Generalized Gross-Pitaevskii Equation Research Articles

Dissipative self-gravitating Bose-Einstein condensates with arbitrary nonlinearity as a model of dark matter halos

We develop a general formalism applying to Newtonian self-gravitating Bose-Einstein condensates. This formalism may find application in the context of dark matter halos. We introduce a generalized Gross-Pitaevskii equation including a source of dissipation (damping) and an arbitrary nonlinearity. Using the Madelung transformation, we derive the hydrodynamic representation of this generalized Gross-Pitaevskii equation and obtain a damped quantum Euler equation involving a friction force proportional and opposite to the velocity and a pressure force associated with an equation of state determined by the nonlinearity present in the generalized Gross-Pitaevskii equation. In the strong friction limit, we obtain a quantum Smoluchowski equation. These equations satisfy an H-theorem for a free energy functional constructed with a generalized entropy. We specifically consider the Boltzmann and Tsallis entropies associated with isothermal and polytropic equations of state. We also consider the entropy associated with the logotropic equation of state. We derive the virial theorem corresponding to the generalized Gross-Pitaevskii equation, damped quantum Euler equation, and quantum Smoluchowski equation. Using a Gaussian ansatz, we obtain a simple equation governing the dynamical evolution of the size of the condensate. We develop a mechanical analogy associated with this gross dynamics. We highlight a specific model of dark matter halos corresponding to a generalized Gross-Pitaevskii equation with a logarithmic nonlinearity and a cubic nonlinearity. It corresponds to a damped quantum Euler equation associated with a mixed entropy combining the Boltzmann and Tsallis entropies. It leads to dark matter halos with an equation of state $P=\rho k_{B} T_{\rm eff}/m+2\pi a_{s}\hbar^{2}\rho^{2}/m^{3}$ presenting a condensed core (BEC/soliton) and an isothermal halo with an effective temperature $T_{\rm eff}$ . We propose that this model provides an effective coarse-grained parametrization of dark matter halos experiencing gravitational cooling. Specific applications of our formalism to dark matter halos will be developed in future papers.

Non-Hermitian coupled-mode theory for incoherently pumped exciton-polariton condensates

The generalized Gross-Pitaevskii equation (gGPE) is an effective phenomenological description for the dynamics of incoherently pumped exciton-polariton condensates. However, a brute force numerical simulation of the gGPE provides little physical insight into condensate formation under arbitrary pumping configurations, and is demanding in terms of computational resources. We introduce in this paper a modal description of polariton condensation under incoherent pumping of arbitrary spatial profile, based on eigenmodes of the non-Hermitian generator of the linearized dynamics. A pump-dependent basis is then introduced to formulate a temporal coupled-mode theory that captures condensate dynamics in the presence of all nonlinear interactions. Simulations using a single set of modes for a given pumping and trapping configuration agree very well with a full integration of the gGPE in diverse dynamical regimes, supporting the validity of this modal description, while also providing a speedup in simulation times.

Polariton condensation threshold investigation through the numerical resolution of the generalized Gross-Pitaevskii equation.

We present a numerical approach for the solution of the dissipative Gross-Pitaevskii equation coupled to the reservoir equation governing the exciton-polaritons Bose-Einstein condensation. It is based on the finite difference method applied to space variables and on the fourth order Range-Kutta algorithm applied to the time variable. Numerical tests illustrate the stability and accuracy of the proposed scheme. Then results on the behavior of the condensate under large Gaussian pumping and around the threshold are presented. We determine the threshold through the particular behavior of the self-energy and characterize it by tracking the establishment time of the steady state.

On stable solitons and interactions of the generalized Gross-Pitaevskii equation with PT- and non-PT-symmetric potentials

We report the bright solitons of the generalized Gross-Pitaevskii (GP) equation with some types of physically relevant parity-time- (PT-) and non-PT-symmetric potentials. We find that the constant momentum coefficient Γ can modulate the linear stability and complicated transverse power-flows (not always from the gain toward loss) of nonlinear modes. However, the varying momentum coefficient Γ(x) can modulate both unbroken linear PT-symmetric phases and stability of nonlinear modes. Particularly, the nonlinearity can excite the unstable linear mode (i.e., broken linear PT-symmetric phase) to stable nonlinear modes. Moreover, we also find stable bright solitons in the presence of non-PT-symmetric harmonic-Gaussian potential. The interactions of two bright solitons are also illustrated in PT-symmetric potentials. Finally, we consider nonlinear modes and transverse power-flows in the three-dimensional (3D) GP equation with the generalized PT-symmetric Scarff-II potential.

Dark soliton pair of ultracold Fermi gases for a generalized Gross-Pitaevskii equation model.

We present the theoretical investigation of dark soliton pair solutions for one-dimensional as well as three-dimensional generalized Gross-Pitaevskii equation (GGPE) which models the ultracold Fermi gas during Bardeen-Cooper-Schrieffer-Bose-Einstein condensates crossover. Without introducing any integrability constraint and via the self-similar approach, the three-dimensional solution of GGPE is derived based on the one-dimensional dark soliton pair solution, which is obtained through a modified F-expansion method combined with a coupled modulus-phase transformation technique. We discovered the oscillatory behavior of the dark soliton pair from the theoretical results obtained for the three-dimensional case. The calculated period agrees very well with the corresponding reported experimental result [Weller et al., Phys. Rev. Lett. 101, 130401 (2008)PRLTAO0031-900710.1103/PhysRevLett.101.130401], demonstrating the applicability of the theoretical treatment presented in this work.

Dark soliton solution of the three-dimensional Gross-Pitaevskii equation with an isotropic harmonic potential and nonlinearity in polytropic approximation

We study the three-dimensional Fermi gas in an isotropic harmonic trap during the Bardeen- Cooper-Schrieffer superfluid to Bose-Einstein condensate (BCS-BEC) crossover, which is modeled by using the generalized Gross-Pitaevskii equation (GGPE) in the polytropic approximation. We analytically solved the 3D GGPE with a coupled modulus-phase transformation without introducing any additional integrability constraint, reaching the dark soliton-like solution. We find that the dark soliton identified undergoes an oscillation with a constant period over the whole BCS-BEC crossover region, although the amplitude of the dark soliton varies with polytropic index, demonstrating the peculiar nonlinear properties for the system modeled by using the 3D GGPE.

Hydrodynamic description of trapped ultracold paraexcitons inCu2O

In this paper we present a theoretical model to describe the dynamics of paraexcitons in cuprous oxide at ultralow temperatures inside potential traps. A possible condensate is described by a generalized Gross-Pitaevskii equation. The noncondensed excitons evolve under a set of hydrodynamic equations, which were derived from a quantum Boltzmann equation. The model takes into account the finite lifetime of the excitons, the pump laser, the Auger-like two-body decay, as well as exciton-exciton and exciton-phonon scattering. The numerical results show the strong influence of the Auger effect on exciton temperatures and densities not only at high pump powers but also at ultralow temperatures. Furthermore, the excitons do not cool down to very low bath temperatures $({T}_{\mathrm{B}}\ensuremath{\ll}0.5\phantom{\rule{0.28em}{0ex}}\mathrm{K})$ under continuous wave excitation. We also compare the results of the theoretical model with experimental data.

Optical control of spin textures in quasi-one-dimensional polariton condensates

We investigate, through polarization-resolved spectroscopy, the spin transport by propagating polariton condensates in a quasi one-dimensional microcavity ridge along macroscopic distances. Under circularly polarized, continuous-wave, non-resonant excitation, a sinusoidal precession of the spin in real space is observed, whose phase depends on the emission energy. The experiments are compared with simulations of the spinor-polariton condensate dynamics based on a generalized Gross-Pitaevskii equation, modified to account for incoherent pumping, decay and energy relaxation within the condensate.

Acoustic black hole in a stationary hydrodynamic flow of microcavity polaritons.

We report an experimental study of superfluid hydrodynamic effects in a one-dimensional polariton fluid flowing along a laterally patterned semiconductor microcavity and hitting a micron-sized engineered defect. At high excitation power, superfluid propagation effects are observed in the polariton dynamics; in particular, a sharp acoustic horizon is formed at the defect position, separating regions of sub- and supersonic flow. Our experimental findings are quantitatively reproduced by theoretical calculations based on a generalized Gross-Pitaevskii equation. Promising perspectives to observe Hawking radiation via photon correlation measurements are illustrated.

Temporal coherence of one-dimensional nonequilibrium quantum fluids

We theoretically investigate the time dependence of the first order coherence function for a one-dimensional driven dissipative non-equilibrium condensate. Simulations on the generalized Gross-Pitaevskii equation (GGPE) show that the characteristic time scale of exponential decay agrees with the linearized Bogoliubov theory in the regime of large interaction energy. For very weak interactions, the temporal correlation deviates from the linear theory, and instead respects the dynamic scaling of the Kardar-Parisi-Zhang universality class. This nonlinear dynamics is found to be quantitatively captured by a noisy Kuramoto-Sivashinsky equation for the phase dynamics.

Parameters controlling matter waves in the one-dimensional generalized Gross–Pitaevskii equation with the varying potential and source

In this paper, the one-dimensional generalized Gross–Pitaevskii (GP) equation with the varying external potential and source which plays an important role in Bose–Einstein condensates is investigated. For the different types of external potentials, based on some powerful transformation methods with different types of expressions in Jacobi elliptic functions or polynomial functions, many types of matter-wave solutions with traveling-wave and non-trivial phases are found for the generalized GP equation with the external potential and source. These matter-wave solutions contain the doubly periodic wave solutions, solitary wave solutions, periodic wave solutions and the bounded rational solutions. Moreover, we analyze the properties of some obtained solutions and the corresponding potentials on the basis of the different parameters including the source amplitude. These results may be useful to explain some nonlinear wave phenomena in Bose–Einstein condensates, and nonlinear optics.

Spatial coherence of weakly interacting one-dimensional nonequilibrium bosonic quantum fluids

We present a theoretical analysis of spatial correlations in a one-dimensional driven-dissipative nonequilibrium condensate. Starting from a stochastic generalized Gross-Pitaevskii equation, we derive a noisy Kuramoto-Sivashinsky equation for the phase dynamics. For sufficiently strong interactions, the coherence decays exponentially in close analogy to the equilibrium Bose gas. When interactions are small on a scale set by the nonequilibrium condition, we find through numerical simulations a crossover between a Gaussian and exponential decay with peculiar scaling of the coherence length on the fluid density and noise strength.

Nonlinear relaxation and selective polychromatic lasing of confined polaritons

Polaritons have often been proposed as promising candidates for the realization of all-optical devices due to the easy manipulation and control of their density and spin. In this paper we present a relaxation mechanism for confined polaritons and its application in a device based on polariton lasing in which the incoming monochromatic beam can be channeled into several polariton lasers at different wavelengths. Polaritons are injected in a 3 $\ensuremath{\mu}$m diameter trap by an incoming beam slightly detuned with respect to a confined state, and the resulting optical bistability is then imprinted on the lower confined states through an efficient relaxation mechanism which combines phonon interaction and bosonic stimulation. Above threshold, all the confined states behave like lasers and the onset of coherence is demonstrated by spectral narrowing. Moreover, the initial polarization of the laser is conserved during the relaxation process allowing for spin logic operations. In this paper we demonstrate that, due to the nonlinear behavior, it is possible to switch ON and OFF the lasing from the confined states of the trap by tailoring the input beam. All the experimental findings are validated and very well reproduced in the framework of the generalized Gross-Pitaevskii equation.

Operation speed of polariton condensate switches gated by excitons

We present a time-resolved photoluminescence (PL) study in real- and momentum-space of a polariton condensate switch in a quasi-1D semiconductor microcavity. The polariton flow across the ridge is gated by excitons inducing a barrier potential due to repulsive interactions. A study of the device operation dependence on the power of the pulsed gate beam obtains a satisfactory compromise for the ON/OFF-signal ratio and -switching time of the order of 0.3 and $\thicksim50$ ps, respectively. The opposite transition is governed by the long-lived gate excitons, consequently the OFF/ON-switching time is $\thicksim200$ ps, limiting the overall operation speed of the device to $\thicksim3$ GHz. The experimental results are compared to numerical simulations based on a generalized Gross-Pitaevskii equation, taking into account incoherent pumping, decay and energy relaxation within the condensate.

Exact soliton solutions of the generalized Gross-Pitaevskii equation based on expansion method

We give a more generalized treatment of the 1D generalized Gross-Pitaevskii equation (GGPE) with variable term coefficients. External harmonic trapping potential is fully considered and the nonlinear interaction term is of arbitrary polytropic index of superfluid wave function. We also eliminate the interdependence between variable coefficients of the equation terms avoiding the restrictions that occur in some other works. The exact soliton solutions of the GGPE are obtained through the delicate combined utilization of modified lens-type transformation and F-expansion method with dominant features like soliton type properties highlighted.

Energy relaxation of exciton-polariton condensates in quasi-one-dimensional microcavities

We present a time-resolved study of energy relaxation and trapping dynamics of polariton condensates in a semiconductor microcavity ridge. The combination of two non-resonant, pulsed laser sources in a GaAs ridge-shaped microcavity gives rise to profuse quantum phenomena where the repulsive potentials created by the lasers allow the modulation and control of the polariton flow. We analyze in detail the dependence of the dynamics on the power of both lasers and determine the optimum conditions for realizing an all-optical polariton condensate transistor switch. The experimental results are interpreted in the light of simulations based on a generalized Gross-Pitaevskii equation, including incoherent pumping, decay and energy relaxation within the condensate.

Controlling the group velocity of colliding atomic Bose-Einstein condensates with Feshbach resonances

We report on a proposal to reduce the group velocity of a small ``laser'' Bose-Einstein condensate upon passing through a larger ``medium'' condensate of the same isotopic species in analogy to slowing of light passing through dispersive media. We make use of ultracold collisions near a magnetic Feshbach resonance, which gives rise to a sharp variation in scattering length with collision energy and thereby changes the group velocity. A generalized Gross-Pitaevskii equation is used to derive an expression for the group velocity in a homogeneous medium as well as for the spatial delay $\ensuremath{\delta}$, through a finite-sized medium BEC with a Thomas-Fermi density distribution. We predict a maximum reduction of the group velocity by a half, which physically corresponds to freely propagating Feshbach molecules. For typical narrow Feshbach resonances and a medium with number density ${10}^{15}$ cm${}^{\ensuremath{-}3}$, up to $85%$ of the lower bound can be achieved, making the effect experimentally observable. We also derive constraints on the experimental realization of our proposal.

Dynamics of a polariton condensate transistor switch

We present a time-resolved study of the logical operation of a polariton condensate transistor switch. Creating a polariton condensate (source) in a GaAs ridge-shaped microcavity with a non-resonant pulsed laser beam, the polariton propagation towards a collector, at the ridge edge, is controlled by a second weak pulse (gate), located between the source and the collector. The experimental results are interpreted in the light of simulations based on the generalized Gross-Pitaevskii equation, including incoherent pumping, decay, and energy relaxation within the condensate.

Dynamics of dark-soliton formation in a polariton quantum fluid

Polariton fluids have revealed huge potentialities in order to investigate the properties of bosonic fluids at the quantum scale. Among those properties, the opportunity to create dark as well as bright solitons has been demonstrated recently. In the present experiments, we image the formation dynamics of oblique dark solitons. They nucleate in the wake of an engineered attractive potential that perturbs the polariton quantum fluid. Thanks to time and phase measurements, we assess quantitatively the formation process. The formation velocity is observed to increase with increasing distance between the flow injection point and the obstacle which modulates the density distribution of the polariton fluid. We propose an explanation in terms of the increased resistance to the flow and of the conditions for the convective instability of dark solitons. By using an iterative solution of the generalized Gross-Pitaevskii equation, we are able to reproduce qualitatively our experimental results.

Turbulence in a Bose-Einstein condensate of dipolar excitons in coupled quantum wells

The nonlinear dynamics of a Bose-Einstein condensate (BEC) of dipolar excitons trapped in an external confining potential in coupled quantum wells is analyzed. It is demonstrated that under typical experimental conditions the dipolar exciton BEC can be described by a generalized Gross-Pitaevskii equation with the local interaction between the excitons, which depends on the exciton distribution function. It is shown that, if the system is pumped at sufficiently high frequencies, a steady turbulent state can be formed.