1. Outline of these 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 Magnetism and pairing in systems with repulsive interactions. Current experiments: paramgnetic Mott state, nonequilibrium dynamics. Dynamics near Fesbach resonance. Competition of Stoner instability and pairing

2. Learning about order from noise Quantum noise studies of ultracold atoms

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

4. 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

5. 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

6. 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

7. 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)

8. 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

9. 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)

10. Quantum noise analysis of time-of-flight experiments with atoms in optical lattices: Hanburry-Brown-Twiss experiments and beyond Theory: Altman et al., PRA (2004) Experiment: Folling et al., Nature (2005); Spielman et al., PRL (2007); Tom et al. Nature (2006)

11. Time of flight experiments Quantum noise interferometry of atoms in an optical lattice Second order coherence Cloud after expansion Cloud before expansion

12. Experiment: Folling et al., Nature (2005) Quantum noise analysis of time-of-flight experiments with atoms in optical lattices

13. Hanburry-Brown-Twiss stellar interferometer

14. 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

15. Second order correlations as Hanburry-Brown-Twiss effect Bosons/Fermions

16. Second order coherence in the insulating state of fermions. Experiment: Tom et al. Nature (2006)

17. Second order correlations as Hanburry-Brown-Twiss effect Bosons/Fermions Tom et al. Nature (2006)Folling et al., Nature (2005)

18. Probing spin order in optical lattices Correlation function measurements after TOF expansion. Extra Bragg peaks appear in the second order correlation function in the AF phase. This reflects doubling of the unit cell by magnetic order.

19. Interference experiments with cold atoms Probing fluctuations in low dimensional systems

20. 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

21. 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

22. 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

23. 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)

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

25. 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

26. 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

27. 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

28. 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

29. 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

30. 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)

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

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

33. integration over x axis DxDx z z integration over x axis z x integration distance D x (pixels) Contrast after integration 0.4 0.2 0 0 102030 middle T low T high T integration over x axis z Experiments with 2D Bose gas Hadzibabic et al., Nature 441:1118 (2006)

34. fit by: integration distance D x Integrated contrast 0.4 0.2 0 0102030 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 0.5 00.10.20.3 0.4 0.3 high Tlow T “Sudden” jump!? Experiments with 2D Bose gas Hadzibabic et al., Nature 441:1118 (2006)

35. Experiments with 2D Bose gas. Proliferation of thermal vortices Hadzibabic et al., Nature (2006) Fraction of images showing at least one dislocation Exponent  0.5 00.10.20.3 0.4 0.3 central contrast The onset of proliferation coincides with  shifting to 0.5! 0 10% 20% 30% central contrast 00.10.20.30.4 high T low T

36. Spin dynamics in 1d systems: Ramsey interference experiments T. Kitagawa et al., PRL 2010, Theory A. Widera, V. Gritsev et al, PRL 2008, Theory + Expt J. Schmiedmayer et al., unpublished expts

37. Working with N atoms improves the precision by. Ramsey interference t 0 1 Atomic clocks and Ramsey interference:

38. Ramsey Interference with BEC Single mode approximation time Amplitude of Ramsey fringes Interactions should lead to collapse and revival of Ramsey fringes

39. 1d systems in microchips Treutlein et.al, PRL 2004, also Schmiedmayer, Van Druten Two component BEC in microchip Ramsey Interference with 1d BEC 1d systems in optical lattices Ramsey interference in 1d tubes: A.Widera et al., B. PRL 100:140401 (2008)

40. Ramsey interference in 1d condensates Collapse but no revivals A. Widera, et al, PRL 2008

41. Spin echo. Time reversal experiments Single mode approximation Predicts perfect spin echo The Hamiltonian can be reversed by changing a 12

42. Spin echo. Time reversal experiments No revival? A. Widera et al., PRL 2008 Experiments done in array of tubes. Strong fluctuations in 1d systems. Single mode approximation does not apply. Need to analyze the full model

43. Interaction induced collapse of Ramsey fringes. Multimode analysis Luttinger model Changing the sign of the interaction reverses the interaction part of the Hamiltonian but not the kinetic energy Time dependent harmonic oscillators can be analyzed exactly Low energy effective theory: Luttinger liquid approach

44. Technical noise could also lead to the absence of echo Need “smoking gun” signatures of many-body decoherece Luttinger liquid provides good agreement with experiments. Only q=0 mode shows complete spin echo Finite q modes continue decay The net visibility is a result of competition between q=0 and other modes Interaction induced collapse of Ramsey fringes. Multimode analysis

45. Probing spin dynamics using distribution functions Distribution contains information about higher order correlation functions For longer segments shot noise is not important. Joint distribution functions for different spin components can also be obtained! Distribution

46. Distribution function of fringe contrast as a probe of many-body dynamics Short segments Long segments Radius = Amplitude Angle = Phase

47. Distribution function of fringe contrast as a probe of many-body dynamics Preliminary results by J. Schmiedmayer’s group Splitting one condensate into two.

48. Short segments Long segments l =20 mm l =110 mm ExptTheory Data: Schmiedmayer et al., unpublished

49. Summary of lecture 2 Quantum noise is a powerful tool for analyzing many body states of ultracold atoms Detection of many-body phases using noise correlations: AF/CDW phases in optical lattices, paired states Experiments with low dimensional systems Interference experiments as a probe of BKT transition in 2D, Luttinger liquid in 1d. Analysis of high order correlations

50. Lecture 3 Introduction. Systems of ultracold atoms. Cold atoms in optical lattices. Bose Hubbard model Bose mixtures in optical lattices Detection of many-body phases using noise correlations Experiments with low dimensional systems Fermions in optical lattices Magnetism Pairing in systems with repulsive interactions Current experiments: Paramagnetic Mott state Experiments on nonequilibrium fermion dynamics Lattice modulation experiments Doublon decay Stoner instability

52. Second order correlations. Experimental issues. Complications we need to consider: - finite resolution of detectors - projection from 3D to 2D plane s – detector resolution Autocorrelation function In Mainz experiments and The signal in is - period of the optical lattice

53. Second order coherence in the insulating state of bosons. Experiment: Folling et al., Nature (2005)

54. Second order correlation function Cloud after expansion Cloud before expansion Quantum noise analysis of time-of-flight experiments with atoms in optical lattices Here and are taken after the expansion time t. Two signs correspond to bosons and fermions.

55. Relate operators after the expansion to operators before the expansion. For long expansion times use steepest descent method of integration Second order real-space correlations after TOF expansion can be related to second order momentum correlations inside the trapped system Quantum noise analysis of time-of-flight experiments with atoms in optical lattices TOF experiments map momentum distributions to real space images

56. Example: Mott state of spinless bosons Only local correlations present in the Mott state G - reciprocal vectors of the optical lattice Quantum noise analysis of time-of-flight experiments with atoms in optical lattices

57. Quantum noise in TOF experiments in optical lattices We get bunching when corresponds to one of the reciprocal vectors of the original lattice. Boson bunching arises from the Bose enhancement factors. A single particle state with quasimomentum q is a supersposition of states with physical momentum q+nG. When we detect a boson at momentum q we increase the probability to find another boson at momentum q+nG.

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

59. Example: Band insulating state of spinless fermions Quantum noise analysis of time-of-flight experiments with atoms in optical lattices Only local correlations present in the band insulator state

60. Example: Band insulating state of spinless fermions Quantum noise analysis of time-of-flight experiments with atoms in optical lattices We get fermionic antibunching. This can be understood as Pauli principle. A single particle state with quasimomentum q is a supersposition of states with physical momentum q+nG. When we detect a fermion at momentum q we decrease the probability to find another fermion at momentum q+nG.