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Self-oscillating pump in a topological dissipative atom–cavity system


Pumps are transport mechanisms in which direct currents result from a cyclic evolution of the potential1,2. As Thouless showed, the pumping process can have topological origins, when considering the motion of quantum particles in spatially and temporally periodic potentials3. However, the periodic evolution that drives these pumps has always been assumed to be imparted from outside, as has been the case in the experimental systems studied so far4,5,6,7,8,9,10,11,12. Here we report on an emergent mechanism for pumping in a quantum gas coupled to an optical resonator, where we observe a particle current without applying a periodic drive. The pumping potential experienced by the atoms is formed by the self-consistent cavity field interfering with the static laser field driving the atoms. Owing to dissipation, the cavity field evolves between its two quadratures13, each corresponding to a different centrosymmetric crystal configuration14. This self-oscillation results in a time-periodic potential analogous to that describing the transport of electrons in topological tight-binding models, such as the paradigmatic Rice–Mele pump15. In the experiment, we directly follow the evolution by measuring the phase winding of the cavity field with respect to the driving field and observing the atomic motion in situ. The observed mechanism combines the dynamics of topological and open systems, and features characteristics of continuous dissipative time crystals.

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Fig. 1: Non-stationary lattice in a dissipative atom–cavity system.
Fig. 2: Emergent dynamics of the intracavity field.
Fig. 3: Self-consistent atomic pump.
Fig. 4: Non-Hermitian spectra.

Data availability

The data to reproduce the figures of this study are available in the data repository of ETH Zurich’s Research Collection ( at


  1. Altshuler, B. & Glazman, L. Pumping electrons. Science 283, 1864–1865 (1999).

    CAS  Article  Google Scholar 

  2. Cohen, D. Quantum pumping in closed systems, adiabatic transport, and the Kubo formula. Phys. Rev. B 68, 155303 (2003).

    Article  ADS  CAS  Google Scholar 

  3. Thouless, D. Quantization of particle transport. Phys. Rev. B 27, 6083–6087 (1983).

    MathSciNet  CAS  Article  ADS  Google Scholar 

  4. Switkes, M., Marcus, C., Campman, K. & Gossard, A. An adiabatic quantum electron pump. Science 283, 1905–1908 (1999).

    CAS  PubMed  Article  ADS  Google Scholar 

  5. Aleiner, I. & Andreev, A. Adiabatic charge pumping in almost open dots. Phys. Rev. Lett. 81, 1286–1289 (1998).

    CAS  Article  ADS  Google Scholar 

  6. Blumenthal, M. et al. Gigahertz quantized charge pumping. Nat. Phys. 3, 343–347 (2007).

    CAS  Article  Google Scholar 

  7. Giazotto, F. et al. A Josephson quantum electron pump. Nat. Phys. 7, 857–861 (2011).

    CAS  Article  Google Scholar 

  8. Lu, H.-I. et al. Geometrical pumping with a Bose–Einstein condensate. Phys. Rev. Lett. 116, 200402 (2016).

    PubMed  PubMed Central  Article  ADS  CAS  Google Scholar 

  9. Nakajima, S. et al. Topological Thouless pumping of ultracold fermions. Nat. Phys. 12, 296–300 (2016).

    CAS  Article  Google Scholar 

  10. Lohse, M., Schweizer, C., Zilberberg, O., Aidelsburger, M. & Bloch, I. A Thouless quantum pump with ultracold bosonic atoms in an optical superlattice. Nat. Phys. 12, 350–354 (2016).

    CAS  Article  Google Scholar 

  11. Lohse, M., Schweizer, C., Price, H. M., Zilberberg, O. & Bloch, I. Exploring 4D quantum Hall physics with a 2D topological charge pump. Nature 553, 55–58 (2018).

    CAS  PubMed  Article  ADS  Google Scholar 

  12. Nakajima, A. et al. Competition and interplay between topology and quasi-periodic disorder in Thouless pumping of ultracold atoms. Nat. Phys. 17, 844–849 (2021).

    CAS  Article  Google Scholar 

  13. Dogra, N. et al. Dissipation-induced structural instability and chiral dynamics in a quantum gas. Science 366, 1496–1499 (2019).

    CAS  PubMed  Article  ADS  Google Scholar 

  14. Li, X. et al. First order phase transition between two centro-symmetric superradiant crystals. Phys. Rev. Res. 3, L012024 (2021).

    CAS  Article  Google Scholar 

  15. Rice, M. & Mele, E. Elementary excitations of a linearly conjugated diatomic polymer. Phys. Rev. Lett. 49, 1455–1459 (1982).

    CAS  Article  ADS  Google Scholar 

  16. Xiao, D., Chang, M.-C. & Niu, Q. Berry phase effects on electronic properties. Rev. Mod. Phys. 82, 1959–2007 (2010).

    MathSciNet  CAS  MATH  Article  ADS  Google Scholar 

  17. Niu, Q. Quantum adiabatic particle transport. Phys. Rev. B 34, 5093–5100 (1986).

    CAS  Article  ADS  Google Scholar 

  18. Wang, L., Troyer, M. & Dai, X. Topological charge pumping in a one-dimensional optical lattice. Phys. Rev. Lett. 111, 026802 (2013).

    PubMed  Article  ADS  CAS  Google Scholar 

  19. Qian, Y., Gong, M. & Zhang, C. Quantum transport of bosonic cold atoms in double-well optical lattices. Phys. Rev. A 84, 013608 (2011).

    Article  ADS  CAS  Google Scholar 

  20. Resta, R. Manifestations of Berry’s phase in molecules and condensed matter. J. Phys. Condens. Matter 12, R107 (2000).

    CAS  Article  ADS  Google Scholar 

  21. Vanderbilt, D. & King-Smith, R. Electric polarization as a bulk quantity and its relation to surface charge. Phys. Rev. B 48, 4442–4455 (1993).

    CAS  Article  ADS  Google Scholar 

  22. Resta, R. Macroscopic polarization in crystalline dielectrics: the geometric phase approach. Rev. Mod. Phys. 66, 899–915 (1994).

    CAS  Article  ADS  Google Scholar 

  23. Resta, R. & Vanderbilt, D. in Physics of Ferroelectrics: Topics in Applied Physics Vol. 105 (ed. Lee, Y. P. et al.) 31–68 (Springer, 2007).

  24. Mivehvar, F., Piazza, F., Donner, T. & Ritsch, H. Cavity QED with quantum gases: new paradigms in many-body physics. Adv. Phys. 70, 1–153 (2021).

    Article  ADS  Google Scholar 

  25. Jenkins, A. Self-oscillation. Phys. Rep. 525, 167–222 (2013).

    MathSciNet  MATH  Article  ADS  Google Scholar 

  26. El-Ganainy, R. et al. Non-Hermitian physics and PT symmetry. Nat. Phys. 14, 11–19 (2018).

    CAS  Article  Google Scholar 

  27. Fruchart, M., Hanai, R., Littlewood, P. B. & Vitelli, V. Non-reciprocal phase transitions. Nature 592, 363–369 (2021).

    CAS  PubMed  Article  ADS  Google Scholar 

  28. Baumann, K., Guerlin, C., Brennecke, F. & Esslinger, T. Dicke quantum phase transition with a superfluid gas in an optical cavity. Nature 464, 1301–1306 (2010).

    CAS  PubMed  Article  ADS  Google Scholar 

  29. Chiacchio, E. R. & Nunnenkamp, A. Dissipation-induced instabilities of a spinor Bose–Einstein condensate inside an optical cavity. Phys. Rev. Lett. 122, 193605 (2019).

    CAS  PubMed  Article  ADS  Google Scholar 

  30. Buča, B. & Jaksch, D. Dissipation induced nonstationarity in a quantum gas. Phys. Rev. Lett. 123, 260401 (2019).

    PubMed  Article  ADS  Google Scholar 

  31. Haroche, S. & Raimond, J.-M. Exploring the Quantum: Atoms, Cavities, and Photons (Oxford Univ. Press, 2006).

  32. Baumann, K., Mottl, R., Brennecke, F. & Esslinger, T. Exploring symmetry breaking at the Dicke quantum phase transition. Phys. Rev. Lett. 107, 140402 (2011).

    CAS  PubMed  Article  ADS  Google Scholar 

  33. Fan, J., Chen, G. & Jia, S. Atomic self-organization emerging from tunable quadrature coupling. Phys. Rev. A 101, 063627 (2020).

    CAS  Article  ADS  Google Scholar 

  34. Zupancic, P. et al. P-band induced self-organization and dynamics with repulsively driven ultracold atoms in an optical cavity. Phys. Rev. Lett. 123, 233601 (2019).

    CAS  PubMed  Article  ADS  Google Scholar 

  35. Mekhov, I. B., Maschler, C. & Ritsch, H. Probing quantum phases of ultracold atoms in optical lattices by transmission spectra in cavity quantum electrodynamics. Nat. Phys. 3, 319–323 (2007).

    CAS  Article  Google Scholar 

  36. Laflamme, C., Yang, D. & Zoller, P. Continuous measurement of an atomic current. Phys. Rev. A 95, 043843 (2017).

    Article  ADS  Google Scholar 

  37. El-Ganainy, R., Khajavikhan, M., Christodoulides, D. N. & Ozdemir, S. K. The dawn of non-Hermitian optics. Commun. Phys. 2, 37 (2019).

    Article  Google Scholar 

  38. Ashida, Y., Gong, Z. & Ueda, M. Non-Hermitian physics. Adv. Phys. 69, 249–435 (2020).

    Article  ADS  Google Scholar 

  39. Booker, C., Buča, B. & Jaksch, D. Non-stationarity and dissipative time crystals: spectral properties and finite-size effects. New J. Phys. 22, 085007 (2020).

    MathSciNet  CAS  Article  ADS  Google Scholar 

  40. Kongkhambut, P. et al. Observation of a continuous time crystal. Science (2022).

  41. Cosme, J. G., Skulte, J. & Mathey, L. Time crystals in a shaken atom–cavity system. Phys. Rev. A 100, 053615 (2019).

    CAS  Article  ADS  Google Scholar 

  42. Keßler, H. et al. Observation of a dissipative time crystal. Phys. Rev. Lett. 127, 043602 (2021).

    PubMed  Article  ADS  Google Scholar 

  43. Landig, R. et al. Quantum phases from competing short- and long-range interactions in an optical lattice. Nature 532, 476–479 (2016).

    CAS  PubMed  Article  ADS  Google Scholar 

  44. Zhang, X. et al. Observation of a superradiant quantum phase transition in an intracavity degenerate Fermi gas. Science 373, 1359–1362 (2021).

    CAS  PubMed  Article  ADS  Google Scholar 

  45. Mivehvar, F., Ritsch, H. & Piazza, F. Superradiant topological Peierls insulator inside an optical cavity. Phys. Rev. Lett. 118, 073602 (2017).

    PubMed  Article  ADS  Google Scholar 

  46. Landini, M. et al. Formation of a spin texture in a quantum gas coupled to a cavity. Phys. Rev. Lett. 120, 223602 (2018).

    CAS  PubMed  Article  ADS  Google Scholar 

  47. Kroeze, R. M., Guo, Y., Vaidya, V. D., Keeling, J. & Lev, B. L. Spinor self-ordering of a quantum gas in a cavity. Phys. Rev. Lett. 121, 163601 (2018).

    CAS  PubMed  Article  ADS  Google Scholar 

  48. Morales, A. et al. Two-mode Dicke model from nondegenerate polarization modes. Phys. Rev. A 100, 013816 (2019).

    CAS  Article  ADS  Google Scholar 

  49. Maschler, C., Mekhov, I. B. & Ritsch, H. Ultracold atoms in optical lattices generated by quantized light fields. Eur. Phys. J. D 46, 545–560 (2008).

    CAS  Article  ADS  Google Scholar 

  50. Resta, R. Quantum-mechanical position operator in extended systems. Phys. Rev. Lett. 80, 1800–1803 (1998).

    CAS  Article  ADS  Google Scholar 

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We thank A. Frank for support with the heterodyne electronics; and O. Zilberberg and N. Spaldin for discussion. We acknowledge funding from SNF project numbers IZBRZ2_186312, 182650 and 175329 (NAQUAS QuantERA) and NCCR QSIT, from EU Horizon2020 ERCadvanced grant TransQ (project number 742579) and ITN grant ColOpt (project number 721465).

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Authors and Affiliations



D.D., A.B. and X.L. prepared the experiment, D.D., A.B., X.L. and S.H. took and analysed the data. D.D. performed the numerical simulations. T.D. and T.E. supervised the work. All authors contributed to discussions of the manuscript.

Corresponding author

Correspondence to Tilman Esslinger.

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Extended data figures and tables

Extended Data Fig. 1 Non-averaged phase diagram and repeated measurement results.

a, Phase diagram using the phase data ϕ(t) from the heterodyne detector by varying cavity detuning ΔC of cavity 1. The two different self-organized phases can be well observed for values around ϕ = 0 and ϕ = π/2. At low transverse beam fields the system shows no self-organization and ϕ is not well defined. Between the two self-organized phases the dynamical phase with varying ϕ(t) is visible. b, Many repetitions of the same trace with constant ΔC = −1.1 MHz. The extent of the instability region varies slightly at each repetition.c, Same data as in b, but shifted in time such that the onset of pumping coincides for all traces. The dashed line as guide to the eye illustrates that the rate at which the phase evolves is robust.

Extended Data Fig. 2 Schematic representation of the theoretical models.

a, Self-consistent loop between cavity field α, optical lattice Vlattice and wavefunction ψ as described by the set of equations Eq. (5) and Eq. (11).b, Minimal model given by the three-level momentum expansion of Eq. (14). The coherent coupling (solid arrows) mixes the condensate mode ψ0 with the spatially modulated ψp,q, which are then mutually coupled by dissipation (dashed arrows).

Extended Data Fig. 3 Comparison of experimental and numerical phase diagrams.

Figures show the amplitude of the intracavity light field dependent on V0 and the cavity detuning Δc for cavity 2. a, Dataset showing in each row a trace of single experimental realization for the given cavity detuning Δc. The transverse beam lattice is linearly ramped to the final transverse lattice strength V0 = 40 Er within 5 ms. b, Corresponding simulation of the experimental results with GPE simulation.

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Dreon, D., Baumgärtner, A., Li, X. et al. Self-oscillating pump in a topological dissipative atom–cavity system. Nature 608, 494–498 (2022).

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