Michiel Snoek

September 21, 2011FINESS 2011Heidelberg

Rigorous mean-field dynamics of lattice Rigorous mean-field dynamics of lattice bosons:bosons:

Quenches from the Mott insulatorQuenches 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:

iii

MF

iii

jijiijjiij

jijiij bbJbbbbbbJbbJ

2*

,,

ˆˆˆˆˆˆˆˆˆˆ

Fisher et al., PRB 40, 546 (1989)Rokhsar and Kotliar, PRB 44, 10 328 (1991)Sheshadri et al., EPL 22, 257 (1993)

i

iiiii

iiji

jiij bbbbU

bbbbJH ˆˆˆˆ2

ˆˆˆˆ,

ib̂ zJJ ij

Decomposition in Fock basis:

Self-consistent solution:

Mott insulator:

Superfluid:

Good agreement with 3D QMC calculations

Exact for infinite dimensions/fully connected lattice

max

0

N

nni nc

0

0

Time-evolution driven by mean-field Hamiltonian:

Non-linear differential equation for the cn(t):

with

MFMFMF Hi t

)()1(

2

)(1)()()()( 1*

1

tcnnnU

tcnttcntJtci

n

nnnt

max

1

*1 )()()(

N

nnn tctcjt

Sciolla & Biroli [PRL 105, 220401 (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, 30006

(2011)

Phase diagram for one particle per site:

U/J

MISF

0Uc /J

We find a dynamical critical interaction Ud:

If Uf > Ud: superfluid order emerges

U/J

MISF

0Ud /J Uc /J

We find a dynamical critical interaction Ud:

Uf > Ud: superfluid order emerges

Uf < Ud: the system remains insulating

U/J

MISF

0Ud /J Uc /J

Equations of motion for n=N/V=1, Nmax = 2:

Mott insulator:

Groundstate for

Steady state:

Stability?

))cos(83)(21(),( 222222 pmJmUmpmH

)sin()21(8

))cos(83)(41(

22222

2222

pmJmpHm

pmJUmHp

0,02 Hm

JUU c )83(

002

2

mm

Contours with H=0 for different Uf :

Dynamical critical interaction:

Uf < Ud : disconnected branches, stable Mott insulator

JU d )83(

Uf > Ud: connected branches, unstable Mott insulator

Numerical verification:

Exponential increase for Uf > Ud

Infinitesimal oscillations for Uf < Ud

Results independent of Nmax

Exponent:

Numerical fits (points)

Analytical expression from linearized equations of motion (line):

))(( fcdf UUUU

Observable using optical lattice systems.

U/J can be quenched by Changing the optical lattice depth Feshbach resonances

Ud 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 Ud separates stable and unstable Mott insulators after a quench.

Observable with ultracold atoms in optical lattices