Dynamic Ultraslow Optical-Matter Wave Analog of an Event Horizon

We investigate theoretically the effects of a dynamically increasing medium index on optical-wave propagation in a rubidium condensate. A long pulsed pump laser coupling a $D2$ line transition produces a rapidly growing internally generated field. This results in a significant optical self-focusing effect and creates a dynamically growing medium index anomaly that propagates ultraslowly with the internally generated field. When a fast probe pulse injected after a delay catches up with the dynamically increasing index anomaly, it is forced to slow down and is prohibited from crossing the anomaly, thereby realizing an ultraslow optical-matter wave analog of a dynamic white-hole event horizon.

Figure 1

Left: energy level diagram with laser couplings. Right: density distribution as a function of $\rho /{\rho }_0$ at the condensate entrance ($z/L=1$, dashed curve), and exit ($z/L=0$, dash-dotted curve). Inset plot: matter-wave index anomaly along the long axis ($\rho =0$) as a function of $z/L$. The backward-propagating mixing wave starts at $z/L=1.0$ where the index anomaly is negligible. The index anomaly reaches maximum at $z/L=0$ after a full length of travel with gain.

Figure 2

Mixing wave intensity (a),(c),(e) and condensate density distribution (b),(d),(f) as a function of $z/L$ (horizontal axis) and $\rho /{\rho }_0$ with and without nonlinear term $|{\epsilon }^{(+)}{|}^2$. (a),(b) Numerical solutions of Eqs. (1) and (2). (c),(d) Results from Eqs. (4) and (5). (e),(f) Numerical solution of Eqs. (1) and (2) without the nonlinear term $|{\epsilon }^{(+)}{|}^2$. $|{\psi }_0{|}^2\approx 4.0\times 10^{12}{\mathrm{cm}}^{-3}$, $\mathrm{\Gamma }/2\pi =6\mathrm{MHz}$, ${\gamma }_1/2\pi =2\mathrm{kHz}$, ${\tau }_P=300\mu \mathrm{s}$, ${\kappa }_0=2.76\times 10^{-6}{\mathrm{m}}^2{\mathrm{s}}^{-1}$, $g/\hslash =4.85\times 10^{-17}{\mathrm{m}}^3{\mathrm{s}}^{-1}$, ${\delta }_L/2\pi =-2\mathrm{GHz}$, $k_{\mathrm{G}}\approx 8\times 10^6{\mathrm{m}}^{-1}$, $g_0=1.3\times 10^{-5}$.

Figure 3

Left panel: probe group velocity $V_g^{(p)}$, indicated by equal altitude lines, as a function of ${\delta }_p$ and $\rho /{\rho }_0$ at (a) entrance ($z/L=1$) and (b) exit ($z/L=0$). Mixing-wave velocity $V_g^{(G)}\approx 1.0\mathrm{m}/\mathrm{s}$ is indicated by the red number. (c) $V_g^{(p)}$ as a function of $z/L$ and ${\delta }_p$ along the condensate’s long axis ($\rho =0$). Right panel: evolution of the probe field as a function of $\zeta /L$ and $\tau$ in the mixing-wave co-moving reference. The dashed vertical line indicates the white event horizon in the moving frame. The solid arrow near the dashed line indicates where the faster probe catches the index anomaly and is subsequently slowed down and reflected.