Steady-state Hall response and quantum geometry of driven-dissipative lattices

We study the effects of the quantum geometric tensor, i.e., the Berry curvature and the Fubini-Study metric, on the steady state of driven-dissipative bosonic lattices. We show that the quantum-Hall-type response of the steady-state wave function in the presence of an external potential gradient depends on all the components of the quantum geometric tensor. Looking at this steady-state Hall response, one can map out the full quantum geometric tensor of a sufficiently flat band in momentum space using a driving field localized in momentum space. We use the two-dimensional Lieb lattice as an example and numerically demonstrate how to measure the quantum geometric tensor.

Figure 1

(a) A schematic structure of the two-dimensional Lieb lattice. Three sublattices, A, B, and C, are indicated respectively by white, black, and gray circles. Three lattice sites, such as the ones indicated within the dashed square, form a unit cell, and the respective hopping amplitudes are indicated. (b) The dispersion of the two-dimensional Lieb lattice as a function of momenta $k_x$ and $k_y$ with parameters $t_1=t_2=2t_1^{\prime }=2t_2^{\prime }$.

Figure 2

Complete quantum geometric tensor of a 2D Lieb lattice. [(a)–(d)] Estimated values from the numerical simulation of the driven-dissipative steady-state using (10); the left-hand side of (10) is estimated from the full numerical simulation of (6) and then (7), and we use (10) to extract the components of the quantum geometric tensor. [(e)–(h)] Ideal values calculated from the definition of the Berry curvature (1) and the Fubini-Study metric (3). All the figures are in units of the lattice spacing squared.