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ABSTRACT Dynamical maps describe general transformations of the state of a physical system—their iteration interpreted as generating a discrete time evolution. Prime examples include

classical nonlinear systems undergoing transitions to chaos. Quantum mechanical counterparts show intriguing phenomena such as dynamical localization on the single-particle level. Here we

extend the concept of dynamical maps to a many-particle context, where the time evolution involves both coherent and dissipative elements: we experimentally explore the stroboscopic dynamics

of a complex many-body spin model with a universal trapped ion quantum simulator. We generate long-range phase coherence of spin by an iteration of purely dissipative quantum maps and

demonstrate the characteristics of competition between combined coherent and dissipative non-equilibrium evolution—the hallmark of a previously unobserved dynamical phase transition. We

assess the influence of experimental errors in the quantum simulation and tackle this problem by developing an efficient error detection and reduction toolbox based on quantum feedback.

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support SIMILAR CONTENT BEING VIEWED BY OTHERS QUANTUM SIMULATION OF SPIN-BOSON MODELS WITH STRUCTURED BATH Article Open access 30 April 2025 UNCOVERING LOCAL INTEGRABILITY IN QUANTUM

MANY-BODY DYNAMICS Article Open access 15 March 2025 OBSERVATION OF A FINITE-ENERGY PHASE TRANSITION IN A ONE-DIMENSIONAL QUANTUM SIMULATOR Article 17 January 2025 REFERENCES * Maze, J. R.

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ACKNOWLEDGEMENTS We gratefully acknowledge support by the Austrian Science Fund (FWF), through the SFB FoQus (FWF Project No. F4002-N16 and F4016-N16) and the START grant Y 581-N16 (S.D.),

by the European Commission (AQUTE), as well as the Institut für Quantenoptik und Quanteninformation GmbH. This research was funded by the Office of the Director of National Intelligence

(ODNI), Intelligence Advanced Research Projects Activity (IARPA), through the Army Research Office grant W911NF-10-1-0284. All statements of fact, opinion or conclusions contained herein are

those of the authors and should not be construed as representing the official views or policies of IARPA, the ODNI or the US Government. M.M. acknowledges support by the CAM research

consortium QUITEMAD S2009-ESP-1594, European Commission PICC: FP7 2007-2013, Grant No. 249958, and the Spanish MICINN grant FIS2009-10061. AUTHOR INFORMATION Author notes * J. T. Barreiro

Present address: Present address: Fakultät für Physik, Ludwig-Maximilians-Universität München and Max-Planck Institute of Quantum Optics, Germany, * P. Schindler and M. Müller: These authors

contributed equally to this work AUTHORS AND AFFILIATIONS * Institut für Experimentalphysik, Universität Innsbruck, Technikerstrasse 25, 6020 Innsbruck, Austria P. Schindler, D. Nigg, J. T.

Barreiro, E. A. Martinez, M. Hennrich, T. Monz & R. Blatt * Departamento de Física Teórica I, Universidad Complutense, Avenida Complutense s/n, 28040 Madrid, Spain M. Müller * Institut

für Theoretische Physik, Universität Innsbruck, Technikerstrasse 25, 6020 Innsbruck, Austria S. Diehl & P. Zoller * Institut für Quantenoptik und Quanteninformation, Österreichische

Akademie der Wissenschaften, Technikerstrasse 21A, 6020 Innsbruck, Austria S. Diehl, P. Zoller & R. Blatt Authors * P. Schindler View author publications You can also search for this

author inPubMed Google Scholar * M. Müller View author publications You can also search for this author inPubMed Google Scholar * D. Nigg View author publications You can also search for

this author inPubMed Google Scholar * J. T. Barreiro View author publications You can also search for this author inPubMed Google Scholar * E. A. Martinez View author publications You can

also search for this author inPubMed Google Scholar * M. Hennrich View author publications You can also search for this author inPubMed Google Scholar * T. Monz View author publications You

can also search for this author inPubMed Google Scholar * S. Diehl View author publications You can also search for this author inPubMed Google Scholar * P. Zoller View author publications

You can also search for this author inPubMed Google Scholar * R. Blatt View author publications You can also search for this author inPubMed Google Scholar CONTRIBUTIONS M.M., P.S., J.T.B.

and S.D. developed the research, based on theoretical ideas conceived with P.Z.; P.S. and D.N. performed the experiments; P.S. and T.M. analysed the data; P.S., J.T.B., D.N., T.M., E.A.M.,

M.H. and R.B. contributed to the experimental set-up; P.S., M.M. and S.D wrote the manuscript, with revisions provided by J.T.B., P.Z. and R.B; all authors contributed to the discussion of

the results and manuscript. CORRESPONDING AUTHORS Correspondence to P. Zoller or R. Blatt. ETHICS DECLARATIONS COMPETING INTERESTS The authors declare no competing financial interests.

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P., Müller, M., Nigg, D. _et al._ Quantum simulation of dynamical maps with trapped ions. _Nature Phys_ 9, 361–367 (2013). https://doi.org/10.1038/nphys2630 Download citation * Received: 07

December 2012 * Accepted: 15 April 2013 * Published: 19 May 2013 * Issue Date: June 2013 * DOI: https://doi.org/10.1038/nphys2630 SHARE THIS ARTICLE Anyone you share the following link with

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