Violation of Bell inequalities in an analogue black hole

TL;DR

This paper analyzes how bipartite and tripartite Bell nonlocality can emerge in a quasi-one-dimensional Bose-Einstein condensate analogue black hole, using a Gaussian-state framework for the three outgoing Bogoliubov modes. By mapping the system to an optical setup (parametric down-conversion plus beam splitter) and employing GKMR pseudo-spins, it computes CHSH and Svetlichny/Mermin parameters from the covariance matrix, highlighting that long-wavelength modes form a superposition of degenerate GHZ-like states whose entanglement resists partial tracing. The results show that zero-temperature entanglement between certain mode pairs can violate Bell inequalities (up to the Cirel'son bound $2 oot 2 ext{ }$), whereas finite temperature erodes nonlocal correlations, and genuine tripartite nonlocality is especially fragile to temperature, though in some regimes the Mermin parameter can still signal nonlocality. These findings reveal robust multipartite quantum correlations in analogue gravity settings, offering guidance for experimental tests and potential applications in continuous-variable quantum information.

Abstract

Signals of entanglement and nonlocality are quantitatively evaluated at zero and finite temperature in an analogue black hole realized in the flow of a quasi one-dimensional Bose-Einstein condensate. The violation of Lorentz invariance inherent to this analog system opens the prospect to observe 3-mode quantum correlations and we study the corresponding violation of bipartite and tripartite Bell inequalities. It is shown that the long wavelength modes of the system are maximally entangled, in the sense that they realize a superposition of continuous variable versions of Greenberger-Horne-Zeilinger states whose entanglement resists partial tracing.

Figures (16)

• Figure 1: Graphical representation of the positive frequency part of the dispersion relation \ref{['eq.excit1']} in the far upstream subsonic (upper plot) and downstream supersonic (lower plot) regions. The background color of the lower plot is greyed for recalling that it concerns the interior of the analogue black hole. In both plots the horizontal dashed line represents the angular frequency $\omega$ of a given excitation. In the upstream region there are two channels of propagation associated to each value of $\omega$. In the downstream region there are four (two) propagation channels when $\omega$ is smaller (larger) than the threshold $\Omega$ defined in Eq. \ref{['threshold']}. The channels are denoted as 0, 1 or 2, with an additional "in" ("out") label indicating if the wave propagates towards (away from) the horizon. The direction of propagation of each channel is marked with an arrow.
• Figure 2: Model optical system equivalent to the analogous black hole. The explicit relationship of the effective modes $f_i$ and $e_i$ ($i=0,1$ and 2) with the physical outgoing modes is given in Eqs. \ref{['eq.apsig9']} and \ref{['eq.apsig10']}. The $f_0$ mode is represented with a dashed line because, contrarily to modes $f_1$ and $f_2$, it is not occupied at zero temperature, see Eqs. \ref{['eq.apsig12']}. The long wavelength transmission coefficient of the beam splitter is denoted as $\Gamma_0$ in Appendix \ref{['app.sigma']}. It plays the role of the grey body factor of the analogous black hole.
• Figure 3: Plot of $B^{(i|2)}$ (solid lines) and $\Lambda^{(i|2)}$ (dashed lines) as functions of $\omega$ for the waterfall configuration with $m_d=2.9$ and different temperatures. We only consider the range of frequencies $\omega<\Omega$ for which the vacua of the outgoing and ingoing modes do not coincide. The value of the temperature is indicated in units of $g\, n_u=m c_u^2$. Upper plot: $i=0$, lower plot $i=1$. Non separability of modes $i$ and $2$ is achieved when $\Lambda^{(i|2)}>0$. Bell inequality is violated when $B^{(i|2)}>2$. The inset in the upper plot is a blow-up of the region $1.8\le B^{(0|2)}\le 2.1$ and $0.1\le \omega/\Omega\le 1$ showing that the reduced state $(0|2)$ does not violate Bell inequality at temperatures $T=0.1$ and 0.2.
• Figure 4: Zero temperature value of the CHSH parameter and of the PPT measure characterizing non separability of modes $i$ ($=0$ and 1) and 2 for the waterfall configuration. The maximal value reached by these quantities over the interval $\omega\in[0,\Omega]$ is plotted as a function of the upper Mach number $m_u$ which (as explained in Appendix \ref{['app.different']}) characterizes a given configuration. The values of $B^{(1|2)}$ for $m_u\le 0.01$ are not indicated because of lack of numerical precision. The upper bounds of $B^{(i|2)}$ and $\Lambda^{(i|2)}$ ($\sqrt{8}$ and 1, respectively) are indicated with filled dots.
• Figure 5: Same as Fig. \ref{['fig3:plus']} for a temperature $T=0.2\, g n_u$. Contrarily to what is observed in the zero temperature case displayed in Fig. \ref{['fig3:plus']}, the maxima of $\Lambda$ and $B$ (1 and $2\sqrt{2}$, respectively) are never reached.