Tobias Micklitz

Free probability provides a framework for describing correlations between noncommuting observables in complex quantum systems whose Hilbert-space states follow maximum-entropy distributions. We examine the robustness of this framework under a minimal deviation from freeness: the coupling of a single ancilla qubit to a Haar-distributed quantum circuit of dimension D_{0}≫1. We find that, even in this setting, the correlation functions predicted by free probability theory receive corrections of order O(1). These modifications persist at long times, when the dynamics of the coupled system is already ergodic. We trace their origin to nonuniformly distributed stationary quantum states, which we characterize analytically and confirm numerically.

We introduce a quasiclassical Green's function approach describing the unitary yet irreversible dynamics of quantum systems effectively acting as their own environment. Combining a variety of concepts of quantum many-body theory, notably the nonlinear σ model of disordered systems, the G Σ formalism for strong correlations, and real-time path integration, the theory is capable of describing a wide range of system classes and disorder models. It extends previous work beyond perturbation theory (in inverse Hilbert-space dimensions), enabling a description of thermalization dynamics from short scattering times, through the onset of ergodicity at an effective “Thouless time,” up to the many-body Heisenberg time. We illustrate the approach with two case studies: (1) a brickwork model of unitarily coupled quantum circuits with and without conserved symmetries and (2) an array of capacitively coupled quantum dots. Using the spectral form factor as a test observable, we find good agreement with numerical simulations. We present our formalism in a self-contained and pedagogical manner, aiming to provide a transferable toolbox for the first-principles description of many-body chaotic quantum systems in regimes of strong entanglement.

We investigate the physics of a small group of quantum states defined above the sharply defined ground state of a chaotic ensemble. This “universality class of the first levels” is realized in the majority of “synthetic” random matrix models but, for all we know, in only one microscopically defined system: low-dimensional gravity. We discuss the physical properties of these states, notably their exceptional rigidity against external perturbations, as quantified by the so-called quantum state fidelity. Examining these structures through the lenses of random matrix and string theory, we highlight their relevance to the physics of low-dimensional holographic principles.

Light scattering in random media is usually considered within the framework of the three-dimensional Anderson universality class, with modifications for the vector nature of electromagnetic waves. We propose that the linear dispersiveness of light introduces topological aspects into the picture. The dynamics of electromagnetic waves follow the same differential equations as those of a spin-1 Weyl semimetal. In the presence of disorder, this equivalence leads to a range of phenomena explored in this paper. These include topological protection against localization when helicity hybridization is weak, the emergence of exotic phases in weakly scattering media, and anomalies in optical transparency in the presence of synthetic ‘magnetic fields’. We argue that some of these effects should be visible and investigated already in weakly disordered optical materials.

The evolution of complex correlated quantum systems such as random circuit networks is governed by the dynamical buildup of both entanglement and entropy. We here introduce a real-time field theory approach—essentially a fusion of the G Σ functional of the SYK model and the field theory of disordered systems—engineered to microscopically describe the full range of such crossover dynamics: from initial product states to a maximum entropy ergodic state. To showcase this approach in the simplest nontrivial setting, we consider a tensor product of coupled random matrices, and compare to exact diagonalization. Published by the American Physical Society 2024

Recently, 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 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. Published by the American Physical Society 2024

We report on a photonic simulator of the critical state forming at the quantum phase transition between topologically distinct Anderson insulator phases. We observe a time-staggered profile in the circular photon polarization, which originates from the interplay of a chiral and sublattice symmetry, and has recently been suggested as a signature for topological Anderson criticality within the setup. We discuss the role of statistical detuning from criticality and show that the controlled breaking of phase coherence removes the signal, revealing its origin in quantum coherence. Published by the American Physical Society 2024

We study two-state discrimination in chaotic quantum systems. Assuming that one of two N-qubit pure states has been randomly selected, the probability to correctly identify the selected state from an optimally chosen experiment involving a subset of N − N B qubits is given by the trace-distance of the states, with N _B qubits partially traced out. In the thermodynamic limit N → ∞ , the average subsystem trace-distance for random pure states makes a sharp, first order transition from unity to zero at f = 1 / 2 , as the fraction f = N B / N of unmeasured qubits is increased. We analytically calculate the corresponding crossover for finite numbers N of qubits, study how it is affected by the presence of local conservation laws, and test our predictions against exact diagonalization of models for many-body chaos.