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Probing interacting systems of cold atoms using interference experiments Vladimir Gritsev Adilet Imambekov Anton Burkov Robert Cherng Anatoli Polkovnikov Ehud Altman Mikhail Lukin Eugene Demler Measuring equilibrium correlation functions using interference experiments Studying non equilibrium dynamics of interacting Bose systems in interference experiments Interference of independent condensates Experiments 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 more 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 Nature 4877 255 1963 Interference of one dimensional condensates Experiments with 1d condensates Sengstock Phillips Weiss Bloch Esslinger Interference of 1d condensates Schmiedmayer et al Nature Physics 2005 2006 Transverse imaging trans imaging long imaging Longitudial imaging Figures courtesy of J Schmiedmayer Interference of one dimensional condensates d Polkovnikov Altman Demler PNAS 103 6125 2006 Amplitude of interference fringes x1 x2 For independent condensates Afr is finite but is random For identical condensates Instantaneous correlation function Interference between 1d condensates at T 0 Luttinger liquid at T 0 K Luttinger parameter L For non interacting bosons For impenetrable bosons and and Luttinger liquid at finite temperature Analysis of can be used for thermometry Interference of two dimensional condensates Experiments Hadzibabic et al Nature 2006 Gati et al PRL 2006 Ly Lx Lx Probe beam parallel to the plane of the condensates Interference of two dimensional condensates Quasi long range order and the BKT transition Ly Lx Above BKT transition Below BKT transition Experiments with 2D Bose gas z Hadzibabic Dalibard et al Nature 441 1118 2006 Time of flight x Typical interference patterns low temperature higher temperature Figures courtesy of Z Hadzibabic and J Dalibard Experiments with 2D Bose gas Hadzibabic et al Nature 441 1118 2006 x integration over x axis z z Contrast after integration 0 4 low T integration middle T 0 2 over x axis z high T integration over x axis Dx 0 z 0 10 20 30 integration distance Dx pixels Experiments with 2D Bose gas Integrated contrast Hadzibabic et al Nature 441 1118 2006 0 4 fit by C2 low T 1 Dx 1 Dx Dx 2 g 0 x dx 1 middle T 0 2 Exponent high T 0 0 10 20 30 integration distance Dx if g1 r decays exponentially with 0 5 0 4 0 3 high T 0 if g1 r decays algebraically or exponentially with a large 0 1 low T 0 2 0 3 central contrast Sudden jump 2 Experiments with 2D Bose gas Hadzibabic et al Nature 441 1118 2006 Exponent c f Bishop and Reppy 0 4 1 0 0 0 5 1 0 1 1 T K 1 2 0 3 high T 0 0 1 low T 0 2 0 3 central contrast He experiments universal jump in the superfluid density Ultracold atoms experiments jump in the correlation function BKT theory predicts 1 4 just below the transition Experiments with 2D Bose gas Proliferation of thermal vortices Hadzibabic et al Nature 441 1118 2006 30 Fraction of images showing at least one dislocation Exponent 20 0 5 10 0 4 low T high T 0 0 0 1 0 2 0 3 central contrast The onset of proliferation coincides with shifting to 0 5 0 4 0 3 0 0 1 0 2 central contrast 0 3 Fundamental noise in interference experiments Amplitude 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 average but the fluctuations as well Shot noise in interference experiments Interference with a finite number of atoms How well can one measure the amplitude of interference fringes in a single shot One atom No Very many atoms Exactly Finite number of atoms Consider higher moments of the interference fringe amplitude and so on Obtain the entire distribution function of Shot noise in interference experiments Polkovnikov Europhys Lett 78 10006 1997 Imambekov Gritsev Demler 2006 Varenna lecture notes Interference of two condensates with 100 atoms in each cloud Number states Coherent states Distribution function of fringe amplitudes for interference of fluctuating condensates Gritsev Altman Demler Polkovnikov Nature Physics 2006 Imambekov Gritsev Demler cond mat 0612011 L is a quantum operator The measured value of will fluctuate from shot to shot Higher moments reflect higher order correlation functions We need the full distribution function of Interference of 1d condensates at T 0 Distribution function of the fringe contrast Narrow distribution for Approaches Gumbel Probability P x K 1 K 1 5 K 3 K 5 distribution Width Wide Poissonian distribution for 0 1 x 2 3 4 Interference of 1d condensates at finite temperature Distribution function of the fringe contrast Luttinger parameter K 5 Interference of 2d condensates at finite temperature Distribution function of the fringe contrast T TKT T 2 3 TKT T 2 5 TKT From visibility of interference fringes to other problems in physics Interference between interacting 1d Bose liquids Distribution function of the interference amplitude is a quantum operator The measured value of will fluctuate from shot to shot How to predict the distribution function of Quantum impurity problem interacting one dimensional electrons scattered on an impurity Conformal 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 2D quantum gravity non intersecting loops Yang Lee singularity Fringe visibility and statistics of random surfaces Fringe visibility h Roughness 2 h d Proof 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 Statistics T Antal et al Phys Rev Lett 87 240601 2001 Non equilibrium coherent dynamics of low dimensional Bose gases probed in interference experiments Studying dynamics using interference experiments Quantum and thermal decoherence Prepare a system by splitting one condensate Take to the regime of zero tunneling Measure time evolution of fringe amplitudes Relative phase dynamics Interference experiments measure only the relative phase Relative phase Earlier work was based on a single mode approximation e g Gardner Zoller Leggett Particle number imbalance Conjugate variables Relative phase dynamics Hamiltonian can be diagonalized in momentum space


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