Measuring Mixed-State Topological Invariant in Open Photonic Quantum Walk

Abstract

Pure-state manifestations of geometric phase are well established and have found applications across essentially all branches of physics, yet their generalization to mixed-state regimes remains largely unexplored experimentally. The Uhlmann geometric phase offers a natural extension of pure-state paradigms and can exhibit a topological character. However, observation of this invariant is impeded by the incompatibility between Uhlmann parallel transport and Hamiltonian dynamics, as well as the difficulty of preparing topologically nontrivial mixed states. To address this challenge, we report an experimentally accessible protocol for directly measuring the mixed-state topological invariant. By engineering controlled nonunitary dynamics in a photonic quantum walk, we prepare topologically nontrivial mixed states from a trivial initial state. Furthermore, by machine-learning the full density matrix in momentum space, we directly extract the quantized geometric phase of the nontrivial mixed states. These results highlight a geometric phase framework that naturally extends to open quantum systems both in and out of thermal equilibrium.

Figures (3)

• Figure 1: Protocol for preparing topologically nontrivial states. (a) Directly preparing a nontrivial ground state with $\Phi_{\text{B}}=\pi$ from a trivial one with $\Phi_{\text{B}}=0$ is not feasible under unitary dynamics. (b) A nontrivial pseudoground state with $\Phi_{\text{U}}=\pi$ can be achieved from a trivial ground state through almost-adiabatic nonunitray dynamics, except for a small density of excitations near the gap-closing point. Red dots on Bloch sphere denote the momentum-resolved density matrices $\rho_k$ across the Brillouin zone, from which the geometric phase is directly extracted.
• Figure 2: Experimental QW setup. The apparatus comprises four functional modules: (a) Photon pairs are produced via spontaneous parametric down-conversion in BBO2, where the idler photon serves as a herald for the signal photon entering the QW module. (b) QW is implemented using a sequence of polarization rotations and polarization-dependent time shifts, implemented with HWPs and calcite beam displacers, respectively. In each step, HWPs are mounted on motorized rotation stages that introduce slow, controlled angle fluctuations to emulate dephasing noise. (c) A Michelson interferometer performs the basis transformation and enables interference measurements between different time bins (sites). (d) The walker’s position distribution is finally obtained via single-photon frequency up-conversion, allowing for site-resolved detection. Abbreviations: BBO, $\beta-\text{BaB}_{2}\text{O}_{4}$; DM, dichroic mirror; PBS, polarizing beam splitter; HWP, half-wave plate; QWP, quarter-wave plate; PMT, photomultiplier tube.
• Figure 3: Experimental strategy and results. (a) Quench trajectory in the phase diagram, where the system is driven from a topologically trivial Hamiltonian (peach-pink region) to a nontrivial one (blue region). The topologically distinct phases are separated by boundaries at which the quasienergy gap closes at $E = 0$ (black dashed lines) or $E = \pi$ (black solid lines). (b) Measured Berry geometric phase (green opaque bars) of the time-evolving pure state under coherent quench dynamics. (c) Realization of dephasing noise through ensemble-averaged dynamics over disordered Hamiltonians. (d) Measured Uhlmann geometric phase (green opaque bars) of the time-evolving mixed state under noisy quench dynamics. In panels (b) and (d), green transparent bars indicate theoretical predictions. Error bars are obtained from Monte Carlo simulations incorporating photon-counting statistics.