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Experimental extraction of the quantum effective action for a non-equilibrium many-body system

An Author Correction to this article was published on 12 August 2020

This article has been updated


On the fundamental level, quantum fluctuations or entanglement lead to complex dynamical behaviour in many-body systems1 for which a description as emergent phenomena can be found within the framework of quantum field theory. A central quantity in these efforts, containing all information about the measurable physical properties, is the quantum effective action2. Though non-equilibrium quantum dynamics can be exactly formulated in terms of the quantum effective action, finding solutions is in general beyond the capabilities of classical computers3. Here, we present a strategy to determine the non-equilibrium quantum effective action4 using analogue quantum simulators, and demonstrate our method experimentally with a quasi-one-dimensional spinor Bose gas out of equilibrium5,6. Spatially resolved snapshots of the complex-valued transversal spin field7 allow us to infer the quantum effective action up to fourth order in an expansion in one-particle irreducible correlation functions at equal times. We uncover a strong suppression of the irreducible four vertex emerging at low momenta in the highly occupied regime far from equilibrium where perturbative descriptions fail8. Our work constitutes a new realm of large-scale analogue quantum computing9, where highly controlled synthetic quantum systems10 provide the means for solving theoretical problems in high-energy and condensed-matter physics with an experimental approach11,12,13,14.

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Fig. 1: Experimental platform and extraction of correlation functions.
Fig. 2: Statistical significance of the four-point 1PI correlator in momentum space.
Fig. 3: Momentum-conserving diagonals of the 1PI correlators.
Fig. 4: Observation of scaling with time of the distribution function as well as the corresponding couplings.

Data availability

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

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This Letter comprises parts of the doctoral thesis work of M.P.30 and T.V.Z., submitted to Heidelberg University, Germany. We thank S. Erne, T. Gasenzer, P. Hauke, C.-M. Schmied, J. Schmiedmayer, T. Schweigler and M. Tarpin for discussions and R. Rosa-Medina for experimental assistance. This work was supported by the DFG Collaborative Research Center SFB1225 (ISOQUANT) and the ERC Advanced Grant Horizon 2020 EntangleGen (project ID 694561), by Deutsche Forschungsgemeinschaft (DFG) under Germany’s Excellence Strategy EXC-2181/1 - 390900948 (the Heidelberg STRUCTURES Excellence Cluster) and the Heidelberg Center for Quantum Dynamics. P.K. acknowledges support from the Studienstiftung des deutschen Volkes.

Author information




The experimental and theoretical concept was developed in discussion among all authors. M.P., P.K., S.L. and A.B. controlled the experimental apparatus. M.P. and T.V.Z. analysed the data. M.P., T.V.Z., P.K., H.S., J.B. and M.K.O. discussed the measurement results. T.V.Z. and J.B. elaborated the equal-time formalism. All authors contributed to the discussion of the results and the writing of the manuscript.

Corresponding author

Correspondence to Maximilian Prüfer.

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The authors declare no competing interests.

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Extended data

Extended Data Fig. 1 Observation of the complex valued field for all times.

We show the distribution of F (as in Fig. 1c; for the relative frequency see colorbar) for all evolution times shown in Fig. 4. The blue circle indicates F = 0.7.

Extended Data Fig. 2 1PI four vertex.

\({\varGamma }_{t}^{(4)}({p}_{1},{p}_{2},{p}_{3},{p}_{4})\) inferred from the data for the same momenta as shown in Fig. 2c. For comparison we show the results for the Gaussian model.

Extended Data Fig. 3 Momentum conserving diagonals.

Momentum conserving diagonals of the four-point 1PI correlators shown in Fig. 3 for all evolution times between 9 s and 18 s. Grey shaded area is the finite statistical bias from a Gaussian model. All error bars shown are 1 s.d. calculated from bootstrap resampling.

Extended Data Fig. 4 Universal dynamics of two-point correlator.

a, Time evolution of two-point correlator for longest times accessible in the experiment. b, A rescaling analysis yields α = β = 0.45 ± 0.05. After rescaling the data collapses on a universal curve. The found dynamics underlines the universal features found in these and previous experiments.

Extended Data Fig. 5 Rescaling of different momentum conserving diagonals.

Momentum conserving diagonals of the 1PI four-point correlator shown in Fig. 3 for all evolution times between 9 s and 18 s rescaled as in Fig. 4b with γ = 0 and β4 = 1/2. Given the finite statistics, we attribute the quality of the scaling collapse shown in Fig. 4b (compared to the shifted momentum diagonals shown here) to a larger signal-to-noise ratio, which is also evidenced by smaller statistical errors. All error bars shown are 1 s.d. calculated from bootstrap resampling.

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Prüfer, M., Zache, T.V., Kunkel, P. et al. Experimental extraction of the quantum effective action for a non-equilibrium many-body system. Nat. Phys. 16, 1012–1016 (2020).

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