Thermodynamics and density fluctuations in a bilayer Hubbard system of ultracold atoms

We measure the equation of state in a bilayer Hubbard system for different ratios of the two tunnelling amplitudes $t_\perp /t$. From the equation of state we deduce the compressibility and observe its dependency on $t_\perp /t$. Moreover, we infer thermodynamic number fluctuations from the equation of state by employing the fluctuation-dissipation theorem. By comparing the thermodynamic with local density fluctuations, we find that non-local density fluctuations in our bilayer Hubbard system become more prominent for higher $t_\perp /t$ in the low filling regime. To validate our measurements, we compare them to Determinant Quantum Monte Carlo simulations of a bilayer Hubbard system with 6$\times$6 lattice sites per layer.