Bipartite and tripartite entanglement in a Bose-Einstein acoustic black hole

We investigate quantum entanglement in an analog black hole realized in the flow of a Bose-Einstein condensate. The system is described by a three-mode Gaussian state and we construct the corresponding covariance matrix at zero and finite temperature. We study associated bipartite and tripartite entanglement measures and discuss their experimental observation. We identify a simple optical setup equivalent to the analog Bose-Einstein black hole which suggests a different way of determining the Hawking temperature and gray-body factor of the system.

Figure 1

Waterfall configuration. The flow is directed from left to right. The downstream classical field $\mathrm{\Phi }(x>0)$ is exactly a plane wave of density $n_d$ and velocity $V_d$. $\mathrm{\Phi }(x<0)$ is the half profile of a dark soliton which is asymptotically a plane wave of density $n_u$ and velocity $V_u$; see Eqs. (4). The far upstream flow is subsonic, with a velocity $0<V_u<c_u$, and the downstream flow is supersonic with a velocity $V_d>c_d>0$, where $c_{\alpha }={(g{n}_{\alpha }/m)}^{1/2}$ is the speed of sound in region $\alpha =u$ (upstream) or $d$ (downstream). The shaded region $x>0$ corresponds to the interior of the analog black hole; the gradient of gray around $x\lesssim 0$ depicts the (ill-defined; see text) position of the horizon. The coordinate $x$ is plotted in units of the upper healing length ${\xi }_u=\hslash /\left(m{c}_u\right)$.

Figure 2

Graphical representation of the positive frequency part of the dispersion relation (5) in the far upstream (left plot) and downstream (right plot) regions. The downstream region (gray background) is the interior of the analog black hole, while the upstream region (white background) is the exterior. In the upstream region, to any given $\omega$ (represented by a horizontal dashed line) correspond two channels of propagation denoted as $0|\mathrm{in}$ and $0|\mathrm{out}$. In the downstream region there are four or two channels, depending if $\omega$ is smaller or larger than $\mathrm{\Omega }$. The arrows indicate the direction of propagation of the corresponding waves, and the channels are labeled 1 or 2, with an additional “in” (or “out”) indicating if the wave propagates towards (or away from) the horizon.

Figure 3

Local mixedness $a_i\left(\omega \right)$ [see Eqs. (60)] for each mode 0, 1, and 2 as functions of the dimensionless quantity $\hslash \omega /\left(g{n}_u\right)$, for a waterfall configuration with $m_u=0.59$. The frequency ${\omega }_{\mathrm{c}}$ indicates the turning point above which $a_0$ becomes lower than $a_1$. The upper-bound frequency $\mathrm{\Omega }$ corresponds to the vanishing of the mode 2 [see Eq. (9)].

Figure 4

Blue continuous curve: Effective temperature $T_{\mathrm{eff}}^{\left(0\right)}$ defined in Eq. (64) plotted as a function of the frequency $\omega$ for a waterfall configuration with $m_u=0.59$. The dashed red line is the Hawking temperature $T_{\mathrm{H}}^{\left(0\right)}$ given by (65).

Figure 5

Residual contangles $G_{\tau }^{\text{res}\left(0\right)}$ (blue), $G_{\tau }^{\text{res}\left(1\right)}$ (green), $G_{\tau }^{\text{res}\left(2\right)}$ (red). The upper-bound frequency $\mathrm{\Omega }$ corresponds to the vanishing of the mode 2. The frequency ${\omega }_{\mathrm{c}}$ is the value above which $a_0$ becomes lower than $a_1$ (see Fig. 3) and coincides with the point above which $G_{\tau }^{\text{res}\left(0\right)}<G_{\tau }^{\text{res}\left(1\right)}$. The inset displays the difference $G_{\tau }^{\text{res}\left(0\right)}-G_{\tau }^{\text{res}\left(1\right)}$ (cyan).

Figure 6

Measure of tripartite entanglement $G_{\tau }^{\mathrm{res}}$ as a function of the (dimensionless) energy $\hslash \omega /\left(g{n}_u\right)$ and of the upstream Mach number $m_u\in [0.05,0.95]$ defined in Eq. (7). The pink curve corresponds to the upper bound frequency $\mathrm{\Omega }$ (9). For a fixed value $m_u$ mode 2 exists only for a frequency $\omega$ lower than $\mathrm{\Omega }\left(m_u\right)$. Beyond this value the tripartite system $\{0,1,2\}$ no longer exists; this is the reason why the corresponding area is left blank. The horizontal red line corresponds to the value $m_u\simeq 0.59$, corresponding to $m_d=2.9$ as realized in the experiment of [10]. The right plot shows the integral $\int G_{\tau }^{\text{res}}d\omega$ over frequencies $\omega \in [0,\mathrm{\Omega }]$ for each value of $m_u$. The red dot pinpoints the numerical estimate of the integral for the specific value $m_u\simeq 0.59$, while the blue dot locates the maximum of the black curve reached for $m_u=0.14$. The light blue horizontal line on the left graph corresponds to this value.

Figure 7

Schematic representation of an optical process equivalent to the Hawking emission in the transonic BEC system we consider. Entanglement is localized in the two-mode squeezed state $f_1|f_2$. The mode $f_0$ being empty is represented by a dashed line.

Figure 8

Hawking temperature as a function of the upstream Mach number in the waterfall configuration. The blue solid line is the result (65), and the red solid line comes from expression (95). The dashed line is the semiclassical expectation (C8).

Figure 9

Evolution of the PPT measure $1-{\nu }_{-}^{\mathrm{PT}}$ (blue), of the Cauchy-Schwarz parameter ${\mathrm{\Delta }}_{\mathrm{CS}}$ (red) and of the Gaussian contangle $G_{\tau }^{\left(0\right|2)}$ (green) for the bipartite system $0|2$ (i.e., the analog Hawking pair) as functions of the dimensionless frequency $\hslash \omega /\left(g{n}_u\right)$ and for different temperatures of the system, denoted by $T_{\mathrm{BEC}}$, ranging from 0 (a) to $1.8g{n}_u$ (f). All the plots are obtained for an upstream Mach number $m_u=0.59$. The dashed blue curves in panels (b)–(f) correspond to the zero-temperature value of $1-{\nu }_{-}^{\mathrm{PT}}$. The gray areas indicate the range of frequencies for which the bipartite system is entangled [see text and Eqs. (105) and (107)]. The purple dots locate the upper-bound frequency $\mathrm{\Omega }$ at which mode 2 vanishes.

Figure 10

PPT measure $1-{\nu }_{-}^{\mathrm{PT}}$ of the Hawking pair $0|2$ plotted as a function of the upstream Mach number $m_u$ and of the frequency $\omega$, for temperatures (a) $T_{\mathrm{BEC}}=0.5g{n}_u$ and (b) $T_{\mathrm{BEC}}=1.8g{n}_u$. The pink curve corresponds to the upper-bound frequency $\mathrm{\Omega }$ (9). For a fixed value $m_u$, i.e., along a horizontal cut on the graph, mode 2 only exists for a frequency $\omega$ lower than $\mathrm{\Omega }\left(m_u\right)$ (see Fig. 2). The dashed black curve corresponds to $1-{\nu }_{-}^{\mathrm{PT}}=0$ and thus delimits the region where the analog Hawking pair is entangled.

Figure 11

Same as Fig. 10 for $T_{\mathrm{BEC}}=5T_{\mathrm{H}}^{\left(2\right)}$.

Figure 12

Evolution of ${\mathrm{\Delta }}_{\mathrm{CS}}$ given by Eq. (107) as a function of the measure of bipartite entanglement $G_{\tau }^{\left(0\right|2)}$ given by expressions (108) and (E23), for the same set of temperatures as in Fig. 9, ranging from $T_{\mathrm{BEC}}=0$ (blue curve) to $T_{\mathrm{BEC}}=1.5g{n}_u$ (red curve), with $m_u=0.59$. When possible, the corresponding temperature for each curve is indicated on the graph (we dropped the factor $g{n}_u$ for readability). Note that for $T_{\mathrm{BEC}}>0$, the curves describe a loop.

Figure 13

Evolution of $1-{\nu }_{-}^{\mathrm{PT}}$ given by Eq. (106) as a function of the measure of bipartite entanglement $G_{\tau }^{\left(0\right|2)}$ given by expressions (108) and (E23), for the same set of temperatures as in Fig. 9, ranging from $T_{\mathrm{BEC}}=0$ (blue curve) to $T_{\mathrm{BEC}}=1.5g{n}_u$ (red curve), with $m_u=0.59$. When possible, the corresponding temperature for each curve is indicated on the graph (we dropped the factor $g{n}_u$ for readability). For each temperature the common maximal value of $1-{\nu }_{-}^{\mathrm{PT}}$ and $G_{\tau }^{\left(0\right|2)}$ is marked with a point.

Figure 14

Same as Figs. 12 and 13 for the GPH parameter $-\mathcal{P}$ defined in Eqs. (109) and (110).

Figure 15

Zero-temperature PPT measure as a function of energy in a waterfall configuration with $m_u=0.59$. The blue solid curve is the same as in Fig. 9. The points with error bars are evaluated from Eq. (111), extracting the value of $|\langle {\widehat{c}}_0{\widehat{c}}_2\rangle |$ from the experimental data of Ref. [10] as discussed in the text. The dashed line represents the result of Eq. (111) obtained by assuming that $|\langle {\widehat{c}}_0{\widehat{c}}_2\rangle |\simeq \frac{1}{2}\sqrt{a_0^2-1}\simeq \sqrt{{\overline{n}}_0({\overline{n}}_0+1)}$, where ${\overline{n}}_0\left(\omega \right)$ is a thermal Bose occupation evaluated at the Hawking temperature $T_{\mathrm{H}}=0.124g{n}_u$ determined in [10].