# Correlations in space and time

If one throws a rock into a calm pond, circular waves spread around the position where it hit the surface. Several seconds later one can still reconstruct the position of the impact from the waves it caused. How well two events (rock impact and waves) at different positions and times are correlated, tells us something about how regular or chaotic the dynamics of a system is. If it was windy and the pond surface wasn't calm but turbulent, the regular waves would disappear much more quickly. The general physical concept behind this simple observation is the one of dynamical response functions. Measuring these objects requires to measure correlations between different positions and times.

In the lab, we can prepare and control quantum systems nearly perfectly and thus study these correlation functions in engineered model systems. The ability to resolve the dynamical response function in such highly tunable systems allows us to study the transition from regular to chaotic behavior. An additional twist is added by the fact that we can also time-reverse the dynamics, i.e. let the waves propagate backwards again. If, before starting the backwards evolution, the wave pattern is displaced slightly, in a regular system, the waves will refocus near the original impact position of the rock, but in a chaotic system, they might not refocus at all...

Thomas Gasenzer

Phone: +49 6221 54 5173

thomas.gasenzer@kip.uni-heidelberg.de

Martin Gärttner

Phone: +49 6221 54 5185

martin.gaerttner@kip.uni-heidelberg.de

Maurus Hans

Phone: +49 6221 54 5177

mhans@kip.uni-heidelberg.de

Philipp Kunkel

Phone: +49 6221 54 5178

philipp.kunkel@kip.uni-heidelberg.de

Stefan Lannig

Phone: +49 6221 54 5178

stefan.lannig@kip.uni-heidelberg.de

Markus Oberthaler

Phone: +49 6221 54 5170

markus.oberthaler@kip.uni-heidelberg.de

Maximilian Prüfer

Phone: +49 6221 54 5178

maximilian.pruefer@kip.uni-heidelberg.de

Christian-Marcel Schmied

Phone: +49 6221 54 5186

christian-marcel.schmied@kip.uni-heidelberg.de

Marius Sparn

Phone: +49 6221 54 5177

marius.sparn@kip.uni-heidelberg.de

Helmut Strobel

Phone: +49 6221 54 9276

helmut.strobel@kip.uni-heidelberg.de

Marc Robin Strohmaier

Phone: +49 6221 54 5178

robin.strohmaier@kip.uni-heidelberg.de

Celia Viermann

Phone: +49 6221 54 5177

celia.viermann@kip.uni-heidelberg.de

## Recent publications

### Observation of universal dynamics in a spinor Bose gas far from equilibrium

HD-KIP 18-143, 2018, Nature, Volume: 563, Issue: 7730 , arXiv:1805.11881 217-220 PDF-FileThe dynamics of quantum systems far from equilibrium represents one of the most challenging problems in theoretical many-body physics. While the evolution is in general intractable in all its details, relevant observables can become insensitive to microscopic system parameters and initial conditions. This is the basis of the phenomenon of universality. Far from equilibrium, universality is identified through the scaling of the spatio-temporal evolution of the system, captured by universal exponents and functions. Theoretically, this has been studied in examples as different as the reheating process in inflationary universe cosmology, the dynamics of nuclear collision experiments described by quantum chromodynamics, or the post-quench dynamics in dilute quantum gases in non-relativistic quantum field theory. However, an experimental demonstration of such scaling evolution in space and time in a quantum many-body system is lacking so far. Here we observe the emergence of universal dynamics by evaluating spatially resolved spin correlations in a quasi one-dimensional spinor Bose-Einstein condensate. For long evolution times we extract the scaling properties from the spatial correlations of the spin excitations. From this we find the dynamics to be governed by transport of an emergent conserved quantity towards low momentum scales. Our results establish an important class of non-stationary systems whose dynamics is encoded in time-independent scaling exponents and functions signaling the existence of non-thermal fixed points. We confirm that the non-thermal scaling phenomenon involves no fine-tuning, by preparing different initial conditions and observing the same scaling behaviour. Our analog quantum simulation approach provides the basis to reveal the underlying mechanisms and characteristics of non-thermal universality classes. One may use this universality to learn, from experiments with ultra-cold gases, about fundamental aspects of dynamics studied in cosmology and quantum chromodynamics.

### Relating out-of-time-order correlations to entanglement via multiple-quantum coherences

HD-KIP 18-11, 2018, Phys. Rev. Lett., arXiv:1706.01616 (120) 040402 PDF-FileOut-of-time-order correlations (OTOCs) characterize the scrambling, or delocalization, of quantum information over all the degrees of freedom of a system and thus have been proposed as a proxy for chaos in quantum systems. Recent experimental progress in measuring OTOCs calls for a more thorough understanding of how these quantities characterize complex quantum systems, most importantly in terms of the buildup of entanglement. Although a connection between OTOCs and entanglement entropy has been derived, the latter only quantifies entanglement in pure systems and is hard to access experimentally. In this work, we formally demonstrate that the multiple-quantum coherence spectra, a specific family of OTOCs well known in NMR, can be used as an entanglement witness and as a direct probe of multiparticle entanglement. Our results open a path to experimentally testing the fascinating idea that entanglement is the underlying glue that links thermodynamics, statistical mechanics, and quantum gravity.

### Universal equilibrium scaling functions at short times after a quench

HD-KIP 17-30, 2017, Phys. Rev. E, Volume: 96, Issue: 2, arXiv:1704.03517 022110 PDF-FileBy analyzing spin-spin correlation functions at relatively short distances, we show that equilibrium near-critical properties can be extracted at short times after quenches into the vicinity of a quantum critical point. The time scales after which equilibrium properties can be extracted are sufficiently short so that the proposed scheme should be viable for quantum simulators of spin models based on ultracold atoms or trapped ions. Our results, analytic as well as numeric, are for one-dimensional spin models, either integrable or nonintegrable, but we expect our conclusions to be valid in higher dimensions as well.

### Experimental observation of the Poincaré-Birkhoff scenario in a driven many-body quantum system

HD-KIP 17-12, 2017, Phys. Rev. A, Volume: 95, Issue: 1 011602 PDF-FileAccessing the connection between classical chaos and quantum many-body systems has been a long-standing experimental challenge. Here, we investigate the onset of chaos in periodically driven two-component Bose-Einstein condensates, whose small quantum uncertainties allow for exploring the phase space with high resolution. By analyzing the uncertainties of time-evolved many-body states, we find signatures of elliptic and hyperbolic periodic orbits generated according to the Poincaré-Birkhoff theorem, and the formation of a chaotic region at increasing driving strengths. The employed fluctuation analysis allows for probing the phase-space structure by use of only short-time quantum dynamics.

### Strongly anomalous non-thermal fixed point in a quenched two-dimensional Bose gas

HD-KIP 16-110, 2016, New J. Phys., arXiv:1611.01163 (19) 093014 PDF-FileUniversal scaling behavior in the relaxation dynamics of an isolated two-dimensional Bose gas is studied by means of semi-classical stochastic simulations of the Gross-Pitaevskii model. The system is quenched far out of equilibrium by imprinting vortex defects into an otherwise phase-coherent condensate. A strongly anomalous non-thermal fixed point is identified, associated with a slowed decay of the defects in the case that the dissipative coupling to the thermal background noise is suppressed. At this fixed point, a large anomalous exponent η ≃ −3 and, related to this, a large dynamical exponent

*z*≃ 5 are identified. The corresponding power-law decay is found to be consistent with three-vortex-collision induced loss. The article discusses these aspects of non-thermal fixed points in the context of phase-ordering kinetics and coarsening dynamics, thus relating phenomenological and analytical approaches to classifying far-from-equilibrium scaling dynamics with each other. In particular, a close connection between the anomalous scaling exponent η, introduced in a quantum-field theoretic approach, and conservation-law induced scaling in classical phase-ordering kinetics is revealed. Moreover, the relation to superfluid turbulence as well as to driven stationary systems is discussed.### Nonlinear dressed states at the miscibility-immiscibility threshold

HD-KIP 15-101, 2015, Physical Review A, Volume: 92, Issue: 5, arXiv:1407.8049 056314 PDF-FileThe dynamical evolution of spatial patterns in a complex system can reveal the underlying structure and stability of stationary states. As a model system we employ a two-component Bose-Einstein condensate at the transition from miscible to immiscible with the additional control of linear interconversion. Excellent agreement is found between the detailed experimental time evolution and the corresponding numerical mean-field computations. Analyzing the dynamics of the system, we find clear indications of stationary states that we term nonlinear dressed states. A steady-state bifurcation analysis reveals a smooth connection of these states with dark-bright soliton solutions of the integrable two-component Manakov model.

### Observation of Scaling in the Dynamics of a Strongly Quenched Quantum Gas

HD-KIP 15-92, 2015, PHYSICAL REVIEW LETTERS,Volume: 115, Issue: 24, arXiv:1509.02173 245301 PDF-FileWe report on the experimental observation of scaling in the time evolution following a sudden quench into the vicinity of a quantum critical point. The experimental system, a two-component Bose gas with coherent exchange between the constituents, allows for the necessary high level of control of parameters as well as the access to time-resolved spatial correlation functions. The theoretical analysis reveals that when quenching the system close to the critical point, the energy introduced by the quench leads to a short-time evolution exhibiting crossover reminiscent of the finite-temperature critical properties in the system’s universality class. Observing the time evolution after a quench represents a paradigm shift in accessing and probing experimentally universal properties close to a quantum critical point and allows in a new way benchmarking of quantum many-body theory with experiments.

### Rabi Flopping Induces Spatial Demixing Dynamics

HD-KIP 11-76, 2011, PHYSICAL REVIEW LETTERS, Volume: 107, Issue: 19, arXiv:1109.5601 193001 PDF-FileWe experimentally investigate the mixing and demixing dynamics of Bose-Einstein condensates in the presence of a linear coupling between two internal states. The observed amplitude reduction of the Rabi oscillations can be understood as a result of demixing dynamics of dressed states as experimentally confirmed by reconstructing the spatial profile of dressed state amplitudes. The observations are in quantitative agreement with numerical integration of coupled Gross-Pitaevskii equations without free parameters, which also reveals the criticality of the dynamics on the symmetry of the system. Our observations demonstrate new possibilities for changing effective atomic interactions and studying critical phenomena.

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