Dirac quantum time mirror

Both metaphysical and practical considerations related to time inversion have intrigued scientists for generations. Physicists have strived to devise and implement time-inversion protocols, in particular different forms of “time mirrors” for classical waves. Here we propose an instantaneous time mirror for quantum systems, i.e., a controlled time discontinuity generating wave function echoes with high fidelities. This concept exploits coherent particle-hole oscillations in a Dirac spectrum in order to achieve population reversal, and can be implemented in systems such as (real or artificial) graphene.

Figure 1

Echo in a Dirac quantum system. The absolute value of an $\hslash$-shaped wave packet is shown in real space (a.u.). The initial wave packet (at $t=0$) becomes completely blurred while propagating until $t=t_0$. A fast quantum time-reversal pulse at $t=t_0$ leads to a nearly perfect echo at $t=2t_0$. The right panel (at $t=3t_0$) shows a second echo after a further subsequent pulse at $t=2.5t_0$. This simulation is done without disorder. The colorscale is the same in all snapshots, normalized to the highest value of the modulus of the initial wave packet.

Figure 2

Quantitative analysis of the echo strength. (a) Correlation $\mathcal{C}\left(t\right)$, Eq. (4), obtained from numerical propagation of the $\hslash$ wave; see Fig. 1. Black circles mark the snapshot times. The solid black curve shows distinct echo peaks for the clean Dirac system, Eq. (1). The insets show sketches of the Dirac-type dispersion with a gap opening at $t=t_0$. Further curves correspond to different disorder types: gap disorder with ${\tau }_{\text{gap}}\approx 0.2t_0$ (red dotted), spatial disorder with ${\tau }_{\text{imp}}\approx 0.8t_0$ (blue), and ${\tau }_{\text{imp}}\approx 0.2t_0$ (green). ${\tau }_{\text{imp}}$ and ${\tau }_{\text{gap}}$ are the respective elastic scattering times (see Sec. pp2 in the appendix). (b) Modulus $\left|A\right(k\left)\right|$ of the transition amplitude, Eq. (3), plotted as a function of parameters $\eta$ and $\mu$.

Figure 3

Effects of disorder on graphene echo. (a) The echo strength measured by the echo fidelity $\tilde{M}\left(t_{\text{echo}}\right)$ is shown as a function of pulse time $t_0$ for various disorder strengths $u_{\text{imp}}$ between $0.002M_0$ and $0.016M_0$, averaged over 50 realizations (see text). The error bars denote the standard error of the mean. An exponential fit is used to extract the decay rate. For higher $u_{\text{imp}}$ an expected saturation sets in, such that only a brief regime of exponential decay is visible. (b) Plot of the fitted decay rates $1/{\tau }_0$ as a function of $u_{\text{imp}}$, Eq. (B5), which is quadratic for weak scattering. The black dotted line is the analytically expected curve, which matches well the fitted quadratic function (blue), until the expected strong scattering saturation sets in.