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Realization of density-dependent Peierls phases to engineer quantized gauge fields coupled to ultracold matter


Gauge fields that appear in models of high-energy and condensed-matter physics are dynamical quantum degrees of freedom due to their coupling to matter fields. Since the dynamics of these strongly correlated systems is hard to compute, it was proposed to implement this basic coupling mechanism in quantum simulation platforms with the ultimate goal to emulate lattice gauge theories. Here, we realize the fundamental ingredient for a density-dependent gauge field acting on ultracold fermions in an optical lattice by engineering non-trivial Peierls phases that depend on the site occupations. We propose and implement a Floquet scheme that relies on breaking time-reversal symmetry by driving the lattice simultaneously at two frequencies that are resonant with the on-site interactions. This induces density-assisted tunnelling processes that are controllable in amplitude and phase. We demonstrate techniques in a Hubbard dimer to quantify the amplitude and to directly measure the Peierls phase with respect to the single-particle hopping. The tunnel coupling features two distinct regimes as a function of the modulation amplitudes, which can be characterized by a \({\Bbb Z}_2\)-invariant. Moreover, we provide a full tomography of the winding structure of the Peierls phase around a Dirac point that appears in the driving parameter space.

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Fig. 1: Experimental set-up and driving scheme.
Fig. 2: Schemes to measure the absolute value and phase of the effective tunnel coupling.
Fig. 3: Gap closing in K1K2 parameter space.
Fig. 4: Dirac point and Peierls phase vortex.

Data availability

All data files are available from the corresponding author upon request. Source data for Figs. 24 and Supplementary Figs. 1d and 7 are provided in the Supplementary information.

Code availability

The source code for the fit of the Ramsey fringes is available from the corresponding author upon request.


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We thank L. Barbiero, A. Bermudez, A. Eckardt, N. Goldman, F. Grusdt, Y. Murakami, M. Rizzi, L. Santos, K. Viebahn, P. Werner and O. Zilberberg for insightful discussions, and K. Viebahn and O. Zilberberg for a careful reading of the manuscript. We acknowledge SNF (projects nos. 169320 and 182650), NCCR-QSIT, QUIC (Swiss State Secretary for Education, Research and Innovation contract no. 15.0019) and ERC advanced grant TransQ (project no. 742579) for funding.

Author information

Authors and Affiliations



The data were measured and analysed by F.G., K.S. and J.M. The theoretical framework and measurement scheme were developed by F.G. All work was supervised by T.E. All authors contributed to planning the experiment, discussions and preparation of the manuscript.

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Correspondence to Tilman Esslinger.

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Supplementary information

Supplementary Information

Supplementary text and Figs. 1–7.

Supplementary Data 1

Source data for Fig. 2.

Supplementary Data 2

Source data for Fig. 3.

Supplementary Data 3

Source data for Fig. 4.

Supplementary Data 4

Source data for Supplementary Fig. 1d.

Supplementary Data 5

Source data for Supplementary Fig. 7.

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Görg, F., Sandholzer, K., Minguzzi, J. et al. Realization of density-dependent Peierls phases to engineer quantized gauge fields coupled to ultracold matter. Nat. Phys. 15, 1161–1167 (2019).

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