Geometry and non-adiabatic responses in non-equilibrium systems

Recent experiments suggest the phenomenon of light induced superconductivity above Tc in several materials, including high Tc cuprate YBCO and fullerene superconductor K3C60. This talk will review the distinct phenomena taking place in these systems. We will begin by discussing recent experiments in the pseudogap phase of YBCO that have been interpreted as the light induced Meissner effect. We will consider an explanation of this phenomenon from the perspective of nonlinear dynamics of the sine-Gordon model triggered by the strong terahertz pump pulse. We will review other features associated with photoinduced superconductivity in YBCO, such as photo-induced edges in reflectivity. We will show how to understand them from the perspective of a Floquet material, in which the pump-induced oscillations of a collective mode lead to the parametric generation of excitation pairs. We will discuss the photoinduced superconductivity in K3C60 that persists over time exceeding the lifetime of collective excitations. In this material, the unusual character of electron-phonon interactions results in enhanced BCS pairing through optical driving and the slow relaxation of superconducting correlations after they have been created.

Advancements in new-generation quantum materials have unveiled a plethora of emergent phases, from unconventional superconductors to Fractional Chern Insulators. Beyond their unique bandstructures, such materials, represented by twisted transition metal dichalcogenides and twisted multilayer graphene, exhibit nearly dispersionless quantum states ("flat bands") with distinctive quantum geometrical properties. A new paradigm, centered on the quantum geometry of flat bands, is gaining momentum in understanding the nature of these phases. Quantum geometry, which quantifies the proximity between adjacent quantum states in Hilbert space, has been instrumental in quantum information science, but had been largely overlooked in solid-state experiments. In this seminar, we will explore the quantum transport formalism through the lens of quantum geometry and demonstrate the construction of novel observables. As a practical application, we propose an innovative experimental approach to measure the ultra-narrow topological band gap using quantum noise measurements. Time permitting, we discuss importance of quantum geometry for developing new-generation Fractional Chern insulators, and the challenge of constructing Fractional Chern insulators supporting non-abelian anyons.

We show that lasing in flat band lattices can be stabilized by means of the geometrical properties of the Bloch states, in settings where the single-particle dispersion is flat in both its real and imaginary parts. We illustrate a general projection method and compute the collective excitations, which display a diffusive behavior ruled by quantum geometry through a peculiar coefficient involving gain, losses and interactions, and entailing resilience against modulational instabilities. Then, we derive an equation of motion for the phase dynamics and identify a Kardar-Parisi-Zhang term of geometric origin. This term is shown to exactly cancel whenever the real and imaginary parts of the laser nonlinearity are proportional to each other, or when the uniform-pairing condition is satisfied. We confirm our results through numerical studies of the $\pi$-flux diamond chain. This work highlights the key role of Bloch geometric effects in nonlinear dissipative systems and KPZ physics, with direct implications for the design of laser arrays with enhanced coherence.

The past few years have seen a revived interest in quantum geometrical characterizations of band structures due to the rapid development of topological insulators and semi-metals. Although the metric tensor has been connected to many geometrical concepts for single bands, the exploration of these concepts to a multi-band paradigm still promises a new field of interest. Formally, multi-band systems, featuring in particular degeneracies, have been related to projective spaces, explaining also the success of relating quantum geometrical aspects of flat band systems, albeit usually in the single band picture. Here, we propose a different route involving Pl\"ucker embeddings to represent arbitrary classifying spaces, being the essential objects that encode $all$ the relevant topology.This paradigm allows for the quantification of geometrical quantities directly in readily manageable vector spaces that a priori do not involve projectors or the need of flat band conditions. As a result, our findings are shown to pave the way for identifying new geometrical objects and defining metrics in arbitrary multi-band systems, especially beyond the single flatband limit, promising a versatile tool that can be applied in contexts that range from response theories to finding quantum volumes and bounds on superfluid densities as well as possible quantum computations."

The Berry curvature can be understood as a consequence of the geometry of quantum states. It materializes, among other experimental consequences related to transport and topology, as an anomalous velocity of wave packets. In non-Hermitian systems, these wave packet dynamics are enriched by additional terms that can be expressed by generalizations of the Berry connection to non-orthogonal eigenstates. Here, we contextualize these anomalous non-Hermitian contributions by showing that they directly arise from the geometry of the underlying quantum states as a higher-order correction to the distance between left and perturbed right eigenstates. By calculating the electric susceptibility for a single-band wave packet and comparing it with the wave packet’s localization, we demonstrate that these terms can, in some circumstances, lead to a violation of fluctuation-dissipation relations in non-Hermitian systems.

A remarkable phenomenon associated with Hermitian topology is the quantum Hall effect – the quantisation of the Hall resistance in terms of a topological invariant. So far, such a clear observable signature of non-Hermitian topology had been lacking. In this talk, I will show that non-trivial, non-Hermitian topology is in one-to-one correspondence with the phenomenon of directional amplification [1-3] in one-dimensional bosonic systems, e.g., cavity arrays. Directional amplification allows to selectively amplify signals depending on their propagation direction and has attracted much attention as key resource for applications, such as quantum information processing. Remarkably, in non-trivial topological phases, the end-to-end gain grows exponentially with the number of sites [1]. Furthermore, we show this effect to be robust against disorder [2] with the amount of tolerated disorder given by the separation between the complex spectrum and the origin. Beyond that, it is possible to restore the bulk-boundary correspondence with the help of the singular value decomposition which has a clear link to directional amplification [3]. In collaboration with the group of Ewold Verhagen at AMOLF, Amsterdam, we experimentally demonstrated the connection between non-Hermitian topology and directional amplification in a cavity optomechanical system [4] by realising a bosonic version of the Kitaev-Majorana chain proposed in [5] which relies on a different notion of non-reciprocity [6]. Furthermore, we show in the experiment that a similar system proposed in [7] can be utilised as a sensor with a sensitivity that grows exponentially with system size [4]. Our work opens up new routes for the design of both phase-preserving and phase-sensitive multimode robust directional amplifiers and sensors based on non-Hermitian topology that can be integrated in scalable platforms such as superconducting circuits, optomechanical systems and nanocavity arrays. [1] Wanjura, Brunelli, Nunnenkamp. Nat Commun 11, 3149 (2020). [2] Wanjura, Brunelli, Nunnenkamp. Phys. Rev. Lett. 127, 213601 (2021)
 [3] Brunelli, Wanjura, Nunnenkamp. SciPost Phys 15, 173 (2023). 
[4] Slim, Wanjura, Brunelli, del Pino, Nunnenkamp, Verhagen. arXiv:2309.05825 (2023); Nature, in press.
[5] McDonald, Pereg-Barnea, Clerk. Phys Rev X 8, 041031 (2018). [6] Wanjura, Slim, del Pino, Brunelli, Verhagen, Nunnenkamp. Nat Phys 19, 1429–1436 (2023). 
[7] McDonald, Clerk. Nat Commun 11, 5382 (2020).

This talk aims to provide an introduction to the notion of Generalized Landau Levels (GLLs) from a geometric point of view. GLLs are Bloch bands that generalize the standard notion of Landau levels of a charged particle in a uniform magnetic field. Beginning with a foundational introduction to quantum geometry---the differential geometry of families of quantum states---we delve into the specific case of Bloch bands, unraveling the inequalities that emerge relating the quantum metric and the Berry curvature, the saturation of which implies holomorphicity and gives rise to the concept of Kähler band. A Kähler band can then be understood as a regular holomorphic curve in complex projective space. The geometry of holomorphic curves shares many properties with that of real curves in Euclidean space. In particular, there is a distinguished moving frame along the curve, the Frenet-Serret frame (unique up to a global phase), whose elements are the GLLs. The frame satisfies the so-called Frenet-Serret equations which, together with the Maurer-Cartan structure equation, allow us not only to derive the quantum geometry of each GLL but also to establish geometric recursion relations among them. The content of these recursion relations is a manifestation of Calabi's rigidity theorem for Kähler immersions into projective space that, in this language, not only establishes the uniqueness, up to a momentum-independent unitary transformation, of a Kähler band with a given Berry curvature profile, but also completely determines the quantum geometry of the GLLs. As a natural consequence, the quantum volume of the quantum metric of the $n$th GLL is exactly quantized to $2n+1$. The discussion finds direct applications to moiré materials, where the $0$th GLL, the Kähler band, and the $1$st GLL are bands which can stabilize fractional Abelian and non-Abelian, respectively, fractional Chern insulating phases.

Quantum geometry is a fundamental concept to quantum states. Recent works pointed out saturating certain quantum geometric bounds allows for a topological Chern band to share many essential features with the lowest Landau level. In this talk, we discuss the generalization of this line to arbitrary higher Landau levels. We derive geometric properties of individual and multiple generalized Landau levels from Landau level analogs. Moreover, we use generalized Landau levels to construct a toy model which captures a large portion of the single-particle Hilbert space of a generic Chern band analogous to the first Landau level. Using this model, we employ large-scale exact diagonalization to identify a single-particle geometric criterion allowing for the non-Abelian Moore-Read phase. We discuss implications of our findings to non-Abelian fractionalization in small angle twisted MoTe2 material.

Intermediate-scale quantum technologies provide new opportunities for scientific discovery, yet they also pose the challenge of identifying suitable problems that can take advantage of such devices in spite of their present-day limitations. In solid-state materials, fractional quantum Hall (FQH) phases continue to attract attention as hosts of emergent geometrical excitations analogous to gravitons, resulting from the non-perturbative interactions between the electrons. However, the direct observation of such excitations remains a challenge. Here, we identify a quasi-one-dimensional model that captures the geometric properties and graviton dynamics of FQH states. We then simulate geometric quench and the subsequent graviton dynamics on the IBM quantum computer using an optimally-compiled Trotter circuit with bespoke error mitigation. Moreover, we develop an efficient, optimal-control-based variational quantum algorithm that can efficiently simulate graviton dynamics in larger systems. Our results open a new avenue for studying the emergence of gravitons in a new class of tractable models on the existing quantum hardware.

We will present our recent results on how quantum geometry affects various physical observables. Focusing on flat band systems, we start by discussing the properties of weakly interacting Bose-Einstein condensates, using the multiband Bogoliubov theory. We highlight how the speed of sound and the excitation fraction are dictated by the underlying geometry, specifically through orbital-position-independent generalizations of the quantum metric, which only reduce to the Fubini-Study metric under special cases. Due to the limitations of the Bogoliubov theory in evaluating the superfluid weight, we employ exact diagonalization techniques to show that the superfluidity in one-dimensional Bose-Hubbard models is governed by a many-body version of quantum metric. We finally discuss various physical platforms where the effects of the quantum geometry can be observed, ranging from ultracold gases to non-Hermitian plasmonic lattices of metallic nanoparticles.

Synthetic dimensions provide a powerful approach for simulating condensed matter physics in cold atoms and photonics, whereby a set of discrete degrees of freedom are coupled together and re-interpreted as lattice sites along an extra artificial dimension. In this talk, I will review how to create a long and controllable synthetic dimension of atomic harmonic trap states by periodically modulating the trapping potential in time. I will discuss an experimental realisation in which we were able to observe Bloch oscillations along the synthetic dimension of harmonic trap states by controlling the detuning between the frequency of the driving potential and that of the harmonic trap. I will also present theoretical proposals for how to realise a 2D quantum Hall system by combining the synthetic dimension with a real spatial dimension and an artificial magnetic field, in order to investigate topological chiral edge states. These works provide an intuitive framework for the manipulation and control of highly-excited trap states, and set the stage for the future exploration of topological physics in this synthetic dimension.

I will start from a brief overview of defining chaos through complexity of adiabatic transformations and the low frequency asymptotical behavior of the spectral functions of physical observables. Then I will focus on dynamical response close to integrability, where the system is neither integrable nor ergodic.I will illustrate how this (KAM) regime can be characterized by universal slow glassy non-ergodic dynamics and that it always preceeds thermalizing/ETH regime. I will argue that the full transition from integrable to ergodic/ETH regimes can only be characterized by a two-parameter scaliing in time.

We propose a circuit architecture for a dissipatively error-corrected GKP qubit. The device consists of a high-impedance LC circuit coupled to a Josephson junction and a resistor via a controllable switch. When the switch is activated via a particular family of stepwise protocols, the resistor absorbs all noise-induced entropy, resulting in dissipative error correction of both phase and amplitude errors. This leads to an exponential increase of qubit lifetime, reaching beyond 10ms in simulations with near-feasible parameters. We show that the lifetime remains exponentially long in the presence of extrinsic noise and device/control imperfections (e.g., due to parasitics and finite control bandwidth) under specific thresholds. In this regime, lifetime is likely only limited by phase slips and quasiparticle tunneling. We show that the qubit can be read out and initialized via measurement of the supercurrent in the Josephson junction. We finally show that the qubit supports native self-correcting single-qubit Clifford gates, where dissipative error-correction of control noise leads to exponential suppression of gate infidelity.

In systems with a real Bloch Hamiltonian band nodes can be characterised by a non-Abelian frame-rotation charge. The ability of these band nodes to annihilate pairwise is path dependent, since by braiding nodes in adjacent gaps the sign of their charges can be changed. Here, we theoretically construct and numerically confirm two concrete methods to experimentally probe these non-Abelian braiding processes and charges in ultracold atomic systems. We consider a coherent superposition of two bands that can be created by moving atoms through the band singularities at some angle in momentum space. Analyzing the dependency on the frame charges and the geometry, we demonstrate an interferometry scheme passing through two band nodes, which reveals the relative frame charges and allows for measuring the multi-gap topological invariant. The second method relies on a single wavepacket probing two nodes sequentially, where the frame charges can be determined from the band populations. Our results present a feasible avenue for measuring non-Abelian charges of band nodes and the experimental verification of braiding procedures directly, which can be applied in a variety of settings including the recently discovered anomalous non-Abelian phases arising under periodic driving.

Floquet engineering of bandstructures via application of coherent time-periodic drives is emerging as a powerful tool for creating new non-equilibrium phases of matter. We show how this approach can be used to induce non-equilibrium correlated states with dynamical spontaneously broken symmetry in two dimensional materials. We propose a method, which we call “Modulated Floquet Parametric Driving” (MFPD), to parametrically excite low frequency collective modes in an interacting many-body system. The method utilizes a Floquet drive at optical frequencies with a modulated amplitude. We demonstrate that it can be used to design plasmonic time-varying media with singular dispersions. Plasmons near resonance with half the modulation frequency exhibit two lines of exceptional points connected by dispersionless states. Above a critical driving strength, resonant plasmon modes become unstable and undergo a continuous transition towards a crystal-like structure stabilized by interactions and nonlinearities. This new state breaks the discrete time translational symmetry of the drive as well as the translational and rotational spatial symmetries of the system and exhibits soft, Goldstone-like phononic excitations. We discuss possible applications of MFPD to other low-energy collective modes.

We show that heat production in slowly driven quantum systems is linked to the topological structure of the driving protocol through the Fubini-Study tensor. Analyzing a minimal model of a spin weakly coupled to a heat bath, we find that dissipation is controlled by the quantum metric and a "quality factor" characterizing the spin's precession. Utilizing these findings, we establish lower bounds on the heating rate in two-tone protocols, such as those employed in topological frequency converters. Notably, these bounds are determined by the topology of the protocol, independent of its microscopic details. Our results bridge topological phenomena and energy dissipation in slowly driven quantum systems, providing a design principle for optimal driving protocols.

In synthetic quantum systems, such as ultracold atomic quantum gases in optical lattices, artificial magnetic fields can be realized effectively by driving the system. Interestingly, the experimentalist controls the corresponding gauge potential (e.g. in the form of Peirls phases) rather than the magnetic flux, so that a time-dependent variation gives rise not only to artificial magnetic, but also to artificial electric fields. In the first half of my presentation, I will argue that the latter can be used to control the system: It can determine the path along which charge is pumped in 2D Chern insulators [1] and it can allow for ultrafast state preparation in bosonic flux ladders by choosing a ramp of the gauge potential that minimizes the overlap between final and initial state [2]. In the second part of my presentation, I will talk about quantum geometry of bosonic Bogoliubov-de Gennes (BdG) systems. Here topological and geometrical features have mainly been studied by utilizing a generalized symplectic version of the Berry curvature and Chern number. In addition, we propose a generalized (symplectic) quantum geometric tensor (SQGT) for these systems, whose imaginary part leads to the previously studied symplectic Berry curvature, while the real part gives rise to a symplectic quantum metric, providing a natural distance measure in the space of bosonic Bogoliubov modes [3]. I will also show that all components of the SQGT can be extracted from excitation rates in response to periodic modulations. [1] B. Wang, F.N. Ünal, A. Eckardt, Phys. Rev. Lett. 120, 243602 (2018) [2] B. Wang, X.-Y. Dong, F.N. Ünal, A. Eckardt, New J. Phys. 23, 063017 (2021) [3] I. Tesfaye and A. Eckardt, arXiv:2405.*****.

Periodically driven systems provide a novel route to control the topology of quantum materials. In particular, Floquet theory allows an effective band description of periodically-driven systems through the Floquet Hamiltonian. Here, we study the time evolution of d-wave superconductors irradiated with intense circularly-polarized laser light. We consider the Floquet t-J model with time-periodic interactions, and investigate its mean-field dynamics by formulating the time-dependent Gutzwiller approximation. We observe the development of the $id_{xy}$-wave pairing amplitude along with the original $d_{x^2-y^2}$-wave order upon gradual increasing of the field amplitude. We further numerically construct the Floquet Hamiltonian for the steady state, with which we identify the system as the fully-gapped d+id superconducting phase with a nonzero Chern number. We explore the low-frequency regime where the perturbative approaches in the previous studies break down, and find that the topological gap of an experimentally-accessible size can be achieved at much lower laser intensities. [1] T. Anan, T. Morimoto, S. Kitamura, arXiv:2309.06069 (to appear in Commun. Phys.)

Predicting the fate of an interacting system in the limit where the electronic bandwidth is quenched is often highly non-trivial. The complex interplay between interactions and quantum fluctuations driven by the band geometry can drive a competition between various ground states, such as charge density wave order and superconductivity. In this work, we study an electronic model of topologically-trivial flat bands with a continuously tunable Fubini-Study metric in the presence of on-site attraction and nearest-neighbor repulsion, using numerically exact quantum Monte Carlo simulations. By varying the electron filling and the spatial extent of the localized flat-band Wannier wavefunctions, we obtain a number of intertwined orders. These include a phase with coexisting charge density wave order and superconductivity, i.e., a supersolid. In spite of the non-perturbative nature of the problem, we identify an analytically tractable limit associated with a `small' spatial extent of the Wannier functions, and derive a low-energy effective Hamiltonian that can well describe our numerical results.

Shortcuts to adiabaticity provide fast protocols for quantum state preparation in which the use of auxiliary counter-diabatic controls circumvents the requirement of slow driving in adiabatic strategies. While their development is well established in simple systems, their engineering and implementation are challenging in many-body quantum systems with many degrees of freedom. We show that the equation for the counterdiabatic term, equivalently the adiabatic gauge potential, is solved by introducing a Krylov basis. The Krylov basis spans the minimal operator subspace in which the dynamics unfolds and provides an efficient way to construct the counterdiabatic term. We apply our strategy to paradigmatic single and many-particle models. The properties of the counterdiabatic term are reflected in the Lanczos coefficients obtained in the course of the construction of the Krylov basis by an algorithmic method. We examine how the expansion in the Krylov basis incorporates many-body interactions in the counterdiabatic term.

Over the last decade, periodic (Floquet) drives have emerged as a useful technique to engineer the properties of quantum systems. Additionally, periodically driven systems can manifest nonequilibrium phases of matter without static counterparts, such as discrete time crystals or anomalous topological insulators. The ability to realize effective Hamiltonians with properties drastically different from those of the non-driven system, makes Floquet engineering a vital ingredient in present-day quantum simulators. However, the manipulation of the states dressed by strong periodic drives remains an outstanding challenge in Floquet engineering. The state-of-the-art in Floquet control is the adiabatic change of parameters. Yet, this requires long protocols conflicting with the limited coherence times in experiments. This talk introduces Floquet Counterdiabatic Driving, a generalization of variational counterdiabatic driving to Floquet systems, that enables transitionless driving of Floquet eigenstates far away from the adiabatic regime. In particular, we present a nonperturbative variational principle to find local approximations to the Floquet adiabatic gauge potential. The Floquet adiabatic gauge potential is also closely related to the response functions of the Floquet system.

Decoherence and imperfect control are crucial challenges for quantum technologies. Common protection strategies rely on noise temporal autocorrelation, which is not optimal if other correlations are present. We develop and demonstrate experimentally a strategy that utilizes the cross-correlation of two noise sources. We achieve a tenfold coherence time extension by destructive interference of cross-correlated noise, improve control fidelity, and surpass the state-of-the-art sensitivity for high frequency quantum sensing, significantly expanding the applicability of noise protection strategies.

Open quantum systems described by a non-Hermitian Hamiltonian exhibit intriguing dynamics due to the topology of their complex energy spectrum. In the vicinity of an exceptional point degeneracy, this topology allows for topological state transport, chiral geometric phases, and eigenvalue braiding. To access these topological features, it is desirable to drive the system adiabatically. With a complex energy spectrum, different eigenstates of the Hamiltonain will have different imaginary parts of their eigenenergy; adiabatic transport conventionally only possible for the eigenstate with lowest loss. Previous work demonstrated such adiabatic evolution for the quantum state with relative gain, yet observed a breakdown in adiabaticity for quantum states with relative loss. I will discuss how we harness the notion of shortcuts to adiabaticity to drive the system faster, avoiding the effects of loss while maintaining trajectories that follow the instantaneous eigenstates. We experimentally investigate the robustness of this control method using a superconducting transmon circuit with engineered dissipation around the second order exceptional point of its effective non-Hermitian Hamiltonian. This method can be extended to explore the more complicated energy structures of higher-order exceptional points.