  The chemical potential u of an electron system is a fundamental property of a
solid. A precise measurement of u plays a crucial role in understanding the
electron interaction and quantum states of matter. However, thermodynamics
measurements in micro and nanoscale samples are challenging because of the
small sample volume and large background signals. Here, we report an optical
readout technique for u of an arbitrary two-dimensional (2D) material. A
monolayer semiconductor sensor is capacitively coupled to the sample. The
sensor optical response determines a bias that fixes its chemical potential to
the band edge and directly reads u of the sample. We demonstrate the technique
in AB-stacked MoTe2/WSe2 moire bilayers. We obtain u with DC sensitivity about
20 ueV/sqrt(Hz), and the compressibility and interlayer electric polarization
using AC readout. The results reveal a correlated insulating state at the
doping density of one hole per moire unit cell, which evolves from a Mott to a
charge-transfer insulator with increasing out-of-plane electric field.
Furthermore, we image u and quantify the spatial inhomogeneity of the sample.
Our work opens the door for high spatial and temporal resolution measurements
of the thermodynamic properties of 2D quantum materials.
