Superfluid to Mott Insulator (2D)

Bose-Hubbard model

Ultracold bosons in a cubic optical lattice are described by H = -J sum(a_i^+ a_j) + (U/2) sum n_i(n_i-1) - mu sum n_i. Tunnelling J favours a delocalised superfluid, on-site repulsion U favours exactly integer filling (a Mott insulator).

Mean-field phase boundary

Decoupling the hopping term with the order parameter psi = <a> gives a single-site problem H = (U/2)n(n-1) - mu n - zJ psi (a + a^+) + zJ psi^2 (z = 6 neighbours), solved here by exact diagonalisation in a 9-state Fock basis and iterated until psi is self-consistent. The Mott lobe of filling n ends where psi first becomes non-zero: zJ/U = (n - mu/U)(mu/U - n + 1)/(1 + mu/U). The n = 1 lobe tip is at zJ/U = 3 - 2 sqrt 2 = 0.1716.

From laser depth to J and U

With lattice depth s = V0/E_R (recoil energy E_R), the standard Wannier-function approximations are J = (4/sqrt pi) E_R s^(3/4) exp(-2 sqrt s) and U = sqrt(8/pi) k a_s E_R s^(3/4), so zJ/U = 6 sqrt2 exp(-2 sqrt s)/(k a_s); k = 2 pi / lambda with lambda = 852 nm. Deeper lattice means smaller J/U, so the system crosses into the Mott lobe.

Checks on this page

(1) The n = 1 lobe tip found by a numerical scan equals 3 - 2 sqrt 2. (2) The self-consistent solver switches psi on at the analytic boundary. (3) The critical depth for 87Rb found by bisection equals the closed form.

Honest simplifications

Mean-field theory overestimates the superfluid region (quantum Monte Carlo puts the 3D critical zJ/U near 0.14 instead of 0.17), ignores the harmonic trap and its shell structure, uses zero temperature, and the Wannier approximations are good only for s above about 5. Site snapshots sample the single-site distribution P(n); they do not include correlations between sites. Experiments find the transition near 13 E_R for the 3D lattice.

• Press Deep lattice: Mott: every site holds exactly one atom, the variance is zero and psi = 0.
• Drag the lattice depth down: the red point leaves the lobe and the sites start to fluctuate.
• Raise the scattering length (Feshbach resonance) to move the transition to shallower lattices.
• Set mu/U between lobes (for example 1.0) to see that no Mott state exists there: the system is superfluid at any depth.