Repulsive Bose-Einstein Condensate Research Articles

Metastability of Repulsive Bose–Einstein Condensate in a Finite Trap and Instability of Ground State Energies

Metastability of Repulsive Bose–Einstein Condensate in a Finite Trap and Instability of Ground State Energies

The instabilities beyond modulational type in a repulsive Bose–Einstein condensate with a periodic potential

The instabilities of the nontrivial phase elliptic solutions in a repulsive Bose–Einstein condensate (BEC) with a periodic potential are investigated. Based on the defocusing nonlinear Schrödinger (NLS) equation with an elliptic function potential, the well-known modulational instability (MI), the more recently identified high-frequency instability, and an unprecedented – to our knowledge – variant of the MI, the so-called isola instability are identified numerically. Upon varying parameters of the solutions, instability transitions occur through suitable bifurcations, such as the Hamiltonian-Hopf one. Specifically, (i) increasing the elliptic modulus k of the solutions, we find that MI switches to the isola instability and the dominant disturbance has twice the elliptic wave’s period, corresponding to a Floquet exponent μ=π2K(k). The isola instability arises from the collision of spectral elements at the origin of the spectral plane. (ii) Upon varying V0, the transition between MI and the high-frequency instability occurs. Differently from the MI and isola instability where the collisions of eigenvalues happen at the origin, the high-frequency instability arises from pairwise collisions of nonzero, imaginary elements of the stability spectrum; (iii) In the limit of sinusoidal potential, we show that MI occurs from a collision of eigenvalues with μ=π2K(k) at the origin; (iv) we also examine the dynamic byproducts of the instability in chaotic fields generated by its manifestation. An interesting observation is that, in addition to MI, the isola instability could also lead to dark localized events in the scalar defocusing NLS equation.

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.

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.

Solitons and soliton interactions in repulsive spinor Bose–Einstein condensates with nonzero background

We characterize the soliton solutions and their interactions for a system of coupled evolution equations of nonlinear Schrödinger (NLS) type that models the dynamics in one-dimensional repulsive Bose–Einstein condensates with spin one, taking advantage of the representation of such model as a special reduction of a $$2\times 2$$ matrix NLS system. Specifically, we study in detail the case in which solutions tend to a nonzero background at space infinities. First we derive a compact representation for the multi-soliton solutions in the system using the Inverse Scattering Transform (IST). We introduce the notion of canonical form of a solution, corresponding to the case when the background as $$x\rightarrow \infty $$ is proportional to the identity. We show that solutions for which the asymptotic behavior at infinity is not proportional to the identity, referred to as being in non-canonical form, can be reduced to canonical form by unitary transformations that preserve the symmetric nature of the solution (physically corresponding to complex rotations of the quantization axes). Then we give a complete characterization of the two families of one-soliton solutions arising in this problem, corresponding to ferromagnetic and to polar states of the system, and we discuss how the physical parameters of the solitons for each family are related to the spectral data in the IST. We also show that any ferromagnetic one-soliton solution in canonical form can be reduced to a single dark soliton of the scalar NLS equation, and any polar one-soliton solution in canonical form is unitarily equivalent to a pair of oppositely polarized displaced scalar dark solitons up to a rotation of the quantization axes. Finally, we discuss two-soliton interactions and we present a complete classification of the possible scenarios that can arise depending on whether either soliton is of ferromagnetic or polar type.

Many-body dark solitons in one-dimensional hard-core Bose gases

The existence and stability of solitonic states in one-dimensional repulsive Bose-Einstein condensates is investigated within a fully many-body framework by considering the limit of infinite repulsion (Tonks-Girardeau gas). A class of stationary, shape-invariant states propagating at constant velocity are explicitly found and compared to the known solution of the Gross-Pitaevskii equation. The typical features attributed to nonlinearity are thus recovered in a purely linear theory, provided the full many-particle physics is correctly accounted for. However, the formation dynamics predicted by the Gross-Pitaevskii approximation considerably differs from the exact many-body evolution.

Study of Zero Temperature Ground State Properties of the Repulsive Bose-Einstein Condensate in an Anharmonic Trap

The zero-temperature ground state properties of experimental 87Rb condensate are studied in a harmonic plus quartic trap [ V(r) = ½mω2r2 + λr4 ]. The anharmonic parameter (λ) is slowly tuned from harmonic to anharmonic. For each choice of λ, the many-particle Schrödinger equation is solved using the potential harmonic expansion method and determines the lowest effective many-body potential. We utilize the correlated two-body basis function, which keeps all possible two-body correlations. The use of van der Waals interaction gives realistic pictures. We calculate kinetic energy, trapping potential energy, interaction energy, and total ground state energy of the condensate in this confining potential, modelled experimentally. The motivation of the present study is to investigate the crucial dependency of the properties of an interacting quantum many-body system on λ. The average size of the condensate has also been calculated to observe how the stability of repulsive condensate depends on anharmonicity. In particular, our calculation presents a clear physical picture of the repulsive condensate in an anharmonic trap.

Multivalley dark solitons in multicomponent Bose-Einstein condensates with repulsive interactions.

We obtain multivalley dark soliton solutions with asymmetric or symmetric profiles in multicomponent repulsive Bose-Einstein condensates by developing the Darboux transformation method. We demonstrate that the width-dependent parameters of solitons significantly affect the velocity ranges and phase jump regions of multivalley dark solitons, in sharp contrast to scalar dark solitons. For double-valley dark solitons, we find that the phase jump is in the range [0,2π], which is quite different from that of the usual single-valley dark soliton. Based on our results, we argue that the phase jump of an n-valley dark soliton could be in the range [0,nπ], supported by our analysis extending up to five-component condensates. The interaction between a double-valley dark soliton and a single-valley dark soliton is further investigated, and we reveal a striking collision process in which the double-valley dark soliton is transformed into a breather after colliding with the single-valley dark soliton. Our analyses suggest that this breather transition exists widely in the collision processes involving multivalley dark solitons. The possibilities for observing these multivalley dark solitons in related Bose-Einstein condensates experiments are discussed.

A bosonic bright soliton in a mixture of repulsive Bose–Einstein condensate and polarized ultracold fermions under the influence of pressure evolution

Repulsive Bose–Einstein condensates, where short-range interaction is included up to the third order by the interaction radius, demonstrate the existence of bright solitons in a narrow interval of parameters. These solitons are studied here for the boson–fermion mixture, where spin-1/2 fermions are considered in the regime of full-spin polarization. The influence of fermions on bosonic bright solitons via the boson–fermion interaction is considered up to the third order by the interaction radius. Fermions themselves are considered within the hydrodynamic model, which includes the kinetic pressure-evolution equation. The interactions between fermions are also considered. The first order by the interaction radius makes a zero contribution to the Euler equation and the kinetic pressure evolution equation for fermions, but the third order by the interaction radius provides nonzero contributions to both equations. The repulsive (attractive) boson–fermion interaction leads to bright (dark) fermionic solitons.

Orbital angular momentum dynamics of Bose-Einstein condensates trapped in two stacked rings

We investigate the stability and dynamics of the orbital angular momentum modes of a repulsive Bose-Einstein condensate trapped in two tunnel-coupled rings in a stack configuration. Within mean-field theory, we derive a two-state model for the system in the case in which we populate equally both rings with a single orbital angular momentum mode and include small perturbations in other modes. Analyzing the fixed point solutions of the model and the associated classical Hamiltonian, we characterize the destabilization of the stationary states and the subsequent dynamics. By populating a single orbital angular momentum mode with an arbitrary population imbalance between the rings, we derive analytically the boundary between the regimes of Josephson oscillations and macroscopic quantum self-trapping and study numerically the stability of these solutions.

Hydrodynamic model of a Bose–Einstein condensate with anisotropic short-range interaction and bright solitons in a repulsive Bose–Einstein condensate

A quantum hydrodynamic model is developed for axial symmetric anisotropic short-range interactions. The quantum stress tensor represents the interaction and is derived up to the third order by the interaction radius. The first order by the interaction radius contains the isotropic part only. This leads to interaction in the Gross–Pitaevskii approximation. Terms existing in the third order by the interaction radius are caused by the isotropic and nonisotropic parts of the interaction, and each of them introduces an interaction constant. Therefore, three interaction constants are involved in the model. Atoms, except for alkali and alkali-earth atoms, can have an anisotropic potential of interaction (this has particularly been demonstrated for the lanthanides). Short-wavelength instability caused by nonlocal terms appears in the Bogoliubov spectrum, and the conditions for stable and unstable behavior are described. Bright solitons in a repulsive Bose–Einstein condensate (BEC) are studied under the influence of anisotropic short-range interaction in the BEC of one species. The area of existence of bright solitons corresponds to the area of instability of the Bogoliubov spectrum. Approximate reduction of nonlocal nonlinearity to quintic nonlinearity in the bright soliton regime is also demonstrated.

Linear and Nonlinear Bullets of the Bogoliubov-de Gennes Excitations.

We report on the focalization of Bogoliubov-de Gennes excitations of the nonlinear Schrödinger equation in the defocusing regime (Gross-Pitaevskii equation for repulsive Bose-Einstein condensates) with a spatially modulated periodic potential. Exploiting the modification of the dispersion relation induced by the modulation, we demonstrate the existence of localized structures of the Bogoliubov-de Gennes excitations, in both the linear and nonlinear regimes (linear and nonlinear "bullets"). These traveling Bogoliubov-de Gennes bullets, localized both spatially and temporally in the comoving reference frame, are robust and propagate remaining stable, without spreading or filamentation. The phenomena reported in this Letter could be observed in atomic Bose-Einstein condensates in the presence of a spatially periodic potential induced by an optical lattice.

Oscillatory instabilities of gap solitons in a repulsive Bose–Einstein condensate

The paper is devoted to numerical study of stability of nonlinear localized modes (“gap solitons”) for the spatially one-dimensional Gross–Pitaevskii equation (1D GPE) with periodic potential and repulsive interparticle interactions. We use the Evans function approach combined with the exterior algebra formulation in order to detect and describe weak oscillatory instabilities. We show that the simplest (“fundamental”) gap solitons in the first and in the second spectral gap undergo oscillatory instabilities for certain values of the frequency parameter (i.e., the chemical potential). The number of unstable eigenvalues and the associated instability rates are described. Several stable and unstable more complex (non-fundamental) gap solitons are also discussed. The results obtained from the Evans function approach are independently confirmed using the direct numerical integration of the GPE.

Lopsided vortex cells in rotating Bose–Einstein condensates

The amplitude of the wavefunction of a repulsive Bose–Einstein condensate in a two-dimensional axisymmetric harmonic trap decreases as we go from the center of the condensate towards its edge. Consequently the wavefunction amplitude at the edge of a vortex cell in a vortex lattice in a rotating repulsive condensate should be greater for the inner cells. Continuity of the wavefunction amplitude between neighboring cells requires then that varies along the edge of each vortex cell, so that it is greater at the points that are closer to the center of the condensate. In other words, the wavefunction amplitude is lopsided in each vortex cell. We show this explicitly via an analytic variational calculation where a simple lopsided wavefunction amplitude is used for each vortex cell. This wavefunction amplitude varies continuously as we go from one cell to the next. The results agree quite well with experimental observations.

Polarization interactions in multi-component defocusing media

We study dark–bright soliton interactions in multi-component media such as nonlinear optical media in the defocusing regime and repulsive Bose–Einstein condensates. This is achieved using the recently developed formalism of the inverse scattering transform for the defocusing multi-component nonlinear Schrödinger equation with non-zero boundary conditions. We show that, generically, these interactions result in a non-trivial polarization shift for the bright components. We compute such polarization shift analytically and compare it to that in focusing two-component nonlinear Schrödinger systems.

Stationary solutions for the nonlinear Schrödinger equation modeling three-dimensional spherical Bose-Einstein condensates in general potentials.

Stationary solutions for the cubic nonlinear Schrödinger equation modeling Bose-Einstein condensates (BECs) confined in three spatial dimensions by general forms of a potential are studied through a perturbation method and also numerically. Note that we study both repulsive and attractive BECs under similar frameworks in order to deduce the effects of the potentials in each case. After outlining the general framework, solutions for a collection of specific confining potentials of physical relevance to experiments on BECs are provided in order to demonstrate the approach. We make several observations regarding the influence of the particular potentials on the behavior of the BECs in these cases, comparing and contrasting the qualitative behavior of the attractive and repulsive BECs for potentials of various strengths and forms. Finally, we consider the nonperturbative where the potential or the amplitude of the solutions is large, obtaining various qualitative results. When the kinetic energy term is small (relative to the nonlinearity and the confining potential), we recover the expected Thomas-Fermi approximation for the stationary solutions. Naturally, this also occurs in the large mass limit. Through all of these results, we are able to understand the qualitative behavior of spherical three-dimensional BECs in weak, intermediate, or strong confining potentials.

Stability and tunneling dynamics of a dark-bright soliton pair in a harmonic trap

We consider a binary repulsive Bose-Einstein condensate in a harmonic trap in one spatial dimension and investigate particular solutions consisting of two dark-bright (DB) solitons. There are two different stationary solutions characterized by the phase difference in the bright component, in-phase and out-of-phase states. We show that above a critical particle number in the bright component, a symmetry breaking bifurcation of the pitchfork type occurs that leads to a new asymmetric solution whereas the parental branch, i.e., the out-of-phase state becomes unstable. These three different states support different small amplitude oscillations, characterized by an almost stationary density of the dark component and a tunneling of the bright component between the two dark solitons. Within a suitable effective double-well picture, these can be understood as the characteristic features of a Bosonic Josephson Junction (BJJ), and we show within a two-mode approach that all characteristic features of the BJJ phase space are recovered. For larger deviations from the stationary states, the simplifying double-well description breaks down due to the feedback of the bright component onto the dark one, causing the solitons to move. In this regime we observe intricate anharmonic and aperiodic dynamics, exhibiting remnants of the BJJ phase space.

Transport of the repulsive Bose–Einstein condensate in a double-well trap: interaction impact and relation to the Josephson effect

Two aspects of the transport of a repulsive Bose–Einstein condensate (BEC) in a double-well trap are inspected: The impact of the interatomic interaction and the analogy with the Josephson effect. The analysis employs a numerical solution of a 3D time-dependent Gross–Pitaevskii equation for a total order parameter covering the whole trap. The population transfer is driven by a time-dependent shift of a barrier separating the left and right wells. The sharp and soft profiles of the barrier velocity are tested. The evolution of the relevant characteristics, involving phase differences and currents, is inspected. It is shown that the repulsive interaction substantially supports the transfer making it possible (i) in a wide velocity interval and (ii) three orders of magnitude faster than in the ideal BEC. The transport can be approximately treated as the dc Josephson effect. The dual origin of the critical barrier velocity (break of the adiabatic following and dc/ac transition) is discussed. Following the calculations, the robustness of the transport (dc) crucially depends on the interaction and barrier velocity profile. Only soft profiles which minimize undesirable dipole oscillations are acceptable.

Dynamics of Bose-Einstein Condensates: Exact Representation and Topological Classification of Coherent Matter Waves

By using the bifurcation theory of dynamical systems, we present the exact representation and topological classification of coherent matter waves in Bose-Einstein condensates (BECs), such as solitary waves and modulate amplitude waves (MAWs). The existence and multiplicity of such waves are determined by the parameter regions selected. The results show that the characteristic of coherent matter waves can be determined by the “angular momentum” in attractive BECs while for repulsive BECs; the waves of the coherent form are all MAWs. All exact explicit parametric representations of the above waves are exhibited and numerical simulations support the result.

Effects of interaction on thermodynamics of a repulsive Bose-Einstein condensate

We report the effects of interaction on thermodynamic properties of a repulsive Bose-Einstein condensate confined in a harmonic trap by using the correlated potential harmonics expansion method. This many-body technique permits the use of a realistic interactomic interaction, which gives rise to the effective long-range interaction of the condensate in terms of the $s$-wave scattering length. We have computed temperature ($T$) dependence of the chemical potential, specific heat, condensate fraction, entropy, pressure, and the average energy per particle of a system containing a large number ($A$) of ${}^{87}$Rb atoms in the Joint Institute for Laboratory Astrophysics (JILA) trap. The repulsion among the interacting bosons results in a small but measurable drop of condensate fraction and critical temperature (${T}_{c}$), compared to those of a noninteracting condensate. These are in agreement with the experiment. Although all thermodynamic quantities have a strong dependence on $A$ and to a smaller extent on the interatomic interaction, our numerical calculation appears to show that a thermodynamic quantity per particle follows a universal behavior as a function of $T/{T}_{c}$. This shows the importance of ${T}_{c}$ for all thermodynamic properties of the condensate. As expected, for $Tg{T}_{c}$, these properties follow those of a trapped noncondensed Bose gas.