In this work, we present a new approach to disordered, periodically driven (Floquet) quantum many-body systems based on flow equations. Specifically, we introduce a continuous unitary flow of Floquet operators in an extended Hilbert space, whose fixed point is both diagonal and time-independent, allowing us to directly obtain the Floquet modes. We first apply this method to a periodically driven Anderson insulator, for which it is exact, and then extend it to driven many-body localized systems within a truncated flow equation ansatz. In particular we compute the emergent Floquet local integrals of motion that characterise a periodically driven many-body localized phase. We demonstrate that the method remains well-controlled in the weakly-interacting regime, and allows us to access larger system sizes than accessible by numerically exact methods, paving the way for studies of two-dimensional driven many-body systems.
We show the emergence of Floquet time crystal (FTC) phases in the Floquet dynamics of periodically driven
p-spin models, which describe a collection of spin-1/2 particles with all-to-all
p-body interactions. Given the mean-field nature of these models, we treat the problem exactly in the thermodynamic limit and show that, for a given
p, these systems can host various robust time-crystalline responses with period
nT, where
T is the period of the drive and
n an integer between 2 and
p. In particular, the case of four-body interactions (
p=4) gives rise to both a usual period-doubling crystal and also a period-quadrupling phase. We develop a comprehensive framework to predict robust subharmonic response in classical area-preserving maps, and use this as a basis to predict the occurrence and characterize the stability of the resulting mean-field FTC phases in the quantum regime. Our analysis reveals that the robustness of the time-crystal behavior is reduced as their period increases, and establishes a connection between the emergence of time crystals, described by eigenstate ordering and robust subharmonic response, and the phenomenology of excited state and dynamical quantum phase transitions. Finally, for the models hosting two or more coexisting time crystal phases, we define protocols where the periodic subharmonic response of the system can be varied in time via the nonperiodic modulation of an external control parameter.
Driven many-body systems exhibit diverse and complex dynamical behaviors. Here, we report the observation of Shapiro steps in periodically driven Josephson junctions between strongly interacting Fermi superfluids of ultracold atoms. The height and the width of the observed quantized plateaus in the current-potential characteristics mirror the external drive frequency and the junction nonlinear response. Direct measurements of the current-phase relationship showcase how Shapiro steps arise from the synchronization between the relative phase of the two reservoirs and the external drive. Such a mechanism is further supported by the detection of periodic phase-slippage processes, in the form of vortex-antivortex pairs. Our results are corroborated by a circuital model and numerical simulations. Our work may open prospects for studying emergent nonequilibrium dynamics in quantum many-body systems under external drives.