Quantum phase transition in a far-from-equilibrium steady state of an XY spin chain

Dissipation-Induced Extended-Localized Transition

A mobility edge (ME), representing the critical energy that distinguishes between extended and localized states, is a key concept in understanding the transition between extended (metallic) and localized (insulating) states in disordered and quasiperiodic systems. Here we explore the impact of dissipation on a quasiperiodic system featuring MEs by calculating steady-state density matrix and analyzing quench dynamics with sudden introduction of dissipation. We demonstrate that dissipation can lead the system into specific states predominantly characterized by either extended or localized states, irrespective of the initial state. Our results establish the use of dissipation as a new avenue for inducing transitions between extended and localized states and for manipulating dynamic behaviors of particles.

Signatures of quantum phases in a dissipative system

Lindbladian formalism, as tuned to dissipative and open systems, has been all-pervasive to interpret non-equilibrium steady states of quantum many-body systems. We study the fate of free fermionic and superconducting phases in a dissipative one-dimensional Kitaev model-where the bath acts both as a source and a sink of fermionic particles with different coupling rates. As a function of these two couplings, we investigate the steady state, its entanglement content, and its approach from varying initial states. Interestingly, we find that the steady state phase diagram retains decipherable signatures of ground state critical physics. We also show that early-time fidelity is a useful marker to find a subclass of phase transitions in such situations. Moreover, we show that the survival of critical signatures at late-times, strongly depend on the thermal nature of the steady state. This connection hints at a correspondence between quantum observables and classical magnetism in the steady state of such systems. Our work uncovers interesting connections between dissipative quantum many-body systems, thermalization of a classical spin and many-body quantum critical phenomena.

Quantum Criticality in Open Quantum Spin Chains with Nonreciprocity

We investigate the impact of nonreciprocity on universality and critical phenomena in open quantum interacting many-body systems. Nonreciprocal open quantum systems often have an exotic spectral sensitivity to boundary conditions, known as the Liouvillian skin effect (LSE). By considering an open quantum XXZ spin chain that exhibits LSE, we demonstrate the existence of a universal scaling regime that is not affected by the presence of the LSE. We resolve the critical exponents, which differ from those of free fermions, via tensor network methods and demonstrate that observables exhibit a universal scaling collapse, irrespective of the reciprocity. We find that the LSE only becomes relevant when a healing length scale ξ_{heal} at the system's edge (which is different from the localization length of the eigenstate of the Liouvillian) exceeds the system size, allowing edge properties to dominate the physics. We expect this result to be a generic feature of nonreciprocal models in the vicinity of a critical point. The driven-dissipative quantum criticality we observe has no classical analog and stems from the existence of multiple dark states.

Universal Subdiffusive Behavior at Band Edges from Transfer Matrix Exceptional Points

We discover a deep connection between parity-time symmetric optical systems and quantum transport in one-dimensional fermionic chains in a two-terminal open system setting. The spectrum of one dimensional tight-binding chain with periodic on-site potential can be obtained by casting the problem in terms of 2×2 transfer matrices. We find that these non-Hermitian matrices have a symmetry exactly analogous to the parity-time symmetry of balanced-gain-loss optical systems, and hence show analogous transitions across exceptional points. We show that the exceptional points of the transfer matrix of a unit cell correspond to the band edges of the spectrum. When connected to two zero temperature baths at two ends, this consequently leads to subdiffusive scaling of conductance with system size, with an exponent 2, if the chemical potential of the baths are equal to the band edges. We further demonstrate the existence of a dissipative quantum phase transition as the chemical potential is tuned across any band edge. Remarkably, this feature is analogous to transition across a mobility edge in quasiperiodic systems. This behavior is universal, irrespective of the details of the periodic potential and the number of bands of the underlying lattice. It, however, has no analog in absence of the baths.

Universal Cooling Dynamics toward a Quantum Critical Point

We investigate the loss of adiabaticity when cooling a many-body quantum system from an initial thermal state toward a quantum critical point. The excitation density, which quantifies the degree of adiabaticity of the dynamics, is found to obey scaling laws in the cooling velocity as well as in the initial and final temperatures of the cooling protocol. The scaling laws are universal, governed by the critical exponents of the quantum phase transition. The validity of these statements is shown analytically for a Kitaev quantum wire coupled to Markovian baths and argued to be valid under rather general conditions. Our results establish that quantum critical properties can be probed dynamically at finite temperature, without even varying the control parameter of the quantum phase transition.

Emergent Kardar-Parisi-Zhang Phase in Quadratically Driven Condensates

In bosonic gases at thermal equilibrium, an external quadratic drive can induce a Bose-Einstein condensation described by the Ising transition, as a consequence of the explicitly broken U(1) phase rotation symmetry down to Z_{2}. However, in physical realizations such as exciton polaritons and nonlinear photonic lattices, thermal equilibrium is lost and the state is rather determined by a balance between losses and external drive. A fundamental question is then how nonequilibrium fluctuations affect this transition. Here, we show that in a two-dimensional driven-dissipative Bose system the Ising phase is suppressed and replaced by a nonequilibrium phase featuring Kardar-Parisi-Zhang (KPZ) physics. Its emergence is rooted in a U(1)-symmetry restoration mechanism enabled by the strong fluctuations in reduced dimensionality. Moreover, we show that the presence of the quadratic drive term enhances the visibility of the KPZ scaling, compared to two-dimensional U(1)-symmetric gases, where it has remained so far elusive.

Subdiffusive Phases in Open Clean Long-Range Systems

We show that a one-dimensional ordered fermionic lattice system with power-law-decaying hopping, when connected to two baths at its two ends with different chemical potentials at zero temperature, features two phases showing subdiffusive scaling of conductance with system size. These phases have no analogues in the isolated system (i.e., in absence of the baths) where the transport is perfectly ballistic. In the open system scenario, interestingly, there occurs two chemical-potential-driven subdiffusive to ballistic phase transitions at zero temperature. We discuss how these phase transitions, to our knowledge, are different from all the known nonequilibrium quantum phase transitions. We provide a clear understanding of the microscopic origin of these phases and argue that the subdiffusive phases are robust against the presence of arbitrary number-conserving many-body interactions in the system. These phases showing subdiffusive scaling of conductance with system size in a two-terminal setup are therefore universal properties of all ordered one-dimensional number-conserving fermionic systems with power-law-decaying hopping at zero temperature.

Long-Range Coherence and Multiple Steady States in a Lossy Qubit Array

We show that a simple experimental setting of a locally pumped and lossy array of two-level quantum systems can stabilize states with strong long-range coherence. Indeed, by explicit analytic construction, we show there is an extensive set of steady-state density operators, from minimally to maximally entangled, despite this being an interacting open many-body problem. Such nonequilibrium steady states arise from a hidden symmetry that stabilizes Bell pairs over arbitrarily long distances, with unique experimental signatures. We demonstrate a protocol by which one can selectively prepare these states using dissipation. Our findings are accessible in present-day experiments.

Resolving a Discrepancy between Liouvillian Gap and Relaxation Time in Boundary-Dissipated Quantum Many-Body Systems

The gap of the Liouvillian spectrum gives the asymptotic decay rate of a quantum dissipative system, and therefore its inverse has been identified as the slowest relaxation time. Contrary to this common belief, we show that the relaxation time due to diffusive transports in a boundary dissipated many-body quantum system is determined not by the gap or low-lying eigenvalues of the Liouvillian but by superexponentially large expansion coefficients for Liouvillian eigenvectors with nonsmall eigenvalues at an initial state. This finding resolves an apparent discrepancy reported in the literature between the inverse of the Liouvillian gap and the relaxation time in dissipative many-body quantum systems.

Uncoupled Majorana fermions in open quantum systems: on the efficient simulation of non-equilibrium stationary states of quadratic Fermi models

A decomposition of the non-equilibrium stationary state of a quadratic Fermi system influenced by linear baths is obtained and used to establish a simulation protocol in terms of tensor states. The scheme is then applied to examine the occurrence of uncoupled Majorana fermions in Kitaev chains subject to baths on the ends. The resulting phase diagram is compared against the topological characterization of the equilibrium chain and the protocol efficiency is studied with respect to this model.

Thermalization in open many-body systems based on eigenstate thermalization hypothesis

We investigate steady states of macroscopic quantum systems under dissipation not obeying the detailed balance condition. We argue that the Gibbs state at an effective temperature gives a good description of the steady state provided that the system Hamiltonian obeys the eigenstate thermalization hypothesis (ETH) and the perturbation theory in the weak system-environment coupling is valid in the thermodynamic limit. We derive a criterion to guarantee the validity of the perturbation theory, which is satisfied in the thermodynamic limit for sufficiently weak dissipation when the Liouvillian is gapped for bulk-dissipated systems, while the perturbation theory breaks down in boundary-dissipated chaotic systems due to the presence of diffusive transports. We numerically confirm these theoretical predictions. This paper suggests a connection between steady states of macroscopic open quantum systems and the ETH.

Exact Nonequilibrium Steady State of Open XXZ/XYZ Spin-1/2 Chain with Dirichlet Boundary Conditions

We investigate a dissipatively driven XYZ spin-1/2 chain in the Zeno limit of strong dissipation, described by the Lindblad master equation. The nonequilibrium steady state is expressed in terms of a matrix product ansatz using novel site-dependent Lax operators. The components of Lax operators satisfy a simple set of linear recurrence equations that generalize the defining algebraic relations of the quantum group U_{q}(sl_{2}). We reveal connection between the nonequilibrium steady state of the nonunitary dynamics and the respective integrable model with edge magnetic fields, described by coherent unitary dynamics.

Spectral Gaps and Midgap States in Random Quantum Master Equations

We discuss the decay rates of chaotic quantum systems coupled to noise. We model both the Hamiltonian and the system-noise coupling by random N×N Hermitian matrices, and study the spectral properties of the resulting Liouvillian superoperator. We consider various random-matrix ensembles, and find that for all of them the asymptotic decay rate remains nonzero in the thermodynamic limit; i.e., the spectrum of the superoperator is gapped as N→∞. For finite N, the probability of finding a very small gap vanishes as P(Δ)∼Δ^{cN}, where c is insensitive to the dissipation strength. A sharp spectral transition takes place as the dissipation strength is increased: for dissipation beyond a critical strength, the slowest-decaying eigenvalues of the Liouvillian correspond to isolated "midgap" states. We give evidence that midgap states exist also for nonrandom system-noise coupling and discuss some experimental implications of the above results.

Critical Behavior of the Quantum Contact Process in One Dimension

The contact process is a paradigmatic classical stochastic system displaying critical behavior even in one dimension. It features a nonequilibrium phase transition into an absorbing state that has been widely investigated and shown to belong to the directed percolation universality class. When the same process is considered in a quantum setting, much less is known. So far, mainly semiclassical studies have been conducted and the nature of the transition in low dimensions is still a matter of debate. Also, from a numerical point of view, from which the system may look fairly simple-especially in one dimension-results are lacking. In particular, the presence of the absorbing state poses a substantial challenge, which appears to affect the reliability of algorithms targeting directly the steady state. Here we perform real-time numerical simulations of the open dynamics of the quantum contact process and shed light on the existence and on the nature of an absorbing state phase transition in one dimension. We find evidence for the transition being continuous and provide first estimates for the critical exponents. Beyond the conceptual interest, the simplicity of the quantum contact process makes it an ideal benchmark problem for scrutinizing numerical methods for open quantum nonequilibrium systems.

Mixed-Order Symmetry-Breaking Quantum Phase Transition Far from Equilibrium

We study the current-carrying steady state of a transverse field Ising chain coupled to magnetic thermal reservoirs and obtain the nonequilibrium phase diagram as a function of the magnetization potential of the reservoirs. Upon increasing the magnetization bias we observe a discontinuous jump of the magnetic order parameter that coincides with a divergence of the correlation length. For steady states with a nonvanishing conductance, the entanglement entropy at zero temperature displays a bias dependent logarithmic correction that violates the area law and differs from the well-known equilibrium case. Our findings show that out-of-equilibrium conditions allow for novel critical phenomena not possible at equilibrium.

Observation of the Crossover from Photon Ordering to Delocalization in Tunably Coupled Resonators

Networks of nonlinear resonators offer intriguing perspectives as quantum simulators for nonequilibrium many-body phases of driven-dissipative systems. Here, we employ photon correlation measurements to study the radiation fields emitted from a system of two superconducting resonators in a driven-dissipative regime, coupled nonlinearly by a superconducting quantum interference device, with cross-Kerr interactions dominating over on-site Kerr interactions. We apply a parametrically modulated magnetic flux to control the linear photon hopping rate between the two resonators and its ratio with the cross-Kerr rate. When increasing the hopping rate, we observe a crossover from an ordered to a delocalized state of photons. The presented coupling scheme is intrinsically robust to frequency disorder and may therefore prove useful for realizing larger-scale resonator arrays.

Uhlmann curvature in dissipative phase transitions

A novel approach based on the Uhlmann curvature is introduced for the investigation of non-equilibrium steady-state quantum phase transitions (NESS-QPTs). Equilibrium phase transitions fall invariably into two markedly non-overlapping categories: classical phase transitions and quantum phase transitions. NESS-QPTs offer a unique arena where such a distinction fades off. We propose a method to reveal and quantitatively assess the quantum character of such critical phenomena. We apply this tool to a paradigmatic class of lattice fermion systems with local reservoirs, characterised by Gaussian non-equilibrium steady states. The relations between the behaviour of the Uhlmann curvature, the divergence of the correlation length, the character of the criticality and the dissipative gap are demonstrated. We argue that this tool can shade light upon the nature of non equilibrium steady state criticality in particular with regard to the role played by quantum vs classical fluctuations.

Solvable Family of Driven-Dissipative Many-Body Systems

Exactly solvable models have played an important role in establishing the sophisticated modern understanding of equilibrium many-body physics. Conversely, the relative scarcity of solutions for nonequilibrium models greatly limits our understanding of systems away from thermal equilibrium. We study a family of nonequilibrium models, some of which can be viewed as dissipative analogues of the transverse-field Ising model, in that an effectively classical Hamiltonian is frustrated by dissipative processes that drive the system toward states that do not commute with the Hamiltonian. Surprisingly, a broad and experimentally relevant subset of these models can be solved efficiently. We leverage these solutions to compute the effects of decoherence on a canonical trapped-ion-based quantum computation architecture, and to prove a no-go theorem on steady-state phase transitions in a many-body model that can be realized naturally with Rydberg atoms or trapped ions.

Stochastic thermodynamics of quantum maps with and without equilibrium

We study stochastic thermodynamics for a quantum system of interest whose dynamics is described by a completely positive trace-preserving (CPTP) map as a result of its interaction with a thermal bath. We define CPTP maps with equilibrium as CPTP maps with an invariant state such that the entropy production due to the action of the map on the invariant state vanishes. Thermal maps are a subgroup of CPTP maps with equilibrium. In general, for CPTP maps, the thermodynamic quantities, such as the entropy production or work performed on the system, depend on the combined state of the system plus its environment. We show that these quantities can be written in terms of system properties for maps with equilibrium. The relations that we obtain are valid for arbitrary coupling strengths between the system and the thermal bath. The fluctuations of thermodynamic quantities are considered in the framework of a two-point measurement scheme. We derive the entropy production fluctuation theorem for general maps and a fluctuation relation for the stochastic work on a system that starts in the Gibbs state. Some simplifications for the probability distributions in the case of maps with equilibrium are presented. We illustrate our results by considering spin 1/2 systems under thermal maps, nonthermal maps with equilibrium, maps with nonequilibrium steady states, and concatenations of them. Finally, and as an important application, we consider a particular limit in which the concatenation of maps generates a continuous time evolution in Lindblad form for the system of interest, and we show that the concept of maps with and without equilibrium translates into Lindblad equations with and without quantum detailed balance, respectively. The consequences for the thermodynamic quantities in this limit are discussed.

Rich complex behaviour of self-assembled nanoparticles far from equilibrium

A profoundly fundamental question at the interface between physics and biology remains open: what are the minimum requirements for emergence of complex behaviour from nonliving systems? Here, we address this question and report complex behaviour of tens to thousands of colloidal nanoparticles in a system designed to be as plain as possible: the system is driven far from equilibrium by ultrafast laser pulses that create spatiotemporal temperature gradients, inducing Marangoni flow that drags particles towards aggregation; strong Brownian motion, used as source of fluctuations, opposes aggregation. Nonlinear feedback mechanisms naturally arise between flow, aggregate and Brownian motion, allowing fast external control with minimal intervention. Consequently, complex behaviour, analogous to those seen in living organisms, emerges, whereby aggregates can self-sustain, self-regulate, self-replicate, self-heal and can be transferred from one location to another, all within seconds. Aggregates can comprise only one pattern or bifurcated patterns can coexist, compete, endure or perish.

Quantum simulations and many-body physics with light

In this review we discuss the works in the area of quantum simulation and many-body physics with light, from the early proposals on equilibrium models to the more recent works in driven dissipative platforms. We start by describing the founding works on Jaynes-Cummings-Hubbard model and the corresponding photon-blockade induced Mott transitions and continue by discussing the proposals to simulate effective spin models and fractional quantum Hall states in coupled resonator arrays (CRAs). We also analyse the recent efforts to study out-of-equilibrium many-body effects using driven CRAs, including the predictions for photon fermionisation and crystallisation in driven rings of CRAs as well as other dynamical and transient phenomena. We try to summarise some of the relatively recent results predicting exotic phases such as super-solidity and Majorana like modes and then shift our attention to developments involving 1D nonlinear slow light setups. There the simulation of strongly correlated phases characterising Tonks-Girardeau gases, Luttinger liquids, and interacting relativistic fermionic models is described. We review the major theory results and also briefly outline recent developments in ongoing experimental efforts involving different platforms in circuit QED, photonic crystals and nanophotonic fibres interfaced with cold atoms.

The thermodynamic cost of driving quantum systems by their boundaries

The laws of thermodynamics put limits to the efficiencies of thermal machines. Analogues of these laws are now established for quantum engines weakly and passively coupled to the environment providing a framework to find improvements to their performance. Systems whose interaction with the environment is actively controlled do not fall in that framework. Here we consider systems actively and locally coupled to the environment, evolving with a so-called boundary-driven Lindblad equation. Starting from a unitary description of the system plus the environment we simultaneously obtain the Lindblad equation and the appropriate expressions for heat, work and entropy-production of the system extending the framework for the analysis of new, and some already proposed, quantum heat engines. We illustrate our findings in spin 1/2 chains and explain why an XX chain coupled in this way to a single heat bath relaxes to thermodynamic-equilibrium while and XY chain does not. Additionally, we show that an XX chain coupled to a left and a right heat baths behaves as a quantum engine, a heater or refrigerator depending on the parameters, with efficiencies bounded by Carnot efficiencies.

Relaxation times of dissipative many-body quantum systems

We study relaxation times, also called mixing times, of quantum many-body systems described by a Lindblad master equation. We in particular study the scaling of the spectral gap with the system length, the so-called dynamical exponent, identifying a number of transitions in the scaling. For systems with bulk dissipation we generically observe different scaling for small and for strong dissipation strength, with a critical transition strength going to zero in the thermodynamic limit. We also study a related phase transition in the largest decay mode. For systems with only boundary dissipation we show a generic bound that the gap cannot be larger than ∼1/L. In integrable systems with boundary dissipation one typically observes scaling of ∼1/L(3), while in chaotic ones one can have faster relaxation with the gap scaling as ∼1/L and thus saturating the generic bound. We also observe transition from exponential to algebraic gap in systems with localized modes.

Identifying a bath-induced bose liquid in interacting spin-boson models

We study the ground state phase diagram of a one-dimensional hard-core bosonic model with nearest-neighbor interactions (XXZ model) where every site is coupled Ohmically to an independent but identical reservoir, hereby generalizing spin-boson models to interacting spin-boson systems. We show that a bath-induced Bose liquid phase can occur in the ground state phase diagram away from half filling. This phase is compressible, gapless, and conducting but not superfluid. At half filling, only a Luttinger liquid and a charge density wave are found. The phase transition between them is of Kosterlitz-Thouless type where the Luttinger parameter takes a nonuniversal value. The applied quantum Monte Carlo method can be used for all open bosonic and unfrustrated spin systems, regardless of their dimension, filling factor, and spectrum of the dissipation as long as the quantum system couples to the bath via the density operators.

Quantum information-geometry of dissipative quantum phase transitions

A general framework for analyzing the recently discovered phase transitions in the steady state of dissipation-driven open quantum systems is still lacking. To fill this gap, we extend the so-called fidelity approach to quantum phase transitions to open systems whose steady state is a Gaussian fermionic state. We endow the manifold of correlation matrices of steady states with a metric tensor g measuring the distinguishability distance between solutions corresponding to a different set of control parameters. The phase diagram can then be mapped out in terms of the scaling behavior of g and connections with the Liouvillean gap and the model correlation functions unveiled. We argue that the fidelity approach, thanks to its differential-geometric and information-theoretic nature, provides insights into dissipative quantum critical phenomena as well as a general and powerful strategy to explore them.

Algebraic versus Exponential Decoherence in Dissipative Many-Particle Systems

The interplay between dissipation and internal interactions in quantum many-body systems gives rise to a wealth of novel phenomena. Here we investigate spin-1/2 chains with uniform local couplings to a Markovian environment using the time-dependent density matrix renormalization group. For the open XXZ model, we discover that the decoherence time diverges in the thermodynamic limit. The coherence decay is then algebraic instead of exponential. This is due to a vanishing gap in the spectrum of the corresponding Liouville superoperator and can be explained on the basis of a perturbative treatment. In contrast, decoherence in the open transverse-field Ising model is found to be always exponential. In this case, the internal interactions can both facilitate and impede the environment-induced decoherence.

Criticality in transport through the quantum Ising chain

We consider thermal transport between two reservoirs coupled by a quantum Ising chain as a model for nonequilibrium physics induced in quantum-critical many-body systems. By deriving rate equations based on exact expressions for the quasiparticle pairs generated during the transport, we observe signatures of the underlying quantum phase transition in the steady-state energy current already at finite and different reservoir temperatures.

Quantum transport efficiency and Fourier's law

We analyze the steady-state energy transfer in a chain of coupled two-level systems connecting two thermal reservoirs. Through an analytic treatment we find that the energy current is independent of the system size, hence violating Fourier's law of heat conduction. The classical diffusive behavior in Fourier's law of heat conduction can be recovered by introducing decoherence to the quantum systems constituting the chain. We relate these results to recent discussions of energy transport in biological light-harvesting systems, and discuss the role of quantum coherence and entanglement.

Theory of quantum energy transfer in spin chains: superexchange and ballistic motion

Quantum energy transfer in a chain of two-level (spin) units, connected at its ends to two thermal reservoirs, is analyzed in two limits: (i) in the off-resonance regime, when the characteristic subsystem excitation energy gaps are larger than the reservoirs frequencies, or the baths temperatures are low and (ii) in the resonance regime, when the chain excitation gaps match populated bath modes. In the latter case, the model is studied using a master equation approach, showing that the dynamics is ballistic for the particular chain model explored. In the former case, we analytically study the system dynamics utilizing the recently developed Energy-Transfer Born-Oppenheimer formalism [L.-A. Wu and D. Segal, Phys. Rev. E 83, 051114 (2011)], demonstrating that energy transfers across the chain in a superexchange (bridge assisted tunneling) mechanism, with the energy current decreasing exponentially with distance. This behavior is insensitive to the chain details. Since at low temperatures the excitation spectrum of molecular systems can be truncated to resemble a spin chain model, we argue that the superexchange behavior obtained here should be observed in widespread systems satisfying the off-resonance condition.

Exact nonequilibrium steady state of a strongly driven open XXZ chain

An exact and explicit ladder-tensor-network ansatz is presented for the nonequilibrium steady state of an anisotropic Heisenberg XXZ spin-1/2 chain which is driven far from equilibrium with a pair of Lindblad operators acting on the edges of the chain only. We show that the steady-state density operator of a finite system of size n is-apart from a normalization constant-a polynomial of degree 2n - 2 in the coupling constant. Efficient computation of physical observables is facilitated in terms of a transfer operator reminiscent of a classical Markov process. In the isotropic case we find cosine spin profiles, 1/n(2) scaling of the spin current, and long-range correlations in the steady state. This is a fully nonperturbative extension of a recent result [Phys. Rev. Lett. 106, 217206 (2011)].

Nonequilibrium phase transition in a periodically driven XY spin chain

We present a general formulation of Floquet states of periodically time-dependent open Markovian quasifree fermionic many-body systems in terms of a discrete Lyapunov equation. Illustrating the technique, we analyze periodically kicked XY spin-½ chain which is coupled to a pair of Lindblad reservoirs at its ends. A complex phase diagram is reported with reentrant phases of long range and exponentially decaying spin-spin correlations as some of the system's parameters are varied. The structure of phase diagram is reproduced in terms of counting nontrivial stationary points of Floquet quasiparticle dispersion relation.

Solvable quantum nonequilibrium model exhibiting a phase transition and a matrix product representation

We study a one-dimensional XX chain under nonequilibrium driving and local dephasing described by the Lindblad master equation. The analytical solution for the nonequilibrium steady state found for particular parameters in a previous study [M. Žnidarič J. Stat. Mech. (2010) L05002] is extended to arbitrary coupling constants, driving, and a homogeneous magnetic field. All one-, two-, and three-point correlation functions are explicitly evaluated. It is shown that the nonequilibrium stationary state is not Gaussian. Nevertheless, in the thermodynamic and weak-driving limit it is only weakly correlated and can be described by a matrix product operator ansatz with matrices of fixed dimension 4. A nonequilibrium phase transition at zero dephasing is also discussed. It is suggested that the scaling of the relaxation time with the system size can serve as a signature of a nonequilibrium phase transition.

Long-range order in nonequilibrium interacting quantum spin chains

We conjecture that nonequilibrium boundary conditions generically trigger long-range order in nonequilibrium steady states of locally interacting quantum chains. Our result is based on large scale density matrix renormalization group simulations of several models of quantum spin-1/2 chains which are driven far from equilibrium by coupling to a pair of unequal Lindblad reservoirs attached locally to the ends of the chain. In particular, we find a phase transition from exponentially decaying to long-range spin-spin correlations in an integrable Heisenberg XXZ chain by changing the anisotropy parameter. Long-range order also typically emerges after breaking the integrability of the model.

Fermionized photons in an array of driven dissipative nonlinear cavities

We theoretically investigate the optical response of a one-dimensional array of strongly nonlinear optical microcavities. When the optical nonlinearity is much larger than both losses and intercavity tunnel coupling, the nonequilibrium steady state of the system is reminiscent of a strongly correlated Tonks-Girardeau gas of impenetrable bosons. Signatures of strong correlations are identified in the transmission spectrum of the system, as well as in the intensity correlations of the transmitted light. Possible experimental implementations in state-of-the-art solid-state devices are discussed.

Quantum nonequilibrium steady states induced by repeated interactions

We study the steady state of a finite XX chain coupled at its boundaries to quantum reservoirs made of free spins that interact one after the other with the chain. The two-point correlations are calculated exactly, and it is shown that the steady state is completely characterized by the magnetization profile and the associated current. Except at the boundary sites, the magnetization is given by the average of the reservoirs' magnetizations. The steady-state current, proportional to the difference in the reservoirs' magnetizations, shows a nonmonotonic behavior with respect to the system-reservoir coupling strength, with an optimal current state for a finite value of the coupling. Moreover, we show that the steady state can be described by a generalized Gibbs state.