The interplay between topology and soliton is a central topic in nonlinear topological physics. So far, most studies have been confined to conservative settings. Here, we explore Thouless pumping of dissipative temporal solitons in a nonconservative one-dimensional optical system with gain and spectral filtering, described by the paradigmatic complex Ginzburg-Landau equation. Two dissipatively induced nonlinear topological phase transitions are identified. First, when varying dissipative parameters across a threshold, the soliton transitions from being trapped in time to quantized drifting. This quantized temporal drift remains robust, even as the system evolves from a single-soliton state into multi-soliton state. Second, a dynamically emergent phase transition is found: the soliton is arrested until a critical point of its evolution, where a transition to topological drift occurs. Both phenomena uniquely arise from the dynamical interplay of dissipation, nonlinearity and topology.
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…
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）
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…
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…
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…
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 …
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…
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'…
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…
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…
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 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 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…
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 study the 2d phase transition of a driven-dissipative system of exciton-polaritons under non-resonant pumping. Stochastic calculations are used to investigate the Berezinskii-Kosterlitz-Thouless-like phase diagram for experimentally realistic parameters, with a special attention to the non-equilibrium features.
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…
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…
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 i…
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 …
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…
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 observe quasi-long range coherence in a two-dimensional condensate of exciton-polaritons. Our measurements are the first to confirm that the spatial correlation algebraically decays with a slow power-law, whose exponent quantitatively behaves as predicted by the Berezinskii-Kosterlitz-Thouless theory. The exciton-polaritons are created by non-resonant optical pumping of a micro-cavity sample with embedded GaAs quantum-wells at liquid helium temperature. Michelson interference is used to measure the coherence of the photons emitted by decaying exciton-polaritons.
We consider strongly interacting systems of effective spins, subject to dissipative spin-flip processes associated with optical pumping. We predict the existence of novel magnetic phases in the steady-state of this system, which emerge due to the competition between coherent and dissipative processes. Specifically, for strongly anisotropic spin-spin interactions, we find ferromagnetic, antiferromagnetic, spin-density-wave, and staggered-XY steady states, which are separated by nonequilibrium phase transitions meeting at a Lifshitz point. These transitions are accompanied by quantum correlation…
We report a theoretical study of a quantum optical model consisting of an array of strongly nonlinear cavities incoherently pumped by an ensemble of population-inverted two-level atoms. Projective methods are used to eliminate the atomic dynamics and write a generalized master equation for the photonic degrees of freedom only, where the frequency-dependence of gain introduces non-Markovian features. In the simplest single cavity configuration, this pumping scheme gives novel optical bistability effects and allows for the selective generation of Fock states with a well-defined photon number. Fo…
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.
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…
Adopting a mean-field Gross-Pitaevskii description for a spinor polariton Bose-Einstein condensates under non-resonant pumping, we investigate the static and dynamical properties of dark-bright solitons. We derive analytically the equation of motion for the center of mass of the dark-bright soliton center, using the Hamiltonian approach. The resulting equation captures how the combination of the open-dissipative character and the spin degrees of freedom of a polariton Bose-Einstein condensate affects the properties of a dark-bright soliton, i.e. the dark-bright soliton relaxes by blending with…
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…
Dissipative solitons are non-decaying out-of-equilibrium entities that result from double balances between gain and loss, as well as nonlinearity and dispersion. Here we describe a scenario where double balances rely on the presence of multiple collective excitation channels in open-dissipative quantum systems. It differs from conventional single-channel scenario for well-known dissipative solitons such as dissipative Kerr solitons, in that the soliton itself arises in a decoupled excitation channel and hence coherent nonlinear excitation dynamics, but its background state corresponds to other…
Topological Thouless pumping and Aharonov-Bohm effect are both fundamental effects enabled by the topological properties of the system. Here, we study both effects together: topological pumping of interacting particles through Aharonov-Bohm rings. This system can prepare highly entangled many-particle states, transport them via topological pumping and interfere them, revealing a fractional flux quantum. The type of the generated state is revealed by non-trivial Aharonov-Bohm interference patterns that could be used for quantum sensing. The reflections induced by the interference result from tr…
We theoretically investigate the dynamics of two dark solitons in a polariton condensate under nonresonant pumping, based on driven dissipative Gross-Pitaevskii equations coupled to the rate equation. In particular, the analytical expression of the effective potential between two dark solitons is given. The resulting equation of motion captures how the open-dissipative character of a polariton Bose-Einstein condensate affects the properties of dynamics of two-dark soliton, i.e., two-dark soliton relax by blending with the background at a finite time. We further simulate the relative motion of …
We study driven-dissipative Bose-Einstein condensates in a two-mode Josephson system, such as a double-well potential, with asymmetrical pumping. We investigate nonlinear effects on the condensate populations and mode transitions. The generalized Gross-Pitaevskii equations are modified in order to treat pumping of only a single mode. We characterize the steady-state solutions in such a system as well as criteria for potential trapping of a condensate mode. There are many possible steady-states, with different density and/or phase profiles. Transitions between different condensate modes can be …
We study the production of photons in a model of three bosonic atomic modes non-linearly coupled to a cavity mode. In absence of external driving and dissipation, the energy levels at different photon numbers assemble into the steps of an energy staircase which can be employed as guidance for preparing multi-photon states. We consider adiabatic photon production, driving the system through a sequence of Landau-Zener transitions in the presence of external coherent light pumping. We also analyse the non-equilibrium dynamics of the system resulting from the competition of the sudden switch of co…
Thouless pumping, not only achieving quantized transport but also immune to moderate disorder, has attracted growing attention in both experiments and theories. Here, we explore how particle-particle interactions affect topological transport in a periodically-modulated and tilted optical lattice. Not limited to wannier states, our scheme ensures a dispersionless quantized transport even for initial Gaussian-like wave packets of interacting bosons which do not uniformly occupy a given band. This is because the tilting potential leads to Bloch oscillations uniformly sampling the Berry curvatures…
Solid state quantum condensates can differ from other condensates, such as Helium, ultracold atomic gases, and superconductors, in that the condensing quasiparticles have relatively short lifetimes, and so, as for lasers, external pumping is required to maintain a steady state. In this chapter we present a non-equilibrium path integral approach to condensation in a dissipative environment and apply it to microcavity polaritons, driven out of equilibrium by coupling to multiple baths, describing pumping and decay. Using this, we discuss the relation between non-equilibrium polariton condensatio…
