arXiv subjects

Julian Sonner

Publications and source records attributed to Julian Sonner.

At least 19 recordsLinked to original sources

arXiv2026

Matrix Holography on an Optical Lattice

We propose an analog quantum-simulation protocol based on a lattice of atomic ensembles and time-averaged Floquet dynamics to construct the four-body potentials that typically appear in bosonic BFSS-like matrix quantum mechanics. In contrast to gate-based digital implementations requiring deep circuits, our approach generates the target model through a fixed set of control stages per Floquet cycle, keeping the number of consecutive unitaries constant with respect to the matrix size $N$. The principal scaling cost in our construction is given by the frequency range required to control the growing lattice. We show that this cost grows at most as $\mathcal{O}(N^3)$, giving a polynomial scaling route to large$-N$, attractive to experimental implementation. Additionally, we demonstrate that the coupling parameter can be tuned beyond $\mathcal{O}(1)$, enabling the simulation of strong-coupling physics necessary for holographic phases.

hep-th↗

arXiv2026

From phase space to Krylov space, one shell at a time

In this work, we develop and study the classical Lanczos algorithm allowing us to define Krylov complexity using the symplectic structure of phase space: Poisson brackets take on the role of the quantum commutators and phase-space integrals furnish the inner product needed to define the Lanczos recursion. We show, using general methods of quantum mechanics in phase space, that the $\hbar \to 0$ limit of the usual quantum mechanical Krylov framework smoothly goes over into the classical one. In theories with well-defined semiclassical limits, we show that classical Krylov complexity accurately approximates quantum complexity at early enough times, and thus is a useful characteristic of early-time chaotic dynamics. We define a Krylov-Ehrenfest time, which quantifies the eventual divergence of classical and quantum complexities, corresponding to a characteristic depth of the Krylov chain, $n\sim n_*(\hbar)$, which in the time domain translates to the well-known scale, $t_*\sim\lambda_K^{-1}\log(1/\hbar)$, in generic chaotic systems. We additionally define microcanonical Krylov complexities, both in the classical and quantum setting, which allows one a fine-grained study of complexity, energy shell by energy shell. We apply this framework to the Lipkin-Meshkov-Glick (LMG) and Feingold-Peres (FP) models, which are collective spin systems known to classicalize in the thermodynamic limit. In particular, while the FP model features spectral chaos for some range of coupling values, the LMG model is known to exhibit early-time saddle-dominated scrambling. Our analysis shows that the instability in LMG is resolved by the microcanonical Krylov complexity, which is controlled by the integrable structure of the Hamiltonian in spectral windows away from the instability, both at early and late times.

cond-mat.stat-mech↗

arXiv2026

Quantum chaos and the holographic principle

Recent years have witnessed tremendous progress in developing a fine-grained low-dimensional holographic correspondence, specifically the construction of quantum mechanical boundary theories as holographic duals of two-dimensional gravity. In these developments, quantum chaos played a crucial role, both as source of universality and as a guiding principle for the matching of bulk and boundary signatures of gravity. In this article we review the construction of the chaos-assisted low-dimensional holographic correspondence for non-experts. We open with an introductory discussion of the two main protagonists of the theory, the SYK model and two-dimensional Jackiw-Teitelboim gravity. Within this framework we will discuss two independent 'bridges' between bulk and boundary physics, one pertaining to early time chaotic instabilities, the other to late time quantum chaos up to and including time scales of the order of the gravitational quantum level spacing. We will demonstrate that the resolution of these fine-grained quantum scales requires the extension of semiclassical gravity by elements of string theory. We conclude with an outlook towards higher dimensional generalizations of the chaotic holographic correspondence.

quant-ph↗

arXiv2026

Three Hamiltonians are Sufficient for Unitary $k$-Design in Temporal Ensemble

Unitary $k$-designs are central to quantum information and quantum many-body physics, as they provide efficient proxies for Haar-random dynamics. We study how chaotic Hamiltonian evolution can generate unitary $k$-designs using the frame potential (FP). Unlike standard approaches based on independently sampled Hamiltonians or fine-tuned evolution times, we consider a quenched temporal ensemble in which the Hamiltonians are sampled once and then held fixed, while randomness enters only through the evolution times. We compare a two-step protocol (2SP), with evolution under two fixed Hamiltonians, to a three-step protocol (3SP) with one additional quench, with three evolution times sampled independently. Analytically and numerically, we show that the 2SP fails to realize general unitary $k$-designs, whereas the 3SP realizes them for arbitrary $k$. The random phases in the 3SP impose stronger index-matching constraints in the FP, eliminating the independent permutation degrees of freedom that still remain in the 2SP. Even with imperfect time averaging, the 3SP reaches a given accuracy within a parametrically shorter time window. These results hold universally across different symmetry classes and persist in both all-to-all and local models with random interactions.

quant-ph↗

arXiv2025

Quantum simulation using Trotterized disorder Hamiltonians in a single-mode optical cavity

All-to-all interacting and disordered many-body systems are notoriously hard to simulate on quantum platforms, as interactions are commonly mediated by auxiliary degrees of freedom that lower the amount of disorder, introducing undesired correlations. In this work, we show how a Trotterization scheme can be effectively utilized to densify the disorder of the model. In particular, we study the statistical properties of the resulting model, as well as Trotterization errors in the simulation that affect the time evolution and dynamical observables. As a concrete example, we propose an implementation via a single-mode cavity QED platform of the complex Sachdev-Ye-Kitaev model. We analyze several features of the effective model, such as the distribution of the effective couplings, the number of interacting sites, state preparation, and the behavior of quantum chaos probes. We conclude this work with a detailed investigation of the robustness of our findings against dissipation, both analytically and numerically.

quant-ph↗

arXiv2025

Realizing Unitary $k$-designs with a Single Quench

We present a single-quench protocol that generates unitary $k$-designs with minimal control. A system first evolves under a random Hamiltonian $H_1$; at a switch time $t_s \geq t_{\mathrm{Th}}$ (the Thouless time), it is quenched to an independently drawn $H_2$ from the same ensemble and then evolves under $H_2$. This single quench breaks residual spectral correlations that prevent strictly time-independent chaotic dynamics from forming higher-order designs. The resulting ensemble approaches a unitary $k$-design using only a single control operation -- far simpler than Brownian schemes with continuously randomized couplings or protocols that apply random quenches at short time intervals. Beyond offering a direct route to Haar-like randomness, the protocol yields an operational, measurement-friendly definition of $t_{\mathrm{Th}}$ and provides a quantitative diagnostic of chaoticity. It further enables symmetry-resolved and open-system extensions, circuit-level single-quench analogs, and immediate applications to randomized measurements, benchmarking, and tomography.

quant-ph↗

arXiv2025

Holography of K-complexity: Switchbacks and Shockwaves

In this paper we study Krylov complexity in the presence of single and multiple operators in the DSSYK model, where we can use the analytical techniques coming from chord diagrammatics. One of the results we obtain is that it showcases the switchback effect, when the appropriate ``triple-scaling limit'' is taken, under which the model becomes dual to semiclassical JT gravity. We build on previous work, where it was shown that, in the continuum limit, Krylov complexity is defined as the sum of expectations value of right and left chord number operators. Here we argue that this property signals the emergence of the geometric nature of this notion of K-complexity. We show that in the regime where DSSYK is dual to semi-classical gravity, the light matter chord corresponds to a shockwave insertion in JT gravity. We identify the geodesic-length dual of the operator complexity and extend the relevant holographic dictionary to describe the details of the matter insertions. Additionally, we define a class of two-sided perturbations of the Lanczos algorithm that allows to analyze the switchback effect. In the appropriate semi-classical limit, this perturbed operator complexity is dual to an ERB length in JT gravity with corresponding shockwave insertions. We thus establish that K-complexity exhibits the expected switchback effect and universal late-time linear growth, consistent with previous findings regarding its geometric nature in the holographic bulk-boundary map.

hep-th↗

arXiv2025

Krylov Complexity

We introduce and review a new complexity measure, called `Krylov complexity', which takes its origins in the field of quantum-chaotic dynamics, serving as a canonical measure of operator growth and spreading. Krylov complexity, underpinned by the Lanczos algorithm, has since evolved into a highly diverse field of its own right, both because of its attractive features as a complexity, whose definition does not depend on arbitrary control parameters, and whose phenomenology serves as a rich and sensitive probe of chaotic dynamics up to exponentially late times, but also because of its relevance to seemingly far-afield subjects such as holographic dualities and the quantum physics of black holes. In this review we give a unified perspective on these topics, emphasizing the robust and most general features of K-complexity, both in chaotic and integrable systems, state and prove theorems on its generic features and describe how it is geometrised in the context of (dual) gravitational dynamics. We hope that this review will serve both as a source of intuition about K-complexity in and of itself, as well as a resource for researchers trying to gain an overview over what is by now a rather large and multi-faceted literature. We also mention and discuss a number of open problems related to K-complexity, underlining its currently very active status as a field of research.

hep-th↗

arXiv2025

Driven Non-Unitary Dynamics of Quantum Critical Systems

We investigate the interplay between unitary and non-unitary driven many-body dynamics in (1+1)-dimensional quantum critical systems described by conformal field theory (CFT). By formulating a coherent state approach, we demonstrate that the growth of entanglement entropy and energy can be found analytically for a class of non-unitary driven CFTs, where the evolution alternates between real and imaginary time evolution, the latter corresponding to postselected weak measurements. We find that non-unitary evolution leads to the emergence of steady states at infinite times for the cases of periodic, quasiperiodic, and random drives. In a special class of drives, for mixed initial states, we uncover purification phase transitions that arise as a result of the competition between unitary evolution and weak measurements. We compare the CFT evolution with the corresponding non-unitary dynamics of critical lattice models, finding remarkable agreement.

cond-mat.str-el↗

arXiv2025

Measuring R\'enyi entropy with an Echo Protocol

We present efficient and practical protocols to measure the second R\'enyi entropy, whose exponential is known as the purity. Our approach is based on expressing the purity in terms of transition probabilities generated by an echo-type forward-backward evolution sequence, making it applicable to quantum many-body systems. Notably, our approach does not rely on random-noise averaging, a feature that can be extended to protocols to measure out-of-time-order correlation functions, as we demonstrate. By way of example, we show that our protocols can be practically implemented in superconducting qubit-based platforms, as well as in cavity-QED trapped ultra-cold gases.

quant-ph↗

arXiv2024

Operator K-complexity in DSSYK: Krylov complexity equals bulk length

In this paper we study the notion of complexity under time evolution in chaotic quantum systems with holographic duals. Continuing on from our previous work, we turn our attention to the issue of Krylov complexity upon the insertion of a class of single-particle operators in the double-scaled SYK model. Such an operator is described by a matter-chord insertion, which splits the theory into left/right sectors, allowing us, via chord-diagram technology, to compute two different notions of complexity associated to the operator insertion: first a Krylov operator complexity, and second the Krylov complexity of a state obtained by an operator acting on the thermofield double state. We will provide both an analytic proof and detailed numerical evidence, that both Krylov complexities arise from a recursively defined basis of states characterized by a constant total chord number. As a consequence, in all cases we are able to establish that Krylov complexity is given by the expectation value of a length operator acting on the Hilbert space of the theory, expressed in terms of basis states, organized by left and right chord number. We find analytic expressions for the semiclassical limit of K-complexity, and study how the size of the operator encodes the scrambling dynamics upon the matter insertion in Krylov language. We furthermore determine the effective Hamiltonian governing the evolution of K-complexity, showing that evolution on the Krylov chain can equivalently be understood as a particle moving in a Morse potential. A particular type of triple scaling limit allows to access the gravitational sector of the theory, in which the geometrical nature of K-complexity is assured by virtue of being a total chord length, in an analogous fashion to what was found in [1] for the K-complexity of the thermofield double state.

hep-th↗

arXiv2024

Quantum simulation of the Sachdev-Ye-Kitaev model using time-dependent disorder in optical cavities

The Sachdev--Ye--Kitaev (SYK) model is a paradigm for extreme quantum chaos, non-Fermi-liquid behavior, and holographic matter. Yet, the dense random all-to-all interactions that characterize it are an extreme challenge for realistic laboratory realizations. Here, we propose a general scheme for densifying the coupling distribution of random disorder Hamiltonians, using a Trotterized cycling through sparse time-dependent disorder realizations. To diagnose the convergence of sparse to dense models, we introduce an information-theory inspired diagnostic. We illustrate how the scheme can come to bear in the realization of the complex SYK$_4$ model in cQED platforms with available experimental resources, using a single cavity mode together with a fast cycling through independent speckle patterns. The simulation scheme applies to the SYK class of models as well as spin glasses, spin liquids, and related disorder models, bringing them into reach of quantum simulation using single-mode cavity-QED setups and other platforms.

quant-ph↗

arXiv2024

Gravity as a mesoscopic system

We employ a probabilistic mesoscopic description to draw conceptual and quantitative analogies between Brownian motion and late-time fluctuations of thermal correlation functions in generic chaotic systems respecting ETH. In this framework, thermal correlation functions of `simple' operators are described by stochastic processes, which are able to probe features of the microscopic theory only in a probabilistic sense. We apply this formalism to the case of semiclassical gravity in AdS$_3$, showing that wormhole contributions can be naturally identified as moments of stochastic processes. We also point out a `Matryoshka doll' recursive structure in which information is hidden in higher and higher moments, and which can be naturally justified within the stochastic framework. We then re-interpret the gravitational results from the boundary perspective, promoting the OPE data of the CFT to probability distributions. The outcome of this study shows that semiclassical gravity in AdS can be naturally interpreted as a mesoscopic description of quantum gravity, and a mesoscopic holographic duality can be framed as a moment-vs-probability-distribution duality.

hep-th↗

arXiv2024

Quantum Chaos in Liouville CFT

Fast scrambling is a distinctive feature of quantum gravity, which by means of holography is closely tied to the behaviour of large$-c$ conformal field theories. We study this phenomenon in the context of semiclassical Liouville theory, providing both insights into the mechanism of scrambling in CFTs and into the structure of Liouville theory, finding that it exhibits a maximal Lyapunov exponent despite not featuring the identity in its spectrum. However, as we show, the states contributing to the relevant correlation function can be thought of as dressed scramblons. At a technical level we we first use the path integral picture in order to derive the Euclidean four-point function in an explicit compact form. Next, we demonstrate its equivalence to a conformal block expansion, revealing an explicit but non-local map between path integral saddles and conformal blocks. By analytically continuing both expressions to Lorentzian times, we obtain two equivalent formulations of the OTOC, which we use to study the onset of chaos in Liouville theory. We take advantage of the compact form in order to extract a Lyapunov exponent and a scrambling time. From the conformal block expansion formulation of the OTOC we learn that scrambling shifts the dominance of conformal blocks from heavy primaries at early times to the lightest primary at late times. Finally, we discuss our results in the context of holography.

hep-th↗

arXiv2024

Hilbert Space Diffusion in Systems with Approximate Symmetries

Random matrix theory (RMT) universality is the defining property of quantum mechanical chaotic systems, and can be probed by observables like the spectral form factor (SFF). In this paper, we describe systematic deviations from RMT behaviour at intermediate time scales in systems with approximate symmetries. At early times, the symmetries allow us to organize the Hilbert space into approximately decoupled sectors, each of which contributes independently to the SFF. At late times, the SFF transitions into the final ramp of the fully mixed chaotic Hamiltonian. For approximate continuous symmetries, the transitional behaviour is governed by a universal process that we call Hilbert space diffusion. The diffusion constant corresponding to this process is related to the relaxation rate of the associated nearly conserved charge. By implementing a chaotic sigma model for Hilbert-space diffusion, we formulate an analytic theory of this process which agrees quantitatively with our numerical results for different examples.

cond-mat.stat-mech↗

arXiv2024

The Influence Functional in open holography: entanglement and Rényi entropies

Open quantum systems are defined as ordinary unitary quantum theories coupled to a set of external degrees of freedom, which are introduced to take on the rôle of an unobserved environment. Here we study examples of open quantum field theories, with the aid of the so-called Feynman- Vernon Influence Functional (IF), including field theories that arise in holographic duality. We interpret the system in the presence of an IF as an open effective field theory, able to capture the effect of the unobserved environment. Our main focus is on computing Rényi and entanglement entropies in such systems, whose description from the IF, or "open EFT", point of view we develop in this paper. The issue of computing the entanglement-Rényi entropies in open quantum systems is surprisingly rich, and we point out how different prescriptions for the IF may be appropriate depending on the application of choice. A striking application of our methods concerns the fine-grained entropy of subsystems when including gravity in the setup, for example when considering the Hawking radiation emitted by black holes. In this case we show that one prescription for the IF leads to answers consistent with unitary evolution, while the other merely reproduces standard EFT results, well known to be inconsistent with unitary global evolution. We establish these results for asymptotically AdS gravity in arbitrary dimensions, and illustrate them with explicit analytical expressions for the IF in the case of matter-coupled JT gravity in two dimensions.

hep-th↗

arXiv2024

Approximate CFTs and Random Tensor Models

A key issue in both the field of quantum chaos and quantum gravity is an effective description of chaotic conformal field theories (CFTs), that is CFTs that have a quantum ergodic limit. We develop a framework incorporating the constraints of conformal symmetry and locality, allowing the definition of ensembles of `CFT data'. These ensembles take on the same role as the ensembles of random Hamiltonians in more conventional quantum ergodic phases of many-body quantum systems. To describe individual members of the ensembles, we introduce the notion of approximate CFT, defined as a collection of `CFT data' satisfying the usual CFT constraints approximately, i.e. up to small deviations. We show that they generically exist by providing concrete examples. Ensembles of approximate CFTs are very natural in holography, as every member of the ensemble is indistinguishable from a true CFT for low-energy probes that only have access to information from semi-classical gravity. To specify these ensembles, we impose successively higher moments of the CFT constraints. Lastly, we propose a theory of pure gravity in AdS$_3$ as a random matrix/tensor model implementing approximate CFT constraints. This tensor model is the maximum ignorance ensemble compatible with conformal symmetry, crossing invariance, and a primary gap to the black-hole threshold. The resulting theory is a random matrix/tensor model governed by the Virasoro 6j-symbol.

hep-th↗

arXiv2024

Quantum Chaos on Edge

In recent years, the physics of many-body quantum chaotic systems close to their ground states has come under intensified scrutiny. Such studies are motivated by the emergence of model systems exhibiting chaotic fluctuations throughout the entire spectrum (the Sachdev-Ye-Kitaev (SYK) model being a renowned representative) as well as by the physics of holographic principles, which likewise unfold close to ground states. Interpreting the edge of the spectrum as a quantum critical point, here we combine a wide range of analytical and numerical methods to the identification and comprehensive description of two different universality classes: the near edge physics of ``sparse'' and the near edge of ``dense'' chaotic systems. The distinction lies in the ratio between the number of a system's random parameters and its Hilbert space dimension, which is exponentially small or algebraically small in the sparse and dense case, respectively. Notable representatives of the two classes are generic chaotic many-body models (sparse) and single particle systems, invariant random matrix ensembles, or chaotic gravitational systems (dense). While the two families share identical spectral correlations at energy scales comparable to the level spacing, the density of states and its fluctuations near the edge are different. Considering the SYK model as a representative of the sparse class, we apply a combination of field theory and exact diagonalization to a detailed discussion of its edge spectrum. Conversely, Jackiw-Teitelboim gravity is our reference model for the dense class, where an analysis of the gravitational path integral and random matrix theory reveal universal differences to the sparse class, whose implications for the construction of holographic principles we discuss.

hep-th↗