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arXiv:1504.03375 · Physical Review A

Equilibration and approximate conservation laws: Dipole oscillations and perfect drag of ultracold atoms in a harmonic trap

Robert Bamler, Achim Rosch

Published Jun 5, 2015• 8 citations
arXivHTMLPDF

Abstract

The presence of (approximate) conservation laws can prohibit the fast relaxation of interacting many-particle quantum systems. We investigate this physics by studying the center-of-mass oscillations of two species of fermionic ultracold atoms in a harmonic trap. If their trap frequencies are equal, a dynamical symmetry (spectrum-generating algebra), closely related to Kohn's theorem, prohibits the relaxation of center-of-mass oscillations. A small detuning $\ensuremath{\delta}\ensuremath{\omega}$ of the trap frequencies for the two species breaks the dynamical symmetry and ultimately leads to a damping of dipole oscillations driven by interspecies interactions. Using memory-matrix methods, we calculate the relaxation as a function of frequency difference, particle number, temperature, and strength of interspecies interactions. When interactions dominate, there is almost perfect drag between the two species and the dynamical symmetry is approximately restored. The drag can either arise from Hartree potentials or from friction. In the latter case (hydrodynamic limit), the center-of-mass oscillations decay with a tiny rate, $1/\ensuremath{\tau}\ensuremath{\propto}{(\ensuremath{\delta}\ensuremath{\omega})}^{2}/\mathrm{\ensuremath{\Gamma}}$, where $\mathrm{\ensuremath{\Gamma}}$ is a single-particle scattering rate.

DOI
10.1103/physreva.91.063604

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The presence of (approximate) conservation laws can prohibit the fast relaxation of interacting many-particle quantum systems. We investigate this physics by studying the center-of-mass oscillations of two species of fermionic ultracold atoms in a harmonic trap. If their trap frequencies are equal, a dynamical symmetry (spectrum-generating algebra), closely related to Kohn's theorem, prohibits the relaxation of center-of-mass oscillations. A small detuning \ensuremathδ\ensuremathω\ensuremath{\delta}\ensuremath{\omega}\ensuremathδ\ensuremathω of the trap frequencies for the two species breaks the dynamical symmetry and ultimately leads to a damping of dipole oscillations driven by interspecies interactions. Using memory-matrix methods, we calculate the relaxation as a function of frequency difference, particle number, temperature, and strength of interspecies interactions. When interactions dominate, there is almost perfect drag between the two species and the dynamical symmetry is approximately restored. The drag can either arise from Hartree potentials or from friction. In the latter case (hydrodynamic limit), the center-of-mass oscillations decay with a tiny rate, 1/\ensuremathτ\ensuremath∝(\ensuremathδ\ensuremathω)2/\ensuremathΓ1/\ensuremath{\tau}\ensuremath{\propto}{(\ensuremath{\delta}\ensuremath{\omega})}^{2}/\mathrm{\ensuremath{\Gamma}}1/\ensuremathτ\ensuremath∝(\ensuremathδ\ensuremathω)2/\ensuremathΓ, where \ensuremathΓ\mathrm{\ensuremath{\Gamma}}\ensuremathΓ is a single-particle scattering rate.
Ultracold quantum-gas mixtures of fermionic atoms with resonant control of interactions offer a unique test-bed to explore few- and many-body quantum states with unconventional properties. The emergence of such strongly correlated systems, for instance, symmetry-broken superfluids, is usually accompanied by hydrodynamic collective behavior. Thus experimental progress in this field naturally requires a deep understanding of hydrodynamic regimes. Here, we report on experiments employing a tunable Fermi-Fermi mixture of Dy 161 and K 40 near quantum degeneracy. We investigate the full spectrum of dipole modes across a Feshbach resonance and characterize the crossover from collisionless to deep hydrodynamic behavior in measurements of frequencies and damping rates. We compare our results with a theoretical model that considers the motion of the mass centers of the two species, and we identify the contributions of friction and mean-field interaction. We show that one oscillating mode exists over the whole range of interactions, exhibiting striking changes of frequency and damping in the deep hydrodynamic regime. We observe the second oscillating mode to split into two purely exponential damping modes. One of these exponential modes shows very fast damping, faster than any other relevant timescale, and is largely insensitive to experimental imperfections. It provides an accurate measure for the interspecies drag effect, which generalizes the concept of spin drag explored in other experiments. We finally characterize the interspecies drag in terms of a microscopic friction coefficient, and we discuss its unitarity-limited universal behavior on top of the resonance.
We study the relaxation of the center of mass or dipole oscillations in the system of interacting fermions confined spatially. With the confinement frequency ${\ensuremath{\omega}}_{\ensuremath{\perp}}$ fixed, the particles were considered to freely move along one (quasi-1D) or two (quasi-2D) spatial dimensions. We have focused on the regime of rare collisions, such that the inelastic collision rate $1/{\ensuremath{\tau}}_{\mathrm{in}}\ensuremath{\ll}{\ensuremath{\omega}}_{\ensuremath{\perp}}$. The dipole oscillation relaxation rate $1/{\ensuremath{\tau}}_{\ensuremath{\perp}}$ is obtained at three different levels: by direct perturbation theory, solving the integral Bethe-Salpeter equation, and applying the memory function formalism. As long as anharmonicity is weak, $1/{\ensuremath{\tau}}_{\ensuremath{\perp}}\ensuremath{\ll}1/{\ensuremath{\tau}}_{\mathrm{in}}$, the three methods are shown to give identical results. In the quasi-2D case, $1/{\ensuremath{\tau}}_{\ensuremath{\perp}}\ensuremath{\ne}0$ at zero temperature. In quasi-1D system, $1/{\ensuremath{\tau}}_{\ensuremath{\perp}}\ensuremath{\propto}{T}^{3}$ if the Fermi energy EF{E}_{F}EF​ lies below the critical value ${E}_{F}<3{\ensuremath{\omega}}_{\ensuremath{\perp}}/4$. Otherwise, unless the system is close to integrability, the rate $1/{\ensuremath{\tau}}_{\ensuremath{\perp}}$ has a temperature dependence similar to that in quasi-2D. In all cases, the relaxation results from the excitation of particle-hole pairs propagating along unconfined directions resulting in the relationship $1/{\ensuremath{\tau}}_{\ensuremath{\perp}}\ensuremath{\propto}1/{\ensuremath{\tau}}_{\mathrm{in}}$, with the inelastic rate $1/{\ensuremath{\tau}}_{\mathrm{in}}\ensuremath{\ne}0$ as the phase-space opens up at finite energy of excitation ${\ensuremath{\omega}}_{\ensuremath{\perp}}$. While $1/{\ensuremath{\tau}}_{\ensuremath{\perp}}\ensuremath{\propto}{\ensuremath{\tau}}_{\mathrm{in}}$ in the hydrodynamic regime, ${\ensuremath{\omega}}_{\ensuremath{\perp}}\ensuremath{\ll}1/{\ensuremath{\tau}}_{\mathrm{in}}$, in the regime of rare collisions, ${\ensuremath{\omega}}_{\ensuremath{\perp}}\ensuremath{\gg}1/{\ensuremath{\tau}}_{\mathrm{in}}$, we obtain the opposite trend $1/{\ensuremath{\tau}}_{\ensuremath{\perp}}\ensuremath{\propto}1/{\ensuremath{\tau}}_{\mathrm{in}}$.
The center-of-mass (c.m.) oscillation of a many-body system in a harmonic trap is known to be independent of the interparticle interaction. However, this is not necessarily the case if the interactions are treated approximately. Here, we prove a simple general criterion for preservation of the c.m. mode: the approximation has to preserve density and momentum. The result equally applies to zero and finite temperatures, as well as to nonequilibrium situations, and to the linear and nonlinear response regimes.
In this thesis we study the effects of the dipole-dipole interaction on the physics of ultracold quantum gases, both bosonic and fermionic, within the theoretical framework provided by the mean-field regime. This kind of interaction takes place in ultracold atomic gases (for instance 52Cr or 164Dy) due to their atomic magnetic dipole moment, and in ultracold molecular gases due to the magnetic or electric dipole moment. \n \nIn the case of quantum gases of bosonic atoms, or Bose-Einstein condensates, the dipole-dipole interaction can be studied within mean-field approximation using the Gross-Pitaevskii equation, which now contains a new non-linear term due to the dipole-dipole interaction. We investigate, on the one hand dipolar condensates confined in harmonic traps, and on the other dipolar condensates confined in toroidal traps. In the harmonic geometry, our focus is on the study of the ground state and the quantized vortex state, where the density profile is characterized as well as some properties leading to the process of vortex formation, such as the critical frequency and the energy barrier that has to be overcome to bring the vortex from the surface to the centre of the gas. We finish the study of dipolar condensates in harmonic traps by dynamically simulating the precession frequency of an off-center vortex in a non-rotating condensate. \n \nIn the toroidal geometry the dipolar effects are strongly magnified when the polarization axis of the dipoles is perpendicular to the trap symmetry axis. In this case, the anisotropic structure of the density can be understood as the response of the system to the double-well effective potential along the ring. We have studied the dynamics of this system when the initial number of atoms in the left and right wells is imbalanced, predicting Josephson and self-trapping oscillations depending on the initial condition. This has led us to name this new system as Self-induced Josephson Junction. We have studied in detail the self-trapping regime and we have seen that the particle flux inversion is closely related to the crossing of vortices across the Josephson junctions. This result opens the door to establishing a more direct connection between the phase-slip regime, widely addressed in superfluid helium, and the self-trapping regime of condensates. \n \nIn the case of quantum gases of fermionic dipolar particles, we have studied how the radial quadrupole mode allows one to distinguish between hydrodynamic and collisionless regimes. We have analytically calculated the frequency of this mode in the mean-field approximation, generalizing the results from the Thomas-Fermi approximation for trapped ideal Fermi gases. On the one hand, we observe that the frequency in the hydrodynamic regime is smaller than in non-dipolar Fermi gases, while in the collisionless regime the frequency is larger or smaller than that corresponding to the non-interacting system depending on the geometry of the harmonic trap. On the other hand, we predict that reducing the trap deformation (aspect ratio) an observable jump in the frequency of the radial quadrupole mode would take place, which would correspond to the transition between the collisionless and hydrodynamic regimes, for instance when the gas undergoes the transition to the superfluid state.
We present a generalization of the fluid-dynamical scheme developed for nuclear physics to the case of two trapped fermion species with pairing interactions. To establish a macroscopic description of the mass and momentum conservation laws, we adopt a generalization of the usual Thomas-Fermi approach that includes the pairing energy. We analyze the equilibrium density and gap profiles for an equal population mixture of harmonically trapped 6Li^{6}\mathrm{Li}6Li atoms for different choices of the local equation of state. We examine slight departures from equilibrium within our formulation, finding that density oscillations can propagate as first sound coupled to pairing vibrations, that in a homogeneous fermion system exhibit a Bogoliubov-like quasiparticle spectrum. In this case, the dispersion relation for the coupled modes displays a rich scenario of stable, unstable, and damped regimes.
Following a ``bottom-up approach'' in understanding many-particle effects and dynamics we provide a systematic ab initio study of the dependence of the breathing dynamics of ultracold bosons in a one-dimensional (1D) harmonic trap on the number of bosons ranging from few to many. To this end, we employ the multilayer multiconfiguration time-dependent Hartree method for bosons (ML-MCTDHB) which has been developed very recently [Kr\"onke, Cao, Vendrell, and Schmelcher, New J. Phys. 15, 063018 (2013)]. The beating behavior for two bosons is found numerically and consequently explained by an analytical approach. Drawing on this, we show how to compute the complete breathing mode spectrum in this case. We examine how the two-mode breathing behavior of two bosons evolves to the single-frequency behavior of the many-particle limit when adding more particles. In the limit of many particles, we numerically study the dependence of the breathing mode frequency on both the interaction strength as well as on the particle number. We provide an estimate for the parameter region where the mean-field description provides a valid approximation.