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arXiv:cond-mat/0207579 · Journal of Physics B Atomic Molecular and Optical Physics

Resonance in Bose Einstein condensate oscillation from a periodic variation in scattering length

Sadhan K Adhikari

Published Feb 25, 2003• 28 citations
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Abstract

Using the explicit numerical solution of the axially symmetric Gross–Pitaevskii equation, we study the oscillation of the Bose–Einstein condensate (BEC) induced by a periodic variation in the atomic scattering length a . When the frequency of oscillation of a is an even multiple of the radial or axial trap frequency, respectively, the radial or axial oscillation of the condensate exhibits resonance with a novel feature. In this nonlinear problem without damping, at resonance in the steady state the amplitude of oscillation passes through a maximum and minimum. Such a growth and decay cycle of the amplitude may keep on repeating. Similar behaviour is also observed in a rotating BEC.

DOI
10.1088/0953-4075/36/6/303

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Using the explicit numerical solution of the axially symmetric Gross–Pitaevskii equation, we study the oscillation of the Bose–Einstein condensate (BEC) induced by a periodic variation in the atomic scattering length a . When the frequency of oscillation of a is an even multiple of the radial or axial trap frequency, respectively, the radial or axial oscillation of the condensate exhibits resonance with a novel feature. In this nonlinear problem without damping, at resonance in the steady state the amplitude of oscillation passes through a maximum and minimum. Such a growth and decay cycle of the amplitude may keep on repeating. Similar behaviour is also observed in a rotating BEC.
Using the explicit numerical solution of the axially symmetric Gross–Pitaevskii equation, we study the oscillation of the Bose–Einstein condensate (BEC) induced by a periodic variation in the atomic scattering length a. When the frequency of oscillation of a is an even multiple of the radial or axial trap frequency, respectively, the radial or axial oscillation of the condensate exhibits resonance with a novel feature. In this nonlinear problem without damping, at resonance in the steady state the amplitude of oscillation passes through a maximum and minimum. Such a growth and decay cycle of the amplitude may keep on repeating. Similar behaviour is also observed in a rotating BEC. 1.
Nonlinear oscillations of a 3D radial symmetric Bose-Einstein condensate under periodic variation in time of the atomic scattering length have been studied analytically and numerically. The time-dependent variational approach is used for the analysis of the characteristics of nonlinear resonances in the oscillations of the condensate. The bistability in oscillations of the BEC width is invistigated. The dependence of the BEC collapse threshold on the drive amplitude and parameters of the condensate and trap is found. Predictions of the theory are confirmed by numerical simulations of the full Gross-Pitaevski equation.
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.
The collective oscillations of the trapped Bose Einstein Condensate (BEC) are studied using the computational model of the Gross Pitaevskii Equation, the nonlinear Schroedinger equation describing the BEC dynamics. Typical examples of the 3 dimensional BEC for both spherical and axial symmetry showed that the collective BEC oscillations can be stimulated by its growth. A frequency analysis of the oscillations is presented. The numerical simulation showed that the collective BEC oscillations are damped by the interaction of the BEC and the noncondensate Bose gas at a nonzero temperature.
We suggest a pseudospectral method for solving the three-dimensional time-dependent Gross–Pitaevskii (GP) equation, and use it to study the resonance dynamics of a trapped Bose–Einstein condensate induced by a periodic variation in the atomic scattering length. When the frequency of oscillation of the scattering length is an even multiple of one of the trapping frequencies along the x , y or z direction, the corresponding size of the condensate executes resonant oscillation. Using the concept of the differentiation matrix, the partial-differential GP equation is reduced to a set of coupled ordinary differential equations, which is solved by a fourth-order adaptive step-size control Runge–Kutta method. The pseudospectral method is contrasted with the finite-difference method for the same problem, where the time evolution is performed by the Crank–Nicholson algorithm. The latter method is illustrated to be more suitable for a three-dimensional standing-wave optical-lattice trapping potential.
We present a numerical study of the dynamical effects following a sudden change of the transverse trapping frequency in an elongated Bose-Einstein condensate, which induces periodic oscillations of the radial density. At early times, we observe an exponential growth of the number of resonant longitudinal phonons, in agreement with the predictions of the Bogoliubov-de Gennes treatment. We then observe an ordered sequence of phenomena induced by the nonlinearities of the Gross-Pitaevskii equation. The first is a loss of the nonseparability of the resonant phonon pairs. This is followed by the saturation of the exponential growth and a strong depletion of condensed atoms. Notably, these effects are well-described by effective 1D dynamics, and are hardly affected by the damping of the radial oscillations. Finally, the atomic spectrum becomes broad, featureless and almost incoherent, in agreement with experimental results obtained in [J.-C. Jaskula et al., Phys. Rev. Lett. 109, 220401 (2012)]. The link between this sequence of events and the preheating scenario in inflationary cosmology is striking.