Outline of these lectures

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 noiseQuantum 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 Einstein-Podolsky-Rosen experiment Measuring spin of a particle in the left detector instantaneously determines its value in the right detector

5. + + S 1 2 q1 q2 - - 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)

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

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

8. e- e- Shot noise in electron transport Proposed by Schottky to measure the electron charge in 1918 Spectral density of the current noise Related to variance of transmitted charge When shot noise dominates over thermal noise 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. Cloud before expansion Cloud after expansion Time of flight experiments Quantum noise interferometry of atoms in an optical lattice Second order coherence

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

13. Hanburry-Brown-Twiss stellarinterferometer

14. Bosons at quasimomentum expand as plane waves with wavevectors Second order coherence in the insulating state of bosons 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 asHanburry-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 asHanburry-Brown-Twiss effect Bosons/Fermions Folling et al., Nature (2005) Tom et al. Nature (2006)

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

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

23. Amplitude of interference fringes, For identical condensates Interference of fluctuating condensates Polkovnikov et al., PNAS (2006); Gritsev et al., Nature Physics (2006) d x1 For independent condensates Afr is finite but Df is random x2 Instantaneous correlation function

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

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

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

27. Distribution function of interference fringe contrast Hofferberth et al., Nature Physics 2009 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 Comparison of theory and experiments: no free parameters Higher order correlation functions can be obtained

28. Distribution function of 2D quantum gravity, non-intersecting loops Yang-Lee singularity Interference between interacting 1d Bose liquids. Distribution function of the interference amplitude 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, …

29. Distribution function of Roughness 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

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

31. Ly Lx Below KT transition Above KT transition Interference of two dimensional condensates.Quasi long range order and the KT transition

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

33. Contrast after integration integration over x axis z 0.4 low T z middle T 0.2 integration over x axis high T z 0 0 Dx 10 20 30 integration distance Dx (pixels) Experiments with 2D Bose gas Hadzibabic et al., Nature 441:1118 (2006) x integration over x axis z

34. 0.4 low T 0.2 Exponent a middle T 0 0 10 20 30 high T if g1(r) decays exponentially with : high T low T 0.5 0.4 if g1(r) decays algebraically or exponentially with a large : central contrast 0.3 “Sudden” jump!? 0 0.1 0.2 0.3 Experiments with 2D Bose gas Hadzibabic et al., Nature 441:1118 (2006) fit by: Integrated contrast integration distance Dx

35. 30% Exponent a 20% 10% low T high T 0 0 0.1 0.2 0.3 0.4 central contrast 0.5 central contrast 0.4 The onset of proliferation coincides with ashifting to 0.5! 0.3 0 0.1 0.2 0.3 Experiments with 2D Bose gas. Proliferation of thermal vortices Hadzibabicet al., Nature (2006) Fraction of images showing at least one dislocation

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

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

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

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

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

41. Spin echo. Time reversal experiments Single mode approximation The Hamiltonian can be reversed by changing a12 Predicts perfect spin echo

42. Spin echo. Time reversal experiments 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 No revival?

43. Interaction induced collapse of Ramsey fringes.Multimode analysis Low energy effective theory: Luttinger liquid approach 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

44. Interaction induced collapse of Ramsey fringes.Multimode analysis 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 Luttinger liquid provides good agreement with experiments. Technical noise could also lead to the absence of echo Need “smoking gun” signatures of many-body decoherece

45. Distribution 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!

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

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

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

49. Summary of lecture 2 • 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 Quantum noise is a powerful tool for analyzing many body states of ultracold atoms

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 nonequilibriumfermion dynamics Lattice modulation experiments Doublon decay Stoner instability