Counterdiabatic driving in the classical ß-Fermi-Pasta-Ulam-Tsingou chain.

Shortcuts to adiabaticity (STAs) have been used to make rapid changes to a system while eliminating or minimizing excitations in the system's state. In quantum systems, these shortcuts allow us to minimize inefficiencies and heating in experiments and quantum computing protocols, but the theory of STAs can also be generalized to classical systems. We focus on one such STA, approximate counterdiabatic (ACD) driving, and numerically compare its performance in two classical systems: a quartic anharmonic oscillator and the ß Fermi-Pasta-Ulam-Tsingou lattice. In particular, we modify an existing variational technique to optimize the approximate driving and then develop classical figures of merit to quantify the performance of the driving. We find that relatively simple forms for the ACD driving can dramatically suppress excitations regardless of system size. ACD driving in classical nonlinear oscillators could have many applications, from minimizing heating in bosonic gases to finding optimal local dressing protocols in interacting field theories.

Dynamical obstruction to localization in a disordered spin chain.

We analyze a one-dimensional XXZ spin chain in a disordered magnetic field. As the main probes of the system's behavior, we use the sensitivity of eigenstates to adiabatic transformations, as expressed through the fidelity susceptibility, in conjunction with the low-frequency asymptotes of the spectral function. We identify a region of maximal chaos-with exponentially enhanced susceptibility-which separates the many-body localized phase from the diffusive ergodic phase. This regime is characterized by slow transport, and we argue that the presence of such slow dynamics highly constrains any possible localization transition in the thermodynamic limit. Rather, the results are more consistent with absence of the localized phase.

Observing Dynamical Quantum Phase Transitions through Quasilocal String Operators.

We analyze signatures of the dynamical quantum phase transitions in physical observables. In particular, we show that both the expectation value and various out of time order correlation functions of the finite length product or string operators develop cusp singularities following quench protocols, which become sharper and sharper as the string length increases. We illustrated our ideas analyzing both integrable and nonintegrable one-dimensional Ising models showing that these transitions are robust both to the details of the model and to the choice of the initial state.

Persistent dark states in anisotropic central spin models.

Long-lived dark states, in which an experimentally accessible qubit is not in thermal equilibrium with a surrounding spin bath, are pervasive in solid-state systems. We explain the ubiquity of dark states in a large class of inhomogeneous central spin models using the proximity to integrable lines with exact dark eigenstates. At numerically accessible sizes, dark states persist as eigenstates at large deviations from integrability, and the qubit retains memory of its initial polarization at long times. Although the eigenstates of the system are chaotic, exhibiting exponential sensitivity to small perturbations, they do not satisfy the eigenstate thermalization hypothesis. Rather, we predict long relaxation times that increase exponentially with system size. We propose that this intermediate chaotic but non-ergodic regime characterizes mesoscopic quantum dot and diamond defect systems, as we see no numerical tendency towards conventional thermalization with a finite relaxation time.

Quantum diffusion in spin chains with phase space methods.

Connecting short-time microscopic dynamics with long-time hydrodynamics in strongly correlated quantum systems is one of the outstanding questions. In particular, it is hard to determine various hydrodynamic coefficients such as the diffusion constant or viscosity starting from a microscopic model: exact quantum simulations are limited to either small system sizes or to short times, which are insufficient to reach asymptotic behavior and so various approximations must be applied. We show that these difficulties, at least for particular models, can be circumvented by using the cluster truncated Wigner approximation (CTWA), which maps quantum Hamiltonian dynamics into classical Hamiltonian dynamics in auxiliary high-dimensional phase space. We apply CTWA to XXZ next-nearest-neighbor spin-1/2 chains and XY spin ladders, and find behavior consisting of short-time spin relaxation which gradually crosses over to emergent diffusive behavior at long times. For a random initial state, we show that CTWA correctly reproduces the whole spin spectral function. Necessary in this construction is sampling from properly fluctuating initial conditions: the Dirac mean-field (variational) ansatz, which neglects such fluctuations, leads to incorrect predictions.

Floquet-Engineering Counterdiabatic Protocols in Quantum Many-Body Systems.

Counterdiabatic (CD) driving presents a way of generating adiabatic dynamics at an arbitrary pace, where excitations due to nonadiabaticity are exactly compensated by adding an auxiliary driving term to the Hamiltonian. While this CD term is theoretically known and given by the adiabatic gauge potential, obtaining and implementing this potential in many-body systems is a formidable task, requiring knowledge of the spectral properties of the instantaneous Hamiltonians and control of highly nonlocal multibody interactions. We show how an approximate gauge potential can be systematically built up as a series of nested commutators, remaining well defined in the thermodynamic limit. Furthermore, the resulting CD driving protocols can be realized up to arbitrary order without leaving the available control space using tools from periodically driven (Floquet) systems. This is illustrated on few- and many-body quantum systems, where the resulting Floquet protocols significantly suppress dissipation and provide a drastic increase in fidelity.

Asymptotic Prethermalization in Periodically Driven Classical Spin Chains.

We reveal a continuous dynamical heating transition between a prethermal and an infinite-temperature stage in a clean, chaotic periodically driven classical spin chain. The transition time is a steep exponential function of the drive frequency, showing that the exponentially long-lived prethermal plateau, originally observed in quantum Floquet systems, survives the classical limit. Even though there is no straightforward generalization of Floquet's theorem to nonlinear systems, we present strong evidence that the prethermal physics is well described by the inverse-frequency expansion. We relate the stability and robustness of the prethermal plateau to drive-induced synchronization not captured by the expansion. Our results set the pathway to transfer the ideas of Floquet engineering to classical many-body systems, and are directly relevant for photonic crystals and cold atom experiments in the superfluid regime.

Replica Resummation of the Baker-Campbell-Hausdorff Series.

We developed a novel perturbative expansion based on the replica trick for the Floquet Hamiltonian governing the dynamics of periodically kicked systems where the kick strength is the small parameter. The expansion is formally equivalent to an infinite resummation of the Baker-Campbell-Hausdorff series in the undriven (nonperturbed) Hamiltonian, while considering terms up to a finite order in the kick strength. As an application of the replica expansion, we analyze an Ising spin 1/2 chain periodically kicked with a magnetic field with a strength h, which has both longitudinal and transverse components. We demonstrate that even away from the regime of high frequency driving, if there is heating, its rate is nonperturbative in the kick strength, bounded from above by a stretched exponential: e^{-const h^{-1/2}}. This guarantees the existence of a very long prethermal regime, where the dynamics is governed by the Floquet Hamiltonian obtained from the replica expansion.

Minimizing irreversible losses in quantum systems by local counterdiabatic driving.

Counterdiabatic driving protocols have been proposed [Demirplak M, Rice SA (2003) J Chem Phys A 107:9937-9945; Berry M (2009) J Phys A Math Theor 42:365303] as a means to make fast changes in the Hamiltonian without exciting transitions. Such driving in principle allows one to realize arbitrarily fast annealing protocols or implement fast dissipationless driving, circumventing standard adiabatic limitations requiring infinitesimally slow rates. These ideas were tested and used both experimentally and theoretically in small systems, but in larger chaotic systems, it is known that exact counterdiabatic protocols do not exist. In this work, we develop a simple variational approach allowing one to find the best possible counterdiabatic protocols given physical constraints, like locality. These protocols are easy to derive and implement both experimentally and numerically. We show that, using these approximate protocols, one can drastically suppress heating and increase fidelity of quantum annealing protocols in complex many-particle systems. In the fast limit, these protocols provide an effective dual description of adiabatic dynamics, where the coupling constant plays the role of time and the counterdiabatic term plays the role of the Hamiltonian.

Schrieffer-Wolff Transformation for Periodically Driven Systems: Strongly Correlated Systems with Artificial Gauge Fields.

We generalize the Schrieffer-Wolff transformation to periodically driven systems using Floquet theory. The method is applied to the periodically driven, strongly interacting Fermi-Hubbard model, for which we identify two regimes resulting in different effective low-energy Hamiltonians. In the nonresonant regime, we realize an interacting spin model coupled to a static gauge field with a nonzero flux per plaquette. In the resonant regime, where the Hubbard interaction is a multiple of the driving frequency, we derive an effective Hamiltonian featuring doublon association and dissociation processes. The ground state of this Hamiltonian undergoes a phase transition between an ordered phase and a gapless Luttinger liquid phase. One can tune the system between different phases by changing the amplitude of the periodic drive.

Universal dynamic scaling in three-dimensional Ising spin glasses.

We use a nonequilibrium Monte Carlo simulation method and dynamical scaling to study the phase transition in three-dimensional Ising spin glasses. The transition point is repeatedly approached at finite velocity v (temperature change versus time) in Monte Carlo simulations starting at a high temperature. This approach has the advantage that the equilibrium limit does not have to be strictly reached for a scaling analysis to yield critical exponents. For the dynamic exponent we obtain z=5.85(9) for bimodal couplings distribution and z=6.00(10) for the Gaussian case. Assuming universal dynamic scaling, we combine the two results and obtain z=5.93±0.07 for generic 3D Ising spin glasses.

Quantum versus classical annealing: insights from scaling theory and results for spin glasses on 3-regular graphs.

We discuss an Ising spin glass where each S=1/2 spin is coupled antiferromagnetically to three other spins (3-regular graphs). Inducing quantum fluctuations by a time-dependent transverse field, we use out-of-equilibrium quantum Monte Carlo simulations to study dynamic scaling at the quantum glass transition. Comparing the dynamic exponent and other critical exponents with those of the classical (temperature-driven) transition, we conclude that quantum annealing is less efficient than classical simulated annealing in bringing the system into the glass phase. Quantum computing based on the quantum annealing paradigm is therefore inferior to classical simulated annealing for this class of problems. We also comment on previous simulations where a parameter is changed with the simulation time, which is very different from the true Hamiltonian dynamics simulated here.

SU(3) semiclassical representation of quantum dynamics of interacting spins.

We present a formalism for simulating quantum dynamics of lattice spin-1 systems by first introducing local hidden variables and then doing semiclassical (truncated Wigner) approximation in the extended phase space. In this way, we exactly take into account the local on-site Hamiltonian and approximately treat spin-spin interactions. In particular, we represent each spin with eight classical SU(3) variables. Three of them represent the usual spin components and five others are hidden variables representing local spin-spin correlations. We compare our formalism with exact quantum dynamics of fully connected spin systems and find very good agreement. As an application, we discuss quench dynamics of a Bose-Hubbard model near the superfluid-insulator transition for a 3D lattice system consisting of 1000 sites.

Measuring a topological transition in an artificial spin-1/2 system.

We present measurements of a topological property, the Chern number (C_{1}), of a closed manifold in the space of two-level system Hamiltonians, where the two-level system is formed from a superconducting qubit. We manipulate the parameters of the Hamiltonian of the superconducting qubit along paths in the manifold and extract C_{1} from the nonadiabatic response of the qubit. By adjusting the manifold such that a degeneracy in the Hamiltonian passes from inside to outside the manifold, we observe a topological transition C_{1}=1→0. Our measurement of C_{1} is quantized to within 2% on either side of the transition.

Universal rephasing dynamics after a quantum quench via sudden coupling of two initially independent condensates.

We consider a quantum quench in which two initially independent condensates are suddenly coupled and study the subsequent "rephasing" dynamics. For weak tunneling couplings, the time evolution of physical observables is predicted to follow universal scaling laws, connecting the short-time dynamics to the long-time nonperturbative regime. We first present a two-mode model valid in two and three dimensions and then move to one dimension, where the problem is described by a gapped sine-Gordon theory. Combining analytical and numerical methods, we compute universal time-dependent expectation values, allowing a quantitative comparison with future experiments.

Weak and strong typicality in quantum systems.

We study the properties of mixed states obtained from eigenstates of many-body lattice Hamiltonians after tracing out part of the lattice. Two scenarios emerge for generic systems: (i) The diagonal entropy becomes equivalent to the thermodynamic entropy when a few sites are traced out (weak typicality); and (ii) the von Neumann (entanglement) entropy becomes equivalent to the thermodynamic entropy when a large fraction of the lattice is traced out (strong typicality). Remarkably, the results for few-body observables obtained with the reduced, diagonal, and canonical density matrices are very similar to each other, no matter which fraction of the lattice is traced out. Hence, for all physical quantities studied here, the results in the diagonal ensemble match the thermal predictions.

Localized phase structures growing out of quantum fluctuations in a quench of tunnel-coupled atomic condensates.

We investigate the relative phase between two weakly interacting 1D condensates of bosonic atoms after suddenly switching on the tunnel coupling. The following phase dynamics is governed by the quantum sine-Gordon equation. In the semiclassical limit of weak interactions, we observe the parametric amplification of quantum fluctuations leading to the formation of breathers with a finite lifetime. The typical lifetime and density of these "quasibreathers" are derived employing exact solutions of the classical sine-Gordon equation. Both depend on the initial relative phase between the condensates, which is considered as a tunable parameter.

Geometric phase contribution to quantum nonequilibrium many-body dynamics.

We study the influence of geometry of quantum systems underlying space of states on its quantum many-body dynamics. We observe an interplay between dynamical and topological ingredients of quantum nonequilibrium dynamics revealed by the geometrical structure of the quantum space of states. As a primary example we use the anisotropic XY ring in a transverse magnetic field with an additional time-dependent flux. In particular, if the flux insertion is slow, nonadiabatic transitions in the dynamics are dominated by the dynamical phase. In the opposite limit geometric phase strongly affects transition probabilities. This interplay can lead to a nonequilibrium phase transition between these two regimes. We also analyze the effect of geometric phase on defect generation during crossing a quantum-critical point.

Entropy of isolated quantum systems after a quench.

A diagonal entropy, which depends only on the diagonal elements of the system's density matrix in the energy representation, has been recently introduced as the proper definition of thermodynamic entropy in out-of-equilibrium quantum systems. We study this quantity after an interaction quench in lattice hard-core bosons and spinless fermions, and after a local chemical potential quench in a system of hard-core bosons in a superlattice potential. The former systems have a chaotic regime, where the diagonal entropy becomes equivalent to the equilibrium microcanonical entropy, coinciding with the onset of thermalization. The latter system is integrable. We show that its diagonal entropy is additive and different from the entropy of a generalized Gibbs ensemble, which has been introduced to account for the effects of conserved quantities at integrability.