Quantum Optics Measurement Scheme for Quantum Geometry and Topological Invariants

We show how a quantum optical measurement scheme based on heterodyne detection can be used to explore geometrical and topological properties of condensed matter systems. Considering a 2D material placed in a cavity with a coupling to the environment, we compute correlation functions of the photons exiting the cavity and relate them to the hybrid light-matter state within the cavity. Different polarizations of the intracavity field give access to all components of the quantum geometric tensor on contours in the Brillouin zone defined by the transition energy. Combining recent results based on the metric-curvature correspondence with the measured quantum metric allows us to characterize the topological phase of the material. Moreover, in systems where $S_z$ is a good quantum number, the procedure also allows us to extract the spin Chern number. As an interesting application, we consider a minimal model for twisted bilayer graphene at the magic angle, and discuss the feasibility of extracting the Euler number.

Figure 1

(a) Proposed heterodyne detection setup with the cavity depicted on the left. The electric field exiting the cavity (red) impinges on a beam splitter along with a coherent laser source (blue) to produce a superimposed signal which is detected at two photodetectors (right and bottom right). (b) Depiction of the hybrid light-matter state arising from the cavity light-matter coupling (parameter $\lambda$). (c) $k$-space contours in a multiband model defined by a fixed excitation energy $\omega$.

Figure 2

Chern numbers and various bounds for (a) the QWZ model and (b) the SOC model with $\mathrm{\Delta }=0.1$ and (c) the result of the three-band model. All quantities of Eq. (10) are plotted as a function of the parameter $u$ which controls the topological state of the models. The hashed regions near the topological transitions roughly indicate the range of $u$ where Eq. (3) is not valid. For computing $I_{\mathrm{lb}}$ and $I_{\mathrm{ub}}$ we approximate $\delta (x)$ by $(1/\sqrt{\pi }|a|)\exp \{-(x/a{)}^2\}$ with $a=0.1$.

Figure 3

Geometrical quantities for TBG plotted as a function of $t^{\prime }/t$. The trivial phase is indicated with a blue shading; whenever one of the plotted quantities drops below the light-blue line, Eq. (6) implies $e_2=0$. (a) Band structure in the trivial phase where the nodal point at $K$ is gapped due to $\xi =0.8$ (encircled points). (b) Trivial phase ($e_2=0$) with $\xi =0.8$. The inset shows the Wilson loop in the moiré Brillouin zone for $t^{\prime }/t=-0.4$ (solid) and $t^{\prime }/t=1$ (dashed) which are gapped for $\xi =0.8$. (c) Nontrivial phase ($|e_2|=1$) with $\xi =0$ and unit winding of the Wilson loop for all values of $t^{\prime }/t$.