We lay out an experiment to realize time-reversal invariant topological insulators in alkali atomic gases. We introduce an original method to synthesize a gauge field in the near field of an atom chip, which effectively mimics the effects of spin-orbit coupling and produces quantum spin-Hall states. We also propose a feasible scheme to engineer sharp boundaries where the hallmark edge states are localized. Our multiband system has a large parameter space exhibiting a variety of quantum phase transitions between topological and normal insulating phases. Because of their remarkable versatility, cold-atom systems are ideally suited to realize topological states of matter and drive the development of topological quantum computing.

Periodically driven quantum Hall system in fixed magnetic field is found to exhibit a series of phases featuring anomalous edge modes with the "wrong" chirality. This leads to pairs of counterpropagating chiral edge modes at each edge, in sharp contrast to stationary quantum Hall systems. We show the pair of Floquet edge modes are protected by the chiral (sublattice) symmetry, and that they are robust against static disorder. The existence of distinctive phases with the same Chern and winding numbers but very different edge state spectra points to the important role played by symmetry in classifying topological properties of driven systems. We further explore the evolution of the edge states with driving using a simplified model, and discuss their experimental signatures.Cyclic time-evolutions of quantum systems are known to have interesting topological properties [1,2]. Several groups recently showed that periodic driving can turn an ordinary band insulator (superconductor) into a Floquet topological insulator (superconductor) [3][4][5][6][7][8][9][10][11]. This provides a powerful way to engineer effective Hamiltonians that stroboscopically mimic stationary topological insulators [4,5,12]. Moreover, a large class of topological phenomena in periodically driven many-body systems are unique and have no stationary counterparts. An early example is Thouless's one-dimensional charge pump, where he showed that the charge transport is quantized and related to a topological invariant [13]. Other topological invariants for the time evolution operator in two and three dimensions have been constructed recently [3,5,10]. Yet a systematic classification of these invariants analogous to the periodic table of symmetry protected topological phases [14,15] is still to be achieved.In this paper, we identify new topological phenomena in a lattice integer quantum Hall (QH) system under cyclic driving with period T . For fixed magnetic field, variations of the driving parameter induce topological phase transitions where the Chern numbers of the quasienergy bands change. We find multiple phases of the driven QH system featuring counter-propagating chiral edge modes at the each edge, and show they are robust against disorder. In particular, there appear "π-modes", pairs of edge modes with opposite chirality at quasienergy π/T . These anomalous edge modes differ from those found previously in other driven two-dimensional (2D) lattice models, where the edge modes at quasienergy π/T all propagate in the same direction and subsequently their number can be inferred either from the Chern number or the winding number [5,10]. Here, these known topological invariants can not predict the number of edge modes of each chirality, but only their difference. For example, we find two phases (phase A and D below) having the same set of Chern and winding numbers but very different edge state spectra. Our analysis suggests that symmetry of the time evolution operator has to be included to fully characterize and understand the topological properties of...

We study the coherent nonlinear tunneling dynamics of a binary mixture of Bose-Einstein condensates in a double-well potential. We demonstrate the existence of a type of mode associated with the "swapping" of the two species in the two wells of the potential. In contrast to the symmetry-breaking macroscopic quantum self-trapping ͑MQST͒ solutions, the swapping modes correspond to the tunneling dynamics that preserves the symmetry of the double-well potential. As a consequence of two distinct types of broken-symmetry MQST phases where the two species localize in different potential wells or coexist in the same well, the corresponding symmetry-restoring swapping modes result in dynamics where the two species either avoid or chase each other. In view of the possibility to control the interaction between the species, the binary mixture offers a very robust system to observe these novel effects as well as the phenomena of Josephson oscillations and modes.