Thouless pumps are topologically nontrivial states of matter with quantized charge transport, which can be realized in atomic gases loaded into an optical lattice. This topological state is analogous to the quantum Hall state. However, contrarily to the exact, extremely precise, and robust quantization of the Hall conductance, the pumped charge is strictly quantized only when the pumping time is a multiple of a characteristic timescale, i.e., the pumping cycle duration. Here, we show instead that the pumped current becomes exactly quantized, independently from the pumping time, if the system is led into a quasiperiodic, incommensurate regime. In this quasiperiodic and topologically nontrivial state, the Bloch bands and the Berry curvature become flat, the pumped charge becomes linear in time, while the current becomes steady, topologically quantized, and proportional to the Chern number. The quantization of the current is exact up to exponentially small corrections. This has to be contrasted with the case of the commensurate (nonquasiperiodic) regime, where the current is not constant, and the pumped charge is quantized only at integer multiples of the pumping cycle.
Nonlinearity-Induced Thouless Pumping in Quasiperiodic Lattices
School of Physical Science and Technology, Lanzhou University（兰州大学物理科学与技术学院）；School of Mathematics and Statistics, Lanzhou University（兰州大学数学与统计学院）；Institute of Fundamental Physics and Quantum Technology, Ningbo University（宁波大学基础物理与量子技术研究所）
A gas of electrons in a one-dimensional periodic potential can be transported even in the absence of a voltage bias if the potential is modulated slowly and periodically in time. Remarkably, the transferred charge per cycle is only sensitive to the topology of the path in parameter space. Although this so-called Thouless charge pump has first been proposed more than thirty years ago, it has not yet been realized. Here we report the first demonstration of topological Thouless pumping using ultracold atoms in a dynamically controlled optical superlattice. We observe a shift of the atomic cloud a…
Breakdown of bosonic Thouless pump due to interaction in a quasiperiodic lattice
Max Planck Institute for the Physics of Complex Systems（马克斯·普朗克复杂系统物理研究所）；Cavendish Laboratory, University of Cambridge（剑桥大学卡文迪许实验室）；Institut für Theoretische Physik, Georg-August-Universität Göttingen（哥廷根大学理论物理研究所）
In Thouless pumping, although non-flat band has no effects on the quantization of particle transport, it induces wave-packet dispersion which hinders the practical applications of Thouless pumping. Indeed, we find that the dispersion mainly arises from the dynamical phase difference between individual Bloch states. Here we propose two efficient schemes to suppress the dispersion in Thouless pumping: a re-localization echo protocol and a high-order tunneling suppression protocol. In the re-localization echo protocol, we reverse the Hamiltonian in the second pumping cycle to cancel the dynamical…
Robustness against perturbations lies at the heart of topological phenomena. If, however, a perturbation such as disorder becomes dominant, it may cause a topological phase transition between topologically non-trivial and trivial phases. Here we experimentally reveal the competition and interplay between topology and quasi-periodic disorder in a Thouless pump realized with ultracold atoms in an optical lattice, by creating a quasi-periodic potential from weak to strong regimes in a controllable manner. We demonstrate a disorder-induced pumping in which the presence of quasi-periodic disorder c…
We explore adiabatic pumping in the presence of periodic drive, finding a new phase in which the topologically quantized pumped quantity is energy rather than charge. The topological invariant is given by the winding number of the micromotion with respect to time within each cycle, momentum, and adiabatic tuning parameter. We show numerically that this pump is highly robust against both disorder and interactions, breaking down at large values of either in a manner identical to the Thouless charge pump. Finally, we suggest experimental protocols for measuring this phenomenon.
An adiabatic cycle around a degeneracy point in the parameter space of a one-dimensional band insulator is known to result in an integer valued noiseless particle transport in the thermodynamic limit. Recently, it was shown that in the case of an infinite bipartite lattice the adiabatic Thouless protocol can be continuously deformed into a fine tuned finite-frequency cycle preserving the properties of noiseless quantized transport. In this paper, we numerically investigate the implementation of such an ideal rapid-cycle Thouless pumping protocol in a one-dimensional optical lattice. It is show…
Dynamics of particle in a resonantly driven quantum well can be interpreted as that of a particle in a crystal-like structure, with the time playing the role of the coordinate. By introducing an adiabatically varied phase in the driving protocol, we demonstrate a realization of the Thouless pumping in such a time crystalline structure. Next, we extend the analysis beyond a single quantum well by considering a driven one-dimensional optical lattice, thereby engineering a 2D time-space crystalline structure. Such a setup allows us to explore adiabatic pumping in the spatial and the temporal dime…
In Thouless pump, the charge transport in a one-dimensional insulator over an adiabatic cycle is topologically quantized. For nonequilibrium initial states, however, interband coherence will induce a previously unknown contribution to Thouless pumping. Though not geometric in nature, this contribution is independent of the time scale of the pumping protocol. In this work, we perform a detailed analysis of our previous finding [Phys. Rev. B 91, 085420 (2015)] in an already available cold-atom setup. We show that initial states with interband coherence can be obtained via a quench of the system'…
Thouless pumping provides one of the simplest manifestations of topology in quantum systems, and has attracted a lot of recent interest, both theoretically and experimentally. Since the seminal works by Thouless and Niu in 1983 and 1984, it is argued that the quantization of the pumped charge is robust against weak disorder, but a clear characterization of the localization properties of the relevant states, and the breakdown of quantized transport in the presence of interaction or out of the adiabatic approximation, has been long debated. Thouless pumping is also the first example of a topolog…
Nonlinear Thouless pumps for bosons exhibit quantized pumping via soliton motion, despite the lack of a meaningful notion of filled bands. However, the theoretical underpinning of this quantization, as well as its relationship to the Chern number, has thus far been lacking. Here we show that for low power solitons, transport is dictated by the Chern number of the band from which the soliton bifurcates. We do this by expanding the discrete nonlinear Schrödinger equation (equivalently, the Gross-Pitaevskii equation) in the basis of Wannier states, showing that the soliton's position is dictated …
More than 30 years ago, Thouless introduced the concept of a topological charge pump that would enable the robust transport of charge through an adiabatic cyclic evolution of the underlying Hamiltonian. In contrast to classical transport, the transported charge was shown to be quantized and purely determined by the topology of the pump cycle, making it robust to perturbations. On a fundamental level, the quantized charge transport can be connected to a topological invariant, the Chern number, first introduced in the context of the integer quantum Hall effect. A Thouless quantum pump may theref…
We investigate the effect of dissipation from a thermal environment on topological pumping in the periodically-driven Rice-Mele model. We report that dissipation can improve the robustness of pumping quantisation in a regime of finite driving frequencies. Specifically, in this regime, a low-temperature dissipative dynamics can lead to a pumped charge that is much closer to the Thouless quantised value, compared to a coherent evolution. We understand this effect in the Floquet framework: dissipation increases the population of a Floquet band which shows a topological winding, where pumping is e…
The sharply quantized transport observed in the integer quantum Hall effect can be explained via a simple one-dimensional model with a time-periodic, adiabatically varying potential in which electronic charge is pumped from one side of the system to the other. This so-called `Thouless pump' captures the topological physics of the quantum Hall effect using the notion of dimensional reduction: The time-varying potential mathematically maps onto a momentum coordinate in a conceptual second dimension. Importantly, this assumes an electronic system in equilibrium and in its ground state, that is, w…
We elucidate the mechanism for instability of topological Thouless pumping in strongly interacting systems from a viewpoint of symmetry-protected topological phases. If the protecting symmetries of the underlying topological phases change between noninteracting fermions and a bosonic system in the strong coupling limit, the symmetry protection argument enforces a gap closing and thereby predicts a breakdown of the topological pumping. We also demonstrate that, even in the weakly interacting regime where the bulk topological pumping is still robust, the interaction effects manifest themselves i…
The study of topological effects in physics is a hot area, and only recently researchers were able to address the important issues of topological properties of interacting quantum systems. But it is still a great challenge to describe multi-particle and interaction effects. Here, we introduce multi-particle Wannier states for interacting systems with co-translational symmetry. We reveal how the shift of multi-particle Wannier state relates to the multi-particle Chern number, and study the two-boson Thouless pumping in an interacting Rice-Mele model. In addition to the bound-state Thouless pump…
The bulk-boundary correspondence relates quantized edge states to bulk topological invariants in topological phases of matter. In one-dimensional symmetry-protected topological systems (SPTs), quantized topological Thouless pumps directly reveal this principle and provide a sound mathematical foundation. Symmetry-protected higher-order topological phases of matter (HOSPTs) also feature a bulk-boundary correspondence, but its connection to quantized charge transport remains elusive. Here we show that quantized Thouless pumps connecting $C_4$-symmetric HOSPTs can be described by a tuple of four …
We derive a topological classification of the steady states of $d$-dimensional lattice models driven by $D$ incommensurate tones. Mapping to a unifying $(d+D)$-dimensional localized model in frequency space reveals anomalous localized topological phases (ALTPs) with no static analog. While the formal classification is determined by $d+D$, the observable signatures of each ALTP depend on the spatial dimension $d$. For each $d$, with $d+D=3$, we identify a quantized circulating current, and corresponding topological edge states. The edge states for a driven wire ($d=1$) function as a quantized, …
Thouless pumping is a mechanism to perform topologically protected transport of particles by adiabatically modulating the Hamiltonian. The transported current is a topological invariant that is intimately related to the integer quantum Hall effect. Most of the previous works focus on topological pumping in linear and square lattices. In this work, we theoretically propose a mechanism to perform topological pumping in arrays of spin chains with complex geometries. To achieve this, we consider an array where the spin chains are coupled through their edges, which allows to split the populations t…
We introduce a classification scheme for symmetry protected topological phases applicable to stationary states of open systems based on a generalization of the many-body polarization. The polarization can be used to probe the topological properties of non-interacting and interacting closed and open systems as well and remains a meaningful quantity even in the presence of moderate particle-number fluctuations. As examples, we discuss two open-system versions of a topological Thouless pump in the steady state of one-dimensional lattices driven by Markovian reservoirs. In the analogous unitary sy…
Photonic waveguide arrays provide an excellent platform for simulating conventional topological systems, and they can also be employed for the study of novel topological phases in photonics systems. However, a direct measurement of bulk topological invariants remains a great challenge. Here we study topological features of generalized commensurate Aubry-André-Harper (AAH) photonic waveguide arrays and construct a topological phase diagram by calculating all bulk Chern numbers, and then explore the bulk-edge correspondence by analyzing the topological edge states and their winding numbers. In c…
Recent progress in synthetic lattice systems has opened the door to novel explorations of topological matter. In particular, photonic devices and ultracold matter waves offer the unique possibility of studying the rich interplay between topological band structures and tunable nonlinearities. In this emerging field of nonlinear topological physics, a recent experiment revealed the quantized motion of localized nonlinear excitations (solitons) upon driving a Thouless pump sequence; the reported observations suggest that the quantized displacement of solitons is dictated by the Chern number of th…
We consider theoretically the nonlinear quantized Thouless pumping of a Bose-Einstein condensate loaded in a two-dimensional dynamical optical lattices. We encountered three different scenarios of the pumping: quasi-linear one occurring for gradually dispersing wave packets, transport carried by a single two-dimensional soliton, and multi-soliton regime when initial wave packet splits into several solitons. The scenario to be realized depends on the number of atoms in the initial wave packet and on the strength of the two-body interactions. The magnitude and direction of the displacement of a …
When it comes to applying the adiabatic theorem in practice, the key question to be answered is how slow "slowly enough" is. This question can be an intricate one, especially for many-body systems, where the limits of slow driving and large system size may not commute. Recently we have shown how the quantum adiabaticity in many-body systems is related to the generalized orthogonality catastrophe [Phys. Rev. Lett. 119, 200401 (2017)]. We have proven a rigorous inequality relating these two phenomena and applied it to establish conditions for the quantized transport in the topological Thouless p…
We propose a two-dimensional (2D) version of Thouless pumping that can be realized by using ultracold atoms in optical lattices. To be specific, we consider a 2D square lattice tight-binding model with an obliquely introduced superlattice. It is demonstrated that quantized particle transport occurs in this system, and that the transport is expressed as a solution of a Diophantine equation. This topological nature can be understood by mapping the Hamiltonian to a three-dimensional (3D) cubic lattice model with a homogeneous magnetic field. We also propose a continuum model with obliquely introd…
We construct an interacting integrable Floquet model featuring quasiparticle excitations with topologically nontrivial chiral dispersion. This model is a fully quantum generalization of an integrable classical cellular automaton. We write down and solve the Bethe equations for the generalized quantum model, and show that these take on a particularly simple form that allows for an exact solution: essentially, the quasiparticles behave like interacting hard rods. The generalized thermodynamics and hydrodynamics of this model follow directly. Although the model is interacting, its unusually simpl…
University of Maryland（马里兰大学）；Institute for Physical Science and Technology, University of Maryland（马里兰大学物理科学与技术研究院）；Condensed Matter Theory Center, Department of Physics, University of Maryland（马里兰大学物理系凝聚态理论中心）
Adiabatic quantum pumping in one-dimensional lattices is extended by adding a tilted potential to probe better topologically nontrivial bands. This extension leads to almost perfectly quantized pumping for an arbitrary initial state selected in a band of interest, including Bloch states. In this approach, the time variable offers not only a synthetic dimension as in the case of the Thouless pumping, but it assists also in the uniform sampling of all momenta due to the Bloch oscillations induced by the tilt. The quantized drift of Bloch oscillations is determined by a one-dimensional time integ…
We study Thouless pumping out of the adiabatic limit. Our findings show that despite its topological nature, this phenomenon is not {generically} robust to non-adiabatic effects. Indeed we find that the Floquet diagonal ensemble value of the pumped charge shows a deviation from the topologically quantized limit which is quadratic in the driving frequency {for a sudden switch-on of the driving}. This is reflected also in the charge pumped in a single period, which shows a non-analytic behaviour on top of an overall quadratic decrease. Exponentially small corrections are recovered only with a ca…
Experiments with bosonic atoms in optical superlattices allow for the interesting possibility to study the adiabatic quantized pumping of bosonic atoms in the presence of interactions. We investigate this exotic phenomenon for bound bosonic pairs in the paradigmatic Su-Schrieffer-Heeger model where the ground state exhibits topological phase transitions due to dimerized hoppings. At unit filling we show that there exist crossovers and phase transitions to bond-order phases of paired bosons known as pair-bond-order phase as a function of attractive interactions. The pair bond order phase is fou…
