Integer-valued topological indices, characterizing nonlocal properties of quantum states of matter, are known to directly predict robust physical properties of equilibrium systems. The Chern number, e.g., determines the quantized Hall conductivity of an insulator. Using non-interacting fermionic atoms in a periodically driven optical lattice, here we demonstrate experimentally that the Chern number determines also the far-from-equilibrium dynamics of a quantum system. Extending a respective proposal to Floquet systems, we measure the linking number that characterizes the trajectories of momentum-space vortices emerging after a strong quench. We observe that it directly corresponds to the ground-state Chern number. This one-to-one relation between a dynamical and a static topological index allows us to experimentally map out the phase diagram of our system. Furthermore, we measure the instantaneous Chern number and show that it remains zero under the unitary dynamics.
Topology plays an important role in modern solid state physics describing intriguing quantum states such as topological insulators. It is an intrinsically non-local property and therefore challenging to access, often studied only via the resulting edge states. Here, we measure the topological index directly from the far-from-equilibrium dynamics of the bulk. We use the mapping of the Chern number to the linking number of dynamical vortex trajectories appearing after a quench to the Hamiltonian of interest. We thereby map out the topological phase diagram of quantum gases in optical lattices via a purely dynamical response. Such relations between two topological indices in static and dynamical properties could be also an important approach for exploring topology in the case of interactions.
We show how the topological number of a static Hamiltonian can be measured from a dynamical quench process. We focus on a two-band Chern insulator in two dimension, for instance, the Haldane model, whose dynamical process can be described by a mapping from the [k_{x},k_{y},t] space to the Bloch sphere, characterized by the Hopf invariant. Such a mapping has been constructed experimentally by measurements in cold atom systems. We show that, taking any two constant vectors on the Bloch sphere, their inverse images of this mapping are two trajectories in the [k_{x},k_{y},t] space, and the linking number of these two trajectories exactly equals the Chern number of the static Hamiltonian. Applying this result to a recent experiment from the Hamburg group, we show that the linking number of the trajectories of the phase vortices determines the phase boundary of the static Hamiltonian.
Periodically driven systems play a prominent role in optical lattices. In these ultracold atomic systems, driving is used to create a variety of interesting behaviours, of which an important example is provided by topological states of matter. Such Floquet topological phases have a richer classification than their equilibrium counterparts. Although there exist analogues of the equilibrium topological phases that are characterised by a Chern number, the corresponding Hall conductivity, and protected edge states, there is an additional possibility. This is a phase that has a vanishing Chern number and no Hall conductivity, but nevertheless hosts anomalous topological edge states (Rudner et al ( 2013 Phys. Rev. X 3 031005 )). Due to experimental difficulties associated with the observation of such a phase, it has not been experimentally realised in optical lattices so far. In this paper, we show that optical lattices prove to be a good candidate for its realisation and observation, because they can be driven in a controlled manner. Specifically, we present a simple shaking protocol that serves to realise this special Floquet phase, discuss the specific properties that it has, and propose a method to experimentally detect this fascinating topological phase that has no counterpart in equilibrium systems.
We present a topological characterization of time-periodically driven two-band models in 2+1 dimensions as Hopf insulators. The intrinsic periodicity of the Floquet system with respect to both time and the underlying two-dimensional momentum space constitutes a map from a three-dimensional torus to the Bloch sphere. As a result, we find that the driven system can be understood by appealing to a Hopf map that is directly constructed from the micromotion of the drive. Previously found winding numbers are shown to correspond to Hopf invariants, which are associated with linking numbers describing the topology of knots in three dimensions. Moreover, after being cast as a Hopf insulator, not only the Chern numbers but also the winding numbers of the Floquet topological insulator become accessible in experiments as linking numbers. We exploit this description to propose a feasible scheme for measuring the complete set of their Floquet topological invariants in optical lattices.
Interactions generically have important effects on topological quantum phases. For a quantum anomalous Hall (QAH) insulator, the presence of interactions can qualitatively change a topological phase diagram, which, however, is typically hard to measure in an experiment. Here we propose a scheme based on quench dynamics to detect the mean-field topological phase diagram of an interacting Chern insulator described by the QAH-Hubbard model, with nontrivial dynamical quantum physics being uncovered. We focus on the dynamical properties of the system at a weak to intermediate Hubbard interaction, which mainly induces a ferromagnetic order under the mean-field level. Remarkably, three characteristic times---
ts,tc, and
t∗---are found in the quench dynamics. The first two (
ts,tc) capture the emergences of dynamical self-consistent particle density and the dynamical topological phase transition, respectively, while the third one (
t∗) gives a linear scaling time on the topological phase boundaries. We show generically that the characteristic times obey
ts>t∗>tc (
t∗<ts<tc) in the repulsive (attractive) interacting regime, when the system is quenched from an initial nearly fully polarized state to the topologically nontrivial regimes. Moreover, the Chern number of the equilibrium phase of postquench interacting Hamiltonian can be determined by any two of the three timescales, providing a dynamical way to determine the equilibrium mean-field topological phases. Experimentally, the measurement of
ts is challenging, while
tc and
t∗ can be directly read out by measuring the spin polarizations of four Dirac points and the time-dependent particle density, respectively, showing the feasibility of the present dynamical scheme. Our work opens a way to detect by quench dynamics the mean-field phase diagram of Chern insulators with interactions.
The classification of topological Floquet systems with time-periodic Hamiltonians transcends that of static systems. For example, spinless fermions in periodically driven two-dimensional lattices are not completely characterized by the Chern numbers of the quasienergy bands, but rather by a set of winding numbers associated with the gaps. We propose a feasible scheme for measuring these winding numbers in a periodically driven optical lattice efficiently and directly. It is based on the construction of a one-parameter family of drives, continuously connecting the Floquet system of interest to a trivial reference system. The winding numbers are then determined by the identification and the tomography of the band-touching singularities occurring on the way. As a by-product, we also propose a method for probing spectral properties of time evolution operators via a time analog of crystallography.