Realization of a Sonic Black Hole Analog in a Bose-Einstein Condensate

We have created an analog of a black hole in a Bose-Einstein condensate. In this sonic black hole, sound waves, rather than light waves, cannot escape the event horizon. A steplike potential accelerates the flow of the condensate to velocities which cross and exceed the speed of sound by an order of magnitude. The Landau critical velocity is therefore surpassed. The point where the flow velocity equals the speed of sound is the sonic event horizon. The effective gravity is determined from the profiles of the velocity and speed of sound. A simulation finds negative energy excitations, by means of Bragg spectroscopy.

Figure 1

The profile of the potential, shown at a time before the harmonic potential (a) has reached the steplike potential (b). The horizontal arrow indicates the motion of the harmonic potential relative to the stationary steplike potential. The dashed line indicates the chemical potential of the condensate. Gravity is in the $-x$ direction. The profile of the potential step is derived from an image of the laser beam. The harmonic potential is derived from the measured frequency of the harmonic magnetic trap.

Figure 2

In situ absorption images of the sonic black hole, with the potentials on. Gravity is in the $-x$ direction. (a) Condensate in the presence of the steplike potential, at 200 ms. The average of 2 images is shown. The dashed (dash-dotted) line indicates the position-dependent profile of the black hole (white hole) horizon. (b) Like (a), at 205 ms. (c),(d) Side views ($y\mathrm{\text{−}}z$ plane) of (a) and (b), imaged simultaneously with (a) and (b). (e) Side view of the condensate in the harmonic potential only. (f) Density profiles, derived from (a) and (b) in blue and green, respectively. The density is averaged over the $y\mathrm{\text{−}}z$ plane (see text). (g) $-v$ (red curve) and $c$ (black curve). These curves are derived from (f), as described in the text. The filled circle and plus indicate the black hole and white hole horizons, respectively. (h) Time evolution of the density profile. The red, green, cyan, blue, magenta, and black curves indicate equal intervals from 200 to 225 ms. Each curve is derived from an average of 2 images.

Figure 3

Time evolution of the flow velocity. (a) The flow velocity versus the speed of sound. The red, green, cyan, blue, and magenta curves show $-v$ at equal intervals from 202.5 to 222.5 ms, derived from adjacent curves in Fig. 2h (see the text). The dashed curve indicates the average $-v$ from 202.5 to 222.5 ms. The black solid curve indicates the average $c$ (see the text). The filled circles (pluses) indicate the black hole (white hole) horizon. The fringes near the velocity maximum are an artifact of the imaging. (b) Hawking temperature and effective surface gravity. The filled and open circles correspond to the entire condensate and the central slice, respectively. The dashed curve corresponds to the average (dashed) velocity curve of (a). The dash-dotted curve corresponds to the average velocity curve of the central slice of the condensate.

Figure 4

3D Gross-Pitaevskii simulation of the experiment. The central axis of the condensate is shown. (a) The density. The filled circle indicates the black hole horizon. (b) The flow velocity versus the speed of sound. The green curve shows –$v$, and the black curve indicates $c$. Here, the simulation is analyzed with the same technique and resolution as in Fig. 3a from the experiment. (c) The energy terms in the flow equation. The thick black curve shows the small quantum pressure term $-{\hslash }^2/2m({\nabla }^2\sqrt{n}/\sqrt{n})$. The green curve shows the interaction term $gn$. The red curve shows the inertial term $m{v}^2/2$. The cyan curve shows the potential term $V$. The latter two terms are off-scale for much of the figure. The dashed line indicates the black hole horizon. (d) The momentum transferred by a Bragg pulse. The solid (dashed) curve corresponds to the supersonic (subsonic) region. Peaks $A$ and $B$ correspond to $A$ and $B$ in (e). (e) The dispersion relation in the supersonic region. The circles result from Bragg spectroscopy. The solid line is the Bogoliubov dispersion relation. The green curve indicates the trapped excitations with negative energy. (f) The real part of the wave function.