Bogoliubov-Čerenkov Radiation in a Bose-Einstein Condensate Flowing against an Obstacle

Abstract

We study the density modulation that appears in a Bose-Einstein condensate flowing with supersonic velocity against an obstacle. The experimental density profiles observed at JILA are reproduced by a numerical integration of the Gross-Pitaevskii equation and then interpreted in terms ofCerenkov emission of Bogoliubov excitations by the defect. The phonon and the single-particle regions of the Bogoliubov spectrum are respectively responsible for a conical wavefront and a fan-shaped series of precursors.PACS numbers: 03.75. Kk, 41.60.Bq TheCerenkov effect was first discovered in the electromagnetic radiation emitted by charged particles traveling through a dielectric medium at a speed larger than the medium's phase velocity [1]. A charge moving at the speed v is in fact able to resonantly excite those modes of the electromagnetic field which satisfy the kinematicCerenkov resonance condition ω em (k) = v · k: part of the kinetic energy of the particle is then emitted asCerenkov radiation, with a peculiar frequency and angular spectrum [2]. Electromagnetic waves in a nondispersive medium of refractive index n have a linear dispersion law relation ω em (k) = ck/n: theCerenkov condition is then satisfied on a conical surface in k-space of aperture cos φ = c/(nv), which corresponds to a conical wavefront of aperture θ = π/2 − φ behind the particle. Thanks to the interplay of interference and propagation, much richer features appear in the spatial and k-space pattern ofCerenkov radiation in dispersive media [3,4] and photonic crystals [5].The concept ofCerenkov radiation can be generalized to any system where a source is uniformly moving through a homogeneous medium at a speed larger than the phase velocity of some elementary excitation to which the source couples. Many systems have been investigated in this perspective, ranging from e.m. waves emitted by the localized nonlinear polarization induced by a strong light pulse travelling in a nonlinear medium [6,7], to the sonic waves generated by an airplane moving at supersonic velocities, to phonons in a polaritonic superfluid [8], and in a broader sense, to the surface waves emitted by a boat moving on the quiet surface of a lake [9]. In this Letter we compare our theoretical results of the density perturbation induced in a Bose-Einstein condensate (BEC) which flows against a localized obstacle at rest with the experimental images taken by the JILA group [10]. Modulo a Galilean transformation, the physics of a moving source in a stationary medium is in fact equivalent to the one of a uniformly moving medium interacting with a stationary defect. The experiment has been performed by letting a BEC expand at hypersonic speed against the lo- The experiment. The experimental results analyzed in this paper have been obtained by the JILA group [10] with a gas of N = 3 × 10 6 Bose-Einstein condensed 87 Rb atoms confined in a cylindrically symmetric harmonic trap of frequencies {ω r , ω z } = 2π{8.3, 5.3} Hz. The BEC is slightly cigar shaped, with the long axis pointing in the...

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