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arXiv:1603.04702 · Europhysics Letters (EPL)

Numerical study of a recent black-hole lasing experiment

Manuele Tettamanti, Sergio L. Cacciatori, Alberto Parola, Iacopo Carusotto

Published Jun 1, 2016• 42 citations
arXivHTMLPDF

Abstract

We theoretically analyse a recent experiment reporting the observation of a self-amplifying Hawking radiation in a flowing atomic condensate (S teinhauer J., Nat. Phys. , 10 (2014) 864). We are able to accurately reproduce the experimental observations using a theoretical model based on the numerical solution of a mean-field Gross-Pitaevskii equation that does not include quantum fluctuations of the matter field. In addition to confirming the black-hole lasing mechanism, our results show that the underlying dynamical instability has a classical hydrodynamic origin and is triggered by a seed of deterministic nature, linked to the non-stationary of the process, rather than by thermal or zero-point fluctuations.

DOI
10.1209/0295-5075/114/60011

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We theoretically analyse a recent experiment reporting the observation of a self-amplifying Hawking radiation in a flowing atomic condensate (S teinhauer J., Nat. Phys. , 10 (2014) 864). We are able to accurately reproduce the experimental observations using a theoretical model based on the numerical solution of a mean-field Gross-Pitaevskii equation that does not include quantum fluctuations of the matter field. In addition to confirming the black-hole lasing mechanism, our results show that the underlying dynamical instability has a classical hydrodynamic origin and is triggered by a seed of deterministic nature, linked to the non-stationary of the process, rather than by thermal or zero-point fluctuations.
We model a sonic black-hole analog in a quasi-one-dimensional Bose-Einstein condensate, using a Gross-Pitaevskii equation matching the configuration of a recent experiment by Steinhauer [Nat. Phys. 10, 864 (2014)]. The model agrees well with important features of the experimental observations, demonstrating their hydrodynamic nature. We find that a zero-frequency bow wave is generated at the inner (white-hole) horizon, which grows in proportion to the square of the background condensate density. The relative motion of the black- and white-hole horizons produces a Doppler shift of the bow wave at the black hole, where it stimulates the emission of monochromatic Hawking radiation. The mechanism is confirmed using temporal and spatial windowed Fourier spectra of the condensate. Mean field behavior similar to that in the experiment can thus be fully explained without the presence of self-amplifying Hawking radiation.
Abstract We present a theoretical study on the origin of some findings of recent experiments on sonic analogs of gravitational black holes (BHs). We focus on the realization of a BH lasing configuration, where the conclusive identification of stimulated Hawking radiation (HR) requires dealing with the implications of the nonstationary character of the setup. To isolate the basic mechanisms responsible for the observed behavior, we use a toy model where nonstationarity can be described in terms of departures from adiabaticity. Our approach allows one to study which aspects of the characterization of BH lasing in static models are still present in a dynamical scenario. In particular, variations in the role of the dynamical instabilities can be traced. Arguments to conjecture the twofold origin of the detected amplification of sound are given: the differential effect of the instabilities on the mean field and on the quantum fluctuations gives some clues to separate a deterministic component from self-amplified HR. The role of classical noise, present in the experimental setup, is also tackled: we discuss the emergence of differences with the effect of quantum fluctuations when various unstable modes are relevant to the dynamics.
The black-hole laser (BHL) effect is the self-amplification of Hawking radiation in the presence of a pair of horizons which act as a resonant cavity. In a flowing atomic condensate, the BHL effect arises in a finite supersonic region, where Bogoliubov-Cherenkov-Landau (BCL) radiation is coherently excited by any static perturbation. Thus, experimental attempts to produce a black-hole laser unavoidably deal with the presence of a strong BCL background, making the observation of the BHL effect still a major challenge in the analog gravity field. Here, we perform a theoretical study of the BHL-BCL crossover using an idealized model where both phenomena can be unambiguously isolated. By drawing an analogy with an unstable pendulum, we distinguish three main regimes according to the interplay between quantum fluctuations and classical stimulation: quantum BHL, classical BHL, and BCL. Based on quite general scaling arguments, the nonlinear amplification of the initial amplitude of the quantum fluctuations up to saturation is identified as the most robust trait of a quantum BHL. A classical BHL behaves instead as a linear quantum amplifier, where the output is proportional to the input. The BCL regime also acts as a linear quantum amplifier, but its gain is exponentially smaller as compared to a classical BHL. In addition, we find that the decrease in the amplification for increasing BCL amplitude or the nonmonotonic dependence of the growth rate with respect to the background parameters are complementary signatures of black-hole lasing. We also identify interesting analog phenomena such as Hawking-stimulated white-hole radiation or quantum BCL-stimulated Hawking radiation. The results of this work not only are of interest for analog gravity, where they help to distinguish each phenomenon and to design experimental setups leading to a clear observation of the BHL effect, but they also open the prospect of finding applications of analog concepts in quantum technologies.
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.
We numerically study an analog black hole with two horizons with parameters similar to a recent experiment. We find that the Hawking radiation exists on a background which contains a density oscillation, a zero-frequency ripple. The Hawking radiation evolves from spontaneous to self-amplifying, while the background ripple grows steadily with no qualitative change. It is seen that the self-amplifying Hawking radiation has a nonzero frequency. This frequency is independent of the reference frame since it is the magnitude of the wave which oscillates, as in a standing wave. The background ripple appears even before the inner horizon is created, in contrast to predictions. Furthermore, we find that technical noise and shot-to-shot variations in the number of atoms are not sufficient to cause the observed correlation function. This work is in agreement with the recent observation of self-amplifying Hawking radiation, and explains some of the features seen. In contrast to some recent works, our study differentiates between the Hawking radiation observed, and the evolution of the background.
For flowing quantum gases, it has been found that at long times an initial black-hole laser (BHL) configuration exhibits only two possible states: the ground state or a periodic self-oscillating state of continuous emission of solitons. So far, all the works on this subject are based on a highly idealized model, quite difficult to implement experimentally. Here we study the instability spectrum and the time evolution of a recently proposed realistic model of a BHL, thus providing a useful theoretical tool for the clear identification of black-hole lasing in future experiments. We further confirm the existence of a well-defined phase diagram at long times, which bespeaks universality in the long-time behavior of a BHL. Additionally, we develop a complementary model in which the same potential profile is applied to a subsonic homogeneous flowing condensate that, despite not forming a BHL, evolves toward the same phase diagram as the associated BHL model. This result reveals an even stronger form of robustness in the long-time behavior with respect to the transient, which goes beyond what has been described in the previous literature.