components are mutually trapped, which enhances energy transfer between them, leading to the formation of quasi-droplet structures under a suitable balance between the back action of the impurity on the boson, the Lee-Huang-Yang term, as well as the intrinsic self-repulsive nonlinearity.

Non-relativistic particles that are effectively confined to two dimensions can in general move on curved surfaces, allowing dynamical phenomena beyond what can be described with scalar potentials or even vector gauge fields. Here we consider a simple case of piecewise uniform curvature: a particle moves on a plane with a spherical extrusion. Depending on the latitude at which the sphere joins the plane, the extrusion can range from an infinitesimal bump to a nearly full sphere that just touches the plane. Free classical motion on this surface of piecewise uniform curvature follows geodesics that are independent of velocity, while quantum mechanical scattering depends on energy. We compare classical, semi-classical, and fully quantum problems, which are all exactly solvable, and show how semi-classical analysis explains the complex quantum differential cross section in terms of interference between two classical trajectories: the sphere on a plane acts as a kind of double slit.

This work extends to higher-order interactions the results of Nguetcho (2017), in which we discussed only on modulational instability in one-dimensional chain made of atoms, harmonically coupled to their nearest neighbors and subjected to an external on-site potential. Here we investigate the competition between cubic–quartic nonlinearities interactions of the nearest-neighbor and substrate’s deformability, and mainly discuss its impact on the modulational instability of the system. This makes it possible to adapt the theoretical model to a real physical system such as atomic chains or DNA lattices. The governing equation, derived from the modified Frenkel–Kontorova model, is an extended nonlinear Schrödinger equation (eNLS) containing a new higher-order nonlinear term. By employing linear stability analysis, the generic properties of the MI gain spectra of the system are demonstrated. In the presence of the new quartic nonlinearity, the combinations of the system’s parameters open a large variety of gain profiles and instability domains that cannot be explored without the quartic nonlinearity. Direct numerical simulations are performed to support our analytical results, and an excellent agreement is found.

Experiments on periodically driven quantum systems have effectively realized quasi-Hamiltonians, in the sense of Floquet theory, that are otherwise inaccessible in static condensed matter systems. Although the Floquet quasi-Hamiltonians are time-independent, however, these continuously driven systems can still suffer from heating due to a secular growth in the expectation value of the time-dependent physical Hamiltonian. Here we use an exact space-time mapping to construct a class of many-body systems with rapid periodic driving which we nonetheless prove to be completely free of heating, by mapping them exactly onto time-independent systems. The absence of heating despite the periodic driving occurs in these cases of harmonically trapped dilute Bose gas because the driving is a certain periodic but anharmonic modulation of the gas's two-body contact interaction, at a particular frequency. Although we prove that the absence of heating is exact within full quantum many-body theory, we then use mean-field theory to simulate 'Floquet heating spectroscopy' and compute the heating rate when the driving frequency is varied away from the critical value for zero heating. In both weakly and strongly non-linear regimes, the heating rate as a function of driving frequency appears to show a number of Fano resonances, suggesting that the exactly proven absence of heating at the critical frequency may be explained in terms of destructive interferences between excitation modes.

We formulate an exact space-time mapping between the N-point correlation functions of two different experiments with open quantum gases. Our formalism extends a quantum-field mapping result for closed systems [Phys. Rev. A 94, 043628 (2016)] to the general case of open quantum systems with the Markovian property. For this, we consider an open many-body system consisting of a D-dimensional quantum gas of bosons or fermions that interacts with a bath under Born-Markov approximation and evolves according to a Lindblad master equation in a regime of loss or gain. Invoking the independence of expectation values on pictures of quantum mechanics and using the quantum fields that describe the gas dynamics, we derive the Heisenberg evolution of any arbitrary N-point function of the system in the regime when the Lindblad generators feature a loss or a gain. Our quantum field mapping for closed quantum systems is rewritten in the Schrodinger picture and then extended to open quantum systems by relating onto each other two different evolutions of the N-point functions of the open quantum system. As a concrete example of the mapping, we consider the mean-field dynamics of a simple dissipative quantum system that consists of a one-dimensional Bose-Einstein condensate being locally bombarded by a dissipating beam of electrons in both cases when the beam amplitude or the waist is steady and modulated.

We investigate the effect of higher-order interactions induced by shape-dependent confinement in the modulational instability (MI) of a binary mixture of Bose-Einstein condensates. For this, we present and compute both analytically and numerically a system of coupled Gross-Pitaevskii equations with residual nonlinearity that rule the dynamics of the mixture. Using the linear stability approach, we obtain the instability criteria of the mixture and find that the MI can be excited in miscible condensates and altered in immiscible condensates due to the effect of residual nonlinearity. Direct numerical calculations are performed to support the analytical predictions, and a good agreement is found. The space-time evolution of the condensate density is displayed in both cases when the mixture is miscible and immiscible, showing the generation of bright solitons for modes predicted to be unstable.

We use the time-dependent variational method to examine the formation of localized patterns in dynamically unstable anharmonic lattices with cubic-quintic nonlinearities and fourth-order dispersion. The governing equation is an extended nonlinear Schrödinger equation known for modified Frankel-Kontorova models of atomic lattices and here derived from an extended Bose-Hubbard model of bosonic lattices with local three-body interactions. In presence of modulated waves, we derive and investigate the ordinary differential equations for the time evolution of the amplitude and phase of dynamical perturbation. Through an effective potential, we find the modulationally unstable domains of the lattice and discuss the effect of the fourth-order dispersion in the dynamics. Direct numerical simulations are performed to support our analytical results, and a good agreement is found. Various types of localized patterns, including breathers and solitonic chirped-like pulses, form in the system as a result of interplay between the cubic-quintic nonlinearities and the second- and fourth-order dispersions.

This article investigates combined effects of nonlinearities and substrate's deformability on modulational instability. For that, we consider a lattice model based on the nonlinear Klein-Gordon equation with an on-site potential of deformable shape. Such a consideration enables to broaden the description of energy-localization mechanisms in various physical systems. We consider the strong-coupling limit and employ semi-discrete approximation to show that nonlinear wave modulations can be described by an extended nonlinear Schrodinger equation containing a fourth-order dispersion component. The stability of modulation of carrier waves is scrutinized and the following findings are obtained analytically. The various domains of gains and instabilities are provided based upon various combinations of the parameters of the system. The instability gains strongly depend on nonlinear terms and on the kind of shape of the substrate. According to the system's parameters, our model can lead to different sets of known equations such as those in a negative index material embedded into a Kerr medium, glass fibers, resonant optical fiber and others. Consequently, some of the results obtained here are in agreement with those obtained in previous works. The suitable combination of nonlinear terms with the deformability of the substrate can be utilized to specifically control the amplitude of waves and consequently to stabilize their propagations. The results of analytical investigations are validated and complemented by numerical simulations. (C) 2017 Elsevier B.V. All rights reserved.

Experiments on trapped quantum gases can probe challenging regimes of quantum many-body dynamics, where strong interactions or nonequilibrium states prevent exact solutions. Here we present a different kind of exact result, which applies even in the absence of actual solutions: a class of space-time mappings of different experiments onto each other. Since our result is an identity relating second-quantized field operators in the Heisenberg picture of quantum mechanics, it is extremely general; it applies to arbitrary measurements on any mixtures of Bose or Fermi gases, in arbitrary initial states. It represents a strong prediction of quantum field theory which can be tested in current laboratories, and whose practical applications include perfect simulation of interesting experiments with other experiments which may be easier to perform.

We investigate the dynamical instability of Bose-Einstein condensates (BECs) with higher-order interactions immersed in an optical lattice with weak driving harmonic potential. For this, we compute both analytically and numerically a modified Gross-Pitaevskii equation with higher-order nonlinearity and external potentials generated by magnetic and optical fields. Using the time-dependent variational approach, we derive the ordinary differential equations for the time evolution of the amplitude and phase of modulational perturbation. Through an effective potential, we obtain the modulational instability condition of BECs and discuss the effect of the higher-order interaction in the dynamics of the condensates in presence of optical potential. We perform direct numerical simulations to support our analytical results, and good agreement is found.

We use the time-dependent variational approach to demonstrate how the modulational and oscillatory instabilities can be generated in Bose–Einstein condensates (BECs) trapped in a periodic optical lattice with weak driving harmonic potential. We derive and analyze the ordinary differential equations for the time evolution of the amplitude and phase of the modulational perturbation, and obtain the instability condition of the condensates through the effective potential. The effect of the optical potential on the dynamics of the BECs is shown. We perform direct numerical simulations to support our theoretical findings, and good agreement is found.

The matter-wave solutions of Bose-Einstein condensates with three-body interaction are examined through the one-dimensional Gross-Pitaevskii equation. By using a modified lens-type transformation and a further extension of the tanh-function method we obtain the exact analytical solutions which describe the propagation of kink-shaped solitons, anti-kink-shaped solitons, and other families of solitary waves. We realize that the shape of a kink solitary wave depends on both the scattering length and the parameter of atomic exchange with the substrate. The stability of the solitary waves is examined using analytical and numerical methods. Our results can also be applied to nonlinear optics in the presence of cubic-quintic media.