Exploring many-body physics with synthetic matter Theory: David Pekker (CalTech) Andrey Maltsev (Landau Institute), Aleksander Prokofiev (Landau Institute), Eugene Demler (Harvard University) Experiment: Immanuel Bloch’s group at MPQ/LMU Supported by NSF, DARPA, AFOSR MURI, ARO MURI

How cold are ultracold atoms? Density of atoms 1013 cm-1 Distance between atoms 300 nm BEC temperature 1mK feV peV neV µeV meV eV keV MeV GeV TeV pK nK µK mK K He N room temperature LHC current experiments with optical lattices 10-11 - 10-10 K

Single atom resolution in optical lattices Bakr et al., Science 2010 density y x

Many-body physics with ultracold atoms Decoupling from external environment - Long coherence times Long intrinsic time scales - Interaction energy and bandwidth ~ 1kHz - System parameters can be changed over this time scale Can achieve highly non equilibrium quantum many-body states New detection methods - Single atom resolution New frontier in many-body physics: quantum many-body dynamics

Outline Observation of the amplitude Higgs mode in the superfluid state of bosons in optical lattices M. Endres, D. Pekker et al., arXiv:1204.518, Nature in press Universal nonlinear semiclassical hydrodynamics of lattice spin models and strongly correlated bosons in optical lattices A. Maltsev, E.D., Annals of Physics 326:1775 (2011) A. Maltsev, A. Prokofiev, E.D., arXiv:1201.6400

Intoroduction: Ultracold atoms in optical lattice. Bose Hubbard model

Bose Hubbard model U t Confining potential tunneling of atoms between neighboring wells repulsion of atoms sitting in the same well Confining potential

Single atom resolved imaging of a Mott insulator Bose Hubbard Model. Phase diagram Weak lattice t<<U superfluid phase n=3 Mott 2 Superfluid n=2 Mott Strong lattice t>>U Mott state 1 n=1 Mott Single atom resolved imaging of a Mott insulator Sherson et al., Nature 2010

Observation of the amplitude Higgs mode in the superfluid state of bosons in optical lattices Experiment: Manuel Endres, Immanuel Bloch and MPQ team Theory: David Pekker (Caltech), Eugene Demler M. Endres, D. Pekker et al., arXiv:1204.518, Nature in press D. Pekker et al., arXiv:1206.1648 Philosophy similar to Grigory Volovik’s “Universe in a helium droplet”: Find condensed matter analogues of high energy particles and phenomena

Collective modes of strongly interacting superfluid bosons Order parameter Breaks U(1) symmetry Phase (Goldstone) mode = gapless Bogoliubov mode Gapped amplitude mode (Higgs mode)

Excitations of the Bose Hubbard model 2 Mott Superfluid n=3 Mott 2 Superfluid n=2 Mott 1 n=1 Mott Softening of the amplitude mode is the defining characteristic of the second order Quantum Phase Transition

Why it is difficult to observe the amplitude mode Stoferle et al., PRL (2004) Peak at U dominates and does not change as the system goes through the SF/Mott transition

Is there a Higgs resonance 2d? D. Podolsky et al., arXiv:1108.5207 Earlier work: S. Sachdev (1999), W. Zwerger (2004)

Exciting the amplitude mode Absorbed energy

Exciting the amplitude mode M. Endres et al., arXiv:1204.518, Nature in press Mott n=1 Mott n=1 Mott n=1

Experiments: full spectrum Manuel Endres, Immanuel Bloch and MPQ team

Gutzwiller model for the amplitude mode Bogoliubov mode comes from the phase and charge degrees of freedom: and Amplitude/Higgs mode comes from and Time dependent mean-field: project dynamics to factorizable Gutzwiller wavefunctions. It is equivalent to Landau-Lifshitz eqs. It gives collective modes but not coupling between them. Threshold for absorption is captured very well

Time dependent cluster mean-field Lattice height 9.5 Er: (1x1 vs 2x2) single amplitude mode excited single amplitude mode excited multiple modes excited? breathing mode breathing mode 2x2 captures width of spectral feature

Absorption spectra. Theory (1x1 calculations) disappearing amplitude mode Breathing mode details at the QCP spectrum remains gapped due to trap

Higgs Drum Modes 1x1 calculation, 20 oscillations Eabs rescaled so peak heights coincide Similar to Higgs mode in compactified dimensions

Universal nonlinear hydrodynamics of Beyond linear analysis of collective modes Universal nonlinear hydrodynamics of lattice spin models and strongly correlated bosons in optical lattices A. Maltsev, E.D., Annals of Physics 326:1775 (2011) A. Maltsev, A. Prokofiev, E.D., arXiv:1201.6400

Bogoliubov mode in weakly interacting gas Probing Bogoliubov mode with light scattering: D. Stamper-Kurn et al., PRL 83:2876 (1999) Detailed study of dispersion Ozeri et al., RMP 77:187 (2005) Dark soliton in BEC, C. Becker et al., Nature (2008) Our goal: study solitons in strongly correlated states of bosons in optical lattices

Equilibration of density inhomogeneity Vbefore(x) Suddenly change the potential. Observe density redistribution Vafter(x) 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

Bose Hubbard model in the hard core limit Spin representation of the hard core limit of the Hubbard Hamiltonian

Quantum magnetism of bosons in optical lattices Theory: Duan et al., PRL (2003) Kuklov, Svistunov, PRL (2003) Expt: S. Trotzky et al., Science (2008)

Semiclassical soliton dynamics: stable regime Character of solitons: KdV type (Gutzwiller wavfunctions) density superfluid velocity t=0 t=10 t=50

Semiclassical soliton dynamics: unstable regime Formation of lump solutions

Universal phase diagram of dynamics in 2d and 3d anisotropic Heisenberg model Particle solitons. Unstable to 2d modulation 2d lamp solutons Hole solitons. Stable to 2d modulation Both particle and hole solitons allowed Particle solitons. Stable to 2d modulation Decay of inhomogeneities to short wavelength oscillations Hole solitons. Unstable to 2d modulation 2d lamp solutions

Semiclassical dynamics of anisotropic Heisenberg Hamiltonian Semiclassical dynamics of anisotropic Heisenberg Hamiltonian. Density step decay.

Effect of parabolic potential Solitons break apart when crossing boundaries of distinct “universality” regions

Summary Observation of the amplitude Higgs mode in the superfluid state of bosons in optical lattices Universal nonlinear semiclassical hydrodynamics of lattice spin models and strongly correlated bosons in optical lattices

How cold are ultracold atoms? Density 1013 cm-1 Distance between atoms 300 nm BEC temperature 1mK feV peV neV µeV meV eV keV MeV GeV TeV pK nK µK mK K He N room temperature LHC first BEC of alkali atoms