Topology and non-equilibrium dynamics in engineered quantum systems

Given that various synthetic topological matter have been realized with ultracold atoms, the atomic system has become one of most promising platforms for examining topological phases of matter in unprecedented environments, such as those with dissipation. Although such environments are ubiquitous in nature, it is still challenging to exploit them to manipulate the property of the quantum system in which emergent phenomena, such as exceptional points/rings, skin effects, localization, and critical phases, occur. In this talk, I report our recent realization of dissipative spin-orbit couplings (SOCs) in bulk and lattices with ultracold fermions. In bulk, we realize non-Hermitian spin-orbit-coupled quantum gases and observe a parity-time symmetry- breaking transition across the exceptional point (EP) as a result of the competition between SOC and dissipation. The realized EP of the non-Hermitian band structure exhibits chiral response when the quantum state changes near the EP. Recently, we generalize this non-Hermitian SOC scheme to two-dimensional Bloch bands, in which the EP emerges at the end of Fermi arc in the presence of dissipation. By using spin-resolved Rabi spectroscopy in this lattice system, we not only identify the exceptional point but also measure a complex energy-gap spectrum showing the evidence of non-Hermitian skin effect in a two-dimensional lattice system.

A key development of modern condensed matter theory has been the discovery of topological phases of quantum matter that are stabilised by underlying symmetries. I shall discuss the fate of symmetry-protected topological phases in settings where the system is coupled to an external bath. I shall describe how topological features can remain in the edge-state spectra of these open systems, contrasting cases of in- and out- of equilibrium. Finally I shall comment on connections to bulk topological invariants describing the open systems.

Topological gauge theories describe the low-energy properties of certain strongly correlated quantum systems through effective weakly interacting models. A prime example is the Chern–Simons theory of fractional quantum Hall states, where anyonic excitations emerge from the coupling between weakly interacting matter particles and a density-dependent gauge field. Although in traditional solid-state platforms such gauge theories are only convenient theoretical constructions, engineered quantum systems enable their direct implementation and provide a fertile playground to investigate their phenomenology without the need for strong interactions. Here, we report the quantum simulation of a topological gauge theory by realizing a one-dimensional reduction of the Chern–Simons theory (the chiral BF theory) in a Bose–Einstein condensate. Using the local conservation laws of the theory, we eliminate the gauge degrees of freedom in favour of chiral matter interactions, which we engineer by synthesizing optically dressed atomic states with momentum-dependent scattering properties. This allows us to reveal the key properties of the chiral BF theory: the formation of chiral solitons and the emergence of an electric field generated by the system itself. Our results expand the scope of quantum simulation to topological gauge theories and open a route to the implementation of analogous gauge theories in higher dimensions

Topological invariants robustly classify gapped quantum systems in equilibrium, and phenomena such as the quantized Hall effect---the progenitor of the von Klitzing constant---are macroscopic refections of these invariants. In addition to dimensionality, the presence or absence of symmetries determines the possible topological invariants. Thus, these invariants remain constant provided that no gaps close and no symmetries are added or removed. For this reason, one might expect the topology of a dynamical quantum system to be similarly robust; this expectation is untrue. Instead as a system undergoes far from equilibrium evolution symmetries come and go, allowing the topology to change as well. We experimentally study these dynamics with ultracold atoms in a 1D bipartite lattice in terms of the Zak phase and a chiral winding number.

A Thouless pump transports an integer amount of charge when pumping adiabatically around a singularity. We study the splitting of such a critical point into two separate critical points by adding a Hubbard interaction. Furthermore, we consider extensions to a spinful Rice-Mele model, namely a staggered magnetic field or an Ising-type spin coupling, further reducing the spin symmetry. The resulting models additionally allow for the transport of a single charge in a two-component system of spinful fermions, whereas in the absence of interactions, zero or two charges are pumped. In the SU(2)-symmetric case, the ionic Hubbard model is visited once along pump cycles that enclose a single singularity. Adding a staggered magnetic field additionally transports an integer amount of spin while the Ising term realizes a pure charge pump. We employ real-time simulations in finite and infinite systems to calculate the adiabatic charge and spin transport, complemented by the analysis of gaps and the many-body polarization to confirm the adiabatic nature of the pump. The resulting charge pumps are expected to be measurable in finite-pumping speed experiments in ultra-cold atomic gases, for which the SU\((2)\) invariant version is the most promising path. We discuss the implications of our results for a related quantum-gas experiment by Walter \textit{et al.} [arXiv:2204.06561].

Interactions in higher Bloch bands provide a remarkable setting for realizing many-body states that spontaneously break time-reversal symmetry. Here, we show that the low-energy physics of interacting bosons in a plaquette pierced by pi-flux shares feature of p-band bosons, thus displaying condensates with chiral orbital order. We analyze the excitations of the condensate in terms of two orbital-like degrees of freedom and identify a gapped collective mode corresponding to the out-of-phase oscillations of the relative density and phase of the two orbitals. We further highlight the chiral nature of the ground state by revealing the cyclotron-like dynamics of the density upon quenching an impurity potential on a single site. This single-plaquette construction can be used as a building block for extended dimerized lattices with local orbital order, as we exemplify by using the Benalcazar-Bernevig-Hughes model. Our results provide a distinct direction to realize interacting orbital-like models with broken time-reversal symmetry, without resorting to higher bands nor to external drives, with direct implications for cold gases and nonlinear photonics.

In this contribution we present results on quantum gas magnification as a microscopy tool for ultracold atoms in optical lattices. We developed a matter wave optics protocol which magnifies the atomic density distribution by almost two orders of magnitude [1], allowing to image the magnified cloud in a single-shot and with sub-lattice resolution by using standard optical imaging techniques. This approach works with 3D systems with many particles per lattice site, because it does not have the limitations of a small depth of focus and of light induced collisions. We demonstrate sub-lattice resolution by observing the dynamics within the lattice sites after a lattice potential quench, and we realize single-site addressing using magnetic resonance techniques. We report on a novel phenomenon that occurs in the regime of many particles per lattice site when introducing a strong force into the system; this makes the correlated tunneling of pairs of atoms the relevant dynamical process and causes the spontaneous formation of a density wave [2]. Quantum gas magnification opens the path for spatially resolved studies of new quantum many-body regimes, like e.g. 3D systems, orbital lattices, and lattice geometries with smaller lattice spacing. We finally discuss the novel multi-frequency scheme for optical lattices with dynamically tunable geometry and present experimental results and possible multi-frequency realizations of tunable 3D lattices and quasicrystals [3]. References [1] L. Asteria et al., Nature 599, 571-575 (2021) [2] H. P. Zahn et al., PRX 12, 021014 (2022) [3] M. N. Kosch et al., https://arxiv.org/abs/2207.03811

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The construction of the generalized Gibbs ensemble, to which isolated integrable quantum many-body systems relax after a quantum quench, is based upon the principle of maximum entropy. In contrast, there are no universal and model-independent laws that govern the relaxation dynamics and stationary states of open quantum systems, which are subjected to Markovian drive and dissipation. Yet, as we show, relaxation of driven-dissipative systems after a quantum quench can, in fact, be determined by a maximum entropy ensemble, if the Liouvillian that generates the dynamics of the system has parity-time symmetry. Focusing on the specific example of a driven-dissipative Kitaev chain, we show that, similarly to isolated integrable systems, the approach to a parity-time symmetric generalized Gibbs ensemble becomes manifest in the relaxation of local observables and the dynamics of subsystem entropies. In contrast, the directional pumping of fermion parity, which is induced by nontrivial non-Hermitian topology of the Kitaev chain, represents a phenomenon that is unique to relaxation dynamics in driven-dissipative systems. Upon increasing the strength of dissipation, parity-time symmetry is broken at a finite critical value, which thus constitutes a sharp dynamical transition that delimits the applicability of the principle of maximum entropy. While we focus here on the specific example of the Kitaev chain, our results generalize readily to all parity-time symmetric and integrable driven-dissipative systems that are described by or can be mapped to noninteracting fermions.

The digital simulation of quantum many-body dynamics, one of the most promising applications of quantum computers, involves Trotterization as a key element. It is an outstanding challenge to formulate a quantum algorithm allowing adaptive Trotter time steps. This is particularly relevant for today's noisy intermediate scale quantum devices, where the minimization of the circuit depth is a central optimization task. Here, we introduce an adaptive Trotterization scheme providing a controlled solution of the quantum many-body dynamics of local observables. Our quantum algorithm outperforms conventional fixed-time step Trotterization schemes in a quantum quench and even allows for a controlled asymptotic long-time error, where Trotterized dynamics generically enter challenging regimes. This adaptive method can also be generalized to protect various other kinds of symmetries, which we illustrate by preserving the local Gauss's law in a lattice gauge theory. We discuss the requirements imposed by experimental resources, and point out that our adaptive Trotterization scheme can be of use also in numerical approaches based on Trotterization such as in time-evolving block decimation methods.

We prove that invisible bands associated with zeros of the single-particle Green's function exist ubiquitously at topological interfaces of 2D Chern insulators, dual to the chiral edge/domain-wall modes. We verify this statement in a repulsive Hubbard model with a topological flat band, using real-space dynamical mean-field theory to study the domain walls of its ferromagnetic ground state. Moreover, our numerical results show that the chiral modes are split into branches due to the interaction, and that the branches are connected by invisible flat bands. Our work provides deeper insight into interacting topological systems.

Topological insulators (TIs) have been successfully demonstrated in photonic, electrical, phononic, acoustic, atomic, and polaritonic systems. The ubiquitousness of topological effects stems from a common origin geometrical frustration of wavefunctions in space and time. In my talk, I will present a unifying semiclassical treatment that extends the modern theory of polarization to any spinor degree of freedom. We dub this approach spinor topology. I will furthermore introduce a corresponding group theoretical approach to obtain high-order topology. If time permits, I will present a dressed Floquet theory that includes AC stark shifts, thus faithfully describing driven interacting systems. Last, I will discuss a classical-to-quantum transition in a driven-dissipative system where a non-Hermitian exceptional point is erased by many-body interactions.

Predicting the fate of topologically protected transport in the strongly correlated regime represents a central challenge within condensed matter physics. On the one hand, free-fermion energy bands and their geometric properties give rise to quantised transport phenomena, such as the quantum Hall effect and its dynamic analogon, the Thouless pump. The quantisation in these systems is considered robust against perturbations that commute with a protecting symmetry. On the other hand, interparticle interactions support strongly correlated states of matter, which often preclude particle transport, exemplified by the Mott transition in the Hubbard model. Will topology prevail in the presence of strong correlations? Here, we systematically probe the response of a topological Thouless pump to Hubbard interactions in an ultracold-atom experiment. We identify three distinct regimes, that is, pair pumping for strongly attractive interactions, quantised pumping for weak and moderate interactions, as well as the breakdown of transport for strong repulsive Hubbard $U$. In the last case, an appropriate modification to the pump trajectory induces resonant tunnelling processes which can reinstantiate quantised transport. Our experiments pave the way for investigating edge effects in interacting topological insulators, as well as interaction-induced topological phases with no counterpart in free-fermion systems.

I will talk about new exotic topologies involving non-Abelian braiding of band nodes in multi-gap systems, and demonstrate new phases in out-of-equilibrium settings with no static counterparts as well as ways for quantum simulators to probe them. I will further introduce new geometric descriptions to study multi-level systems.

The study of interacting quantum Hall states and their exotic anyonic excitations poses a major challenge in current experimental and theoretical research. Quantum simulators, in particular ultracold atoms in optical lattices, provide a promising platform to realize, manipulate, and understand such systems with a high controllability. In many cases, the experimental platforms use lattices, so that a theoretical understanding of the connection between continuum quantum Hall states and their lattice analogs - called Chern insulators - is desirable. First realizations of the $\nu=\frac{1}{2}$-Laughlin state for a few particles are now within reach. The next goal is now to find suitable observables to directly probe the topological properties of such states. In this talk, I will present results showing that at $\nu=\frac{1}{2}$ three-legged systems are sufficient to realize and probe properties of central interest in interacting topological systems, like the central charge [1]. Furthermore, I will propose new directions available using ultracold experiments, discussing two systems in some detail. At filling factor $\nu=1$ we found a bosonic lattice analog of the Pfaffian state [2], which was long studied as a candidate state for the fermionic $\nu=\frac{5}{2}$ plateau. Yet another direction enabled by cold atom experiments involves fermionic Hofstadter-Hubbard models. There, we propose regimes exhibiting Stoner-type ferromagnetism and spin textures reminiscent of skyrmions in the continuum [3]. [1] F.A.P., S. Mardazad, A. Bohrdt, U. Schollwöck, and F. Grusdt, Phys. Rev. B 106, L081108 (2022) [2] F.A.P., M. Buser, J. Léonard, M. Aidelsburger, U. Schollwöck, and F. Grusdt, Phys. Rev. B 103, L161101 (2021) [3] F.A.P., M. Kurttutan, A. Bohrdt, U. Schollwöck, and F. Grusdt, in preparation

Recent experiments demonstrated deeply subwavelength lattices using atoms with $N$ internal states Raman-coupled with lasers of wavelength $\lambda$ [1]. The resulting unit cell was $\lambda/2N$, an $N$-fold reduction compared to the usual $\lambda/2$ periodicity of an optical lattice. For resonant Raman coupling, this lattice consists of $N$ independent sinusoidal potentials (with period $\lambda/2$) displaced by $\lambda/2N$ from each other. Atoms in each of these lattice potentials are described by dressed states formed from superpositions of the atomic bare states. We show that detuning from Raman resonance leads to tunneling between these potentials spaced at the subwavelength scale with a range up to $N$ sites, and with tunable amplitudes and phases. Periodic modulation of the detuning couples the $s$- and $p$-bands of the lattice, creating a pair of coupled subwavelength Rice--Mele chains that operate as a novel topological charge pump. [1] R. P. Anderson, D. Trypogeorgos, A. Valdés-Curiel, Q.-Y. Liang, J. Tao, M. Zhao, T. Andrijauskas, G. Juzeli\={u}nas, and I. B. Spielman, Realization of a deeply subwavelength adiabatic optical lattice, Phys. Rev. Research 2, 013149 (2020).

Topological spaces have numerous applications for quantum matter with protected chiral edge modes related to an integer-valued Chern number, which also characterizes the global response of a spin-1/2 particle to a magnetic field. Such spin-1/2 models can also describe topological Bloch bands in lattice Hamiltonians. We introduce a geometrical approach on the Bloch sphere of a Spin-1/2 particle that allows us to discuss transport and electromagnetic responses. Then, we introduce interactions in a system of spin-1/2s to reveal a class of topological states with rational-valued topological numbers for each spin providing a geometrical and physical interpretation related to curvatures and quantum entanglement. We study a driving protocol in time to reveal the stability of the fractional topological numbers towards various forms of interactions in the adiabatic limit. We elucidate a correspondence of the one-half topological spin response in topological matter, such as topological semimetals on the honeycomb lattice and topological p-wave superconducting wires coupled through a Coulomb force. These spheres also find practical applications through a quantum dynamo effect when rolling the spin from north to south pole.

The fractional quantum Hall effect appears in two-dimensional systems and gives rise to interesting physics, such as anyons. Here, we show that anyons and the fractional quantum Hall effect can also be realized in fractal dimensions. We do this by constructing fractional quantum Hall models on different fractals and demonstrating that quasiparticles created in the models display anyonic charge and braiding statistics. We also discuss how the properties of fractional quantum Hall systems on fractals differ from the two-dimensional case. Finally, we show that a Hofstadter type Hamiltonian produces fractional quantum Hall physics and anyons on a small fractal lattice. References: PRResearch 2, 023401 (2020); PRA 105, L021302 (2022); PRB 105, 085152 (2022).

We propose a Floquet scheme for the accurate quantum simulation of the Hamiltonian underpinning Kitaev’s toric code. The approach used employs a hybrid analog-digital strategy which fully exploits the commutativity of different terms in Kitaev’s Hamiltonian, enabling high simulation accuracy. We further show that the tools developed provide an efficient protocol for preparing ground states with high fidelity, and confirm that the states prepared exhibit typical signatures of topological order. We conclude by proposing a proof-of-principle implementation of a topological device and the toolbox to manipulate it. Our scheme finds a natural implementation in lattices of superconducting qubits where individual control and tuneable couplings are accessible.

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Floquet engineering, i.e., periodic modulation of a system’s parameters, has proven as a powerful experimental tool for the realization of quantum systems with exotic properties that have no static analog. In particular, the so-called anomalous Floquet phase displays topological properties even if the Chern number of the bulk band vanishes.[1] Our experimental system consists of bosonic atoms in a periodically driven honeycomb lattice. Depending on the driving parameters several out-of-equilibrium topological phases can be realized, including an anomalous phase [2]. Recently, we have added programmable optical potentials to study the edge dynamics in such topological systems. We are investigating the real-space evolution of an initially localized wavepacket close to the edge after releasing it from a tightly-focused optical tweezer. We observe the chiral nature of the edge state, even in the anomalous Floquet phase, thereby directly revealing the topological nature of this phase. [1] Rudner, et al. Anomalous edge states and the bulk-edge correspondence for periodically-driven two dimensional systems, Phys. Rev. X 3, 031005 (2013) [2] Wintersperger, et al. Realization of an anomalous Floquet topological system with ultracold atoms. Nat. Phys. 16, 1058–1063 (2020)

We study the Loschmidt echo for quenches in open one-dimensional lattice models with symmetry protected topological phases. For quenches where dynamical quantum phase transitions do occur we find that cusps in the bulk return rate at critical times tc are associated with sudden changes in the boundary contribution. For our main example, the Su-Schrieffer-Heeger model, we show that these sudden changes are related to the periodical appearance of two eigenvalues close to zero in the dynamical Loschmidt matrix. We demonstrate, furthermore, that the structure of the Loschmidt spectrum is linked to the periodic creation of long-range entanglement between the edges of the system. Generalisations to more complicated systems and higher dimensions are also discussed.

While in non-interacting 1D systems a complete classification of topological insulators can be made through arguments based on chiral invariance, the appearance of topological insulators induced by interacting processes represents a less understood playground. I will first discuss the case of one-dimensional fermions where the competition between local and non-local interactions gives rise to an effective Peierls instability able to stabilize a bond-order-wave insulator. Numerical analyses unveil that such phase turns out to be characterized by a chirally symmetric topological sector remained elusive in previous finite-size numerical studies. In presence of interactions breaking the chiral invariance the analogy with non-interacting models is totally lost. I will show that this is the case of fermions interacting through correlated hopping processes that breaks the particle-hole invariance. Here, a numerical analysis demonstrates that inversion and time-reversal symmetries are still able to stabilize a crystalline topological order analogous to the one characterizing higher-order-topological-insulators

Non-Hermitian “Hamiltonians” occur in the effective description of various physical settings ranging from classical photonics to quantum materials. Using simple examples, I will discuss topological aspects of such systems related to the non-Hermitian concept of exceptional degeneracies at which both eigenvalues and eigenvectors coalesce. I will also discuss how the bulk-boundary correspondence is modified in non-Hermitian systems and how this might be harnessed in novel sensor devices.

Physically, one tends to think of non-Hermitian systems in terms of gain and loss: the decay or amplification of a mode is given by the imaginary part of its energy. Here, we introduce an alternative avenue to the realm of non-Hermitian physics, which involves neither gain nor loss. Instead, complex eigenvalues emerge from the amplitudes and phase differences of waves backscattered from the boundary of insulators. We show that for any strong topological insulator in a Wigner-Dyson class, the reflected waves are characterized by a reflection matrix exhibiting the non-Hermitian skin effect. This leads to an unconventional Goos-Hänchen effect: due to non-Hermitian topology, waves undergo a lateral shift upon reflection, even at normal incidence. Going beyond systems with gain and loss vastly expands the set of experimental platforms that can access non-Hermitian physics and show signatures associated with non-Hermitian topology.

The recent progress in engineering topological band structures in optical-lattice systems makes it promising to study fractional Chern insulator states in these systems. Here we consider a realistic finite system of a few repulsively interacting bosons on a square lattice with magnetic flux and sharp edges, as it can be realized in quantum-gas microscopes. We investigate under which conditions a fractional Chern insulator state corresponding to the Laughlin-like state at filling 1/2 can be stabilized and its fractional excitations probed. Using numerical simulations, we find an incompressible bulk density at the expected filling for systems, whose linear extent is as small as 6-8 sites. This is a promising result, since such small systems are favorable with respect to the required adiabatic state preparation. Moreover, we also see very clear signatures of excitations with fractional charge in response both to static pinning potentials and dynamical flux insertion. Since the compressible edges, which are found to feature chiral currents, can serve as a reservoir, these observations are robust against changes in the total particle number. Our results suggest that signatures of both a fractional Chern insulator state and its fractional excitations can be found under realistic experimental conditions.