Publisher Correction: Classical analogue to driven quantum bits based on macroscopic pendula (Scientific Reports, (2023), 13, 1, (18386), 10.1038/s41598-023-45118-y)

Lorenz H., Kohler S., Parafilo A., Correction to: Scientific Reports, published online 26 October 2023 The original version of this Article contained errors in Figure 2 where the gray data curves were incorrectly captured in panels (a) and (b). The original Figure 2 and accompanying legend appear below. (Figure presented.) Near resonance Rabi oscillations between the two pendula with mean frequency (Formula presented.) mHz, frequency difference (Formula presented.) mHz and modulation frequency (Formula presented.) mHz. At (Formula presented.) pendulum 1 was deflected at maximally attracting lower and no upper magnets. Individual oscillations are not visible owing to the time axis covering 45 minutes. (a, b) Deflections (Formula presented.) and (Formula presented.) of the two pendula for the pivot distances (Formula presented.) mm and (Formula presented.) mm resulting in Rabi frequencies of (Formula presented.) mHz versus (Formula presented.) mHz. (c, d) Effective frequency (Formula presented.) and visibility (Formula presented.) of the Rabi oscillations for (Formula presented.) mm. The solid lines represent model predictions. The original Article has been corrected.

Classical analogue to driven quantum bits based on macroscopic pendula

Lorenz H., Kohler S., Parafilo A., Quantum mechanics increasingly penetrates modern technologies but, due to its non-deterministic nature seemingly contradicting our classical everyday world, our comprehension often stays elusive. Arguing along the correspondence principle, classical mechanics is often seen as a theory for large systems where quantum coherence is completely averaged out. Surprisingly, it is still possible to reconstruct the coherent dynamics of a quantum bit (qubit) by using a classical model system. This classical-to-quantum analogue is based on wave mechanics, which applies to both, the classical and the quantum world. In this spirit we investigate the dynamics of macroscopic physical pendula with a modulated coupling. As a proof of principle, we demonstrate full control of our one-to-one analogue to a qubit by realizing Rabi oscillations, Landau-Zener transitions and Landau-Zener-Stückelberg-Majorana interferometry. Our classical qubit demonstrator can help comprehending and developing useful quantum technologies.

Experimental signature of initial quantum coherence on entropy production

Hernández-Gómez S., Gherardini S., Belenchia A., We report on the experimental quantification of the contribution to non-equilibrium entropy production stemming from the quantum coherence content in the initial state of a qubit exposed to both coherent driving and dissipation. Our experimental demonstration builds on the exquisite experimental control of the spin state of a nitrogen-vacancy defect in diamond and is underpinned, theoretically, by the formulation of a generalized fluctuation theorem designed to track the effects of quantum coherence. Our results provide significant evidence of the possibility to pinpoint the genuinely quantum mechanical contributions to the thermodynamics of non-equilibrium quantum processes in an open quantum systems scenario.

Entanglement asymmetry as a probe of symmetry breaking

Ares F., Murciano S., Symmetry and symmetry breaking are two pillars of modern quantum physics. Still, quantifying how much a symmetry is broken is an issue that has received little attention. In extended quantum systems, this problem is intrinsically bound to the subsystem of interest. Hence, in this work, we borrow methods from the theory of entanglement in many-body quantum systems to introduce a subsystem measure of symmetry breaking that we dub entanglement asymmetry. As a prototypical illustration, we study the entanglement asymmetry in a quantum quench of a spin chain in which an initially broken global U(1) symmetry is restored dynamically. We adapt the quasiparticle picture for entanglement evolution to the analytic determination of the entanglement asymmetry. We find, expectedly, that larger is the subsystem, slower is the restoration, but also the counterintuitive result that more the symmetry is initially broken, faster it is restored, a sort of quantum Mpemba effect, a phenomenon that we show to occur in a large variety of systems.

Collective Excitations of a Strongly Correlated Nonequilibrium Photon Fluid across the Insulator-Superfluid Phase Transition

Caleffi F., We develop a Gutzwiller theory for the nonequilibrium steady states of a strongly interacting photon fluid driven by a non-Markovian incoherent pump. In particular, we explore the collective modes of the system across the out-of-equilibrium insulator-superfluid transition of the system, characterizing the diffusive Goldstone mode in the superfluid phase and the excitation of particles and holes in the insulating one. Observable features in the pump-and-probe optical response of the system are highlighted. Our predictions are experimentally accessible to state-of-the-art circuit-QED devices and open the way for the study of novel driven-dissipative many-body scenarios with no counterparts at equilibrium.

Nonstabilizerness via Perfect Pauli Sampling of Matrix Product States

Lami G., We introduce a novel approach to evaluate the nonstabilizerness of an N-qubits matrix product state (MPS) with bond dimension χ. In particular, we consider the recently introduced stabilizer Rényi entropies (SREs). We show that the exponentially hard evaluation of the SREs can be achieved by means of a simple perfect sampling of the many-body wave function over the Pauli string configurations. The sampling is achieved with a novel MPS technique, which enables us to compute each sample in an efficient way with a computational cost O(Nχ3). We benchmark our method over randomly generated magic states, as well as in the ground-state of the quantum Ising chain. Exploiting the extremely favorable scaling, we easily have access to the nonequilibrium dynamics of the SREs after a quantum quench.

Multipartite entanglement in the measurement-induced phase transition of the quantum Ising chain

Paviglianiti A., External monitoring of quantum many-body systems can give rise to a measurement-induced phase transition characterized by a change in behavior of the entanglement entropy from an area law to an unbounded growth. In this paper, we show that this transition extends beyond bipartite correlations to multipartite entanglement. Using the quantum Fisher information, we investigate the entanglement dynamics of a continuously monitored quantum Ising chain. Multipartite entanglement exhibits the same phase boundaries observed for the entropy in the postselected no-click trajectory. Instead, quantum jumps give rise to a more complex behavior that still features the transition, but adds the possibility of having a third phase with logarithmic entropy but bounded multipartiteness.

Mott-enhanced exciton condensation in a Hubbard bilayer

Giuli S., Amaricci A., We study the conditions to realize an excitonic condensed phase in an electron-hole bilayer system with local Hubbard-like interactions at half-filling, where we can address the interplay with Mott localization. Using dynamical mean-field theory, we find that an excitonic state is stable in a sizable region of a phase diagram spanned by the intralayer (U) and interlayer (V) interactions. The latter term is expected to favor the excitonic phase which is indeed found in a slice of the phase diagram with V>U. Remarkably, we find that, when U is large enough, the excitonic region extends also for U>V, in contrast with naïve expectations. The extended stability of the excitonic phase can be linked to in-layer Mott localization and interlayer spin correlations. Using a mapping to a model with attractive interlayer coupling, we fully characterize the condensate phase in terms of its superconducting counterpart, thereby addressing its coherence and correlation length.

Transport and Entanglement across Integrable Impurities from Generalized Hydrodynamics

Rylands C., Quantum impurity models (QIMs) are ubiquitous throughout physics. As simplified toy models they provide crucial insights for understanding more complicated strongly correlated systems, while in their own right are accurate descriptions of many experimental platforms. In equilibrium, their physics is well understood and have proven a testing ground for many powerful theoretical tools, both numerical and analytical, in use today. Their nonequilibrium physics is much less studied and understood. However, the recent advancements in nonequilibrium integrable quantum systems through the development of generalized hydrodynamics (GHD) coupled with the fact that many archetypal QIMs are in fact integrable presents an enticing opportunity to enhance our understanding of these systems. We take a step towards this by expanding the framework of GHD to incorporate integrable interacting QIMs. We present a set of Bethe-Boltzmann type equations which incorporate the effects of impurity scattering and discuss the new aspects which include entropy production. These impurity GHD equations are then used to study a bipartioning quench wherein a relevant backscattering impurity is included at the location of the bipartition. The density and current profiles are studied as a function of the impurity strength and expressions for the entanglement entropy and full counting statistics are derived.

Nonequilibrium Full Counting Statistics and Symmetry-Resolved Entanglement from Space-Time Duality

Bertini B., Owing to its probabilistic nature, a measurement process in quantum mechanics produces a distribution of possible outcomes. This distribution - or its Fourier transform known as full counting statistics (FCS) - contains much more information than say the mean value of the measured observable, and accessing it is sometimes the only way to obtain relevant information about the system. In fact, the FCS is the limit of an even more general family of observables - the charged moments - that characterize how quantum entanglement is split in different symmetry sectors in the presence of a global symmetry. Here we consider the evolution of the FCS and of the charged moments of a U(1) charge truncated to a finite region after a global quantum quench. For large scales these quantities take a simple large-deviation form, showing two different regimes as functions of time: while for times much larger than the size of the region they approach a stationary value set by the local equilibrium state, for times shorter than region size they show a nontrivial dependence on time. We show that, whenever the initial state is also U(1) symmetric, the leading order in time of FCS and charged moments in the out-of-equilibrium regime can be determined by means of a space-time duality. Namely, it coincides with the stationary value in the system where the roles of time and space are exchanged. We use this observation to find some general properties of FCS and charged moments out of equilibrium, and to derive an exact expression for these quantities in interacting integrable models. We test this expression against exact results in the Rule 54 quantum cellular automaton and exact numerics in the XXZ spin-1/2 chain.

Non-Gaussian fluctuations of a probe coupled to a Gaussian field

Démery V., The motion of a colloidal probe in a complex fluid, such as a micellar solution, is usually described by the generalized Langevin equation, which is linear. However, recent numerical simulations and experiments have shown that this linear model fails when the probe is confined and that the intrinsic dynamics of the probe is actually nonlinear. Noting that the kurtosis of the displacement of the probe may reveal the nonlinearity of its dynamics also in the absence confinement, we compute it for a probe coupled to a Gaussian field and possibly trapped by a harmonic potential. We show that the excess kurtosis increases from zero at short times, reaches a maximum, and then decays algebraically at long times, with an exponent which depends on the spatial dimensionality and on the features and correlations of the dynamics of the field. Our analytical predictions are confirmed by numerical simulations of the stochastic dynamics of the probe and the field where the latter is represented by a finite number of modes.

Many-Body Magic Via Pauli-Markov Chains - From Criticality to Gauge Theories

Tarabunga P.S., Tirrito E., Chanda T., We introduce a method to measure many-body magic in quantum systems based on a statistical exploration of Pauli strings via Markov chains. We demonstrate that sampling such Pauli-Markov chains gives ample flexibility in terms of partitions where to sample from: in particular, it enables the efficient extraction of the magic contained in the correlations between widely separated subsystems, which characterizes the nonlocality of magic. Our method can be implemented in a variety of situations. We describe an efficient sampling procedure using tree tensor networks, that exploit their hierarchical structure leading to a modest O(log N) computational scaling with system size. To showcase the applicability and efficiency of our method, we demonstrate the importance of magic in many-body systems via the following discoveries: (a) for one-dimensional systems, we show that long-range magic displays strong signatures of conformal quantum criticality (Ising, Potts, and Gaussian), overcoming the limitations of full state magic; (b) in two-dimensional Z2 lattice gauge theories, we provide conclusive evidence that magic is able to identify the confinement-deconfinement transition, and displays critical scaling behavior even at relatively modest volumes. Finally, we discuss an experimental implementation of the method, which relies only on measurements of Pauli observables.

Sequences of resource monotones from modular Hamiltonian polynomials

Arias R., De Boer J., Di Giulio G., Keski-Vakkuri E., We introduce two infinite sequences of entanglement monotones, which are constructed from expectation values of polynomials in the modular Hamiltonian. These monotones yield infinite sequences of inequalities that must be satisfied in majorizing state transitions. We demonstrate this for information erasure, deriving an infinite sequence of "Landauer inequalities"for the work cost, bounded by linear combinations of expectation values of powers of the modular Hamiltonian. These inequalities give improved lower bounds for the work cost in finite-dimensional systems, and depend on more details of the erased state than just on its entropy and variance of modular Hamiltonian. Similarly one can derive lower bounds for marginal entropy production for a system coupled to an environment. These infinite sequences of entanglement monotones also give rise to relative quantifiers that are monotonic in more general processes, namely those involving so-called σ majorization with respect to a fixed point full rank state σ; such quantifiers are called resource monotones. As an application to thermodynamics, one can use them to derive finite-dimension corrections to the Clausius inequality. Finally, in order to gain some intuition for what (if anything) plays the role of majorization in field theory, we compare pairs of states in discretized theories at criticality and study how majorization depends on the size of the bipartition with respect to the size of the entire chain.

Complexity of spin configuration dynamics due to unitary evolution and periodic projective measurements

Casagrande H.P., Xing B., We study the Hamiltonian dynamics of a many-body quantum system subjected to periodic projective measurements, which leads to probabilistic cellular automata dynamics. Given a sequence of measured values, we characterize their dynamics by performing a principal component analysis (PCA). The number of principal components required for an almost complete description of the system, which is a measure of complexity we refer to as PCA complexity, is studied as a function of the Hamiltonian parameters and measurement intervals. We consider different Hamiltonians that describe interacting, noninteracting, integrable, and nonintegrable systems, including random local Hamiltonians and translational invariant random local Hamiltonians. In all these scenarios, we find that the PCA complexity grows rapidly in time before approaching a plateau. The dynamics of the PCA complexity can vary quantitatively and qualitatively as a function of the Hamiltonian parameters and measurement protocol. Importantly, the dynamics of PCA complexity present behavior that is considerably less sensitive to the specific system parameters for models which lack simple local dynamics, as is often the case in nonintegrable models. In particular, we point out a figure of merit that considers the local dynamics and the measurement direction to predict the sensitivity of the PCA complexity dynamics to the system parameters.

Quantum integrability vs experiments: correlation functions and dynamical structure factors

Lencsés M., Integrable Quantum Field Theories can be solved exactly using bootstrap techniques based on their elastic and factorisable S-matrix. While knowledge of the scattering amplitudes reveals the exact spectrum of particles and their on-shell dynamics, the expression of the matrix elements of the various operators allows the reconstruction of off-shell quantities such as two-point correlation functions with a high level of precision. In this review, we summarise results relevant to the contact point between theory and experiment providing a number of quantities that can be computed theoretically with great accuracy. We concentrate on universal amplitude ratios which can be determined from the measurement of generalised susceptibilities, and dynamical structure factors, which can be accessed experimentally e.g. via inelastic neutron scattering or nuclear magnetic resonance. Besides an overview of the subject and a summary of recent advances, we also present new results regarding generalised susceptibilities in the tricritical Ising universality class.

Unified role of Green's function poles and zeros in correlated topological insulators

Blason A., Green's function zeros, which can emerge only if correlation is strong, have been for long overlooked and believed to be devoid of any physical meaning, unlike Green's function poles. Here, we prove that Green's function zeros instead contribute on the same footing as poles to determine the topological character of an insulator. The key to the proof, worked out explicitly in two dimensions but easily extendable in three dimensions, is to express the topological invariant in terms of a quasiparticlethermal Green's function matrix G∗(iϵ,k)=1/[iϵ-H∗(ϵ,k)], with Hermitian H∗(ϵ,k), by filtering out the positive-definite quasiparticle residue. In that way, the topological invariant is easily found to reduce to the Thouless, Kohmoto, Nightingale, and den Nijs formula for quasiparticles described by the noninteracting Hamiltonian H∗(0,k). Since the poles of the quasiparticle Green's function G∗(ϵ,k) on the real frequency axis correspond to poles and zeros of the physical-particle Green's function G(ϵ,k), both of them equally determine the topological character of an insulator.

Thermodynamic uncertainty relations for systems with broken time reversal symmetry: The case of superconducting hybrid systems

Taddei F., We derive bounds to the thermodynamic uncertainty relations in the linear-response regime for steady-state transport in two-terminal systems when time reversal symmetry is broken. We find that such bounds are different for charge and heat currents and depend on the details of the system, through the Onsager coefficients, and on the ratio between applied voltage and temperature difference. As a function of such a ratio, the bounds can take any positive values. The bounds are then calculated for a hybrid coherent superconducting system using the scattering approach, and the concrete case of an Andreev interferometer is explored. Interestingly, we find that the bound on the charge current is always smaller than 2 when the system operates as a heat engine, while the bound on the heat current is always larger than 2 when the system operates as a refrigerator.

Deep learning nonlocal and scalable energy functionals for quantum Ising models

Costa E., Density functional theory (DFT) is routinely employed in material science and quantum chemistry to simulate weakly correlated electronic systems. Recently, deep learning (DL) techniques have been adopted to develop promising functionals for the strongly correlated regime. DFT can be applied to quantum spin models too, but functionals based on DL have not been developed yet. Here, we investigate DL-based DFTs for random quantum Ising chains, both with nearest-neighbor and up to next-nearest-neighbor couplings. Our neural functionals are trained and tested on data produced via the Jordan-Wigner transformation, exact diagonalization, and tensor-network methods. An economical gradient-descent optimization is used to find the ground-state properties of previously unseen Hamiltonian instances. Notably, our nonlocal functionals drastically improve upon the common local density approximations, and they are designed to be scalable, allowing us to simulate chain sizes much larger than those used for training. The prediction accuracy is analyzed, paying attention to the spatial correlations of the learned functionals and to the role of quantum criticality. Our findings indicate a suitable strategy to extend the reach of other computational methods with a controllable accuracy.

Quantum effects on the synchronization dynamics of the Kuramoto model

Delmonte A., Romito A., The Kuramoto model serves as a paradigm for describing spontaneous synchronization in a system of classical interacting rotors. In this paper, we extend this model to the quantum domain by coupling quantum interacting rotors to external baths following the Caldeira-Leggett approach. Studying the mean-field model in the overdamped limit using Feynman-Vernon theory, we show how quantum mechanics modifies the phase diagram. Specifically, we demonstrate that quantum fluctuations hinder the emergence of synchronization, albeit not entirely suppressing it. We examine the phase transition into the synchronized phase at various temperatures, revealing that classical results are recovered at high temperatures while a quantum phase transition occurs at zero temperature. Additionally, we derive an analytical expression for the critical coupling, highlighting its dependence on the model parameters, and examine the differences between classical and quantum behavior.

First-order photon condensation in magnetic cavities: A two-leg ladder model

Bacciconi Z., Andolina G.M., Chanda T., Chiriacò G., Schirò M., We consider a model of free fermions in a ladder geometry coupled to a non-uniform cavity mode via Peierls substitution. Since the cavity mode generates a magnetic field, no-go theorems on spontaneous photon condensation do not apply, and we indeed observe a phase transition to a photon condensed phase characterized by finite circulating currents, alternatively referred to as the equilibrium superradiant phase. We consider both square and triangular ladder geometries, and characterize the transition by studying the energy structure of the system, light-matter entanglement, the properties of the photon mode, and chiral currents. The transition is of first order and corresponds to a sudden change in the fermionic band structure as well as the number of its Fermi points. Thanks to the quasi-one dimensional geometry we scrutinize the accuracy of (mean field) cavity-matter decoupling against large scale density-matrix renormalization group simulations. We find that light-matter entanglement is essential for capturing corrections to matter properties at finite sizes and for the description of the correct photon state. The latter remains Gaussian in the the thermodynamic limit both in the normal and photon condensed phases.