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arXiv:1408.2442 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Suppression of infrared instability in transsonic flows by condensation of zero-frequency short wave length phonons

Xavier Busch, Florent Michel, Renaud Parentani

Published Nov 7, 2014• 10 citations

Abstract

We analyze the peculiar infrared instability that characterizes stationary inhomogeneous flows when their velocity crosses the sound speed by decreasing values. For definiteness, we work in the context of one-dimensional atomic Bose condensates. These flows are unstable under ultra-low real frequency perturbations because of the unbounded mode amplification near the sonic horizon. This results in a condensation of low-frequency phonons which produces a spatially structured flow in the supersonic domain. Numerical simulations reveal that this zero-frequency undulation suppresses the instability when its spatial extension is infinite, and when its phase is near that of a ``shadow soliton'' solution attached to the sonic horizon. These phenomena are akin to the condensation of rotons in flowing superfluid $^{4}\mathrm{He}$ when exceeding the Landau velocity. They also pertain to shallow water waves propagating on transcritical flows.

DOI: 10.1103/physrevd.90.105005

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We analyze the peculiar infrared instability that characterizes stationary inhomogeneous flows when their velocity crosses the sound speed by decreasing values. For definiteness, we work in the context of one-dimensional atomic Bose condensates. These flows are unstable under ultra-low real frequency perturbations because of the unbounded mode amplification near the sonic horizon. This results in a condensation of low-frequency phonons which produces a spatially structured flow in the supersonic domain. Numerical simulations reveal that this zero-frequency undulation suppresses the instability when its spatial extension is infinite, and when its phase is near that of a ``shadow soliton'' solution attached to the sonic horizon. These phenomena are akin to the condensation of rotons in flowing superfluid

^{4}\mathrm{He}

when exceeding the Landau velocity. They also pertain to shallow water waves propagating on transcritical flows.

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.

We study solutions of the one-dimensional Gross-Pitaevskii equation to better understand dynamical instabilities occurring in flowing atomic condensates. Whereas transonic stationary flows can be fully described in simple terms, time-dependent flows exhibit a wide variety of behaviors. When the sound speed is crossed once, we observe that flows analogous to black holes obey something similar to the so-called no hair theorem since their late time profile is stationary and uniquely fixed by parameters entering the Hamiltonian and conserved quantities. For flows analogous to white holes, at late time one finds a macroscopic undulation in the supersonic side which has either a fixed amplitude or a widely varying one, signaling a quasiperiodic emission of solitons on the subsonic side. When considering flows which cross the sound speed twice, we observe various scenarios which can be understood from the above behaviors and from the hierarchy of the growth rates of the dynamical instabilities characterizing such flows.

We study double-barrier interfaces separating regions of asymptotically subsonic and supersonic flow of Bose-condensed atoms. These setups contain at least one black hole sonic horizon from which the analogue of Hawking radiation should be generated and emitted against the flow in the subsonic region. Multiple coherent scattering by the double-barrier structure strongly modulates the transmission probability of phonons, rendering it very sensitive to their frequency. As a result, resonant tunneling occurs with high probability within a few narrow frequency intervals. This gives rise to highly non-thermal spectra with sharp peaks. We find that these peaks are mostly associated with decaying resonances and only occasionally with dynamical instabilities. Even at achievable non-zero temperatures, the radiation peaks can be dominated by spontaneous emission, i.e. enhanced zero-point fluctuations, and not, as is often the case in analogue models, by stimulated emission.

We have created an analog of a black hole in a Bose-Einstein condensate. In this sonic black hole, sound waves, rather than light waves, cannot escape the event horizon. A steplike potential accelerates the flow of the condensate to velocities which cross and exceed the speed of sound by an order of magnitude. The Landau critical velocity is therefore surpassed. The point where the flow velocity equals the speed of sound is the sonic event horizon. The effective gravity is determined from the profiles of the velocity and speed of sound. A simulation finds negative energy excitations, by means of Bragg spectroscopy.