Abstract
Topological gauge theories describe the low-energy properties of certain strongly correlated quantum systems through effective weakly interacting models1,2. A prime example is the Chern–Simons theory of fractional quantum Hall states, where anyonic excitations emerge from the coupling between weakly interacting matter particles and a density-dependent gauge field3. Although in traditional solid-state platforms such gauge theories are only convenient theoretical constructions, engineered quantum systems enable their direct implementation and provide a fertile playground to investigate their phenomenology without the need for strong interactions4. Here, we report the quantum simulation of a topological gauge theory by realizing a one-dimensional reduction of the Chern–Simons theory (the chiral BF theory5,6,7) in a Bose–Einstein condensate. Using the local conservation laws of the theory, we eliminate the gauge degrees of freedom in favour of chiral matter interactions8,9,10,11, which we engineer by synthesizing optically dressed atomic states with momentum-dependent scattering properties. This allows us to reveal the key properties of the chiral BF theory: the formation of chiral solitons and the emergence of an electric field generated by the system itself. Our results expand the scope of quantum simulation to topological gauge theories and open a route to the implementation of analogous gauge theories in higher dimensions12.
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Data availability
The datasets supporting this study are available from the corresponding authors upon request.
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Acknowledgements
We thank M. Ballu and J. Sanz for preparatory work on the Raman laser system; and M. Dalmonte, G. Juzeliūnas, P. Öhberg, L. Santos, G. Valentí-Rojas, and the ICFO Quantum Gases Experiment and Quantum Optics Theory groups for discussions. We acknowledge funding from the European Research Council under the European Union’s Framework Programme for Research and Innovation Horizon 2020 (H2020-ERC-CoG SuperComp, grant agreement No. 101003295), the Government of Spain (LIGAS projects PID2020-112687GB-C21 [MCIN/AEI/10.13039/501100011033] at ICFO and PID2020-112687GB-C22 [MCIN/AEI/10.13039/501100011033] at UAB, and Severo Ochoa CEX2019-000910-S [MCIN/AEI/10.13039/501100011033] at ICFO), Deutsche Forschungsgemeinschaft (Research Unit FOR2414, Project No. 277974659), Fundación Ramón Areces (project CODEC), Fundació Cellex, Fundació Mir-Puig, and Generalitat de Catalunya (ERDF Operational Program of Catalunya, Project QUASICAT/QuantumCat Ref. No. 001-P-001644, AGAUR and CERCA program). A.F. acknowledges support from La Caixa Foundation (ID 100010434, PhD fellowship LCF/BQ/DI18/11660040) and the European Union (H2020-Marie Skłodowska-Curie grant agreement No. 713673), C.S.C. from the European Union (H2020-Marie Skłodowska-Curie grant agreement No. 713729), E.N. from the European Union (H2020-Marie Skłodowska-Curie grant agreement No. 101029996 ToPIKS), C.R.C. from an ICFO-MPQ Cellex postdoctoral fellowship, R.R. from the European Union (H2020-Marie Skłodowska-Curie grant agreement No. 101030630 UltraComp), A.C. from the UAB Talent Research programme, and L.T. from the Government of Spain and ESF "Investing in your future" (RYC-2015-17890 [MCIN/AEI/10.13039/501100011033]).
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A.F., C.S.C., E.N. and R.R. took and analysed the data. A.F., C.S.C., E.N., C.R.C. and R.R. prepared the experiment. A.C. and L.T. developed the theory. C.S.C. performed the numerical simulations. L.T. supervised the work. All authors contributed extensively to the interpretation of the results and to the preparation of the manuscript.
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Extended data figures and tables
Extended Data Fig. 1 Calibration of mechanical momentum by spin ejection spectroscopy.
a, Dispersion relation of the dressed states \(| \,-\,\rangle \) and \(| \,+\,\rangle \), and of the bare states \(\left|\uparrow \right\rangle ,\left|\downarrow \right\rangle \), and \(\left|{\rm{aux}}\right\rangle \) in the rotating frame. Due to the Raman coupling, states \(\left|\uparrow \right\rangle \) and \(\left|\downarrow \right\rangle \) are shifted by 2kR in momentum and δ0 in frequency. We exploit the \(| \,-\,\rangle \) to \(\left|{\rm{aux}}\right\rangle \) transition, of frequency f0 + Δf, to extract the mechanical momentum kmech of the cloud. b, Radio-frequency spectroscopy of the \(| \,-\,\rangle \) to \(\left|{\rm{aux}}\right\rangle \) transition for various values of the two-photon Raman detuning δ0 with ħΩ/ER = 4.5(3), corresponding to the parameters of the expansion experiments of Fig. 4b. c, Value of the generalized detuning \(\hbar \widetilde{\delta }/{E}_{R}=\hbar {\delta }_{0}/{E}_{R}-4{k}_{x}/{k}_{R}\) vs. δ0 (circles) extracted from the measured frequency difference \(\Delta f=(\widetilde{\Omega }-\widetilde{\delta })/2\). Solid line: theory prediction for kmech/kR = 0. Shaded area: uncertainty in the theoretical value caused by the uncertainties in δ0 and Ω. We conclude that the preparation procedure for the experiments of Fig. 4 imparts negligible mechanical momentum to the cloud.
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Frölian, A., Chisholm, C.S., Neri, E. et al. Realizing a 1D topological gauge theory in an optically dressed BEC. Nature 608, 293–297 (2022). https://doi.org/10.1038/s41586-022-04943-3
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DOI: https://doi.org/10.1038/s41586-022-04943-3
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