Generation of linear waves in Bose-Einstein condensate flow past an obstacle
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Generation of Linear Waves in Bose–Einstein Condensate Flow Past an Obstacle Yu. G. Gladush and A. M. Kamchatnov Institute of Spectroscopy, Russian Academy of Sciences, Troitsk, Moscow oblast, 142190 Russia e-mail: [email protected] e-mail: [email protected] Received March 6, 2007
Abstract—A theory of linear wave patterns developing in Bose–Einstein condensate flow past an obstacle is developed. The results obtained characterize the wave crestline geometry and the far-field dependence of the wave amplitude on coordinates. The theoretical predictions agree with the results of previous numerical simulations and provide a qualitative explanation of experiments on the flow of a Bose–Einstein condensate released from a trap past an obstacle. PACS numbers: 03.75.Kk DOI: 10.1134/S1063776107090075
1. INTRODUCTION After the discovery of Bose–Einstein condensation, considerable attention has been given to various excitations of Bose–Einstein condensates (e.g., see [1]). As an example, consider waves in a Bose–Einstein condensate (BEC) with repulsive interaction. When the condensate occupies a large volume inside the trap, its unperturbed state can be approximately treated as uniform, with constant density n0, if the excitation wavelength is much smaller than the BEC size. It was shown in [2] that excitations propagating in a condensate of this kind can be described by the dispersion relation ω(k ) =
2 2
k 2 2 c s k + ⎛ --------⎞ , ⎝ 2m ⎠
(1)
In the large-k limit, it reduces to the free-particle dispersion relation for quantum particles of mass m: 2
k ω ≈ -------- , 2m
gn --------0 m
ξ = -------------------- = ---------------- . 2mc s 2mgn 0
2
(3)
Repulsive interatomic interaction implies that g > 0 and cs is real. In the long-wavelength limit, relation (1) describes acoustic waves with velocity cs : k
0.
(6)
2
is the speed of sound in the limit of k 0, m is the atomic mass, and the effective coupling constant g is expressed in terms of the s-wave scattering length as as
ω ≈ c s k,
(5)
In the mean-field approximation, nonlinear dynamics of a nonuniform BEC can be described by the Gross–Pitaevskii equation [1] ∂ψ 2 i ------- = – -------∆ψ + V ( r )ψ + g ψ ψ, ∂t 2m
(2)
4π a g = -----------------s . m
∞.
The intermediate regime between these limits corresponds to wavelengths comparable to the healing length
where cs =
k
(4)
(7)
where the order parameter ψ(r) is the condensate wavefunction and V(r) is an external (e.g., confining trap) potential. In the region where the condensate can be approximately treated as uniform, the potential V is constant with similar accuracy. Therefore, it can be eliminated from the equation by introducing an inessential phase factor into the definition of ψ. Solutions to the equation linearized about the uniform state with n0 = |ψ|2 are plane waves with dispersion relation (1). Nonlinear excitations (vortices and dark solitons) can be described by taking into account the nonlinearity of the last term in (7). Experimental observations of l
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