Applied Physics B

, Volume 113, Issue 1, pp 27–39 | Cite as

Single-site- and single-atom-resolved measurement of correlation functions

  • M. Endres
  • M. Cheneau
  • T. Fukuhara
  • C. Weitenberg
  • P. Schauß
  • C. Gross
  • L. Mazza
  • M. C. Bañuls
  • L. Pollet
  • I. Bloch
  • S. Kuhr
Article

Abstract

Correlation functions play an important role for the theoretical and experimental characterization of many-body systems. In solid-state systems, they are usually determined through scattering experiments, whereas in cold gases systems, time-of-flight, and in situ absorption imaging are the standard observation techniques. However, none of these methods allow the in situ detection of spatially resolved correlation functions at the single-particle level. Here, we give a more detailed account of recent advances in the detection of correlation functions using in situ fluorescence imaging of ultracold bosonic atoms in an optical lattice. This method yields single-site- and single-atom-resolved images of the lattice gas in a single experimental run, thus gaining direct access to fluctuations in the many-body system. As a consequence, the detection of correlation functions between an arbitrary set of lattice sites is possible. This enables not only the detection of two-site correlation functions but also the evaluation of non-local correlations, which originate from an extended region of the system and are used for the characterization of quantum phases that do not possess (quasi-)long-range order in the traditional sense.

Notes

Acknowledgments

We thank Jacob Sherson for his contribution to the experimental setup. We acknowledge helpful discussions with Ehud Altman, Emanuele Dalla Torre, Matteo Rizzi, Ignacio Cirac, Andrew Daley, Peter Zoller, Steffen Patrick Rath, Wolfgang Simeth and Wilhelm Zwerger. This work was supported by MPG, DFG, EU (NAMEQUAM, AQUTE, Marie Curie Fellowship to M.C.), and JSPS (Postdoctoral Fellowship for Research Abroad to T.F.). LM acknowledges the economical support from Regione Toscana, POR FSE 2007–2013. DMRG simulations were performed using code released within the PwP project (http://www.qti.sns.it).

References

  1. 1.
    I. Bloch, J. Dalibard, S. Nascimbène, Quantum simulations with ultracold quantum gases. Nat. Phys. 8, 267–276 (2012)CrossRefGoogle Scholar
  2. 2.
    M. Greiner, O. Mandel, T. Esslinger, T.W. Hänsch, I. Bloch, Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms. Nature 415, 39–44 (2002)ADSCrossRefGoogle Scholar
  3. 3.
    B. Paredes, A. Widera, V. Murg, O. Mandel, S. Fölling, J.I. Cirac, G.V. Shlyapnikov, T. Hänsch, I. Bloch, Tonks-Girardeau gas of ultracold atoms in an optical lattice. Nature 429, 277–281 (2004)ADSCrossRefGoogle Scholar
  4. 4.
    I. Bloch, J. Dalibard, W. Zwerger, Many-body physics with ultracold gases. Rev. Mod. Phys. 80, 885–964 (2008)ADSCrossRefGoogle Scholar
  5. 5.
    R. Jördens, N. Strohmaier, K. Günter, H. Moritz, T. Esslinger, A Mott insulator of fermionic atoms in an optical lattice. Nature 455, 204–207 (2008)ADSCrossRefGoogle Scholar
  6. 6.
    U. Schneider, L. Hackermüller, S. Will, T. Best, I. Bloch, T.A. Costi, R.W. Helmes, D. Rasch, A. Rosch, Metallic and insulating phases of repulsively interacting fermions in a 3D optical lattice. Science 322, 1520–1525 (2008)ADSCrossRefGoogle Scholar
  7. 7.
    M. Randeria, W. Zwerger, M. Zwierlein (eds.), The BCS-BEC Crossover and the Unitary Fermi Gas, vol 836. Lecture Notes in Physics (Springer, 2012)Google Scholar
  8. 8.
    Y.-I. Shin, C.H. Schunck, A. Schirotzek, W. Ketterle, Phase diagram of a two-component Fermi gas with resonant interactions. Nature 451, 689–693 (2008)ADSCrossRefGoogle Scholar
  9. 9.
    N. Gemelke, X. Zhang, C.-L. Hung, C. Chin, In situ observation of incompressible Mott-insulating domains in ultracold atomic gases. Nature 460, 995–998 (2009)ADSCrossRefGoogle Scholar
  10. 10.
    S. Nascimbène, N. Navon, K.J. Jiang, F. Chevy, C. Salomon, Exploring the thermodynamics of a universal Fermi gas. Nature 463, 1057–1060 (2010)ADSCrossRefGoogle Scholar
  11. 11.
    J.T. Stewart, J.P. Gaebler, D.S. Jin, Using photoemission spectroscopy to probe a strongly interacting Fermi gas. Nature 454, 744–747 (2008)ADSCrossRefGoogle Scholar
  12. 12.
    P.T. Ernst, S. Götze, J.S. Krauser, K. Pyka, D.-S. Lühmann, D. Pfannkuche, K. Sengstock, Probing superfluids in optical lattices by momentum-resolved Bragg spectroscopy. Nat. Phys. 6, 56–61 (2009)CrossRefGoogle Scholar
  13. 13.
    W.S. Bakr, J.I. Gillen, A. Peng, S. Fölling, M. Greiner, A quantum gas microscope for detecting single atoms in a Hubbard-regime optical lattice. Nature 462, 74–77 (2009)ADSCrossRefGoogle Scholar
  14. 14.
    W.S. Bakr, A. Peng, M.E. Tai, R. Ma, J. Simon, J.I. Gillen, S. Fölling, L. Pollet, M. Greiner, Probing the superfluid-to-Mott insulator transition at the single-atom level. Science 329, 547–550 (2010)ADSCrossRefGoogle Scholar
  15. 15.
    J.F. Sherson, C. Weitenberg, M. Endres, M. Cheneau, I. Bloch, S. Kuhr, Single-atom-resolved fluorescence imaging of an atomic Mott insulator. Nature 467, 68–72 (2010)ADSCrossRefGoogle Scholar
  16. 16.
    M.P.A. Fisher, P.B. Weichman, G. Grinstein, D.S. Fisher, Boson localization and the superfluid-insulator transition. Phys. Rev. B 40, 546–570 (1989)ADSCrossRefGoogle Scholar
  17. 17.
    D. Jaksch, C. Bruder, J.I. Cirac, C. Gardiner, P. Zoller, Cold Bosonic atoms in optical lattices. Phys. Rev. Lett. 81, 3108–3111 (1998)ADSCrossRefGoogle Scholar
  18. 18.
    S. Sachdev, Quantum Phase Transitions, 2nd edn. (Cambridge University Press, Cambridge, 2011). ISBN 0521514681Google Scholar
  19. 19.
    M. Endres, T. Fukuhara, D. Pekker, M. Cheneau, P. Schauss, C. Gross, E. Demler, S. Kuhr, I. Bloch, The ’Higgs’ amplitude mode at the two-dimensional superfluid/Mott insulator transition. Nature 487, 454–458 (2012)ADSCrossRefGoogle Scholar
  20. 20.
    L. Pollet, N. Prokof’ev, Higgs mode in a two-dimensional superfluid. Phys. Rev. Lett. 109, 010401 (2012)ADSCrossRefGoogle Scholar
  21. 21.
    J. Simon, W.S. Bakr, R. Ma, M.E. Tai, P.M. Preiss, M. Greiner, Quantum simulation of antiferromagnetic spin chains in an optical lattice. Nature 472, 307–312 (2011)ADSCrossRefGoogle Scholar
  22. 22.
    W.S. Bakr, P.M. Preiss, M.E. Tai, R. Ma, J. Simon, M. Greiner, Orbital excitation blockade and algorithmic cooling in quantum gases. Nature 480, 500–503 (2011)ADSCrossRefGoogle Scholar
  23. 23.
    C. Weitenberg, M. Endres, J.F. Sherson, M. Cheneau, P. Schauß, T. Fukuhara, I. Bloch, S. Kuhr, Single-spin addressing in an atomic Mott insulator. Nature 471, 319–324 (2011)ADSCrossRefGoogle Scholar
  24. 24.
    T. Fukuhara, A. Kantian, M. Endres, M. Cheneau, P. Schauss, S. Hild, D. Bellem, U. Schollwöck, T. Giamarchi, C. Gross, I. Bloch, S. Kuhr, Quantum dynamics of a mobile spin impurity. Nat. Phys. 9, 235–241 (2013)Google Scholar
  25. 25.
    M. Endres, M. Cheneau, T. Fukuhara, C. Weitenberg, P. Schauss, C. Gross, L. Mazza, M.C. Banuls, L. Pollet, I. Bloch, S. Kuhr, Observation of correlated particle-hole pairs and string order in low-dimensional Mott insulators. Science 334, 200–203 (2011)ADSCrossRefGoogle Scholar
  26. 26.
    M. Cheneau, P. Barmettler, D. Poletti, M. Endres, P. Schauss, T. Fukuhara, C. Gross, I. Bloch, C. Kollath, S. Kuhr, Light-cone-like spreading of correlations in a quantum many-body system. Nature 481, 484–487 (2012)ADSCrossRefGoogle Scholar
  27. 27.
    R. Grimm, M. Weidemüller, Y.B. Ovchinnikov, Optical dipole traps for neutral atoms. Adv. Atom. Mol. Opt. Phys. 42, 95 (2006)ADSCrossRefGoogle Scholar
  28. 28.
    Z. Hadzibabic, P. Krüger, M. Cheneau, S.P. Rath, J. Dalibard, The trapped two-dimensional Bose gas: from Bose–Einstein condensation to Berezinskii–Kosterlitz–Thouless physics. New J. Phys. 10, 045006 (2008)ADSCrossRefGoogle Scholar
  29. 29.
    C. Weitenberg, Single-Atom Resolved Imaging and Manipulation in an Atomic Mott Insulator. PhD thesis, Ludwig-Maximilians-Universität München, 2011Google Scholar
  30. 30.
    J. Weiner, V. Bagnato, S. Zilio, P. Julienne, Experiments and theory in cold and ultracold collisions. Rev. Mod. Phys. 71, 1–85 (1999)ADSCrossRefGoogle Scholar
  31. 31.
    B. Capogrosso-Sansone, S. Söyler, N. Prokof’ev, B. Svistunov, Monte Carlo study of the two-dimensional Bose–Hubbard model. Phys. Rev. A 77, 015602 (2008)ADSCrossRefGoogle Scholar
  32. 32.
    F. Anfuso, A. Rosch, Fragility of string orders. Phys. Rev. B 76, 085124 (2007)ADSCrossRefGoogle Scholar
  33. 33.
    den M. Nijs, K. Rommelse, Preroughening transitions in crystal surfaces and valence-bond phases in quantum spin chains. Phys. Rev. B 40, 4709 (1989)ADSCrossRefGoogle Scholar
  34. 34.
    H. Kruis, I. McCulloch, Z. Nussinov, J. Zaanen, Geometry and the hidden order of Luttinger liquids: the universality of squeezed space. Phys. Rev. B 70, 075109 (2004)ADSCrossRefGoogle Scholar
  35. 35.
    D. Pérez-García, M. Wolf, M. Sanz, F. Verstraete, J.I. Cirac, String order and symmetries in quantum spin lattices. Phys. Rev. Lett. 100, 167202 (2008)ADSCrossRefGoogle Scholar
  36. 36.
    J.B. Kogut, An introduction to lattice gauge theory and spin systems. Rev. Mod. Phys. 51, 659–713 (1979)MathSciNetADSCrossRefGoogle Scholar
  37. 37.
    Dalla E.G. Torre, E. Berg, E. Altman, Hidden order in 1d bose insulators. Phys. Rev. Lett. 97, 260401 (2006)CrossRefGoogle Scholar
  38. 38.
    E. Berg, Dalla E. Torre, T. Giamarchi, E. Altman, Rise and fall of hidden string order of lattice bosons. Phys. Rev. B 77, 245119 (2008)ADSCrossRefGoogle Scholar
  39. 39.
    E. Kim, G. Fáth, J. Sólyom, D. Scalapino, Phase transitions between topologically distinct gapped phases in isotropic spin ladders. Phys. Rev. B 62, 14965–14974 (2000)ADSCrossRefGoogle Scholar
  40. 40.
    F. Anfuso, A. Rosch, String order and adiabatic continuity of Haldane chains and band insulators. Phys. Rev. B 75, 144420 (2007)ADSCrossRefGoogle Scholar
  41. 41.
    F. Verstraete, M. Martín-Delgado, J. Cirac, Diverging entanglement length in gapped quantum spin systems. Phys. Rev. Lett. 92, 087201 (2004)ADSCrossRefGoogle Scholar
  42. 42.
    F. Verstraete, M. Popp, J. Cirac, Entanglement versus correlations in spin systems. Phys. Rev. Lett. 92, 027901 (2004)ADSCrossRefGoogle Scholar
  43. 43.
    M. Popp, F. Verstraete, M. Martín-Delgado, J. Cirac, Localizable entanglement. Phys. Rev. A 71, 042306 (2005)MathSciNetADSCrossRefGoogle Scholar
  44. 44.
    L. Venuti, M. Roncaglia, Analytic relations between localizable entanglement and string correlations in spin systems. Phys. Rev. Lett. 94, 207207 (2005)ADSCrossRefGoogle Scholar
  45. 45.
    J. García-Ripoll, M. Martin-Delgado, J. Cirac, Implementation of spin hamiltonians in optical lattices. Phys. Rev. Lett. 93, 250405 (2004)ADSCrossRefGoogle Scholar
  46. 46.
    E. Kapit, E. Mueller, Even-odd correlation functions on an optical lattice. Phys. Rev. A 82, 013644 (2010)ADSCrossRefGoogle Scholar
  47. 47.
    F. Gerbier, Boson Mott insulators at finite temperatures. Phys. Rev. Lett. 99, 120405 (2007)ADSCrossRefGoogle Scholar
  48. 48.
    V. Kashurnikov, B. Svistunov, Exact diagonalization plus renormalization-group theory: accurate method for a one-dimensional superfluid-insulator-transition study. Phys. Rev. B 53, 11776–11778 (1996)ADSCrossRefGoogle Scholar
  49. 49.
    T.D. Kühner, S.R. White, H. Monien, One-dimensional Bose–Hubbard model with nearest-neighbor interaction. Phys. Rev. B 61, 12474–12489 (2000)ADSCrossRefGoogle Scholar
  50. 50.
    M. Zwolak, G. Vidal, Mixed-state dynamics in one-dimensional quantum lattice systems: a time-dependent superoperator renormalization algorithm. Phys. Rev. Lett. 93, 207205 (2004)ADSCrossRefGoogle Scholar
  51. 51.
    M. Endres, Probing correlated quantum many-body systems at the single-particle level. PhD thesis, Ludwig-Maximilians-Universität München (2013)Google Scholar
  52. 52.
    S.P. Rath, W. Simeth, M. Endres, W. Zwerger, Non-local order in Mott insulators, Duality and Wilson Loops. Ann. Phys. 334, 256–271 (2013)Google Scholar
  53. 53.
    T. Kühner, H. Monien, Phases of the one-dimensional Bose–Hubbard model. Phys. Rev. B 58, R14741–R14744 (1998)ADSCrossRefGoogle Scholar
  54. 54.
    R. Kubo, Generalized cumulant expansion method. J. Phys. Soc. Jpn. 17, 1100–1120 (1962)MathSciNetADSCrossRefMATHGoogle Scholar
  55. 55.
    P. Schauß, M. Cheneau, M. Endres, T. Fukuhara, S. Hild, A. Omran, T. Pohl, C. Gross, S. Kuhr, I. Bloch, Observation of spatially ordered structures in a two-dimensional Rydberg gas. Nature 490, 87–91 (2012)ADSCrossRefGoogle Scholar
  56. 56.
    J. Honer, H. Weimer, T. Pfau, H. Büchler, Collective many-body interaction in Rydberg dressed atoms. Phys. Rev. Lett. 105, 160404 (2010)ADSCrossRefGoogle Scholar
  57. 57.
    G. Pupillo, A. Micheli, M. Boninsegni, I. Lesanovsky, P. Zoller, Strongly correlated gases of Rydberg-dressed atoms: quantum and classical dynamics. Phys. Rev. Lett. 104, 223002 (2010)ADSCrossRefGoogle Scholar
  58. 58.
    L. Amico, A. Osterloh, V. Vedral, Entanglement in many-body systems. Rev. Mod. Phys. 80, 517–576 (2008)MathSciNetADSCrossRefMATHGoogle Scholar
  59. 59.
    A. Daley, H. Pichler, J. Schachenmayer, P. Zoller, Measuring entanglement growth in quench dynamics of bosons in an optical lattice. Phys. Rev. Lett. 109, 020505 (2012)ADSCrossRefGoogle Scholar
  60. 60.
    H. Pichler, L. Bonnes, A.J. Daley, A.M. Läuchli, P. Zoller, Thermal vs. entanglement entropy: a measurement protocol for fermionic atoms with a quantum gas microscope. New J. Phys. 15, 063003 (2013) Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • M. Endres
    • 1
  • M. Cheneau
    • 1
  • T. Fukuhara
    • 1
  • C. Weitenberg
    • 1
    • 2
  • P. Schauß
    • 1
  • C. Gross
    • 1
  • L. Mazza
    • 1
    • 3
  • M. C. Bañuls
    • 1
  • L. Pollet
    • 4
  • I. Bloch
    • 1
    • 4
  • S. Kuhr
    • 1
    • 5
  1. 1.Max-Planck-Institut für QuantenoptikGarchingGermany
  2. 2.Laboratoire Kastler Brossel, CNRS, UPMCEcole Normale SupérieureParisFrance
  3. 3.Scuola Normale SuperiorePisaItaly
  4. 4.Ludwig-Maximilians-UniversitätMunichGermany
  5. 5.Department of Physics, SUPAUniversity of StrathclydeGlasgowUK

Personalised recommendations