Universal High-Frequency Behavior of Periodically Driven Systems: from Dynamical Stabilization to Floquet Engineering

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.Comment: 84 pages, 25 figures, 4 appendice

Parametric instability in periodically driven Luttinger liquids

We analyze the properties of a Luttinger liquid under the influence of a periodic driving of the interaction strength. Irrespective of the details the driven system develops an instability due to a parametric resonance. For slow and fast driving, however, we identify intermediate long-lived meta-stable states at constant time-averaged internal energies. Due to the instability perturbations in the fermionic density are amplified exponentially leading to the buildup of a superlattice. The momentum distribution develops a terrace structure due to scattering processes that can be associated with the absorption of quanta of the driving frequency.Comment: 7 pages, 4 figure

Bose-Fermi Mixtures: A Mean-Field Study

We study the low-energy properties of the spinful Bose-Fermi Hubbard model on a simple cubic lattice for attractive fermion-fermion and repulsive boson-boson interactions at half-filling for the fermions and unit-filling for the bosons. Recent DMFT results predict a variety of phases, including superfluids, charge density waves, mixed states and supersolids. We develop a self-consistent mean-field scheme which allows for all orders to appear and determine the ground state of the system by comparing the corresponding energies. In the double superfluid phase, we consider possibilities for exotic extended s-, p- and d-wave boson-assisted superfluidity

Glassy Phase of Optimal Quantum Control

We study the problem of preparing a quantum many-body system from an initial to a target state by optimizing the fidelity over the family of bang-bang protocols. We present compelling numerical evidence for a universal spin-glass-like transition controlled by the protocol time duration. The glassy critical point is marked by a proliferation of protocols with close-to-optimal fidelity and with a true optimum that appears exponentially difficult to locate. Using a machine learning (ML) inspired framework based on the manifold learning algorithm t-SNE, we are able to visualize the geometry of the high-dimensional control landscape in an effective low-dimensional representation. Across the transition, the control landscape features an exponential number of clusters separated by extensive barriers, which bears a strong resemblance with replica symmetry breaking in spin glasses and random satisfiability problems. We further show that the quantum control landscape maps onto a disorder-free classical Ising model with frustrated nonlocal, multibody interactions. Our work highlights an intricate but unexpected connection between optimal quantum control and spin glass physics, and shows how tools from ML can be used to visualize and understand glassy optimization landscapes.Comment: Modified figures in appendix and main text (color schemes). Corrected references. Added figures in SI and pseudo-cod