Measuring Hopf links and Hopf invariants in a quenched topological Raman lattice

TL;DR

This paper discusses experimental methods to measure Hopf links and invariants in a quenched topological Raman lattice, linking dynamical topological features to the system's equilibrium phase diagram.

Contribution

It introduces a practical approach to measure Hopf links and invariants in a 2D topological lattice via quench dynamics and Bloch state tomography.

Findings

Hopf links can be measured through Bloch state tomography.

Dynamical Hopf invariants are experimentally accessible.

Hopf invariants correspond to the Chern number of the post-quench Hamiltonian.

Abstract

In a recent experimental work [Z. Wu et al., Science 354, 83 (2016)], the PKU-USTC group realized a two-dimensional two-band quantum anomalous Hall model on a square Raman lattice. By quenching the atom-laser detuning of such a Raman lattice, the time-dependent Bloch vectors for each quasimomentum points define a Hopf mapping from quasimomentum-time space $(k_x,k_y,t)\in T^3$ to the Bloch sphere $S^2$. The Hopf links between the preimages of any two Bloch vectors on $S^2$ can be measured experimentally through Bloch state tomography using spin-resolved time-of-flight measurement together with suitable radio-frequency manipulations. The dynamical Hopf invariants, which are quasimomentum-time integrations of Chern-Simons densities, can also be extracted experimentally through certain quench processes. As the Hopf invariant equals the Chern number of the post-quench Hamiltonian, the measurements of Hopf links and Hopf invariants provide an alternative way of understanding the topological phase diagram of the equilibrium system.