Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1003.0371 · Physical Review A

Collective excitation frequencies and stationary states of trapped dipolar Bose-Einstein condensates in the Thomas-Fermi regime

Rick van Bijnen, Nicholas G. Parker, S. J. J. M. F. Kokkelmans, A. Martin, Duncan H J O’Dell

Published Sep 15, 2010• 40 citations
arXivHTMLPDF

Abstract

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 $s$-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.

DOI
10.1103/physreva.82.033612

Related papers

Exact solution of the Thomas-Fermi equation for a trapped Bose-Einstein condensate with dipole-dipole interactions

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

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 $s$-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 $s$-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.

arXiv:cond-mat/0311100•Physical Review A•Mar 15, 2005•143 citations

Exact Hydrodynamics of a Trapped Dipolar Bose-Einstein Condensate

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

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.

arXiv:cond-mat/0308096•Physical Review Letters•Jun 21, 2004•169 citations

Quadrupole and scissors modes and nonlinear mode coupling in trapped two-component Bose-Einstein condensates

Kenichi Kasamatsu, Makoto Tsubota, Masahito Ueda

We theoretically investigate quadrupolar collective excitations in two-component Bose-Einstein condensates and their nonlinear dynamics associated with harmonic generation and mode coupling. Under the Thomas-Fermi approximation and the quadratic polynomial ansatz for density fluctuations, the linear analysis of the superfluid hydrodynamic equations predicts excitation frequencies of three normal modes constituted from monopole and quadrupole oscillations, and those of three scissors modes. These six modes are bifurcated into in-phase and out-of-phase modes by the intercomponent interaction, yielding the nonlinear dynamics that are absent in a single-component condensate. We obtain analytically the resonance conditions for the second-harmonic generation in terms of the trap aspect ratio and the strength of intercomponent interaction. The numerical simulation of the coupled Gross-Pitaevskii equations vindicates the validity of the analytical results and reveals the dynamics of the second-harmonic generation and nonlinear mode coupling that lead to nonlinear oscillations of the condensate with damping and recurrence reminiscent of the Fermi-Pasta-Ulam problem.

arXiv:cond-mat/0309539•Physical Review A•Apr 29, 2004•26 citations

Transition states and thermal collapse of dipolar Bose-Einstein condensates

Andrej Junginger, Manuel Kreibich, Jörg Main +1 more

We investigate thermally excited, dipolar Bose-Einstein condensates (BECs). Quasiparticle excitations of the atomic cloud cause density fluctuations which can induce the collapse of the condensate if the interparticle interaction is attractive. Within a variational approach, we identify the collectively excited stationary states of the gas which form transition states on the way to the BEC's collapse. We analyze transition states with different $m$-fold rotational symmetry and identify the one which mediates the collapse. The latter's symmetry depends on the trap aspect ratio of the external trapping potential, which determines the shape of the BEC. Moreover, we present the collapse dynamics of the BEC and calculate the corresponding decay rate using transition-state theory. We observe that the thermally induced collapse mechanism is important near the critical scattering length, where the lifetime of the condensate can be significantly reduced. Our results are valid for an arbitrary strength of the dipole-dipole interaction. Specific applications are discussed for the elements ${}^{52}$Cr, ${}^{164}$Dy, and ${}^{168}$Er with which dipolar BECs have been experimentally realized.

arXiv:1309.6746•Physical Review A•Oct 11, 2013•10 citations

Ground-state and collective modes of a spin-polarized dipolar Bose-Einstein condensate in a harmonic trap

Igor Sapina, T. Dahm, Nils Schopohl

We report results for the Thomas-Fermi ground state of a spin-polarized dipolar interacting Bose-Einstein condensate for the case when the external magnetic field $\mathbf{B}$ is not orientated parallel to a principal axis but is aligned parallel to a symmetry plane of a harmonic anisotropic trap. For a dipole interaction strength parameter ${\ensuremath{\varepsilon}}_{D}\ensuremath{\ne}0$ the release energy of the condensate depends on the trap orientation angle ${\ensuremath{\vartheta}}_{T}$ between the principal axis ${\mathbf{e}}_{z,T}$ of the trap and the field $\mathbf{B}$. From the quasiclassical Josephson equation of macroscopic quantum physics we determine the low-lying eigenfrequencies of small-amplitude collective modes of the condensate density for various trap frequencies ${\ensuremath{\omega}}_{a}$ and trap orientation angles ${\ensuremath{\vartheta}}_{T}$. For the special case of a spherical harmonic trap with trap frequency $\ensuremath{\omega}$ it is rigorously shown for $\ensuremath{-}\frac{1}{2}<{\ensuremath{\varepsilon}}_{D}<1$ that a pure $s$-wave symmetry breather excitation of the condensate density exists, that oscillates at a constant frequency ${\ensuremath{\Omega}}_{s}=\sqrt{5}\ensuremath{\omega}$ around the ground-state cloud, despite the well-known fact that the shape of the ground-state cloud of a spin-polarized dipolar condensate is for ${\ensuremath{\varepsilon}}_{D}\ensuremath{\ne}0$ not isotropic. For ${\ensuremath{\vartheta}}_{T}\ensuremath{\ne}0$ the small-amplitude modes of the particle density with isotropic and quadrupolar symmetry consist of two groups. There exist four modes that are combinations of basis functions, with $s$-wave, ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$- and ${d}_{{z}^{2}}$-wave, and ${d}_{\mathit{xz}}$-wave symmetry, and two modes that are combinations of basis functions with ${d}_{\mathit{yz}}$- and ${d}_{\mathit{xy}}$-wave symmetry. A characteristic difference in the dependence of the frequencies of these six collective modes on the dipole interaction strength parameter ${\ensuremath{\varepsilon}}_{D}$ for prolate and oblate harmonic triaxial traps, respectively, is suggested to be used as an experimental method to measure the $s$-wave scattering length ${a}_{s}$ of the atoms.

arXiv:1006.4734•Physical Review A•Nov 18, 2010•16 citations

Structure formation during the collapse of a dipolar atomic Bose-Einstein condensate

Nicholas G. Parker, Christopher Ticknor, A. Martin +1 more

We investigate the collapse of a trapped dipolar Bose-Einstein condensate. This is performed by numerical simulations of the Gross-Pitaevskii equation and the novel application of the Thomas-Fermi hydrodynamic equations to collapse. We observe regimes of both global collapse, where the system evolves to a highly elongated or flattened state depending on the sign of the dipolar interaction, and local collapse, which arises due to dynamically unstable phonon modes and leads to a periodic arrangement of density shells, disks, or stripes. In the adiabatic regime, where ground states are followed, collapse can occur globally or locally, while in the nonadiabatic regime, where collapse is initiated suddenly, local collapse commonly occurs. We analyze the dependence on the dipolar interactions and trap geometry, the length and time scales for collapse, and relate our findings to recent experiments.

arXiv:0810.2028•Physical Review A•Jan 21, 2009•79 citations
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 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 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 theoretically investigate quadrupolar collective excitations in two-component Bose-Einstein condensates and their nonlinear dynamics associated with harmonic generation and mode coupling. Under the Thomas-Fermi approximation and the quadratic polynomial ansatz for density fluctuations, the linear analysis of the superfluid hydrodynamic equations predicts excitation frequencies of three normal modes constituted from monopole and quadrupole oscillations, and those of three scissors modes. These six modes are bifurcated into in-phase and out-of-phase modes by the intercomponent interaction, yielding the nonlinear dynamics that are absent in a single-component condensate. We obtain analytically the resonance conditions for the second-harmonic generation in terms of the trap aspect ratio and the strength of intercomponent interaction. The numerical simulation of the coupled Gross-Pitaevskii equations vindicates the validity of the analytical results and reveals the dynamics of the second-harmonic generation and nonlinear mode coupling that lead to nonlinear oscillations of the condensate with damping and recurrence reminiscent of the Fermi-Pasta-Ulam problem.
We investigate thermally excited, dipolar Bose-Einstein condensates (BECs). Quasiparticle excitations of the atomic cloud cause density fluctuations which can induce the collapse of the condensate if the interparticle interaction is attractive. Within a variational approach, we identify the collectively excited stationary states of the gas which form transition states on the way to the BEC's collapse. We analyze transition states with different mmm-fold rotational symmetry and identify the one which mediates the collapse. The latter's symmetry depends on the trap aspect ratio of the external trapping potential, which determines the shape of the BEC. Moreover, we present the collapse dynamics of the BEC and calculate the corresponding decay rate using transition-state theory. We observe that the thermally induced collapse mechanism is important near the critical scattering length, where the lifetime of the condensate can be significantly reduced. Our results are valid for an arbitrary strength of the dipole-dipole interaction. Specific applications are discussed for the elements 52{}^{52}52Cr, 164{}^{164}164Dy, and 168{}^{168}168Er with which dipolar BECs have been experimentally realized.
We report results for the Thomas-Fermi ground state of a spin-polarized dipolar interacting Bose-Einstein condensate for the case when the external magnetic field B\mathbf{B}B is not orientated parallel to a principal axis but is aligned parallel to a symmetry plane of a harmonic anisotropic trap. For a dipole interaction strength parameter ${\ensuremath{\varepsilon}}_{D}\ensuremath{\ne}0$ the release energy of the condensate depends on the trap orientation angle ${\ensuremath{\vartheta}}_{T}$ between the principal axis ez,T{\mathbf{e}}_{z,T}ez,T​ of the trap and the field B\mathbf{B}B. From the quasiclassical Josephson equation of macroscopic quantum physics we determine the low-lying eigenfrequencies of small-amplitude collective modes of the condensate density for various trap frequencies ${\ensuremath{\omega}}_{a}$ and trap orientation angles ${\ensuremath{\vartheta}}_{T}$. For the special case of a spherical harmonic trap with trap frequency $\ensuremath{\omega}$ it is rigorously shown for $\ensuremath{-}\frac{1}{2}<{\ensuremath{\varepsilon}}_{D}<1$ that a pure sss-wave symmetry breather excitation of the condensate density exists, that oscillates at a constant frequency ${\ensuremath{\Omega}}_{s}=\sqrt{5}\ensuremath{\omega}$ around the ground-state cloud, despite the well-known fact that the shape of the ground-state cloud of a spin-polarized dipolar condensate is for ${\ensuremath{\varepsilon}}_{D}\ensuremath{\ne}0$ not isotropic. For ${\ensuremath{\vartheta}}_{T}\ensuremath{\ne}0$ the small-amplitude modes of the particle density with isotropic and quadrupolar symmetry consist of two groups. There exist four modes that are combinations of basis functions, with sss-wave, ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$- and dz2{d}_{{z}^{2}}dz2​-wave, and dxz{d}_{\mathit{xz}}dxz​-wave symmetry, and two modes that are combinations of basis functions with dyz{d}_{\mathit{yz}}dyz​- and dxy{d}_{\mathit{xy}}dxy​-wave symmetry. A characteristic difference in the dependence of the frequencies of these six collective modes on the dipole interaction strength parameter ${\ensuremath{\varepsilon}}_{D}$ for prolate and oblate harmonic triaxial traps, respectively, is suggested to be used as an experimental method to measure the sss-wave scattering length as{a}_{s}as​ of the atoms.
We investigate the collapse of a trapped dipolar Bose-Einstein condensate. This is performed by numerical simulations of the Gross-Pitaevskii equation and the novel application of the Thomas-Fermi hydrodynamic equations to collapse. We observe regimes of both global collapse, where the system evolves to a highly elongated or flattened state depending on the sign of the dipolar interaction, and local collapse, which arises due to dynamically unstable phonon modes and leads to a periodic arrangement of density shells, disks, or stripes. In the adiabatic regime, where ground states are followed, collapse can occur globally or locally, while in the nonadiabatic regime, where collapse is initiated suddenly, local collapse commonly occurs. We analyze the dependence on the dipolar interactions and trap geometry, the length and time scales for collapse, and relate our findings to recent experiments.