Relating Out-of-Time-Order Correlations to Entanglement via Multiple-Quantum Coherences

Fast Quantum State Preparation and Bath Dynamics Using Non-Gaussian Variational Ansatz and Quantum Optimal Control

Fast preparation of quantum many-body states is essential for myriad quantum algorithms and metrological applications. Here, we develop a new pathway for fast, nonadiabatic preparation of quantum many-body states that combines quantum optimal control with a variational Ansatz based on non-Gaussian states. We demonstrate our approach on the spin-boson model, a single spin interacting with the bath. We use a multipolaron Ansatz to prepare near-critical ground states. For one mode, we achieve a reduction in infidelity of up to ≈60 (≈10) times compared to linear (optimized local adiabatic) ramps; for many modes, we achieve a reduction in infidelity of up to ≈5 times compared to nonadiabatic linear ramps. Further, we show that the typical control quantity, the leakage from the variational manifold, provides only a loose bound on the state's fidelity. Instead, in analogy to the bond dimension of matrix product states, we suggest a controlled convergence criterion based on the number of polarons. Finally, motivated by the possibility of realizations in trapped ions, we study the dynamics of a system with bath properties going beyond the paradigm of (sub- and/or super-) Ohmic couplings.

Quantum information scrambling in two-dimensional Bose-Hubbard lattices

It is a well-understood fact that the transport of excitations throughout a lattice is intimately governed by the underlying structures. Hence, it is only natural to recognize that the dispersion of information also has to depend on the lattice geometry. In the present work, we demonstrate that two-dimensional lattices described by the Bose-Hubbard model exhibit information scrambling for systems as little as two hexagons. However, we also find that the out-of-time-ordered correlator (OTOC) shows the exponential decay characteristic for quantum chaos only for a judicious choice of local observables. More generally, the OTOC is better described by Gaussian-exponential convolutions, which alludes to the close similarity of information scrambling and decoherence theory.

Witnessing quantum chaos using observational entropy

We study observation entropy (OE) for the quantum kicked top model, whose classical counterpart possesses different phases: regular, mixed, or chaotic, depending on the strength of the kicking parameter. We show that OE grows logarithmically with coarse-graining length beyond a critical value in the regular phase, while OE growth is much faster in the chaotic regime. In the dynamics, we demonstrate that the short-time growth rate of OE acts as a measure of the chaoticity in the system, and we compare our results with out-of-time-ordered correlators (OTOC). Moreover, we show that in the deep quantum regime, the results obtained from OE are much more robust compared to OTOC results. Finally, we also investigate the long-time behavior of OE to distinguish between saddle-point scrambling and true chaos, where the former shows large persistent fluctuations compared to the latter.

1H multiple quantum NMR in alternating quasi-one-dimensional spin chains of hambergite

Multiple quantum (MQ) dynamics was investigated in quasi-one-dimensional ¹H zigzag spin chains in hambergite (Be₂BO₃OH) single crystals. Due to the non-linear arrangement of the spins, dipolar coupling strengths alternate along the chain. To solve the problem of MQ NMR experiments taking too much time due to extremely long ¹H spin-lattice relaxation times, the samples were exposed to gamma irradiation to produce the defects accelerating the relaxation. The influence of the radiation dose was investigated. The experimental dependencies of MQ coherence intensities on the MQ excitation time in alternating spin chains were obtained and compared with the theory for inhomogeneous spin chains with nearest neighbor interactions developed earlier. The correspondence of the observed MQ dynamics to the alternating spin chain was demonstrated.

Strong Quantum Metrological Limit from Many-Body Physics

Surpassing the standard quantum limit and even reaching the Heisenberg limit using quantum entanglement, represents the Holy Grail of quantum metrology. However, quantum entanglement is a valuable resource that does not come without a price. The exceptional time overhead for the preparation of large-scale entangled states raises disconcerting concerns about whether the Heisenberg limit is fundamentally achievable. Here, we find a universal speed limit set by the Lieb-Robinson light cone for the quantum Fisher information growth to characterize the metrological potential of quantum resource states during their preparation. Our main result establishes a strong precision limit of quantum metrology accounting for the complexity of many-body quantum resource state preparation and reveals a fundamental constraint for reaching the Heisenberg limit in a generic many-body lattice system with bounded one-site energy. It enables us to identify the essential features of quantum many-body systems that are crucial for achieving the quantum advantage of quantum metrology, and brings an interesting connection between many-body quantum dynamics and quantum metrology.

Quantum transitions, ergodicity, and quantum scars in the coupled top model

We consider an interacting collective spin model known as coupled top (CT), exhibiting a rich variety of phenomena related to quantum transitions, ergodicity, and formation of quantum scars, discussed in our previous work [Mondal, Sinha, and Sinha, Phys. Rev. E 102, 020101(R) (2020)2470-004510.1103/PhysRevE.102.020101]. In this work, we present a detailed analysis of the different type of transitions in the CT model, and find their connection with the underlying collective spin dynamics. Apart from the quantum scarring phenomena, we also identify another source of deviation from ergodicity due to the presence of nonergodic multifractal states. The degree of ergodicity of the eigenstates across the energy band is quantified from the relative entanglement entropy as well as multifractal dimensions, which can be probed from nonequilibrium dynamics. Finally, we discuss the detection of nonergodic behavior and different types of quantum scars using "out-of-time-order correlators," which has relevance in the recent experiments.

Quantum Fisher Information from Randomized Measurements

The quantum Fisher information (QFI) is a fundamental quantity of interest in many areas from quantum metrology to quantum information theory. It can in particular be used as a witness to establish the degree of multiparticle entanglement in quantum many-body systems. In this work, we use polynomials of the density matrix to construct monotonically increasing lower bounds that converge to the QFI. Using randomized measurements we propose a protocol to accurately estimate these lower bounds in state-of-the-art quantum technological platforms. We estimate the number of measurements needed to achieve a given accuracy and confidence level in the bounds, and present two examples of applications of the method in quantum systems made of coupled qubits and collective spins.

Dynamical route to ergodicity and quantum scarring in kicked coupled top

Unlike classical systems, understanding ergodicity from phase space mixing remains unclear for interacting quantum systems due to the absence of phase space trajectories. By considering an interacting spin model known as kicked coupled top, we elucidate the manifestation of phase space dynamics on local ergodic behavior of its quantum counterpart and quantum scarring phenomena. A transition to chaos occurs by increasing the kicking strength, and in the mixed phase space, the islands of regular motions within the chaotic sea clearly exhibit deviation from ergodicity, which we quantify from entanglement entropy and survival probability. Interestingly, the reminiscence of unstable orbits and fixed points can be identified as scars in quantum states, exhibiting athermal behavior and violation of Berry's conjecture for ergodic states. We also discuss the detection of quantum scars by a recently developed method of "out-of-time-order correlators," which has experimental relevance.

Experimental Detection of the Correlation Rényi Entropy in the Central Spin Model

We propose and experimentally measure an entropy that quantifies the volume of correlations among qubits. The experiment is carried out on a nearly isolated quantum system composed of a central spin coupled and initially uncorrelated with 15 other spins. Because of the spin-spin interactions, information flows from the central spin to the surrounding ones forming clusters of multispin correlations that grow in time. We design a nuclear magnetic resonance experiment that directly measures the amplitudes of the multispin correlations and use them to compute the evolution of what we call correlation Rényi entropy. This entropy keeps growing even after the equilibration of the entanglement entropy. We also analyze how the saturation point and the timescale for the equilibration of the correlation Rényi entropy depend on the system size.

Entanglement entropy and out-of-time-order correlator in the long-range Aubry-André-Harper model

We investigate the nonequilibrium dynamics of entanglement entropy and out-of-time-order correlator (OTOC) of noninteracting fermions at half-filling starting from a product state to distinguish the delocalized, multifractal (in the limit of nearest neighbor hopping), localized and mixed phases hosted by the quasiperiodic Aubry-André-Harper (AAH) model in the presence of long-range hopping. For sufficiently long-range hopping strength a secondary logarithmic behavior in the entanglement entropy is found in the mixed phases whereas the primary behavior is a power-law the exponent of which is different in different phases. The saturation value of entanglement entropy in the delocalized, multifractal and mixed phases depends linearly on system size whereas in the localized phase (in the short-range regime) it is independent of system size. The early-time growth of OTOC shows very different power-law behaviors in the presence of nearest neighbor hopping and long-range hopping. The late time decay of OTOC leads to noticeably different power-law exponents in different phases. The spatial profile of OTOC and its system-size dependence also provide distinct features to distinguish phases. In the mixed phases the spatial profile of OTOC shows two different dependences on space for small and large distances respectively. Interestingly the spatial profile contains large fluctuations at the special locations related to the quasiperiodicity parameter in the presence of multifractal states.

Energy-Resolved Information Scrambling in Energy-Space Lattices

Weakly interacting Fermi gases simulate spin lattices in energy space, offering a rich platform for investigating information spreading and spin coherence in a large many-body quantum system. We show that the collective spin vector can be determined as a function of energy from the measured spin density, enabling general energy-space resolved protocols. We measure an out-of-time-order correlation function in this system and observe the energy dependence of the many-body coherence.

Hybrid quantum-classical approach to enhanced quantum metrology

Quantum metrology plays a fundamental role in many scientific areas. However, the complexity of engineering entangled probes and the external noise raise technological barriers for realizing the expected precision of the to-be-estimated parameter with given resources. Here, we address this problem by introducing adjustable controls into the encoding process and then utilizing a hybrid quantum-classical approach to automatically optimize the controls online. Our scheme does not require any complex or intractable off-line design, and it can inherently correct certain unitary errors during the learning procedure. We also report the first experimental demonstration of this promising scheme for the task of finding optimal probes for frequency estimation on a nuclear magnetic resonance (NMR) processor. The proposed scheme paves the way to experimentally auto-search optimal protocol for improving the metrology precision.

Detecting Out-of-Time-Order Correlations via Quasiadiabatic Echoes as a Tool to Reveal Quantum Coherence in Equilibrium Quantum Phase Transitions

We propose a new dynamical method to connect equilibrium quantum phase transitions and quantum coherence using out-of-time-order correlations (OTOCs). Adopting the iconic Lipkin-Meshkov-Glick and transverse-field Ising models as illustrative examples, we show that an abrupt change in coherence and entanglement of the ground state across a quantum phase transition is observable in the spectrum of multiple quantum coherence intensities, which are a special type of OTOC. We also develop a robust protocol to obtain the relevant OTOCs using quasi-adiabatic quenches through the ground state phase diagram. Our scheme allows for the detection of OTOCs without time reversal of coherent dynamics, making it applicable and important for a broad range of current experiments where time reversal cannot be achieved by inverting the sign of the underlying Hamiltonian.

1H NMR in a quasi-one-dimensional zig-zag spin chain of hambergite, Be2BO3(OH)

One-dimensional spin chains are important testbeds for the investigation of quantum many-body systems. Here, we introduce a new quasi-one-dimensional chain of spin-1/2 ¹H nuclei in hambergite (Be₂BO₃(OH)) crystal. The distinctive feature of hambergite is the zig-zag arrangement of the spins in the chains compared to the systems reported earlier. A single crystal of hambergite was investigated experimentally using ¹H nuclear magnetic resonance at different orientations in the external magnetic field. The experimental spectra were compared with the Van Vleck's second moments calculated from the structural data. For the orientation of the chain along the external magnetic field the resonance resembles that of the linear spin chains and the calculated contributions to the second moment from spins within one chain of hambergite amount to 96%. For other orientations the spectra demonstrate features specific to the zig-zag chain. Our results show that ¹H nuclei in hambergite crystals could serve a good model of a well-isolated zig-zag spin chain.

Chaos and quantum scars in a coupled top model

We consider a coupled top model describing two interacting large spins, which is studied semiclassically as well as quantum mechanically. This model exhibits a variety of interesting phenomena such as a quantum phase transition (QPT), a dynamical transition, and excited-state quantum phase transitions above a critical coupling strength. Both classical dynamics and entanglement entropy reveal ergodic behavior at the center of the energy density band for an intermediate range of coupling strength above QPT, where the level spacing distribution changes from Poissonian to Wigner-Dyson statistics. Interestingly, in this model we identify quantum scars as reminiscent of unstable collective dynamics even in the presence of an interaction. The statistical properties of such scarred states deviate from the ergodic limit corresponding to the random matrix theory and violate Berry's conjecture. In contrast to ergodic evolution, the oscillatory behavior in the dynamics of the unequal time commutator and survival probability is observed as the dynamical signature of a quantum scar, which can be relevant for its detection.

Transition from quantum chaos to localization in spin chains

Recent years have seen an increasing interest in quantum chaos and related aspects of spatially extended systems, such as spin chains. However, the results are strongly system dependent: generic approaches suggest the presence of many-body localization, while analytical calculations for certain system classes, here referred to as the "self-dual case," prove adherence to universal (chaotic) spectral behavior. We address these issues studying the level statistics in the vicinity of the latter case, thereby revealing transitions to many-body localization as well as the appearance of several nonstandard random-matrix universality classes.

Positive quantum Lyapunov exponents in experimental systems with a regular classical limit

Quantum chaos refers to signatures of classical chaos found in the quantum domain. Recently, it has become common to equate the exponential behavior of out-of-time order correlators (OTOCs) with quantum chaos. The quantum-classical correspondence between the OTOC exponential growth and chaos in the classical limit has indeed been corroborated theoretically for some systems and there are several projects to do the same experimentally. The Dicke model, in particular, which has a regular and a chaotic regime, is currently under intense investigation by experiments with trapped ions. We show, however, that for experimentally accessible parameters, OTOCs can grow exponentially also when the Dicke model is in the regular regime. The same holds for the Lipkin-Meshkov-Glick model, which is integrable and also experimentally realizable. The exponential behavior in these cases are due to unstable stationary points, not to chaos.

Localization as an Entanglement Phase Transition in Boundary-Driven Anderson Models

The Anderson localization transition is one of the most well studied examples of a zero temperature quantum phase transition. On the other hand, many open questions remain about the phenomenology of disordered systems driven far out of equilibrium. Here we study the localization transition in the prototypical three-dimensional, noninteracting Anderson model when the system is driven at its boundaries to induce a current carrying nonequilibrium steady state. Recently we showed that the diffusive phase of this model exhibits extensive mutual information of its nonequilibrium steady-state density matrix. We show that this extensive scaling persists in the entanglement and at the localization critical point, before crossing over to a short-range (area-law) scaling in the localized phase. We introduce an entanglement witness for fermionic states that we name the mutual coherence, which, for fermionic Gaussian states, is also a lower bound on the mutual information. Through a combination of analytical arguments and numerics, we determine the finite-size scaling of the mutual coherence across the transition. These results further develop the notion of entanglement phase transitions in open systems, with direct implications for driven many-body localized systems, as well as experimental studies of driven-disordered systems.

Emergent Prethermalization Signatures in Out-of-Time Ordered Correlations

How a many-body quantum system thermalizes-or fails to do so-under its own interaction is a fundamental yet elusive concept. Here we demonstrate nuclear magnetic resonance observation of the emergence of prethermalization by measuring out-of-time ordered correlations. We exploit Hamiltonian engineering techniques to tune the strength of spin-spin interactions and of a transverse magnetic field in a spin chain system, as well as to invert the Hamiltonian sign to reveal out-of-time ordered correlations. At large fields, we observe an emergent conserved quantity due to prethermalization, which can be revealed by an early saturation of correlations. Our experiment not only demonstrates a new protocol to measure out-of-time ordered correlations, but also provides new insights in the study of quantum thermodynamics.

Experimental Implementation of Efficient Quantum Pseudorandomness on a 12-Spin System

Quantum pseudorandomness, also known as unitary designs, comprises a powerful resource for emergent quantum technologies. Although in theory pseudorandom unitary operators can be constructed efficiently, realizing these objects in realistic physical systems is a challenging task. Here, we demonstrate experimental generation and detection of quantum pseudorandomness on a 12-qubit nuclear magnetic resonance system. We first apply random sequences to the interacting nuclear spins, leading to random quantum evolutions that can quickly form unitary designs. Then, in order to probe the growth of quantum pseudorandomness during the time evolutions, we propose the idea of using the system's multiple-quantum coherence distribution as an indicator. Based on this indicator, we measure the spreading of quantum coherences and find that substantial quantum pseudorandomness has been achieved at the 12-qubit scale. This may open up a path to experimentally explore quantum randomness on forthcoming large-scale quantum processors.

Operator Size at Finite Temperature and Planckian Bounds on Quantum Dynamics

It has long been believed that dissipative timescales τ obey a "Planckian" bound τ≳(ℏ/k_{B}T) in strongly coupled quantum systems. Despite much circumstantial evidence, however, there is no known τ for which this bound is universal. Here we define operator size at a finite temperature, and conjecture such a τ: the timescale over which small operators become large. All known many-body theories are consistent with this conjecture. This proposed bound explains why previously conjectured Planckian bounds do not always apply to weakly coupled theories, and how Planckian timescales can be relevant to both transport and chaos.

Unifying scrambling, thermalization and entanglement through measurement of fidelity out-of-time-order correlators in the Dicke model

Scrambling is the process by which information stored in local degrees of freedom spreads over the many-body degrees of freedom of a quantum system, becoming inaccessible to local probes and apparently lost. Scrambling and entanglement can reconcile seemingly unrelated behaviors including thermalization of isolated quantum systems and information loss in black holes. Here, we demonstrate that fidelity out-of-time-order correlators (FOTOCs) can elucidate connections between scrambling, entanglement, ergodicity and quantum chaos (butterfly effect). We compute FOTOCs for the paradigmatic Dicke model, and show they can measure subsystem Rényi entropies and inform about quantum thermalization. Moreover, we illustrate why FOTOCs give access to a simple relation between quantum and classical Lyapunov exponents in a chaotic system without finite-size effects. Our results open a path to experimental use FOTOCs to explore scrambling, bounds on quantum information processing and investigation of black hole analogs in controllable quantum systems.

The scrambling of quantum information in a many-body system leads to the emergence of statistical mechanics and chaotic behaviour. Here the authors establish quantitative relationships between experimentally-measureable correlators, the Rényi entropy and Lyapunov exponents in the Dicke model.

Single-scan acquisition of multiple multidimensional spectra

Multidimensional coherent spectroscopy is a powerful tool for understanding the ultrafast dynamics of complex quantum systems. To fully characterize the nonlinear optical response of a system, multiple pulse sequences must be recorded and quantitatively compared. We present a new single-scan method that enables rapid and parallel acquisition of all unique pulse sequences corresponding to first- and third-order degenerate wave-mixing processes. Signals are recorded with shot-noise limited detection, enabling acquisition times of ~2 minutes with ~100 zs phase stability and ~8 orders of dynamic range, in a collinear geometry, on a single-pixel detector. We demonstrate this method using quantum well excitons, and quantitative analysis reveals new insights into the bosonic nature of excitons. This scheme may enable rapid and scalable analysis of unique chemical signatures, metrology of optical susceptibilities, nonperturbative coherent control, and the implementation of quantum information protocols using multidimensional spectroscopy.