📝 Abstract

We have created an analogue 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 step-like 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.

💡 Analysis

We have created an analogue 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 step-like 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.

📄 Content

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Realization of a sonic black hole analogue in a Bose-Einstein condensate

Oren Lahav, Amir Itah, Alex Blumkin, Carmit Gordon, Shahar Rinott, Alona Zayats, and Jeff Steinhauer

Technion – Israel Institute of Technology, Haifa, Israel

We have created an analogue 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 step-like 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.

The event horizon is a boundary around the black hole, enclosing the region from which even light cannot escape. It has been suggested that an analogue of a black hole could be created in a variety of quantum mechanical [1-6] or classical [7-9] systems. In the case of a quantum fluid such as the Bose-Einstein condensate studied here [3], it is sound waves, rather than light waves, which cannot escape. This sonic black hole contains regions of subsonic flow, as well as regions of supersonic flow. Since a phonon cannot propagate against the supersonic flow, the boundary between the subsonic and supersonic regions marks the event horizon of the sonic black hole. The analogy was later extended to include excitations with a non-linear dispersion relation, in addition to phonons [10-12]. 2

The experimental challenge is to create a steady flow which exceeds the speed of sound [1,3,13,14]. Consider a phonon with momentum k = . In the reference frame of the moving fluid, the phonon has energy kc E

= , where c is the speed of sound. In the laboratory frame, by a Galilean transformation [15], this energy becomes v k ⋅ +

′

E E , where v is the flow velocity. For the case of supersonic flow (v > c), E′ can be zero, resulting in the unstable production of phonons. This instability is thought to prevent the supersonic flow required to realize a sonic black hole, a phenomenon referred to as the Landau critical velocity [3,13,15]. By momentum conservation however, the production of such phonons requires an additional body such as an impurity particle [16] or a container with a rough wall [15]. This body provides momentum in the opposite direction to the flow. Thus, we have arranged an experimental apparatus which does not supply much momentum in this direction, allowing for supersonic flow during the timescale of the experiment [3]. The free flow required to overcome the Landau critical velocity also helps prevent the production of quantized vortices, which usually limit the flow to speeds much lower than the speed of sound [17].

Suggested schemes for forming a sonic black hole in a condensate include a Laval nozzle [18, 19], flow along a ring or a long, thin condensate [3, 20], a gradient in the coupling constant [21,22], a soliton [2,23], an expanding condensate [24], and repulsive potential maxima [5,25]. We achieve the black hole horizon by a step-like potential combined with a harmonic potential, as shown in Fig. 1. We translate the harmonic potential to the left as indicated by the horizontal arrow, moving the condensate towards the stationary 3

step. While crossing the step, the condensate accelerates to supersonic speeds. Thus, the region to the left of the step is supersonic, and the region to the right is subsonic. There is therefore a black hole horizon at the location of the step.

The condensate consists of 1 × 105 87Rb atoms in the F = 2, mF = 2 state, and is initially prepared in the harmonic part of the potential, a magnetic trap with oscillation frequencies of 26 Hz and 10 Hz in the radial and axial (y) directions respectively. The x- coordinate of the minimum of the harmonic trap is controlled by adjusting the trap frequencies, which adjusts the sag due to gravity (in the -x direction). The step-like potential is created by a large diameter, red-detuned laser beam with a Gaussian profile (1/e2 radius of 56 µm, wavelength 812 nm). Half of this beam is blocked, so that the boundary between the dark and light regions forms the potential step of height kHz 0.2 / 0

h V . Initally, the condensate is located to the right of the step, as shown in Fig. 1. Starting at t = 0, the harmonic potential is accelerated until it reaches the constant velocity of roughly 0.3 mm s-1. The condensate then passes over the potential step, as shown in Figs. 2a and 2b. We observe no increase in the thermal fraction in this process.
Furthermore, it is seen that the density of the condensate is much smaller to the left of the potential step, whose location appro

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