1. Nonequilibrium dynamics of bosons in optical lattices $$ NSF, AFOSR MURI, DARPA, RFBR Harvard-MIT Eugene Demler Harvard University

2. Local resolution in optical lattices Gemelke et al., Nature 2009 Density profiles in optical lattice: from superfluid to Mott states Nelson et al., Nature 2007 Imaging single atoms in an optical lattice

3. Nonequilibrium dynamics of ultracold atoms Trotzky et al., Science 2008 Observation of superexchange in a double well potential Palzer et al., arXiv:1005.3545 Interacting gas expansion in optical lattice Strohmaier et al., PRL 2010 Doublon decay in fermionic Hubbard model J

4. Dynamics and local resolution in systems of ultracold atoms Dynamics of on-site number statistics for a rapid SF to Mott ramp Bakr et al., Science 2010 Single site imaging from SF to Mott states

5. This talk Formation of soliton structures in the dynamics of lattice bosons collaboration with A. Maltsev (Landau Institute)

6. Formation of soliton structures in the dynamics of lattice bosons

7. Equilibration of density inhomogeneity V before (x) V after (x) Suddenly change the potential. Observe density redistribution Strongly correlated atoms in an optical lattice: appearance of oscillation zone on one of the edges Semiclassical dynamics of bosons in optical lattice: Kortweg- de Vries equation Instabilities to transverse modulation

8. U t t Bose Hubbard model Hard core limit - projector of no multiple occupancies Spin representation of the hard core bosons Hamiltonian

9. Anisotropic Heisenberg Hamiltonian We will be primarily interested in 2d and 3d systems with initial 1d inhomogeneity Semiclassical equations of motion Time-dependent variational wavefunction Landau-Lifshitz equations

10. Equations of Motion Gradient expansion Density relative to half filling Phase gradient superfluid velocity Mass conservation Josephson relation Expand equations of motion around state with small density modulation and zero superfluid velocity

11. From wave equation to solitonic excitations

12. First non-linear expansion Separate left- and right-moving parts Equations of Motion Left moving part. Zeroth order solution Right moving part. Zeroth order solution

13. Assume that left- and right-moving parts separate before nonlinearities become important Left-moving part Right-moving part

14. Breaking point formation. Hopf equation Left-moving part Right-moving part Singularity at finite time T 0 Density below half filling Regions with larger density move faster

15. Dispersion corrections Left moving part Right moving part Competition of nonlinearity and dispersion leads to the formation of soliton structures Mapping to Kortweg - de Vries equations In the moving frame and after rescaling when

16. Soliton solutions of Kortweg - de Vries equation Solitons preserve their form after interactions Velocity of a soliton is proportional to its amplitude To solve dynamics: decompose initial state into solitons Solitons separate at long times Competition of nonlinearity and dispersion leads to the formation of soliton structures

17. Decay of the step Left moving part Right moving part Below half-filling steepness decreases steepness increases

18. From increase of the steepness To formation of the oscillation zone

19. Decay of the step Above half-filling

20. Half filling. Modified KdV equation Particle type solitonsHole type solitons Particle-hole solitons

21. Stability to transverse fluctuations

22. Dispersion Non-linear waves Kadomtsev-Petviashvili equation Planar structures are unstable to transverse modulation if

23. Kadomtsev-Petviashvili equation Stable regime. N-soliton solution. Plane waves propagating at some angles and interacting Unstable regime. “Lumps” – solutions localized in all directions. Interactions between solitons do not produce phase shits.

24. Summary and outlook $$ RFBR, NSF, AFOSR MURI, DARPA Harvard-MIT Solitons beyond longwavelength approximation. Quantum solitons Beyond semiclassical approximation. Emission on Bogoliubov modes. Dissipation. Transverse instabilities. Dynamics of lump formation Multicomponent generalizations. Matrix KdV Formation of soliton structures in the dynamics of lattice bosons within semiclassical approximation.