bosons and spin lattice models - Harvard University
Transcript of bosons and spin lattice models - Harvard University
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Dynamics of strongly correlatedDynamics of strongly correlated bosons and spin lattice models
Observation of the amplitude Higgs modein the superfluid state of bosons in optical lattices
Experiment: Manuel Endres, Immanuel Bloch and MPQ teamTheory: David Pekker (Caltech), Eugene Demler
Universal nonlinear semiclassical hydrodynamics of lattice spin models and strongly correlated bosons in optical latticesTheory: Andrey Maltsev, Aleksander Prokofiev (Landau Institute),
Eugene DemlerEugene Demler
Supported by NSF, DARPA, AFOSR MURI, ARO MURI
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BEC of weakly interacting gas
Probing Bogoliubov mode with light scattering: D. Stamper‐Kurn et al., PRL 83:2876 (1999)
Detailed study of dispersionOzeri et al., RMP 77:187 (2005)
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Solitons in BEC. Gross‐Pitaevskii regimeSoliton train in BEC, Khawaja et al., Nature (2002), PRL (2002)
Dark soliton in BEC, C. Becker et al., Nature (2008)
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Strongly correlated bosons in optical lattice
Collective modesN li h d d i (i l di li )Nonlinear hydrodynamics (including solitons)
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Observation of the amplitude Higgs modein the superfluid state of bosonsin the superfluid state of bosons in optical lattices
Experiment: Manuel Endres, Immanuel Bloch and MPQ teamTheor Da id Pekker (Caltech) E gene DemlerTheory: David Pekker (Caltech), Eugene Demler
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Collective modes of strongly interactingsuperfluid bosons
Order parameter Breaks U(1) symmetry
superfluid bosons
Phase (Goldstone) mode = gapless Bogoliubov mode
Gapped amplitude mode (Higgs mode)
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Excitations of the Bose Hubbard modelExcitations of the Bose Hubbard model
U
22
1nn=3
Superfluid
Mott
2
1
n=2
p
Mott
Mott Superfluid
0
Mottn=1
Softening of the amplitude mode is the defining characteristicSoftening of the amplitude mode is the defining characteristicof the second order Quantum Phase Transition
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Why it is difficult to observe the amplitude mode
Stoferle et al., PRL (2004)Stoferle et al., PRL (2004)
Peak at U dominates and does not change as the system goes through the SF/Mott transition
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Exciting the amplitude mode
Absorbed energy
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Exciting the amplitude modeManuel Endres, Immanuel Bloch and MPQ team
Mottn=1 Mottn=1 Mottn=1
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Absorption spectrum. Theoryh h ( )Lattice height 9.5 Er: (1x1 vs 2x2)
breathing mode
single amplitude mode excited multiple modes
excited?single amplitude mode excited
avg. over number of atoms in trap2x2 captures width of spectral feature
e c ted
breathing mode
mode excited
p pfine structure due to deficiency of method?
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Experiments: full spectrump pManuel Endres, Immanuel Bloch and MPQ team
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Absorption spectra. Theory (1x1 calculations)
disappearing amplitude mode
B thi dBreathing mode
details at the QCP
spectrum remains gapped due to trap
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Higgs Drum Modes1x1 calculation, 20 oscillationsEabs rescaled so peak heights coincide
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Beyond linear analysis of collective modes
Universal nonlinear hydrodynamics of y ylattice spin models and strongly correlated bosons in optical latticesbosons in optical lattices
A d M lt Al k d P k fi (L d I tit t )Andrey Maltsev, Aleksander Prokofiev (Landau Institute), Eugene Demler
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Equilibration of density inhomogeneityV ( )Vbefore(x)
Suddenly change th t ti l
Vafter(x)
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 equationKortweg- de Vries equation
Instabilities to transverse modulation
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Bose Hubbard model in the hard core limitU
tt tt
Hard core limit
‐ projector of no multiple occupancies
Spin representation of the hard core bosons Hamiltonian
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Quantum magnetism of bosons in optical Q g plattices
Duan et al., PRL (2003)Kuklov, Svistunov, PRL (2003)
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Universal phase diagram of dynamics in 2d and 3d anisotropic Heisenberg modelin 2d and 3d anisotropic Heisenberg model
Particle solitons. Unstable to 2d modulation
Hole solitons.Stable to 2d modulation
Unstable to 2d modulation2d lamp solutons
Stable to 2d modulation
Both particle and holesolitons allowed
Particle solitons.Stable to 2d modulation
Decay of inhomogeneitiesto short wavelength
ill ti
Stable to 2d modulation
Hole solitonsoscillations Hole solitons. Unstable to 2d modulation2d lamp solutions
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Semiclassical soliton dynamics: stable regimeCharacter of solitons: KdV type
d itdensity
superfluid velocity
t=0 t=10 t=50
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Semiclassical soliton dynamics: unstable regime
t 0 t=40t=0 t=40
Formation of lump solutions
t=80 t=90
Formation of lump solutions
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Semiclassical dynamics of anisotropic Heisenberg Hamiltonian. Density step decay.Heisenberg Hamiltonian. Density step decay.
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Dynamics of strongly correlatedDynamics of strongly correlated bosons and spin lattice modelsObservation of the amplitude Higgs modein the superfluid state of bosons in optical lattices
Experiment: Manuel Endres, Immanuel Bloch and MPQ teamTheory: David Pekker (Caltech), Eugene Demler
Universal nonlinear semiclassical hydrodynamics of lattice spin models and strongly correlated bosons in optical latticesin optical lattices
Theory: Andrey Maltsev, Aleksander Prokofiev (Landau Institute), Eugene DemlerEugene Demler