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 University of Maryland Sankar Das Sarma University of Maryland Eugene Demler Harvard Detection and characterization of many-body quantum phases

2. Outline Measuring correlation functions in intereference experiments 1. Interference of independent condensates 2. Interference of interacting 1D systems 3. Interference of 2D systems 4. Full counting statistics of intereference experiments. Connection to quantum impurity problem Quantum noise interferometry in time of flight experiments References: 1. Polkovnikov, Altman, Demler, cond-mat/0511675 Gritsev, Altman, Demler, Polkovnikov, cond-mat/0602475 2.Altman, Demler, Lukin, PRA 70:13603 (2004) Scarola, Das Sarma, Demler, cond-mat/0602319

3. Measuring correlation functions in intereference experiments

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

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

6. Amplitude of interference fringes,, contains information about phase fluctuations within individual condensates y x Interference of one dimensional condensates x1x1 d Experiments: Schmiedmayer et al., Nature Physics 1 (05) x2x2

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

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

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

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

11. Interference of two dimensional condensates. Quasi long range order and the KT transition LyLy LxLx Below KT transitionAbove KT transition Theory: Polkovnikov, Altman, Demler, cond-mat/0511675

12. x z Time of flight low temperaturehigher temperature Typical interference patterns Experiments with 2D Bose gas Hadzibabic, Stock, Dalibard, et al.

13. 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, Stock, Dalibard, et al.

14. 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, Stock, Dalibard, et al.

15. Exponent  central contrast 0.5 00.10.20.3 0.4 0.3 high Tlow T He experiments: universal jump in the superfluid density T (K) 1.0 1.1 1.2 1.0 0 c.f. Bishop and Reppy Experiments with 2D Bose gas Hadzibabic, Stock, Dalibard, et al. Ultracold atoms experiments: jump in the correlation function. KT theory predicts a =1/4 just below the transition

16. Experiments with 2D Bose gas. Proliferation of thermal vortices Hadzibabic, Stock, Dalibard, et al. 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 Z. Hadzibabic et al., in preparation

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

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

19. 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

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

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

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

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

24. Outlook One and two dimensional systems with tunneling. Competition of single particle tunneling, quantum, and thermal fluctuations. Full counting statistics of the phase and the amplitude Full counting statistics of interference between fluctuating 2D condensates Time dependent evolution of distribution functions Expts: Shmiedmayer et al. (1d), Oberthaler et al. (2d) Extensions, e.g. spin counting in Mott states of multicomponent systems Ultimate goal: Creation, characterization, and manipulation of quantum many-body states of atoms

25. Quantum noise interferometry in time of flight experiments

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

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

28. 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)

29. Hanburry-Brown-Twiss stellar interferometer

30. 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

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. 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 Mott domains

33. Potential applications of quantum noise intereferometry Detection of magnetically ordered Mott states Detection of paired states of fermions Altman et al., PRA 70:13603 (2004) Experiment: Greiner et al. PRL

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