Detecting Non-Bloch Topological Invariants in Quantum Dynamics

Non-Bloch topological invariants preserve the bulk-boundary correspondence in non-Hermitian topological systems, and are a key concept in the contemporary study of non-Hermitian topology. Here we report the dynamic detection of non-Bloch topological invariants in single-photon quantum walks, revealed through the biorthogonal chiral displacement, and crosschecked with the dynamic spin textures in the generalized quasimomentum-time domain following a quantum quench. Both detection schemes are robust against symmetry-preserving disorders, and yield consistent results with theoretical predictions. Our experiments are performed far away from any boundaries, and therefore underline non-Bloch topological invariants as intrinsic properties of the system that persist in the thermodynamic limit. Our work sheds new light on the experimental investigation of non-Hermitian topology.

Figure 1

Detecting non-Bloch winding numbers. (a) Topological phase diagrams under the non-Bloch and (b) Bloch band theories. Different topological phases are characterized by the non-Bloch (Bloch) winding numbers ${\nu }_{\beta }$ ($\nu$) as functions of the coin parameters $({\theta }_1,{\theta }_2)$ with a fixed $\gamma =0.2746$. Symbols along the blue and red lines indicate coin parameters we adopt along the two different paths on the phase diagram. While the GBZs along the blue line remain the same, those along the red line differ in radius. Black stars correspond to the coin parameters of $U^i$ for the quench processes. Green dashed lines in (a) indicate locations of the non-Bloch exceptional points. (c) The measured average chiral displacement $\overline{C}$ (blue symbols) for six-step quantum walks, along the blue path in the phase diagram with a fixed ${\theta }_2=0.4\pi$. The vertical dashed and dash-dotted lines indicate the locations of non-Bloch and Bloch topological phase boundaries, respectively. The blue (cyan) solid line corresponds to numerical simulations for six-step (twenty-step) quantum walks, and the solid black line indicates the numerically calculated ${\nu }_{\beta }$. (d) The measured $\mathrm{\Delta }{\overline{n}}_z$ (blue symbols) along the blue path for six-step quantum walks. The blue solid line represents results from numerical simulations with the same number of steps. The gray symbols and lines show the experimental and numerical results of $\mathrm{\Delta }{\overline{n}}_z$ respectively, calculated using the Bloch-band theory. Error bars are due to the statistical uncertainty in photon-number counting.

Figure 2

Experimental (lower layer) and numerical results (upper layer) of dynamic spin textures. (a) The dynamic spin textures $\mathit{n}(k_{\beta },t)$ for the same quench process projected into the GBZ and (b) BZ. For either case, the quench takes place between ${\nu }^i=1$ and ${\nu }^f=0$, with $U^i$ characterized by $({\theta }_1=3\pi /4,{\theta }_2=0.4\pi )$, and $U^f$ by $({\theta }_1=0.528\pi ,{\theta }_2=0.4\pi )$. Spin textures are colored according to the value of $n_z(k_{\beta },t)$. For the non-Bloch dynamics, the momentum-dependent periods of the oscillatory spin dynamics are illustrated by the solid black lines in (a). By contrast, spin dynamics in (b) are nonoscillatory, due to the complex quasienergy spectra. Note that, due to the nature of discrete-time quantum walks, the experimental data are taken only at integer time steps, while numerical simulations are performed at smaller time intervals. Numerically simulated spin textures (upper layer) at integer time steps are highlighted by the red dashed boxes, to compare with the corresponding experimental data in the lower layer. See Supplemental Material [42] for a detailed error analysis.

Figure 3

Measurements across non-Bloch exceptional points. (a) Measured $\overline{C}$ and (b) $\mathrm{\Delta }{\overline{n}}_z$ for six-step quantum walks, with varying ${\theta }_1$ along the red lines in Fig. 1. Theoretical results are shown in red solid lines, and experimental results are presented by red symbols. The vertical dashed and dash-dotted gray lines indicate, respectively, the locations of non-Bloch and Bloch topological phase boundaries. The vertical dashed green lines indicate the locations of the non-Bloch exceptional points. The solid black line in (a) denotes the numerically calculated non-Bloch winding number. The gray solid line and squares in (b) indicate the theoretical and measured $\mathrm{\Delta }{\overline{n}}_z$ according to the Bloch band theory. Error bars are due to the statistical uncertainty in photon-number counting.

Figure 4

Impact of disorder. (a) Time- and disorder-averaged biorthogonal chiral displacement and (b) $\mathrm{\Delta }{\overline{n}}_z$ for six-step quantum walks under static disorder. The coin parameters are randomly chosen in the interval $[{\theta }_i-\pi /20,{\theta }_i+\pi /20]$ ($i=1$, 2) at each spatial location, where ${\theta }_1$ is scanned along the blue line in Fig. 1. Coin parameters do not change with time steps. (c) Time- and disorder-averaged biorthogonal chiral displacement and (d) $\mathrm{\Delta }{\overline{n}}_z$ for six-step quantum walks under dynamic disorder. The symbols and shaded regions respectively indicate the mean values of experimental measurements and the range of the standard deviations averaged over 10 different disorder configurations for each set of $({\theta }_1,{\theta }_2)$. The vertical lines indicating the non-Bloch and Bloch phase boundaries are the same as those in Figs. 1 and 1.