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arXiv:0901.3767 · Physical Review A

Transcritical flow of a Bose-Einstein condensate through a penetrable barrier

Antin M. Leszczyszyn, Gennady A. El, Yu. G. Gladush, A. M. Kamchatnov

Published Jun 3, 2009• 47 citations
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Abstract

The problem of the transcritical flow of a Bose-Einstein condensate through a wide repulsive penetrable barrier is studied analytically using the combination of the locally steady ``hydraulic'' solution of the one-dimensional Gross-Pitaevskii equation and the solutions of the Whitham modulation equations describing the resolution of the upstream and downstream discontinuities through dispersive shocks. It is shown that within the physically reasonable range of parameters, the downstream dispersive shock is attached to the barrier and effectively represents the train of very slow dark solitons, which can be observed in experiments. The rate of the soliton emission, the amplitudes of the solitons in the train, and the drag force are determined in terms of the Bose-Einstein condensate oncoming flow velocity and the strength of the potential barrier. Good agreement with direct numerical solutions is demonstrated. Connection with recent experiments is discussed.

DOI
10.1103/physreva.79.063608

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The problem of the transcritical flow of a Bose-Einstein condensate through a wide repulsive penetrable barrier is studied analytically using the combination of the locally steady ``hydraulic'' solution of the one-dimensional Gross-Pitaevskii equation and the solutions of the Whitham modulation equations describing the resolution of the upstream and downstream discontinuities through dispersive shocks. It is shown that within the physically reasonable range of parameters, the downstream dispersive shock is attached to the barrier and effectively represents the train of very slow dark solitons, which can be observed in experiments. The rate of the soliton emission, the amplitudes of the solitons in the train, and the drag force are determined in terms of the Bose-Einstein condensate oncoming flow velocity and the strength of the potential barrier. Good agreement with direct numerical solutions is demonstrated. Connection with recent experiments is discussed.
We experimentally study the fluid flow induced by a broad, penetrable barrier moving through an elongated dilute gaseous Bose-Einstein condensate. The barrier is created by a laser beam swept through the condensate, and the resulting dipole potential can be either attractive or repulsive. We examine both cases and find regimes of stable and unstable fluid flow: At slow speeds of the barrier, the fluid flow is steady due to the superfluidity of the condensate. At intermediate speeds, we observe an unsteady regime in which the condensate gets filled with dark solitons. At faster speeds, soliton formation completely ceases, and a remarkable absence of excitation in the condensate is seen again.
We derive and analyze shock-wave solutions of hydrodynamic equations describing repulsively interacting one-dimensional Bose gas. We also use the number-conserving Bogolubov approach to verify accuracy of the Gross-Pitaevskii equation in shock wave problems. We show that quantum corrections to dynamics of shocks (dark-shock-originated solitons) in a Bose-Einstein condensate are negligible (important) for a realistic set of system parameters. We point out possible signatures of a Bose-Einstein condensate---Tonks crossover in shock dynamics. Our findings can be directly verified in different experimental setups.
Acoustic horizons in Bose--Einstein condensates are usually characterized through long-wavelength Bogoliubov phonons. We study a nonlinear counterpart: whether a one-dimensional dark-soliton branch can sustain upstream laboratory motion in a stationary transonic flow. The mechanism is local at leading order. A regular dark soliton has a bounded fluid-frame velocity, limited by the local sound speed; therefore, in the supersonic region, the background flow exceeds the largest upstream velocity available to the soliton branch. The sonic point is thus the upstream edge of the local dark-soliton branch, rather than a hard wall or a soliton geodesic surface. We construct stationary transonic Gross--Pitaevskii backgrounds and evolve the full order parameter in an open, nonperiodic domain. The simulations show upstream propagation on the subsonic side, finite-depth stalling on the subsonic side, and downstream advection for defects initialized in the supersonic region with upstream velocity relative to the fluid. Convergence, branch-consistency, local-density, phase-jump, and a dense scan of dimensionless upstream attempts support the soliton-like interpretation. The result is a branch-existence constraint, not a rigorous lower bound on arbitrary density minima of the Gross--Pitaevskii field.
Abstract The problem of wave breaking during its propagation in the Bose–Einstein condensate to a stationary medium is considered for the case when the initial profile at the breaking instant can be approximated by a power function of the form (–x)1/n. The evolution of the wave is described by the Gross–Pitaevskii equation so that a dispersive shock wave is formed as a result of breaking; this wave can be represented using the Gurevich–Pitaevskii approach as a modulated periodic solution to the Gross–Pitaevskii equation, and the evolution of the modulation parameters is described by the Whitham equations obtained by averaging the conservation laws over fast oscillations in the wave. The solution to the Whitham modulation equations is obtained in closed form for n = 2, 3, and the velocities of the dispersion shock wave edges for asymptotically long evolution times are determined for arbitrary integers n > 1. The problem considered here can be applied for describing the generation of dispersion shock waves observed in experiments with the Bose–Einstein condensate.
The problem of the flow of a Bose—Einstein condensate in a channel under the action of a piston is considered. Problems of this kind are topical in connection with experiments on condensate flow control in quasi-one-dimensional (cigar-shaped) traps, in wh ich the repulsive potential produced by a laser beam focused across the trap acts as a piston. A dispersive shock wave characterized by rapid oscillations of the condensate density and flow velocity is shown to be formed in the condensate flow after some instant of time for an arbitrary law of piston motion. The Whitham averaging method is used to obtain a solution for the main parameters of the dispersive shock wave in the case of a uniformly accelerated piston motion. The evolution of the dispersive shock wave immediately after the breaking time, when the dispersionless solution is well approximated by a cubic parabola for the coordinate dependence of the density, is analyzed in the case of an arbitrary piston motion. Comparison shows good agreement of the numerical calculation with the approximate analytical theory. The developed theory complements the previously considered case of a piston moving with a constant velocity and is important for describing the condensate transport in atomic chips.