Quantum simulation of antiferromagnetic spin chains in an optical lattice

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Understanding exotic forms of magnetism in quantum mechanical systems is a central goal of modern condensed matter physics, with implications for systems ranging from high-temperature superconductors to spintronic devices. Simulating magnetic materials in the vicinity of a quantum phase transition is computationally intractable on classical computers, owing to the extreme complexity arising from quantum entanglement between the constituent magnetic spins. Here we use a degenerate Bose gas of rubidium atoms confined in an optical lattice to simulate a chain of interacting quantum Ising spins as they undergo a phase transition. Strong spin interactions are achieved through a site-occupation to pseudo-spin mapping. As we vary a magnetic field, quantum fluctuations drive a phase transition from a paramagnetic phase into an antiferromagnetic phase. In the paramagnetic phase, the interaction between the spins is overwhelmed by the applied field, which aligns the spins. In the antiferromagnetic phase, the interaction dominates and produces staggered magnetic ordering. Magnetic domain formation is observed through both in situ site-resolved imaging and noise correlation measurements. By demonstrating a route to quantum magnetism in an optical lattice, this work should facilitate further investigations of magnetic models using ultracold atoms, thereby improving our understanding of real magnetic materials.

At a glance


  1. Spin model and its phase diagram.
    Figure 1: Spin model and its phase diagram.

    a, An antiferromagnetic one-dimensional Ising chain in longitudinal (hz) and transverse(hx) magnetic fields exhibits two phases at zero temperature39, 40: an antiferromagnetic phase in weak fields, and a paramagnetic phase in strong fields. These phases are separated by a second-order phase transition (red line), except at the multicritical point (hz, hx) = (1, 0), where the absence of quantum fluctuations produces a classical first-order transition. The region accessible in our experiment is highlighted in blue. b, Here the Hamiltonian may be decomposed into a constraint term (shaded red) that prevents adjacent ‘down’-spins, and field terms (shaded blue) that drive the phase transition. See text for details.

  2. Tilted Hubbard model and mapping to spin model.
    Figure 2: Tilted Hubbard model and mapping to spin model.

    Consider first the middle row. a, When a Mott insulator is tilted by E per lattice site, it maintains unity occupancy until E reaches the on-site interaction energy U. b, As the energy cost to tunnel, Δ = EU, vanishes, an atom can tunnel to the neighbouring site if the atom on that site has not itself tunnelled. Otherwise, the tilt E inhibits tunnelling, producing a strong constraint. c, Tilting further, the system undergoes a transition into a doubly degenerate staggered phase. d, Here we show how this system maps onto interacting spin-½ particles, whose two spin states correspond to the two possible locations of each atom. The tunnelling constraint forbids adjacent down spins, realizing a spin–spin interaction. Top row: the initial Mott insulator corresponds to a paramagnet (a), the state at resonant tilt to an entangled (critical) spin configuration (b), and staggered ordering at large tilt to an antiferromagnet (c). Bottom row: parity-sensitive site-resolved imaging results in bright paramagnetic (a; o, odd), and dark antiferromagnetic (c; e, even) domains.

  3. Probing the paramagnet to antiferromagnet phase transition.
    Figure 3: Probing the paramagnet to antiferromagnet phase transition.

    A, Single-shot images as the tilt is swept through the phase transition in 250ms. A, a, n = 1 (bright) and n = 2 (centre dark) Mott insulator shells in the paramagnetic (PM) phase. A, b, ‘Inverted’ shells characteristic of the staggered ordering of the antiferromagnetic (AF) phase. A, c, Several chains (red rectangles in a and b) of the n = 1 shell at various times during the sweep (left to right, in ms: 0, 50, 100, 150, 175, 250), showing antiferromagnetic domain formation. B, Demonstration of reversibility by tuning across the transition and back. Data points are podd, red line is tilt E/U, both versus time. C, A closer look at the paramagnetic to antiferromagnetic quantum phase transition in an n = 1 shell. Error bars, 1σ statistical errors in the region-averaged mean podd. Blue curve is a guide for the eye. Insets, state of the system at the beginning and end of the sweep. D, One-dimensional noise correlation measurement, with peaks at P = h/a in the paramagnetic phase (a), and additional peaks at P = h/2a in the antiferromagnetic phase (b).

  4. Effect of harmonic confinement.
    Figure 4: Effect of harmonic confinement.

    a, For a ramp (in E/U; see Fig. 3B) across the transition and back in 500ms, harmonic confinement broadens the transition region, inducing two rows (red and black data points) separated by seven lattice sites to go dark at different applied tilts. Red and black dashed lines are guides to the eye. b, Energy spectrum (black lines) of a one-dimensional Ising chain in a longitudinal field gradient (see Supplementary Information). Avoided crossings of the ground state correspond to single spin-flips. c, After nulling the confinement, the two rows undergo the transition together. d, Energy spectrum for a homogeneous chain. A single avoided crossing drives the many-body transition, whose gap scales inversely with system size. Solid and dashed blue arrows denote forward- and subsequent reverse-tilt ramps in all panels. Error bars are 1σ statistical uncertainties in the region-averaged mean of podd.

  5. Site-resolved transition in near-homogeneous Ising model.
    Figure 5: Site-resolved transition in near-homogeneous Ising model.

    Shown is podd for six individual lattice sites forming a contiguous one-dimensional chain, plotted against the tilt, for both forward (black) and reverse (red) ramps of 250ms. The spins are observed to undergo the transition at the same applied field to within the curve width, set by quantum fluctuations. Typical 1σ statistical error bar is shown. The reverse curve demonstrates the preparation of the highest-energy state of the restricted spin Hamiltonian, exhibiting paramagnetic ordering on the antiferromagnetic side of the transition, and antiferromagnetic ordering on the paramagnetic side.

  6. Dynamics of antiferromagnetic domain formation.
    Figure 6: Dynamics of antiferromagnetic domain formation.

    a, Within a single six-site chain with low disorder, the mean dark-domain length is plotted against tilt for forward (black) and reverse (red) ramps, as it grows to nearly the chain length. The reverse ramp produces antiferromagnetic domains on the paramagnetic side of the transition—corresponding to the highest-energy states of the spin Hamiltonian (inset). Arrows denote ramp direction. b, Within the same chain, podd (blue) and dark domain length (black) are plotted against ramp time from E/U = 0.7 to E/U = 1.2. Error bars for dark domain lengths are 1σ statistical uncertainties arising from the number of detected domains of each length; those for podd are 1σ statistical uncertainties in the mean.


  1. Balents, L. Spin liquids in frustrated magnets. Nature 464, 199208 (2010)
  2. Binder, K. & Young, A. P. Spin glasses: experimental facts, theoretical concepts, and open questions. Rev. Mod. Phys. 58, 801976 (1986)
  3. Kitaev, A. Anyons in an exactly solved model and beyond. Ann. Phys. 321, 2111 (2006)
  4. Sachdev, S. Quantum Phase Transitions (Cambridge Univ. Press, 2001)
  5. Anderson, P. W. The resonating valence bond state in La2CuO4 and superconductivity. Science 235, 11961198 (1987)
  6. Rüegg, C. et al. Quantum magnets under pressure: controlling elementary excitations in TlCuCl3. Phys. Rev. Lett. 100, 205701 (2008)
  7. Coldea, R. et al. Quantum criticality in an Ising chain: experimental evidence for emergent E8 symmetry. Science 327, 177180 (2010)
  8. Lewenstein, M. et al. Ultracold atomic gases in optical lattices: mimicking condensed matter physics and beyond. Adv. Phys. 56, 243379 (2007)
  9. Bloch, I., Dalibard, J. & Zwerger, W. Many-body physics with ultracold gases. Rev. Mod. Phys. 80, 885964 (2008)
  10. Greiner, M., Mandel, O., Esslinger, E., Haensch, T. W. & Bloch, I. Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms. Nature 415, 3944 (2002)
  11. Sachdev, S. Quantum magnetism and criticality. Nature Phys. 4, 173185 (2008)
  12. Zhang, X., Hung, C.-L., Tung, S.-K., Gemelke, N. & Chin, C. Exploring quantum criticality based on ultracold atoms in optical lattices. Preprint at left fencehttp://arXiv.org/abs/1101.0284right fence (2010)
  13. Hung, C.-L., Zhang, X., Gemelke, N. & Chin, C. Observation of scale invariance and universality in two-dimensional Bose gases. Nature 470, 236239 (2011)
  14. Bakr, W. S., Gillen, J. I., Peng, A., Foelling, S. & Greiner, M. A quantum gas microscope for detecting single atoms in a Hubbard-regime optical lattice. Nature 462, 7477 (2009)
  15. Bakr, W. S. et al. Probing the superfluid-to-Mott insulator transition at the single-atom level. Science 329, 547550 (2010)
  16. Sherson, J. F. et al. Single-atom-resolved fluorescence imaging of an atomic Mott insulator. Nature 467, 6872 (2010)
  17. Weitenberg, C. et al. Single-spin addressing in an atomic Mott insulator. Nature 471, 319324 (2011)
  18. Jo, G. et al. Itinerant ferromagnetism in a Fermi gas of ultracold atoms. Science 325, 15211524 (2009)
  19. Friedenauer, A., Schmitz, H., Glueckert, J. T., Porras, D. & Schaetz, T. Simulating a quantum magnet with trapped ions. Nature Phys. 4, 757761 (2008)
  20. Kim, K. et al. Quantum simulation of frustrated Ising spins with trapped ions. Nature 465, 590593 (2010)
  21. Ni, K.-K. et al. A high phase-space-density gas of polar molecules. Science 322, 231235 (2008)
  22. Saffman, M. & Walker, T. G. Quantum information with Rydberg atoms. Rev. Mod. Phys. 82, 23132363 (2010)
  23. Lukin, M. D. et al. Dipole blockade and quantum information processing in mesoscopic atomic ensembles. Phys. Rev. Lett. 87, 037901 (2001)
  24. Büchler, H. P. et al. Strongly correlated 2D quantum phases with cold polar molecules: controlling the shape of the interaction potential. Phys. Rev. Lett. 98, 060404 (2007)
  25. Weimer, H., Müller, M. & Lesanovsky, I. A Rydberg quantum simulator. Nature Phys. 6, 382388 (2010)
  26. Lee, P. J. et al. Sublattice addressing and spin-dependent motion of atoms in a double-well lattice. Phys. Rev. Lett. 99, 020402 (2007)
  27. Soltan-Panahi, P. et al. Multi-component quantum gases in spin-dependent hexagonal lattices. Nature Phys. advance online publication. doi:10.1038/nphys1916 (13 February 2011)
  28. Fölling, S. et al. Direct observation of second-order atom tunnelling. Nature 448, 10291032 (2007)
  29. Fisher, M. P. A., Weichman, P. B., Grinstein, G. & Fisher, D. S. Boson localization and the superfluid-insulator transition. Phys. Rev. B 40, 546570 (1989)
  30. Jaksch, D., Bruder, C., Cirac, J. I., Gardiner, C. W. & Zoller, P. Cold bosonic atoms in optical lattices. Phys. Rev. Lett. 81, 31083111 (1998)
  31. Sachdev, S., Sengupta, K. & Girvin, S. M. Mott insulators in strong electric fields. Phys. Rev. B 66, 075128 (2002)
  32. Duan, L. M., Demler, E. & Lukin, M. D. Controlling spin exchange interactions of ultracold atoms in optical lattices. Phys. Rev. Lett. 91, 090402 (2003)
  33. Trotzky, S. et al. Time-resolved observation and control of superexchange interactions with ultracold atoms in optical lattices. Science 319, 295299 (2008)
  34. Capogrosso-Sansone, B., Soeyler, S. G., Prokof'ev, N. V. & Svistunov, B. V. Critical entropies for magnetic ordering in bosonic mixtures on a lattice. Phys. Rev. A 81, 053622 (2010)
  35. Weld, D. M. et al. Spin gradient thermometry for ultracold atoms in optical lattices. Phys. Rev. Lett. 103, 245301 (2009)
  36. Medley, P., Weld, D., Miyake, H., Pritchard, D. E. & Ketterle, W. Spin gradient demagnetization cooling of ultracold atoms. Preprint at left fencehttp://arxiv.org/abs/1006.4674right fence (2010)
  37. McKay, D. & DeMarco, B. Cooling in strongly correlated optical lattices: prospects and challenges. Preprint at left fencehttp://arxiv.org/abs/1010.0198right fence (2010)
  38. García-Ripoll, J. J., Martin-Delgado, M. A. & Cirac, J. I. Implementation of spin Hamiltonians in optical lattices. Phys. Rev. Lett. 93, 250405 (2004)
  39. Novotny, M. A. & Landau, D. P. Zero temperature phase diagram for the d = 1 quantum Ising antiferromagnet. J. Magn. Magn. Mater. 54-57, 685686 (1986)
  40. Ovchinnikov, A. A., Dmitriev, D. V., Krivnov, V. Y. & Cheranovskii, V. O. Antiferromagnetic Ising chain in a mixed transverse and longitudinal magnetic field. Phys. Rev. B 68, 214406 (2003)
  41. Imry, Y. & Ma, S.-k. Random-field instability of the ordered state of continuous symmetry. Phys. Rev. Lett. 35, 13991401 (1975)
  42. Dziarmaga, J. Dynamics of a quantum phase transition in the random Ising model: logarithmic dependence of the defect density on the transition rate. Phys. Rev. B 74, 064416 (2006)
  43. Vidal, G., Latorre, J. I., Rico, E. & Kitaev, A. Entanglement in quantum critical phenomena. Phys. Rev. Lett. 90, 227902 (2003)
  44. Altman, E., Demler, E. & Lukin, M. D. Probing many-body states of ultracold atoms via noise correlations. Phys. Rev. A 70, 013603 (2004)
  45. Fölling, S. et al. Spatial quantum noise interferometry in expanding ultracold atom clouds. Nature 434, 481484 (2005)
  46. Sørensen, A. S. et al. Adiabatic preparation of many-body states in optical lattices. Phys. Rev. A 81, 061603 (2010)
  47. Bañuls, M. C., Cirac, J. I. & Hastings, M. B. Strong and weak thermalization of infinite nonintegrable quantum systems. Phys. Rev. Lett. 106, 050405 (2011)
  48. Plötz, P., Schlagheck, P. & Wimberger, S. Effective spin model for interband transport in a Wannier-Stark lattice system. Eur. Phys. J. D doi:10.1140/epjd/e2010-10554-7 (2010)
  49. Pielawa, S., Kitagawa, T., Berg, E. & Sachdev, S. Correlated phases of bosons in tilted, frustrated lattices. Preprint at left fencehttp://arxiv.org/abs/1101.2897right fence (2010)

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Author information


  1. Department of Physics, Harvard University, 17 Oxford Street, Cambridge, Massachusetts 02138, USA

    • Jonathan Simon,
    • Waseem S. Bakr,
    • Ruichao Ma,
    • M. Eric Tai,
    • Philipp M. Preiss &
    • Markus Greiner


All authors contributed to the construction of the experiment, the collection and analysis of the data, and the writing of the manuscript.

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The authors declare no competing financial interests.

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  1. Supplementary Information (547K)

    The file contains Supplementary Discussions, Supplementary Table 1, Supplementary Figures 1-3 with legends and additional references.

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