Post on 21-Dec-2015
Nonequilibrium dynamics of bosons in optical lattices
$$ NSF, AFOSR MURI, DARPA, RFBR Harvard-MIT
Eugene Demler Harvard University
Local resolution in optical lattices
Gemelke et al., Nature 2009
Density profiles inoptical lattice: from superfluid to Mott states
Nelson et al., Nature 2007
Imaging single atomsin an optical lattice
Nonequilibrium dynamics of ultracold atoms
Trotzky et al., Science 2008Observation of superexchange in a double well potential
Palzer et al., arXiv:1005.3545Interacting gas expansionin optical lattice
Strohmaier et al., PRL 2010Doublon decay in fermionic Hubbard model
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Dynamics and local resolution insystems of ultracold atoms
Dynamics of on-site number statistics for a rapid SF to Mott ramp
Bakr et al.,Science 2010
Single site imagingfrom SF to Mott states
This talk
Formation of soliton structures in the dynamics of lattice bosons
collaboration with A. Maltsev (Landau Institute)
Equilibration of density inhomogeneityVbefore(x)
Vafter(x)
Suddenly change the potential.Observe densityredistribution
Strongly correlated atoms in an optical lattice:appearance of oscillation zone on one of the edges
Semiclassical dynamicsof bosons in optical lattice:Kortweg- de Vries equation
Instabilities to transverse modulation
U
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Bose Hubbard model
Hard core limit
- projector of no multiple occupancies
Spin representation of the hard core bosons Hamiltonian
Anisotropic Heisenberg Hamiltonian
We will be primarily interested in 2d and 3d systems with initial 1d inhomogeneity
Semiclassical equations of motionTime-dependentvariational wavefunction
Landau-Lifshitzequations
Equations of Motion Gradient expansion
Density relative to half filling
Phase gradient superfluid velocity
Massconservation
Josephsonrelation
Expand equations of motion around state with smalldensity modulation and zero superfluid velocity
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
Assume that left- and right-moving partsseparate before nonlinearities become important
Left-moving part
Right-moving part
Breaking point formation. Hopf equation
Left-moving part
Right-moving part
Singularity at finite time T0
Density below half filling Regions with larger density move faster
Dispersion corrections
Left moving part
Right moving part
Competition of nonlinearity and dispersion leads tothe formation of soliton structures
Mapping to Kortweg - de Vries equationsIn the moving frame and after rescaling
when
when
Soliton solutions of Kortweg - de Vries equation
Solitons preserve their form after interactionsVelocity of a soliton is proportional to its amplitude
To solve dynamics: decompose initial state into solitonsSolitons separate at long times
Competition of nonlinearity and dispersion leads tothe formation of soliton structures
Decay of the step
Left moving part
Right moving part
Below half-filling
steepness decreases
steepness increases
Half filling. Modified KdV equation
Particle type solitons Hole type solitons
Particle-hole solitons
Stability to transverse fluctuations
Dispersion
Non-linear waves
Kadomtsev-Petviashvili equation
Planar structures are unstable to transverse modulation if
Kadomtsev-Petviashvili equation
Stable regime. N-soliton solution. Plane waves propagatingat some angles and interacting
Unstable regime.“Lumps” – solutions localized in all directions.Interactions between solitons do not produce phase shits.
Summary and outlook
$$ RFBR, NSF, AFOSR MURI, DARPAHarvard-MIT
Solitons beyond longwavelength approximation. Quantum solitons
Beyond semiclassical approximation. Emission on Bogoliubovmodes. Dissipation.
Transverse instabilities. Dynamics of lump formation
Multicomponent generalizations. Matrix KdV
Formation of soliton structures in the dynamics of lattice bosons within semiclassical approximation.