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arXiv:0908.4313 · Physical Review A

Aligned dipolar Bose-Einstein condensate in a double-well potential: From cigar shaped to pancake shaped

Muhammad Asaduzzaman, D. Blume

Published Nov 24, 2009• 32 citations
arXivHTMLPDF

Abstract

We consider a Bose-Einstein condensate, which is characterized by long-range and anisotropic dipole-dipole interactions and vanishing $s$-wave scattering length, in a double-well potential. The properties of this system are investigated as functions of the height of the barrier that splits the harmonic trap into two halves, the number of particles (or dipole-dipole strength) and the aspect ratio $\ensuremath{\lambda}$, which is defined as the ratio between the axial and longitudinal trapping frequencies ${\ensuremath{\omega}}_{z}$ and ${\ensuremath{\omega}}_{\ensuremath{\rho}}$. The phase diagram is determined by analyzing the stationary mean-field solutions. Three distinct regions are found: a region where the energetically lowest lying stationary solution is symmetric, a region where the energetically lowest lying stationary solution is located asymmetrically in one of the wells, and a region where the system is mechanically unstable. For sufficiently large aspect ratio $\ensuremath{\lambda}$ and sufficiently high barrier height, the system consists of two connected quasi-two-dimensional sheets with density profiles whose maxima are located either at $\ensuremath{\rho}=0$ or away from $\ensuremath{\rho}=0$. The stability of the stationary solutions is investigated by analyzing the Bogoliubov--de Gennes excitation spectrum and the dynamical response to small perturbations. These studies reveal unique oscillation frequencies and distinct collapse mechanisms. The results derived within the mean-field framework are complemented by an analysis based on a two-mode model.

DOI
10.1103/physreva.80.053622

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We consider a Bose-Einstein condensate, which is characterized by long-range and anisotropic dipole-dipole interactions and vanishing sss-wave scattering length, in a double-well potential. The properties of this system are investigated as functions of the height of the barrier that splits the harmonic trap into two halves, the number of particles (or dipole-dipole strength) and the aspect ratio \ensuremathλ\ensuremath{\lambda}\ensuremathλ, which is defined as the ratio between the axial and longitudinal trapping frequencies \ensuremathωz{\ensuremath{\omega}}_{z}\ensuremathωz​ and \ensuremathω\ensuremathρ{\ensuremath{\omega}}_{\ensuremath{\rho}}\ensuremathω\ensuremathρ​. The phase diagram is determined by analyzing the stationary mean-field solutions. Three distinct regions are found: a region where the energetically lowest lying stationary solution is symmetric, a region where the energetically lowest lying stationary solution is located asymmetrically in one of the wells, and a region where the system is mechanically unstable. For sufficiently large aspect ratio \ensuremathλ\ensuremath{\lambda}\ensuremathλ and sufficiently high barrier height, the system consists of two connected quasi-two-dimensional sheets with density profiles whose maxima are located either at \ensuremathρ=0\ensuremath{\rho}=0\ensuremathρ=0 or away from \ensuremathρ=0\ensuremath{\rho}=0\ensuremathρ=0. The stability of the stationary solutions is investigated by analyzing the Bogoliubov--de Gennes excitation spectrum and the dynamical response to small perturbations. These studies reveal unique oscillation frequencies and distinct collapse mechanisms. The results derived within the mean-field framework are complemented by an analysis based on a two-mode model.
We study the Josephson oscillation and self-trapping dynamics of a cigar-shaped dipolar Bose-Einstein condensate of 52{}^{52}52Cr atoms polarized along the symmetry axis of an axially symmetric double-well potential using the numerical solution of a mean-field model, for dominating repulsive contact interaction (large positive scattering length aaa) over an anisotropic dipolar interaction. Josephson-type oscillation emerges for small and very large numbers of atoms, whereas self-trapping is noted for an intermediate number of atoms. The dipolar interaction pushes the system away from self-trapping towards Josephson oscillation. We consider a simple two-mode description for a qualitative understanding of the dynamics.
The effect of dipolar orientation with respect to the condensate plane on the mean-field dynamics of dipolar Bose-Einstein condensates in a pancake-shaped confinement is discussed. The stability of a quasi-two-dimensional condensate, with respect to the tilting angle, is found to be different from a two-dimensional layer of dipoles, indicating the relevance of the transverse extension while characterizing two-dimensional dipolar systems. An anisotropic excitation spectrum exhibiting a highly tunable, rotonlike minimum can arise entirely from the dipole-dipole interactions, by tilting the dipoles. At the magic angle and in the absence of contact interactions, the long-wavelength excitations are not phononlike and always unstable. The post-roton-instability dynamics, in contrast to phonon instability, in a uniform condensate, is featured by a transient, defect-free, stripe pattern, which eventually undergoes local collapses, and driving the condensate back into the stable regime can make them sustained for longer. Hopping between stripes has been observed before it melts into a uniform state in the presence of dissipation. Finally, we discuss a class of solutions, in which a quasi-two-dimensional condensate is self-trapped in one direction, as well as a regime of interaction parameters, including attractive short-range interactions, at which a two-dimensional anisotropic soliton can be stabilized, and we show that a chromium condensate with a relatively small number of atoms is well suited for this.
We consider the stability and dynamics of multiple dark solitons in cigar-shaped Bose-Einstein condensates. Our study is motivated by the fact that multiple matter-wave dark solitons may naturally form in such settings as per our recent work [Phys. Rev. Lett. 101, 130401 (2008)]. First, we study the dark soliton interactions and show that the dynamics of well-separated solitons (i.e., ones that undergo a collision with relatively low velocities) can be analyzed by means of particle-like equations of motion. The latter take into regard the repulsion between solitons (via an effective repulsive potential) and the confinement and dimensionality of the system (via an effective parabolic trap for each soliton). Next, based on the fact that stationary, well-separated dark multisoliton states emerge as a nonlinear continuation of the appropriate excited eigenstates of the quantum harmonic oscillator, we use a Bogoliubov-de Gennes analysis to systematically study the stability of such structures. We find that for a sufficiently large number of atoms, multiple soliton states are dynamically stable, while for a small number of atoms, we predict a dynamical instability emerging from resonance effects between the eigenfrequencies of the soliton modes and the intrinsic excitation frequencies of the condensate. Finally, we present experimental realizations of multisoliton states including a three-soliton state consisting of two solitons oscillating around a stationary one and compare the relevant results to the predictions of the theoretical mean-field model.
By means of mean-field theory, we have studied the structure and excitation spectrum of a purely dipolar Bose gas in a pancake shaped trap where the confinement in the xxx-yyy plane is provided by a highly anharmonic potential, resulting in an almost uniform confinement in the plane. We show that the stable condensates are characterized by marked radially structured density profiles. The stability diagram is calculated by independently varying the strength of the interaction and the trap geometry. By computing the Bogoliubov excitation spectrum near the instability line we show that soft ``angular'' rotons are responsible for the collapse of the system. The free expansion of the cloud after the trap is released is also studied by means of time-dependent calculations, showing that a prolate cigar-shaped condensate is dynamically stabilized during the expansion, which would otherwise collapse. Dipolar condensates rotating with sufficiently high angular velocity show the formation of multiply quantized giant vortices, while the condensates acquire a ring-shaped form.
We derive rigorous one- and two-dimensional mean-field equations for cigar- and pancake-shaped dipolar Bose-Einstein condensates with arbitrary polarization angle. We show how the dipolar interaction modifies the contact interaction of the strongly confined atoms. In addition, our equations introduce a nonlocal potential, which is anisotropic for pancake-shaped condensates. We propose to observe this anisotropy via measurement of the condensate aspect ratio. We also derive analytically approximate density profiles from our equations. Both the numerical solutions of our reduced mean-field equations and the analytical density profiles agree well with numerical solutions of the full Gross-Pitaevskii equation while being more efficient to compute.
We study a Bose-Einstein condensate of 52Cr^{52}\mathrm{Cr}52Cr atoms confined in a toroidal trap with a variable strength of sss-wave contact interactions. We analyze the effects of the anisotropic nature of the dipolar interaction by considering the magnetization axis to be perpendicular to the trap symmetry axis. In the absence of a central repulsive barrier, when the trap is purely harmonic, the effect of reducing the scattering length is a tuning of the geometry of the system from a pancake-shaped condensate when it is large to a cigar-shaped condensate for small scattering lengths. For a condensate in a toroidal trap, the interaction in combination with the central repulsive Gaussian barrier produces an azimuthal dependence of the particle density for a fixed radial distance. We find that along the magnetization direction the density decreases as the scattering length is reduced but presents two symmetric density peaks in the perpendicular axis. For even lower values of the scattering length we observe that the system undergoes a dipolar-induced symmetry breaking phenomenon. The whole density becomes concentrated in one of the peaks, resembling an origin-displaced cigar-shaped condensate. In this context we also analyze stationary vortex states and their associated velocity fields, finding that these also show a strong azimuthal dependence for small scattering lengths. The expectation value of the angular momentum along the zzz direction provides a qualitative measure of the difference between the velocity in the different density peaks.