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arXiv:cond-mat/0308096 · Physical Review Letters

Exact Hydrodynamics of a Trapped Dipolar Bose-Einstein Condensate

Duncan H J O’Dell, S. Giovanazzi, Claudia Eberlein

Published Jun 21, 2004• 169 citations
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Abstract

We present exact results in the Thomas-Fermi regime for the statics and dynamics of a harmonically trapped Bose-Einstein condensate that has dipole-dipole interactions in addition to the usual s-wave contact interactions. Remarkably, despite the nonlocal and anisotropic nature of the dipolar interactions, the density profile in a general time-dependent harmonic trap is an inverted parabola. The evolution of the condensate radii is governed by local, ordinary differential equations, and as an example we calculate the monopole and quadrupole shape oscillation frequencies.

DOI
10.1103/physrevlett.92.250401

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We present exact results in the Thomas-Fermi regime for the statics and dynamics of a harmonically trapped Bose-Einstein condensate that has dipole-dipole interactions in addition to the usual s-wave contact interactions. Remarkably, despite the nonlocal and anisotropic nature of the dipolar interactions, the density profile in a general time-dependent harmonic trap is an inverted parabola. The evolution of the condensate radii is governed by local, ordinary differential equations, and as an example we calculate the monopole and quadrupole shape oscillation frequencies.
We derive an exact solution to the Thomas-Fermi equation for a Bose-Einstein condensate (BEC) which has dipole-dipole interactions as well as the usual sss-wave contact interaction, in a harmonic trap. Remarkably, despite the nonlocal anisotropic nature of the dipolar interaction the solution is an inverted parabola, as in the pure sss-wave case, but with a different aspect ratio. We explain in detail the mathematical tools necessary to describe dipolar BECs with or without cylindrical symmetry. Various properties such as electrostriction and stability are discussed.
We present a general method for obtaining the exact static solutions and collective excitation frequencies of a trapped Bose-Einstein condensate (BEC) with dipolar atomic interactions in the Thomas-Fermi regime. The method incorporates analytic expressions for the dipolar potential of an arbitrary polynomial density profile, thereby reducing the problem of handling nonlocal dipolar interactions to the solution of algebraic equations. We comprehensively map out the static solutions and excitation modes, including non-cylindrically-symmetric traps, and also the case of negative scattering length where dipolar interactions stabilize an otherwise unstable condensate. The dynamical stability of the excitation modes gives insight into the onset of collapse of a dipolar BEC. We find that global collapse is consistently mediated by an anisotropic quadrupolar collective mode, although there are two trapping regimes in which the BEC is stable against quadrupole fluctuations even as the ratio of the dipolar to sss-wave interactions becomes infinite. Motivated by the possibility of a fragmented condensate in a dipolar Bose gas due to the partially attractive interactions, we pay special attention to the scissors modes, which can provide a signature of superfluidity, and identify a long-range restoring force which is peculiar to dipolar systems. As part of the supporting material for this paper we provide the computer program used to make the calculations, including a graphical user interface.
We calculate the critical rotation frequency at which a vortex state becomes energetically favorable over the vortex-free ground state in a harmonically trapped Bose-Einstein condensate whose atoms have dipole-dipole interactions as well as the usual sss-wave contact interactions. In the Thomas-Fermi (hydrodynamic) regime, dipolar condensates in oblate cylindrical traps (with the dipoles aligned along the axis of symmetry of the trap) tend to have lower critical rotation frequencies than their purely sss-wave contact interaction counterparts. The converse is true for dipolar condensates in prolate traps. Quadrupole excitations and center of mass motion are also briefly discussed as possible competing mechanisms to a vortex as means by which superfluids with partially attractive interactions might carry angular momentum.
We have employed the theory of harmonically trapped dipolar Bose-Einstein condensates to examine the influence of a uniform magnetic field that rotates at an arbitrary angle to its own orientation. This is achieved by semi-analytically solving the dipolar superfluid hydrodynamics of this system within the Thomas-Fermi approximation and by allowing the body frame of the condensate's density profile to be tilted with respect to the symmetry axes of the nonrotating harmonic trap. This additional degree of freedom manifests itself in the presence of previously unknown stationary solution branches for any given dipole tilt angle. We also find that the tilt angle of the stationary state's body frame with respect to the rotation axis is a nontrivial function of the trapping geometry, rotation frequency, and dipole tilt angle. For rotation frequencies of at least an order of magnitude higher than the in-plane trapping frequency, the stationary state density profile is almost perfectly equivalent to the profile expected in a time-averaged dipolar potential that effectively vanishes when the dipoles are tilted along the ``magic angle'' 54.7 deg. However, by linearizing the fully time-dependent superfluid hydrodynamics about these stationary states, we find that they are dynamically unstable against the formation of collective modes, which we expect would result in turbulent decay.
We have employed the theory of harmonically trapped dipolar Bose-Einstein condensates to examine the influence of a uniform magnetic field that rotates at an arbitrary angle to its own orientation. This is achieved by semi-analytically solving the dipolar superfluid hydrodynamics of this system within the Thomas-Fermi approximation and by allowing the body frame of the condensate's density profile to be tilted with respect to the symmetry axes of the nonrotating harmonic trap. This additional degree of freedom manifests itself in the presence of previously unknown stationary solution branches for any given dipole tilt angle. We also find that the tilt angle of the stationary state's body frame with respect to the rotation axis is a nontrivial function of the trapping geometry, rotation frequency, and dipole tilt angle. For rotation frequencies of at least an order of magnitude higher than the in-plane trapping frequency, the stationary state density profile is almost perfectly equivalent to the profile expected in a time-averaged dipolar potential that effectively vanishes when the dipoles are tilted along the “magic angle” 54.7 deg. However, by linearizing the fully time-dependent superfluid hydrodynamics about these stationary states, we find that they are dynamically unstable against the formation of collective modes, which we expect would result in turbulent decay.
The quest for quantum degenerate Fermi gases interacting through the anisotropic and long-range dipole-dipole interaction is an exciting and fast developing branch within the cold-atom research program. Recent experimental progress in trapping, cooling, and controlling polar molecules with large electric dipole moments has, therefore, motivated much theoretical effort. In a recent letter, we have briefly discussed the application of a variational time-dependent Hartree-Fock approach to study theoretically both the static and the dynamic properties of such a system in a cylinder-symmetric harmonic trap. We focused on the hydrodynamic regime, where collisions ensure the equilibrium locally. Here we present a detailed theory extended to encompass the general case of a harmonic trap geometry without any symmetry. After deriving the equations of motion for the gas, we explore their static solutions to investigate key properties like the aspect ratios in both real and momentum space as well as the stability diagram. We find that, despite the lack of symmetry of the trap, the momentum distribution remains cylinder symmetric. The equations of motion are then used to study the low-lying hydrodynamic excitations, where, apart from the quadrupole and monopole modes, the radial quadrupole mode is also investigated. Furthermore, we study the time-of-flight dynamics as it represents an important diagnostic tool for quantum gases. We find that the real-space aspect ratios are inverted during the expansion, while that in momentum space becomes asymptotically unity. In addition, anisotropic features of the dipole-dipole interaction are discussed in detail. These results could be particularly useful for future investigations of strongly dipolar heteronuclear polar molecules deep in the quantum degenerate regime.