Entanglement in nonequilibrium systems is considered. A general definition for entanglement measure is introduced, which can be applied for characterizing the level of entanglement produced by arbitrary operators. Applying this definition to reduced density matrices makes it possible to measure the entanglement in nonequilibrium as well as in equilibrium statistical systems. An example of a multimode Bose-Einstein condensate is discussed.
Evolutional entanglement production is defined as the amount of entanglement produced by the evolution operator. This quantity is analyzed for systems whose Hamiltonians are characterized by spin operators. The evolutional entanglement production at the initial stage grows quadratically in time. For longer times, it oscillates, being quasiperiodic or periodic depending on the system parameters.
In recent years it has been found that quantum systems can posses entanglement in equilibrium thermal states provided temperature is low enough. In the present work we explore a possibility of having entanglement in nonequilibrium stationary states. We show analytically that, in a simple one-dimensional spin chain, there is entanglement even at highest attainable energies; that is, starting from an equilibrium state at infinite temperature, a sufficiently strong driving can induce entanglement, even in the thermodynamic limit. We also show that dissipative dephasing, on the other hand, destroy…
We study entanglement properties in a nonequilibrium steady state of a free-fermion chain, that emerges after connecting two half-chains prepared at different temperatures. The entanglement negativity and the Rényi mutual information between two adjacent intervals scale logarithmically in the system size, with prefactors that we calculate analytically as a function of the bath temperatures. In particular, we show that the negativity and the Rényi mutual information with index $α=1/2$ are described by different prefactors, and thus the two quantities provide inequivalent information about the s…
We investigate the size scaling of the entanglement entropy (EE) in nonequilibrium steady states (NESSs) of a one-dimensional open quantum system with a random potential. It models a mesoscopic conductor, composed of a long quantum wire (QWR) with impurities and two electron reservoirs at zero temperature. The EE at equilibrium obeys the logarithmic law. However, in NESSs far from equilibrium the EE grows anomalously fast, obeying the `quasi volume law,' although the conductor is driven by the zero-temperature reservoirs. This anomalous behavior arises from both the far from equilibrium condit…
Entanglement production, generated by an evolution operator, is considered. The measure of entanglement production, introduced earlier by one of the authors, is employed. As an illustration, a two-qubit register is studied and the corresponding measure of evolutional entanglement production is calculated. Such two-qubit registers can be realized by atomic systems in a double well or by trapped atoms with two coherent modes.
We investigate nonequilibrium entanglement generation in a chain of harmonic oscillators with time-dependent linear coupling. We use optimal control theory to determine the coupling modulation that leads to maximum logarithmic negativity for a pair of opposite oscillators and show that it corresponds to a synchronization of the eigenmodes of the chain. We further analytically relate the maximum attainable entanglement to the irreversible work done to produce it, thus bridging nonequilibrium entanglement production and nonequilibrium thermodynamics.
We study the entanglement and the Bell nonlocality of a coupled two-qubit system, in which each qubit is coupled with one individual environment. We study how the nonequilibrium environments (with different temperatures or chemical potentials) influence the entanglement and the Bell nonlocality. The nonequilibrium environments can have constructive effects on the entanglement and the Bell nonlocality. Nonequilibrium thermodynamic cost can sustain the thermal energy or particle current and enhance the entanglement and the Bell nonlocality. However, the nonequilibrium conditions (characterized b…
We study the nonequilibrium dynamics of a quasiperiodic quantum Ising chain after a sudden change in the strength of the transverse field at zero temperature. In particular we consider the dynamics of the entanglement entropy and the relaxation of the magnetization. The entanglement entropy increases with time as a power-law, and the magnetization is found to exhibit stretched-exponential relaxation. These behaviors are explained in terms of anomalously diffusing quasiparticles, which are studied in a wave packet approach. The nonequilibrium magnetization is shown to have a dynamical phase tra…
We investigate analytically and numerically the steady-state entanglement and coherence of two coupled qubits each interacting with a local boson or fermion reservoir, based on the Bloch-Redfield master equation beyond the secular approximation. We find that there is non-vanishing steady-state coherence in the nonequilibrium scenario, which grows monotonically with the nonequilibrium condition quantified by the temperature difference or chemical potential difference of the two baths. The steady-state entanglement in general is a non-monotonic function of the nonequilibrium condition as well as…
Results on heat current, entropy production rate and entanglement are reported for a quantum system coupled to two different temperature heat reservoirs. By applying a temperature gradient, different quantum states can be found with exactly the same amount of entanglement but different purity degrees and heat currents. Furthermore, a nonequilibrium enhancement-suppression transition behavior of the entanglement is identified.
Einstein-Podolsky-Rosen steering describes a quantum correlation in addition to entanglement and Bell nonlocality. However, conceptually different from entanglement and Bell nonlocality, quantum steering has an asymmetric definition. Motivated by the asymmetric definition of quantum steering, we study the steerability of two-interacting qubits, which have asymmetric energy levels, coupled with asymmetric environments. The asymmetric (nonequilibrium) environments are two environments with different temperatures or chemical potentials. The Bloch-Redfield equation is applied to study the dynamics…
We study the steady state entanglement and correlations of an open system comprised of two coupled fermions in the equilibrium or nonequilibrium environments and distill the nonequilibrium contribution to the quantum correlations. We show that in the equilibrium condition, the steady-state quantum correlations exhibit non-monotonic behavior, while in the nonequilibrium case, the monotonicity is determined by many parameters. The entanglement vanishes abruptly upon the increase of the temperature (bias) and chemical potential bias, it witnesses a critical chemical potential above which the conc…
The dynamics of two spins coupled to bosonic baths at different temperatures is studied. The analytical solution for the reduced density matrix of the system in the Markovian and Post-Markovian case with exponential memory kernel is found. The dynamics and temperature dependence of spin-spin entanglement is analyzed.
The dynamics of a simple spin chain (2 spins) coupled to bosonic baths at different temperatures is studied. The analytical solution for the reduced density matrix of the system is found. The dynamics and temperature dependence of spin-spin entanglement is analyzed. It is shown that the system converges to a steady-state. If the energy levels of the two spins are different, the steady-state concurrence assumes its maximum at unequal bath temperatures. It is found that a difference in local energy levels can make the steady-state entanglement more stable against high temperatures.
We propose schemes for generating spatially-separated spin entanglement in systems of two quantum dots with onsite Coulomb repulsion weakly coupled to a joint electron reservoir. An enhanced probability for the formation of spin entanglement is found in nonequilibrium situations with one extra electron on each dot, either in the transient state after rapid changes of the gate voltage, or in the steady state with applied bias voltage. In both cases so-called Werner states with spin singlet fidelity exceeding 1/2 are generated, which indicates entanglement.
We study the process of heat transfer through an entangled pair of two-level system, demonstrating the role of quantum correlations in this nonequilibrium process. While quantum correlations generally degrade with increasing the temperature bias, introducing spatial asymmetry leads to an intricate behavior: Connecting the qubits unequally to the reservoirs one finds that quantum correlations persist and increase with the temperature bias when the system is more weakly linked to the hot reservoir. In the reversed case, linking the system more strongly to the hot bath, the opposite, more natural…
The dynamics of a chain of three spins coupled at both ends to separate bosonic baths at different temperatures is studied. An exact analytical so- lution of the master equation in the Born-Markov approximation for the reduced density matrix of the chain is constructed. It is shown that for long times the reduced density matrix converges to the non-equilibrium steady- state. Dynamical and steady state properties of the concurrence between the first and the last spin are studied.
The dynamics of simple qubit systems in a chain configuration coupled at both ends to separate bosonic baths at different temperatures is studied. An exact analytical solution of the master equation in the Born-Markov approximation for the reduced density matrix of the qubit system is constructed. The unique non-equilibrium stationary state for the long time behavior of the reduced density matrix in obtained. Dynamical and steady state properties of the concurrence between the first and the last spin are studied.
Making use of the master equation and effective Hamiltonian approach, we investigate the steady state entanglement in a three-qubit $XX$ model. Both symmetric and nonsymmetric qubit-qubit couplings are considered. The system (the three qubits) is coupled to two bosonic baths at different temperatures. We calculate the steady state by the effective Hamiltonian approach and discuss the dependence of the steady state entanglement on the temperatures and couplings. The results show that for symmetric qubit-qubit couplings, the entanglements between the nearest neighbor are equal, independent of th…
In this work, we extend the recently established system-bath entanglement theorem (SBET) [J. Chem. Phys. 152, 034102 (2020)] to the nonequilibrium scenario, in which an arbitrary system couples to multiple Gaussian baths environments at different temperatures. While the existing SBET connects the entangled system-bath response functions to those of local systems, the extended theory is concerned with the nonequilibrium steady-state quantum transport current through molecular junctions. The new theory is established on the basis of the generalized Langevin equation, with a close relation to non…
It is shown that a nonequilibrium environment can be instrumental in suppressing decoherence between distinct decoherence free subspaces in quantum registers. The effect is found in the framework of exact coherent-product solutions for model registers decohering in a bath of degenerate harmonic modes, through couplings linear in bath coordinates. These solutions represent a natural nonequilibrium extension of the standard solution for a decoupled initial register state and a thermal environment. Under appropriate conditions, the corresponding reduced register distribution can propagate in an u…
We study in detail the generation and relaxation of quantum coherences (entanglement) in a system of coupled polariton traps. By exploiting a Lie algebraic based super-operator technique we provide an analytical exact solution for the Markovian dissipative dynamics (Master equation) of such system which is valid for arbitrary cluster size, polariton-polariton interaction strength, temperature and initial state. Based on the exact solution of the Master equation at T=0K, we discuss how dissipation affects the quantum entanglement dynamics of coupled polariton systems.
The possibility of maintaining entanglement in a quantum system at finite, even high, temperatures -- the so-called `hot entanglement' -- has obvious practical interest, but also requires closer theoretical scrutiny. Since quantum entanglement in a system evolves in time and is continuously subjected to environmental degradation, a nonequilibrium description by way of open quantum systems is called for. To identify the key issues and the contributing factors that may permit `hot entanglement' to exist, or the lack thereof, we carry out a model study of two spatially-separated, coupled oscillat…
The random magnetic field produced by nuclear spins has long been viewed as the dominating source of decoherence in the quantum-dot based spins. Here we obtain in both exact and analytical manner the dynamics of spin qubits coupled to nuclear spin environments via the hyperfine interaction, going beyond the weak system-bath interaction and Markovian approximation. We predict that the detailed-balance breaking produced by chemical potential gradient in nuclear baths leads to the rapid oscillations of populations, quantum coherence and entanglement, which are absent in the conventional case (i.e…
A general description of entanglement is suggested as an action realized by an arbitrary operator over given disentangled states. The related entanglement measure is defined. Because of its generality, this definition can be employed for any physical systems, pure or mixed, equilibrium or nonequilibrium, and characterized by any type of operators, whether these are statistical operators, field operators, spin operators, or anything else. Entanglement of any number of parts from their total ensemble forming a multiparticle composite system can be determined. Interplay between entanglement and o…
Entanglement plays a prominent role in the study of condensed matter many-body systems: Entanglement measures not only quantify the possible use of these systems in quantum information protocols, but also shed light on their physics. However, exact analytical results remain scarce, especially for systems out of equilibrium. In this work we examine a paradigmatic one-dimensional fermionic system that consists of a uniform tight-binding chain with an arbitrary scattering region near its center, which is subject to a DC bias voltage at zero temperature. The system is thus held in a current-carryi…
This is the second of a series of three papers examining how viable it is for entanglement to be sustained at high temperatures for quantum systems in thermal equilibrium (Case A), in nonequilibrium (Case B) and in nonequilibrium steady state conditions (Case C). The system we analyze here consists of two coupled quantum harmonic oscillators each interacting with its own bath described by a scalar field, set at temperatures $T_1 > T_2$. For \textit{constant bilinear inter-oscillator coupling} studied here (Case C1) owing to the Gaussian nature, the problem can be solved exactly at arbitrary te…
The problem of entanglement produced by an arbitrary operator is formulated and a related measure of entanglement production is introduced. This measure of entanglement production satisfies all properties natural for such a characteristic. A particular case is the entanglement produced by a density operator or a density matrix. The suggested measure is valid for operations over pure states as well as over mixed states, for equilibrium as well as nonequilibrium processes. Systems of arbitrary nature can be treated, described either by field operators, spin operators, or any other kind of operat…
General nonequilibrium quantum transport equations are derived for a coupled system of charge carriers, Dirac spin, isospin (or valley spin), and pseudospin, such as either one of the band, layer, impurity, and boundary pseudospins. Limiting cases are obtained for one, two or three different kinds of spin ocurring in a system. We show that a characteristic integer number $N_{s}$ determines the formal form of spin quantum transport equations, irrespective of the type of spins or pseudospins, as well as the maximal entanglement entropy. The results may shed a new perspective on the mechanism lea…
Entanglement pre-thermalization (EP) is a quasi-stationary nonequilibrium state of a composite system in which each individual subsystem looks thermal but the entire system remains nonthermal due to quantum entanglement between subsystems. We theoretically study the dynamics of EP following a coherent split of a one-dimensional harmonic potential in which two interacting bosons are confined. This problem is equivalent to that of an interaction quench between two harmonic oscillators. We show that this simple model captures the bare essentials of EP; that is, each subsystem relaxes to an approx…
Many-body localized systems in which interactions and disorder come together defy the expectations of quantum statistical mechanics: In contrast to ergodic systems, they do not thermalize when undergoing nonequilibrium dynamics. What is less clear, however, is how topological features interplay with many-body localized phases as well as the nature of the transition between a topological and a trivial state within the latter. In this work, we numerically address these questions, using a combination of extensive tensor network calculations, specifically DMRG-X, as well as exact diagonalization, …
We show the existence of an entangled nonequilibrium state at very high temperatures when two linearly coupled harmonic oscillators are parametrically driven and dissipate into two independent heat baths. This result has a twofold meaning: first, it fundamentally shifts the classical-quantum border to temperatures as high as our experimental ability allows us, and second, it can help increase by at least one order of magnitude the temperature at which current experimental setups are operated.
We quantify the quantum correlations between two accelerated detectors coupled to a scalar field in a cavity. It has been realized that an accelerated detector will experience a thermal bath, which is termed the Unruh effect. We examine the similarities and differences for quantum correlations regarding either temperature or acceleration. As the accelerations (resp. temperatures) of the detectors increase, the entanglement decreases to zero at some instant but the mutual information (resp. discord) can be amplified. As the accelerations increase but are in opposite directions, the quantum corr…
We find that the dynamical phase transition (DPT) in nearest-neighbor bipartite entanglement of time-evolved states of the anisotropic infinite quantum XY spin chain, in a transverse time-dependent magnetic field, can be quantitatively characterized by the dynamics of an information-theoretic quantum correlation measure, namely, quantum work-deficit (QWD). We show that only those nonequilibrium states exhibit entanglement resurrection after death, on changing the field parameter during the DPT, for which the cumulative bipartite QWD is above a threshold. The results point to an interesting int…
A measure of quantum correlation defined from an information-theoretic perspective, namely, quantum discord, is applied to study the time-evolved nonequilibrium state of the infinite anisotropic quantum XY spin chain in a transverse time-dependent field. In particular, we probe whether the collapse and revival of nearest-neighbor entanglement of the state seen with a varying initial applied field strength, at a fixed evolution time, may be predicted from the behavior of the quantum correlation measure. For this quantum many-body system, realizable with currently available technology, we find t…
The quest for nonequilibrium quantum phase transitions is often hampered by the tendency of driving and dissipation to give rise to an effective temperature, resulting in classical behavior. Could this be different when the dissipation is engineered to drive the system into a nontrivial quantum coherent steady state? In this work we shed light on this issue by studying the effect of disorder on recently-introduced dissipation-induced Chern topological states, and examining the eigenmodes of the Hermitian steady state density matrix or entanglement Hamiltonian. We find that, similarly to equili…
