1. Learning about order from noise Quantum noise studies of ultracold atoms Eugene Demler Harvard University Collaborators: Takuya Kitagawa, Susanne Pielawa, Adilet Imambekov, Ehud Altman, Vladimir Gritsev, Anatoli Polkovnikov, Mikhail Lukin Experiments: Bloch et al., Dalibard et al., Schmiedmayer et al.

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

3. Probabilistic nature of quantum mechanics “Spooky action at a distance” Bohr-Einstein debate: EPR thought experiment (1935) Aspect’s experiments with correlated photon pairs: tests of Bell’s inequalities (1982) Analysis of correlation functions can be used to rule out hidden variables theories + - + - 12 S

4. Second order coherence: HBT experiments Classical theory Hanburry Brown and Twiss (1954) Used to measure the angular diameter of Sirius Quantum theory Glauber (1963) For bosons For fermions HBT experiments with matter

5. Shot noise in electron transport Shot noise Schottky (1918) Variance of transmitted charge e-e- e-e- Measurements of fractional charge 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)

6. Analysis of quantum noise: powerful experimental tool Can we use it for cold atoms?

7. Quantum noise in interference experiments with independent condensates Outline Goal: new methods of detection of quantum many-body phases of ultracold atoms Quantum noise analysis of time-of-flight experiments with atoms in optical lattices: HBT experiments and beyond Quantum noise in Ramsey interference experiments

8. Interference experiments with cold atoms Analysis of thermal and quantum noise in low dimensional systems

9. 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

10. x z Time of flight Experiments with 2D Bose gas Hadzibabic, Kruger, Dalibard et al., Nature 441:1118 (2006) Experiments with 1D Bose gas Hofferberth et al., Nature Physics 4:489 (2008)

11. 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

12. 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)

13. Fluctuations in 1d BEC Thermal fluctuations Thermally energy of the superflow velocity Quantum fluctuations Weakly interacting atoms

14. 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

15. 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, Varenna lecture notes, c-m/0703766 We need the full distribution function of

16. Distribution function of interference fringe contrast Hofferberth et al., Nature Physics 4:489 (2008) 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

17. 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

18. 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 Fringe visibility Roughness Analysis of sine-Gordon models of the type

19. Time-of-flight experiments with atoms in optical lattices

20. Atoms in optical lattices Theory: Jaksch et al. PRL (1998) Experiment: Greiner et al., Nature (2001) and many more Motivation: quantum simulations of strongly correlated electron systems including quantum magnets and unconventional superconductors

21. Antiferromagnetic and superconducting Tc of the order of 100 K Atoms in optical lattice Antiferromagnetism and pairing at sub-micro Kelvin temperatures Fermionic Hubbard model From high temperature superconductors to ultracold atoms

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

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

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

25. Hanburry-Brown-Twiss stellar interferometer

26. 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

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

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

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

30. Detection of fermion pairing Quantum noise analysis of TOF images is more than HBT interference

31. Second order interference from the BCS superfluid n(r) n(r’) n(k) k BCS BEC k F Theory: Altman et al., PRA 70:13603 (2004)

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

33. 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

34. Quantum noise as a probe of non-equilibrium dynamics Ramsey interferometry and many-body decoherence

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

36. Two component BEC. Single mode approximation Interaction induced collapse of Ramsey fringes time Ramsey fringe visibility Experiments in 1d tubes: A. Widera et al. PRL 100:140401 (2008)

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

38. Spin echo. Time reversal experiments No revival? Expts: 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

39. 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

40. Time-dependent harmonic oscillator Explicit quantum mechanical wavefunction can be found From the solution of classical problem We solve this problem for each momentum component See e.g. Lewis, Riesengeld (1969) Malkin, Man’ko (1970)

41. Interaction induced collapse of Ramsey fringes in one dimensional systems Fundamental limit on Ramsey interferometry 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 Decoherence due to many-body dynamics of low dimensional systems How to distinquish decoherence due to many-body dynamics?

42. Single mode analysis Kitagawa, Ueda, PRA 47:5138 (1993) Multimode analysis evolution of spin distribution functions T. Kitagawa, S. Pielawa, A. Imambekov, et al. Interaction induced collapse of Ramsey fringes

43. Summary Experiments with ultracold atoms provide a new perspective on the physics of strongly correlated many-body systems. Quantum noise is a powerful tool for analyzing many body states of ultracold atoms Thanks to: Harvard-MIT

45. Interference experiments with cold atoms Analysis of thermal and quantum noise in low dimensional systems Theory: For review see Imambekov et al., Varenna lecture notes, c-m/0612011 Experiment 2D: Hadzibabic, Kruger, Dalibard, Nature 441:1118 (2006) 1D: Hofferberth et al., Nature Physics 4:489 (2008)

46. Time-of-flight experiments with atoms in optical lattices Theory: Altman, Demler, Lukin, PRA 70:13603 (2004) Experiment: Folling et al., Nature 434:481 (2005); Spielman et al., PRL 98:80404 (2007); Tom et al. Nature 444:733 (2006); Guarrera et al., PRL 100:250403 (2008)

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

48. 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)

49. 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: BKT transition Experiments with 2D Bose gas Hadzibabic et al., Nature 441:1118 (2006)

50. Experiments with 2D Bose gas. Proliferation of thermal vortices Hadzibabic et al., Nature 441:1118 (2006) The onset of proliferation coincides with  shifting to 0.5! Fraction of images showing at least one dislocation 0 10% 20% 30% central contrast 00.10.20.30.4 high T low T

51. Fringe contrast in two dimensions. Evolution of distribution function Experiments: Kruger, Hadzibabic, Dalibard