Probing Complex Quantum Dynamics through
Out-of-time-ordered Correlators

We describe in this work two methods to control multiqubit Ising interactions in system of trapped ions. In the first method, we use adiabatic control of these interactions to show how that a single-step and high-fidelity Toffoli gate can be realized in a linear crystal of ions [1]. The adiabatic activation limits undesired entanglement between the qubits and the motion of the ions, reducing one of the largest infidelity sources. The high fidelities (>99%) obtained also for large crystals could make the gate competitive with gate-decomposed, multistep variants of the N-qubit Toffoli gate, at the expense of requiring ground-state cooling of the ion crystal. In the second method [2], we describe how to engineer interactions in one- and two-dimensional trapped-ion quantum simulators. Here we consider the use of optical tweezers to engineer the sound-wave spectrum of trapped ion crystals. We show that this approach allows us to tune the interactions and connectivity of the ion qubits beyond the power-law interactions accessible in current setups. We illustrate this method with examples of a controlled power-law interaction in one dimension and lattice of spins with frustration in two-dimensions, considering realistic experimental parameters. We present also the prospects for the simulation of chaotic dynamics.

We study a one dimensional quantum XY spin chain driven by a local noisy spin impurity with finite correlation time, along the transverse field direction. We recover the celebrated Zeno crossover and we show that entanglement can be used as a proxy for the heating and strong-measurement regimes. We compute the entanglement entropy of a block of spins and we observe that its velocity spreading decreases at strong dissipation, as a result of the Zeno effect. Upon increasing the correlation time of the noise, the location of the Zeno crossover shifts at stronger dissipation rates opening up a broader heating phase. We offer insight on the mechanisms underlying the dynamics of the entanglement entropy by monitoring different time traces of the local transverse magnetisation profile. Our results aim at starting a complementary viewpoint on the field of dissipative quantum impurities, based on a theoretical quantum information perspective.

Lieb-Robinson bounds quantify the maximal speed of information spreading in nonrelativistic quantum systems. We discuss the relation of Lieb-Robinson bounds to out-of-time order correlators, which correspond to different norms of commutators $C(r,t)=|A_i(t),B_{i+r}|$ of local operators. Using an exact Krylov space-time evolution technique, we calculate these two different norms of such commutators for the spin-1/2 Heisenberg chain with interactions decaying as a power law $1/r^\alpha$ with distance $r$. Our numerical analysis shows that both norms (operator norm and normalized Frobenius norm) exhibit the same asymptotic behavior, namely, a linear growth in time at short times and a power-law decay in space at long distance, leading asymptotically to power-law light cones for $\alpha<1$ and to linear light cones for $\alpha>1$. The asymptotic form of the tails of $C(r,t)~\dfrac{t}{r^\alpha}$ is described by short-time perturbation theory, which is valid at short times and long distances.

Many-body Localization (MBL) generalizes the concept of Anderson localization to the interacting case and has emerged as a novel paradigm for ergodicity breaking phenomenon in generic many-body systems subject to strong disorder. In this talk, I will discuss the phenomenon of MBL from the Fock-space point of view [1]. We will focus our attention on the MBL transition and relate it to the eigenstates structure in the Fock space. I will show that the many-body fractal dimension $D_q$ is non-self-averaging and directly relate it to the jump of $D_q$ as well as of the localization length of the integrals of motion at the transition. In particular, I will show that the MBL transition can be seen as a transition between ergodic to non-ergodic but extended states in the Fock space. Finally, I will provide analytical considerations which underline the connection between the avalanche theory of delocalization [3-4] and the putative jump in the fractal dimension [2] at the transition. References: [1] GDT et al., PRB 104, 024202 (2021). [2] N. Macè et al., PRL 123, 180601 (2019). [3] T. Thiery et al., PRL 121, 140601 (2018) [4] P. T. Dumitrescu et al., PRB 99, 094205 (2019)

We study a model of free fermions on a chain with dynamics generated by random unitary gates acting on nearest neighbour bonds and present an exact calculation of an out-of-time-ordered correlator (OTOC). The OTOC obeys a diffusion equation and is understood in terms of evolving strings of operators. This is in sharp contrast to the chaotic case [1], where the OTOC obeys a biased diffusion equation, with its front propagating ballistically. This diffusive operator spreading agrees with the $\sim\sqrt{t}$ entanglement growth and differs from the Anderson localization for the case of static disorder and the ballistic behaviour observed in the clean case. [1] A. Nahum, S. Vijay, and J. Haah, Phys. Rev. X 8, 021014 (2018).

The fully frustrated ladder – a quasi-1D geometrically frustrated spin one half Heisenberg model – is non-integrable with local conserved quantities on rungs of the ladder, inducing the local fragmenta- tion of the Hilbert space into sectors composed of singlets and triplets on rungs. We explore the far- from-equilibrium dynamics of this model through the entanglement entropy and out-of-time-ordered correlators (OTOC). The post-quench dynamics of the entanglement entropy is highly anomalous as it shows clear non-damped revivals that emerge from short connected chunks of triplets. We find that the maximum value of the entropy follows from a picture where coherences between different fragments co-exist with perfect thermalization within each fragment. This means that the eigenstate thermalization hypothesis holds within all sufficiently large Hilbert space fragments. The OTOC shows short distance oscillations arising from short coupled fragments, which become decoherent at longer distances, and a sub-ballistic spreading and long distance exponential decay stemming from an emergent length scale tied to fragmentation.

We consider isolated many-body quantum systems in the integrable and chaotic domain, and investigate how the equilibration time and the temporal fluctuations after equilibration of various observables depend on system size and degree of chaos. We compare the behaviors of the survival probability, participation ratio, spin autocorrelation function, connected spin-spin correlation function, and out-of-time ordered correlators. If dynamical manifestations of spectral correlations in the form of the correlation hole (“ramp”) are taken into account, the time for equilibration scales exponentially with system size, while if they are neglected, the scaling is better described by a power law with system size. Our results also suggest that temporal fluctuations can decay exponentially with system size in both regimes, integrable and chaotic.

Fabricio S. Lozano-Negro, Pablo R. Zangara, Horacio M. Pastawski The metal-insulator transition (MIT), which includes Anderson localization and Mott insulators as extreme regimes, has received renewed interest as the many-body effects often constitute a limitation for the handling of quantum interference. This resulted in the field dubbed many-body localization (MBL), intensively studied theoretically and experimentally as understanding the appearance of equilibration and thermalization becomes relevant in dealing with finite systems. Here, we propose a new observable to study this transition in a spin chain under the ``disorder'' of a Harper-Hofstadter-Aubry-André on-site potential. This quantity, which we call zeroth-order gradient entanglement (ZOGE) is extracted from the fundamental Fourier mode of a family of out-of-time-ordered correlators (OTOCs). These are just Loschmidt Echoes, where a field gradient is applied before the time reversal. In the absence of many-body interactions, the ZOGE coincides with the inverse participation ratio of a Fermionic wave function. By adding an Ising interaction to an XY Hamiltonian, one can explore the MBL phase diagram of the system. Close to the critical region, the excitation dynamics is consistent with a diffusion law. However, for weak disorder, quantum diffusion prevails while for strong disorder the excitation dynamics is sub-diffusive.

I will present a detailed numerical study of the spectra and eigenstates in triangular billiards. Depending on the number theoretical properties of the triangle angles, the dynamics of classical triangle billiards may display various properties from strong to weak mixing and ergodicity. However, the Lyapunov exponents are zero and the dynamics is not chaotic. According to the quantum chaos conjecture, the statistical properties of the spectra of chaotic quantum systems follow random matrix theory predictions. The goal of this investigation is to establish if random matrix theory applies to purely mixing systems. The most commonly used spectral statistics like the mode-fluctuations, level spacing distributions and ratios, the number variance, the spectral form factor will be presented for various triangle billiards with different mixing properties. Furthermore, I will present an analysis of the eigenstates of the triangle billiards, based on the recently developed methodology using localization measures defined in the Poincare-Husimi representation, mapping the quantum eigenfunctions onto the classical phases space.

Disordered interacting spin chains that undergo a many-body localization transition are characterized by two limiting behaviors where the dynamics are chaotic and integrable. However, the transition region between them is not fully understood yet. We propose here a possible finite-size precursor of a critical point that shows a typical finite-size scaling and distinguishes between two different dynamical phases. The kurtosis excess of the diagonal fluctuations of the full one-dimensional momentum distribution from its microcanonical average is maximum at this singular point in the paradigmatic disordered J1-J2 model. For system sizes accessible to exact diagonalization, both the position and the size of this maximum scale linearly with the system size. Furthermore, we show that this singular point is found at the same disorder strength at which the Thouless and the Heisenberg energies coincide. Below this point, the spectral statistics follow the universal random matrix behavior up to the Thouless energy. Above it, no traces of chaotic behavior remain, and the spectral statistics are well described by a generalized semi-Poissonian model, eventually leading to the integrable Poissonian behavior. We provide, thus, an integrated scenario for the many-body localization transition, conjecturing that the critical point in the thermodynamic limit, if it exists, should be given by this value of disorder strength.

Quantum information stored in local operators spreads over other degrees of freedom of the system during time evolution, known as scrambling. This process is conveniently characterized by the out-of-time-order commutators (OTOC), whose time dependence reveals salient aspects of the system's dynamics. Here we study the spatially local spin correlation function i.e., the expectation value of spin commutator and the corresponding OTOC of Dirac--Weyl systems in 1, 2, and 3 spatial dimensions. The OTOC can be written as the square of the expectation value of the commutator and the variance of the commutator. In principle, the problem features two energy scales, the chemical potential, and the high energy cutoff. We find that only the latter is dominant, therefore the time evolution is separated into only two different regions. The spin correlation function grows linearly with time initially and decays as $t^{-2}$ for late times. The OTOC reveals a universal $t^2$ initial growth from both the commutator and the variance while its late time decay, $t^{-2}$ originates from the variance of the commutator. This late time decay is identified as a characteristic signature of Dirac--Weyl fermions. These features remain present also at finite temperatures. Our results indicate that Dirac--Weyl systems are slow information scramblers and are essential when additional channels for scrambling, i.e. interaction or disorder are analyzed.

Competition between unitary dynamics that scrambles quantum information non-locally and local measurements that probe and collapse the quantum state can result in a measurement-induced entanglement phase transition. Here we introduce analytically tractable models of measurement-induced criticality in large-N Brownian hybrid circuit model composed of qubits [1]. The system is initially entangled with an equal sized reference, and the subsequent hybrid system dynamics either partially preserves or totally destroys this entanglement depending on the measurement rate. Our approach can access a variety of entropic observables, which are represented as a path integral coupling four replicas with twisted boundary conditions. Saddle-point analysis reveals a second-order phase transition corresponding to replica permutation symmetry breaking below a critical measurement rate. The transition is mean-field-like and we characterize the critical properties near the transition in terms of a simple Ising field theory in 0+1 dimensions. We also extend these solvable models to study the effects of power-law long-range couplings on measurement-induced phases. In one dimension, the long-range coupling is irrelevant for $α>3/2$, with $α$ being the power-law exponent. For $α<3/2$ the long-range coupling becomes relevant, leading to a nontrivial dynamical exponent at the measurement-induced phase transition. More interestingly, for $α<1$ the entanglement pattern receives a sub-volume correction for both area-law and volume-law phases. The volume-law phase with such a sub-volume correction realizes a novel quantum error correcting code whose code distance scales as $L^{2−2α}$ [2]. [1] Measurement-induced purification in large-N hybrid Brownian circuits - Gregory Bentsen*, Subhayan Sahu*, Brian Swingle. Phys. Rev. B 104, 094304 (2021), ArXiv:2104.07688. [2] Entanglement Phases in large-N hybrid Brownian circuits with long-range couplings - Subhayan Sahu*, Shao-Kai Jian*, Gregory Bentsen, Brian Swingle. ArXiv:2109.00013.

Inspired by the work done in [1,2], we have studied the extreme events of the eigenstates intensities of three parameter-dependent quantum maps: standard map, perturbed cat map, and kicked Harper map. In order to expand the extreme statistics in semiclassical systems, we have considered not only the very chaotic regime but also states from near-integrable and mixed regimes. Interesting results have been obtained from the computation of the kurtosis of the intensities distributions. It is expected from [3-5] that the high semiclassical intensities due to the scarring of eigenstates by classical periodic orbits that undergo generic bifurcations must be present. However, we will show that quantum phases also play a role. Ref: [1] S.C.L. Srivastava, A. Lakshminarayan, Chaos, Solitons & Fractals, 74, 67-78 (2015) [2] A. Lakshminarayan, S. Tomsovic, O. Bohigas, S.N. Majumdar, Phys. Rev. Lett., 100, 44-103 (2008) [3] J.P. Keating, Nonlinearity, 4, 309-341 (1991) [4] M.V. Berry, J.P. Keating, H. Schomerus, Proc. R. Soc. Lond. A, 456, 1659- 1668 (2000) [5] J.P. Keating, S.D. Prado, Proc. R. Soc. Lond. A, 457, 1855-1872 (2001)

We investigate the delocalization of quantum information in the nonequilibrium dynamics of the $XY$ spin chain with asymptotically decaying interactions $\sim 1/r^{\alpha}$. As a figure of merit, we employ the tripartite mutual information (TMI), whose sign indicates if quantum information is predominantly shared globally. Interestingly, the sign of the TMI distinguishes regimes of the exponent $\alpha$ that are known for different behavior of information propagation. While an effective causal region bounds the propagation of information, if interactions decay sufficiently fast, this information is mainly delocalized, which leads to the necessity of global measurements. Furthermore, the results indicate that mutual information is monogamous for all possible partitionings in this case, implying that quantum entanglement is the dominant correlation. If interactions decay sufficiently slow, though information can propagate (quasi-)instantaneously, it is mainly accessible by local measurements at early times. Furthermore, it takes some finite time until correlations start to become monogamous, which suggests that apart from entanglement, other nonmonogamous correlations are sizeable at early times. Our findings give new insights into the dynamics, and structure of quantum information in many-body systems with long-range interactions, and might get verified on state-of-the-art experimental platforms.

Quantum information scrambling refers to the rapid spreading of local information over an entire system, through the generation of global entanglement. The study of this effect has recently emerged as a new field that goes beyond quantum chaos and thermalization. However, experimental investigations of information scrambling suffer from fake positive signals from various sources, e.g., decoherence or operational errors. In this talk, I will present our approach to this long-standing problem. Using a novel quantum butterfly effect, our protocol can unambiguously single out the effect of genuine scrambling from the noisy background.