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arXiv:1712.05456 · Europhysics Letters (EPL)

Slow scrambling in sonic black holes

Gabriel Menezes, Jamir Marino

Published Mar 1, 2018• 12 citations
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Abstract

We study from the perspective of quantum information scrambling an acoustic black hole modeled by two semi-infinite, stationary, one-dimensional condensates, connected by a spatial step-like discontinuity, and flowing, respectively, at subsonic and supersonic velocities. We develop a simple analytical treatment based on the Bogoliubov theory of quantum fluctuations which is sufficient to derive an analogue Hawking emission, and we compute out-of-time order correlations (OTOCs) of the Bose density field. We find that a large class of sonic black holes presents slow scrambling contrary to their astrophysical counterparts. This manifests itself in a power law growth of OTOCs in contrast to the exponential increase in time expected for fast scramblers.

DOI
10.1209/0295-5075/121/60002

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We study from the perspective of quantum information scrambling an acoustic black hole modeled by two semi-infinite, stationary, one-dimensional condensates, connected by a spatial step-like discontinuity, and flowing, respectively, at subsonic and supersonic velocities. We develop a simple analytical treatment based on the Bogoliubov theory of quantum fluctuations which is sufficient to derive an analogue Hawking emission, and we compute out-of-time order correlations (OTOCs) of the Bose density field. We find that a large class of sonic black holes presents slow scrambling contrary to their astrophysical counterparts. This manifests itself in a power law growth of OTOCs in contrast to the exponential increase in time expected for fast scramblers.
We consider simple one dimensional models of acoustic black holes formed by Bose-Einstein condensates where the flow is stepwise homogeneous. We shall concentrate on the case in which an extended sonic region is present, i.e. a region where the flow velocity equals exactly the speed of sound. These models can be thought of as acoustic black holes with a thick horizon. Particular attention will be devoted to the number of created particles and to the density-density correlation functions, these latter being at present time the experimental tool to identify Hawking like radiation in acoustic black holes.
We study acoustic white holes in a steadily flowing atomic Bose–Einstein condensate. A white hole configuration is obtained when the flow velocity goes from a super-sonic value in the upstream region to a sub-sonic one in the downstream region. The scattering of phonon wavepackets on a white hole horizon is numerically studied in terms of the Gross–Pitaevskii equation of mean-field theory: dynamical stability of the acoustic white hole is found, as well as a signature of a nonlinear back-action of the incident phonon wavepacket onto the horizon. The correlation pattern of density fluctuations is numerically studied by means of the truncated-Wigner method, which includes quantum fluctuations. Signatures of the white hole radiation of correlated phonon pairs by the horizon are characterized; analogies and differences with Hawking radiation from acoustic black holes are discussed. In particular, a short wavelength feature is identified in the density correlation function, whose amplitude steadily grows in time since the formation of the horizon. The numerical observations are quantitatively interpreted by means of an analytical Bogoliubov theory of quantum fluctuations for a white hole configuration within the step-like horizon approximation.
In a previous article, we studied stationary solutions to the dynamics of a Bose-Einstein condensate corresponding to acoustic (or Unruh) black or white holes, namely, configurations where the flow becomes supersonic, creating a horizon for phonons. In this paper, we consider again the Gross-Pitaevskii equation, but looking for stationary numerical solutions in the case where the couplings are position dependent in a prescribed manner. Initially, we consider a two-dimensional quantum gas in a funnel-like spatial metric. We then reinterpret this solution as a solution in a flat metric but with spatially dependent coupling and external potential. In these solutions, the local speed of sound and the magnitude of flow velocity cross, indicating the existence of a supersonic region and therefore of sonic analogs of black and white holes and wormholes. We discuss the numerical techniques used. We also study phase (and density) fluctuations in these solutions and derive approximate acoustic metric tensors. For certain external potentials, we find uniform-density acoustic-black-hole configurations and obtain their Hawking temperature.
Abstract Out-of-time-order correlators (OTOCs) that capture maximally chaotic properties of a black hole are determined by scattering processes near the horizon. This prompts the question to what extent OTOCs display chaotic behaviour in horizonless microstate geometries. This question is complicated by the fact that Lyapunov growth of OTOCs requires nonzero temperature, whereas constructions of microstate geometries have been mostly restricted to extremal black holes. In this paper, we compute OTOCs for a class of extremal black holes, namely maximally rotating BTZ black holes, and show that on average they display “slow scrambling”, characterized by cubic (rather than exponential) growth. Superposed on this average power-law growth is a sawtooth pattern, whose steep parts correspond to brief periods of Lyapunov growth associated to the nonzero temperature of the right-moving degrees of freedom in a dual conformal field theory. Next we study the extent to which these OTOCs are modified in certain “superstrata”, horizonless microstate geometries corresponding to these black holes. Rather than an infinite throat ending on a horizon, these geometries have a very deep but finite throat ending in a cap. We find that the superstrata display the same slow scrambling as maximally rotating BTZ black holes, except that for large enough time intervals the growth of the OTOC is cut off by effects related to the cap region, some of which we evaluate explicitly.
We extend the Gross-Pitaevskii equation to incorporate the effect of quantum fluctuations onto the flow of a weakly interacting Bose-Einstein condensate. Applying this framework to an analog black hole in a quasi-one-dimensional, transonic flow, we investigate how acoustic Hawking radiation back-reacts on the background condensate. Our results point to the emergence of stationary density and velocity undulations in the supersonic region (analogous to the black hole interior) and enable to evaluate the change in upstream and downstream Mach numbers caused by Hawking radiation. These findings provide new insight into the interplay between quantum fluctuations and analog gravity in Bose-Einstein condensates.
Simulation of a sonic black-hole/white-hole pair in a (2+1)-dimensional Bose-Einstein condensate shows formation of superfluid vortices through dynamical instabilities seeded by initial quantum noise. The instabilities saturate in a quasi-steady state of superfluid turbulence within the supersonic region, from which sound waves are emitted in qualitative resemblance to Hawking radiance. The power spectrum of the radiation from the slowly decaying two-dimensional sonic black hole is strongly non-thermal, however.