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We discuss many-body fermionic and bosonic systems subject to dissipative particle losses in arbitrary spatial dimensions $d$, within the Keldysh path-integral formulation of the quantum master equation. This open quantum dynamics represents a generalisation of classical reaction-diffusion dynamics to the quantum realm. We first show how initial conditions can be introduced in the Keldysh path integral via boundary terms. We then study binary annihilation reactions $A+A\to\emptyset$, for which we derive a Boltzmann-like kinetic equation. The ensuing algebraic decay in time for the particle density depends on the particle statistics. In order to model possible experimental implementations with cold atoms, for fermions in $d=1$ we further discuss inhomogeneous cases involving the presence of a trapping potential. In this context, we quantify the irreversibility of the dynamics studying the time evolution of the system entropy for different quenches of the trapping potential. We find that the system entropy features algebraic decay for confining quenches, while it saturates in deconfined scenarios., Comment: 43+4 pages, 9+1 figures for main text and Appendix respectively

We study the probability distribution of the first return time to the initial state of a quantum many-body system subject to stroboscopic projective measurements. We show that this distribution can be interpreted as a continuation of the canonical partition function of a spin chain with non-interacting domains at equilibrium, which is entirely characterised by the Loschmidt amplitude of the quantum many-body system. This allows us to show that this probability may decay either algebraically or exponentially asymptotically in time, depending on whether the spin model displays a ferromagnetic or a paramagnetic phase. We illustrate this idea on the example of the return time of $N$ adjacent fermions in a tight-binding model, revealing a rich phase behaviour, which can be tuned by scaling the probing time with $N$. Our analytical predictions are corroborated by exact numerical computations., Comment: 6 pages, 4 figures, 12 pages supplemental material

We investigate quantum reaction-diffusion systems in one-dimension with bosonic particles that coherently hop in a lattice, and when brought in range react dissipatively. Such reactions involve binary annihilation ($A + A \to \emptyset$) and coagulation ($A + A \to A$) of particles at distance $d$. We consider the reaction-limited regime, where dissipative reactions take place at a rate that is small compared to that of coherent hopping. In classical reaction-diffusion systems, this regime is correctly captured by the mean-field approximation. In quantum reaction-diffusion systems, for non-interacting fermionic systems, the reaction-limited regime recently attracted considerable attention because it has been shown to give universal power law decay beyond mean-field for the density of particles as a function of time. Here, we address the question whether such universal behavior is present also in the case of the non-interacting Bose gas. We show that beyond mean-field density decay for bosons is possible only for reactions that allow for destructive interference of different decay channels. Furthermore, we study an absorbing-state phase transition induced by the competition between branching $A\to A+A$, decay $A\to \emptyset$ and coagulation $A+A\to A$. We find a stationary phase-diagram, where a first and a second-order transition line meet at a bicritical point which is described by tricritical directed percolation. These results show that quantum statistics significantly impact on both the stationary and the dynamical universal behavior of quantum reaction-diffusion systems., Comment: 22 pages: 15 pages main text, 6 figures, 5 pages appendices, 3 pages references

We consider the quantum reaction-diffusion dynamics in $d$ spatial dimensions of a Fermi gas subject to binary annihilation reactions $A+A \to \emptyset$. These systems display collective nonequilibrium long-time behavior, which is signalled by an algebraic decay of the particle density. Building on the Keldysh formalism, we devise a field theoretical approach for the reaction-limited regime, where annihilation reactions are scarce. By means of a perturbative expansion of the dissipative interaction, we derive a description in terms of a large-scale universal kinetic equation. Our approach shows how the time-dependent generalized Gibbs ensemble assumption, which is often employed for treating low-dimensional nonequilibrium systems, emerges from systematic diagrammatics. It also allows to exactly compute -- for arbitrary spatial dimension -- the decay exponent of the particle density. The latter is based on the large-scale description of the quantum dynamics and it differs from the mean-field prediction even in dimension larger than one. We moreover consider spatially inhomogeneous setups involving an external potential. In confined systems the density decay is accelerated towards the mean-field algebraic behavior, while for deconfined scenarios the power-law decay is replaced by a slower non-algebraic decay., Comment: 8+10 pages, 3+2 figures for main text and Supplemental Material respectively

We investigate the quantum reaction-diffusion dynamics of fermionic particles which coherently hop in a one-dimensional lattice and undergo annihilation reactions. The latter are modelled as dissipative processes which involve losses of pairs $2A \to \emptyset$, triplets $3A \to \emptyset$, and quadruplets $4A \to \emptyset$ of neighbouring particles. When considering classical particles, the corresponding decay of their density in time follows an asymptotic power-law behavior. The associated exponent in one dimension is different from the mean-field prediction whenever diffusive mixing is not too strong and spatial correlations are relevant. This specifically applies to $2A\to \emptyset$, while the mean-field power-law prediction just acquires a logarithmic correction for $3A \to \emptyset$ and is exact for $4A \to \emptyset$. A mean-field approach is also valid, for all the three processes, when the diffusive mixing is strong, i.e., in the so-called reaction-limited regime. Here, we show that the picture is different for quantum systems. We consider the quantum reaction-limited regime and we show that for all the three processes power-law behavior beyond mean field is present as a consequence of quantum coherences, which are not related to space dimensionality. The decay in $3A\to \emptyset$ is further, highly intricate, since the power-law behavior therein only appears within an intermediate time window, while at long times the density decay is not power-law. Our results show that emergent critical behavior in quantum dynamics has a markedly different origin, based on quantum coherences, to that applying to classical critical phenomena, which is, instead, solely determined by the relevance of spatial correlations., Comment: 17 pages: 10 pages main text, 5 pages appendices and 2 pages bibliography. 6 figures: 4 figures in main text, 2 figures in the appendices

At large scales of space and time, the nonequilibrium dynamics of local observables in extensive many-body systems is well described by hydrodynamics. At the Euler scale, one assumes that each mesoscopic region independently reaches a state of maximal entropy under the constraints given by the available conservation laws. Away from phase transitions, maximal entropy states show exponential correlation decay, and independence of fluid cells might be assumed to subsist during the course of time evolution. We show that this picture is incorrect: under ballistic scaling, regions separated by macroscopic distances develop long-range correlations as time passes. These correlations take a universal form that only depends on the Euler hydrodynamics of the model. They are rooted in the large-scale motion of interacting fluid modes, and are the dominant long-range correlations developing in time from long-wavelength, entropy-maximised states. They require the presence of interaction and at least two different fluid modes, and are of a fundamentally different nature from well-known long-range correlations occurring from diffusive spreading or from quasi-particle excitations produced in far-from-equilibrium quenches. We provide a universal theoretical framework to exactly evaluate them, an adaptation of the macroscopic fluctuation theory to the Euler scale. We verify our exact predictions in the hard-rod gas, by comparing with numerical simulations and finding excellent agreement., Comment: 6 pages, 1 figure

We consider the quantum nonequilibrium dynamics of systems where fermionic particles coherently hop on a one-dimensional lattice and are subject to dissipative processes analogous to those of classical reaction-diffusion models. Particles can either annihilate in pairs, $A+A \to \emptyset$, coagulate upon contact, $A+A \to A$, and possibly also branch, $A \to A+A$. In classical settings, the interplay between these processes and particle diffusion leads to critical dynamics as well as to absorbing-state phase transitions. Here, we analyze the impact of coherent hopping and of quantum superposition, focusing on the so-called reaction-limited regime. Here, spatial density fluctuations are quickly smoothed out due to fast hopping, which for classical systems is described by a mean-field approach. By exploiting the time-dependent generalized Gibbs ensemble method, we demonstrate that quantum coherence and destructive interference play a crucial role in these systems and are responsible for the emergence of locally protected dark states and collective behavior beyond mean-field. This can manifest both at stationarity and during the relaxation dynamics. Our results highlight fundamental differences between classical nonequilibrium dynamics and their quantum counterpart and show that quantum effects indeed change collective universal behavior., Comment: 6+11 pages, 2+2 figures for main text and supplemental material, respectively. v3: close to published version. Selected as "Editors' Suggestion" in Physical Review Letters

We introduce a new universal framework describing fluctuations and correlations in quantum and classical many-body systems, at the Euler hydrodynamic scale of space and time. The framework adapts the ideas of the conventional macroscopic fluctuation theory (MFT) to systems that support ballistic transport. The resulting "ballistic MFT" (BMFT) is solely based on the Euler hydrodynamics data of the many-body system. Within this framework, mesoscopic observables are classical random variables depending only on the fluctuating conserved densities, and Euler-scale fluctuations are obtained by deterministically transporting thermodynamic fluctuations via the Euler hydrodynamics. Using the BMFT, we show that long-range correlations in space generically develop over time from long-wavelength inhomogeneous initial states in interacting models. This result, which we verify by numerical calculations, challenges the long-held paradigm that at the Euler scale, fluid cells may be considered uncorrelated. We also show that the Gallavotti-Cohen fluctuation theorem for non-equilibrium ballistic transport follows purely from time-reversal invariance of the Euler hydrodynamics. We check the validity of the BMFT by applying it to integrable systems, and in particular the hard-rod gas, with extensive simulations that confirm our analytical results., Comment: v1: 47 pages, 5 figures. v2: 48 pages, 5 figures; generalized proof of the GCFT and the conclusion slightly expanded. v3: 80 pages, 5 figures; single column ver

We introduce and investigate an open many-body quantum system in which kinetically constrained coherent and dissipative processes compete. The form of the incoherent dissipative dynamics is inspired by that of epidemic spreading or cellular-automaton-based computation related to the density-classification problem. It features two non-fluctuating absorbing states as well as a $\mathcal{Z}_2$-symmetric point in parameter space. The coherent evolution is governed by a kinetically constrained $\mathcal{Z}_2$-symmetric many-body Hamiltonian which is related to the quantum XOR-Fredrickson-Andersen model. We show that the quantum coherent dynamics can stabilize a fluctuating state and we characterize the transition between this active phase and the absorbing states. We also identify a rather peculiar behavior at the $\mathcal{Z}_2$-symmetric point. Here the system approaches the absorbing-state manifold with a dynamics that follows a power-law whose exponent continuously varies with the relative strength of the coherent dynamics. Our work shows how the interplay between coherent and dissipative processes as well as symmetry constraints may lead to a highly intricate non-equilibrium evolution and may stabilize phases that are absent in related classical problems., Comment: 10 pages, 4 figures

We investigate the dynamics of a non-interacting spin system, undergoing coherent Rabi oscillations, in the presence of stochastic resetting. We show that resetting generally induces long-range quantum and classical correlations both in the emergent dissipative dynamics and in the non-equilibrium stationary state. Moreover, for the case of conditional reset protocols -- where the system is reinitialized to a state dependent on the outcome of a preceding measurement -- we show that, in the thermodynamic limit, the spin system can feature collective behavior which results in a phenomenology reminiscent of that occurring in non-equilibrium phase transitions. The discussed reset protocols can be implemented on quantum simulators and quantum devices that permit fast measurement and readout of macroscopic observables, such as the magnetisation. Our approach does not require the control of coherent interactions and may therefore highlight a route towards a simple and robust creation of quantum correlations and collective non-equilibrium states, with potential applications in quantum enhanced metrology and sensing., Comment: 6+6 pages, 3 figures

We consider Markovian open quantum systems subject to stochastic resetting, which means that the dissipative time evolution is reset at randomly distributed times to the initial state. We show that the ensuing dynamics is non-Markovian and has the form of a generalized Lindblad equation. Interestingly, the statistics of quantum-jumps can be exactly derived. This is achieved by combining techniques from the thermodynamics of quantum-jump trajectories with the renewal structure of the resetting dynamics. We consider as an application of our analysis a driven two-level and an intermittent three-level system. Our findings show that stochastic resetting may be exploited as a tool to tailor the statistics of the quantum-jump trajectories and the dynamical phases of open quantum systems., Comment: 37 pages and 7 Figures

We consider closed quantum many-body systems subject to stochastic resetting. This means that their unitary time evolution is interrupted by resets at randomly selected times. When a reset takes place the system is reinitialized to a state chosen from a set of reset states conditionally on the outcome of a measurement taken immediately before resetting. We construct analytically the resulting non-equilibrium stationary state, thereby establishing an explicit connection between quantum quenches in closed systems and the emergent open system dynamics induced by stochastic resetting. We discuss as an application the paradigmatic transverse-field quantum Ising chain. We show that signatures of its ground-state quantum phase transition are visible in the steady state of the reset dynamics as a sharp crossover. Our findings show that a controlled stochastic resetting dynamics allows to design non-equilibrium stationary states of quantum many-body systems, where uncontrolled dissipation and heating can be prevented. These states can thus be created on demand and exploited, e.g., as a resource for quantum enhanced sensing on quantum simulator platforms., Comment: 23 pages. 6 pages main text and 17 pages supplemental material

We derive an exact formula for the scaled cumulant generating function of the time-integrated current associated to an arbitrary ballistically transported conserved charge. Our results rely on the Euler-scale description of interacting, many-body, integrable models out of equilibrium given by the generalized hydrodynamics, and on the large deviation theory. Crucially, our findings extend previous studies by accounting for inhomogeneous and dynamical initial states in interacting systems. We present exact expressions for the first three cumulants of the time-integrated current. Considering the non-interacting limit of our general expression for the scaled cumulant generating function, we further show that for the partitioning protocol initial state our result coincides with previous results of the literature. Given the universality of the generalized hydrodynamics, the expression obtained for the scaled cumulant generating function is applicable to any interacting integrable model obeying the hydrodynamic equations, both classical and quantum., Comment: 35 pages, 1 figure

We study the dynamics of the statistics of the energy transferred across a point along a quantum chain which is prepared in the inhomogeneous initial state obtained by joining two identical semi-infinite parts thermalized at two different temperatures. In particular, we consider the transverse field Ising and harmonic chains as prototypical models of non-interacting fermionic and bosonic excitations, respectively. Within the so-called hydrodynamic limit of large space-time scales we first discuss the mean values of the energy density and current, and then, aiming at the statistics of fluctuations, we calculate exactly the scaled cumulant generating function of the transferred energy. From the latter, the evolution of the associated large deviation function is obtained. A natural interpretation of our results is provided in terms of a semi-classical picture of quasi-particles moving ballistically along classical trajectories. Similarities and differences between the transferred energy scaled cumulant and the large deviation functions in the cases of non-interacting fermions and bosons are discussed., Comment: 29 pages, 8 figures

We devise an iterative scheme for numerically calculating dynamical two-point correlation functions in integrable many-body systems, in the Eulerian scaling limit. Expressions for these were originally derived in Ref. [1] by combining the fluctuation-dissipation principle with generalized hydrodynamics. Crucially, the scheme is able to address non-stationary, inhomogeneous situations, when motion occurs at the Euler-scale of hydrodynamics. In such situations, in interacting systems, the simple correlations due to fluid modes propagating with the flow receive subtle corrections, which we test. Using our scheme, we study the spreading of correlations in several integrable models from inhomogeneous initial states. For the classical hard-rod model we compare our results with Monte-Carlo simulations and observe excellent agreement at long time-scales, thus providing the first demonstration of validity for the expressions derived in Ref. [1]. We also observe the onset of the Euler-scale limit for the dynamical correlations., Comment: Submission to SciPost

Confinement of excitations induces quasilocalized dynamics in disorder-free isolated quantum many-body systems in one spatial dimension. This occurrence is signalled by severe suppression of quantum correlation spreading and of entanglement growth, long-time persistence of spatial inhomogeneities, and long-lived coherent oscillations of local observables. In this work, we present a unified understanding of these dramatic effects. The slow dynamical behavior is shown to be related to the Schwinger effect in quantum electrodynamics. We demonstrate that it is quantitatively captured for long time scales by effective Hamiltonians exhibiting Stark localization of excitations and weak growth of the entanglement entropy for arbitrary coupling strength. This analysis explains the phenomenology of real-time string dynamics investigated in a number of lattice gauge theories, as well as the anomalous dynamics observed in quantum Ising chains after quenches. Our findings establish confinement as a robust mechanism for hindering the approach to equilibrium in translationally-invariant quantum statistical systems with local interactions., Comment: 7+12 pages, 3+7 figures. v2: close to published version. Selected as "Editors' Suggestion" in Physical Review B

We study the statistics of large deviations of the intensive work done in an interaction quench of a one-dimensional Bose gas with a large number N of particles, system size L and fixed density. We consider the case in which the system is initially prepared in the non-interacting ground state and a repulsive interaction is suddenly turned on. For large deviations of the work below its mean value, we show that the large deviation principle holds by means of the quench action approach. Using the latter, we compute exactly the so-called rate function, and study its properties analytically. In particular, we find that fluctuations close to the mean value of the work exhibit a marked non- Gaussian behavior, even though their probability is always exponentially suppressed below it as L increases. Deviations larger than the mean value, instead, exhibit an algebraic decay, whose exponent can not be determined directly by large-deviation theory. Exploiting the exact Bethe ansatz representation of the eigenstates of the Hamiltonian, we calculate this exponent for vanishing particle density. Our approach can be straightforwardly generalized to quantum quenches in other interacting integrable systems., Comment: 24 Pages, 6 figures

The laws of thermodynamics require any initial macroscopic inhomogeneity in extended many-body systems to be smoothed out by the time evolution through the activation of transport processes. In generic, non-integrable quantum systems, transport is expected to be governed by a diffusion law, whereas a sufficiently strong quenched disorder can suppress it completely due to many-body localization of quantum excitations. Here we show that the confinement of quasi-particles can also lead to transport suppression even if the dynamics are generated by homogeneous Hamiltonians. We demonstrate this in the quantum Ising chain with transverse and longitudinal magnetic fields in the paradigmatic case of the evolution of domain-wall states. We perform extensive numerical simulations of the dynamics which turn out to be in excellent agreement with an effective analytical description valid within both weak and strong confinement regimes. Our results show that the energy flow from "hot" to "cold" regions of the chain is suppressed for all accessible times. We argue that this phenomenon is connected with the presence of atypical states in the many-body energy spectrum which violate the eigenstate thermalization hypothesis, as recently reported in the literature., Comment: 6+5 pages, 2+5 figures

We consider two semi-infinite quantum Ising chains initially at thermal equilibrium at two different temperatures and subsequently joined by an interaction between their end points. Transport properties such as the heat current are determined by the dynamics of the left- and right-moving fermionic quasi-particles which characterize the ensuing unitary dynamics. Within the so-called semi-classical space-time scaling limit we extend known results by determining the full space and time dependence of the density and current of energy and of fermionic quasi-particles. Upon approaching the edge of the propagating front, these quantities as well as the two-point correlation function display qualitatively different behaviors depending on the transverse field of the chain being critical or not. While in the latter case corrections to the leading behavior are described, as expected, by the Airy kernel, in the former a novel scaling form emerges with universal features., Comment: 29 pages, 11 figures