Skip to main content

Thank you for visiting nature.com. 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.

Quantum field simulator for dynamics in curved spacetime

Abstract

In most cosmological models, rapid expansion of space marks the first moments of the Universe and leads to the amplification of quantum fluctuations1. The description of subsequent dynamics and related questions in cosmology requires an understanding of the quantum fields of the standard model and dark matter in curved spacetime. Even the reduced problem of a scalar quantum field in an explicitly time-dependent spacetime metric is a theoretical challenge2,3,4,5, and thus a quantum field simulator can lead to insights. Here we demonstrate such a quantum field simulator in a two-dimensional Bose–Einstein condensate with a configurable trap6,7 and adjustable interaction strength to implement this model system. We explicitly show the realization of spacetimes with positive and negative spatial curvature by wave-packet propagation and observe particle-pair production in controlled power-law expansion of space, using Sakharov oscillations to extract amplitude and phase information of the produced state. We find quantitative agreement with analytical predictions for different curvatures in time and space. This benchmarks and thereby establishes a quantum field simulator of a new class. In the future, straightforward upgrades offer the possibility to enter unexplored regimes that give further insight into relativistic quantum field dynamics.

This is a preview of subscription content, access via your institution

Access options

Buy article

Get time limited or full article access on ReadCube.

$32.00

All prices are NET prices.

Fig. 1: Curvature in space and time realized in a BEC.
Fig. 2: Configurable density distribution for hyperbolic and spherical geometry.
Fig. 3: Correlation function of fluctuations before and after the ramp.
Fig. 4: Propagation of correlations after the end of the expansion.
Fig. 5: Expansion histories extracted by heterodyne detection.

Data availability

The datasets generated and analysed during the current study are available from the corresponding author.

Code availability

The conclusions of this study do not depend on code or algorithms beyond standard numerical evaluations.

References

  1. Weinberg, S. Cosmology (Oxford Univ. Press, 2008).

  2. Schrödinger, E. The proper vibrations of the expanding universe. Physica 6, 899–912 (1939).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. Parker, L. Quantized fields and particle creation in expanding universes. I. Phys. Rev. 183, 1057–1068 (1969).

    Article  ADS  MATH  Google Scholar 

  4. Birrell, N. D. & Davies, P. C. W. Quantum Fields in Curved Space (Cambridge Monographs on Mathematical Physics, Cambridge Univ. Press, 1982).

  5. Mukhanov, V. & Winitzki, S. Introduction to Quantum Effects in Gravity (Cambridge Univ. Press, Cambridge, 2007).

  6. Saint-Jalm, R. et al. Dynamical symmetry and breathers in a two-dimensional Bose gas. Phys. Rev. X 9, 021035 (2019).

    CAS  Google Scholar 

  7. Gauthier, G. et al. in Advances in Atomic, Molecular, and Optical Physics Vol. 70 (eds Dimauro, L. F. et al.) Ch. 1, 1–101 (Academic Press, 2021).

  8. Unruh, W. G. Experimental black-hole evaporation? Phys. Rev. Lett. 46, 1351–1353 (1981).

    Article  ADS  Google Scholar 

  9. Unruh, W. G. Sonic analogue of black holes and the effects of high frequencies on black hole evaporation. Phys. Rev. D 51, 2827–2838 (1995).

    Article  ADS  MathSciNet  CAS  Google Scholar 

  10. Garay, L. J., Anglin, J. R., Cirac, J. I. & Zoller, P. Sonic analog of gravitational black holes in Bose–Einstein condensates. Phys. Rev. Lett. 85, 4643–4647 (2000).

    Article  ADS  CAS  Google Scholar 

  11. Visser, M., Barceló, C. & Liberati, S. Analogue models of and for gravity. Gen. Relativ. Gravit. 34, 1719–1734 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  12. Novello, M., Visser, M. & Volovik, G. E. (eds) Artificial Black Holes (World Scientific Publishing, 2002).

  13. Barceló, C., Liberati, S. & Visser, M. Probing semiclassical analog gravity in Bose–Einstein condensates with widely tunable interactions. Phys. Rev. A 68, 053613 (2003).

    Article  ADS  Google Scholar 

  14. Fedichev, P. O. & Fischer, U. R. “Cosmological” quasiparticle production in harmonically trapped superfluid gases. Phys. Rev. A 69, 033602 (2004).

    Article  ADS  Google Scholar 

  15. Jain, P., Weinfurtner, S., Visser, M. & Gardiner, C. W. Analog model of a Friedmann–Robertson–Walker universe in Bose–Einstein condensates: application of the classical field method. Phys. Rev. A 76, 033616 (2007).

    Article  ADS  Google Scholar 

  16. Schützhold, R. Recreating fundamental effects in the laboratory?. Adv. Sci. Lett. 2, 121–132 (2009).

    Article  Google Scholar 

  17. Prain, A., Fagnocchi, S. & Liberati, S. Analogue cosmological particle creation: quantum correlations in expanding Bose–Einstein condensates. Phys. Rev. D 82, 105018 (2010).

    Article  ADS  Google Scholar 

  18. Barceló, C., Liberati, S. & Visser, M. Analogue gravity. Living Rev. Relativ. 14, 3 (2011).

    Article  ADS  MATH  Google Scholar 

  19. Jacquet, M. J., Weinfurtner, S. & König, F. The next generation of analogue gravity experiments. Phil. Trans. R Soc. A 378, 20190239 (2020).

    Article  ADS  CAS  Google Scholar 

  20. Philbin, T. G. et al. Fiber-optical analog of the event horizon. Science 319, 1367–1370 (2008).

    Article  ADS  CAS  Google Scholar 

  21. Weinfurtner, S., Tedford, E. W., Penrice, M. C. J., Unruh, W. G. & Lawrence, G. A. Measurement of stimulated Hawking emission in an analogue system. Phys. Rev. Lett. 106, 021302 (2011).

    Article  ADS  Google Scholar 

  22. Carusotto, I., Fagnocchi, S., Recati, A., Balbinot, R. & Fabbri, A. Numerical observation of Hawking radiation from acoustic black holes in atomic Bose–Einstein condensates. New J. Phys. 10, 103001 (2008).

    Article  ADS  Google Scholar 

  23. Lahav, O. et al. Realization of a sonic black hole analog in a Bose–Einstein condensate. Phys. Rev. Lett. 105, 240401 (2010).

    Article  ADS  Google Scholar 

  24. Steinhauer, J. Observation of self-amplifying Hawking radiation in an analogue black-hole laser. Nat. Phys. 10, 864–869 (2014).

    Article  CAS  Google Scholar 

  25. Eckel, S., Kumar, A., Jacobson, T., Spielman, I. B. & Campbell, G. K. A rapidly expanding Bose–Einstein condensate: an expanding universe in the lab. Phys. Rev. X 8, 021021 (2018).

    CAS  Google Scholar 

  26. Muñoz de Nova, J. R., Golubkov, K., Kolobov, V. I. & Steinhauer, J. Observation of thermal Hawking radiation and its temperature in an analogue black hole. Nature 569, 688–691 (2019).

    Article  ADS  Google Scholar 

  27. Wittemer, M. et al. Phonon pair creation by inflating quantum fluctuations in an ion trap. Phys. Rev. Lett. 123, 180502 (2019).

    Article  ADS  CAS  Google Scholar 

  28. Banik, S. et al. Accurate determination of Hubble attenuation and amplification in expanding and contracting cold-atom universes. Phys. Rev. Lett. 128, 090401 (2022).

    Article  ADS  CAS  Google Scholar 

  29. D'Errico, C. et al. Feshbach resonances in ultracold 39K. New J. Phys. 9, 223 (2007).

    Article  ADS  Google Scholar 

  30. Jaskula, J.-C. et al. Acoustic analog to the dynamical Casimir effect in a Bose–Einstein condensate. Phys. Rev. Lett. 109, 220401 (2012).

    Article  ADS  Google Scholar 

  31. Hung, C.-L., Gurarie, V. & Chin, C. From cosmology to cold atoms: observation of Sakharov oscillations in a quenched atomic superfluid. Science 341, 1213–1215 (2013).

    Article  ADS  CAS  Google Scholar 

  32. Chen, C.-A., Khlebnikov, S. & Hung, C.-L. Observation of quasiparticle pair production and quantum entanglement in atomic quantum gases quenched to an attractive interaction. Phys. Rev. Lett. 127, 060404 (2021).

    Article  ADS  CAS  Google Scholar 

  33. Steinhauer, J. et al. Analogue cosmological particle creation in an ultracold quantum fluid of light. Nat. Commun. 13, 2890 (2022).

    Article  ADS  CAS  Google Scholar 

  34. Tolosa-Simeón, M. et al. Curved and expanding spacetime geometries in Bose–Einstein condensates. Phys. Rev. A 106, 033313 (2022).

  35. Gross, C. et al. Atomic homodyne detection of continuous-variable entangled twin-atom states. Nature 480, 219–223 (2011).

    Article  ADS  CAS  Google Scholar 

  36. Sakharov, A. D. The initial stage of an expanding Universe and the appearance of a nonuniform distribution of matter. Sov. Phys. JETP 22, 241–249 (1966).

    ADS  Google Scholar 

  37. Grishchuk, L. P. Cosmological Sakharov oscillations and quantum mechanics of the early Universe. Phys. Uspekhi 55, 210 (2012).

    Article  ADS  CAS  Google Scholar 

  38. Giorgini, S., Pitaevskii, L. P. & Stringari, S. Condensate fraction and critical temperature of a trapped interacting Bose gas. Phys. Rev. A 54, R4633 (1996).

    Article  ADS  CAS  Google Scholar 

  39. Berges, J., Floerchinger, S. & Venugopalan, R. Dynamics of entanglement in expanding quantum fields. J. High Energy Phys. 2018, 145 (2018).

  40. Robertson, S., Michel, F. & Parentani, R. Controlling and observing nonseparability of phonons created in time-dependent 1D atomic Bose condensates. Phys. Rev. D 95, 065020 (2017).

    Article  ADS  Google Scholar 

  41. Kunkel, P. et al. Detecting entanglement structure in continuous many-body quantum systems. Phys. Rev. Lett. 128, 020402 (2022).

    Article  ADS  CAS  Google Scholar 

  42. Gibbons, G. W. & Hawking, S. W. Cosmological event horizons, thermodynamics, and particle creation. Phys. Rev. D 15, 2738–2751 (1977).

    Article  ADS  MathSciNet  Google Scholar 

  43. Jacobson, T. Thermodynamics of spacetime: the Einstein equation of state. Phys. Rev. Lett. 75, 1260–1263 (1995).

    Article  ADS  MathSciNet  CAS  MATH  Google Scholar 

  44. Jacobson, T. Entanglement equilibrium and the Einstein equation. Phys. Rev. Lett. 116, 201101 (2016).

    Article  ADS  MathSciNet  Google Scholar 

  45. Fischer, U. R. & Schützhold, R. Quantum simulation of cosmic inflation in two-component Bose–Einstein condensates. Phys. Rev. A 70, 063615 (2004).

    Article  ADS  Google Scholar 

  46. Schmidt-May, A. & von Strauss, M. Recent developments in bimetric theory. J. Phys. A 49, 183001 (2016).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  47. Hans, M. et al. High signal to noise absorption imaging of alkali atoms at moderate magnetic fields. Rev. Sci. Instrum. 92, 023203 (2021).

    Article  CAS  Google Scholar 

  48. Volovik, G. E. The Universe in a Helium Droplet (Oxford Univ. Press, 2009).

  49. Bilić, N. & Tolić, D. FRW universe in the laboratory. Phys. Rev. D 88, 105002 (2013).

    Article  ADS  Google Scholar 

  50. Sánchez-Kuntz, N., Parra-López, Á., Tolosa-Simeón, M., Haas, T. & Floerchinger, S. Scalar quantum fields in cosmologies with 2 + 1 spacetime dimensions. Phys. Rev. D 105, 105020 (2022).

    Article  ADS  MathSciNet  Google Scholar 

Download references

Acknowledgements

We thank S. Brunner and F. Schmutte for discussions. This work is supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy EXC 2181/1 - 390900948 (the Heidelberg STRUCTURES Excellence Cluster) and under SFB 1225 ISOQUANT - 273811115, as well as FL 736/3-1. N.L. acknowledges support by the Studienstiftung des Deutschen Volkes. N.S.-K. is supported by the Deutscher Akademischer Austauschdienst (DAAD, German Academic Exchange Service) under the Länderbezogenes Kooperationsprogramm mit Mexiko: CONACYT Promotion, 2018 (57437340). Á.P.-L. is supported by the MIU (Ministerio de Universidades, Spain) fellowship FPU20/05603 and the MICINN (Ministerio de Ciencia e Innovación, Spain) project PID2019-107394GB-I00 (AEI/FEDER,UE).

Author information

Authors and Affiliations

Authors

Contributions

The experimental concept was developed in discussion among all authors. M.H., E.K., N.L., M.K.O., M.S., H.S. and C.V. controlled the experimental apparatus, discussed the measurement results and analysed the data. S.F., T.H., Á.P.-L., N.S.-K. and M.T.-S. elaborated the theoretical framework. All authors contributed to the discussion of the results and the writing of the manuscript.

Corresponding author

Correspondence to Celia Viermann.

Ethics declarations

Competing interests

The authors declare no competing interests.

Peer review

Peer review information

Nature thanks Silke Weinfurtner and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Additional information

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

Extended data figures and tables

Extended Data Fig. 1 Theoretical prediction for density-contrast correlation functions in real space.

The initial temperature is taken to be 40 nK, the final speed of sound is 1.2 µm ms−1 and the ramp goes from as = 350aB to as = 50aB with scale factor a(t) tγ for a decelerated expansion with γ = 0.5 and a duration ∆t = 1.5 ms (left) and ∆t = 3.0 ms (right). The fields are convoluted with a Gaussian of σ = 0.8 µm corresponding to the optical resolution of the experiment. Different colours correspond to different hold times.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Viermann, C., Sparn, M., Liebster, N. et al. Quantum field simulator for dynamics in curved spacetime. Nature 611, 260–264 (2022). https://doi.org/10.1038/s41586-022-05313-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41586-022-05313-9

Comments

By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.

Search

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