We consider classical and quantum propagators for two different time intervals. If these propagators follow one another in a Fibonacci sequence we get a discrete quasiperiodic system. A theorem due to Nielsen provides a novel conserved quantity for this system. The Nielsen quantity controls the transition between commutative and non-commutative propagation in time. The quasiperiodically kicked oscillator moreover is dominated by quasiperiodic analogues of the Floquet theorem.
Time-periodic (Floquet) drive can give rise to novel symmetry breaking and topological phases of matter. Recently, we showed that a quintessential Floquet topological phase known as the anomalous Floquet-Anderson insulator is stable to noise on the timing of its Floquet drive. Here, we perturb the anomalous Floquet-Anderson insulator at a single incommensurate frequency, resulting in a quasiperiodic 2-tone drive. Our numerics indicate that a robust topological phase survives at weak noise with topological pumping that is more stable than the case of white noise. Within the topological phase, w…
Recent work [M. H. Kolodrubetz et al, PRL 120, 150601] has demonstrated that periodically driven one-dimensional fermionic systems can support quantized energy pumping resulting from an adiabatic modulation of a second parameter. In this work, we explore this topological Floquet-Thouless energy pump in the quasiperiodic driving regime where the parametric driving occurs at finite frequency. We show that quantization of energy pumping persists for finite ramping frequencies, as long as they are incommensurate with the driving frequency, and the system remains localized by spatial disorder. Thus…
The Nielsen-Ninomiya theorem is a fundamental theorem on the realization of chiral fermions in static lattice systems in high-energy and condensed matter physics. Here we extend the theorem in dynamical systems, which include the original Nielsen-Ninomiya theorem in the static limit. In contrast to the original theorem, which is a no-go theorem for bulk chiral fermions, the new theorem permits them due to bulk topology intrinsic to dynamical systems. The theorem is based on duality enabling a unified treatment of periodically driven systems and non-Hermitian ones. We also present the extended …
Time-quasiperiodic Majoranas are generalizations of Floquet Majoranas in time-quasiperiodic superconducting systems. We show that in a system driven at $d$ mutually irrational frequencies, there are up to $2^d$ types of such Majoranas, coexisting despite spatial overlap and lack of time-translational invariance. Although the quasienergy spectrum is dense in such systems, the time-quasiperiodic Majoranas can be stable and robust against resonances due to localization in the periodic-drives induced synthetic dimensions. This is demonstrated in a time-quasiperiodic Kitaev chain driven at two freq…
We study the mechanisms responsible for quantum diffusion in the quasiperiodic kicked rotor. We report experimental measurements of the diffusion constant on the atomic version of the system and develop a theoretical approach (based on the Floquet theorem) explaining the observations, especially the ``sub-Fourier'' character of the resonances observed in the vicinity of exact periodicity, i.e. the ability of the system to distinguish two neighboring driving frequencies in a time shorter than the inverse of the difference of the two frequencies.
We study the time-dependent circuit complexity of the periodically driven transverse field Ising model using Nielsen's geometric approach. In the high-frequency driving limit the system is known to exhibit non-equilibrium phase transitions governed by the amplitude of the driving field. We analytically compute the complexity in this regime and show that it clearly distinguishes between the different phases, exhibiting a universal linear behavior at early times. We also evaluate the time averaged complexity, provide evidence of non-analytic behavior at the critical points, and discuss its origi…
We study the stroboscopic dynamics of a spin-$S$ object subjected to $δ$-function kicking in the transverse magnetic field which is generated following the Fibonacci sequence. The corresponding classical Hamiltonian map is constructed in the large spin limit, $S \rightarrow \infty$. Upon evolving such a map for large kicking strength and time period, the phase space appears to be chaotic; interestingly, however, the geodesic distance increases linearly with the stroboscopic time implying that the Lyapunov exponent is zero. We derive the Sutherland invariant for the underlying $SO(3)$ matrix go…
Family-Vicsek dynamical scaling and Kardar-Parisi-Zhang-like superdiffusive growth of surface roughness in a driven one-dimensional quasiperiodic model
Simulating Hamiltonian dynamics is one of the most fundamental and significant tasks for characterising quantum materials. Recently, a series of quantum algorithms employing block-encoding of Hamiltonians have succeeded in providing efficient simulation of time-evolution operators on quantum computers. While time-independent Hamiltonians can be simulated by the quantum eigenvalue transformation (QET) or quantum singularvalue transformation with the optimal query complexity in time $t$ and desirable accuracy $\varepsilon$, generic time-dependent Hamiltonians face at larger query complexity and …
We investigate the driven quantum phase transition between the oscillating motion and the classical nearly free rotations of the Josephson pendulum coupled to a harmonic oscillator in the presence of dissipation. We refer to this as the Josephson-Rabi model. This model describes the standard setup of circuit quantum electrodynamics, where typically a transmon device is embedded in a superconducting cavity. We find that by treating the system quantum mechanically this transition occurs at higher drive powers than expected from an all-classical treatment, which is a consequence of the quasiperio…
