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

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 wotk 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 and photons: 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

15. Strongly correlated many-body systems of photons

16. Linear geometrical optics

17. 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

18. Outline of these lectures Introduction. Systems of ultracold atoms. Bogoliubov theory. Spinor condensates. Cold atoms in optical lattices. Bose Hubbard model and extensions Bose mixtures in optical lattices Quantum magnetism of ultracold atoms. Current experiments: observation of superexchange Fermions in optical lattices Magnetism and pairing in systems with repulsive interactions. Current experiments: Mott state Detection of many-body phases using noise correlations Experiments with low dimensional systems Interference experiments. Analysis of high order correlations Non-equilibrium dynamics Emphasis of these lectures: Detection of many-body phases Dynamics

19. Ultracold atoms

20. 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 (e.g. Grimm et al.) BEC of 52 Cr (Pfau et al.) BEC of 84 Sr (e.g. Grimm et al.), 87 Sr and 88 Sr (e.g. Ye et al.) BEC of 168 Yb, 170 Yb, 172 Yb, 174 Yb, 176 Yb Quantum degenerate fermions 171 Yb 173 Yb (Takahashi et al.)

21. 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

22. 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

23. 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

24. 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

25. Bogoliubov theory of weakly interacting BEC. Collective modes

26. BEC of spinless bosons. Bogoliubov theory We consider a uniform system first For non-interacting atoms at T=0 all atoms are in k=0 state. Mean field equations Minimizing with respect to N 0 we find - boson annihilation operator at momentum p, - strength of contact s-wave interaction - volume of the system - chemical potential

27. BEC of spinless bosons. Bogoliubov theory We now expand around the nointeracting solution for small U 0. From the definition of Bose operators When N 0 >>1 we can treat b 0 as a c-number and replace by - means that p,-p pairs should be counted only once n 0 = N 0 /V - density and m = n 0 U 0 Mean-field Hamiltonian

28. Bogoliubov transformation Bosonic commutation relations are preserved when

29. Bogoliubov transformation Mean-field Hamiltonian becomes

30. Bogoliubov transformation Cancellation of non-diagonal terms requires To satisfy take Solution of these equations Mean-field Hamiltonian

31. Bogoliubov modes Dispersion of collective modes Define healing length from Long wavelength limit,, sound dispersion Short wavelength limit,, free particles Sound velocity

32. Probing the dispersion of BEC by off-resonant light scattering For details see cond-mat/0005001 Treat optical field as classical Excitation rate out of the ground state |g> Dynamical structure factor For small q we find

33. PRL 83:2876 (1999)

34. Ω Ω PRL 88:60402 (2002)

36. Rev. Mod. Phys. 77:187 (2005)

38. Gross-Pitaevskii equation

39. Hamiltonian of interacting bosons Commutation relations Equations of motion This is operator equation. We can take classical limit by assuming that all atoms condense into the same state. The last equation becomes an equation on the wavefunction

40. Gross-Pitaevskii equation Analysis of fluctuations on top of the mean-field GP equations leads to the Bogoliubov modes

41. Two-component mixtures

42. Two-component Bose mixture Consider mean-field (all particles in the condensate) Repulsive interactions. Miscible and immiscible regimes System with a finite density of both species is unstable to phase separation

43. What happens if we prepare a mixture of two condensates in the immiscible regime? Two component GP equation Assume equal densities Analyze fluctuations Bose mixture: dynamics of phase separation

44. Define density fluctuation phase fluctuation Equations of motion: charge conservation and Josephson relation Equation on collective modes

45. Bose mixture: dynamics of phase separation Here When we get imaginary frequencies Most unstable mode Imaginary frequencies indicate exponential growth of fluctuations, i.e. instability. The most unstable mode sets the length for pattern formation. Note that when. This is required by spin conservation.

46. PRL 82:2228 (1999)

48. Spinor condensates F=1

49. Spinor condensates. F=1 Three component order parameter: m F =-1,0,+1 Contact interaction depends on relative spin orientation When g 2 >0 interaction is antiferromagnetic. Example 23 Na When g 2 <0 interaction is antiferromagnetic. Example 87 Rb

50. F=1 spinor condensates. Hamiltonian - spin operators for F=1 Total F z is conserved so linear Zeeman term should (usually) be understood as Lagrange multiplier that controls F z. Quadratic Zeeman effect causes the energy of m F =0 state to be lower than the energy of m F =-1,+1 states. The antiferromagnetic interaction (g 2 >0) favors the nematic (polar) state (m F =0 and its rotations). The ferromagnetic interaction (g 2 <0) favors spin polarized state (m F =+1) and its rotations).

51. Phase diagram of F=1 spinor condensates g 2 >0 Antiferromagnetic g 2 <0 Ferromagnetic Shaded region: mixture of all three states. There is XY component of the spin. Shaded region: mixture of m F =-1,+1 states. This state lowers interaction energy.

52. Nature 396:345 (1998) Using magnetic field gradient to explore the phase diagram of F=1 atoms with AF interactions 23 Na

53. Nature 443:312 (2006)

55. Magnetic dipolar interactions in ultracold atoms

56. Magnetic dipolar interactions in spinor condensates Comparison of contact and dipolar interactions. Typical value a=100a B q For 87 Rb m = m B and e =0.007 For 52 Cr m =6 m B and e =0.16 Bose condensation of 52 Cr. T. Pfau et al. (2005) Review: Menotti et al., arXiv 0711.3422

57. Magnetic dipolar interactions in spinor condensates Interaction of F=1 atoms Ferromagnetic Interactions for 87 Rb Spin-depenent part of the interaction is small. Dipolar interaction may be important (D. Stamper-Kurn) a 2 -a 0 = -1.07 a B A. Widera, I. Bloch et al., New J. Phys. 8:152 (2006)

58. Spontaneously modulated textures in spinor condensates Fourier spectrum of the fragmented condensate Vengalattore et al. PRL (2008)

59. Energy scales: importance of Larmor precession S-wave Scattering Spin independent (g 0 n = kHz) Spin dependent (g s n = 10 Hz) Dipolar Interaction Anisotropic (g d n=10 Hz) Long-ranged Magnetic Field Larmor Precession (100 kHz) Quadratic Zeeman (0-20 Hz) B F Reduced Dimensionality Quasi-2D geometry

60. Stability of systems with static dipolar interactions Ferromagnetic configuration is robust against small perturbations. Any rotation of the spins conflicts with the “head to tail” arrangement Large fluctuation required to reach a lower energy configuration

61. XY components of the spins can lower the energy using modulation along z. Z components of the spins can lower the energy using modulation along x X Dipolar interaction averaged after precession “Head to tail” order of the transverse spin components is violated by precession. Only need to check whether spins are parallel Strong instabilities of systems with dipolar interactions after averaging over precession X

62. Instabilities of F=1 Rb (ferromagnetic) condensate due to dipolar interactions Theory: unstable modes in the regime corresponding to Berkeley experiments. Cherng, Demler, PRL (2009) Experiments. Vengalattore et al. PRL (2008)

63. Berkeley Experiments: checkerboard phase

64. Instabilities of magnetic dipolar interactions: general analysis a – angle between magnetic field and normal to the plane d n – layer thickness q measures the strength of quadratic Zeeman effect

65. Instabilities of collective modes

66. Magnetoroton softening Q measures the strength of quadratic Zeeman effect

67. 87 Rb condensate: magnetic supersolid ?

68. Phase diagram of 4He Possible supersolid phase in 4 He A.F. Andreev and I.M. Lifshits (1969): Melting of vacancies in a crystal due to strong quantum fluctuations. Also G. Chester (1970); A.J. Leggett (1970) T. Schneider and C.P. Enz (1971). Formation of the supersolid phase due to softening of roton excitations

69. Resonant period as a function of T

70. Interlayer coherence in bilayer quantum Hall systems at n=1 Hartree-Fock predicts roton softening and transition into a state with both interlayer coherence and stripe order. Transport experiments suggest first order transition into a compressible state. L. Brey and H. Fertig (2000) Eisenstein, Boebinger et al. (1994)

71. Low energy effective model for ferromagnetic condensates Mass superflow current caused by spin texture Topological spin winding (Pontryagin index) Magnetic dipolar interaction. Favors spin skyrmions Requires net zero spin winding

72. Magnetic crystal phases in F=1 87 Rb ferromagnetic condensates Experimental constraints: 1.Dihedral symmetry of the magnetic crystal 2.Rectangular lattice 3.Non-planar spin orientation Magnetic crystal lattices optimized within each class Optimal spin configuration. Cherng, Demler, unpublished

73. Spin correlation functions for spin components parallel and perpendicular to the magnetic field Magnetic crystal phases in F=1 87 Rb ferromagnetic condensates

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

75. 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

76. 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

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

78. Band structure

82. Semiclassical dynamics in the lattice 1.Band index is constant 2. 3. Bloch oscillations Consider a uniform and constant force

83. PRL 76:4508 (1996)

85. 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 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. Analogously state will only be affected by, which gives the potential. Decomposing hyperfine states we find

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