Time-dependent study of a black-hole laser in a flowing atomic condensate

We numerically study the temporal evolution of a black-hole laser configuration displaying a pair of black- and white-hole horizons in a flowing atomic condensate. This configuration is initially prepared starting from a homogeneous flow via a suitable space-dependent change of the interaction constant and the evolution is then followed up to long times. Depending on the values of the system parameters, the system typically either converges to the lowest-energy solution by evaporating away the horizons or displays a continuous and periodic coherent emission of solitons. By making a physical comparison with optical laser devices, we identify the latter regime of continuous emission of solitons as the proper black-hole laser effect.

Figure 1

Scheme of the theoretical model here considered for studying analog black-hole laser configurations, where the stationary GP wave function is a plane wave of the form $\mathrm{\Psi }\left(x\right)=\sqrt{n_0}{e}^{iqx}$. Upper panel: scheme of the spatial profile of the constant coupling $g\left(x\right)$ (solid blue) and the external potential $V\left(x\right)$ (dashed red), which are piecewise functions satisfying Eq. (2). Lower panel: scheme of the velocity profile. The local speed of sound $c\left(x\right)=\sqrt{g\left(x\right)n_0/m}$ (solid blue line) crosses twice the uniform flow velocity $v\left(x\right)=v=\hslash q/m>0$, directed in the rightwards direction (red dashed line). The size of the internal supersonic region between the two horizons is $X$ and the supersonic sound speed is $c_2<v$.

Figure 2

Density profile of the $n=0,1,2$ nonlinear stationary solutions of the GP equation (16), for parameters $v=0.9,c_2=0.5$, and $X=10$. Dashed vertical black lines represent the limits of the internal supersonic region. Units are taken such that $\hslash =m=c_1$ and $\rho \left(x\right)$ is in units of $n_0$.

Figure 3

Sound (solid blue) and flow velocity (dashed red) profiles corresponding to the $n=0,1,2$ stationary GP solutions of Fig. 2. Units are taken such that $\hslash =m=c_1$.

Figure 4

Snapshots of the condensate density (thick solid blue line) at different evolution times (indicated in the panels) for a configuration with $v=0.75,c_2=0.3$, and $X=2$ that eventually reaches the $n=0$ stationary solution (thin solid gray line). The vertical dashed lines mark the boundaries of the supersonic region. The complete time evolution is shown in Movie 1 [36]. Upper row: evolution of the system when the initial linear instability grows in the direction of the $n=0$. Central and lower row: evolution of the system when the initial linear instability grows in the opposite direction. The complete time evolution is shown in Movie 2 [36]. Units are taken such that $\hslash =m=c_1$ and the density is in units of $n_0$.

Figure 5

Similar to Fig. 4 but now for a system with $v=0.75,c_2=0.5$, and $X=5$, where the unstable mode presents a nonzero real part of the frequency so it oscillates while growing. The complete evolution is shown in Movie 3 [36].

Figure 6

Analog of Fig. 4 but for a system with $v=0.75,c_2=0.6$, and $X=10$ so there are two unstable modes. Upper and central rows: evolution of the system when the linear unstable mode $n=1$ grows towards the $n=1$ solution (depicted in a thin solid gray line in the upper row) and remains there for some time until it collapses and evolves towards the $n=0$ ground state (thin solid gray line in the central row). The complete evolution is displayed in Movie 4 [36]. Lower row: evolution of the system when the linear unstable mode $n=1$ grows in the opposite direction, emitting some waves and solitons before reaching the $n=0$ solution (thin solid gray line). The complete evolution is shown in Movie 5 [36].

Figure 7

Phase diagram in the $(c_2,v)$ plane for different values of $X$. The blue dots represents numerical simulations that end in the $n=0$ ground state (labeled as GS in the legend). The black crosses represents simulations in which CES has been observed. The solid curves from bottom to top represent the lower boundary of the linear instability regions defined in Eq. (7) for $n=0,1/2,1,3/2...$ The dashed oblique straight line is the upper boundary of the region where the flow is everywhere subsonic. Units are taken such that $\hslash =m=c_1$.

Figure 8

Snapshots of the condensate density (thick solid blue line) at different evolution times (indicated in the panels) for a configuration with $v=0.95,c_2=0.4$, and $X=2$ that eventually reaches the CES regime. The thin solid gray line in the background represents the $n=0$ ground-state solution. After some transient, the continuous emission of solitons begins. Upper row: initial evolution of the system when the initial linear instability sets in. The vertical black line marks the $x=0$ position. Lower row: periodic evolution at late times after the system has reached the CES regime. The vertical dashed lines mark the boundaries of the supersonic region. At $t=1940$, a soliton emerges near the black hole (subsonic-supersonic) interface at $x<0$. This soliton cannot travel upstream and then it bounces back, traveling to the downstream region and finally crossing the supersonic region (see plot at $t=1950$). After this soliton has left around $t=1968$, the density grows again and a new soliton starts growing near the interface. Finally, at $t=1982$, the system recovers the same state as at $t=1940$ and the process repeats. The complete time evolution is shown in Movie  7 [36]. Units are taken such that $\hslash =m=c_1$ and the density is in units of $n_0$.

Figure 9

Main panel: plot of $t_{\mathrm{sol}}$, the time lapse between the arrival of consecutive solitons, as a function of the soliton number: after the emission of some solitons, the system reaches a regime of periodic soliton emission of period. Inset: time evolution of the density at a given point $\rho (x_0,t)$ with $x_0=50$. The density minima correspond to the periodic passage of solitons. System parameters: $v=0.95,c_2=0.4$, and $X=2$. Units are taken such that $\hslash =m=c_1$ and the density is in units of $n_0$.

Figure 10

Soliton emission period $T$ as a function of $c_2$ for (from top to bottom) $v=0.8,0.85,0.9,0.95$ and for fixed $X=2$. Units are taken such that $\hslash =m=c_1$.