Abstract Chaos sets a fundamental limit to quantum-information processing schemes. We study the onset of chaos in spatially extended quantum many-body systems that are relevant to quantum optical devices. We consider an extended version of the Tavis–Cummings model on a finite chain. By studying level-spacing statistics, adjacent gap ratios, and spectral form factors, we observe the transition from integrability to chaos as the hopping between the Tavis–Cummings sites is increased above a finite value. The results are obtained by means of exact numerical diagonalization which becomes notoriously hard for extended lattice geometries. In an attempt to circumvent these difficulties, we identify a minimal single-site quantum impurity model that successfully captures the spectral properties of the lattice model. This approach is intended to be adaptable to other lattice models with large local Hilbert spaces.
Abstract Chaos sets a fundamental limit to quantum-information processing schemes. We study the onset of chaos in spatially extended quantum many-body systems that are relevant to quantum optical devices. We consider an extended version of the Tavis–Cummings model on a finite chain. By studying level-spacing statistics, adjacent gap ratios, and spectral form factors, we observe the transition from integrability to chaos as the hopping between the Tavis–Cummings sites is increased above a finite value. The results are obtained by means of exact numerical diagonalization which becomes notoriously hard for extended lattice geometries. In an attempt to circumvent these difficulties, we identify a minimal single-site quantum impurity model that successfully captures the spectral properties of the lattice model. This approach is intended to be adaptable to other lattice models with large local Hilbert spaces.
Integrable quantum systems of finite size are generically robust against weak enough integrability-breaking perturbations, but become quantum chaotic and thermalizing if the integrability breaking is strong enough. We argue that the onset of quantum chaos can be described as a Fock-space delocalization process, with the eigenstates of the integrable system being taken as the Fock states. The integrability-breaking perturbation introduces hopping in this Fock space, and chaos sets in when this hopping delocalizes the many-body eigenstates in this space. Depending on the range of the dominant Fock-space hopping, delocalization can occur either through a crossover or via a transition that becomes sharp in the appropriate large-system dynamic limit. In either case, the perturbation strength at the onset of chaos scales to zero in the usual thermodynamic limit, with a size dependence that we estimate analytically and compute numerically for a few specific models. We also identify two intermediate finite-size-dependent regimes: There is generally an intermediate nonchaotic regime in which integrability is broken strongly enough to produce some systemwide many-body resonances but not enough to thermalize the system. In spatially extended systems (but not in quantum dots) there is also a crossover or transition between chaotic regimes where the ratio of the system size to the mean free path of the quasiparticles of the integrable system is small versus large compared to unity.
We solve a minimal model for an ergodic phase in a spatially extended quantum many-body system.The model consists of a chain of sites with nearest-neighbor coupling under Floquet time evolution.Quantum states at each site span a q-dimensional Hilbert space, and time evolution for a pair of sites is generated by a q 2 × q 2 random unitary matrix.The Floquet operator is specified by a quantum circuit of depth two, in which each site is coupled to its neighbor on one side during the first half of the evolution period and to its neighbor on the other side during the second half of the period.We show how dynamical behavior averaged over realizations of the random matrices can be evaluated using diagrammatic techniques and how this approach leads to exact expressions in the large-q limit.We give results for the spectral form factor, relaxation of local observables, bipartite entanglement growth, and operator spreading.
We study chaotic many-body quantum dynamics in a minimal model with spatial structure and local interactions. It has a time-independent Hamiltonian, in contrast to quantum circuits and Brownian models, and is simple at the single-site level, in contrast to Sachdev-Ye-Kitaev chains. It is analytically tractable for large local Hilbert space dimension and weak intersite coupling. In this limit, we show that energy dynamics is described by a classical master equation and is diffusive. We also show that the spectral form factor can be expressed exactly in terms of the solution to this master equation. For a two-site system, we obtain closed-form expressions for both the two-point correlator of energy density and the spectral form factor, in essentially perfect agreement with numerical simulations. For an L -site system we show at late times how a linear ramp emerges in the spectral form factor, as universally expected from level repulsion in chaotic quantum systems. Conversely, at earlier times we identify two distinct mechanisms for an increase of the spectral form factor above its ramp value. One of these is associated with energy diffusion and is effective until the Thouless time, which varies as L 2 . The other involves contributions like those that would appear if the system were composed of many uncoupled subsystems: They generate a large enhancement of the spectral form factor and are suppressed on a timescale varying as ( ln L ) 2 . Besides being exact for the limit considered, we believe our approach provides the natural approximation even for small local Hilbert space dimension and strong intersite coupling. We present a numerical study of a spin-half chain, finding an early-time enhancement of the spectral form factor which is qualitatively similar to that in our solvable model.
We study spectral statistics in spatially extended chaotic quantum many-body systems, using simple lattice Floquet models without time-reversal symmetry. Computing the spectral form factor K(t) analytically and numerically, we show that it follows random matrix theory (RMT) at times longer than a many-body Thouless time, t_{Th}. We obtain a striking dependence of t_{Th} on the spatial dimension d and size of the system. For d>1, t_{Th} is finite in the thermodynamic limit and set by the intersite coupling strength. By contrast, in one dimension t_{Th} diverges with system size, and for large systems there is a wide window in which spectral correlations are not of RMT form. Lastly, our Floquet model exhibits a many-body localization transition, and we discuss the behavior of the spectral form factor in the localized phase.
Phenomena in quantum chaotic systems such as spectral fluctuations are known to be described remarkably well by the random matrix theory (RMT) in small energy intervals and over large time scales. However, in exchange for analytical tractability, RMT has been constructed to leave out an important element of realistic physical systems { the notion of locality. Since quantum information propagates at a finite speed in a spatially-extended quantum chaotic system, there exists time scales before which RMT is effective. How do we describe the non-equilibrium quantum dynamics at intermediate timescales which are much longer than the local relaxation time but too short for RMT to apply? What are the signatures of many-body quantum chaotic systems beyond the description of RMT? In this thesis, we address these questions by analysing minimal models that incorporate a notion of locality and at the same time retain some analytical tractability of RMT. More specifically, we study Floquet quantum circuits that are random in space but periodic in time. These circuits consist of local and random quantum gates that act on a q-state `spin' at each site. After a brief review of classical chaos, quantum chaos, and quantum circuits in Chapter 1, we present three main results. In Chapter 2, we formulate a diagrammatical approach for computing ensemble averages of observables over random unitary circuits, and derive exact results on spectral statistics and dynamical observables for a Floquet circuit in the large-q limit. In Chapter 3, we further investigate the spectral statistics in spatiallyextended many-body quantum chaotic systems by computing the spectral form factor K(t) analytically and numerically. We show that K(t) follows the RMT description at times longer than the Thouless time tTh and identify the scaling of tTh at the large-q limit of a Floquet circuit. In Chapter 4, we discuss eigenstate correlations for spatially-extended many-body quantum chaotic systems, in terms of the statistical properties of matrix elements of local observables. While the eigenstate thermalization hypothesis (ETH) is known to give an excellent description of these quantities, the phenomenon of scrambling and the butterfly effect imply structure beyond ETH. We determine analytically and verify numerically the universal form of this structure at long distances and small eigenvalue separations for Floquet systems.
Systems of interacting bosons in triple-well potentials are of significant theoretical and experimental interest. They are explored in contexts that range from quantum phase transitions and quantum dynamics to semiclassical analysis. Here, we systematically investigate the onset of quantum chaos in a triple-well model that moves away from integrability as its potential gets tilted. Even in its deepest chaotic regime, the system presents features reminiscent of integrability. Our studies are based on level spacing distribution and spectral form factor, structure of the eigenstates, and diagonal and off-diagonal elements of observables in relationship to the eigenstate thermalization hypothesis. With only three sites, the system's eigenstates are at the brink of becoming fully chaotic, so they do not yet exhibit Gaussian distributions, which resonates with the results for the observables.