We discuss several classes of integrable Floquet systems, i.e. systems which do not exhibit chaotic behavior even under a time dependent perturbation. The first class is associated with finite-dimensional Lie groups and infinite-dimensional generalization thereof. The second class is related to the row transfer matrices of the 2D statistical mechanics models. The third class of models, called here "boost models", is constructed as a periodic interchange of two Hamiltonians - one is the integrable lattice model Hamiltonian, while the second is the boost operator. The latter for known cases coincides with the entanglement Hamiltonian and is closely related to the corner transfer matrix of the corresponding 2D statistical models. We present several explicit examples. As an interesting application of the boost models we discuss a possibility of generating periodically oscillating states with the period different from that of the driving field. In particular, one can realize an oscillating state by performing a static quench to a boost operator. We term this state a ``Quantum Boost Clock''. All analyzed setups can be readily realized experimentally, for example in cold atoms.
We discuss several classes of integrable Floquet systems, i.e. systems which do not exhibit chaotic behavior even under a time dependent perturbation. The first class is associated with finite-dimensional Lie groups and infinite-dimensional generalization thereof. The second class is related to the row transfer matrices of the 2D statistical mechanics models. The third class of models, called here "boost models", is constructed as a periodic interchange of two Hamiltonians - one is the integrable lattice model Hamiltonian, while the second is the boost operator. The latter for known cases coincides with the entanglement Hamiltonian and is closely related to the corner transfer matrix of the corresponding 2D statistical models. We present several explicit examples. As an interesting application of the boost models we discuss a possibility of generating periodically oscillating states with the period different from that of the driving field. In particular, one can realize an oscillating state by performing a static quench to a boost operator. We term this state a ``Quantum Boost Clock''. All analyzed setups can be readily realized experimentally, for example in cold atoms.
We give a general overview of the high-frequency regime in periodically driven systems and identify three distinct classes of driving protocols in which the infinite-frequency Floquet Hamiltonian is not equal to the time-averaged Hamiltonian. These classes cover systems, such as the Kapitza pendulum, the Harper–Hofstadter model of neutral atoms in a magnetic field, the Haldane Floquet Chern insulator and others. In all setups considered, we discuss both the infinite-frequency limit and the leading finite-frequency corrections to the Floquet Hamiltonian. We provide a short overview of Floquet theory focusing on the gauge structure associated with the choice of stroboscopic frame and the differences between stroboscopic and non-stroboscopic dynamics. In the latter case, one has to work with dressed operators representing observables and a dressed density matrix. We also comment on the application of Floquet Theory to systems described by static Hamiltonians with well-separated energy scales and, in particular, discuss parallels between the inverse-frequency expansion and the Schrieffer–Wolff transformation extending the latter to driven systems.
Analogue and digital Hamiltonian simulation represent two distinct approaches to making one quantum system replicate the physical properties of another, with consequences ranging from practical applications to complexity theory and quantum gravity. They are rooted in two distinct regimes of quantum dynamics: time-independent Hamiltonians for the former; fully controllable time-dependent quantum circuits in the latter. Time-periodic Hamiltonians represent an intermediate regime. In this work, we establish a theory of Floquet simulation where we use periodically driven Hamiltonians to synthesise time-independent ones.
We show that a set of interactions S can Floquet-simulate every Hamiltonian in the Lie algebra Lie(S), and consequently we give a complete and constructive characterisation of Floquet-universal Hamiltonians that are able to produce any target Hamiltonian. Notably, our construction uses only O(1) local interaction strengths and ratios thereof, avoiding the impractical multi-scale local interaction strengths often required in analogue simulation via time-independent Hamiltonians. Instead, the construction requires a range of driving frequencies which scales polynomially with system size for important classes such as k-local lattice Hamiltonians. This has immediate complexity-theoretic implications, including numerous BQP-completeness and QMA-hardness results for natural physical properties and quantities studied in Floquet physics.
Controlling and manipulating complex many-body quantum systems will be a key ingredient \nfor the development of next-generation technologies. While the realisation of a \nuniversal quantum machine is still out of reach, in recent years experimental systems of \nultracold atoms have already evolved into a vivid field of research for quantum simulation. \nCrucially, such systems even allow for the successful quantum engineering of targeted \nmany-body systems by means of coherent periodic driving. The essential properties of \nthese Floquet systems encompass two main aspects: fast driving facilitates the simulation \nof effective static systems, and interactions lead to unique heating effects as energy is only \nconserved modulo the driving frequency. Within this thesis we theoretically study both \nof these aspects in respective model systems. \nIn part I of this thesis, we investigate the dynamics of excitations of a bosonic Mott \ninsulator in a designed one-dimensional Floquet system. Here, periodic driving in combination \nwith breaking all mirror symmetries of the system can induce directed motion of \nparticles. In the limit of small excitation densities, the effectively non-interacting quantum \nratchet determines the motion of holes and doublons in the Mott insulator and can in fact \nbe used to manipulate the dynamics of such. This little quantum machine can also be \nused to drive particles against an external force, where transport is possible but requires \nthe fulfilment of a commensurability condition for long times. \nIn part II, we discuss the role of interactions for periodically driven systems by means \nof a Floquet version of the Boltzmann equation. Starting from the Keldysh approach, \nwe develop this semiclassical formalism based on a clear separation of time scales. The \nresult is a description of the dynamics and the scattering of Floquet quasiparticles in \nsuch systems. Here, the property of discrete energy violation is naturally encoded in our \nformalism predicting the heating of interacting Floquet systems to infinite temperatures in \nthe long-time limit. As a first application of this approach, we investigate a cold atom setup \nrealising the Haldane model by means of periodic shaking. While homogeneous systems \nheat up globally, a confining potential evokes thermoelectric transport effects resulting \nfrom spatially dependent heating characteristics. Moreover, we show that the interplay \nof intrinsic heating, macroscopic diffusion and non-trivial topological properties of the \nHaldane model lead to an anomalous Floquet-Nernst effect, which describes anomalous \nparticle transport as the result of developing temperature gradients. \nIn part III, we elaborate on the quantum simulator aspect of ultracold atoms by providing \na theoretical framework for a possible simulation of a topological edge state in a \none-dimensional optical lattice. In this case, the one-dimensional Dirac equation with \nspatially varying mass is important, which captures the topological properties of a corresponding \nsystem of the BDI symmetry class. We analytically discuss such system and \ninvestigate the role of mean-field interaction effects. We also identify the emergence of \ndynamical instabilities in a realisation with bosonic atoms.
We present two directions of research in statistical mechanics of nonequilibrium onedimensional quantum systems. One is related to isolated integrable models, the other one to exactly solvable dissipatively driven spin chains. The device enabling their analysis is identified as the quantum group symmetry.
In the framework of isolated systems we focus on the concept of integrable Floquet driven systems. In particular, we show how to build such systems out of the basic constituents of integrability structure. This allows substantiated and conclusive statements,for instance, about the transport phenomena. Despite their inherent relation, the presented integrable periodically driven models originate in two different ideas. On the one hand we have Trotterisations of integrable
spin chains, concieved in an effort to better understand the dynamics of spin models. Then, there are systems originating in the attempts to solve quantum field theory in an algebraically closed form, for example, the quantum Hirota equation. The bulk of our consideration of isolated quantum systems consists of the construction of extensive conservation laws that either (i) constitute the generalised Gibbs ensemble and the hydrodynamic description of thermalisation after a quantum quench, or (ii) prevent the decay of current autocorrelations and thus characterise ideal transport in the system. The exact way in which these charges determine the dynamics stems from the symmetries of the quantum group representations.
The second part of the exposition is dedicated to open quantum systems. Again we distinguish two formally related settings. The first one is that of a boundary driven quantum cellular automaton, our particular example being the integrable Trotterisation of the Heisenberg magnet. Starting from a repeated interaction protocol, in which the system under scrutiny is repeatedly coupled to the environment, we introduce the Kraus map as a general form of a dissipative time evolution of the density matrix. We then solve for its unique nonequilibrium steady state, using integrability structure of the model. The other setting is that of a dissipatively boundary driven spin chain in the continuous time. Here we present the recently developed formalism of inhomogeneous Lax structure. Using it we demonstrate the solvability of the XXZ and XYZ spin chains, acted upon by the Lindblad operators that polarise the boundary spins in arbitrary directions. The ansatz for the steady state is particularly interesting, since it exhibits a
previously unknown integrability structure, differing from site to site in the spin chain. This structure can independently produce nontrivial conservation laws in an isolated spin
chain with arbitrary boundary fields.
The coherent dynamics of many body quantum system is nowadays an experimental reality: by means of the cold atoms in optical lattices, many Hamiltonians and time-dependent perturbations can be engineered. In this Thesis we discuss what happens in these systems when a periodic perturbation is applied. Thanks to Floquet theory, we can see that -- if the Floquet spectrum obeys certain continuity conditions possible in the thermodynamic limit-- dephasing among Floquet quasi-energies makes local observables relax to a periodic steady regime described by an effective density matrix: the Floquet diagonal ensemble (FDE). By means of numerical examples on the Quantum Ising Chain and the Lipkin model, we discuss the properties of the FDE focusing on the difference among ergodic and regular quantum dynamics and on how this reflects on the thermal properties (T=∞) of the asymptotic condition. We verify thermalization in the classically ergodic Lipkin model and we demonstrate that this effect is induced by the Floquet states being delocalized and obeying Eigenstate Thermalization Hypothesis.We discuss also, in the Ising chain case, the work probability distribution, whose asymptotic condition is not described by the form (Generalized Gibbs Ensemble) that FDE acquires for local obserbvables because of integrability. Dephasing makes some correlations invisible in the local observables, but they are still present in the system. We consider also the linear response limit: when the amplitude of the perturbation is vanishingly small, the Floquet diagonal ensemble is not sufficient to describe the asymptotic condition given by LRT. For every small but finite amplitude, there are quasi-degeneracies in the Floquet spectrum giving rise to pre-relaxation to the condition predicted by Linear Response; these phenomena are strictly related to energy absorption and boundedness of the spectrum.
We study a system of one-dimensional interacting quantum particles subjected to a time-periodic potential linear in space. After discussing the cases of driven one- and two-particle systems, we derive the analogous results for the many-particle case in the presence of a general interaction two-body potential and the corresponding Floquet Hamiltonian. When the undriven model is integrable, the Floquet Hamiltonian is shown to be integrable too. We determine the micromotion operator and the expression for a generic time evolved state of the system. We discuss various aspects of the dynamics of the system both at stroboscopic and intermediate times, in particular the motion of the center of mass of a generic wave packet and its spreading over time. We also discuss the case of accelerated motion of the center of mass, obtained when the integral of the coefficient strength of the linear potential on a time period is nonvanishing, and we show that the Floquet Hamiltonian gets in this case an additional static linear potential. We also discuss the application of the obtained results to the Lieb-Liniger model.