Skip to main content

Thank you for visiting You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Floquet approach to 2 lattice gauge theories with ultracold atoms in optical lattices


Quantum simulation has the potential to investigate gauge theories in strongly interacting regimes, which are currently inaccessible through conventional numerical techniques. Here, we take a first step in this direction by implementing a Floquet-based method for studying \({\Bbb Z}_2\) lattice gauge theories using two-component ultracold atoms in a double-well potential. For resonant periodic driving at the on-site interaction strength and an appropriate choice of the modulation parameters, the effective Floquet Hamiltonian exhibits \({\Bbb Z}_2\) symmetry. We study the dynamics of the system for different initial states and critically contrast the observed evolution with a theoretical analysis of the full time-dependent Hamiltonian of the periodically driven lattice model. We reveal challenges that arise due to symmetry-breaking terms and outline potential pathways to overcome these limitations. Our results provide important insights for future studies of lattice gauge theories based on Floquet techniques.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.

Fig. 1: 1D \({\Bbb Z}_2\) lattice gauge theory coupled to matter.
Fig. 2: Driving scheme for \({\Bbb Z}_2\) LGTs on a double well.
Fig. 3: Dynamics of the matter–gauge system prepared initially in an eigenstate of the electric field \({\hat{\boldsymbol \tau }}^{\boldsymbol{x}}\).
Fig. 4: Dynamics of the matter–gauge system prepared initially in an eigenstate of the gauge field \({\hat{\boldsymbol \tau }}^{\boldsymbol{z}}\).
Fig. 5: Finite-frequency corrections to the effective Floquet Hamiltonian (equation (4)).

Data availability

The data that support the plots within this paper and other findings of this study are available from the corresponding author on reasonable request.

Code availability

The code that supports the plots within this paper are available from the corresponding author on reasonable request.


  1. 1.

    Wilson, K. G. Confinement of quarks. Phys. Rev. D 10, 2445–2459 (1974).

    ADS  Google Scholar 

  2. 2.

    Kogut, J. B. An introduction to lattice gauge theory and spin systems. Rev. Mod. Phys. 51, 659–713 (1979).

    ADS  MathSciNet  Google Scholar 

  3. 3.

    Wen, X.-G. Quantum Field Theory of Many-Body Systems (Oxford University Press, 2004).

  4. 4.

    Levin, M. & Wen, X.-G. Colloquium: photons and electrons as emergent phenomena. Rev. Mod. Phys. 77, 871–879 (2005).

    ADS  Google Scholar 

  5. 5.

    Lee, P. A., Nagaosa, N. & Wen, X.-G. Doping a Mott insulator: physics of high temperature superconductivity. Rev. Mod. Phys. 78, 17–85 (2006).

    ADS  Google Scholar 

  6. 6.

    Ichinose, I. & Matsui, T. Lattice gauge theory for condensed matter physics: ferromagnetic superconductivity as its example. Mod. Phys. Lett. B 28, 1430012 (2014).

    ADS  Google Scholar 

  7. 7.

    Aoki, S. et al. Review of lattice results concerning low-energy particle physics. Eur. Phys. J. C 77, 112 (2017).

    ADS  Google Scholar 

  8. 8.

    Troyer, M. & Wiese, U.-J. Computational complexity and fundamental limitations to fermionic quantum Monte Carlo simulations. Phys. Rev. Lett. 94, 170201 (2005).

    ADS  Google Scholar 

  9. 9.

    Alford, M. G., Schmitt, A., Rajagopal, K. & Schäfer, T. Color superconductivity in dense quark matter. Rev. Mod. Phys. 80, 1455–1515 (2008).

    ADS  Google Scholar 

  10. 10.

    Buyens, B., Verstraete, F. & Acoleyen, K. V. Hamiltonian simulation of the Schwinger model at finite temperature. Phys. Rev. D 94, 085018 (2016).

    ADS  Google Scholar 

  11. 11.

    Bañuls, M. C. et al. Towards overcoming the Monte Carlo sign problem with tensor networks. EPJ Web Conf. 137, 04001 (2017).

    Google Scholar 

  12. 12.

    Silvi, P., Rico, E., Dalmonte, M., Tschirsich, F. & Montangero, S. Finite-density phase diagram of a (1 + 1) − d non-abelian lattice gauge theory with tensor networks. Quantum 1, 9 (2017).

    Google Scholar 

  13. 13.

    Gazit, S., Randeria, M. & Vishwanath, A. Emergent Dirac fermions and broken symmetries in confined and deconfined phases of Z 2 gauge theories. Nat. Phys. 13, 484–490 (2017).

    Google Scholar 

  14. 14.

    Weimer, H., Müller, M., Lesanovsky, I., Zoller, P. & Büchler, H. P. A Rydberg quantum simulator. Nat. Phys. 6, 382–388 (2010).

    Google Scholar 

  15. 15.

    Blatt, R. & Roos, C. F. Quantum simulations with trapped ions. Nat. Phys. 8, 277–284 (2012).

    Google Scholar 

  16. 16.

    Gross, C. & Bloch, I. Quantum simulations with ultracold atoms in optical lattices. Science 357, 995–1001 (2017).

    ADS  Google Scholar 

  17. 17.

    Romero, G., Solano, E. & Lamata, L. in Quantum Simulations with Photons and Polaritons (ed. Angelakis, D.) 153–180 (Springer, 2017).

  18. 18.

    Tagliacozzo, L., Celi, A., Zamora, A. & Lewenstein, M. Optical abelian lattice gauge theories. Ann. Phys. 330, 160–191 (2013).

    ADS  MathSciNet  MATH  Google Scholar 

  19. 19.

    Wiese, U.-J. Ultracold quantum gases and lattice systems: quantum simulation of lattice gauge theories. Ann. Phys. 525, 777–796 (2013).

    MathSciNet  MATH  Google Scholar 

  20. 20.

    Zohar, E., Cirac, J. I. & Reznik, B. Quantum simulations of lattice gauge theories using ultracold atoms in optical lattices. Rep. Prog. Phys. 79, 014401 (2015).

    ADS  MathSciNet  Google Scholar 

  21. 21.

    Dalmonte, M. & Montangero, S. Lattice gauge theory simulations in the quantum information era. Contemp. Phys. 57, 388–412 (2016).

    ADS  Google Scholar 

  22. 22.

    Notarnicola, S. et al. Discrete Abelian gauge theories for quantum simulations of QED. J. Phys. A 48, 30FT01 (2015).

    MathSciNet  MATH  Google Scholar 

  23. 23.

    Kasper, V., Hebenstreit, F., Jendrzejewski, F., Oberthaler, M. K. & Berges, J. Implementing quantum electrodynamics with ultracold atomic systems. New J. Phys. 19, 023030 (2017).

    ADS  Google Scholar 

  24. 24.

    Kuno, Y., Sakane, S., Kasamatsu, K., Ichinose, I. & Matsui, T. Quantum simulation of (1 + 1)-dimensional U(1) gauge-Higgs model on a lattice by cold Bose gases. Phys. Rev. D 95, 094507 (2017).

    ADS  Google Scholar 

  25. 25.

    Zhang, J. et al. Quantum simulation of the universal features of the polyakov loop. Phys. Rev. Lett. 121, 223201 (2018).

    ADS  Google Scholar 

  26. 26.

    Aidelsburger, M., Nascimbène, S. & Goldman, N. Artificial gauge fields in materials and engineered systems. C. R. Phys. 19, 394–432 (2018).

    ADS  Google Scholar 

  27. 27.

    Clark, L. W. et al. Observation of density-dependent gauge fields in a bose-einstein condensate based on micromotion control in a shaken two-dimensional lattice. Phys. Rev. Lett. 121, 030402 (2018).

    ADS  Google Scholar 

  28. 28.

    Anderlini, M. et al. Controlled exchange interaction between pairs of neutral atoms in an optical lattice. Nature 448, 452–456 (2007).

    ADS  Google Scholar 

  29. 29.

    Trotzky, S. et al. Time-resolved observation and control of superexchange interactions with ultracold atoms in optical lattices. Science 319, 295–299 (2008).

    ADS  Google Scholar 

  30. 30.

    Dai, H.-N. et al. Four-body ring-exchange interactions and anyonic statistics within a minimal toric-code Hamiltonian. Nat. Phys. 13, 1195–1200 (2017).

    Google Scholar 

  31. 31.

    Klco, N. et al. Quantum-classical computation of schwinger model dynamics using quantum computers. Phys. Rev. A 98, 032331 (2018).

    ADS  Google Scholar 

  32. 32.

    Martinez, E. A. et al. Real-time dynamics of lattice gauge theories with a few-qubit quantum computer. Nature 534, 516–519 (2016).

    ADS  Google Scholar 

  33. 33.

    Barbiero, L. et al. Coupling ultracold matter to dynamical gauge fields in optical lattices: from flux-attachment to 2 lattice gauge theories. Preprint at (2018).

  34. 34.

    Zohar, E., Farace, A., Reznik, B. & Cirac, J. I. Digital quantum simulation of 2 lattice gauge theories with dynamical fermionic matter. Phys. Rev. Lett. 118, 070501 (2017).

    ADS  Google Scholar 

  35. 35.

    Horn, D., Weinstein, M. & Yankielowicz, S. Hamiltonian approach to Z(N) lattice gauge theories. Phys. Rev. D 19, 3715–3731 (1979).

    ADS  Google Scholar 

  36. 36.

    Ju, H. & Balents, L. Finite-size effects in the Z 2 spin liquid on the kagome lattice. Phys. Rev. B 87, 195109 (2013).

    ADS  Google Scholar 

  37. 37.

    González-Cuadra, D., Dauphin, A., Grzybowski, P. R., Lewenstein, M. & Bermudez, A. Symmetry-breaking topological insulator in the 2 Bose-Hubbar dmodel. Phys. Rev. B 99, 045139 (2019).

    ADS  Google Scholar 

  38. 38.

    Kitaev, A. Y. Fault-tolerant quantum computation by anyons. Ann. Phys. 303, 2–30 (2003).

    ADS  MathSciNet  MATH  Google Scholar 

  39. 39.

    Keilmann, T., Lanzmich, S., McCulloch, I. & Roncaglia, M. Statistically induced phase transitions and anyons in 1d optical lattices. Nat. Commun. 2, 361 (2011).

    ADS  Google Scholar 

  40. 40.

    Greschner, S. & Santos, L. Anyon hubbard model in one-dimensional optical lattices. Phys. Rev. Lett. 115, 053002 (2015).

    ADS  Google Scholar 

  41. 41.

    Bermudez, A. & Porras, D. Interaction-dependent photon-assisted tunneling in optical lattices: a quantum simulator of strongly correlated electrons and dynamical gauge fields. New. J. Phys. 17, 103021 (2015).

    ADS  Google Scholar 

  42. 42.

    Sträter, C., Srivastava, S. C. L. & Eckardt, A. Floquet realization and signatures of one-dimensional anyons in an optical lattice. Phys. Rev. Lett. 117, 205303 (2016).

    ADS  Google Scholar 

  43. 43.

    Goldman, N., Dalibard, J., Aidelsburger, M. & Cooper, N. R. Periodically driven quantum matter: the case of resonant modulations. Phys. Rev. A 91, 033632 (2015).

    ADS  Google Scholar 

  44. 44.

    Goldman, N. & Dalibard, J. Periodically driven quantum systems: effective hamiltonians and engineered gauge fields. Phys. Rev. X 4, 031027 (2014).

    Google Scholar 

  45. 45.

    Bukov, M., D’Alessio, L. & Polkovnikov, A. Universal high-frequency behavior of periodically driven systems: from dynamical stabilization to Floquet engineering. Adv. Phys. 64, 139–226 (2015).

    ADS  Google Scholar 

  46. 46.

    Eckardt, A. Colloquium: atomic quantum gases in periodically driven optical lattices. Rev. Mod. Phys. 89, 011004 (2017).

    ADS  MathSciNet  Google Scholar 

  47. 47.

    Ma, R. et al. Photon-assisted tunneling in a biased strongly correlated bose gas. Phys. Rev. Lett. 107, 095301 (2011).

    ADS  Google Scholar 

  48. 48.

    Chen, Y.-A. et al. Controlling correlated tunneling and superexchange interactions with ac-driven optical lattices. Phys. Rev. Lett. 107, 210405 (2011).

    ADS  Google Scholar 

  49. 49.

    Meinert, F., Mark, M. J., Lauber, K., Daley, A. J. & Nägerl, H.-C. Floquet engineering of correlated tunneling in the bose-hubbard model with ultracold atoms. Phys. Rev. Lett. 116, 205301 (2016).

    ADS  Google Scholar 

  50. 50.

    Görg, F. et al. Realization of density-dependent Peierls phases to engineer quantized gauge fields coupled to ultracold matter. Nat. Phys. (2019).

  51. 51.

    Keay, B. J. et al. Dynamic localization, absolute negative conductance, and stimulated, multiphoton emission in sequential resonant tunneling semiconductor superlattices. Phys. Rev. Lett. 75, 4102–4105 (1995).

    ADS  Google Scholar 

  52. 52.

    Lignier, H. et al. Dynamical control of matter-wave tunneling in periodic potentials. Phys. Rev. Lett. 99, 220403 (2007).

    ADS  Google Scholar 

  53. 53.

    Sias, C. et al. Observation of photon-assisted tunneling in optical lattices. Phys. Rev. Lett. 100, 63 (2008).

    Google Scholar 

  54. 54.

    Mukherjee, S. et al. Modulation-assisted tunneling in laser-fabricated photonic Wannier–Stark ladders. New J. Phys. 17, 115002 (2015).

    ADS  Google Scholar 

  55. 55.

    Scarola, V. W. & Sarma, S. D. Quantum phases of the extended bose-hubbard hamiltonian: possibility of a supersolid state of cold atoms in optical lattices. Phys. Rev. Lett. 95, 033003 (2005).

    ADS  Google Scholar 

  56. 56.

    Banerjee, D. et al. Atomic quantum simulation of U(N) and SU(N) non-Abelian lattice gauge theories. Phys. Rev. Lett. 110, 125303 (2013).

    ADS  Google Scholar 

  57. 57.

    Kühn, S., Cirac, J. I. & Bañuls, M.-C. Quantum simulation of the Schwinger model: a study of feasibility. Phys. Rev. A 90, 042305 (2014).

    ADS  Google Scholar 

Download references


We acknowledge insightful discussions with M. Dalmonte, A. Trombettoni and M. Lohse. This work was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under project no. 277974659 via Research Unit FOR 2414 and under project no. 282603579 via DIP, the European Commission (UQUAM grant no. 5319278) and the Nanosystems Initiative Munich (NIM, grant no. EXC4). The work was further funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy—EXC-2111—390814868. Work in Brussels was supported by the FRS-FNRS (Belgium) and the ERC Starting Grant TopoCold. F.G. additionally acknowledges support by the Gordon and Betty Moore Foundation under the EPIQS programme and from the Technical University of Munich—Institute for Advanced Study, funded by the German Excellence Initiative and the European Union FP7 under grant agreement 291763, from the DFG grant no. KN 1254/1-1, and DFG TRR80 (Project F8). F.G. and E.D. acknowledge funding from Harvard-MIT CUA, AFOSR-MURI Quantum Phases of Matter (grant no. FA9550-14-1-0035), AFOSR-MURI: Photonic Quantum Matter (award no. FA95501610323) and DARPA DRINQS programme (award no. D18AC00014).

Author information




C.S., F.G., N.G. and M.A. planned the experiment and performed theoretical calculations. C.S. and M.B. performed the experiment and analysed the data with M.A. All authors discussed the results and contributed to the writing of the paper.

Corresponding author

Correspondence to Monika Aidelsburger.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary information

Supplementary Information

Supplementary Figs. 1–13, text and references.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Schweizer, C., Grusdt, F., Berngruber, M. et al. Floquet approach to 2 lattice gauge theories with ultracold atoms in optical lattices. Nat. Phys. 15, 1168–1173 (2019).

Download citation

Further reading


Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing