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

Journal name:
Nature Physics
Volume:
11,
Pages:
859–864
Year published:
DOI:
doi:10.1038/nphys3421
Received
Accepted
Published online

Abstract

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

Figures

  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.

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Affiliations

  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

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

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