1. Probing many-body systems of ultracold atoms E. Altman (Weizmann), A. Aspect (CNRS, Paris), M. Greiner (Harvard), V. Gritsev (Freiburg), S. Hofferberth (Harvard), A. Imambekov (Yale), T. Kitagawa (Harvard), M. Lukin (Harvard), S. Manz (Vienna), I. Mazets (Vienna), D. Petrov (CNRS, Paris), T. Schumm (Vienna), J. Schmiedmayer (Vienna) Eugene Demler Harvard University Collaboration with experimental group of I. Bloch

2. Outline Density ripples in expanding low-dimensional condensates Review of earlier work Analysis of density ripples spectrum 1d systems 2d systems Phase sensitive measurements of order parameters in many body system of ultra-cold atoms Phase sensitive experiments in unconventional superconductors Noise correlations in TOF experiments From noise correlations to phase sensitive measurements

3. Density ripples in expanding low-dimensional condensates

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

5. Density fluctuations in 1D condensates In-situ observation of density fluctuations is difficult. Density fluctuations in confined clouds are suppressed by interactions. Spatial resolution is also a problem. When a cloud expands, interactions are suppressed and density fluctuations get amplified. Phase fluctuations are converted into density ripples

6. Density ripples in expanding anisotropic 3d condensates Dettmer et al. PRL 2001 Hydrodynamics expansion is dominated by collisions Complicated relation between original fluctuations and final density ripples.

7. Density ripples in expanding anisotropic 3d condensates

8. Fluctuations in 1D condensates and density ripples New generation of low dimensional condensates. Tight transverse confinement leads to essentially collision-less expansion. Assuming ballistic expansion we can find direct relation between density ripples and fluctuations before expansion. A pair of 1d condensates on a microchip. J. Schmiedmayer et al. 1d tubes created with optical lattice potentials I. Bloch et al.

9. Density ripples: Bogoliubov theory Expansion during time t Density after expansion Density correlations

10. Density ripples: Bogoliubov theory Spectrum of density ripples

11. Density ripples: Bogoliubov theory Maxima at Minima at The amplitude of the spectrum is dependent on temperature and interactions Non-monotonic dependence on momentum. Matter-wave near field diffraction: Talbot effect Concern: Bogoliubov theory is not applicable to low dimensional condensates. Need extensions beyond mean-field theory

12. Density ripples: general formalism Free expansion of atoms. Expansion in different directions factorize We are interested in the motion along the original trap. For 1d systems

13. Quasicondensates One dimensional systems with Two dimensional systems below BKT transition Factorization of higher order correlation functions One dimensional quasicondensate, Mora and Castin (2003)

14. Density ripples in 1D for weakly interacting Bose gas Thermal correlation length T/ m =1, 0.67, 0.3, 0 Different times of flight. T/ m =0.67 A single peak in the spectrum after

15. Density ripples in expanding cloud: Time-evolution of g 2 (x,t) Sufficient spatial resolution required to resolve oscillations in g 2

16. Density ripples in expanding cloud: Time-evolution of g 2 (x,t) for hard core bosons T=0. Expansion times T/ m =1 “Antibunching” at short distances is rapidly suppressed during expansion Finite temperature

17. Density ripples in 2D Quasicondensates in 2D below BKT transition For weakly interacting Bose gas is a universal dimensionless function Below Berezinsky-Kosterlitz-Thouless transition at h c =1/4

18. Density ripples in 2D Expansion times Fixed time of flight. Different temperatures 87 Rb t = 4, 8, 12 ms h = 0.1, 0.15, 0.25

19. Applications of density ripples Thermometry at low temperatures T/ m =1, 0.67, 0.3, 0 Probe of roton softening Analysis of non-equilibrium states?

20. Phase sensitive measurements of order parameters in many-body systems of ultracold atoms

21. d-wave pairing Fermionic Hubbard model Possible phase diagram of the Hubbard model D.J.Scalapino Phys. Rep. 250:329 (1995)

22. Non-phase sensitive probes of d-wave pairing: dispersion of quasiparticles ++ - - Quasiparticle energies Superconducting gap Normal state dispersion of quasiparticles Low energy quasiparticles correspond to four Dirac nodes Observed in: Photoemission Raman spectroscopy T-dependence of thermodynamic and transport properties, c V, k, l L STM and many other probes

23. Phase sensitive probe of d-wave pairing in high Tc superconductors Superconducting quantum interference device (SQUID) F Van Harlingen, Leggett et al, PRL 71:2134 (93)

24. From noise correlations to phase sensitive measurements in systems of ultra-cold atoms

25. Quantum noise analysis in time of flight experiments Second order coherence

26. Second order coherence in the insulating state of bosons. Hanburry-Brown-Twiss experiment Experiment: Folling et al., Nature 434:481 (2005) Theory: Altman et al., PRA 70:13603 (2004)

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

28. 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)

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

30. Fermion pairing in an optical lattice Second Order Interference In the TOF images Normal State Superfluid State Measures the absolute value of the Cooper pair wavefunction. Not a phase sensitive probe

31. P-wave molecules How to measure the non-trivial symmetry of y (p)? We want to measure the relative phase between components of the molecule at different wavevectors

32. Two particle interference Beam splitters perform Rabi rotation Coincidence count Coincidence count is sensitive to the relative phase between different components of the molecule wavefunction Questions: How to make atomic beam splitters and mirrors? Phase difference includes phase accumulated during free expansion. How to control it?

33. Bragg + Noise G k -k p -p G Bragg pulse is applied in the beginning of expansion Coincidence count Assuming mixing between k and p states only Common mode propagation after the pulse. We do not need to worry about the phase accumulated during the expansion.

34. G k -k p -p G Many-body BCS state BCS wavefunction Strong Bragg pulse: mixing of many momentum eigenstates Noise correlations Interference term is sensitive to the phase difference between k and p parts of the Cooper pair wavefunction and to the phases of Bragg pulses

35. Noise correlations in the BCS state Interference between different components of the Cooper pair

36. Noise correlations in the BCS state Compare to V 0 t controls Rabi angle b Bragg pulse phases control c ’s

37. Systems with particle-hole correlations D-density wave state Suggested as a competing order in high Tc cuprates Phase sensitive probe of DDW order parameter

38. G k -k p -p G Summary Density ripples in expanding low-dimensional condensates Phase sensitive measurements of order parameters in many body systems of ultra-cold atoms T/ m =1, 0.67, 0.3, 0 Different times of flight. T/ m =0.67

39. Detection of spin superexchange interactions and antiferromagnetic states Spin noise analysis Bruun, Andersen, Demler, Sorensen, PRL (2009)

40. Spin shot noise as a probe of AF order Measure net spin in a part of the system. Laser beam passes through the sample. Photons experience phase shift determined by the net spin. Use homodyne to measure phase shift Average magnetization zero Shot to shot magnetization fluctuations reflect spin correlations

41. Spin shot noise as a probe of AF order High temperatures. Every spin fluctuates independently Low temperatures. Formation of antiferromagnetic correlations Suppression of spin fluctuations due to spin superexchange interactions can be observed at temperatures well above the Neel ordering transition

44. Two particle interference Beam splitters perform Rabi rotation Molecule wavefunction

45. Two particle interference Coincidence count Coincidence count is sensitive to the relative phase between different components of the molecule wavefunction Phase difference includes phase accumulated during free expansion