#### Transcript Learning about order from noise Quantum noise studies of

Outline of these lectures • • • • Introduction. Systems of ultracold atoms. Cold atoms in optical lattices. Bose Hubbard model. Equilibrium and dynamics Bose mixtures in optical lattices. Quantum magnetism of ultracold atoms. • Detection of many-body phases using noise correlations • Experiments with low dimensional systems Interference experiments. Analysis of high order correlations • Fermions in optical lattices Magnetism and pairing in systems with repulsive interactions. Current experiments: paramgnetic Mott state, nonequilibrium dynamics. • Dynamics near Fesbach resonance. Competition of Stoner instability and pairing Learning about order from noise Quantum noise studies of ultracold atoms Quantum noise Classical measurement: collapse of the wavefunction into eigenstates of x Histogram of measurements of x Probabilistic nature of quantum mechanics Bohr-Einstein debate on spooky action at a distance Einstein-Podolsky-Rosen experiment Measuring spin of a particle in the left detector instantaneously determines its value in the right detector Aspect’s experiments: tests of Bell’s inequalities + + 1 - q1 S q2 2 - S Correlation function Classical theories with hidden variable require Quantum mechanics predicts B=2.7 for the appropriate choice of q‘s and the state Experimentally measured value B=2.697. Phys. Rev. Let. 49:92 (1982) Hanburry-Brown-Twiss experiments Classical theory of the second order coherence Hanbury Brown and Twiss, Proc. Roy. Soc. (London), A, 242, pp. 300-324 Measurements of the angular diameter of Sirius Proc. Roy. Soc. (London), A, 248, pp. 222-237 Quantum theory of HBT experiments Glauber, Quantum Optics and Electronics (1965) HBT experiments with matter For bosons Experiments with neutrons Ianuzzi et al., Phys Rev Lett (2006) Experiments with electrons Kiesel et al., Nature (2002) For fermions Experiments with 4He, 3He Westbrook et al., Nature (2007) Experiments with ultracold atoms Bloch et al., Nature (2005,2006) Shot noise in electron transport Proposed by Schottky to measure the electron charge in 1918 e- e- Spectral density of the current noise Related to variance of transmitted charge When shot noise dominates over thermal noise Poisson process of independent transmission of electrons Shot noise in electron transport Current noise for tunneling across a Hall bar on the 1/3 plateau of FQE Etien et al. PRL 79:2526 (1997) see also Heiblum et al. Nature (1997) Quantum noise analysis of time-of-flight experiments with atoms in optical lattices: Hanburry-Brown-Twiss experiments and beyond Theory: Altman et al., PRA (2004) Experiment: Folling et al., Nature (2005); Spielman et al., PRL (2007); Tom et al. Nature (2006) Time of flight experiments Cloud before expansion Cloud after expansion Quantum noise interferometry of atoms in an optical lattice Second order coherence Quantum noise analysis of time-of-flight experiments with atoms in optical lattices Experiment: Folling et al., Nature (2005) Hanburry-Brown-Twiss stellar interferometer 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 Second order correlations as Hanburry-Brown-Twiss effect Bosons/Fermions Second order coherence in the insulating state of fermions. Experiment: Tom et al. Nature (2006) Second order correlations as Hanburry-Brown-Twiss effect Bosons/Fermions Folling et al., Nature (2005) Tom et al. Nature (2006) Probing spin order in optical lattices Correlation function measurements after TOF expansion. Extra Bragg peaks appear in the second order correlation function in the AF phase. This reflects doubling of the unit cell by magnetic order. Interference experiments with cold atoms Probing fluctuations in low dimensional systems 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 Experiments with 2D Bose gas Hadzibabic, Dalibard et al., Nature 2006 z Time of flight x Experiments with 1D Bose gas Hofferberth et al. Nat. Physics 2008 Interference of two independent condensates r’ r Assuming ballistic expansion 1 r+d d 2 Phase difference between clouds 1 and 2 is not well defined Individual measurements show interference patterns They disappear after averaging over many shots Interference of fluctuating condensates d Polkovnikov et al., PNAS (2006); Gritsev et al., Nature Physics (2006) Amplitude of interference fringes, x1 x2 For independent condensates Afr is finite but Df is random For identical condensates Instantaneous correlation function Fluctuations in 1d BEC Thermal fluctuations Thermally energy of the superflow velocity Quantum fluctuations Interference between Luttinger liquids Luttinger liquid at T=0 K – Luttinger parameter For non-interacting bosons For impenetrable bosons Finite temperature Experiments: Hofferberth, Schumm, Schmiedmayer and and Distribution function of fringe amplitudes for interference of fluctuating condensates Gritsev, Altman, Demler, Polkovnikov, Nature Physics 2006 Imambekov, Gritsev, Demler, PRA (2007) is a quantum operator. The measured value of will fluctuate from shot to shot. L Higher moments reflect higher order correlation functions We need the full distribution function of Distribution function of interference fringe contrast Hofferberth et al., Nature Physics 2009 Quantum fluctuations dominate: asymetric Gumbel distribution (low temp. T or short length L) Thermal fluctuations dominate: broad Poissonian distribution (high temp. T or long length L) Intermediate regime: double peak structure Comparison of theory and experiments: no free parameters Higher order correlation functions can be obtained Interference between interacting 1d Bose liquids. Distribution function of the interference amplitude 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 Distribution function of Mapping between fringe visibility and the problem of surface roughness for fluctuating random surfaces. Relation to 1/f Noise and Extreme Value Statistics h( ) 2 Roughness h( ) d 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 KT transition Ly Lx Above KT transition Below KT 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 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 a 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!? 2a Experiments with 2D Bose gas. Proliferation of thermal vortices Hadzibabic et al., Nature (2006) 30% Fraction of images showing at least one dislocation Exponent a 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 a shifting to 0.5! 0.4 0.3 0 0.1 0.2 central contrast 0.3 Spin dynamics in 1d systems: Ramsey interference experiments A. Widera, V. Gritsev et al, PRL 2008, Theory + Expt T. Kitagawa et al., PRL 2010, Theory J. Schmiedmayer et al., unpublished expts Ramsey interference 1 0 Atomic clocks and Ramsey interference: Working with N atoms improves the precision by . t Ramsey Interference with BEC Single mode approximation Amplitude of Ramsey fringes Interactions should lead to collapse and revival of Ramsey fringes time Ramsey Interference with 1d BEC 1d systems in optical lattices Ramsey interference in 1d tubes: A.Widera et al., B. PRL 100:140401 (2008) 1d systems in microchips Two component BEC in microchip Treutlein et.al, PRL 2004, also Schmiedmayer, Van Druten Ramsey interference in 1d condensates A. Widera, et al, PRL 2008 Collapse but no revivals Spin echo. Time reversal experiments Single mode approximation The Hamiltonian can be reversed by changing a12 Predicts perfect spin echo Spin echo. Time reversal experiments A. Widera et al., PRL 2008 No revival? Experiments done in array of tubes. Strong fluctuations in 1d systems. Single mode approximation does not apply. Need to analyze the full model Interaction induced collapse of Ramsey fringes. Multimode analysis Low energy effective theory: Luttinger liquid approach Luttinger model Changing the sign of the interaction reverses the interaction part of the Hamiltonian but not the kinetic energy Time dependent harmonic oscillators can be analyzed exactly Interaction induced collapse of Ramsey fringes. Multimode analysis Only q=0 mode shows complete spin echo Finite q modes continue decay The net visibility is a result of competition between q=0 and other modes Luttinger liquid provides good agreement with experiments. Technical noise could also lead to the absence of echo Need “smoking gun” signatures of many-body decoherece Distribution Probing spin dynamics using distribution functions Distribution contains information about higher order correlation functions For longer segments shot noise is not important. Joint distribution functions for different spin components can also be obtained! Distribution function of fringe contrast as a probe of many-body dynamics Short segments Radius = Amplitude Angle = Phase Long segments Distribution function of fringe contrast as a probe of many-body dynamics Splitting one condensate into two. Preliminary results by J. Schmiedmayer’s group Short segments Long segments l =20 mm l =110 mm Expt Theory Data: Schmiedmayer et al., unpublished Summary of lecture 2 • Detection of many-body phases using noise correlations: AF/CDW phases in optical lattices, paired states • Experiments with low dimensional systems Interference experiments as a probe of BKT transition in 2D, Luttinger liquid in 1d. Analysis of high order correlations Quantum noise is a powerful tool for analyzing many body states of ultracold atoms Lecture 3 • Fermions in optical lattices Magnetism Pairing in systems with repulsive interactions Current experiments: Paramagnetic Mott state • Experiments on nonequilibrium fermion dynamics Lattice modulation experiments Doublon decay Stoner instability Second order correlations. Experimental issues. Autocorrelation function Complications we need to consider: - finite resolution of detectors - projection from 3D to 2D plane s – detector resolution - period of the optical lattice In Mainz experiments The signal in and is Second order coherence in the insulating state of bosons. Experiment: Folling et al., Nature (2005) Quantum noise analysis of time-of-flight experiments with atoms in optical lattices Cloud before expansion Cloud after expansion Second order correlation function Here and are taken after the expansion time t. Two signs correspond to bosons and fermions. Quantum noise analysis of time-of-flight experiments with atoms in optical lattices Relate operators after the expansion to operators before the expansion. For long expansion times use steepest descent method of integration TOF experiments map momentum distributions to real space images Second order real-space correlations after TOF expansion can be related to second order momentum correlations inside the trapped system Quantum noise analysis of time-of-flight experiments with atoms in optical lattices Example: Mott state of spinless bosons Only local correlations present in the Mott state G - reciprocal vectors of the optical lattice Quantum noise in TOF experiments in optical lattices We get bunching when corresponds to one of the reciprocal vectors of the original lattice. Boson bunching arises from the Bose enhancement factors. A single particle state with quasimomentum q is a supersposition of states with physical momentum q+nG. When we detect a boson at momentum q we increase the probability to find another boson at momentum q+nG. Interference of an array of independent condensates Hadzibabic et al., PRL 93:180403 (2004) Smooth structure is a result of finite experimental resolution (filtering) 3 1.4 2.5 1.2 2 1 1.5 0.8 1 0.6 0.5 0.4 0 -0.5 0.2 -1 0 -1.5 0 200 400 600 800 1000 1200 -0.2 0 200 400 600 800 1000 1200 Quantum noise analysis of time-of-flight experiments with atoms in optical lattices Example: Band insulating state of spinless fermions Only local correlations present in the band insulator state Quantum noise analysis of time-of-flight experiments with atoms in optical lattices Example: Band insulating state of spinless fermions We get fermionic antibunching. This can be understood as Pauli principle. A single particle state with quasimomentum q is a supersposition of states with physical momentum q+nG. When we detect a fermion at momentum q we decrease the probability to find another fermion at momentum q+nG.