Takuya Kitagawa

Topological Characterization of Periodically-Driven Quantum Systems

Topological properties of physical systems can lead to robust behaviors that are insensitive to microscopic details. Such topologically robust phenomena are not limited to static systems but can also appear in driven quantum systems. In this paper, we show that the Floquet operators of periodically driven systems can be divided into topologically distinct (homotopy) classes, and give a simple physical interpretation of this classification in terms of the spectra of Floquet operators. Using this picture, we provide an intuitive understanding of the well-known phenomenon of quantized adiabatic pumping. Systems whose Floquet operators belong to the trivial class simulate the dynamics generated by time-independent Hamiltonians, which can be topologically classified according to the schemes developed for static systems. We demonstrate these principles through an example of a periodically driven two--dimensional hexagonal lattice model which exhibits several topological phases. Remarkably, one of these phases supports chiral edge modes even though the bulk is topologically trivial.

Direct measurement of the Zak phase in topological Bloch bands

Geometric phases that characterize the topological properties of Bloch bands play a fundamental role in the band theory of solids. Here we report on the measurement of the geometric phase acquired by cold atoms moving in one-dimensional optical lattices. Using a combination of Bloch oscillations and Ramsey interferometry, we extract the Zak phase—the Berry phase gained during the adiabatic motion of a particle across the Brillouin zone—which can be viewed as an invariant characterizing the topological properties of the band. For a dimerized lattice, which models polyacetylene, we measure a difference of the Zak phase’ Zak D 0:97(2) for the two possible polyacetylene phases with different dimerization. The two dimerized phases therefore belong to different topological classes, such that for a filled band, domain walls have fractional quantum numbers. Our work establishes a new general approach for probing the topological structure of Bloch bands in optical lattices.

Transport Properties of Non-Equilibrium Systems Under the Application of Light: Photo-Induced Quantum Hall Insulators Without Landau Levels

In this paper, we study transport properties of non-equilibrium systems under the application of light in many-terminal measurements, using the Floquet picture. We propose and demonstrate that the quantum transport properties can be controlled in materials such as graphene and topological insulators, via the application of light. Remarkably, under the application of off-resonant light, topological transport properties can be induced; these systems exhibits quantum Hall effects in the absence of a magnetic field with a near quantization of the Hall conductance, realizing so-called quantum Hall systems without Landau levels first proposed by Haldane.

Exploring topological phases with quantum walks

The quantum walk was originally proposed as a quantum-mechanical analog of the classical random walk, and has since become a powerful tool in quantum information science. In this paper, we show that discrete-time quantum walks provide a versatile platform for studying topological phases, which are currently the subject of intense theoretical and experimental investigations. In particular, we demonstrate that recent experimental realizations of quantum walks with cold atoms, photons, and ions simulate a nontrivial one-dimensional topological phase. With simple modifications, the quantum walk can be engineered to realize all of the topological phases, which have been classified in one and two dimensions. We further discuss the existence of robust edge modes at phase boundaries, which provide experimental signatures for the nontrivial topological character of the system.

Prethermalization revealed by the relaxation dynamics of full distribution functions

We detail the experimental observation of the non-equilibrium many-body phenomenon prethermalization. We study the dynamics of a rapidly and coherently split one-dimensional Bose gas. An analysis based on the use of full quantum mechanical probability distributions of matter wave interference contrast reveals that the system evolves towards a quasi-steady state. This state, which can be characterized by an effective temperature, is not the final thermal equilibrium state. We compare the evolution of the system to an integrable Tomonaga-Luttinger liquid model and show that the system dephases to a prethermalized state rather than undergoing thermalization towards a final thermal equilibrium state.

Interferometric Approach to Measuring Band Topology in 2D Optical Lattices

Recently, optical lattices with nonzero Berrys phases of Bloch bands have been realized. New approaches for measuring Berrys phases and topological properties of bands with experimental tools appropriate for ultracold atoms need to be developed. In this Letter, we propose an interferometric method for measuring Berrys phases of two-dimensional Bloch bands. The key idea is to use a combination of Ramsey interference and Bloch oscillations to measure Zak phases, i.e., Berrys phases for closed trajectories corresponding to reciprocal lattice vectors. We demonstrate that this technique can be used to measure the Berry curvature of Bloch bands, the π Berrys phase of Dirac points, and the first Chern number of topological bands. We discuss several experimentally feasible realizations of this technique, which make it robust against low-frequency magnetic noise.

Topological phenomena in quantum walks: elementary introduction to the physics of topological phases

Discrete quantum walks are dynamical protocols for controlling a single quantum particle. Despite of its simplicity, quantum walks display rich topological phenomena and provide one of the simplest systems to study and understand topological phases. In this article, we review the physics of discrete quantum walks in one and two dimensions in light of topological phenomena and provide elementary explanations of topological phases and their physical consequence, namely the existence of boundary states. We demonstrate that quantum walks are versatile systems that simulate many topological phases whose classifications are known for static Hamiltonians. Furthermore, topological phenomena appearing in quantum walks go beyond what has been known in static systems; there are phenomena unique to quantum walks, being an example of periodically driven systems, that do not exist in static systems. Thus the quantum walks not only provide a powerful tool as a quantum simulator for static topological phases but also give unique opportunity to study topological phenomena in driven systems.

Anomalous Expansion of Attractively Interacting Fermionic Atoms in an Optical Lattice

Fermion Behavior in an Optical Lattice Due to their extreme tunability, optical lattices loaded with fermions and bosons are expected to act as quantum simulators, answering complicated many-body physics questions beyond the reach of theory and computation. Some of these many-body states, such as the Mott insulator and the superfluid, have been achieved in bosonic optical lattices by simply changing the characteristic depth of the lattice potential wells. Now, Hackermüller et al. (p. 1621) describe an unusual effect in an optical lattice loaded with fermions: When the strength of the attraction between the fermions is increased adiabatically, instead of contracting, the gas expands in order to preserve entropy. Thermodynamic effects cause an ultracold potassium gas to expand unexpectedly when the attraction between atoms is increased. The interplay of thermodynamics and quantum correlations can give rise to counterintuitive phenomena in many-body systems. We report on an isentropic effect in a spin mixture of attractively interacting fermionic atoms in an optical lattice. As we adiabatically increase the attraction between the atoms, we observe that the gas expands instead of contracting. This unexpected behavior demonstrates the crucial role of the lattice potential in the thermodynamics of the fermionic Hubbard model.

Multimode Dynamics and Emergence of a Characteristic Length Scale in a One-Dimensional Quantum System

We study the nonequilibrium dynamics of a coherently split one-dimensional Bose gas by measuring the full probability distribution functions of matter-wave interference. Observing the system on different length scales allows us to probe the dynamics of excitations on different energy scales, revealing two distinct length-scale-dependent regimes of relaxation. We measure the crossover length scale separating these two regimes and identify it with the prethermalized phase-correlation length of the system. Our approach enables a direct observation of the multimode dynamics characterizing one-dimensional quantum systems.

Cooling through Optimal Control of Quantum Evolution

Nonadiabatic unitary evolution with tailored time-dependent Hamiltonians can prepare systems of cold-atomic gases with various desired properties such as low excess energies. For a system of two one-dimensional quasicondensates coupled with a time-varying tunneling amplitude, we show that the optimal protocol, for maximizing any figure of merit in a given time, is bang-bang, i.e., the coupling alternates between only two values through a sequence of sudden quenches. Minimizing the energy of one of the quasicondensates with such a nonadiabatic protocol, and then decoupling it at the end of the process, can result in effective cooling beyond the current state of the art. Our cooling method can be potentially applied to arbitrary systems through an integration of the experiment with simulated annealing computations.