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Nonequilibrium dynamics of bosons in optical lattices

Nonequilibrium dynamics of bosons in optical lattices. Eugene Demler Harvard University. Harvard-MIT. $$ NSF, AFOSR MURI, DARPA, RFBR. Local resolution in optical lattices. Nelson et al., Nature 2007. Imaging single atoms in an optical lattice. Gemelke et al., Nature 2009.

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Nonequilibrium dynamics of bosons in optical lattices

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

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

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

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

  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. Vbefore(x) Vafter(x) Equilibration of density inhomogeneity 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 Hard core limit - projector of no multiple occupancies Bose Hubbard model 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. Phase gradient superfluid velocity Equations of Motion Gradient expansion Density relative to half filling 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. Equations of Motion Separate left- and right-moving parts First non-linear expansion 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. Density below half filling Regions with larger density move faster Right-moving part Left-moving part Breaking point formation. Hopf equation Singularity at finite time T0

  15. when when 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

  16. Soliton solutions of Kortweg - de Vries equation Competition of nonlinearity and dispersion leads to the formation of soliton structures 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

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

  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 solitons Hole type solitons Particle-hole solitons

  21. Stability to transverse fluctuations

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

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