Attractive Bose-Einstein Condensate Research Articles

Mass concentration near the boundary for attractive Bose–Einstein condensates in bounded domains

We are devoted to studying the ground states of trapped attractive Bose–Einstein condensates in a bounded domain Ω⊂R2, which can be described by minimizers of L2-critical constraint Gross–Pitaevskii energy functional. It has been shown that there is a threshold a∗>0 such that minimizers exist if and only if the interaction strength a satisfies a<a∗. In present paper, we prove that when the trapping potential V(x) attains its flattest global minimum only at the boundary of Ω, the mass of minimizers must concentrate near the boundary of Ω as a↗a∗. This result extends the work of Luo and Zhu (2019).

Mass Concentration Behavior of Attractive Bose–Einstein Condensates with Sinusoidal Potential in a Circular Region

Mass Concentration Behavior of Attractive Bose–Einstein Condensates with Sinusoidal Potential in a Circular Region

Blowup of two-dimensional attractive Bose–Einstein condensates at the critical rotational speed

We study the ground states of a two-dimensional focusing non-linear Schrödinger equation with rotation and harmonic trapping. When the strength of the interaction approaches a critical value from below, the system collapses to a profile obtained from the optimizer of a Gagliardo– Nirenberg interpolation inequality. This was established before in the case of fixed rotation frequency. We extend the result to rotation frequencies approaching, or even equal to, the critical frequency at which the centrifugal force compensates the trap. We prove that the blow-up scenario is to leading order unaffected by such a strong deconfinement mechanism. In particular, the blow-up profile remains independent of the rotation frequency.

Impurity induced modulational instability in Bose-Einstein condensates

By means of linear stability analysis (LSA) and direct numerical simulations of the coupled Gross-Pitaevskii (GP) equations, we address the impurity induced modulational instability (MI) and the associated nonlinear dynamics in Bose-Einstein condensates (BECs). We explore the dual role played by the impurities within the BECs—the instigation of MI and the dissipation of the initially generated solitary waves. Because of the impurities, the repulsive BECs are even modulationally unstable and this tendency towards MI increases with increasing impurity fraction and superfluid-impurity coupling strength. However, the tendency of a given BEC towards the MI decreases with the decreasing mass of the impurity atoms while the sign of the superfluid-impurity interaction plays no role. The above results are true even for attractive BECs except for a weak superfluid-impurity coupling, where the MI phenomenon is marginally suppressed by the presence of impurities. The dissipation of the solitons reduces their lifetime and is eminent for a larger impurity fraction and strong superfluid-impurity strength respectively.

Anisotropic cosmological exact solutions of Petrov type D of a mixture of dark energy and an attractive Bose-Einstein condensate

Anisotropic cosmological exact solutions of Petrov type D of a mixture of dark energy and an attractive Bose-Einstein condensate

Vortex states of Bose-Einstein condensates with attractive interactions

In this paper, we mainly investigate complex-valued vortex states of trapped attractive Bose-Einstein condensates (BECs), which can be described by an $ L^2- $critical constraint minimization problem. We prove that both the existence and nonexistence of vortex states may occur at the threshold $ a^*(m) $ depending on the value of $ V(0) $. When there is no vortex states at the threshold $ a^*(m) $, the limit behavior of vortex states as $ a \nearrow a^*(m) $ is also investigated if the trapping potential $ V(x) $ vanishes on the unit circle $ B(0,1) $ in $ {{\mathbb{R}}}^2 $.

Role of Higher-Order Interactions on the Modulational Instability of Bose-Einstein Condensate Trapped in a Periodic Optical Lattice

In this paper, we investigate the impact of higher-order interactions on the modulational instability (MI) of Bose-Einstein Condensates (BECs) immersed in an optical lattice potential. We derive the new variational equations for the time evolution of amplitude, phase of modulational perturbation, and effective potential for the system. Through effective potential techniques, we find that high density attractive and repulsive BECs exhibit new character with direct impact over the MI phenomenon. Results of intensive numerical investigations are presented and their convergence with the above semi analytical approach is brought out.

Anharmonicity-induced criticality of the ground-state properties of attractive Bose–Einstein condensate

Anharmonicity-induced criticality of the ground-state properties of attractive Bose–Einstein condensate

Dynamics of Bose-Einstein condensates under anharmonic trap

The dynamics of weakly interacting three-dimensional Bose-Einstein condensates (BECs), trapped in external axially symmetric plus anharmonic distortion potential are studied. Within a variational approach and time-dependent Gross-Pitaevskii equation, the coupled condensate width equations are derived. By modulating anharmonic distortion of the trapping potential, nonlinear features are studied numerically and illustrated analytically, such as mode coupling of oscillation modes, and resonances. Furthermore, the stability of attractive interaction BEC in both repulsive and attractive anharmonic distortion is examined. We demonstrate that a small repulsive and attractive anharmonic distortion is effective in reducing (extending) the condensate stability region since it decreases (increases) the critical number of atoms in the trapping potential.

Existence and uniqueness of ground states for attractive Bose–Einstein condensates in box-shaped traps

We consider ground states of two-dimensional Bose–Einstein condensates in box-shaped trapping potentials Vext(x) with inhomogeneous attractive interactions am(x), which can be described equivalently by minimizers of Gross–Pitaevskii energy functional in bounded domains. In this paper, we prove that there is a threshold a* &amp;gt; 0 such that minimizers exist for 0 &amp;lt; a &amp;lt; a* and the minimizer does not exist for any a &amp;gt; a*. However, if a = a*, it is shown that whether minimizers exist depends sensitively on the asymptotic behaviors of m(x) near its maximum points. Moreover, based on a detailed analysis on the limit behavior of minimizers as a ↗ a*, we prove local uniqueness of minimizers under some suitable assumptions on m(x).

Limit behavior of attractive Bose-Einstein condensates passing an obstacle

In this paper, we mainly investigate ground states of trapped attractive Bose-Einstein condensates (BEC) passing an obstacle in the plane, which can be described by an L2-critical constraint minimization problem in an exterior domain Ω=R2∖ω, where the bounded convex domain ω⊂R2 with smooth boundary denotes the region of the obstacle. It is shown that minimizers (i.e. ground states) exist, if and only if the interaction strength a satisfies a<a⁎=‖Q‖22, where Q>0 is the unique positive radial solution of Δu−u+u3=0 in R2. If the trapping potential V(x) attains its global minima only along the whole boundary ∂Ω, the limit behavior of minimizers is analyzed as a↗a⁎ by employing the Pohozaev identity and the delicate energy analysis, where the mass concentration occurs at the flattest critical point of V(x) on ∂Ω.

The Nonexistence of Vortices for Rotating Bose–Einstein Condensates with Attractive Interactions

This article is devoted to studying the model of two-dimensional attractive Bose-Einstein condensates in a trap $V(x)$ rotating at the velocity $\Omega $. This model can be described by the complex-valued Gross-Pitaevskii energy functional. It is shown that there exists a critical rotational velocity $0<\Omega^*:=\Omega^*(V)\leq \infty$, depending on the general trap $V(x)$, such that for any rotational velocity $0\leq \Omega <\Omega ^*$, minimizers (i.e., ground states) exist if and only if $a<a^*=\|w\|^2_2$, where $a>0$ denotes the absolute product for the number of particles times the scattering length, and $w>0$ is the unique positive solution of $\Delta w-w+w^3=0$ in $\mathbb{R}^2$. If $V(x)=|x|^2$ and $ 0<\Omega <\Omega^*(=2)$ is fixed, we prove that, up to a constant phase, all minimizers must be real-valued, unique and free of vortices as $a \nearrow a^*$, by analyzing the refined limit behavior of minimizers and employing the non-degenerancy of $w$.

Sharp threshold of global existence for nonlinear Schrödinger equation with partial confinement

This paper deals with the nonlinear Schrödinger equation with partial confinement, which may model the attractive Bose–Einstein condensate under a partial trap potential. By using the variational characteristic of the classic nonlinear scalar field equation and the Hamilton conservation, we get the threshold for global existence of the Cauchy problem on mass. Moreover by a novel scaling argument, we prove that this threshold is sharp. In addition, by a numerical computation, we obtain the numerical result of the threshold.

Nonlocal energy functionals and long range interactions in collapsing condensates in a box

The collapse of attractive Bose–Einstein condensates in a box with tunable interatomic interactions was studied experimentally recently. Not only were remarkably stable remnant condensates observed, but furthermore they often seem to involve two stable plateaus. We suggest that these plateaus correspond in fact to two minima of the energy, the attractive atomic interactions being long-range. We show in detail that all the experimental data for these remnant condensates can be accounted for by minimising a nonlocal variational energy involving a long-range interatomic potential that is attractive everywhere and whose range is of the order of the magnetic length.

Limitation of the Lee–Huang–Yang interaction in forming a self-bound state in Bose–Einstein condensates

The perturbative Lee–Huang–Yang (LHY) interaction proportional to n3∕2, where n is the density, creates an infinitely repulsive potential at the center of a Bose–Einstein condensate (BEC) with net attraction, which stops the collapse to form a self-bound state in a dipolar BEC and in a binary BEC. However, recent microscopic calculations of the non-perturbative beyond-mean-field (BMF) interaction indicate that the LHY interaction is valid only for very small values of gas parameter x. We show that a realistic non-perturbative BMF interaction can stop collapse and form a self-bound state only in a weakly attractive binary BEC with small x values (x⪅0.01), whereas the perturbative LHY interaction stops collapse for all attractions. We demonstrate these aspects using an analytic BMF interaction with appropriate weak-coupling LHY and strong coupling limits.

Sharp conditions of global existence for nonlinear Schrödinger equation with a harmonic potential

Abstract The Cauchy problem of nonlinear Schrödinger equation with a harmonic potential for describing the attractive Bose-Einstein condensate under the magnetic trap is considered. We give some sufficient conditions of global existence and finite time blow up of solutions by introducing a family of potential wells. Some different sharp conditions for global existence, and some invariant sets of solutions are also obtained here.

Primordial Universe with radiation and Bose–Einstein condensate

In this work we derive a scenario {in which} the early universe consists of {radiation fluid} and Bose-Einstein condensate. The possibility of gravitational self-interaction due to an attractive Bose-Einstein condensate is analyzed. The classical behavior of the scale factor of the universe is determined by a parameter associated with the Bose-Einstein fluid with bouncing or Big Crunch solutions. After we proceed to compute the finite-norm wave packet solutions to the Wheeler-DeWitt equation. The behavior of the scale factor is studied by applying the many-worlds interpretation of quantum mechanics. The quantum cosmological model is free from the singularities.

Ground states of two-component attractive Bose–Einstein condensates I: Existence and uniqueness

We study ground states of two-component Bose–Einstein condensates (BEC) with trapping potentials in R2, where the intraspecies interaction (−a1,−a2) and the interspecies interaction −β are both attractive, i.e, a1, a2 and β are all positive. The existence and non-existence of ground states are classified completely by investigating equivalently the associated L2-critical constraint variational problem. The uniqueness and symmetry-breaking of ground states are also analyzed under different types of trapping potentials as β↗β⁎=a⁎+(a⁎−a1)(a⁎−a2), where 0<ai<a⁎:=‖w‖22 (i=1,2) is fixed and w is the unique positive solution of Δw−w+w3=0 in R2. The semi-trivial limit behavior of ground states is tackled in the companion paper [12].

Attractive Bose-Einstein condensates in anharmonic traps: Accurate numerical treatment and the intriguing physics of the variance

The dynamics of attractive bosons trapped in one dimensional anharmonic potentials is investigated. Particular emphasis is put on the variance of the position and momentum many-particle operators. Coupling of the center-of-mass and relative-motion degrees-of-freedom necessitates an accurate numerical treatment. The multiconfigurational time-dependent Hartree for bosons (MCTDHB) method is used, and high convergence of the energy, depletion and occupation numbers, and position and momentum variances is proven numerically. We demonstrate for the ground state and out-of-equilibrium dynamics, for condensed and fragmented condensates, for small systems and en route to the infinite-particle limit, that intriguing differences between the density and variance of an attractive Bose-Einstein condensate emerge. Implications are briefly discussed.

A constrained variational problem arising in attractive Bose–Einstein condensate with ellipse-shaped potential

We consider a minimization problem for the variational functional associated with a Gross–Pitaevskii equation arising in the study of an attractive Bose–Einstein condensate. Under an ellipse-shaped trapping potential, that is, the bottom of the trapping potential is an ellipse, we prove that any minimizer of the minimization problem blows up at one of the endpoints of the major axis of the ellipse if the parameter associated to the attractive interaction strength approaches a critical value.