Bulk detection of time-dependent topological transitions in quenched chiral models

The topology of one-dimensional chiral systems is captured by the winding number of the Hamiltonian eigenstates. Here we show that this invariant can be read out by measuring the mean chiral displacement of a single-particle wave function that is connected to a fully localized one via a unitary and translation-invariant map. Remarkably, this implies that the mean chiral displacement can detect the winding number even when the underlying Hamiltonian is quenched between different topological phases. We confirm experimentally these results in a quantum walk of structured light.

Figure 1

MCD dynamics on simple models. (a) ${\mathrm{SSH}}_4$ model with chiral symmetry broken before $t_c=20/a$, and restored after that [tunnelings: $a=b=c=\frac{d}{2}$, staggering: $\beta =a\theta (t_c-t)$]. (b) SSH-LR model quenched from $\nu =0$ to 1 to 2 [tunnelings: $a=b$, $c=0$, and $d/a$ is quenched from 0.5, to $-0.5$, to 1.5 at times $t_{c,1}=20/a$ and $t_{c,2}=60/a$]. In both panels, the thick line is a moving average of the underlying data over a window of duration $10/a$, which removes the fast oscillations.

Figure 2

Quantum walk winding numbers: (a) Dependence of the winding number on the parameter $\delta$ for protocols $U$ and $\tilde{U}$, as indicated in the legend. [(b), (c)] On the Poincaré sphere, we plot eigenstates $\mathbf{n}\left(q\right)$ of the evolution operator $U$, in two cases $\delta =\pi$ and $\delta =\pi /4$, which have windings 1 and 0, respectively. These states are positioned on a single plane, perpendicular to the vector ${\mathbf{v}}_{\mathrm{\Gamma }}$ ($\mathrm{\Gamma }={\mathbf{v}}_{\mathrm{\Gamma }}·\mathit{\sigma }$).

Figure 3

Experimental setup and chiral probability distributions. (a) In our experiment, the input beam is prepared with a polarizer ($P$), and then it performs a QW generated by either $U$ or $\tilde{U}$ (see main text). Finally, the chiral polarization components are analyzed with a QWP (W) and a polarizer, and the light intensity is recorded on a camera (C) placed in the focal plane of a lens (L). (b) Representative intensity distribution recorded by the camera. (c) Probability distributions $P_{-}\left(m\right)$ for a QW with a quench (at the fifth step; see the red dashed line) from the chiral protocol $U\left(\pi \right)$ to the chiral protocol $U(2\pi /5)$. Experimental results (left) are compared to numerical simulations (right).

Figure 4

MCD in quenched QWs. MCD in quantum walks governed by protocol $V_1$ from steps 1 to 5 and by protocol $V_2$ from steps 6 to 13. The winding number $\nu$ of each protocol is shown with red lines. (a) $V_1=T(3\pi /4)S$ is nonchiral (therefore its winding is not defined), $V_2=U\left(\pi \right)$. (b) $V_1=U\left(\pi \right)$, $V_2=U(2\pi /5)$. (c) To study quenches between nontrivial phases, we chose here $V_1=\tilde{U}(7\pi /4)$ and $V_2=\tilde{U}\left(\pi \right)$, having winding numbers 2 and 1, respectively. (d) Initial and final evolutions are swapped with respect to panel (c). In all plots, experimental data (orange dots) are compared with theoretical simulations (blue squares). In panels (c) and (d), an extra plate implementing the operator $T[({\delta }_2-{\delta }_1)/2]$ was inserted between the fifth and sixth steps.