Michiel Snoek September 21, 2011 FINESS 2011 Heidelberg Rigorous mean-field dynamics of lattice bosons: Quenches from the Mott insulator Quenches from the Mott insulator

Out-of-equilibrium many-body quantum mechanics:  Theoretically very challenging  Experimentally feasible with ultracold atoms:  Decoupled from the environment  Highly tunable  New questions: thermalization

 Gutzwiller mean-field theory: decoupling of the hopping term  Mean-field eigenstates are product states over the lattice sites: Fisher et al., PRB 40, 546 (1989) Rokhsar and Kotliar, PRB 44, (1991) Sheshadri et al., EPL 22, 257 (1993)

 Decomposition in Fock basis:  Self-consistent solution:  Mott insulator:  Superfluid:  Good agreement with 3D QMC calculations  Exact for infinite dimensions/fully connected lattice

 Time-evolution driven by mean-field Hamiltonian:  Non-linear differential equation for the c n (t):  with

Sciolla & Biroli [PRL 105, (2010)]:  Hamiltonian is invariant under lattice site permutations  Ground states are invariant under permutations.  Dynamics driven by a classical Hamiltonian.  Gutzwiller dynamics is exact M. Snoek, EPL 95, (2011)

 Phase diagram for one particle per site: U/J MI SF 0 U c /J

 We find a dynamical critical interaction U d :  If U f > U d : superfluid order emerges U/J MI SF 0 U d /J U c /J

 We find a dynamical critical interaction U d :  U f > U d : superfluid order emerges  U f < U d : the system remains insulating U/J MI SF 0 U d /J U c /J

 Equations of motion for n=N/V=1, N max = 2:  Mott insulator:  Groundstate for  Steady state:  Stability?

 Contours with H=0 for different U f :  Dynamical critical interaction:  U f < U d : disconnected branches, stable Mott insulator  U f > U d : connected branches, unstable Mott insulator

 Numerical verification:  Exponential increase for U f > U d  Infinitesimal oscillations for U f < U d  Results independent of N max

 Exponent:  Numerical fits (points)  Analytical expression from linearized equations of motion (line):

 Observable using optical lattice systems.  U/J can be quenched by  Changing the optical lattice depth  Feshbach resonances  U d expected to shift, but positive  Trapping potential obscures transition:  Particle transport after the quench  Wedding cake structure: external source of superfluid order.

 Gutzwiller mean-field dynamics is exact on the fully connected lattice and therefore a controlled mean-field method.  A dynamical critical interaction U d separates stable and unstable Mott insulators after a quench.  Observable with ultracold atoms in optical lattices