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Measuring correlation functions in interacting systems of cold atoms

Measuring correlation functions in interacting systems of cold atoms. Detection and characterization of many-body quantum phases. Anatoli Polkovnikov Harvard/Boston University Ehud Altman Harvard/Weizmann Vladimir Gritsev Harvard Mikhail Lukin Harvard

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Measuring correlation functions in interacting systems of cold atoms

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  1. Measuring correlation functions in interacting systems of cold atoms Detection and characterization of many-body quantum phases Anatoli PolkovnikovHarvard/Boston University Ehud AltmanHarvard/Weizmann Vladimir GritsevHarvard Mikhail Lukin Harvard Vito Scarola University of Maryland Sankar Das Sarma University of Maryland Eugene DemlerHarvard

  2. 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 • References: • 1. Polkovnikov, Altman, Demler,cond-mat/0511675 • Gritsev, Altman, Demler, Polkovnikov, cond-mat/0602475 • 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 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

  6. 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

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

  8. 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

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

  10. 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

  11. 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

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

  13. Experiments with 2D Bose gas Hadzibabic, 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

  14. Experiments with 2D Bose gas Hadzibabic, 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

  15. Experiments with 2D Bose gas Hadzibabic, 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

  16. 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 Hadzibabic, Stock, Dalibard, et al. Fraction of images showing at least one dislocation 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. 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)

  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. 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

  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 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

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

  23. 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, …

  24. Outlook Full counting statistics of interference between fluctuating 2D condensates 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 Expts: Shmiedmayer et al. (1d), Oberthaler et al. (2d) Time dependent evolution of distribution functions 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. Mott insulator Superfluid t/U Atoms in an optical lattice.Superfluid to Insulator transition Greiner et al., Nature 415:39 (2002)

  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 stellarinterferometer

  30. 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

  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 Altman et al., PRA 70:13603 (2004) Detection of magnetically ordered Mott states Detection of paired states of fermions 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

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