Black-hole lasing in coherently coupled two-component atomic condensates

We study theoretically the black-hole lasing phenomenon in a flowing one-dimensional, coherently coupled two-component atomic Bose–Einstein condensate whose constituent atoms interact via a spin-dependent $s$-wave contact interaction. We show by a numerical analysis the onset of the dynamical instability in the spin branch of the excitations, once a finite supersonic region is created in this branch. We study both a spatially homogeneous geometry and a harmonically trapped condensate. Experimental advantages of the two-component configuration are pointed out, with an eye towards studies of backreaction phenomena.

Figure 1

Spin-mode dispersion relations: nonlasing configuration. (a) Downstream and upstream regions: $\mathrm{\Omega }/gn=-0.16, g_{ab}/g=0.8$. (b) Cavity region: $\mathrm{\Omega }/gn=-0.12, g_{ab}/g=0.8$.

Figure 2

Spin-mode dispersion relations: lasing configuration. (a) Downstream and upstream regions: $\mathrm{\Omega }/gn=-0.12, g_{ab}/g=0.8$. (b) Cavity region: $\mathrm{\Omega }/gn=-0.01, g_{ab}/g=0.8$.

Figure 3

Different schemes for the coherent coupling of the atomic levels. (a) Two-level model. (b) $\mathrm{\Lambda }$-structure model. (c) Scheme of $\mathrm{\Lambda }$ model implemented with an elongated atomic cloud in a three-tube inhomogeneous potential. (d) Here, the hopping strength between $a-b$ and $b-c$ tubes can be controlled via the distance between the clouds.

Figure 4

Time evolution of the wave-packet in real space. The dashed lines enclose the cavity region. (a) Initial condition of the simulation. (b), (c) State of the spin density $\delta n$ of the system, respectively at $\hslash t/2m{\xi }^2=800$ and 3400, for the nonlasing conditions defined by the set of parameters given in the text and corresponding to the dispersion plotted in Figs. 1 and 1. (d), (e) State of the spin density $\delta n$ of the system, respectively at $\hslash t/2m{\xi }^2=800$ and 5000, for the lasing condition defined by the set of parameters given in the text and corresponding to the dispersion plotted in Fig. 2. The arrows indicate the group velocity $v^{\prime }$ of the different emerging wave packets.

Figure 5

Nonlasing case: evolution in time of the spatial Fourier transform of the spin density of the system in the different regions. (a) Downstream region. The wave-vector components of the d-R packet given as initial condition for the simulation appear, together with the d-L component due to its reflection from the WH. (b) Cavity region. The cav-R packet transmitted through the WH appears. The cav-L component due to the reflection of the cav-R packet at the BH is indicated by an arrow but is hardly visible in the figure. (c) Upstream region. The u-R packet transmitted through the cavity appears. System parameters are the same as in Figs. 4 and 4.

Figure 6

Nonlasing case: Spatial Fourier transform of the spin density of the system at different instants of time, showing the spectral content of the transmitted and reflected packets in the downstream, upstream, and cavity regions. The dashed lines identify the central wave-vector expected from the dispersion relations shown in Figs. 1 and 1. (a) Downstream region, $\hslash t/2m{\xi }^2=0$: initial condition. (b) Downstream region, $\hslash t/2m{\xi }^2=800$: wave packet reflected by the WH. (c) Upstream region, $\hslash t/2m{\xi }^2=800$: wave packet transmitted through the cavity. (d) Cavity region, $\hslash t/2m{\xi }^2=500$: wave packets transmitted through the WH and reflected by the BH. System parameters are the same as in Figs. 4, 4, and 5.

Figure 7

Lasing case: evolution in time of spatial Fourier transform of spin density of the system in the different regions. (a) Downstream region. The wave components of the d-R packet given as initial condition for the simulation appear, together with the d-L contributes due to its reflection from the WH. At late times, the leakage of the self-amplified radiation from the cavity appears in the d-L modes. (b) Cavity region. At early times, the cav-R1 packet transmitted through the WH appears, while the cav-L1 component due to its reflection at the BH is barely visible. At late times the onset of the exponential amplification of the unstable modes is evident. (c) Upstream region. At the early times, the u-R packet transmitted through the cavity appears while, at the late times, it is evident the leakage in the u-R mode of the self-amplified radiation from the cavity. (d), (e) Magnified views of the regions indicated by dashed white rectangles in panel (b). In panel (d), the early times excitation of the cav-R2 and cav-L2 modes is now visible. In panel (e) the focus is on the unstable mode. System parameters are the same as in Figs. 4 and 4.

Figure 8

Lasing case at early times: spatial Fourier transform of the spin density of the system at different instants of time and in different regions, showing the spectral content of the different wave packets. The dashed lines identify the central wave vector expected from the analysis of the subsonic and supersonic dispersion relations in Figs. 2 and 2. (a) Downstream region, $\hslash t/2m{\xi }^2=0$: initial condition. (b) Downstream region, $\hslash t/2m{\xi }^2=500$: reflected packet by the WH. (c) Upstream region, $\hslash t/2m{\xi }^2=500$: wave packet transmitted through the cavity. (d) Cavity region, $\hslash t/2m{\xi }^2=500$: in contrast to the nonlasing case, the negative norm components are now present and visible. System parameters are the same as in Figs. 4, 4, and 7.

Figure 9

Lasing case at late times: (a), (c), (e) Spatial Fourier transform of spin density at $\hslash t/2m{\xi }^2=5000$, in the downstream, upstream, and cavity regions, respectively. The pictures show the dominant unstable modes in the cavity and their leakage in the downstream and upstream regions. (b), (d), (f)–(i) Time evolution of the unstable modes in three different regions clearly shows the exponential character of their amplification. Note that panel (i) shows the $k\xi =0$ component instead of the $k\xi =0.04$ component just because the wave-vector resolution in the simulation is equal to 0.08. System parameters are the same as in Figs. 4, 4, 7, and 8.

Figure 10

Noise amplification in a lasing configuration with $g=0.8g_{ab}$ and ${\mathrm{\Omega }}^{\prime }=-0.015$ (${\mathrm{\Omega }}^{\prime }=-0.5$) inside (outside) the cavity. Spatial profiles of the spin density at different times (a) $t^{\prime }=0$, (b) $t^{\prime }=3000$, (c) $t^{\prime }=4100$ starting from an initially noisy configuration. (d) Spin-mode dispersion relations inside the cavity for the chosen parameters.

Figure 11

The rate ${\mathrm{\Gamma }}^{\prime }$ of the exponential amplification $exp\left({\mathrm{\Gamma }}^{\prime }{t}^{\prime }\right)$ of the spin excitations inside the cavity. The value of $\mathrm{\Omega }/gn$ at which the supersonic-subsonic transition occurs in the cavity is indicated by the dotted line. The inset shows a magnified view of the region within the dashed rectangle.

Figure 12

Spatial profile of $\mathrm{\Omega }/\hslash {\omega }_0$ (in units of $\chi$) used in the simulations.

Figure 13

(a)–(g) Different time snapshots of black-hole lasing in a harmonically trapped condensate. The insets show magnified views of the lasing region. The values of the parameters are given in the text. (h) Lin-log plot of the time evolution of the standard deviation of the modulation in the density (yellow) and spin (blue) components with respect to their equilibrium values.