Measuring correlation functions in interacting systems of cold atoms

1. Measuring correlation functions in interacting systems of cold atoms Anatoli Polkovnikov Harvard/Boston University Ehud Altman Harvard/Weizmann Vladimir Gritsev Harvard Mikhail Lukin Harvard Vito Scarola Maryland Sankar Das Sarma Maryland Eugene Demler Harvard Thanks to: J. Schmiedmayer, M. Oberthaler, V. Vuletic, M. Greiner, M. Oshikawa, Z. Hadzibabic

2. Correlation functions in condensed matter physics Most experiments in condensed matter physics measure correlation functions Example: neutron scattering measures spin and density correlation functions Neutron diffraction patterns for MnO Shull et al., Phys. Rev. 83:333 (1951)

3. This talk: Detection of many-body quantum phases by measuring correlation functions

4. Outline Measuring correlation functions inintereferenceexperiments 1. Interference of independent condensates 2. Interference of interacting 1Dsystems 3. Interference of 2D systems 4. Full counting statistics of intereference experiments. Connection to quantum impurity problem Quantum noiseinterferometryin time of flight experiments 1. Detection of magnetically ordered Mott states in optical lattices 2. Observation of fermion pairing

5. Measuring correlation functions in intereference experiments

6. Interference of two independent condensates Andrews et al., Science 275:637 (1997)

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

8. x y Interference of one dimensional condensates Experiments: Schmiedmayer et al., Nature Physics 1 (05) d Amplitude of interference fringes, , contains information about phase fluctuations within individual condensates x1 x2

9. L Interference amplitude and correlations Polkovnikov, Altman, Demler,cond-mat/0511675 For identical condensates Instantaneous correlation function

10. L For non-interacting bosons and For impenetrable bosons and Analysis of can be used for thermometry Interference between Luttinger liquids Luttinger liquid at T=0 K – Luttinger parameter Luttinger liquid at finite temperature

11. For large imaging angle, , Rotated probe beam experiment Luttinger parameter K may be extracted from the angular dependence of q

12. Interference between two-dimensional BECs at finite temperature. Kosteritz-Thouless transition

13. Ly Lx Lx Interference of two dimensional condensates Experiments: Stock, Hadzibabic, Dalibard, et al., cond-mat/0506559 Gati, Oberthaler, et al., cond-mat/0601392 Probe beam parallel to the plane of the condensates

14. Ly Lx Below KT transition Above KT transition Interference of two dimensional condensates.Quasi long range order and the KT transition Theory: Polkovnikov, Altman, Demler, cond-mat/0511675

15. Experiments with 2D Bose gas low temperature higher temperature Haddzibabic, Stock, Dalibard, et al. Time of flight z x Typical interference patterns

16. Experiments with 2D Bose gas Haddzibabic, Stock, Dalibard, et al. 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) x integration over x axis z

17. Experiments with 2D Bose gas Haddzibabic, Stock, Dalibard, et al. 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 fit by: Integrated contrast integration distance Dx

18. Experiments with 2D Bose gas Haddzibabic, Stock, Dalibard, et al. c.f. Bishop and Reppy 0.5 1.0 0.4 T (K) 0 1.1 1.0 1.2 0.3 0 0.1 0.2 0.3 Exponent a high T low T central contrast Ultracold atoms experiments: jump in the correlation function. KT theory predicts a=1/4 just below the transition He experiments: universal jump in the superfluid density

19. 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 Haddzibabic, Stock, Dalibard, et al. Fraction of images showing at least one dislocation Z. Hadzibabic et al., in preparation

20. Rapidly rotating two dimensional condensates Time of flight experiments with rotating condensates correspond to density measurements Interference experiments measure single particle correlation functions in the rotating frame

21. Full counting statistics of interference between two interacting one dimensional Bose liquids Gritsev, Altman, Demler, Polkovnikov, cond-mat/0602475

22. L Explicit expressions for are available but cumbersome Fendley, Lesage, Saleur, J. Stat. Phys. 79:799 (1995) Higher moments of interference amplitude is a quantum operator. The measured value of will fluctuate from shot to shot. Can we predict the distribution function of ? Higher moments Changing to periodic boundary conditions (long condensates)

23. 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

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

25. 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 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 Bethe ansatz solution for a quantum impurity Interference amplitude and spectral determinant

26. Evolution of the distribution function Narrow distribution for . Distribution width approaches Wide Poissonian distribution for

27. correspond to vacuum eigenvalues of Q operators of CFT Bazhanov, Lukyanov, Zamolodchikov, Comm. Math. Phys.1996, 1997, 1999 2D quantum gravity, non-intersecting loops on 2D lattice Yang-Lee singularity From interference amplitudes to conformal field theories 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, …

28. Quantum noise interferometry in time of flight experiments

29. Mott insulator Superfluid t/U Atoms in an optical lattice.Superfluid to Insulator transition Greiner et al., Nature 415:39 (2002)

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

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

32. Hanburry-Brown-Twiss stellarinterferometer

33. 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

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

35. Effect of parabolic potential on the second order coherence Experiment: Spielman, Porto, et al., Theory: Scarola, Das Sarma, Demler, cond-mat/0602319 Width of the correlation peak changes across the Transition, reflecting the evolution of the Mott domains

36. Applications of quantum noise interferometry Spin order in Mott states of atomic mixtures

37. Probing spin order of bosons Correlation Function Measurements Extra Bragg peaks appear in the second order correlation function in the AF phase

38. Applications of quantum noise interferometry Detection of fermion pairing

39. Fermions with repulsive interactions U t t Possible d-wave pairing of fermions

40. Positive U Hubbard model Possible phase diagram. Scalapino, Phys. Rep. 250:329 (1995) Antiferromagnetic insulator D-wave pairing of fermions

41. n(r’) k F n(r) Second order interference from a BCS superfluid n(k) k BCS BEC

42. Momentum correlations in paired fermions Theory: Altman et al., PRA 70:13603 (2004) Experiment: Greiner et al., PRL 94:110401 (2005)

43. 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

44. Conclusions Interference of extended condensates can be used to probe correlation functions in one and two dimensional systems Noise interferometry is a powerful tool for analyzing quantum many-body states in optical lattices