Experimental Measurement of the Divergent Quantum Metric of an Exceptional Point

The geometry of Hamiltonian’s eigenstates is encoded in the quantum geometric tensor (QGT), containing both the Berry curvature, central to the description of topological matter, and the quantum metric. So far, the full QGT has been measured only in Hermitian systems, where the role of the quantum metric is mostly limited to corrections. On the contrary, in non-Hermitian systems, and, in particular, near exceptional points, the quantum metric is expected to diverge and to often play a dominant role, for example, in the enhanced sensing and in wave packet dynamics. In this Letter, we report the first experimental measurement of the quantum metric in a non-Hermitian system. The specific platform under study is an organic microcavity with exciton-polariton eigenstates, which demonstrate exceptional points. We measure the quantum metric’s divergence, and we determine the scaling exponent $n=-1.01\pm 0.08$, which is in agreement with the theoretical description of second-order exceptional points.

Figure 1

Reflectivity of the organic microcavity. (a) Scheme of the microcavity sample. (b) Structure of the DPAVBi molecule. (c),(d) Reflectivity as a function of wave vector $k_x$ and $k_y$ (respectively) and energy, exhibiting anticrossing along $k_y$. (e) Reflectivity as a function of the in-plane polar angle $\varphi$ and energy $E$ for $|k|=|k^{*}|$ (EP wave vector). (f) Real and imaginary parts of the mode energies (dots with error bars, experiment; lines, theory).

Figure 2

Stokes vector components of the lowest-energy eigenstates (experiment and theory). (a)–(c) Experiment ($S_1$, $S_2$, $S_3$); (d)–(f) theory ($S_1$, $S_2$, $S_3$). A region where the pseudospin could not be extracted experimentally is hatched.

Figure 3

Quantum metric of an exceptional point. 2D maps of the trace of the quantum metric in the vicinity of an EP marked as a star: (a) experiment; (b) theory. The gray region covers the discontinuity of the wave function (branch cut). (c) A log-log plot of the experimentally extracted quantum metric $g_{qq}$ near $k_x=0$ (black dots) and near the EP (red dots) and its fit (red line), giving the scaling exponent $n=-1.01\pm 0.08$. The metric is extracted along the white dashed lines shown in (a).