Though topological aspects of energy bands are known to play a key role in quantum transport in solid-state systems, the implications of Floquet band topology for transport in momentum space (i.e., acceleration) have not been explored so far. Using a ratchet accelerator model inspired by existing cold-atom experiments, here we characterize a class of extended Floquet bands of one-dimensional driven quantum systems by Chern numbers, reveal topological phase transitions therein, and theoretically predict the quantization of adiabatic transport in momentum space. Numerical results confirm our theory and indicate the feasibility of experimental studies.

Recent years have seen a strong interest in topological effects within periodically driven systems.In this work, we explore topological effects in two closely related 2-dimensional driven systems described by Floquet operators possessing chiral symmetry (CS). Our numerical and analytical results suggest the following. Firstly, the CS is associated with the existence of the anomalous counter-propagating (ACP) modes reported recently. Specifically, we show that a particular form of CS protects the ACP modes occurring at quasienergies of ±π. We also find that these modes are only present along selected boundaries, suggesting that they are a weak topological effect. Secondly, we find that CS can give rise to protected 0 and π quasienergy modes, and that the number of these modes may increase without bound as we tune up certain system parameters. Like the ACP modes, these 0 and π modes also appear only along selected boundaries and thus appear to be a weak topological effect. To our knowledge, this work represents the first detailed study of weak topological effects in periodically driven systems. Our findings add to the still-growing knowledge on driven topological systems.

Recently the creation of novel topological states of matter by a periodic driving field has attracted great attention. To motivate further experimental and theoretical studies, we investigate interesting * Electronic address: phygj@nus.edu.sg arXiv:1404.7248v1 [cond-mat.quant-gas]

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Recent studies have established that, in addition to the well-known kicked-Harper model (KHM), an on-resonance double-kicked rotor (ORDKR) model also has Hofstadter's butterfly Floquet spectrum, with strong resemblance to the standard Hofstadter spectrum that is a paradigm in studies of the integer quantum Hall effect. Earlier it was shown that the quasienergy spectra of these two dynamical models (i) can exactly overlap with each other if an effective Planck constant takes irrational multiples of 2π and (ii) will be different if the same parameter takes rational multiples of 2π. This work makes detailed comparisons between these two models, with an effective Planck constant given by 2πM/N, where M and N are coprime and odd integers. It is found that the ORDKR spectrum (with two periodic kicking sequences having the same kick strength) has one flat band and N-1 nonflat bands with the largest bandwidth decaying in a power law as ~K(N+2), where K is a kick strength parameter. The existence of a flat band is strictly proven and the power-law scaling, numerically checked for a number of cases, is also analytically proven for a three-band case. By contrast, the KHM does not have any flat band and its bandwidths scale linearly with K. This is shown to result in dramatic differences in dynamical behavior, such as transient (but extremely long) dynamical localization in ORDKR, which is absent in the KHM. Finally, we show that despite these differences, there exist simple extensions of the KHM and ORDKR model (upon introducing an additional periodic phase parameter) such that the resulting extended KHM and ORDKR model are actually topologically equivalent, i.e., they yield exactly the same Floquet-band Chern numbers and display topological phase transitions at the same kick strengths. A theoretical derivation of this topological equivalence is provided. These results are also of interest to our current understanding of quantum-classical correspondence considering that the KHM and ORDKR model have exactly the same classical limit after a simple canonical transformation.