Measuring correlation functions in interacting systems of cold atoms Anatoli Polkovnikov Harvard/Boston University Ehud Altman Harvard/Weizmann Vladimir Gritsev Harvard Mikhail Lukin Harvard Vito Scarola Maryland Sankar Das Sarma Maryland Eugene Demler Harvard Thanks to: J. Schmiedmayer, M. Oberthaler, V. Vuletic, M. Greiner, M. Oshikawa, Z. Hadzibabic

Correlation functions in condensed matter physics Most experiments in condensed matter physics measure correlation functions Example: neutron scattering measures spin and density correlation functions Neutron diffraction patterns for MnO Shull et al., Phys. Rev. 83:333 (1951)

This talk: Detection of many-body quantum phases by measuring correlation functions

Outline Measuring correlation functions in intereference experiments 1. Interference of independent condensates 2. Interference of interacting 1D systems 3. Interference of 2D systems 4. Full counting statistics of intereference experiments. Connection to quantum impurity problem Quantum noise interferometry in time of flight experiments 1. Detection of magnetically ordered Mott states in optical lattices 2. Observation of fermion pairing

Measuring correlation functions in intereference experiments

Interference of two independent condensates Andrews et al., Science 275:637 (1997)

Interference of two independent condensates 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

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 y x

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

Interference between Luttinger liquids Luttinger liquid at T=0 L K – Luttinger parameter For non-interacting bosons and For impenetrable bosons and Luttinger liquid at finite temperature Analysis of can be used for thermometry

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

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

Interference of two dimensional condensates Experiments: Stock, Hadzibabic, Dalibard, et al., cond-mat/0506559 Gati, Oberthaler, et al., cond-mat/0601392 Lx Ly Lx Probe beam parallel to the plane of the condensates

Interference of two dimensional condensates Interference of two dimensional condensates. Quasi long range order and the KT transition Ly Lx Theory: Polkovnikov, Altman, Demler, cond-mat/0511675 Above KT transition Below KT transition

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

Experiments with 2D Bose gas Haddzibabic, Stock, Dalibard, et al. x z integration over x axis integration distance Dx (pixels) Contrast after integration 0.4 0.2 10 20 30 middle T z low T high T integration over x axis z integration over x axis Dx z

Experiments with 2D Bose gas Haddzibabic, Stock, Dalibard, et al. 0.4 0.2 10 20 30 low T middle T high T fit by: Integrated contrast Exponent a central contrast 0.5 0.1 0.2 0.3 0.4 high T low T “Sudden” jump!? integration distance Dx if g1(r) decays exponentially with : if g1(r) decays algebraically or exponentially with a large :

Experiments with 2D Bose gas Haddzibabic, Stock, Dalibard, et al. Exponent a T (K) 1.0 1.1 1.2 c.f. Bishop and Reppy 0.5 0.1 0.2 0.3 0.4 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

Experiments with 2D Bose gas Experiments with 2D Bose gas. Proliferation of thermal vortices Haddzibabic, Stock, Dalibard, et al. 10% 20% 30% central contrast 0.1 0.2 0.3 0.4 high T low T Fraction of images showing at least one dislocation Exponent a 0.5 0.1 0.2 0.3 0.4 central contrast The onset of proliferation coincides with a shifting to 0.5! Z. Hadzibabic et al., in preparation

Rapidly rotating two dimensional condensates Time of flight experiments with rotating condensates correspond to density measurements Interference experiments measure single particle correlation functions in the rotating frame

Full counting statistics of interference between two interacting one dimensional Bose liquids Gritsev, Altman, Demler, Polkovnikov, cond-mat/0602475

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) Explicit expressions for are available but cumbersome Fendley, Lesage, Saleur, J. Stat. Phys. 79:799 (1995)

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

Relation between quantum impurity problem and interference of fluctuating condensates Normalized amplitude of interference fringes Distribution function of fringe amplitudes Relation to the impurity partition function Distribution function can be reconstructed from using completeness relations for the Bessel functions

Bethe ansatz solution for a quantum impurity 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 Interference amplitude and spectral determinant 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

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

From interference amplitudes to conformal field theories correspond to vacuum eigenvalues of Q operators of CFT Bazhanov, Lukyanov, Zamolodchikov, Comm. Math. Phys.1996, 1997, 1999 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, … Yang-Lee singularity 2D quantum gravity, non-intersecting loops on 2D lattice

Quantum noise interferometry in time of flight experiments

Atoms in an optical lattice. Superfluid to Insulator transition Greiner et al., Nature 415:39 (2002) t/U Superfluid Mott insulator What if the ground state cannot be described by single particle matter waves? Current experiments are reaching such regimes. A nice example is the experiment by Markus and collaborators in Munich where cold bosons on an optical lattice were tuned across the SF-Mott transition. Explain experiment. SF well described by a wavefunction in which the zero momentum state is macroscopically occupied. Mott state is rather well described by definite RS occupations.

Time of flight experiments Quantum noise interferometry of atoms in an optical lattice Second order coherence

Second order coherence in the insulating state of bosons 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)

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 coherence in the insulating state of bosons 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)

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 the Mott domains

Applications of quantum noise interferometry Spin order in Mott states of atomic mixtures

Probing spin order of bosons Correlation Function Measurements Extra Bragg peaks appear in the second order correlation function in the AF phase

Applications of quantum noise interferometry Detection of fermion pairing

Fermions with repulsive interactions Possible d-wave pairing of fermions

Positive U Hubbard model Possible phase diagram. Scalapino, Phys. Rep. 250:329 (1995) Antiferromagnetic insulator D-wave pairing of fermions

Second order interference from a BCS superfluid n(k) n(r) n(r’) k F k BCS BEC

Momentum correlations in paired fermions Theory: Altman et al., PRA 70:13603 (2004) Experiment: Greiner et al., PRL 94:110401 (2005)

Fermion pairing in an optical lattice Second Order Interference In the TOF images Normal State Superfluid State measures the Cooper pair wavefunction One can identify unconventional pairing

Conclusions Interference of extended condensates can be used to probe correlation functions in one and two dimensional systems Noise interferometry is a powerful tool for analyzing quantum many-body states in optical lattices