1. Strongly correlated many-body systems: from electronic materials to ultracold atoms to photons Eugene Demler Harvard University Thanks to: E. Altman, I. Bloch, A. Burkov, H.P. Büchler, I. Cirac, D. Chang, R. Cherng, V. Gritsev, B. Halperin, W. Hofstetter, A. Imambekov, M. Lukin, G. Morigi, A. Polkovnikov, G. Pupillo, G. Refael, A.M. Rey, O. Romero-Isart, J. Schmiedmayer, R. Sensarma, D.W. Wang, P. Zoller, and many others

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

3. “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

4. Strongly correlated electron systems Quantum Hall systems kinetic energy suppressed by magnetic field One dimensional electron systems non-perturbative effects of interactions in 1d High temperature superconductors, Heavy fermion materials, Organic conductors and superconductors many puzzling non-Fermi liquid properties

5. Bose-Einstein condensation of weakly interacting atoms Scattering length is much smaller than characteristic interparticle distances. Interactions are weak

6. Strongly correlated systems of cold atoms Low dimensional systems Optical lattices Feshbach resonances

7. Linear geometrical optics

8. 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., arXive:0712.1817 Strongly correlated systems of photons

9. We understand well: many-body systems of non-interacting or weakly interacting particles. For example, electron systems in semiconductors and simple metals, When the interaction energy is smaller than the kinetic energy, perturbation theory works well We do not understand: many-body systems with strong interactions and correlations. For example, electron systems in novel materials such as high temperature superconductors. When the interaction energy is comparable or larger than the kinetic energy, perturbation theory breaks down. Many surprising new phenomena occur, including unconventional superconductivity, magnetism, fractionalization of excitations

10. Ultracold atoms have energy scales of 10 -6 K, compared to 10 4 K for electron systems We will also get new systems useful for applications in quantum information and communications, high precision spectroscopy, metrology By engineering and studying strongly interacting systems of cold atoms we should get insights into the mysterious properties of novel quantum materials

11. Strongly interacting systems of ultracold atoms and photons: NOT the analogue simulators These are independent physical systems with their own “personalities”, physical properties, and theoretical challenges

12. Focus of these lectures: challenges of new strongly correlated systems Part I Detection and characterization of many body states Quantum noise analysis and interference experiments Part II New challenges in quantum many-body theory: non-equilibrium coherent dynamics

13. Quantum noise studies of ultracold atoms Part I

14. Introduction. Historical review Quantum noise analysis of the time of flight experiments with utlracold atoms (HBT correlations and beyond) Quantum noise in interference experiments with independent condensates Quantum noise analysis of spin systems Outline of part I

15. Quantum noise Classical measurement: collapse of the wavefunction into eigenstates of x Histogram of measurements of x

16. Probabilistic nature of quantum mechanics Bohr-Einstein debate on spooky action at a distance Measuring spin of a particle in the left detector instantaneously determines its value in the right detector Einstein-Podolsky-Rosen experiment

17. Aspect’s experiments: tests of Bell’s inequalities S Correlation function Classical theories with hidden variable require Quantum mechanics predicts B=2.7 for the appropriate choice of q ‘s and the state Experimentally measured value B=2.697. Phys. Rev. Let. 49:92 (1982) + - + - 12 q1q1 q2q2 S

18. Hanburry-Brown-Twiss experiments Classical theory of the second order coherence Measurements of the angular diameter of Sirius Proc. Roy. Soc. (London), A, 248, pp. 222-237 Hanbury Brown and Twiss, Proc. Roy. Soc. (London), A, 242, pp. 300-324

19. Shot noise in electron transport e-e- e-e- When shot noise dominates over thermal noise Spectral density of the current noise Proposed by Schottky to measure the electron charge in 1918 Related to variance of transmitted charge Poisson process of independent transmission of electrons

20. Shot noise in electron transport Current noise for tunneling across a Hall bar on the 1/3 plateau of FQE Etien et al. PRL 79:2526 (1997) see also Heiblum et al. Nature (1997)

21. Quantum noise analysis of the time of flight experiments with utlracold atoms (HBT correlations and beyond) Theory: Altman, Demler, Lukin, PRA 70:13603 (2004) Experiments: Folling et al., Nature 434:481 (2005); Greiner et al., PRL 94:110401 (2005); Tom et al. Nature 444:733 (2006); see also Hadzibabic et al., PRL 93:180403 (2004) Spielman et al., PRL 98:80404 (2007); Guarrera et al., preprint (2007)

22. 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); Ketterle et al., PRL (2006)

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

24. Superfluid to insulator transition in an optical lattice M. Greiner et al., Nature 415 (2002) t/U Superfluid Mott insulator

25. Why study ultracold atoms in optical lattices?

26. t U t Fermionic atoms in optical lattices Experiments with fermions in optical lattice, Kohl et al., PRL 2005

27. Antiferromagnetic and superconducting Tc of the order of 100 K Atoms in optical lattice Antiferromagnetism and pairing at sub-micro Kelvin temperatures Same microscopic model

28. Positive U Hubbard model Possible phase diagram. Scalapino, Phys. Rep. 250:329 (1995) Antiferromagnetic insulator D-wave superconductor

29. Atoms in optical lattice Same microscopic model Quantum simulations of strongly correlated electron systems using ultracold atoms Detection?

30. Quantum noise analysis as a probe of many-body states of ultracold atoms

31. Time of flight experiments Quantum noise interferometry of atoms in an optical lattice Second order coherence

32. Second order coherence in the insulating state of bosons. Hanburry-Brown-Twiss experiment Experiment: Folling et al., Nature 434:481 (2005)

33. Hanburry-Brown-Twiss stellar interferometer

34. Hanburry-Brown-Twiss interferometer

35. Quantum theory of HBT experiments For bosons For fermions Glauber, Quantum Optics and Electronics (1965) HBT experiments with matter Experiments with 4He, 3He Westbrook et al., Nature (2007) Experiments with neutrons Ianuzzi et al., Phys Rev Lett (2006) Experiments with electrons Kiesel et al., Nature (2002) Experiments with ultracold atoms Bloch et al., Nature (2005,2006)

36. Second order coherence in the insulating state of bosons. Hanburry-Brown-Twiss experiment Experiment: Folling et al., Nature 434:481 (2005)

37. Second order coherence in the insulating state of bosons Bosons at quasimomentum expand as plane waves with wavevectors First order coherence : Oscillations in density disappear after summing over Second order coherence : Correlation function acquires oscillations at reciprocal lattice vectors

38. Second order coherence in the insulating state of bosons. Hanburry-Brown-Twiss experiment Experiment: Folling et al., Nature 434:481 (2005)

39. Interference of an array of independent condensates Hadzibabic et al., PRL 93:180403 (2004) Smooth structure is a result of finite experimental resolution (filtering)

40. Quantum theory of HBT experiments For bosons For fermions

41. Second order coherence in the insulating state of fermions. Hanburry-Brown-Twiss experiment Experiment: Tom et al. Nature 444:733 (2006)

42. How to detect antiferromagnetism

43. Probing spin order in optical lattices Correlation Function Measurements Extra Bragg peaks appear in the second order correlation function in the AF phase

44. How to detect fermion pairing Quantum noise analysis of TOF images: beyond HBT interference

45. Second order interference from the BCS superfluid n(r) n(r’) n(k) k BCS BEC k F

46. Momentum correlations in paired fermions Greiner et al., PRL 94:110401 (2005)

47. Fermion pairing in an optical lattice Second Order Interference In the TOF images Normal State Superfluid State measures the Cooper pair wavefunction One can identify unconventional pairing

48. How to see a “cat” state in the collapse and revival experiments Quantum noise analysis of TOF images: beyond HBT interference

49. Collapse and revival experiments with bosons in an optical lattice Coherent state in each well at t=0 Increase the height of the optical lattice abruptly Time evolution within each well Initial superfluid state Individual wells isolated

50. Collapse and revival experiments with bosons in an optical lattice At revival times t r =h/U, 2h/U, … all number states are in phase again The collapse occurs due to the loss of coherence between different number states collapse t c =h/NU revival t r =h/U time coherence

51. Collapse and revival experiments with bosons in an optical lattice Greiner et al., Nature 419:51 (2002) Dynamical evolution of the interference pattern after jumping the optical lattice potential

52. Collapse and revival experiments with bosons in an optical lattice Greiner et al., Nature 419:51 (2002) Quantum dynamics of a coherent states Cat state at t=t r /2

53. How to see a “cat” state in collapse and revival experiments pairing-like correlations no first order coherence Properties of the “cat” state: In the time of flight experiments this should lead to correlations between and Define correlation function (overlap original and flipped images) Romero-Isart et al., unpublished

54. How to see a “cat” state in collapse and revival experiments Exact diagonalization of the 1d lattice system. U/t=3, N=2

55. Collapse and revival experiments with bosons in an optical lattice Greiner et al., Nature 419:51 (2002) Dynamical evolution of the interference pattern after jumping the optical lattice potential Perfect correlations “hiding” in the image

56. Quantum noise in interference experiments with independent condensates

57. 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

58. Nature 4877:255 (1963)

59. x z Time of flight Experiments with 2D Bose gas Hadzibabic, Dalibard et al., Nature 441:1118 (2006) Experiments with 1D Bose gas S. Hofferberth et al. arXiv0710.1575

60. Interference of two independent condensates 1 2 r r+d d r’ Clouds 1 and 2 do not have a well defined phase difference. However each individual measurement shows an interference pattern

61. 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, Altman, Demler, PNAS 103:6125(2006)

62. Fluctuations in 1d BEC Thermal fluctuations Thermally energy of the superflow velocity Quantum fluctuations For a review see Shlyapnikov et al., J. Phys. IV France 116, 3-44 (2004)

63. 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

64. 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, cond-mat/0612011 We need the full distribution function of

65. Distribution function of interference fringe contrast Experiments: Hofferberth et al., arXiv0710.1575 Theory: Imambekov et al., cond-mat/0612011 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

66. 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, cond-mat/0612011 We need the full distribution function of

67. L Change to periodic boundary conditions (long condensates) Explicit expressions for are available but cumbersome Fendley, Lesage, Saleur, J. Stat. Phys. 79:799 (1995) Calculating distribution function of interference fringe amplitudes Method I: mapping to quantum impurity problem

68. Impurity in a Luttinger liquid Expansion of the partition function in powers of g Partition function of the impurity contains correlation functions taken at the same point and at different times. Moments of interference experiments come from correlations functions taken at the same time but in different points. Euclidean invariance ensures that the two are the same

69. Relation between quantum impurity problem and interference of fluctuating condensates Distribution function of fringe amplitudes Distribution function can be reconstructed from using completeness relations for the Bessel functions Normalized amplitude of interference fringes Relation to the impurity partition function

70. is related to a Schroedinger equation Dorey, Tateo, J.Phys. A. Math. Gen. 32:L419 (1999) Bazhanov, Lukyanov, Zamolodchikov, J. Stat. Phys. 102:567 (2001) Spectral determinant Bethe ansatz solution for a quantum impurity can be obtained from the Bethe ansatz following Zamolodchikov, Phys. Lett. B 253:391 (91); Fendley, et al., J. Stat. Phys. 79:799 (95) Making analytic continuation is possible but cumbersome Interference amplitude and spectral determinant

71. Interference of 1d condensates at T=0. Distribution function of the fringe contrast Narrow distribution for. Approaches Gumbel distribution. Width Wide Poissonian distribution for

72. When K>1, is related to Q operators of CFT with c<0. This includes 2D quantum gravity, non- intersecting loop model on 2D lattice, growth of random fractal stochastic interface, high energy limit of multicolor QCD, … Yang-Lee singularity 2D quantum gravity, non-intersecting loops on 2D lattice correspond to vacuum eigenvalues of Q operators of CFT Bazhanov, Lukyanov, Zamolodchikov, Comm. Math. Phys.1996, 1997, 1999 From interference amplitudes to conformal field theories