Topology and non-equilibrium dynamics in engineered quantum systems

We describe a driven system coupled to a single cavity mode or a collection of modes forming an Ohmic boson bath. When the system Hamiltonian changes in time, this induces a dynamical field in the bosonic modes having resonant frequencies with the driving velocity. This field opposes the change of the external driving field in a way reminiscent of the Faraday effect in electrodynamics, justifying the term ‘quantum dynamo effect’. For the specific situation of a periodically driven spin-1/2 with adiabatic ground state on the Bloch sphere, we show that the work done by rolling the spin from north to south pole can efficiently be converted into a coherent displacement of the resonant bosonic modes. The effect thus corresponds to a work-to-work conversion and allows to interpret this transmitted energy into the bath as work. We study this effect, its performance and limitations in detail for a driven spin-1/2 in the presence of a radial magnetic field addressing a relation with topological systems through the formation of an effective charge in the core of the sphere. We show that the dynamo effect is directly related to the dynamically measured topology of this spin-1/2 and thus in the adiabatic limit provides a topologically protected method to convert driving work into a coherent field in the reservoir. The quantum dynamo model is realizable in mesoscopic and atomic systems.

Non-Hermitian (NH) lattice Hamiltonians display a unique kind of energy gap and extreme sensitivity to changes of boundary conditions—the so-called NH skin effect. Due to the NH skin effect, the separation between edge and bulk states is blurred and the bulk-boundary correspondence in its conventional form breaks down [1]. Despite considerable efforts to accommodate the NH skin effect into a modified bulk-boundary correspondence, a formulation for point-gapped spectra has remained elusive. In this contribution, I will show how to restore the bulk-boundary correspondence for the most paradigmatic class of NH lattice models, namely single-band models without symmetries. This is achieved by presenting an alternative route to the classification of NH topological phases, where the focus is shifted (i) from effective NH Hamiltonians to NH Hamiltonians obtained from the unconditional dynamics of driven-dissipative arrays of cavities, and (ii) from the eigen-decomposition to the singular value decomposition as the main tool for studying their bandstructure. The class of NH Hamiltonians that reveal the bulk-boundary correspondence are unconditional NH Hamiltonians, which do not neglect quantum jumps altogether but instead retain a contribution from fluctuation-dissipation processes by averaging over the quantum state of the system. Concretely, the desired NH Hamiltonian is implemented in one-dimensional driven-dissipative cavity arrays, in which Hermiticity-breaking terms — in the form of non-reciprocal hopping amplitudes, gain and loss — are explicitly modelled via coupling to (engineered and non-engineered) reservoirs. I will show that this approach introduces extra constraints to the NH Hamiltonian, neglected so far, which determine the following major changes to the topological characterization: First, the complex spectrum is not invariant under complex energy shifts, which removes the arbitrariness in the definition of the topological invariant. Second, topologically non-trivial Hamiltonians are only a strict subset of those with a point gap; this implies that the NH skin effect does not have a topological origin. Third, topological phase transitions are accompanied by the closure of a real-valued gap, defined in terms of the singular values. I will then show how to reinstate the bulk-boundary correspondence in terms of the singular value decomposition, instead of the eigen-decomposition, and explain its physical significance. The NH bulk-boundary correspondence takes the following simple form: An integer value $\nu\in \mathbb{Z}$ of the winding number defined on the complex spectrum of the system under periodic boundary conditions corresponds to $\vert \nu\vert$ exponentially small singular values, associated with singular vectors that are exponentially localized at the system edge under open boundary conditions and vice versa; the sign of $\nu$ determines at which edge the vectors localize. Non-trivial topology manifests as directional amplification with gain exponential in system size, which is the hallmark of NH topology [2]. I will explain the physical relevance of this peculiar behaviour. More details in: M. Brunelli, C. C. Wanjura, and A. Nunnenkamp, arXiv:2207.12427 (2022). References: [1] E. J. Bergholtz, J. C. Budich, and F. K. Kunst, Rev. Mod. Phys. 93, 015005 (2021). 
 [2] C. C. Wanjura, M. Brunelli, and A. Nunnenkamp, Nature Communications 11, 3149 (2020). 


Ultra-cold atoms in optical lattices have demonstrated utility for simulating various condensed matter phenomena as well as realizing paradigmatic models. However, conventional optical lattices for ultra-cold atoms rely on the AC Stark shift to produce a potential proportional to the local optical intensity. As a direct result, the lattice period cannot be smaller than half the optical wavelength $\lambda$. Recently, two techniques have emerged to create deeply sub-wavelength lattices; both can be understood in terms of “dressed states” created by coupling internal atomic states with one- or two- photon optical fields. Here we focus on a scheme, relying on sequentially coupling $N$ internal atomic states using two photon Raman transitions. This results in an adiabatic potential for each of the $N$ dressed states, displaced by $\lambda/2N$ from each other. We show that adding temporal modulation to the detuning from Raman resonance can couple the $s$ and $p$ bands of adjacent lattice sites belonging to different dressed states. In the tight-binding limit, this gives rise to a pair of coupled Rice-Mele (RM) chains with new regimes of topological charge pumping. The present study opens new possibilities in studying the topological properties of subwavelength optical lattices induced by periodic driving.

We show that there exists a disorder free localization transition in two dimensions in a generic model of a lattice gauge theory, namely the U(1) quantum link model, relevant to frustrated magnets. We study the nature of localization transition using a percolation model from which we show that in a certain regime of the Hamiltonian, the disorder free localization transition is a continuous transition whose universality class we determine by calculating exact critical exponents. We also calculate spectral features of such a localized system deep in the localized phase using a cluster expansion approach. We show that such a localized system has sharp peaks in spatially averaged high temperature spectral functions even in the infinite size limit, whose positions we exactly estimate analytically. Our results highlight unique features of disorder free localization which distinguish it from conventional many body localization in disordered systems as well as otherwise expected high temperature paramagnetic response in frustrated magnets.

We investigate the topological phases of two interacting super-conducting wires in one-dimension, and propose topological invariants directly measurable from ground state correlation-functions. These numbers remain powerful tools in the presence of couplings and interactions. We show with \emph{density matrix renormalization group} that the \emph{double critical Ising (DCI)} phase discovered in \cite{Herviou_2016} { is a fractional topological phase with gapless Majorana modes in the bulk and one-half topological invariant per wire. Using both numerics and quantum field theoretical methods we show that the effect of an inter-wire hopping amplitude $t_{\bot}$ on the phase diagram reveals that the \emph{DCI} phase is stable at length scales below $\sim 1/t_{\bot}$. For large inter-wire hopping instead we show the emergence of two integer topological phases hosting one edge mode per boundary, shared between both wires. At large interactions the two wires are described by Mott physics, with the $t_{\bot}$ hopping resulting in a paramagnetic order.

The $\alpha$-$\mathcal{T}_3$ lattice is a two-dimensional three-band crossing model. It has a hopping parameter α which, on tuning from 0 to 1, results in a continuous evolution of its low energy Dirac-Weyl Hamiltonian from pseudospin 1/2 to pseudospin-1. The system also has an $\alpha$-dependent Berry phase whose effect has been observed in various physical quantities. We have studied the Floquet states of $\alpha$-$\mathcal{T}_3$ lattice irradiated by intense circularly polarized radiation in both low and high frequency regimes. At low frequency driving, we obtain the quasienergy bands close to the Dirac points which exhibit a strong dependence on $\alpha$ unlike the static bands. The quasienergy bands have no valley degeneracy and electron-hole symmetry for $0<\alpha<1$. The analytical expressions of the quasienergy gaps show that they are related to the $\alpha$-dependent Berry phase, which is acquired by the quasiparticles on traversing closed loops around the Dirac points triggered by circularly polarized light. The tunability of Berry phase in this system makes it an ideal platform to probe the effect of geometric phase through generation of quasienergy bands. In high frequency regime, we obtain an effective static Hamiltonian within the Floquet formalism. We observe that the effective Hamiltonian does not preserve time-reversal symmetry and gives rise to Floquet topological insulator phases. The three-fold degeneracy at the Dirac point is lifted for all values of $\alpha$. Topological phase transitions occur at $\alpha=1/\sqrt{2}$ characterized by a change in Chern number of the valence (conduction) band from 1(-1) to 2(-2). The point of transition is independent of the polarization of light (except linear) within the high-frequency approximation. Hence, this system offers a Floquet topological insulator phase with higher Chern number than those of conventional two-band systems like graphene, HgTe/CdTe quantum wells, etc.

Periodically driven clean noninteracting systems are known to host several interesting topological phases. Particularly, for high frequency driving, they have been found to host the analogues of equilibrium topological phases, like the Haldane phase. However, upon lowering the driving frequencies these systems have been found to host anomalous phases with robust edge modes despite all Chern numbers being zero. Moreover, theoretical works have shown that adding disorder to such anomalous phases leads to quantized charge pumping through the edge modes even when all bulk states become localised. We investigate the fate of these phases in presence of electron-electron interactions of the Falicov-Kimball type.

We consider the effects of so-called Frustrated Boundary Conditions (FBC) on quantum spin chains, namely periodic BC with an odd number of sites. In absence of external fields, FBC allow for the direct determination of correlation functions that signal a spontaneous symmetry breaking, such as the spontaneous magnetization. When paired with anti-ferromagnetic interactions, FBC introduce geometrical frustration into the system and the ground state develops properties which differ from those present with other boundary conditions, thus brining striking, yet puzzling, evidence that certain boundary conditions can affect the bulk properties of a 1D system. We argue that FBC introduce long-range order in the system, similar to that enjoyed by SPT phases, and add a sizeble amount of complexity to the ground state. Our results prove that even the weakest form of geometrical frustration can deeply affect a system's properties and pave a way for a bottom-up approach to better understand the effects of frustration and their exploitations also for technological purposes.

We propose a three-step periodic drive protocol to engineer two-dimensional~(2D) Floquet quadrupole superconductors and three-dimensional~(3D) Floquet octupole superconductors hosting zero-dimensional Majorana corner modes~(MCMs), based on unconventional $d$-wave superconductivity. Remarkably, the driven system conceives four phases with only $0$ MCMs, no MCMs, only anomalous $\pi$ MCMs, and both regular $0$ and anomalous $\pi$ MCMs. To circumvent the subtle issue of characterizing $0$ and $\pi$ MCMs separately, we employ the periodized evolution operator to architect the dynamical invariants, namely quadrupole and octupole motion in 2D and 3D, respectively, that can distinguish different higher order topological phases unambiguously. Furthermore, we extend our study using the periodic harmonic drive and generalize the definitions of the dynamical quadrupolar moment for this drive.

The quantum Jarzynski equality and the Crooks relation are fundamental laws connecting equilibrium processes with nonequilibrium fluctuations. They are promising tools to benchmark quantum devices and to measure free energy differences. While they are well established theoretically and also experimental realizations for few-body systems already exist, their experimental verification in the quantum many-body regime has remained an outstanding challenge. Here, we present results for nonequilibrium protocols in systems with up to sixteen interacting degrees of freedom obtained on trapped ion and superconducting qubit quantum computers, which verify the quantum Jarzynski equality and the Crooks relation in the many-body regime. To achieve this, we overcome present-day limitations in the preparation of thermal ensembles and in the measurement of work distributions on noisy intermediate-scale quantum devices. We discuss the accuracy to which the Jarzynski equality holds on different quantum computing platforms subject to platform-specific errors. Our analysis reveals a novel dissipative nonequilibrium regime, where a fast unitary drive compensates for dissipation and restores the validity of Jarzynski's equality. Our insights provide a new way approach to analyze errors in many-body quantum simulators.

We propose a method to address the existence of topological edge modes in one-dimensional (1D) nonlinear lattices, by deforming the edge modes of linearized models into solutions of the fully nonlinear system. For large enough nonlinearites, the energy of the modied edge modes may eventually shift out of the gap, leading to their delocalisation in the bulk. We identify a class of nonlinearities satisfying a generalised chiral symmetry where this mechanism is forbidden, and the nonlinear edge states are protected by a topological order parameter. Different behaviours of the edge modes are then found and explained by the interplay between the nature of the nonlinarities and the topology of the linearized models.

Mobile charge carriers in high-$T_c$ superconductors, i.e. holes doped into a Mott-insulator, leave behind a string of displaced spins when they move through the antiferromagnetic spin background. Here we study the thermal deconfinement of holes in a many-body setting using classical $\mathbb{Z}_2$ lattice gauge theories as a strongly simplified model of such strings. The confined phase is characterized by localized hole pairs connected by (short) strings while deconfinement implies a global net of strings spanning over the entire lattice. We probe the deconfinement phase transition using classical Monte Carlo and percolation-inspired order parameters. In two dimensions, we show that for small hole doping, there is a thermal deconfinement phase transition. For large hole doping, we find strong indications that holes are always confined in the thermodynamic limit. The Hamiltonian in two dimensions is designed from scratch to be experimentally realistic in Rydberg atom array experiments. In three dimensions, in contrast to the two dimensional case, a thermal deconfinement phase transition exists for arbitrary hole doping. We map out the phase diagram and calculate the critical exponents of the phase transition. Our results provide new insights into the physics of the deconfinement of holes and can be tested experimentally using Rydberg atom array experiments.

The non-Hermitian skin effect refers to the accumulation of an extensive number of eigenstates at the edges or corners of a system. Here, we introduce a new type of skin effect, which is generated by the accumulation of chiral edge modes in a two-dimensional unitary system. We show that removing a boundary hopping from the real-space time-evolution operator stops the propagation of edge modes in anomalous Floquet topological insulators. This is in contrast to the behavior of periodically-driven Chern insulators, in which boundary modes continue propagating. By evaluating the local density of states, we found that the resulting non-Hermitian skin effect is critical, i.e. scale-invariant, due to the nonzero coupling between the bulk and the edge. Further, it is a consequence of nontrivial topology, which we show by introducing a real-space topological invariant. The latter predicts the appearance of a skin effect in an anomalous Floquet topological insulator phase, and its absence for either trivial or Chern phases. Our work opens the possibility to realize topological, non-Hermitian skin effects via periodic driving, as well as a new direction to establish the bulk-boundary correspondence for non-Hermitian topological classifications.

A mobile quantum impurity resonantly coupled to a BEC forms a quasiparticle termed Bose polaron. Bose polaron is a generic concept central to describing many phenomena in numerous condensed matter systems. Ultracold atomic mixtures have emerged as a promising platform to study polaron physics with unprecedented accuracy and control. Especially, the strength of the impurity-boson interactions is tunable via Feshbach resonances, allowing access to interaction regimes beyond the capacity of conventional solid-state platforms. In particular, in the repulsive side of a Feshbach resonance, the impurity-boson potential admits a bound state, which significantly affects the properties of the interacting system in the many-body limit. At strong coupling when an impurity-boson dimer exists, the impurity can bind a diverging number of non-interacting bosons, resulting in an unbound ground state energy. In this case, interboson interactions are crucial to stabilize the Bose polaron, resulting in a strongly correlated state. To describe the strong coupling Bose polaron across unitarity we employ a Fock-Coherent-State (FCS) ansatz, which takes an arbitrary number of bound state occupations into account ($n=0,1,2,\cdots$) as well as includes a coherent state of excitations on top. In this manner, we extend the standard coherent state ansatz which has proven to be powerful in describing the dynamics and quasiparticle properties of Bose polarons. We study polaron dynamics at strong coupling, and we find that the multi-body resonances predicted by the coherent state ansatz dissolve when adding the interboson repulsive interaction, which was absent in the prior mean-field analysis of the Bose polaron.

The dynamical behaviour of strongly correlated many-body systems out of equilibrium is notoriously hard to describe both analytically and numerically. In recent years dual-unitary circuits have emerged as paradigmatic examples of chaotic many-body systems in which a variety of dynamical quantities can be computed exactly. However, in many respects dual-unitary circuits display behaviour that differs strikingly from the phenomenology observed numerically in more generic models. We investigate the behaviour of OTOCs in a broad class of perturbed dual-unitary circuits and show that even arbitrarily weak perturbations lead to the emergence of a diffusively broadening operator front at late times. This can be captured by a simple approximation in which the only characteristic of the gate entering is its operator entanglement.

Abstract: We investigate the dynamical effects of a magnetic flux quench in the Su-Schrieffer-Heeger model in a one-dimensional ring geometry. We show that even when the system is initially in the half-filled insulating state, the flux quench induces a time-dependent current that eventually reaches a finite stationary value. Such persistent current, which exists also in the thermodynamic limit, cannot be captured by the linear response theory and is the hallmark of nonlinear dynamical effects occurring in the presence of dimerization. Moreover, we show that for a range of values of dimerization strength and initial flux, the system exhibits dynamical quantum phase transitions, even though the quench is performed within the same topological class of the model. Reference: L. Rossi, and F. Dolcini, Phys. Rev. B 106, 045410 (2022)

Motivated by recent experimental progress, we consider a mixture of two species of cold atoms, non-interacting Fermions in a 2D optical lattice, embedded in a bath of weakly interacting bosons that are in a BEC. The 2D lattice is driven time-periodically such that the Floquet-Bands inhibit Floquet-Chern insulating behavior. We show that by careful engineering of the BEC bath, the steady state of the system can have an effective temperature that is low enough to observe quantized charge pumping, which is absent for generic ohmic bath environments. The results suggest a strategy for robust state-preparation in quantum simulators and might also generalize as a strategy to fight Floquet-heating in interacting systems.

Recent atomic physics experiments and numerical works have reported complementary signatures of the emergence of a topological quantum spin liquid (QSL) in arrays of Rydberg atoms with blockade interactions. To elucidate the origin of this QSL phase, we introduce an exact relation between an Ising-Higgs lattice gauge theory on the kagome lattice and the Rydberg blockaded models on the Ruby lattice. Based on this relation, we argue that the observed QSL phase is linked to the deconfined phase of an analytically solvable lattice gauge theory. We show that our model hosts the QSL phase in a broad region of the parameter space, thus opening up further possibilities to realize QSLs in different experimental settings with Rydberg dressing techniques.

Isaac Tesfaye, Botao Wang, André Eckardt Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany Mimicking fermionic Chern insulators with bosons has drawn a lot of interest in experiments by using, for example, cold atoms [1,2] or photons [3]. Here we present a scheme to prepare and probe a bosonic Chern insulator analog by using (a) an ensemble of randomized bosonic states and (b) an initial Mott state configuration. By applying a staggered superlattice, we can identify the lowest band with individual lattice sites. The delocalization over this band in quasimomentum space is then achieved by introducing on-site disorder or local random phases (a). Switching off the interactions and adiabatically decreasing the superlattice then gives rise to a bosonic Chern insulator, whose topologically non-trivial property is further confirmed from the Laughlin-type quantized charge pumping. Adding to this, we propose a detection scheme allowing for the observation of the bosonic quantized charge pump using a feasible number of experimental snapshots. Our protocol provides a useful tool to realize and probe topological states of matter in quantum gases or photonic systems. [1] Aidelsburger, Monika, et al. "Measuring the Chern number of Hofstadter bands with ultracold bosonic atoms." Nature Physics 11.2 (2015): 162-166. [2] Cooper, N. R., J. Dalibard, and I. B. Spielman. "Topological bands for ultracold atoms." Reviews of modern physics 91.1 (2019): 015005. [3] Ozawa,Tomoki, et. al. “Topological photonics.” Rev. of Mod. Phys. 91.1 (2019): 015006

Quantum spin chains, readily implemented in current state-of-the-art quantum simulators, have been shown to be a versatile platform to simulate lattice gauge theories, featuring rich physical phenomena ranging from string breaking to meson collisions. Yet, many questions concerning such phenomena still remain open. In particular, previous microscopic studies suggest a dichotomy for the fate of the confining string: its fission can occur relatively fast or be dramatically delayed. Here we aim to provide a unified account of the aforementioned scenarios in terms of an underlying dynamical phase transition. As a first step, we map the problem of string breaking for a short string to an impurity diffusion problem in Fock space. This effective description captures accurately the decay of the string in the regime of weak transverse fields. Next, we generalize such a description to a spin-boson model to approximate the breaking of a long string, which is effectively represented as a few level impurity immersed in a weakly interacting meson bath. We find that, within the considered limit, there is a localization-delocalization transition that separates a phase with a long-lived (prethermal) string from a fast string-breaking phase. This transition is identified through the scaling of the inverse participation ratio of the impurity modes, captured by universal scaling exponents and functions. Our description thus sheds light on possible universal aspects of nonequilibrium string breaking dynamics on the lattice and connects this phenomenon to the physics of quantum impurity models.

The interplay between dissipation, topology, and sensitivity to boundary conditions has recently attracted tremendous amounts of attention at the level of effective non-Hermitian descriptions. Here we exactly solve a quantum mechanical Lindblad master equation describing a dissipative topological Su-Schrieffer-Heeger (SSH) chain of fermions for both open boundary condition (OBC) and periodic boundary condition (PBC). We find that the extreme sensitivity on the boundary conditions associated with the non-Hermitian skin effect is directly reflected in the rapidities governing the time evolution of the density matrix giving rise to a Liouvillian skin effect. This leads to several intriguing phenomena including boundary sensitive damping behavior, steady state currents in finite periodic systems, and diverging relaxation times in the limit of large systems. We illuminate how the role of topology in these systems differs in the effective non-Hermitian Hamiltonian limit and the full master equation framework. Reference: F. Yang, Q.-D. Jiang, and E. J. Bergholtz, Phys. Rev. Research 4, 023160 (2022)