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arXiv:1612.08268 · Physical Review A

Nonlinear mixing of collective modes in harmonically trapped Bose-Einstein condensates

Takahiro Mizoguchi, Shohei Watabe, Tetsuro Nikuni

Published Mar 21, 2017• 0 citations
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Abstract

We study nonlinear mixing effects among quadrupole modes and scissors modes in a harmonically trapped Bose-Einstein condensate. Using a perturbative technique in conjunction with a variational approach with a Gaussian trial wave function for the Gross-Pitaevskii equation, we find that mode mixing occurs selectively. Our perturbative approach is useful in gaining a qualitative understanding of the recent experiment [M. Yamazaki et al., J. Phys. Soc. Jpn. 84, 44001 (2015)], exhibiting a beating phenomenon of the scissors mode as well as a modulation phenomenon of the low-lying quadrupole mode by the high-lying quadrupole mode frequency. Within the second-order treatment of the nonlinear mode coupling terms, our approach predicts all the spectral peaks obtained by the numerical simulation of the Gross-Pitaevskii equation.

DOI
10.1103/physreva.95.033623

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We study nonlinear mixing effects among quadrupole modes and scissors modes in a harmonically trapped Bose-Einstein condensate. Using a perturbative technique in conjunction with a variational approach with a Gaussian trial wave function for the Gross-Pitaevskii equation, we find that mode mixing occurs selectively. Our perturbative approach is useful in gaining a qualitative understanding of the recent experiment [M. Yamazaki et al., J. Phys. Soc. Jpn. 84, 44001 (2015)], exhibiting a beating phenomenon of the scissors mode as well as a modulation phenomenon of the low-lying quadrupole mode by the high-lying quadrupole mode frequency. Within the second-order treatment of the nonlinear mode coupling terms, our approach predicts all the spectral peaks obtained by the numerical simulation of the Gross-Pitaevskii equation.
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 study the scissors modes of a harmonically trapped Bose-Einstein condensate under the influence of a synthetic magnetic field, which induces rigid rotational components in the velocity field. Our investigation reveals that the scissors mode, excited in the plane perpendicular to the synthetic magnetic field, becomes coupled to the quadrupole modes of the condensate, giving rise to typical beating effects. Moreover, the two scissors modes excited in the vertical planes are also coupled together by the synthetic magnetic field, resulting in intriguing gyroscope dynamics. Our analytical results, derived from a spinor hydrodynamic theory, are further validated through numerical simulations of the three-dimensional Gross-Pitaevskii equation. These predictions for the condensates subject to a synthetic magnetic field are experimentally accessible with current cold-atom setups and hold promise for potential applications in quantum sensing.
We analytically and numerically study nonlinear dynamics in Bose–Einstein condensates (BECs) induced either by a harmonic modulation of the interaction or by the geometry of the trapping potential. To analytically describe BEC dynamics, we use a perturbative expansion based on the Poincaré–Lindstedt analysis of a Gaussian variational ansatz, whereas in the numerical approach we use numerical solutions of both a variational system of equations and the full time-dependent Gross–Pitaevskii equation. The harmonic modulation of the atomic s-wave scattering length of a BEC of 7 Li was achieved recently via Feshbach resonance, and such a modulation leads to a number of nonlinear effects, which we describe within our approach: mode coupling, higher harmonics generation and significant shifts in the frequencies of collective modes. In addition to the strength of atomic interactions, the geometry of the trapping potential is another key factor for the dynamics of the condensate, as well as for its collective modes. The asymmetry of the confining potential leads to important nonlinear effects, including resonances in the frequencies of collective modes of the condensate. We study in detail such geometric resonances and derive explicit analytic results for frequency shifts for the case of an axially symmetric condensate with two- and three-body interactions. Analytically obtained results are verified by extensive numerical simulations.
We report the first experimental observation of Beliaev coupling between collective excitations of a Bose-Einstein condensed gas. Beliaev coupling is not predicted by the Gross-Pitaevskii equation and so this experiment tests condensate theory beyond the mean field approximation. Measurements of the amplitude of a high frequency scissors mode show that the Beliaev process transfers energy to a lower-lying mode and then back and forth between these modes, unlike Landau processes which lead to a monotonic decrease in amplitude. To enhance the Beliaev process we adjusted the geometry of the magnetic trapping potential to give a frequency ratio of 2 to 1 between the two scissors modes.
The experimental discovery of dilute ultracold quantum gases in 1995 has instigated a broad interest in ultracold atoms and molecules, and paved the way for extensive studies of a wide range of both experimental and theoretical topics. Various theories were developed to describe Bose-Einstein condensates (BECs) for different ranges of temperature and interaction strengths. This thesis focuses on studying the mean-field theory of the Gross-Pitaevskii (GP) equation which provides a good description for the macroscopic wave function when both quantum and thermal fluctuations are negligible. This means that the range of validity is restricted to small interaction strengths at zero temperature. Within the GP theory we study analytically and numerically the nonlinear dynamics of BECs which are induced by a harmonic modulation of either the interaction strength or the harmonic trapping potential. At first a detailed numerical investigation reveals that solving the partial differential equation of Gross and Pitaevskii can be reduced to solving ordinary differential equations for the condensate widths within a Gaussian variational approach. Despite this tremendous simplification the prominent nonlinearity of the underlying GP equation is inherited by the variational equations, which thus allow to describe a rich variety of nonlinear phenomena in different experimental setups. In order to describe them analytically we transfer the Poincar\'e-Lindstedt method, which represents a well-established tool in the field of nonlinear dynamics, to the Gaussian variational approach. Modulation of some of the parameters of a BEC can give rise to prominent nonlinear features, such as shifts in the frequencies of collective oscillation modes, higher harmonics generation, and resonant mode coupling. As a first application we consider a recent Bose-Einstein condensate of Li-7, which has been excited experimentally by a harmonic modulation of the atomic s-wave scattering length with the help of a Feshbach resonance. Combining an analytical perturbative approach with extensive numerical simulations, we analyze the emerging nonlinear dynamics. In particular, we present the resulting shift of collective excitation spectra close to parametric resonances where the driving frequency coincides with a collective mode frequency. Another application of experimental importance is to study geometric resonances in BECs with two-body and three-body contact interactions, where changing the geometry of the trapping potential leads to commensurate collective excitation frequencies with emerging nonlinear effects such as frequency shifts and resonant mode coupling. In this context we also show that a small repulsive three-body interaction is able to extend the stability region of the condensate for an attractive two-body interaction as it increases the critical number of atoms in the trap. Apart from a periodic modulation of a system parameter the dipole mode frequency can be excited by introducing an abrupt change in the potential, where the center of mass oscillates according to the Kohn theorem back and forth in the trapping potential with the natural trap frequency irrespective of both the strength and the type of the two-particle interaction. To this end, we study the collective excitation modes of a Bose-Einstein condensate in a harmonic Ioffe- Pritchard in the vicinity of a Feshbach resonance for experimentally realistic parameters of a Rb-85 BEC, where the dipole mode is excited in z-direction. A linearization of the underlying equations of motions shows that the dipole mode frequency decreases when the bias magnetic field approaches the Feshbach resonance, so the Kohn theorem is violated. Finally within a variational approach we discuss the physical properties of a BEC in an axially-symmetric harmonic trap with a single vortex in the center. At first we analyze the equilibrium configuration and determine the vortex size as well as the Thomas- Fermi radii of the condensate in the Thomas-Fermi limit. Then we calculate the critical rotation frequency for the emergence of the vortex and compare our findings with the literature. Finally, we investigate how the presence of the vortex changes the collective excitation frequencies and we discuss the free expansion of the BEC. All these results indicate how nonlinear dynamics properties could be made clearly observable in future experiments on the basis of our results.
We explore the nonlinear coupling of the three scissors modes of an anisotropic Bose-Einstein condensate. We show that only when the frequency of one of the scissors modes is twice the frequency of another scissors mode, these two modes can be resonantly coupled and a down-conversion can occur. We perform the calculation variationally using a gaussian trial wave function. This enables us to obtain simple analytical results that describe the oscillation and resonance behavior of the two coupled modes.