Detecting topological invariants in chiral symmetric insulators via losses

We show that the bulk winding number characterizing one-dimensional topological insulators with chiral symmetry can be detected from the displacement of a single particle, observed via losses. Losses represent the effect of repeated weak measurements on one sublattice only, which interrupt the dynamics periodically. When these do not detect the particle, they realize negative measurements. Our repeated measurement scheme covers both time-independent and periodically driven (Floquet) topological insulators, with or without spatial disorder. In the limit of rapidly repeated, vanishingly weak measurements, our scheme describes non-Hermitian Hamiltonians, as the lossy Su-Schrieffer-Heeger model of Rudner and Levitov, [Phys. Rev. Lett. 102, 065703 (2009)]. We find, contrary to intuition, that the time needed to detect the winding number can be made shorter by decreasing the efficiency of the measurement. We illustrate our results on a discrete-time quantum walk, and propose ways of testing them experimentally.