A random sampling method to test weak Eigenstate Thermalization*

Isolated quantum systems are observed to come to thermal equilibrium, even in the absence of an external bath. According to the weak Eigenstate Thermalization Hypothesis (wETH), this is because the vast majority of eigenstates within a microcanonical energy shell are thermal. This is in contrast to the Eigenstate Thermalization Hypothesis (ETH) in which all eigenstates within the energy shell must be thermal. To test the validity of the wETH for systems too large for exact diagonalization, We developed a new method to statistically sample eigenstates. We calculate experimentally relevant expectation values and their standard deviation. We assess wETH by repeating this calculation with a series of systems increasing in size. We apply our method to systems of interacting ultracold-Rydberg atoms that are believed to violate ETH.