QUANTUM MANY-BODY SYSTEMS OF ULTRACOLD ATOMS Eugene Demler Harvard University Grad students: A. Imambekov (->Rice), Takuya Kitagawa Postdocs: E. Altman (->Weizmann), A. Polkovnikov (->U. Boston) A.M. Rey (->U. Colorado), V. Gritsev (-> U. Fribourg), D. Pekker (-> Caltech), R. Sensarma (-> JQI Maryland) Collaborations with experimental groups of I. Bloch (MPQ), T. Esslinger (ETH), J.Schmiedmayer (Vienna) Supported by NSF, DARPA OLE, AFOSR MURI, ARO MURI

keVMeVGeVTeVfeVpeVµeVmeVeV pKnKµKmKK neV room temperature LHC He N current experiments K How cold are ultracold atoms? first BEC of alkali atoms

Bose-Einstein condensation of weakly interacting atoms Scattering length is much smaller than characteristic interparticle distances. Interactions are weak Density cm -1 Typical distance between atoms 300 nm Typical scattering length 10 nm

New Era in Cold Atoms Research Focus on Systems with Strong Interactions Atoms in optical lattices Feshbach resonances Low dimensional systems Systems with long range dipolar interactions Rotating systems

Feshbach resonance Greiner et al., Nature (2003); Ketterle et al., (2003) Ketterle et al., Nature 435, (2005)

One dimensional systems Strongly interacting regime can be reached for low densities One dimensional systems in microtraps. Thywissen et al., Eur. J. Phys. D. (99); Hansel et al., Nature (01); Folman et al., Adv. At. Mol. Opt. Phys. (02) 1D confinement in optical potential Weiss et al., Science (05); Bloch et al., Esslinger et al.,

Atoms in optical lattices Theory: Jaksch et al. PRL (1998) Experiment: Kasevich et al., Science (2001); Greiner et al., Nature (2001); Phillips et al., J. Physics B (2002) Esslinger et al., PRL (2004); and many more …

Antiferromagnetic and superconducting Tc of the order of 100 K Atoms in optical lattice Antiferromagnetism and pairing at nano Kelvin temperatures Same microscopic model Quantum simulations with ultracold atoms

Strongly correated systems Atoms in optical latticesElectrons in Solids Simple metals Perturbation theory in Coulomb interaction applies. Band structure methods work Strongly Correlated Electron Systems Band structure methods fail. Novel phenomena in strongly correlated electron systems: Quantum magnetism, phase separation, unconventional superconductivity, high temperature superconductivity, fractionalization of electrons …

Strongly correlated systems of ultracold atoms should also be useful for applications in quantum information, high precision spectroscopy, metrology By studying strongly interacting systems of cold atoms we expect to get insights into the mysterious properties of novel quantum materials: Quantum Simulators BUT Strongly interacting systems of ultracold atoms : are NOT direct analogues of condensed matter systems These are independent physical systems with their own “personalities”, physical properties, and theoretical challenges

Cold atoms in optical lattices Bose Hubbard model. Superfluid to Mott transition Looking for Higgs particle in the Bose Hubbard model Quantum magnetism with ultracold atoms in optical lattices Low dimensional condensates Observing quasi-long range order in interference experiments Observation of prethermolization First lecture: experiments with ultracold bosons

Second lecture: Ultracold fermions Fermions in optical lattices. Fermi Hubbard model. Current state of experiments Lattice modulation experiments Doublon lifetimes Strongly interacting fermions in continuum. Stoner instability

Bose Hubbard model Ultracold Bose atoms in optical lattices

Bose Hubbard model tunneling of atoms between neighboring wellsrepulsion of atoms sitting in the same well U t In the presence of confining potential we also need to include Typically

Bose Hubbard model. Phase diagram M.P.A. Fisher et al., PRB (1989) Mottn=1 n=2 n=3 Superfluid Mott Weak lattice Superfluid phaseStrong lattice Mott insulator phase

Bose Hubbard model Hamiltonian eigenstates are Fock states 01 Set. Away from level crossings Mott states have a gap. Hence they should be stable to small tunneling.

Bose Hubbard Model. Phase diagram Particle-hole excitation Mott insulator phase Mottn=1 n=2 n=3 Superfluid Mott Tips of the Mott lobes z- number of nearest neighbors, n – filling factor

Gutzwiller variational wavefunction Normalization Kinetic energy z – number of nearest neighbors Interaction energy favors a fixed number of atoms per well. Kinetic energy favors a superposition of the number states.

Bose Hubbard Model. Phase diagram Mottn=1 n=2 n=3 Superfluid Mott Note that the Mott state only exists for integer filling factors. For even when atoms are localized, make a superfluid state.

Nature 415:39 (2002)

Optical lattice and parabolic potential Jaksch et al., PRL 81:3108 (1998) Parabolic potential acts as a “cut” through the phase diagram. Hence in a parabolic potential we find a “wedding cake” structure Mottn=1 n=2 n=3 Superfluid Mott

Quantum gas microscope Bakr et al., Science 2010 x y density

Nature 2010

The Higgs (amplitude) mode in a trapped 2D superfluid on a lattice Theory: David Pekker, Eugene Demler Experiments: Manuel Endres, Takeshi Fukuhara, Marc Cheneau, Peter Schauss, Christian Gross, Immanuel Bloch, Stefan Kuhr Sherson et. al. Nature 2010 Cold Atoms (Munich) Elementary Particles LHC)

Order parameter Phase (Goldstone) mode = gapless Bogoliubov mode Breaks U(1) symmetry Gapped amplitude mode (Higgs mode) Collective modes of strongly interacting superfluid bosons Figure from Bissbort et al. (2010)

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

Is there a Higgs mode in 2D ? Danger from scattering on phase modes In 2D: infrared divergence Different susceptibility has no divergence Higgs   S. Sachdev, Phys. Rev. B 59, (1999) W. Zwerger, Phys. Rev. Lett. 92, (2004) N. Lindner and A. Auerbach, Phys. Rev. B 81, (2010) Podolsky, Auerbach, Arovas, Phys. Rev. B 84, (2011) Higgs neutron scattering lattice modulation spectroscopy

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 Bissbort et al., PRL(2010)

Exciting the amplitude mode Absorbed energy

Mottn=1Mottn=1Mottn=1 Exciting the amplitude mode Manuel Endres, Immanuel Bloch and MPQ team

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

Similar to Landau-Lifshitz equations in magnetism Time dependent mean-field: Gutzwiller Threshold for absorption is captured very well Keep two states per site only

Plaquette Mean Field “Better Gutzwiller” Variational wave functions better captures local physics – better describes interactions between quasi-particles Equivalent to MFT on plaquettes

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

Comparison of experiments and Gutzwiller theories Experiment2x2 Clusters Key experimental facts: “gap” disappears at QCP wide band band spreads out deep in SF Single site Gutzwiller Captures gap Does not capture width Plaquette Gutzwiller Captures gap Captures most of the width

Beyond Gutzwiller: Scaling at low frequencies signature of Higgs/Goldstone mode coupling External drive couples vacuum to Higgs Higgs can be excited only virtually Higgs decays into a pair of Goldstone modes with conservation of energy Matrix element w 2 /w=w Density of states w Fermi’s golden rule: w 2 x w = w 3 vacuum 2 Goldstones Higgs w

Open question: observing discreet modes Breathing mode disappearing amplitude mode details at the QCP spectrum remains gapped due to trap

Higgs Drum Modes 1x1 calculation, 20 oscillations E abs rescaled so peak heights coincide

Quantum magnetism with ultracold atoms in optical lattices

t t Two component Bose mixture in optical lattice Two component Bose Hubbard model Example:. Mandel et al., Nature (2003) We consider two component Bose mixture in the n=1 Mott state with equal number of and atoms. We need to find spin arrangement in the ground state.

Quantum magnetism of bosons in optical lattices Duan et al., PRL (2003) Ferromagnetic Antiferromagnetic

In the regime of deep optical lattice we can treat tunneling as perturbation. We consider processes of the second order in t We can combine these processes into anisotropic Heisenberg model Two component Bose Hubbard model

Two component Bose mixture in optical lattice. Mean field theory + Quantum fluctuations 2 nd order line Hysteresis 1 st order Altman et al., NJP (2003)

Two component Bose Hubbard model + infinitely large U aa and U bb New feature: coexistence of checkerboard phase and superfluidity

Exchange Interactions in Solids antibonding bonding Kinetic energy dominates: antiferromagnetic state Coulomb energy dominates: ferromagnetic state

Questions: Detection of topological order Creation and manipulation of spin liquid states Detection of fractionalization, Abelian and non-Abelian anyons Melting spin liquids. Nature of the superfluid state Realization of spin liquid using cold atoms in an optical lattice Theory: Duan, Demler, Lukin PRL (03) H = - J x  i x  j x - J y  i y  j y - J z  i z  j z Kitaev model Annals of Physics (2006)

Superexchange interaction in experiments with double wells Theory: A.M. Rey et al., PRL 2008 Experiments: S. Trotzky et al., Science 2008

J J Use magnetic field gradient to prepare a stateObserve oscillations between and states Observation of superexchange in a double well potential Theory: A.M. Rey et al., PRL 2008 Experiments: S. Trotzky et al. Science 2008

Reversing the sign of exchange interaction Preparation and detection of Mott states of atoms in a double well potential

Comparison to the Hubbard model

Basic Hubbard model includes only local interaction Extended Hubbard model takes into account non-local interaction Beyond the basic Hubbard model

Probing low dimensional condensates with interference experiments Quasi long range order Prethermalization

Interference of independent condensates Experiments: Andrews et al., Science 275:637 (1997) Theory: Javanainen, Yoo, PRL 76:161 (1996) Cirac, Zoller, et al. PRA 54:R3714 (1996) Castin, Dalibard, PRA 55:4330 (1997) and many more

x z Time of flight Experiments with 2D Bose gas Hadzibabic, Dalibard et al., Nature 2006 Experiments with 1D Bose gas Hofferberth et al. Nat. Physics 2008

Interference of two independent condensates 1 2 r r+d d r’ Phase difference between clouds 1 and 2 is not well defined Assuming ballistic expansion Individual measurements show interference patterns They disappear after averaging over many shots

x1x1 d Amplitude of interference fringes, Interference of fluctuating condensates For identical condensates Instantaneous correlation function For independent condensates A fr is finite but Df is random x2x2 Polkovnikov et al., PNAS (2006); Gritsev et al., Nature Physics (2006)

Matter-wave interferometry phase, contrast FDF of phase and contrast

Matter-wave interferometry contrast phase, contrast FDF of phase and contrast phase Plot as circular statistics

Matter-wave interferometry: repeat many times Plot i>100 contrast i phase phase, contrast FDF of phase and contrast accumulate statistics Calculate average contrast

Fluctuations in 1d BEC Thermal fluctuations Thermally energy of the superflow velocity Quantum fluctuations

For impenetrable bosons and Interference between Luttinger liquids Luttinger liquid at T=0 K – Luttinger parameter Finite temperature Experiments: Hofferberth, Schumm, Schmiedmayer For non-interacting bosons and

Distribution function of fringe amplitudes for interference of fluctuating condensates L is a quantum operator. The measured value of will fluctuate from shot to shot. Higher moments reflect higher order correlation functions Gritsev, Altman, Demler, Polkovnikov, Nature Physics 2006 Imambekov, Gritsev, Demler, PRA (2007) We need the full distribution function of

Distribution function of interference fringe contrast Hofferberth et al., Nature Physics 2009 Comparison of theory and experiments: no free parameters Higher order correlation functions can be obtained Quantum fluctuations dominate : asymetric Gumbel distribution (low temp. T or short length L) Thermal fluctuations dominate: broad Poissonian distribution (high temp. T or long length L) Intermediate regime : double peak structure

Quantum impurity problem: interacting one dimensional electrons scattered on an impurity Conformal field theories with negative central charges: 2D quantum gravity, non-intersecting loop model, growth of random fractal stochastic interface, high energy limit of multicolor QCD, … Interference between interacting 1d Bose liquids. Distribution function of the interference amplitude Distribution function of Yang-Lee singularity 2D quantum gravity, non-intersecting loops

Fringe visibility and statistics of random surfaces Mapping between fringe visibility and the problem of surface roughness for fluctuating random surfaces. Relation to 1/f Noise and Extreme Value Statistics Roughness Distribution function of

Interference of two dimensional condensates LyLy LxLx LxLx Experiments: Hadzibabic et al. Nature (2006) Probe beam parallel to the plane of the condensates Gati et al., PRL (2006)

Interference of two dimensional condensates. Quasi long range order and the BKT transition LyLy LxLx Below BKT transitionAbove BKT transition

x z Time of flight low temperaturehigher temperature Typical interference patterns Experiments with 2D Bose gas Hadzibabic, Dalibard et al., Nature 441:1118 (2006)

integration over x axis DxDx z z integration over x axis z x integration distance D x (pixels) Contrast after integration middle T low T high T integration over x axis z Experiments with 2D Bose gas Hadzibabic et al., Nature 441:1118 (2006)

fit by: integration distance D x Integrated contrast low T middle T high T if g 1 (r) decays exponentially with : if g 1 (r) decays algebraically or exponentially with a large : Exponent  central contrast high Tlow T “Sudden” jump!? Experiments with 2D Bose gas Hadzibabic et al., Nature 441:1118 (2006)

Experiments with 2D Bose gas. Proliferation of thermal vortices Hadzibabic et al., Nature (2006) Fraction of images showing at least one dislocation Exponent  central contrast The onset of proliferation coincides with  shifting to 0.5! 0 10% 20% 30% central contrast high T low T

Quantum dynamics of split one dimensional condensates Prethermalization Theory: Takuya Kitagawa et al., PRL (2010) Experiments: D. Smith, J. Schmiedmayer, et al. New J. Phys. (2011) arXiv:

Relaxation to equilibrium Bolzmann equation Thermalization: an isolated interacting systems approaches thermal equilibrium at long times (typically at microscopic timescales). All memory about the initial conditions except energy is lost. U. Schneider et al., arXiv:

Prethermalization We observe irreversibility and approximate thermalization. At large time the system approaches stationary solution in the vicinity of, but not identical to, thermal equilibrium. The ensemble therefore retains some memory beyond the conserved total energy…This holds for interacting systems and in the large volume limit. Prethermalization in ultracold atoms, theory: Eckstein et al. (2009); Moeckel et al. (2010), L. Mathey et al. (2010), R. Barnett et al.(2010) Heavy ions collisions QCD

Measurements of dynamics of split condensate

Theoretical analysis of dephasing Luttinger liquid model

Luttinger liquid model of phase dynamics

For each k-mode we have simple harmonic oscillators

Segment size is smaller than the fluctuation lengthscale At long times the difference between the two regime occurs for Phase diffusion vs Contrast Decay Segment size is longer than the fluctuation lengthscale

Length dependent phase dynamics “Short segments” = phase diffusion “Long segments” = contrast decay 110 µm 61µm 41µm 30 µm 20µm 10µm 15 ms

Energy distribution Energy stored in each mode initially Equipartition of energy For 2d also pointed out by Mathey, Polkovnikov in PRA (2010) At t=0 system is in a squeezed state with large number fluctuations The system should look thermal like after different modes dephase. Effective temperature is not related to the physical temperature

Do we have thermal-like distributions at longer times? Comparison of experiments and LL analysis

Prethermalization Interference contrast is described by thermal distributions but at temperature much lower than the initial temperature

Testing Prethermalization

Cold atoms in optical lattices Bose Hubbard model. Superfluid to Mott transition Looking for Higgs particle in the Bose Hubbard model Quantum magnetism with ultracold atoms in optical lattices Low dimensional condensates Observing quasi-long range order in interference experiments Observation of prethermolization First lecture: experiments with ultracold bosons

Beyond Gutzwiller: Scaling at low frequencies signature of Higgs/Goldstone mode coupling Excite virtual Higgs excitation Virtual Higgs decays into a pair of Goldstone excitations Matrix element of Higgs to Goldstone coupling scales as w 2 Phase space scales as 1/w Fermi’s golden rule: (w 2 ) 2 x (1/w) = w 3