Hawking time crystals

Abstract

We propose a time crystal based on a quantum black-hole laser, where the genuinely spontaneous character of the symmetry breaking stems from the self-amplification of spontaneous Hawking radiation. The resulting Hawking time crystal (HTC) is characterized by the periodic dependence of the out-of-time density-density correlation function, while equal-time observables are time-independent because they embody averages over different realizations with a random oscillation phase. The HTC can be regarded as a nonlinear periodic analogue of the Andreev-Hawking effect, exhibiting anticorrelation bands resulting from the spontaneous, quantum emission of pairs of dispersive waves and solitons into the upstream and downstream regions. Remarkably, the time-crystal formation is understood in terms of two time operators: one associated to the initial black-hole laser and another associated to the final spontaneous Floquet state.

Figures (10)

• Figure 1: (a) Sound (solid blue) and flow (dashed red) velocity profile of the FPBHL at $t=0$. Horizontal dashed blue line represents the initial homogeneous condensate $\Psi_0(x)=e^{ivx}$. The shaded area indicates the supersonic lasing cavity where the coupling constant is quenched for $t\geq 0$ so that $c(x)=c_2$. A perturbation of amplitude $A$ (dotted magenta) is added to the initial GP condition as classical seed for the unstable mode (dashed-dotted green). (b) Velocity dependence of $\Gamma$ (solid blue) and $\omega$ (dashed red) for $c_2=0.4,L=2$. Vertical line indicates $v=v_c$.
• Figure 2: Expectation values for a FPBHL with $v=0.95,c_2=0.4,L=2$, computed via TW method. (a)-(c) Ensemble-averaged density $n(x,t)$ for initial classical amplitudes $A=0.05,0.005,0$, respectively. (d) Snapshots of $n(x,t)$ at the times indicated by horizontal dashed line in (a)-(c). Black line is the theoretical prediction $n_{\rm{HTC}}(x)$, Eq. (\ref{['eq:Hteo']}). (e)-(g) ETCF $g^{(2)}(x,x',t)$ for (a)-(c), evaluated at the times shown in (d). (h) Theoretical ETCF $g_{\rm{HTC}}^{(2)}(x,x',\tau=0)$, Eq. (\ref{['eq:Hteo']}). Dashed lines indicate the expected correlation bands, Eqs. (\ref{['eq:HawkingParallelOOT']}), (\ref{['eq:NonLinearAndreedHawkingOOT']}). (i)-(j) OTCF $g^{(2)}(x,x',\tau;t)$ for $A=0$, evaluated at $\tau=10,20$ and fixed $t=870$. (k) $\textrm{Re}\, \mathcal{G}(t,t')$. Inset: 1D profile along the black line in the main plot. Dashed cyan is the theoretical prediction $\textrm{Re}\, \mathcal{G}_{\rm{HTC}}(\tau)=\cos\omega\tau$. l) Spatial structure of the ETCF for a flat-profile black hole with $v=0.95,c_2=0.4$. Magenta (white) solid lines indicate the Hawking (Andreev) correlation bands.
• Figure 3: (a)-(c) Histogram: phase-shift $\phi_0$ for the TW ensembles of Figs. \ref{['fig:ClassicalQuantum']}a-c. Horizontal line with error bars: uniform distribution and its statistical uncertainty. (d) Schematic obtention of $\phi_0$. Blue line: mean-field trajectory. Red line: particular TW realization. (e)-(g) Histogram: lasing amplitude $X$ at $t=0$ for (a)-(c). Dots with error bars: expected Gaussian distribution (\ref{['eq:GaussianWignerLaser']}) and its statistical uncertainty. (h) Mean $X_C$ (black dots) and statistical deviation $\Delta X_Q$ (blue squares) as a function of $A$, and their theoretical predictions (solid red and horizontal dashed black lines, respectively). Inset: Zoom of $\Delta X_Q$.
• Figure 4: (a)-(b) Schematic representation of the dispersion relation (\ref{['eq:DispersionRelation']}) for a subsonic ($c>v$) and supersonic ($c<v$) flow. (c) Spacetime diagram of the outgoing modes in an analogue black hole. Vertical line indicates the horizon at $x=0$. (d) Same as (c) but for an HTC, where the vertical gray band now represents the finite lasing cavity.
• Figure 5: (a)-(d) Spatial profile of the real and imaginary part of the BdG components $u_I,v_I,u_S,v_S$, respectively. Solid lines are numerically obtained by finite methods and dashed lines are the analytical result from solving the BdG scattering problem [see Eq. (\ref{['eq:BdGScat']}) and ensuing discussion].