# Quantum Chaos 2020 Seminars

During 2020, together with Pieter Claeys we organized a series of online talks about topics related to **Quantum Chaos** in its various forms, including (but not limiting to)

Manifestations of chaos in quantum systems

Quantum information scrambling

Ergodicity and thermalization in closed many-body quantum systems

Quantum simulations of complex quantum dynamics

Talks were given by senior researchers as well as students and postdocs.

Be sure to check out other series of online talks: Virtual AMO Seminars, Quarantine Thermo, Quantum Matter seminars.

**To suscribe to our ****mailing list**** to get updates and announcements of the upcoming seminars, send an email to ****qchaos2020+subscribe@googlegroups.com**

## Past talks

**BONUS SEMINAR(S)**

**Pablo Poggi **(University of New Mexico)

*Quantifying the sensitivity to errors in noisy quantum simulators*

**Pieter Claeys **(Cambridge University)

*Thermalization and scrambling in dual-unitary circuit models *

Thursday December 3rd -** ****Link**

**Quantifying the sensitivity to errors in analog quantum simulation**

Quantum simulators are widely seen as one of the most promising near-term applications of quantum technologies. However, it remains unclear to what extent a noisy device can output reliable results in the presence of unavoidable imperfections. Here we propose a framework to characterize the performance of quantum simulators by linking the robustness of measured quantum expectation values to the spectral properties of the output observable, which in turn can be associated with its macroscopic or microscopic character. We show that, under general assumptions and on average over all states, imperfect devices are able to reproduce the dynamics of macroscopic observables accurately, while the relative error in the expectation value of microscopic observables is much larger on average. We experimentally demonstrate the universality of these features in a state-of-the-art quantum simulator and show that the predicted behavior is generic for a highly accurate device, without assuming any detailed knowledge about the nature of the imperfections.

**Th****ermalization and scrambling in dual-unitary circuit models **

Quantum lattice models with time evolution governed by local unitary quantum circuits can serve as a minimal model for the study of general unitary dynamics governed by local interactions. Although such circuit dynamics exhibit many of the features expected of generic many-body dynamics, exact results generally require the presence of randomness in the circuit. After a short introduction to unitary circuit models, we discuss the class of dual-unitary circuits characterized by an underlying space-time symmetry. We present exactly solvable models for ergodic and non-ergodic thermalization and prethermalization for correlations and the scrambling of out-of-time-order correlators.

**Monika Schleier-Smith **(Stanford)

*Optically Programmable Interactions for Quantum Simulation*

Tuesday November 24th -** ****Link**

The dream of the quantum engineer is to have an “arbitrary waveform generator” for designing quantum states and Hamiltonians. Motivated by this vision, I will report on advances in optical control of long-range interactions among cold atoms. By coupling atoms to light in an optical resonator, we generate tunable non-local Heisenberg interactions, characterizing the resulting phases and dynamics by real-space imaging. Notable observations include interaction-based protection of spin coherence and photon-mediated spin-mixing—a new mechanism for generating correlated atom pairs. I will present recent results on optically programming the distance-dependence of the spin-spin couplings, with prospects for studies of fast scrambling. I will also touch on a complementary approach of Rydberg dressing for optical control of local interactions, and discuss broader prospects in quantum simulation, quantum optimization, and quantum metrology.

**Diego Wisniacki **(University of Buenos Aires)

*Signatures of quantum chaos transition in small systems and more *

Thursday November 19th - **Link**

The usual way to detect and gauge the integrability to chaos transition in quantum systems is based on the spectral distributions. I will discuss the possibility to use other indicators as the long-time regime of the Out-of-time-order correlators or the mean value of the purity. More interestingly, even in the case of systems with extremely small Hilbert space, such measures are able to reproduce the whole integrable to chaos transition. Finally, I will show the implications for quantum control experiments.

**Fiona Burnell** (University of Minnesota)

*Exact models of many-body quantum scars *

Thursday November 12th - **Link**

There are a few well-known ways for quantum mechanical, many-body systems to avoid coming to thermal equilibrium. For example, we know of two classes of systems -- integrable systems, and many-body localized systems -- for which conservation laws prevent any eigenstate from reaching (conventional) thermal equilibrium. More recently, a much more subtle type of non-thermal quantum phenomenon has been discovered, dubbed many-body quantum scars. In these systems, a small number of eigenstates (and hence a small number of initial conditions) have non-thermal behavior, while most initial states will approach thermal equilibrium in the usual way. I will give a general picture of how and when this phenomenon arises, and discuss several examples of systems exhibiting exact quantum many-body scars.

Classical chaotic dynamics exhibit extreme sensitivity to initial conditions -- known as the butterfly effect. The problem in quantum mechanics, however, is much more subtle. Conventional approaches usually address quantum chaos in the energy domain, e.g., spectral correlations. Recently developments invented different diagnostics to reveal the quantum butterfly effect in the time domain. The quantum butterfly effect has a much richer structure than its classical counterpart. In this talk, I will tell a coherent story and introduce various types of butterflies in the quantum world: The famous Lorenz-Butterfly with circuit complexity; a Bradbury-Butterfly known as a novel correlator, the out-of-time correlator; and an Anti-Butterfly which can heal damaged information from the past.

Recent work has shown that unitary circuits subject to repeated projective measurements can undergo an entanglement transition as a function of the measurement rate. This transition is generally understood in terms of a competition between the scrambling effects of unitary dynamics and the disentangling effects of measurements. I will discuss how, surprisingly, entanglement transitions are possible even in the absence of unitary dynamics in “measurement only” models. I will talk about the entanglement phase diagrams in these models, and also present results on measures of locality under non-unitary dynamics. Finally, I will discuss a recent proposal to measure entanglement without issues of postselection in a class of non-unitary circuits using ideas of spacetime duality.

**Shreya Vardhan ** (MIT)

*A universal approximation for entanglement entropies of equilibrated pure states*

Thursday October 22th - **Link**

When a pure state in a non-integrable quantum many-body system is evolved to late times, we expect it to thermalize—that is, we expect its macroscopic properties to resemble those of an equilibrium density matrix. However, the entanglement entropies of such a state must obey certain constraints coming from unitarity, which are not obeyed by an equilibrium density matrix. In this talk, I will explain an approximation method that leads to a simple universal expression for the entanglement entropies of an equilibrated pure state in any quantum many-body system. This expression is independent of the details of the initial state and hence reflects thermalization, while also being manifestly consistent with unitarity. I will also discuss how this method can be applied to equilibrated pure states in gravitational systems, such as those involving black holes, where it can be used to address the information loss paradox of Hawking.

**Daniel Ranard ** (Stanford University)

*Bounding the spread of quantum information, and emergent classicality in dynamics of large systems*

Thursday October 15th - **Link**

How can information about a single subsystem spread through a many-body environment? We show that whenever a subsystem interacts with an environment, for almost everywhere in the environment, any locally accessible information about the subsystem must be approximately classical, i.e. obtainable from some fixed measurement. The result strengthens the earlier result of arXiv:1310.8640. It may also be seen as a new consequence of the principles of no-cloning or monogamy of entanglement. The proof offers a constructive optimization procedure for determining the effective "measurement" on the subsystem induced by the dynamics. Alternatively, under channel-state duality, these results characterize the marginals of multipartite states. *Talk based primarily on arXiv:2001.01507.*

**Lennart Dabelow ** (Bielefeld University)

*Typicality approach to perturbed quantum many-body relaxation *

Thursday October 8th - **Link**

We develop an analytical prediction for the relaxation of isolated many-body quantum systems subject to weak-to-moderate perturbations. Provided that the unperturbed behavior is known, we employ a typicality approach modeling the essential characteristics of the perturbation operator to describe the time evolution of expectation values in the perturbed system. In particular, the prediction identifies two decisive parameters of the perturbation: its overall strength and its energy range or band width. The theory provides a unified framework for such diverse phenomena as prethermalization, quantum quenches, or the relaxation of system-bath compounds. We demonstrate its wide applicability by comparison with various experimental and numerical examples.

**Annabelle Bohrdt ** (Technical University Munich)

*Probing dynamics in quantum simulators *

Thursday October 1st - **Link**

In this talk I will discuss new possibilities to probe the dynamics of quantum many-body systems, in particular the Bose-Hubbard model with and without disorder.

Out-of-time-ordered (OTO) correlation functions have been proposed to describe the distribution or “scrambling” of information across a quantum state. We investigate both time-ordered and OTO correlation functions in the non-integrable, one-dimensional Bose-Hubbard model at high temperatures where well-defined quasiparticles cease to exist. We propose an interferometric scheme to approach the challenge of measuring these correlation functions in real space and time. Performing numerical simulations based on matrix product operators, we observe a linear light-cone spreading of quantum information in the OTO correlators. In contrast with the fast spreading of information, the thermalization of the system takes parametrically longer due to the slow diffusion of conserved quantities. Adding strong disorder can inhibit thermalization, leading to a many-body localized (MBL) phase. Beyond the absence of transport, the MBL phase has distinctive signatures, such as slow dephasing and logarithmic entanglement growth. We experimentally probe these signatures in a system of coupled superconducting qubits.

**Adolfo del Campo** (Donostia International Physics Center)

*Quantum Chaos versus Decoherence *

Thursday September 24th - **Link**

Quantum chaos imposes universal spectral signatures that govern the thermofield dynamics of a many-body system in isolation. The fidelity between the initial and time-evolving thermofield double states exhibits as a function of time a decay, dip, ramp and plateau. Sources of decoherence give rise to a non-unitary evolution and result in information loss. Energy dephasing gradually suppresses quantum noise fluctuations and the dip associated with spectral correlations. Decoherence further delays the appearance of the dip and shortens the span of the linear ramp associated with chaotic behavior. The interplay between signatures of quantum chaos and information loss is determined by the competition among the decoherence, dip and plateau characteristic times, as demonstrated in the stochastic Sachdev-Ye-Kitaev model.

**Samuel Garratt** (Oxford University)

Thursday September 17th - **Link**

*Many-body quantum chaos and local pairing of Feynman histories *

In this talk I will discuss work on many-body quantum dynamics using Floquet quantum circuits in one space dimension as simple examples of systems with local couplings that support ergodic phases. Physical properties can be expressed in terms of multiple sums over Feynman histories, which for these models are many-body orbits in Fock space. A natural simplification of these sums is the diagonal approximation, where the only terms that are retained are ones in which each orbit is paired with a partner that carries the complex conjugate weight. I examine for Floquet quantum circuits when the diagonal approximation holds, its consequences in calculations of physical properties, and the nature of the leading corrections to it. I show that properties are dominated at long times by contributions to orbit sums in which each orbit is paired locally with a conjugate, as in the diagonal approximation, but that in large systems these dominant contributions consist of many spatial domains, with distinct local pairings in neighbouring domains. The existence of these domains is reflected in deviations of the spectral form factor from RMT predictions, and of matrix element correlations from ETH predictions; deviations of both kinds diverge with system size. I will demonstrate that this physical picture of orbit-pairing domains has a precise correspondence in the spectral properties of a transfer matrix that acts in the space direction to generate the ensemble-averaged spectral form factor.

Joint work with John Chalker: arXiv:2008.01697

**Alvaro Alhambra **(Max Planck Institute for Quantum Optics)

Thursday September 10th - **Link**

*Equilibration of quantum systems, eigenstate thermalization and linear response functions*

In this talk we will explore the equilibration of closed non-integrable (or chaotic) quantum systems, when driven both close and far from thermal equilibrium. First, we give analytical arguments constraining the behavior of dynamical two-point correlation functions, which model time evolution after weak perturbations away from equilibrium. We show how dissipation emerges in this regime, and, based on ETH, give evidence for the relevant timescale governing it. We connect these results to the problem of equilibration after quenches far away from equilibrium, revisiting ideas from Srednicki (J. Phys. A 32 (1999) 1163). These suggest that certain regimes of far from equilibrium behavior are well approximated by linear response functions, an idea reminiscent of “fluctuation-dissipation” theorems. In doing so, we give arguments constraining the initial timescale of equilibration after generic quantum quenches, consistent with numerical results in spin chains.

Joint work with Jonathon Riddell and Luis Pedro Garcia-Pintos.

**Lea Santos **(Yeshiva University)

Thursday August 13th - **Link**

*Indicators of many-body quantum chaos and time scales for equilibration *

In this talk, various indicators of quantum chaos will be presented and compared, including level statistics, structure of eigenstates, matrix elements of observables, and out-of-time ordered correlators. Particular attention will be given to the correlation hole, which is a dynamical manifestation of spectral correlations. We use it to discuss the time scales involved in the relaxation process of isolated many-body quantum systems. While there is consensus on what equilibration and thermalization mean in these systems, there is no agreement on how long they take to reach equilibrium.

**Nicole Yunger Halpern **(Harvard-Smithsonian ITAMP)

Thursday August 6th - **Link**

*Entropic uncertainty relations for quantum-information scrambling *

How violently do two quantum operations disagree? Different subfields of physics answer differently, featuring different notions of operator incompatibility: (i) In pure quantum information theory, uncertainty relations are cast in terms of entropies. These entropic uncertainty relations constrain measurement outcomes. (ii) Condensed matter and high-energy physics feature interacting quantum many-body systems, such as spin chains. A local perturbation, such as a Pauli operator on one side of a chain, spreads through many-body entanglement. The perturbation comes to overlap, and to disagree, with probes localized on the opposite side of the system. This disagreement signals that information about the perturbation has scrambled, or become hidden in highly nonlocal correlations. I will unite these two notions of quantum operator disagreement, presenting an entropic uncertainty relation for quantum-information scrambling. The relation can be tested experimentally with superconducting qubits, trapped ions, and quantum dots.

NYH, Bartolotta, and Pollack, Communications Physics 2, 92 (2019). https://www.nature.com/articles/s42005-019-0179-8 .

**Gonzalo Álvarez **(Centro Atómico Bariloche - CNEA)

Thursday July 30th - **Link**

*Decoherence scaling transition in the dynamics of quantum information scrambling *

Reliable processing of quantum information for developing quantum technologies requires precise control of out-of-equilibrium many-body systems. This is a highly challenging task as the fragility of quantum states to external perturbations increases with the system-size. In this talk, I will report on a series of experimental quantum simulations that allow to quantify the sensitivity of a controlled Hamiltonian evolution to perturbations that drive the system away from the targeted evolution [1,2]. Based on out-of-time order correlations, we demonstrate that the decay-rate of the process fidelity increases with the effective number K of correlated qubits as K^α [2]. As a function of the perturbation strength, we observed a sharp decoherence scaling transition of the exponent α between two distinct dynamical regimes. In the limiting case below the critical perturbation strength, there is not inherent limit to the number of qubits that can be controlled with high fidelity. This may indicate that reliable control of large quantum systems might be possible if the perturbation can be kept below this critical threshold.

[1] Gonzalo A. Álvarez, Dieter Suter, and Robin Kaiser. “Localization-delocalization transition in the dynamics of dipolar-coupled nuclear spins”. Science **349**, 846 (2015).

[2] Federico D. Dominguez, Maria Cristina Rodriguez, Robin Kaiser, Dieter Suter, Gonzalo A. Álvarez. “Decoherence scaling transition in the dynamics of quantum information scrambling”. arXiv:2005.12361 (2020).

**Pavel Kos **(University of Ljubljana)

Thursday July 23rd - **Link**

*Solvable Chaotic Many-Body Quantum Systems*

In my talk, I will discuss a class of solvable chaotic many-body quantum models. These locally interacting models have a special property, called dual-unitarity, meaning that the evolution propagator in the space direction is also unitary. This property allows us to exactly answer some questions about the chaotic evolution of these systems.

Firstly, I will briefly address the results regarding spectral form factor, which is a spectral indicator of chaos. Secondly, I will focus on a dynamical indicator of chaos: operator entanglement entropy, which measures the complexity of the evolving operators in the Heisenberg picture. Thirdly, I will discuss the spatio-temporal correlation functions in these models.

[1] B. Bertini, P. Kos and T. Prosen, Exact Spectral Form Factor in a Minimal Model of Many-Body Quantum Chaos, Phys. Rev. Lett. 121, 264101 (2018).

[2] B. Bertini, P. Kos and T. Prosen, Operator Entanglement in Local Quantum Circuits I: Chaotic Dual-Unitary Circuits, SciPost Phys. 8, 67 (2020).

[3] B. Bertini, P. Kos and T. Prosen, Exact Correlation Functions for Dual-Unitary Lattice Models in 1 + 1 Dimensions, Phys. Rev. Lett. 123, 210601 (2019).

[4] P. Kos, B. Bertini and T. Prosen, Correlations in Perturbed Dual-Unitary Circuits: Efficient Path-Integral Formula, arXiv:2006.07304 (2020).

**Ana Maria Rey **(University of Colorado - JILA)

Thursday July 16th - **Link**

*Entanglement dynamics and scrambling in a trapped ion quantum magnet*

In this talk I will report on our current effort to develop protocols that can quantify the build-up of quantum correlations and storage of quantum information in a planar crystal of trapped ions [1]. Using a pair of lasers, we couple the spins to the vibrational modes (phonons) of the crystal. The phonons mediate interactions between the spins which we use to generate entanglement starting from easily prepared uncorrelated states[2]. I will also discuss a measurement scheme, implemented by using a manybody echo sequence that reverses the Hamiltonian dynamics, and gives experimental access to out-of-time-order correlations (OTOCs)[3,4]. Measuring OTOCs in controllable atomic laboratories can not only have a great impact on quantum information processing and quantum enhanced metrology, but also opens a path for future tests of the holographic duality between quantum and gravitational systems.

[1] R. J. Lewis-Swan, Safavi-Naini, A. , Kaufman, A. M. , and Rey, A. M. , “Dynamics of quantum information”, *Nat. Rev. Phys.*, vol. 1, pp. 627-634, 2019.

[2] A. Safavi-Naini, Lewis-Swan, R. J. , Bohnet, J. G. , Gaerttner, M. , Gilmore, K. A. , Jordan, J. E. , Cohn, J. , Freericks, J. K. , Rey, A. M. , and Bollinger, J. J. , “Verification of a many-ion simulator of the Dicke model through slow quenches across a phase transition”, *Physical Review Letters*, vol. 121, p. 040503, 2018.

[3] M. Gärttner, Bohnet, J. G. , Safavi-Naini, A. , Wall, M. L. , Bollinger, J. J. , and Rey, A. M. , “Measuring out-of-time-order correlations and multiple quantum spectra in a trapped-ion quantum magnet”, *Nature Physics*, vol. 13, 2017.

[4] R. J. Lewis-Swan, Safavi-Naini, A. , Bollinger, J. J. , and Rey, A. M. , “Unifying scrambling, thermalization and entanglement through measurement of fidelity out-of-time-order correlators in the Dicke model”, *Nat. Comm.*, vol. 10, no. 1581, 2019.

**Juan Diego Urbina **(University of Regensburg)

Thursday July 9th - **Link**

*Universal signatures of many-body chaos and integrability: semiclassical theory of quantum interference in Fock space*

After initial insights from Einstein, Pauli, Dirac and other brilliant minds, the concepts and techniques of semiclassical quantization for single-particle systems reached its modern form with the pioneering work of Michael Berry and Martin Gutzwiller in the 70's. In a nutshell, the semiclassical program boils down to an asymptotic analysis of Feynman path integral when $\hbar \to 0$ providing a representation of quantum mechanical amplitudes as coherent sums over contributions from classical processes, and thus attempts to use very non-trivially classical information in order to predict and analyze quantum mechanical phenomena including aspects like entanglement and interference. In the semiclassical framework, the quantum signatures of classical phase space morphology from integrability to chaos can be precisely addressed. This connection is the definitional subject of Quantum Chaos, and has lead to deep insights, from the role of interfering periodic orbits to explain the transition from Poissonian to Wigner-Dyson universal spectral fluctuations, to the prediction of weak localization and proximity effects in mesoscopic trasport, and many more.

During the last years, running in parallel with the interest on quantum-chaos-related signatures in systems far away from the semiclassical regime, the extension of the semiclassical program to many body quantum systems has been achieved by lifting its ideas and methods into Fock space. This has paved the way to connect the well-defined notions of chaos and integrability at the level of whatever plays the role of classical limit, with the elusive and fascinating consequences of many body interference. Examples of many body semiclassics include then he saturation of OTOCs beyond the scrambling time, the emergence of discrete many body spectrum due to interfering of contributions associated with different mean-field solutions, the Fock space versions of coherent backscattering and spin echo, and the possibility of very precise wavepacket propagation beyond the Eherenfest time where Truncated Wigner methods break down.

I will present a survey of the development, applications, insights and open aspects of the semiclassical program in quantum many body systems.

**Charlie Nation **(University College London)

Thursday July 2nd - **Link**

*Thermalization dynamics and the emergence of Brownian motion in chaotic quantum systems*

The question of how statistical physics emerges in a closed quantum system is a long-standing problem, beginning with von-Neumann and Schrodinger. In this talk we will see that a description of such systems in terms of random matrix theory, and more generally chaotic wavefunctions, leads to a consistent description of the decay to thermal equilibrium. We will further see that classical behaviour can be derived in relevant limits, such that thermalization dynamics can be described by an effective model of Brownian motion. Further, the action of successive measurements on otherwise closed chaotic systems is shown to render the dynamics unchanged on average, thus allowing a description of thermalization in terms of quantum trajectories. We will additionally see that this theoretical approach may be exploited to characterize quantum devices; allowing for the inference of the density of states and effective Hilbert space dimension of devices such as quantum simulators. The question of how statistical physics emerges in a closed quantum system is a long-standing problem, beginning with von-Neumann and Schrodinger. In this talk we will see that a description of such systems in terms of random matrix theory, and more generally chaotic wavefunctions, leads to a consistent description of the decay to thermal equilibrium. We will further see that classical behaviour can be derived in relevant limits, such that thermalization dynamics can be described by an effective model of Brownian motion. Further, the action of successive measurements on otherwise closed chaotic systems is shown to render the dynamics unchanged on average, thus allowing a description of thermalization in terms of quantum trajectories. We will additionally see that this theoretical approach may be exploited to characterize quantum devices; allowing for the inference of the density of states and effective Hilbert space dimension of devices such as quantum simulators.

**Silvia Pappalardi **(SISSA)

Thursday June 25th - **Link**

*Entanglement dynamics and chaos in systems with collective and long-range interactions *

It is widely recognized that entanglement generation and dynamical chaos are intimately related in semiclassical models. In this talk I will describe a unifying framework which directly connects the bipartite and multipartite entanglement growth to the quantifiers of classical and quantum chaos. In the semiclassical regime, the dynamics of the von Neumann entanglement entropy, the spin squeezing, the quantum Fisher information and the out-of-time-order square commutator are governed by the divergence of nearby phase-space trajectories via the local Lyapunov spectrum.

This constitutes the underlying mechanism for the counterintuitive slow (logarithmic) growth of entanglement in quantum spin systems with slowly-decaying interactions, shown by several numerical simulations. The standard quasiparticle contribution is shown to get suppressed as the interaction range is sufficiently increased. All our analytical results agree with numerical computations, both for two paradigmatic models of quantum chaos (the Dicke model and the kicked top) and for the quantum Ising chains with long-range couplings.

The talk will be based on the following references:

1. Origin of the slow growth of entanglement entropy in long-range interacting spin systems, A Lerose, S Pappalardi - Physical Review Research, 2020

2. Bridging entanglement dynamics and chaos in semiclassical systems, A Lerose, S Pappalardi - arXiv preprint arXiv:2005.03670, 2020

Abstract: I will define a speed of quantum information propagation in strongly chaotic systems and show how it relates to the speed of operator spreading and the rate of entanglement growth. I will also discuss how these timescales depend on temperature and pose some puzzles for future work. Based on 2005.10814 and 1908.06993. * *

**Anatoli Polkovnikov **(Boston University)

Thursday June 11th - **Link**

*Detecting quantum chaos through sensitivity of eigenstates to adiabatic transformations *

In this talk I will discuss how one can use sensitivity of eigenstates to adiabatic deformations of the Hamiltonian as a very sensitive probe of emergence of quantum chaos. This probe can be equivalently viewed as a detector of the operator spreading at exponentially long times. Using several examples I will illustrate that the properly normalized norm of the adiabatic gauge potential, the generator of adiabatic transformations, can detect exponentially small in the system size integrability breaking perturbations. Surprisingly, using this probe and across many models, we found existence of a very robust and universal intermediate regime where the system is chaotic but non-ergodic, i.e. not-satisfying ETH. This regime is characterized by exponentially long in the system size relaxation times and the lack of complete thermalization.