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Quantum field simulator for dynamics in curved spacetime


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.

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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.

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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.


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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).

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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.

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Correspondence to Celia Viermann.

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

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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.

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Viermann, C., Sparn, M., Liebster, N. et al. Quantum field simulator for dynamics in curved spacetime. Nature 611, 260–264 (2022).

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