Observation of Bose–Einstein condensation in a strong synthetic magnetic field

Journal name:
Nature Physics
Year published:
Published online


Extensions of Berrys phase and the quantum Hall effect have led to the discovery of new states of matter with topological properties. Traditionally, this has been achieved using magnetic fields or spin–orbit interactions, which couple only to charged particles. For neutral ultracold atoms, synthetic magnetic fields have been created that are strong enough to realize the Harper–Hofstadter model. We report the first observation of Bose–Einstein condensation in this system and study the Harper–Hofstadter Hamiltonian with one-half flux quantum per lattice unit cell. The diffraction pattern of the superfluid state directly shows the momentum distribution of the wavefunction, which is gauge-dependent. It reveals both the reduced symmetry of the vector potential and the twofold degeneracy of the ground state. We explore an adiabatic many-body state preparation protocol via the Mott insulating phase and observe the superfluid ground state in a three-dimensional lattice with strong interactions.

At a glance


  1. Observation of Bose-Einstein condensation in the Harper-Hofstadter model.
    Figure 1: Observation of Bose–Einstein condensation in the Harper–Hofstadter model.

    a, Spatial structure of the cubic lattice with the synthetic vector potential—(dashed) x-bonds feature a spatially dependent tunnelling phase, whereas tunnelling along (solid) y-links is the normal tunnelling. The synthetic magnetic field generates a lattice unit cell that is twice as large as the bare cubic lattice (green diamond). b, The band structure of the lowest band shows a twofold degeneracy of the ground state. The magnetic Brillouin zone (green diamond) has half the area of the original Brillouin zone. Owing to the twofold degeneracy, the primitive cell of the band structure is even smaller (doubly reduced Brillouin zone, brown square). These lattice symmetries are both revealed in time-of-flight pictures (shown in gj) showing the momentum distribution of the wavefunction. cf, Schematics of the momentum peaks of a superfluid. The dominant momentum peak (filled circle) is equal to the quasimomentum of the ground state. Owing to the spatial periodicity of the wavefunction, additional momentum peaks (open circles) appear, separated by reciprocal lattice vectors (green arrows) or vectors connecting degenerate states in the band structure (brown arrows). gj, Time-of-flight images. The superfluid ground state of the normal cubic lattice is shown in g compared with different repetitions of the same sequence for the superfluid ground state of the HH lattice hj. In h, only one minimum of the band structure is filled, directly demonstrating the symmetry in our chosen gauge. The number of momentum components in i,j is doubled again owing to population of both degenerate ground states. The micromotion of the Floquet Hamiltonian is illustrated in e,f,i,j as a periodically shifted pattern in the x direction, analogous to a Bloch oscillation. All diffraction images have a field of view of 631μm × 631μm and were taken at a lattice depth of 11Er and 2.7kHz Raman coupling, with at least 30ms hold in the HH lattice.

  2. Population imbalance of the two ground states of the HH Hamiltonian with 1/2 flux.
    Figure 2: Population imbalance of the two ground states of the HH Hamiltonian with 1/2 flux.

    a, Band mapping sequence adiabatically connecting quasimomentum to free space momentum. The Raman beams were ramped down from the initial strength of 1.4Er to zero in 0.88ms, followed by a linear ramp down of the lattice beams from 11Er to zero in 0.43ms. b, The histogram shows the relative population imbalance of the two degenerate minima. Equal population in the two diffraction peaks is suggestive of domain formation due to spontaneous symmetry breaking, but can also be driven by lattice noise and technical fluctuations. The data consists of 30 shots taken after a hold time of 29.4ms in the HH lattice. The inset shows a raw image for the band mapping of the 1/2 flux superfluid with two degenerate ground states compared with a topologically trivial superfluid with one ground state (see text). The Brillouin zones of the cubic lattice (grey) and the gauge (green) are overlaid for clarity.

  3. Decay of Bose-Einstein condensates in modulated lattices.
    Figure 3: Decay of Bose–Einstein condensates in modulated lattices.

    The figure compares the decay of the 1/2 flux HH superfluid (red circles) against the decay of the AM superfluid (blue squares). Note that the lower visibility of the HH superfluid is due to the peak doubling, which at the same condensate fraction leads to lower visibility. Exponential fits to the decay of the visibility of the diffraction patterns give lifetimes of 142 ± 15ms and 71 ± 8ms, respectively. Data were taken with an 11Er cubic lattice with either 2.7kHz Raman coupling or 20% amplitude modulation, and start after a 10ms hold time after switching on the final Hamiltonian using the non-adiabatic procedure (see Methods). Uncertainty is given by the statistical error of the mean of five repetitions of the experiment, added in quadrature to uncertainty in the peak visibility fitting.

  4. Experimental sequences for two different state preparation protocols.
    Figure 4: Experimental sequences for two different state preparation protocols.

    The non-adiabatic scheme switches suddenly from a standard 2D lattice to the HH lattice. The adiabatic protocol uses a quantum phase transition to the 3D Mott insulator as an intermediate step. For this, lattice beams are adiabatically ramped up in all three directions to 20Er to enter the Mott insulating phase, after which the hyperfine state is flipped with a 0.29ms radiofrequency sweep, as done in the non-adiabatic scheme. The lattices are then ramped down to their final values while the Raman lattice is ramped up in 35ms. Lifetimes of both methods are given in the top corner of each figure; the lifetime of the adiabatic approach is measured from the end of the 35ms ramp. Errors are statistical and given by the exponential fit to the decay of the peak visibility.

  5. HH model with strong interactions.
    Figure 5: HH model with strong interactions.

    Lifetime of the visibility of the diffraction pattern versus z-lattice depth. The top axis shows the Hubbard interaction parameter U. All lifetimes are measured from the end of the 35ms ramp exiting the Mott insulator. Uncertainty is given by the statistical error of the mean of five repetitions of the experiment, added in quadrature to uncertainty in the peak visibility fitting.


  1. Chen, X., Gu, Z.-C., Liu, Z.-X. & Wen, X.-G. Symmetry-protected topological orders in interacting bosonic systems. Science 338, 16041606 (2012).
  2. Wang, C., Potter, A. C. & Senthil, T. Classification of interacting electronic topological insulators in three dimensions. Science 343, 629631 (2014).
  3. Willett, R. et al. Observation of an even-denominator quantum number in the fractional quantum Hall effect. Phys. Rev. Lett. 59, 17761779 (1987).
  4. Fu, L. & Kane, C. L. Superconducting proximity effect and Majorana fermions at the surface of a topological insulator. Phys. Rev. Lett. 100, 096407 (2008).
  5. Deng, M. T. et al. Anomalous zero-bias conductance peak in a Nb–InSb nanowire–Nb hybrid device. Nano Lett. 12, 64146419 (2012).
  6. Nadj-Perge, S. et al. Observation of Majorana fermions in ferromagnetic atomic chains on a superconductor. Science 346, 602607 (2014).
  7. Madison, K. W., Chevy, F., Wohlleben, W. & Dalibard, J. Vortex formation in a stirred Bose–Einstein condensate. Phys. Rev. Lett. 84, 806809 (2000).
  8. Abo-Shaeer, J., Raman, C., Vogels, J. & Ketterle, W. Observation of vortex lattices in Bose–Einstein condensates. Science 292, 476479 (2001).
  9. Gemelke, N., Sarajlic, E. & Chu, S. Rotating few-body atomic systems in the fractional quantum Hall regime. Preprint at http://arXiv.org/abs/1007.2677 (2010).
  10. Jaksch, D. & Zoller, P. Creation of effective magnetic fields in optical lattices: the Hofstadter butterfly for cold neutral atoms. New J. Phys. 5, 56 (2003).
  11. Kolovsky, A. R. Creating artificial magnetic fields for cold atoms by photon-assisted tunneling. Europhys. Lett. 93, 20003 (2011).
  12. Lin, Y.-J., Compton, R. L., Jimenez-Garcia, K., Porto, J. V. & Spielman, I. B. Synthetic magnetic fields for ultracold neutral atoms. Nature 462, 628632 (2009).
  13. Aidelsburger, M. et al. Realization of the Hofstadter Hamiltonian with ultracold atoms in optical lattices. Phys. Rev. Lett. 111, 185301 (2013).
  14. Miyake, H., Siviloglou, G. A., Kennedy, C. J., Burton, W. C. & Ketterle, W. Realizing the Harper Hamiltonian with laser-assisted tunneling in optical lattices. Phys. Rev. Lett. 111, 185302 (2013).
  15. Struck, J. et al. Engineering Ising-XY spin-models in a triangular lattice using tunable artificial gauge fields. Nature Phys. 9, 738743 (2013).
  16. Jotzu, G. et al. Experimental realization of the topological Haldane model with ultracold fermions. Nature 515, 237240 (2014).
  17. Harper, P. G. Single band motion of conduction electrons in a uniform magnetic field. Proc. Phys. Soc. A 68, 874878 (1955).
  18. Azbel, M. Y. Energy spectrum of a conduction electron in a magnetic field. Sov. Phys. JETP 19, 634645 (1964).
  19. Hofstadter, D. R. Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields. Phys. Rev. B 14, 22392249 (1976).
  20. Aidelsburger, M. et al. Measuring the Chern number of Hofstadter bands with ultracold bosonic atoms. Nature Phys. 11, 162166 (2015).
  21. Goldman, N., Dalibard, J., Aidelsburger, M. & Cooper, N. R. Periodically driven quantum matter: The case of resonant modulations. Phys. Rev. A 91, 033632 (2015).
  22. Bukov, M. & Polkovnikov, A. Stroboscopic versus nonstroboscopic dynamics in the Floquet realization of the Harper–Hofstadter Hamiltonian. Phys. Rev. A 90, 043613 (2014).
  23. Bilitewski, T. & Cooper, N. R. Scattering theory for Floquet–Bloch states. Phys. Rev. A 91, 033601 (2015).
  24. Choudhury, S. & Mueller, E. J. Transverse collisional instabilities of a Bose–Einstein condensate in a driven one-dimensional lattice. Phys. Rev. A 91, 023624 (2015).
  25. Aidelsburger, M. et al. Experimental realization of strong effective magnetic fields in an optical lattice. Phys. Rev. Lett. 107, 255301 (2011).
  26. Atala, M. et al. Observation of chiral currents with ultracold atoms in bosonic ladders. Nature Phys. 10, 588593 (2014).
  27. Stuhl, B. K., Lu, H.-I., Aycock, L. M., Genkina, D. & Spielman, I. B. Visualizing edge states with an atomic Bose gas in the quantum Hall regime. Preprint at http://arXiv.org/abs/1502.02496 (2015).
  28. Lim, L.-K., Hemmerich, A. & Smith, C. M. Artificial staggered magnetic field for ultracold atoms in optical lattices. Phys. Rev. A 81, 023404 (2010).
  29. Möller, G. & Cooper, N. R. Condensed ground states of frustrated Bose–Hubbard models. Phys. Rev. A 82, 063625 (2010).
  30. Powell, S., Barnett, R., Sensarma, R. & Das Sarma, S. Bogoliubov theory of interacting bosons on a lattice in a synthetic magnetic field. Phys. Rev. A 83, 013612 (2011).
  31. Polini, M., Fazio, R., MacDonald, A. H. & Tosi, M. P. Realization of fully frustrated Josephson-junction arrays with cold atoms. Phys. Rev. Lett. 95, 010401 (2005).
  32. Kohmoto, M. Topological invariant and the quantization of the Hall conductance. Ann. Phys. 160, 343354 (1985).
  33. Lin, Y.-J. et al. A synthetic electric force acting on neutral atoms. Nature Phys. 7, 531534 (2011).
  34. Polak, T. P. & Zaleski, T. A. Time-of-flight patterns of ultracold bosons in optical lattices in various Abelian artificial magnetic field gauges. Phys. Rev. A 87, 033614 (2013).
  35. Ozawa, T., Price, H. M. & Carusotto, I. The momentum-space Harper–Hofstadter model. Preprint at http://arXiv.org/abs/1411.1203 (2014).
  36. Parker, C. V., Ha, L.-C. & Chin, C. Direct observation of effective ferromagnetic domains of cold atoms in a shaken optical lattice. Nature Phys. 9, 769774 (2013).
  37. Haller, E. et al. Inducing transport in a dissipation-free lattice with super Bloch oscillations. Phys. Rev. Lett. 104, 200403 (2010).
  38. Alberti, A., Ferrari, G., Ivanov, V., Chiofalo, M. & Tino, G. Atomic wave packets in amplitude-modulated vertical optical lattices. New J. Phys. 12, 065037 (2010).
  39. Braun, S. et al. Negative absolute temperature for motional degrees of freedom. Science 339, 5255 (2013).
  40. Zurek, W. H., Dorner, U. & Zoller, P. Dynamics of a quantum phase transition. Phys. Rev. Lett. 95, 105701 (2005).
  41. Umucalılar, R. O. & Oktel, M. Ö. Phase boundary of the boson Mott insulator in a rotating optical lattice. Phys. Rev. A 76, 055601 (2007).
  42. Dubček, T. et al. Weyl points in three-dimensional optical lattices: Synthetic magnetic monopoles in momentum space. Phys. Rev. Lett. 114, 225301 (2015).
  43. Kennedy, C. J., Siviloglou, G. A., Miyake, H., Burton, W. C. & Ketterle, W. Spin–orbit coupling and quantum spin Hall effect for neutral atoms without spin flips. Phys. Rev. Lett. 111, 225301 (2013).
  44. Lewenstein, M., Sanpera, A. & Ahufinger, V. Ultracold Atoms in Optical Lattices: Simulating Quantum Many-Body Systems (Oxford Univ. Press, 2012).
  45. Goldman, N., Juzeliūnas, G., Öhberg, P. & Spielman, I. B. Light-induced gauge fields for ultracold atoms. Rep. Prog. Phys. 77, 126401 (2014).
  46. Cooper, N. R. & Dalibard, J. Reaching fractional quantum Hall states with optical flux lattices. Phys. Rev. Lett. 110, 185301 (2013).

Download references

Author information


  1. MIT-Harvard Center for Ultracold Atoms, Research Laboratory of Electronics, Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

    • Colin J. Kennedy,
    • William Cody Burton,
    • Woo Chang Chung &
    • Wolfgang Ketterle


All authors contributed to experimental work, data analysis and manuscript preparation.

Competing financial interests

The authors declare no competing financial interests.

Corresponding author

Correspondence to:

Author details

Supplementary information

PDF files

  1. Supplementary information (450 KB)

    Supplementary information

Additional data