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Vladimir Gritsev Harvard Adilet Imambekov Harvard

Harvard-MIT CUA. Interference of fluctuating condensates. Vladimir Gritsev Harvard Adilet Imambekov Harvard Anton Burkov Harvard Robert Cherng Harvard Anatoli Polkovnikov Harvard/Boston University

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Vladimir Gritsev Harvard Adilet Imambekov Harvard

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  1. Harvard-MIT CUA Interference of fluctuating condensates Vladimir GritsevHarvard Adilet Imambekov Harvard Anton Burkov Harvard Robert Cherng Harvard Anatoli PolkovnikovHarvard/Boston University Ehud AltmanHarvard/Weizmann Mikhail LukinHarvard Eugene Demler Harvard

  2. Outline Measuring correlation functions inintereferenceexperiments 1. Interference of independent condensates 2. Interference of interacting 1Dsystems 3. Interference of 2D systems 4. Full distribution function of fringe visibility in intereference experiments. Connection to quantum impurity problem Studying decoherence in interference experiments. Effects of finite temperature Distribution function of magnetization for lattice spin systems

  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. trans. imaging long. imaging Interference of one dimensional condensates Experiments: Schmiedmayer et al., Nature Physics (2005,2006) Reduction of the contrast due to fluctuations Figures courtesy of J. Schmiedmayer

  7. x y Interference of one dimensional condensates d Amplitude of interference fringes, , contains information about phase fluctuations within individual condensates x1 x2

  8. L Interference amplitude and correlations Polkovnikov, Altman, Demler,PNAS 103:6125 (2006) For identical condensates Instantaneous correlation function

  9. 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

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

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

  12. Ly Lx Lx Interference of two dimensional condensates Experiments: Stock, Hadzibabic, Dalibard, et al., PRL 95:190403 (2005) Probe beam parallel to the plane of the condensates

  13. 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, PNAS 103:6125 (2006)

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

  15. 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) Experiments with 2D Bose gas Hadzibabic et al., Nature 441:1118 (2006) x integration over x axis z

  16. 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 Experiments with 2D Bose gas Hadzibabic et al., Nature 441:1118 (2006) fit by: Integrated contrast integration distance Dx

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

  18. 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 et al., Nature 441:1118 (2006) Fraction of images showing at least one dislocation

  19. Full distribution function of fringe amplitudes for interference experiments between two 1d condensates Gritsev, Altman, Demler, Polkovnikov, Nature Physics 2:705(2006)

  20. 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)

  21. 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

  22. 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

  23. is related to the 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

  24. Evolution of the distribution function Narrow distribution for . Approaches Gumbel distribution. Width Wide Poissonian distribution for

  25. Gumbel distribution and 1/f noise For K>>1 Fringe visibility Roughness 1/f Noise and Extreme Value StatisticsT.Antal et al. Phys.Rev.Lett. 87, 240601(2001)

  26. Gumbel Distribution in Statistics Describes Extreme Value Statistics, appears in climate studies, finance, etc. Stock performance: distribution of “best performers” for random sets chosen from S&P500 Distribution of largest monthly rainfall over a period of 291 years at Kew Gardens

  27. Distribution function for open boundary conditions, finite temperature, 2D systems, … A. Imambekov et al. Periodic boundary conditions Open boundary conditions Finite temperature Partition function of classical plasma 1. High temperature expansion (expansion in 1/K) 2. Non-perturbative solution

  28. Periodic versus Open boundary conditions K=5 periodic open

  29. Distribution functions at finite temperature L K=5 For K>>1, crossover at

  30. Time of flight Interference between fluctuating 2d condensates. Distribution function of the interference amplitude • Imambekov et al. • preprint K=5 aspect ratio 1 (Lx=Ly). K=3 K=2, at BKT transition

  31. 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, …

  32. Condensate decoherence at finite temperature probed with interference experiments

  33. Studying dynamics using interference experiments.Thermal decoherence Prepare a system by splitting one condensate Take to the regime of zero tunneling Measure time evolution of fringe amplitudes

  34. Finite temperature phase dynamics Temperature leads to phase fluctuations within individual condensates Interference experiments measure only the relative phase

  35. A collection of harmonic oscillators with Relative phase dynamics Hamiltonian can be diagonalized in momentum space Conjugate variables Need to solve dynamics of harmonic oscillators at finite T Coherence

  36. Relative phase dynamics High energy modes, , quantum dynamics Low energy modes, , classical dynamics Combining all modes Quantum dynamics Classical dynamics For studying dynamics it is important to know the initial width of the phase

  37. Relative phase dynamics Naive estimate

  38. J Relative phase dynamics Separating condensates at finite rate Instantaneous Josephson frequency Adiabatic regime Instantaneous separation regime Adiabaticity breaks down when Charge uncertainty at this moment Squeezing factor

  39. Relative phase dynamics Quantum regime 1D systems 2D systems Classical regime 1D systems 2D systems

  40. Probing spin systems using distribution function of magnetization

  41. Higher moments of contain information about higher order correlation functions Probing spin systems using distribution function of magnetization R. Cherng, E. Demler, cond-mat/0609748 Magnetization in a finite system Average magnetization

  42. x-Ferromagnet polarized 1 or g ? ? ? Probing spin systems using distribution function of magnetization

  43. x-Ferromagnet polarized 1 or g ? Distribution Functions ? ?

  44. No static magnetization Using noise to detect spin liquids Spin liquids have no broken symmetries No sharp Bragg peaks Algebraic spin liquids have long range spin correlations A Noise in magnetization exceeds shot noise

  45. Conclusions Interference of extended condensates can be used to probe equilibrium correlation functions in one and two dimensional systems Analysis of time dependent decoherence of a pair of condensates can be used to probe low temperature dissipation Measurements of the distribution function of magnetization provide a new way of analyzing correlation functions in spin systems realized with cold atoms

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