Probing interacting systems of cold atoms using interference experimentsVladimir Gritsev, Adilet Imambekov, Anton Burkov, Robert Cherng, Anatoli Polkovnikov, Ehud Altman, Mikhail Lukin, Eugene DemlerMeasuring equilibrium correlation functions using interference experimentsStudying non-equilibrium dynamics of interacting Bose systems in interference experimentsInterference of independent condensatesExperiments: Andrews et al., Science 275:637 (1997)Theory: Javanainen, Yoo, PRL 76:161 (1996)Cirac, Zoller, et al. PRA 54:R3714 (1996)Castin, Dalibard, PRA 55:4330 (1997)and many moreInterference of two independent condensates12rr+ddr’Clouds 1 and 2 do not have a well defined phase difference.However each individual measurement shows an interference patternNature 4877:255 (1963)Interference of one dimensional condensatesInterference of 1d condensates: Schmiedmayer et al., Nature Physics (2005,2006)long. imagingtrans.imagingLongitudialimagingTransverse imagingFigures courtesy of J. SchmiedmayerExperiments with 1d condensates: Sengstock , Phillips, Weiss, Bloch, Esslinger, …x1dAmplitude of interference fringes, Interference of one dimensional condensatesFor identical condensatesInstantaneous correlation functionFor independent condensates Afris finite but ∆φ is random x2Polkovnikov, Altman, Demler, PNAS 103:6125 (2006)For impenetrable bosons and Interference between 1d condensates at T=0Luttinger liquid at T=0K – Luttinger parameterLuttinger liquid at finite temperatureFor non-interacting bosons andAnalysis of can be used for thermometry LInterference of two dimensional condensatesLyLxLxExperiments: Hadzibabic et al. Nature (2006)Probe beam parallel to the plane of the condensatesGati et al., PRL (2006)Interference of two dimensional condensates.Quasi long range order and the BKT transitionLyLxBelow BKT transitionAbove BKT transitionxzTime offlightlow temperature higher temperatureTypical interference patternsExperiments with 2D Bose gasHadzibabic, Dalibard et al., Nature 441:1118 (2006)Figures courtesy of Z. Hadzibabic and J. Dalibardintegrationover x axisDxzzintegrationover x axiszxintegration distance Dx(pixels)Contrast afterintegration0.40.20010 20 30middle Tlow Thigh Tintegrationover x axiszExperiments with 2D Bose gasHadzibabic et al., Nature 441:1118 (2006)fit by:integration distance DxIntegrated contrast0.40.200 10 20 30low Tmiddle Thigh Tif g1(r) decays exponentially with : if g1(r) decays algebraically or exponentially with a large : Exponent αcentral contrast0.50 0.1 0.2 0.30.40.3high T low T[ ]α22121~),0(1~∫xDxDdxxgDCx“Sudden” jump!?Experiments with 2D Bose gasHadzibabic et al., Nature 441:1118 (2006)Exponent αcentral contrast0.50 0.1 0.2 0.30.40.3high T low THe experiments:universal jump in the superfluid densityT (K)1.01.11.21.00c.f. Bishop and ReppyExperiments with 2D Bose gasHadzibabic et al., Nature 441:1118 (2006) Ultracold atoms experiments:jump in the correlation function.BKT theory predicts α=1/4 just below the transitionExperiments with 2D Bose gas. Proliferation of thermal vortices Hadzibabic et al., Nature 441:1118 (2006) Fraction of images showing at least one dislocationExponent α0.50 0.1 0.2 0.30.40.3central contrastThe onset of proliferation coincides with αshifting to 0.5!010%20%30%central contrast0 0.1 0.2 0.3 0.4high Tlow TFundamental noise in interference experimentsAmplitude of interference fringes is a quantum operator. The measured value of the amplitude will fluctuate from shot to shot. We want to characterize not only the averagebut the fluctuations as well.Shot noise in interference experimentsInterference with a finite number of atoms. How well can one measure the amplitude of interference fringes in a single shot?One atom: NoVery many atoms: ExactlyFinite number of atoms: ?Consider higher moments of the interference fringe amplitude,, and so onObtain the entire distribution function ofShot noise in interference experimentsInterference of two condensates with 100 atoms in each cloudCoherent statesNumber statesPolkovnikov, Europhys. Lett. 78:10006 (1997)Imambekov, Gritsev, Demler, 2006 Varenna lecture notesDistribution function of fringe amplitudes for interference of fluctuating condensatesLis a quantum operator. The measured value of will fluctuate from shot to shot.Higher moments reflect higher order correlation functionsGritsev, Altman, Demler, Polkovnikov, Nature Physics (2006)Imambekov, Gritsev, Demler, cond-mat/0612011We need the full distribution function of0 1 2 3 4 Probability P(x)x K=1 K=1.5 K=3 K=5Interference of 1d condensates at T=0. Distribution function of the fringe contrastNarrow distributionfor .Approaches Gumbeldistribution. WidthWide Poissoniandistribution forInterference of 1d condensates at finite temperature. Distribution function of the fringe contrastLuttinger parameter K=5Interference of 2d condensates at finite temperature. Distribution function of the fringe contrastT=TKTT=2/3 TKTT=2/5 TKTFrom visibility of interference fringes to other problems in physicsQuantum impurity problem: interacting one dimensionalelectrons scattered on an impurityConformal field theories with negative central charges: 2D quantum gravity, non-intersecting loop model, growth of random fractal stochastic interface, high energy limit of multicolor QCD, …Interference between interacting 1d Bose liquids.Distribution function of the interference amplitudeis a quantum operator. The measured value of will fluctuate from shot to shot.How to predict the distribution function of Yang-Lee singularity2D quantum gravity,non-intersecting loopsFringe visibility and statistics of random surfaces)(ϕhProof of the Gumbel distribution of interfernece fringe amplitude for 1d weakly interacting bosons relied on the known relation between 1/f Noise and Extreme Value StatisticsT.Antal et al. Phys.Rev.Lett. 87, 240601(2001)Fringe visibilityRoughnessϕϕdh2)(∫=Non-equilibrium coherentdynamics of low dimensional Bose gases probed in interference experimentsStudying dynamics using interference experiments.Quantum and thermal decoherencePrepare a system by splitting one condensateTake to the regime of zero tunnelingMeasure time evolutionof fringe amplitudesRelative phase dynamicsInterference experiments measure only the relative phaseRelative phaseParticle number imbalanceConjugate variablesEarlier work was