1. Eugene Demler Harvard University Strongly correlated many-body systems: from electronic materials to ultracold atoms

2. “Conventional” solid state materials Bloch theorem for non-interacting electrons in a periodic potential

3. B V H I d First semiconductor transistor EFEF Metals EFEF Insulators and Semiconductors Consequences of the Bloch theorem

4. “Conventional” solid state materials Electron-phonon and electron-electron interactions are irrelevant at low temperatures kxkx kyky kFkF Landau Fermi liquid theory: when frequency and temperature are smaller than E F electron systems are equivalent to systems of non-interacting fermions Ag

5. Strongly correlated electron systems Quantum Hall systems kinetic energy suppressed by magnetic field Heavy fermion materials many puzzling non-Fermi liquid properties High temperature superconductors Unusual “normal” state, Controversial mechanism of superconductivity, Several competing orders UCu 3.5 Pd 1.5 CeCu 2 Si 2

6. What is the connection between strongly correlated electron systems and ultracold atoms?

7. Bose-Einstein condensation of weakly interacting atoms Scattering length is much smaller than characteristic interparticle distances. Interactions are weak Density 10 13 cm -1 Typical distance between atoms 300 nm Typical scattering length 10 nm

8. 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

9. Feshbach resonance and fermionic condensates Greiner et al., Nature (2003); Ketterle et al., (2003) Ketterle et al., Nature 435, 1047-1051 (2005)

10. 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.,

11. 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 …

12. Strongly correlated 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 …

13. 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

14. New Phenomena in quantum many-body systems of ultracold atoms Long intrinsic time scales - Interaction energy and bandwidth ~ 1kHz - System parameters can be changed over this time scale Decoupling from external environment - Long coherence times Can achieve highly non equilibrium quantum many-body states New detection methods Interference, higher order correlations Nonequilibrium dynamics

15. Dynamics of many-body quantum systems Big Bang and Inflation Cosmic microwave background radiation. Manifestation of quantum fluctuations during inflation Heavy Ion collisions at RHIC Signatures of quark-gluon plasma?

16. Paradigms for equilibrium states of many-body systems Broken symmetry phases (magnetism, pairing, etc.) Order parameters RG flows and fixed points (e.g. Landau Fermi liquids) Topological states Effective low energy theories Classical and quantum critical points Scaling Do we get any collective (universal?) phenomena in the case of nonequilibrium dynamics? Theoretical work on many-body nonequilibrium dynamics of ultracold atoms: E. Altman, J.S. Caux, A. Cazalilla, K. Collath, A.J. Daley, T. Giamarchi, V. Gritsev, T.L. Ho, A. Iucci, L. Levitov, M. Lewenstein, A.Muramatsu, A. Polkovnikov, S. Sachdev, P. Zoller and many more

17. Strongly correlated many-body systems of photons

18. Linear geometrical optics

19. Strong optical nonlinearities in nanoscale surface plasmons Akimov et al., Nature (2007) Strongly interacting polaritons in coupled arrays of cavities M. Hartmann et al., Nature Physics (2006) Crystallization (fermionization) of photons in one dimensional optical waveguides D. Chang et al., Nature Physics (2008) Strongly correlated systems of photons

20. Outline of these lectures Introduction. Magnetic and optical trapping of ultracold atoms. Cold atoms in optical lattices. Bose Hubbard model. Equilibrium and dynamics Bose mixtures in optical lattices Quantum magnetism of ultracold atoms. Detection of many-body phases using noise correlations Experiments with low dimensional systems Interference experiments. Analysis of high order correlations Fermions in optical lattices Magnetism and pairing in systems with repulsive interactions. Current experiments: paramagnetic Mott state, nonequilibrium dynamics. Dynamics near Fesbach resonance. Competition of Stoner instability and pairing Detection of many-body phases Nonequilibrium dynamics Emphasis of these lectures:

21. Ultracold atoms

22. Most common bosonic atoms: alkali 87 Rb and 23 Na Most common fermionic atoms: alkali 40 K and 6 Li Ultracold atoms Other systems: BEC of 133 Cs (Innsbruck) BEC of 52 Cr (Stuttgart) BEC of 84 Sr (Innsbruck), and 88 Sr (Boulder) BEC of 168 Yb, 170 Yb, 172 Yb, 174 Yb, 176 Yb (Kyoto) Degenerate fermions 171 Yb, 173 Yb (Kyoto), 87 Sr (Boulder)

23. Single valence electron in the s-orbital and Nuclear spin Magnetic properties of individual alkali atoms Zero field splitting between and states For 23 Na A HFS = 1.8 GHz and for 87 Rb A HFS = 6.8 GHz Total angular momentum (hyperfine spin) Hyperfine coupling mixes nuclear and electron spins

24. Magnetic properties of individual alkali atoms Effect of magnetic field comes from electron spin g s =2 and m B =1.4 MHz/G When fields are not too large one can use (assuming field along z) The last term describes quadratic Zeeman effect q=h 390 Hz/G 2

25. Magnetic trapping of alkali atoms Magnetic trapping of neutral atoms is due to the Zeeman effect. The energy of an atomic state depends on the magnetic field. In an inhomogeneous field an atom experiences a spatially varying potential. Example: Potential: Magnetic trapping is limited by the requirement that the trapped atoms remain in weak field seeking states. For 23 Na and 87 Rb there are three states

26. Optical trapping of alkali atoms Based on AC Stark effect - polarizability Typically optical frequencies. Potential: Dipolar moment induced by the electric field Far-off-resonant optical trap confines atoms regardless of their hyperfine state

27. Ultracold atoms in optical lattices. Band structure. Semiclasical dynamics.

28. Optical lattice The simplest possible periodic optical potential is formed by overlapping two counter-propagating beams. This results in a standing wave Averaging over fast optical oscillations (AC Stark effect) gives Combining three perpendicular sets of standing waves we get a simple cubic lattice This potential allows separation of variables

29. Optical lattice For each coordinate we have Matthieu equation Eigenvalues and eigenfunctions are known In the regime of deep lattice we get the tight-binding model - bandgap - recoil energy Lowest band

30. Optical lattice Effective Hamiltonian for non-interacting atoms in the lowest Bloch band nearest neighbors

31. Band structure

34. State dependent optical lattices How to use selection rules for optical transitions to make different lattice potentials for different internal states. Fine structure for 23 Na and 87 Rb Analogously state will only be affected by, which gives the potential. Decomposing hyperfine states we find The right circularly polarized light couples to two excited levels P 1/2 and P 3/2. AC Stark effects have opposite signs and cancel each other for the appropriate frequency. At this frequency AC Stark effect for the state comes only from polarized light and gives the potential.

35. PRL 91:10407 (2003) State dependent lattice

37. Atoms in optical lattices. Bose Hubbard model

38. 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

39. Bose Hubbard model. Phase diagram M.P.A. Fisher et al., PRB (1989) 2 1 0 Mottn=1 n=2 n=3 Superfluid Mott Weak lattice Superfluid phaseStrong lattice Mott insulator phase

40. 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.

41. Bose Hubbard Model. Phase diagram Particle-hole excitation Mott insulator phase 2 1 0 Mottn=1 n=2 n=3 Superfluid Mott Tips of the Mott lobes z- number of nearest neighbors, n – filling factor

42. 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.

43. Gutzwiller variational wavefunction Transition takes place when coefficient before becomes negative. For large n this corresponds to Take the middle of the Mott plateau Expand to order Example: stability of the Mott state with n atoms per site

44. Bose Hubbard Model. Phase diagram 2 1 0 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.

45. Bose Hubbard model Experiments with 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); many more …

46. Nature 415:39 (2002)

47. 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. 2 1 0 Mottn=1 n=2 n=3 Superfluid Mott

48. Nature 2009

49. arXiv:1006.3799

50. Nonequilibrium dynamics of Bose Hubbard model

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

52. Moving condensate in an optical lattice. Dynamical instability v Theory: Niu et al. PRA (01), Smerzi et al. PRL (02) Experiment: Fallani et al. PRL (04)

53. Linear stability analysis: States with p> p /2 are unstable Classical limit of the Hubbard model. Discreet Gross-Pitaevskii equation Current carrying states r Dynamical instability Amplification of density fluctuations unstable

54. Dynamical instability. Gutzwiller approximation Wavefunction Time evolution Phase diagram. Integer filling We look for stability against small fluctuations Altman et al., PRL 95:20402 (2005)

55. The first instability develops near the edges, where N=1 U=0.01 t J=1/4 Gutzwiller ansatz simulations (2D) Optical lattice and parabolic trap. Gutzwiller approximation

56. PRL (2007)

57. Beyond semiclassical equations. Current decay by tunneling phase site index j Current carrying states are metastable. They can decay by thermal or quantum tunneling Thermal activation Quantum tunneling site index j Thermal phase slips observed by DeMarco et al., Nature (2008) Quantum phase slips observed by Ketterle et al., PRL (2007) Polkovnikov et al., Phys. Rev. A (2005)

58. Engineering magnetic systems using cold atoms in an optical lattice

59. 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.

60. Quantum magnetism of bosons in optical lattices Duan, Demler, Lukin, PRL (2003) Ferromagnetic Antiferromagnetic

61. 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

62. 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)

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

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

65. 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)

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

67. 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

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

69. Comparison to the Hubbard model

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

72. Summary of lecture I Introduction. Systems of ultracold atoms. Cold atoms in optical lattices. Bose Hubbard model. Equilibrium and dynamics Bose mixtures in optical lattices. Quantum magnetism of ultracold atoms.

73. Outline of future lectures Introduction. Systems of ultracold atoms. Cold atoms in optical lattices. Bose Hubbard model. Equilibrium and dynamics Bose mixtures in optical lattices. Quantum magnetism of ultracold atoms. Detection of many-body phases using noise correlations Experiments with low dimensional systems Interference experiments. Analysis of high order correlations Fermions in optical lattices Dynamics near Fesbach resonance. Competition of Stoner instability and pairing