Hawking Radiation from an Acoustic Black Hole on an Ion Ring

In this Letter we propose to simulate acoustic black holes with ions in rings. If the ions are rotating with a stationary and inhomogeneous velocity profile, regions can appear where the ion velocity exceeds the group velocity of the phonons. In these regions phonons are trapped like light in black holes, even though we have a discrete field theory and a nonlinear dispersion relation. We study the appearance of Hawking radiation in this setup and propose a scheme to detect it.

Figure 1

Schematic depiction of the ion rotation with velocity $v(\theta )$ and the phononic excitations with velocity $\pm c(\theta )$, that depend on the varying ion spacing. The external force $F^e(\theta )$ localized at the transition between the super- and subsonic region (de-)accelerates the ions. The inset shows a typical velocity profile with a black hole like region for $v(\theta )>c(\theta )$.

Figure 2

Values of $\delta {\theta }_k(t)$ for propagation backwards in time starting from the final wave function in Eq. (7) with $s=5$. We use $\sigma v_{\min }T=2\pi \times 0.375$, and $N=1000$. The final negative frequency pulses are depicted in green (solid line), the initial negative frequency pulses in blue (dash-dotted lines), and the initial positive frequency pulses in red (dashed lines). We depict the discrete norm distribution and dispersion relation as a continuous curve here. (a) Final ($t=0$) and initial ($t=-0.67T$) norm distribution of $\delta {\theta }_k(t)$; $e^2/4\pi {ϵ}_0=1.2591/(2N)m{L}^3{T}^{-2}$. (b) Final ($t=0$) and initial ($t=-0.51T$) norm distribution of $\delta {\theta }_k(t)$ showing Bloch oscillation; $e^2/4\pi {ϵ}_0=2.0004/(2N)m{L}^3{T}^{-2}$. (c) Phononic group velocity in the flat subsonic region as a function of $k$ for full interactions (blue dashed line) and nearest-neighbor interactions only (green solid line); $e^2$ as in (a).

Figure 3

Correlations $C_{ij}(t)$ at time $t$ with real space lab frame positions. We consider $N=1000$ ions, $\sigma v_{\min }T=2\pi \times 0.25$, and $\tau =0.05T$. (a) Only nearest-neighbor interactions are considered with $e^2/4\pi {ϵ}_0=1.127/(2N)m{L}^3{T}^{-2}$ and $t=0.5T$. (b) Full Coulomb interactions are considered with $e^2/4\pi {ϵ}_0=0.2453/(2N)m{L}^3{T}^{-2}$ and $t=0.6T$.