Quantum backreaction in an analog black hole

TL;DR

The work develops a self-consistent, perturbative framework for quantum backreaction in a Bose-Einstein condensate, extending the Gross-Pitaevskii description to include the influence of quantum fluctuations on the background flow. By combining an amplitude-phase formalism with Bogoliubov theory, it derives backreaction equations that couple the classical condensate to quadratic quantum averages, applicable to time- and space-dependent flows in arbitrary dimensions. When applied to a 1D transonic flow mimicking an analog black hole, the approach predicts stationary density and velocity undulations in the supersonic region and small, Hawking-radiation–driven modifications of the upstream and downstream Mach numbers. These results illuminate the interplay between quantum fluctuations and analog gravity in Bose-Einstein condensates and pave the way for more detailed studies of horizon dynamics under backreaction.

Abstract

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.

Figures (6)

• Figure 1: Effect of backreaction on the asymptotic Mach numbers for the waterfall configuration. The results are plotted as functions of the upstream asymptotic Mach number (computed within the Gross-Pitaevskii approximation) $M_{\rm GP}^u$. Upper plot: the solid lines represent the rescaled relative modifications $\delta M_u/M_{\rm GP}^u$ and $\delta M_d/M_{\rm GP}^d$ of the upstream and downstream Mach numbers. The dashed lines terminated by circles present the result obtained when discarding the contribution of the Hawking radiation. Lower plot: relative variation of the Mach numbers computed by removing the usual beyond mean field contributions and incorporating only the backreaction induced by Hawking radiation. The values of the functions for $M_{\rm GP}^u=1$ are marked with full circles and correspond to expressions \ref{['eq.asymWu']} and \ref{['eq.asymWd']}.
• Figure 2: Upper plot: density profile of the waterfall configuration. Lower plot: density profile of the $\delta$-peak configuration. In both cases the order parameter takes the form of a plane wave downstream ($x>0$), and of a fraction of a gray soliton upstream ($x<0$). The downstream region is shaded to underline that is corresponds to the interior of the analog black hole. Note that, according to the convention used in the main text, all the quantities in this plot should be written with an index "GP", since they correspond to a (stationary) solution of Gross-Pitaevskii equation \ref{['eq16']}. These indices are omitted here for legibility.
• Figure 3: Upper plot: dispersion relation \ref{['eq.m1']} in the asymptotic upstream subsonic region. Lower plot: dispersion relation in the downstream supersonic region. This plot is shaded to emphasize that this region corresponds to the interior of the analog black hole. The horizontal dashed line is the angular frequency $\omega$ of a given excitation. The colored points mark the corresponding excitation channels, of wavevectors $q_{0|{\rm out}}(\omega)$, $q_{0|{\rm in}}(\omega)$, $q_{1|{\rm in}}(\omega)$ etc. The arrows indicate the direction of propagation of the different channels. The wave vector $\kappa_d$ corresponds to a zero energy channel, see Sec. \ref{['sec.analog.stat.1D']}.
• Figure 4: Same as Fig. \ref{['fig:W']} for the $\delta$-peak configuration.
• Figure 5: Hawking induced modifications $\delta M_\alpha^{(H)}$ of the asymptotic Mach numbers in a flat profile partially imitating a waterfall configuration.