Fig. 2: Lyapunov exponent from bulk dynamics.

Measured polarization-averaged growth rates \(\bar{\lambda }(v)\) for a unitary quantum walk with γ = 0 in (a) and a non-unitary quantum walk with γ = 0.1 in (b). Red triangles with error bars are the experimental data and blue triangles are from numerical simulations. The horizontal dashed line indicates the threshold values below which experimental data were no longer reliable due to photon loss. To construct \(\bar{\lambda }\), we initialize the walker in the state \(\left|0\right\rangle \otimes \left|H\right\rangle\) (\(\left|0\right\rangle \otimes \left|V\right\rangle\)), evolve it up to ten steps under the parameters (θ₁ = 4.3, θ₂ = 2.175, W = 0), and projectively measure the horizontally (vertically) polarized photon distribution following the last time step. Note that the system is in a topologically non-trivial phase under the chosen parameters. We construct λ_H and λ_V from these polarization-resolved distributions, from which \(\bar{\lambda }\) is calculated. c, d The polarization-resolved photon distribution after the last time step t = 10, for the dynamics in a and b, respectively. For each bar, the blue (top) and red (bottom) portions are respectively the numerical results for the horizontal-polarization-resolved and vertical-polarization-resolved photon distributions, initialized in \(\left|0\right\rangle \otimes \left|H\right\rangle\) and \(\left|0\right\rangle \otimes \left|V\right\rangle\), respectively. The white dots are the experimental measurements for the vertical-polarization-resolved photon distribution, and black dots are the experimental results for the sum of the polarization-resolved distributions. Error bars are due to the statistical uncertainty in photon-number-counting.