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We consider a free fermionic chain with monitoring of the particle density on a single site of the chain and study the entanglement dynamics of quantum jump trajectories. We show that the entanglement entropy grows in time towards a stationary state which display volume law scaling of the entropy, in stark contrast with both the unitary dynamics after a local quench and the no-click limit corresponding to full post-selection. We explain the extensive entanglement growth as a consequence of the peculiar distribution of quantum jumps in time, which display superpoissonian waiting time distribution characterised by a bunching of quantum jumps followed by long dark intervals where no-clicks are detected, akin to the distribution of fluorescence light in a driven atom. We show that the presence of dark intervals is the key feature to explain the effect and that by increasing the number of sites which are monitored the volume law scaling gives away to the Zeno effect and its associated area law., Comment: 11 pages, 8 figures

These Lecture Notes discuss the recent theoretical advances in the understanding of open quantum many-body physics in platforms where both dissipative and coherent processes can be tuned and controlled to a high degree. We start by reviewing the theoretical frameworks and methods used to describe and tackle open quantum many-body systems. We then discuss the use of dissipative processes to engineer many-body stationary states with desired properties and the emergence of dissipative phase transitions arising out of the competition between coherent evolution and dissipation. We review the dynamics of open quantum many body systems in the presence of correlated many-body dissipative processes, such as heating and many-body losses. Finally we provide a different perspective on open quantum many-body systems by looking at stochastic quantum trajectories, relevant for the case in which the environment represents a monitoring device, and the associated measurement-induced phase transitions., Comment: Lecture Notes, 185 pages, 40 figure

Recently, there has been interest in the dynamics of monitored quantum systems using linear jump operators related to the creation or annihilation of particles. Here we study the dynamics of the entanglement entropy under quantum jumps that induce local particle losses in a model of free fermions with hopping and $\mathbb{Z}_2$ pairing. We explore the different steady-state entanglement regimes by interpolating between monitored free fermions with U(1) symmetry and $\mathbb{Z}_2$ fermions. In the absence of pairing, the U(1) symmetric model approaches the vacuum at long times, with the entanglement entropy showing non-monotonic behavior over time that we capture with a phenomenological quasiparticle ansatz. In this regime, quantum jumps play a key role, and we highlight this by exactly computing their waiting-time distribution. On the other hand, the interplay between losses and pairing in the $\mathbb{Z}_2$ case gives rise to quantum trajectories with entangled steady-states. We show that by tuning the several system parameters, a measurement-induced entanglement transition occurs where the entanglement entropy scaling changes from logarithmic to area-law. We compare this transition with the one derived in the no-click limit and observe qualitative agreement in most of the phase diagram. Furthermore, the statistics of entanglement gain and loss are analyzed to better understand the impact of the linear jump operators., Comment: 15 pages, 11 figures

Many-body systems which saturate the quantum bound on chaos are attracting interest across a wide range of fields. Notable examples include the Sachdev-Ye-Kitaev model and its variations, all characterised by some form or randomness and all to all couplings. Here we study many-body quantum chaos in a quantum impurity model showing Non-Fermi-Liquid physics, the overscreened multichannel $SU(N)$ Kondo model. We compute exactly the low-temperature behavior of the out-of time order correlator in the limit of large $N$ and large number of channels $K$, at fixed ratio $\gamma=K/N$. Due to strong correlations at the impurity site the spin fractionalizes in auxiliary fermions and bosons. We show that all the degrees of freedom of our theory acquire a Lyapunov exponent which is linear in temperature as $T\rightarrow 0$, with a prefactor that depends on $\gamma$. Remarkably, for $N=K$ the impurity spin displays maximal chaos, while bosons and fermions only get up to half of the maximal Lyapunov exponent. Our results highlights two new features: a non-disordered model which is maximally chaotic due to strong correlations at its boundary and a fractionalization of quantum chaos., Comment: 16 pages, 7 figures

Poor man's Majorana Bound States (MBS) arise in minimal Kitaev chains when the parameters are fine-tuned to a sweet spot. We consider an interacting two-site Kitaev chain coupled to a single-mode cavity and show that the sweet spot condition can be controlled with the cavity frequency and the hopping between sites. Furthermore, we demonstrate that photon-mediated effective interactions can be used to screen intrinsic interactions, improving the original quality of the MBS. We describe experimental signatures in the cavity transmission to detect their presence and quality. Our work proposes a new way to tune poor man's MBS in a quantum dot array coupled to a cavity., Comment: 5 pages, 4 figures and supplementary material

Efficient numerical methods are still lacking to probe the unconventional dynamics of quantum many-body systems under non-unitary evolution. In this work, we use Faber polynomials to numerically simulate both the dynamics of non-Hermitian systems and the quantum jumps unravelling of the Lindblad dynamics. We apply the method to the non-interacting and interacting Hatano-Nelson models evolving from two different setups: i) a N\'eel state, and ii) a domain wall. In the first case, we study how interactions preserve the initial magnetic order against the skin effect. In the second example, we present numerical evidence of the existence of an effective hydrodynamic description for the domain-wall melting problem in the non-interacting limit. Additionally, we investigate both the conditional and unconditional dynamics of the quantum jump unravelling in two quantum spin chains, which exhibit either the non-Hermitian or the Liouvillian skin effect. This numerical method inherently generalises the well-established method based on Chebyshev polynomials to accommodate non-Hermitian scenarios., Comment: 36 pages, 13 figures, Resubmission to SciPost, minor changes

Isolated quantum many-body systems which thermalize under their own dynamics are expected to act as their own thermal baths, thereby bringing their local subsystems to thermal equilibrium. Here we show that the infinite-dimensional limit of a quantum lattice model, as described by Dynamical Mean-Field theory (DMFT), provides a natural framework to understand this self-consistent thermalization process. Using the Fermi-Hubbard model as working example, we demonstrate that the emergence of a self-consistent bath thermalising the system is characterized by a sharp thermalization front, moving balistically and separating the initial condition from the long-time thermal fixed point. We characterize the full DMFT dynamics through an effective temperature for which we derive a travelling-wave equation of the Fisher-Kolmogorov-Petrovsky-Piskunov (FKPP) type. This equation allows to predict the asymptotic shape of the front and its velocity, which match perfectly the full DMFT numerics. Our results provide a new angle to understand the onset of quantum thermalisation in closed isolated systems., Comment: 18 pages, 4 figures

We study the dynamics of a quantum dot coupled to a metallic bath and subject to continuous monitoring of its charge density. The dynamics averaged over measurement noise is described by a dissipative Anderson impurity model with local Markovian dephasing, that we solve using an extension of the Non-Crossing Approximation in the vectorized Hilbert space. We show that the decay time scale of an initially polarised spin which is suddenly coupled to the bath and to the monitoring protocol displays a crossover from Kondo screening, with a lifetime controlled by interactions, to Quantum Zeno effect, with a lifetime which decreases with bare dissipation as the dephasing or monitoring rate is increased. Using a Schrieffer-Wolff transformation on the Lindbladian we derive an effective model for the long-time dynamics which is described at weak dissipation by a non-Hermitian Kondo model with complex-valued spin-spin exchange. As the dephasing is increased heating due to doublon production takes over and control the spin decay., Comment: 19 pages, 7 figures

Collective effects, such as Dicke superradiant emission, can enhance the performance of a quantum device. Here, we study the heat current flowing between a cold and a hot bath through an ensemble of $N$ qubits, which are collectively coupled to the thermal baths. We find a regime where the collective coupling leads to a quadratic scaling of the heat current with $N$ in a finite-size scenario. Conversely, when approaching the thermodynamic limit, we prove that the collective scenario exhibits a parametric enhancement over the non-collective case. We then consider the presence of a third uncontrolled {\it parasitic} bath, interacting locally with each qubit, that models unavoidable couplings to the external environment. Despite having a non-perturbative effect on the steady-state currents, we show that the collective enhancement is robust to such an addition. Finally, we discuss the feasibility of realizing such a Dicke heat valve with superconducting circuits. Our findings indicate that in a minimal realistic experimental setting with two superconducting qubits, the collective advantage offers an enhancement of approximately $10\%$ compared to the non-collective scenario., Comment: 17 pages, comments are welcome

We investigate the periodically driven dynamics of many-body systems, either classical or quantum, finite-dimensional or mean-field, displaying an unbounded phase-space. We find that the inclusion of a smooth periodic drive atop an otherwise ergodic dynamics leads to a long-lived prethermalization, even at moderate driving frequencies. In specific asymptotic limits, we compute the corresponding prethermal Hamiltonian from an analytical perturbation scheme., Comment: 4 + 11 pages

Monitored quantum many-body systems display a rich pattern of entanglement dynamics, which is unique to this non-unitary setting. This work studies the effect of quantum jumps on the entanglement dynamics beyond the no-click limit corresponding to a deterministic non-Hermitian evolution. We consider two examples, a monitored SSH model and a quantum Ising chain, for which we show the jumps have remarkably different effects on the entanglement despite having the same statistics as encoded in their waiting-time distribution. To understand this difference, we introduce a new metric, the statistics of entanglement gain and loss due to jumps and non-Hermitian evolution. This insight allows us to build a simple stochastic model of a random walk with partial resetting, which reproduces the entanglement dynamics, and to dissect the mutual role of jumps and non-Hermitian evolution on the entanglement scaling. We demonstrate that significant deviations from the no-click limit arise whenever quantum jumps strongly renormalize the non-Hermitian dynamics, as in the case of the SSH model at weak monitoring or in the Ising chain at large transverse field. On the other hand, we show that the weak monitoring phase of the Ising chain leads to a robust sub-volume logarithmic phase due to weakly renormalized non-Hermitian dynamics., Comment: 12 + 5 pages, 7 + 5 figures. Major overhaul of the text

We develop a diagrammatic Monte Carlo method for the real-time dynamics of dissipative quantum impurity models. These are small open quantum systems with interaction and local Markovian dissipation, coupled to a large quantum bath. Our algorithm sample the hybridization expansion formulated on a single real-time contour, rather than on the double Keldysh one, as it naturally arises in the thermofield/vectorized representation of the Lindblad dynamics. We show that local Markovian dissipation generally helps the convergence of the diagrammatic Monte Carlo sampling by reducing the sign problem, thus allowing to reach longer time scales as compared to the conventional unitary case. We apply our method to an Anderson impurity model in presence of local dephasing and discuss its effect on the charge and spin dynamics of the impurity., Comment: 19 pages, 11 figures

Understanding the influence of measurements on the properties of many-body systems is a fundamental problem in quantum mechanics and for quantum technologies. This paper explores how a finite density of stochastic local measurement modifies a given state's entanglement structure. Considering various measurement protocols, we explore the typical quantum correlations of their associated projected ensembles arising from the ground state of the quantum Ising model. Using large-scale numerical simulations, we demonstrate substantial differences among inequivalent measurement protocols. Surprisingly, we observe that forced on-site measurements can enhance both bipartite and multipartite entanglement. We present a phenomenological toy model and perturbative calculations to analytically support these results. Furthermore, we extend these considerations to the non-Hermitian Ising model, naturally arising in optically monitored systems, and we show that its qualitative entanglement features are not altered by a finite density of projective measurements. Overall, these results reveal a complex phenomenology where local quantum measurements do not simply disentangle degrees of freedom, but may actually strengthen the entanglement in the system., Comment: 13 pages, 5 figures; removed residual comment from proofreading

We consider a one-dimensional topological superconductor hosting Majorana bound states at its ends coupled to a single mode cavity. In the strong light-matter coupling regime, electronic and photonic degrees of freedom hybridize resulting in the formation of polaritons. We find the polariton spectrum by calculating the cavity photon spectral function of the coupled electron-photon system. In the topological phase the lower in energy polariton modes are formed by the bulk-Majorana transitions coupled to cavity photons and are also sensitive to the Majorana parity. In the trivial phase the lower polariton modes emerge due to the coupling of the bulk-bulk transitions across the gap to photons. Our work demonstrates the formation of polaritons in topological superconductors coupled to photons that contain information on the features of the Majorana bound states., Comment: 9 pages, 5 figures; references added

We present the spin-wave theory of the excitation spectrum and quench dynamics of the non-Hermitian transverse-field Ising model. The complex excitation spectrum is obtained for a generic hypercubic lattice using the linear approximation of the Holstein-Primakoff transformation together with the complex bosonic Bogolyubov transformation. In the one-dimensional case, our result compares very well with the exact quasiparticle dispersion relation obtained via a fermionic representation of the problem, at least in the regime of large dissipation and transverse field. When applied to the quench dynamics we show however that the linear spin-wave approximation breaks down and the bosonic theory is plagued by a divergence at finite times. We understand the origin of this instability using a single mode approximation. While limited to short times, we show that this approach allows us to characterize the dynamics arising from the quench of the dissipative term and the structure of the Lieb-Robinson light-cone of the propagation quantum correlations. Furthermore, for the one-dimensional case, the linear spin-wave dynamics shows good agreement with the exact fermionic solution, both for the local magnetization and the spin-spin correlations., Comment: 17 pages, 15 figures

Interspersing unitary dynamics with local measurements results in measurement-induced phases and transitions in many-body quantum systems. When the evolution is driven by a local Hamiltonian, two types of transitions have been observed, characterized by an abrupt change in the system size scaling of entanglement entropy. The critical point separates the strongly monitored area-law phase from a volume law or a sub-extensive, typically logarithmic-like one at low measurement rates. Identifying the key ingredients responsible for the entanglement scaling in the weakly monitored phase is the key purpose of this work. For this purpose, we consider prototypical one-dimensional spin chains with local monitoring featuring the presence/absence of U(1) symmetry, integrability, and interactions. Using exact numerical methods, the system sizes studied reveal that the presence of interaction is always correlated to a volume-law weakly monitored phase. In contrast, non-interacting systems present sub-extensive scaling of entanglement. Other characteristics, namely integrability or U(1) symmetry, do not play a role in the character of the entanglement phase., Comment: 4 pages, 4 figures

We consider a one-dimensional system of non-interacting fermions featuring both boundary driving and continuous monitoring of the bulk particle density. Due to the measurements, the expectation values of the local density and current operators are random variables whose average behavior is described by a well studied Lindblad master equation. By means of exact numerical computations, we go beyond the averaged dynamics and study their full probability distribution functions, focusing on the late-time stationary regime. We find that, contrary to the averaged values, the spatial profiles of the median density and current are non-trivial, exhibiting qualitative differences as a function of the monitoring strength. At weak monitoring, the medians are close to the means, displaying diffusive spatial profiles. At strong monitoring, we find that the median density and current develop a domain-wall and single-peak profile, respectively, which are suggestive of a Zeno-like localization in typical quantum trajectories. While we are not able to identify a sharp phase transition as a function of the monitoring rate, our work highlights the usefulness of characterizing typical behavior beyond the averaged values in the context of monitored many-body quantum dynamics., Comment: 10 pages, 6 figures; v3: minor revision

Understanding the nature of entanglement growth in many-body systems is one of the fundamental questions in quantum physics. Here, we study this problem by characterizing the entanglement fluctuations and distribution of $(d+1)$ qubit lattice evolved under a random unitary circuit. Focusing on Clifford gates, we perform extensive numerical simulations of random circuits in $1\le d\le 4$ dimensions. Our findings demonstrate that properties of growth of bipartite entanglement entropy are characterized by the roughening exponents of a $d$-dimensional membrane in a $(d+1)$ elastic medium., Comment: 4 pages

We study the properties of a monitored ensemble of atoms driven by a laser field and in the presence of collective decay. The properties of the quantum trajectories describing the atomic cloud drastically depend on the monitoring protocol and are distinct from those of the average density matrix. By varying the strength of the external drive, a measurement-induced phase transition occurs separating two phases with entanglement entropy scaling sub-extensively with the system size. Incidentally, the critical point coincides with the superradiance transition of the trajectory-averaged dynamics. Our setup is implementable in current light-matter interaction devices, and most notably, the monitored dynamics is free from the post-selection measurement problem, even in the case of imperfect monitoring., Comment: 7 + 4 pages, 3 + 4 figures

Universal quantum computing requires nonstabilizer (magic) quantum states. Quantifying the nonstabilizerness and relating it to other quantum resources is vital for characterizing the complexity of quantum many-body systems. In this work, we prove that a quantum state is a stabilizer if and only if all states belonging to its Clifford orbit have a flat probability distribution on the computational basis. This implies, in particular, that multifractal states are nonstabilizers. We introduce multifractal flatness, a measure based on the participation entropy that quantifies the wave-function distribution flatness. We demonstrate that this quantity is analytically related to the stabilizer entropy of the state and present several examples elucidating the relationship between multifractality and nonstabilizerness. In particular, we show that the multifractal flatness provides an experimentally and computationally viable nonstabilizerness certification. Our work unravels the direct relation between the nonstabilizerness of a quantum state and its wave-function structure., Comment: 6 pages, Comments are welcome!

We study the dynamics of a quantum $p$-spin glass model starting from initial states defined in microcanonical shells, in a classical regime. We compute different chaos estimators, such as the Lyapunov exponent and the Kolmogorov-Sinai entropy, and find a marked maximum as a function of the energy of the initial state. By studying the relaxation dynamics and the properties of the energy landscape we show that the maximal chaos emerges in correspondence with the fastest spin relaxation and the maximum complexity, thus suggesting a qualitative picture where chaos emerges as the trajectories are scattered over the exponentially many saddles of the underlying landscape. We also observe hints of ergodicity breaking at low energies, indicated by the correlation function and a maximum of the fidelity susceptibility., Comment: 14+17 pages, 8 figures

Reservoir engineering is a powerful approach for using controlled driven-dissipative dynamics to prepare target quantum states and phases. In this work, we study a paradigmatic model that can realize a Mott insulator of photons in its steady-state. We show that, while in some regimes its steady state approximates a Mott-insulating ground state, this phase can become unstable through a non-equilibrium transition towards a coherent yet non-classical limit-cycle phase, driven by doublon excitations. This instability is completely distinct from the ground-state Mott-insulator to superfluid transition. This difference has dramatic observable consequences and leads to an intrinsic fragility of the steady-state Mott phase: a fast pump compared to losses is required to sustain the phase, but also determines a small critical hopping. We identify unique features of the steady-state Mott phase and its instability, that distinguish them from their ground-state counterpart and can be measured in experiments., Comment: 13 pages, 5 figures

We consider a model of free fermions in a ladder geometry coupled to a nonuniform 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., Comment: 28 pages, 9 figures. Submission to SciPost. Comments are welcome

The interplay between unitary dynamics and local quantum measurements results in unconventional non-unitary dynamical phases and transitions. In this paper we investigate the dynamics of $(d+1)$-dimensional hybrid stabilizer circuits, for $d=1,2,3$. We characterize the measurement-induced phases and their transitions using large-scale numerical simulations focusing on entanglement measures, purification dynamics, and wave-function structure. Our findings demonstrate the measurement-induced transition in $(d+1)$ spatiotemporal dimensions is conformal and close to the percolation transition in $(d+1)$ spatial dimensions., Comment: 19pp, 16 figs, comments welcome!

We consider the dynamics of the non-Hermitian Su-Schrieffer-Heeger model arising as the no-click limit of a continuously monitored free fermion chain where particles and holes are measured on two sublattices. The model has $\mathcal{PT}$-symmetry, which we show to spontaneously break as a function of the strength of measurement backaction, resulting in a spectral transition where quasiparticles acquire a finite lifetime in patches of the Brillouin zone. We compute the entanglement entropy's dynamics in the thermodynamic limit and demonstrate an entanglement transition between volume-law and area-law scaling, which we characterize analytically. Interestingly we show that the entanglement transition and the $\mathcal{PT}$-symmetry breaking do not coincide, the former occurring when the entire decay spectrum of the quasiparticle becomes gapped., Comment: version 3 , 4 figures, footnote

We study the dissipative dynamics of a fermionic superfluid in presence of two-body losses. We use a variational approach for the Lindblad dynamics and obtain dynamical equations for Anderson's pseudo-spins where dissipation enters as a complex pairing interaction as well as effective, density-dependent, single particle losses which break the conservation of the pseudo-spin norm. We show that this latter has key consequences on the dynamical behavior of the system. In the case of a sudden switching of two-body losses we show that the superfluid order parameter decays much faster than then particle density at short times and eventually slows-down, setting into a power-law decay at longer time scales driven by the depletion of the system. We then consider a quench of pairing interaction, leading to coherent oscillations in the unitary case, followed by the switching of the dissipation. We show that losses wash away the dynamical BCS synchronization by introducing not only damping but also a renormalization of the frequency of coherent oscillations, which depends non-linearly from the rate of the two-body losses., Comment: 5 pages, 3 figures. Supplementary Material: 6 pages, 3 figures

We study the dynamics of a quantum emitter coupled to a two-dimensional photonic crystal featuring a finite bandwidth with sharp edges and a Van-Hove singularity. We study the effect of strong system-bath coupling and non-Markovianity of the photonic environment using a nonperturbative approach based on the recently introduced NCA dynamical map for open quantum systems. We show that several characteristic features of the dynamics near a photonic band-edge such as the freezing of spontaneous emission and the maximum light-matter entanglement, get strongly modified in presence of counter-rotating terms in the system-bath coupling, beyond the rotating-wave approximation. Furthermore, by computing the spectral function of the quantum emitter we comment on the role played by atom-photon bound-state and show that this acquires a much larger lifetime once the rotating-wave approximation is relaxed., Comment: 12 pages, 10 figures

Energy transport in quantum many-body systems with well defined quasiparticles has recently attracted interest across different fields, including out of equilibrium conformal field theories, one dimensional quantum lattice models and holographic matter. Here we study energy transport between \emph{strange quantum baths} without quasiparticles, made by two Sachdev-Ye-Kitaev (SYK) models at temperatures $T_L\neq T_R$ and connected by a Fermi-Liquid system. We obtain an exact expression for the nonequilibrium energy current, valid in the limit of large bath and system size and for any system-bath coupling $V$. We show that the peculiar criticality of the SYK baths has direct consequences on the thermal conductance, which above a temperature $T^*(V)\sim V^4$ is parametrically enhanced with respect to the linear-$T$ behavior expected in systems with quasiparticles. Interestingly, below $T^*(V)$ the linear thermal conductance behavior is restored, yet transport is not due to quasiparticles. Rather the system gets strongly renormalized by the strange bath and becomes Non-Fermi-Liquid and maximally chaotic. Finally, we discuss the full nonequilibrium energy current and show that its form is compatible with the structure $\mathcal{J}=\Phi(T_L)-\Phi(T_R)$, with $\Phi(T)\sim T^{\gamma}$ and power law crossing over from $\gamma=3/2$ to $\gamma=2$ below $T^*$., Comment: 5 pages, 4 figures + supplementary material

We investigate entanglement dynamics in continuously monitored open quantum systems featuring current-carrying non-equilibrium states. We focus on a prototypical one-dimensional model of boundary-driven non-interacting fermions with monitoring of the local density, whose average Lindblad dynamics features a well-studied ballistic to diffusive crossover in transport. Here we analyze the dynamics of the fermionic negativity, mutual information, and purity along different quantum trajectories. We show that monitoring this boundary-driven system enhances its entanglement negativity at long times, which otherwise decays to zero in absence of measurements. This result is in contrast with the case of unitary evolution where monitoring suppresses entanglement production. For small values of $\gamma$, the stationary-state negativity shows a logarithmic scaling with system size, transitioning to an area-law scaling as $\gamma$ is increased beyond a critical value. Similar critical behavior is found in the mutual information, while the late-time purity shows no apparent signature of a transition, being $O(1)$ for all values of $\gamma$. Our work unveils the double role of weak monitoring in current-driven open quantum systems, simultaneously damping transport and enhancing entanglement., Comment: 10 pages, comments are welcome

Controlling the topological properties of quantum matter is a major goal of condensed matter physics. A major effort in this direction has been devoted to using classical light in the form of Floquet drives to manipulate and induce states with non-trivial topology. A different route can be achieved with cavity photons. Here we consider a prototypical model for topological phase transition, the one-dimensional Su-Schrieffer-Heeger (SSH) model, coupled to a single mode cavity. We show that quantum light can affect the topological properties of the system, including the finite-length energy spectrum hosting edge modes and the topological phase diagram. In particular we show that depending on the lattice geometry and the strength of light-matter coupling one can either turn a trivial phase into a topological one or viceversa using quantum cavity fields. Furthermore, we compute the polariton spectrum of the coupled electron-photon system, and we note that the lower polariton branch disappears at the topological transition point. This phenomenon can be used to probe the phase transition in the SSH model., Comment: 12 pages, 5 figures

Non-Hermitian quantum many-body systems are attracting widespread interest for their exotic properties, including unconventional quantum criticality and topology. Here we study how quantum information and correlations spread under a quantum quench generated by a prototypical non-Hermitian spin chain. Using the mapping to fermions we solve exactly the problem and compute the entanglement entropy and the correlation dynamics in the thermodynamic limit. Depending on the quench parameters, we identify two dynamical phases. One is characterized by rapidly saturating entanglement and correlations. The other instead presents a logarithmic growth in time, and correlations spreading faster than the Lieb-Robinson bound, with collapses and revivals giving rise to a modulated light-cone structure. Here, in the long-time limit, we compute analytically the entanglement entropy that we show to scale logarithmically with the size of the cut, with an effective central charge that we obtain in closed form. Our results provide an example of an exactly solvable non-Hermitian many-body problem that shows rich physics including entanglement and spectral transitions., Comment: 4.5 + 11 pag

We study a driven-dissipative Bose-Hubbard model in presence of two-particle losses and an incoherent single-particle drive on each lattice site, leading to a finite-density stationary state. Using dynamical mean-field theory (DMFT) and an impurity solver based on exact diagonalization of the associated Lindbladian, we investigate the regime of strong two-particle losses. Here, a stationary-state quantum Zeno effect emerges, as can be seen in the on-site occupation and spectral function. We show that DMFT captures this effect through its self-consistent bath. We show that, in the deep Zeno regime, the bath structure simplifies, with the occupation of all bath sites except one becoming exponentially suppressed. As a result, an effective dissipative hard-core Bose-Hubbard dimer model emerges, where the auxiliary bath site has single-particle dissipation controlled by the Zeno dissipative scale., Comment: 14 pages, 8 figures. With respect to version 1 we've carried out a minor revision of the document

We discuss and compare two recently proposed toy models for anomalous transport and Griffiths effects in random systems near the Many-Body Localization transitions: the random dephasing model, which adds thermal inclusions in an Anderson Insulator as local Markovian dephasing channels that heat up the system, and the random Gaussian Orthogonal Ensemble (GOE) approach which models them in terms of ensembles of random regular graphs. For these two settings we discuss and compare transport and dissipative properties and their statistics. We show that both types of dissipation lead to similar Griffiths-like phenomenology, with the GOE bath being less effective in thermalising the system due to its finite bandwidth. We then extend these models to the case of a quasi-periodic potential as described by the Andr\'e-Aubry-Harper model coupled to random thermal inclusions, that we show to display, for large strength of the quasiperiodic potential, a similar phenomenology to the one of the purely random case. In particular, we show the emergence of subdiffusive transport and broad statistics of the local density of states, suggestive of Griffiths like effects arising from the interplay between quasiperiodic localization and random coupling to the baths., Comment: 31 pages

We put forward a phenomenological theory for entanglement dynamics in monitored quantum many-body systems with well-defined quasiparticles. Within this theory entanglement is carried by ballistically propagating non-Hermitian quasiparticles which are stochastically reset by the measurement protocol with rate given by their finite inverse lifetime. We write down a renewal equation for the statistics of the entanglement entropy and show that depending on the spectrum of quasiparticle decay rates different entanglement scaling can arise and even sharp entanglement phase transitions. When applied to a Quantum Ising chain where the transverse magnetization is measured by quantum jumps, our theory predicts a critical phase with logarithmic scaling of the entanglement, an area law phase and a continuous phase transition between them, with an effective central charge vanishing as a square root at the transition point. We compare these predictions with with exact numerical calculations on the same model and find an excellent agreement., Comment: 5+7 pages, 3 figures + Erratum

We discuss the theory of linear and non-linear spectroscopy of an rf-SQUID coupled to a Josephson spectrometer. Recent experimental measurements on this system have shown a strongly non-linear absorption lineshape, whose current peak maximum undergoes a forward-backward bending transition depending on the value of the rf-SQUID phase. We show that this transition can be qualitatively understood by mapping the dynamics of the driven rf-SQUID onto a generalized Duffing oscillator, with tunable drive and non-linearity, undergoing a bifurcation. Finally we show that in order to quantitatively reproduce the experimental data reported in arXiv:2106.02632, it is crucial to include the feedback from the load-line, leading to an additional source of non-linearity., Comment: 13 pages, 7 figures

We study the nonequilibrium quench dynamics of a mixed Sachdev-Ye-Kitaev model, with competing two bodies random interactions leading to maximally chaotic Non-Fermi Liquid dynamics and a single body term which dominates at low temperatures and leads to Fermi liquid behavior. For different quench protocols, including sudden switching of two-body interaction and double quench protocols, we solve the large $N$ real-time Dyson equation on the Keldysh contour and compute the dynamics of Green's functions from which we obtain effective temperature and relaxation rates. We show that the model thermalizes to a finite temperature equilibrium and that depending on the value of the quench parameters the effective temperature can be below or above the Fermi-Liquid to Non-Fermi Liquid crossover scale, which can then be accessed through the nonequilibrium dynamics. We identify quench protocols for which the heating dynamics slow down significantly, an effect that we interpret as a signature of prethermalization., Comment: 14 pages, 13 figures. Updated figures and bibliography

In several cases, open quantum systems can be successfully described using master equations relying on Born-Markov approximations, but going beyond these approaches has become often necessary. In this work, we introduce the NCA and NCA-Markov dynamical maps for open quantum systems, which go beyond these master equations replacing the Born approximation with a self-consistent approximation, known as non-crossing approximation (NCA). These maps are formally similar to master equations, but allow to capture non-perturbative effects of the environment at a moderate extra numerical cost. To demonstrate their capabilities, we apply them to the spin-boson model at zero temperature for both a Ohmic and a sub-Ohmic environment, showing that they can both qualitatively capture its strong-coupling behaviour, and be quantitatively correct beyond standard master equations., Comment: 30 pages, 8 figures

We study a model of non-interacting spinless fermions coupled to local dephasing and boundary drive and described within a Lindblad master equation. The model features an interplay between infinite temperature thermalization due to bulk dephasing and a non-equilibrium stationary state due to the boundary drive and dissipation. We revisit the linear and non-linear transport properties of the model, featuring a crossover from ballistic to diffusive scaling, and compute the spectral and occupation properties encoded in the single particle Green's functions, that we compute exactly using the Lindblad equations of motion in spite of the \emph{interacting} nature of the dephasing term. We show that the distribution function in the bulk of the system becomes frequency independent and flat, consistent with infinite temperature thermalization, while near the boundaries it retains strong non-equilibrium features that reflect the continuous injection and depletion of particles due to driving and dissipation., Comment: 11 pages

We investigate measurement-induced phase transitions in the Quantum Ising chain coupled to a monitoring environment. We compare two different limits of the measurement problem, the stochastic quantum-state diffusion protocol corresponding to infinite small jumps per unit of time and the no-click limit, corresponding to post-selection and described by a non-Hermitian Hamiltonian. In both cases we find a remarkably similar phenomenology as the measurement strength $\gamma$ is increased, namely a sharp transition from a critical phase with logarithmic scaling of the entanglement to an area-law phase, which occurs at the same value of the measurement rate in the two protocols. An effective central charge, extracted from the logarithmic scaling of the entanglement, vanishes continuously at the common transition point, although with different critical behavior possibly suggesting different universality classes for the two protocols. We interpret the central charge mismatch near the transition in terms of noise-induced disentanglement, as suggested by the entanglement statistics which displays emergent bimodality upon approaching the critical point. The non-Hermitian Hamiltonian and its associated subradiance spectral transition provide a natural framework to understand both the extended critical phase, emerging here for a model which lacks any continuous symmetry, and the entanglement transition into the area law., Comment: 12 pages, 6 figures, comments are welcome

We investigate signatures of a self-trapping transition in the driven-dissipative Bose Hubbard dimer, in presence of incoherent pump and single-particle losses. For fully symmetric couplings the stationary state density matrix is independent of any Hamiltonian parameter, and cannot therefore capture the competition between hopping-induced delocalization and the interaction-dominated self-trapping regime. We focus instead on the exact quantum dynamics of the particle imbalance after the system is prepared in a variety of initial states, and on the frequency-resolved spectral properties of the steady state, as encoded in the single-particle Green's functions. We find clear signatures of a localization-delocalization crossover as a function of hopping to interaction ratio. We further show that a finite a pump-loss asymmetry restores a delocalization crossover in the steady-state imbalance and leads to a finite intra-dimer dissipation., Comment: 16 pages, 12 figures. With respect to version 1 we've added a new section "VIII. Discussion" and we've carried out a minor revision of the other sections

We develop a trajectory-based approach for excited-state molecular dynamics simulations of systems subject to an external periodic drive. We combine the exact-factorization formalism, allowing to treat electron-nuclear systems in nonadiabatic regimes, with the Floquet formalism for time-periodic processes. The theory is developed starting with the molecular time-dependent Schroedinger equation with inclusion of an external periodic drive that couples to the system dipole moment. With the support of the Floquet formalism, quantum dynamics is approximated by combining classical-like, trajectory-based, nuclear evolution with electronic dynamics represented in the Floquet basis. The resulting algorithm, which is an extension of the coupled-trajectory mixed quantum-classical scheme for periodically driven systems, is applied to a model study, exactly solvable, with different field intensities.

We present calculations of the time-evolution of the driven-dissipative XYZ model using the infinite Projected Entangled Pair Operator (iPEPO) method, introduced by [A. Kshetrimayum, H. Weimer and R. Or\'us, Nat. Commun. 8, 1291 (2017)]. We explore the conditions under which this approach reaches a steady state. In particular, we study the conditions where apparently converged calculations may become unstable with increasing bond dimension of the tensor-network ansatz. We discuss how more reliable results could be obtained., Comment: Submission to SciPost

It is well known that by repeatedly measuring a quantum system it is possible to completely freeze its dynamics into a well defined state, a signature of the quantum Zeno effect. Here we show that for a many-body system evolving under competing unitary evolution and variable-strength measurements the onset of the Zeno effect takes the form of a sharp phase transition. Using the Quantum Ising chain with continuous monitoring of the transverse magnetization as paradigmatic example we show that for weak measurements the entanglement produced by the unitary dynamics remains protected, and actually enhanced by the monitoring, while only above a certain threshold the system is sharply brought into an uncorrelated Zeno state. We show that this transition is invisible to the average dynamics, but encoded in the rare fluctuations of the stochastic measurement process, which we show to be perfectly captured by a non-Hermitian Hamiltonian which takes the form of a Quantum Ising model in an imaginary valued transverse field. We provide analytical results based on the fermionization of the non-Hermitian Hamiltonian in supports of our exact numerical calculations., Comment: 14 pages, 10 figures

We discuss the problem of gauge fixing for strongly correlated electrons coupled to quantum light, described by projected low-energy models such as those obtained within tight-binding methods. Drawing from recent results in the field of quantum optics, we present a general approach to write down quantum light-matter Hamiltonian in either dipole or Coulomb gauge which are explicitly connected by a unitary transformation, thus ensuring gauge equivalence even after projection. The projected dipole gauge Hamiltonian features a linear light-matter coupling and an instantaneous self-interaction for the electrons, similar to the structure in the full continuum theory. On the other hand, in the Coulomb gauge the photon field enters in a highly non-linear way, through phase factors that dress the electronic degrees of freedom. We show that our approach generalises the well-known Peierls approximation, to which it reduces when only local, on-site orbital contributions to light-matter coupling are taken into account. As an application, we study a two-orbital model of interacting electrons coupled to a uniform cavity mode, recently studied in the context of excitonic superradiance and associated no-go theorems. Using both gauges we recover the absence of superradiant phase in the ground state and show that excitations on top of it, described by polariton modes, contain instead non-trivial light-matter entanglement. Our results highlight the importance of treating the non-linear light-matter interaction of the Coulomb gauge non-perturbatively, to obtain a well-defined ultrastrong coupling limit and to not spoil gauge equivalence., Comment: 20 pages, 6 figures, references added

In this paper we study the out-of-equilibrium phase diagram of the quantum version of Derrida's Random Energy Model, which is the simplest model of mean-field spin glasses. We interpret its corresponding quantum dynamics in Fock space as a one-particle problem in very high dimension to which we apply different theoretical methods tailored for high-dimensional lattices: the Forward-Scattering Approximation, a mapping to the Rosenzweig-Porter model, and the cavity method. Our results indicate the existence of two transition lines and three distinct dynamical phases: a completely many-body localized phase at low energy, a fully ergodic phase at high energy, and a multifractal "bad metal" phase at intermediate energy. In the latter, eigenfunctions occupy a diverging volume, yet an exponentially vanishing fraction of the total Hilbert space. We discuss the limitations of our approximations and the relationship with previous studies., Comment: 21 pages, 13 figures

Open quantum many body systems describe a number of experimental platforms relevant for quantum simulations, ranging from arrays of superconducting circuits to ultracold atoms in optical lattices. Their theoretical understanding is hampered by their large Hilbert space and by their intrinsic nonequilibrium nature, limiting the applicability of many traditional approaches. In this work we extend the nonequilibrium bosonic Dynamical Mean Field Theory (DMFT) to Markovian open quantum systems. Within DMFT, a Lindblad master equation describing a lattice of dissipative bosonic particles is mapped onto an impurity problem describing a single site embedded in its Markovian environment and coupled to a self-consistent field and to a non-Markovian bath, where the latter accounts for finite lattice connectivity corrections beyond Gutzwiller mean-field theory. We develop a non-perturbative approach to solve this bosonic impurity problem, which treats the non-Markovian bath in a non-crossing approximation. As a first application, we address the steady-state of a driven-dissipative Bose-Hubbard model with two-body losses and incoherent pump. We show that DMFT captures hopping-induced dissipative processes, completely missed in Gutzwiller mean-field theory, which crucially determine the properties of the normal phase, including the redistribution of steady-state populations, the suppression of local gain and the emergence of a stationary quantum-Zeno regime. We argue that these processes compete with coherent hopping to determine the phase transition towards a non-equilibrium superfluid, leading to a strong renormalization of the phase boundary at finite-connectivity. We show that this transition occurs as a finite-frequency instability, leading to an oscillating-in-time order parameter, that we connect with a quantum many-body synchronization transition of an array of quantum van der Pol oscillators., Comment: 29 pages, 19 figures, including appendices

We generalize the theory of flow equations to open quantum systems focusing on Lindblad master equations. We introduce and discuss three different generators of the flow that transform a linear non-Hermitian operator into a diagonal one. We first test our dissipative flow equations on a generic matrix and on a physical problem with a driven-dissipative single fermionic mode. We then move to problems with many fermionic modes and discuss the interplay between coherent (disordered) dynamics and localized losses. Our method can also be applied to non-Hermitian Hamiltonians., Comment: 22 pages plus appendices, 9 figures. Typos in Sec. 2.2 and appendix A corrected

Robust engineering of phonon squeezed states in optically excited solids has emerged as a promising tool to control and manipulate their properties. However, in contrast to quantum optical systems, detection of phonon squeezing is subtle and elusive, and an important question is what constitutes an unambiguous signature of it. The state of the art involves observing oscillations at twice the phonon frequency in time resolved measurements of the out of equilibrium phonon fluctuation. Using Keldysh formalism we show that such a signal is a necessary but not a sufficient signature of a squeezed phonon, since we identify several mechanisms that do not involve squeezing and yet which produce similar oscillations. We show that a reliable detection requires a time and frequency resolved measurement of the phonon spectral function.

We study a driven-dissipative model of spins one-half (qubits) on a lattice with nearest-neighbor interactions. Focusing on the role of spatially extended spin-spin correlations in determining the phases of the system, we characterize the spatial structure of the correlations in the steady state, as well as their temporal dynamics. In dimension one we use essentially exact matrix-product-operator simulations on large systems, and pushing these calculations to dimension two, we obtain accurate results on small cylinders. We also employ an approximation scheme based on solving the dynamics of the mean field dressed by the feedback of quantum fluctuations at leading order. This approach allows us to study the effect of correlations in large lattices with over one hundred thousand spins, as the spatial dimension is increased up to five. In dimension two and higher we find two new states that are stabilized by quantum correlations and do not exist in the mean-field limit of the model. One of these is a steady state with mean magnetization values that lie between the two bistable mean-field values, and whose correlation functions have properties reminiscent of both. The correlation length of the new phase diverges at a critical point, beyond which we find emerging a new limit cycle state with the magnetization and correlators oscillating periodically in time., Comment: 21 pages, 25 figures. v2 includes some clarifications

We introduce and study a toy model for anomalous transport and Griffiths effects in one dimensional quantum disordered isolated systems near the Many-Body Localization (MBL) transitions. The model is constituted by a collection of 1d tight-binding chains with on-site random energies, locally coupled to a weak GOE-like perturbation, which mimics the effect of thermal inclusions due to delocalizing interactions by providing a local broadening of the Poisson spectrum. While in absence of such a coupling the model is localized as expected for the one dimensional Anderson model, increasing the coupling with the GOE perturbation we find a delocalization transition to a conducting one driven by the proliferation of quantum avalanches which does not fit the standard paradigm of Anderson localization. In particular an intermediate Griffiths region emerges, where exponentially distributed insulating segments coexist with a few, rare resonances. Typical correlations decay exponentially fast, while average correlations decay as stretched exponential and diverge with the length of the chain, indicating that the conducting inclusions have a fractal structure and that the localization length is broadly distributed at the critical point. This behavior is consistent with a Kosterlitz-Thouless-like criticality of the transition. Transport and relaxation are dominated by rare resonances and rare strong insulating regions, and show anomalous behaviors strikingly similar to those observed in recent simulations and experiments in the bad metal delocalized phase preceding MBL. In particular, we find sub-diffusive transport and power-laws decay of the return probability at large times, with exponents that gradually change as one moves across the intermediate region. Concomitantly, the a.c. conductivity vanishes near zero frequency with an anomalous power-law., Comment: 20 pages, 15 figures