Mesoscopic effects in quantum phases of ultracold quantum gases in optical lattices*

We present a wide array of quantum measures on numerical solutions of one-dimensional Bose- and Fermi-Hubbard Hamiltonians for finite-size systems with open boundary conditions. Specifically, for the Bose-Hubbard Hamiltonian we calculate number, quantum depletion, local von Neumann entropy, generalized entanglement or Q measure, fidelity, and fidelity susceptibility; for the Fermi-Hubbard Hamiltonian we also calculate the pairing correlations, magnetization, charge-density correlations, and antiferromagnetic structure factor. Our numerical method is imaginary time propagation via time-evolving block decimation. As part of our study we provide a careful comparison of canonical versus grand canonical ensembles and Gutzwiller versus entangled simulations. The most striking effect of finite size occurs for bosons: we observe a strong blurring of the tips of the Mott lobes accompanied by higher depletion, and show how the location of the first Mott lobe tip approaches the thermodynamic value as a function of system size.