Relaxation times of dissipative many-body quantum systems

Lieb-Schultz-Mattis Theorem in Open Quantum Systems

The Lieb-Schultz-Mattis (LSM) theorem provides a general constraint on quantum many-body systems and plays a significant role in the Haldane gap phenomena and topological phases of matter. Here, we extend the LSM theorem to open quantum systems and establish a general theorem that restricts the steady state and spectral gap of Liouvillians based solely on symmetry. Specifically, we demonstrate that the unique gapped steady state is prohibited when translation invariance and U(1) symmetry are simultaneously present for noninteger filling numbers. As an illustrative example, we find that no dissipative gap is open in the spin-1/2 dissipative Heisenberg model, while a dissipative gap can be open in the spin-1 counterpart-an analog of the Haldane gap phenomena in open quantum systems. Furthermore, we show that the LSM constraint manifests itself in a quantum anomaly of the dissipative form factor of Liouvillians. We also find the LSM constraints due to symmetry intrinsic to open quantum systems, such as Kubo-Martin-Schwinger symmetry. Our work leads to a unified understanding of phases and phenomena in open quantum systems.

Non-Markovian master equation for quantum transport of fermionic carriers

We propose a simple, yet feasible, model for quantum transport of fermionic carriers across tight-binding chain connecting two reservoirs maintained at arbitrary temperatures and chemical potentials. The model allows for elementary derivation of the master equation for the reduced single particle density matrix (SPDM) in a closed form in both Markov and Born approximations. In the Markov approximation the master equation is solved analytically, whereas in the Born approximation the problem is reduced to an algebraic equation for the SPDM in the Redfield form. The non-Markovian equation is shown to lead to resonant transport similar to Landauer's conductance. It is shown that in the deep non-Markovian regime the transport current can be matched with that obtained by the non-equilibrium Green's function method.

Symmetrized Liouvillian Gap in Markovian Open Quantum Systems

Markovian open quantum systems display complicated relaxation dynamics. The spectral gap of the Liouvillian characterizes the asymptotic decay rate toward the steady state, but it does not necessarily give a correct estimate of the relaxation time because the crossover time to the asymptotic regime may be too long. We here give a rigorous upper bound on the transient decay of autocorrelation functions in the steady state by introducing the symmetrized Liouvillian gap. The standard Liouvillian gap and the symmetrized one are identical in an equilibrium situation but differ from each other in the absence of the detailed balance condition. It is numerically shown that the symmetrized Liouvillian gap always gives a correct upper bound on the decay of the autocorrelation function, but the standard Liouvillian gap does not.

Noise-Induced Quantum Synchronization

Synchronization is a widespread phenomenon in science and technology. Here, we study noise-induced synchronization in a quantum spin chain subjected to local Gaussian white noise. We demonstrate stable (anti)synchronization between the endpoint magnetizations of a quantum XY model with transverse field of arbitrary length. Remarkably, we show that noise applied to a single spin suffices to reach stable (anti)synchronization, and find that the two synchronized end spins are entangled. We additionally determine the optimal noise amplitude that leads to the fastest synchronization along the chain, and further compare the optimal synchronization speed to the fundamental Lieb-Robinson bound for information propagation.

Rise and Fall, and Slow Rise Again, of Operator Entanglement under Dephasing

The operator space entanglement entropy, or simply "operator entanglement" (OE), is an indicator of the complexity of quantum operators and of their approximability by matrix product operators (MPOs). We study the OE of the density matrix of 1D many-body models undergoing dissipative evolution. It is expected that, after an initial linear growth reminiscent of unitary quench dynamics, the OE should be suppressed by dissipative processes as the system evolves to a simple stationary state. Surprisingly, we find that this scenario breaks down for one of the most fundamental dissipative mechanisms: dephasing. Under dephasing, after the initial "rise and fall," the OE can rise again, increasing logarithmically at long times. Using a combination of MPO simulations for chains of infinite length and analytical arguments valid for strong dephasing, we demonstrate that this growth is inherent to a U(1) conservation law. We argue that in an XXZ spin model and a Bose-Hubbard model the OE grows universally as 1/4log_{2}t at long times and as 1/2log_{2}t for a Fermi-Hubbard model. We trace this behavior back to anomalous classical diffusion processes.

Transport and spectral properties of the XX+XXZ diode and stability to dephasing

We study the transport and spectral property of a segmented diode formed by an XX+XXZ spin chain. This system has been shown to become an ideal rectifier for spin current for large enough anisotropy. Here we show numerical evidence that the system in reverse bias has signatures pointing toward the existence of three different transport regimes depending on the value of the anisotropy: ballistic, diffusive, and insulating. In forward bias we observe two regimes, ballistic and diffusive. The system in forward and reverse bias shows significantly different spectral properties, with distribution of rapidities converging toward different functions. In the presence of dephasing the system becomes diffusive, rectification is significantly reduced, the relaxation gap increases, and the spectral properties in forward and reverse bias tend to converge. For large dephasing the relaxation gap decreases again as a result of quantum Zeno physics.

Liouvillian Skin Effect: Slowing Down of Relaxation Processes without Gap Closing

It is highly nontrivial to what extent we can deduce the relaxation behavior of a quantum dissipative system from the spectral gap of the Liouvillian that governs the time evolution of the density matrix. We investigate the relaxation processes of a quantum dissipative system that exhibits the Liouvillian skin effect, which means that the eigenmodes of the Liouvillian are localized exponentially close to the boundary of the system, and find that the timescale for the system to reach a steady state depends not only on the Liouvillian gap Δ, but also on the localization length ξ of the eigenmodes. In particular, we show that the longest relaxation time τ that is maximized over initial states and local observables is given by τ∼Δ^{-1}(1+L/ξ) with L being the system size. This implies that the longest relaxation time can diverge for L→∞ without gap closing.

Joint statistics of work and entropy production along quantum trajectories

In thermodynamics, entropy production and work quantify irreversibility and the consumption of useful energy, respectively, when a system is driven out of equilibrium. For quantum systems, these quantities can be identified at the stochastic level by unravelling the system's evolution in terms of quantum jump trajectories. We here derive a general formula for computing the joint statistics of work and entropy production in Markovian driven quantum systems, whose instantaneous steady states are of Gibbs form. If the driven system remains close to the instantaneous Gibbs state at all times, then we show that the corresponding two-variable cumulant generating function implies a joint detailed fluctuation theorem so long as detailed balance is satisfied. As a corollary, we derive a modified fluctuation-dissipation relation (FDR) for the entropy production alone, applicable to transitions between arbitrary steady states, and for systems that violate detailed balance. This FDR contains a term arising from genuinely quantum fluctuations, and extends an analogous relation from classical thermodynamics to the quantum regime.

Solving the Liouvillian Gap with Artificial Neural Networks

We propose a machine-learning inspired variational method to obtain the Liouvillian gap, which plays a crucial role in characterizing the relaxation time and dissipative phase transitions of open quantum systems. By using "spin bi-base mapping," we map the density matrix to a pure restricted-Boltzmann-machine (RBM) state and transform the Liouvillian superoperator to a rank-two non-Hermitian operator. The Liouvillian gap can be obtained by a variational real-time evolution algorithm under this non-Hermitian operator. We apply our method to the dissipative Heisenberg model in both one and two dimensions. For the isotropic case, we find that the Liouvillian gap can be analytically obtained and in one dimension even the whole Liouvillian spectrum can be exactly solved using the Bethe ansatz method. By comparing our numerical results with their analytical counterparts, we show that the Liouvillian gap could be accessed by the RBM approach efficiently to a desirable accuracy, regardless of the dimensionality and entanglement properties.

Scheme for automatic differentiation of complex loss functions with applications in quantum physics

The past few years have seen a significant transfer of tools from machine learning to solve quantum physics problems. Automatic differentiation is one standard algorithm used to efficiently compute gradients of loss functions for generic neural networks. In this work we show how to extend automatic differentiation to the case of complex loss function in a way that can be readily implemented in existing frameworks and which is compatible with the common case of real loss functions. We then combine this tool with matrix product states and apply it to compute the ground state and the steady state of a close and an open quantum system. Compared to the traditional density matrix renormalization group algorithm, complex automatic differentiation allows both straightforward GPU accelerations as well as generalizations to different types of tensor and neural networks.

Vaporization Dynamics of a Dissipative Quantum Liquid

We investigate the stability of a Luttinger liquid, upon suddenly coupling it to a dissipative environment. Within the Lindblad equation, the environment couples to local currents and heats the quantum liquid up to infinite temperatures. The single particle density matrix reveals the fractionalization of fermionic excitations in the spatial correlations by retaining the initial noninteger power law exponents, accompanied by an exponential decay in time with an interaction dependent rate. The spectrum of the time evolved density matrix is gapped, which collapses gradually as -ln(t). The von Neumann entropy crosses over from the early time -tln(t) behavior to ln(t) growth for late times. The early time dynamics is captured numerically by performing simulations on spinless interacting fermions, using several numerically exact methods. Our results could be tested experimentally in bosonic Luttinger liquids.

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.

Frustration and inhomogeneous environments in relaxation of open chains with Ising-type interactions

Frustration can contribute to very slow relaxation times in large open chains, as in spin glasses and in biopolymers. However, frustration may not be sufficient to produce broken ergodicity in finite systems. Here we employ a system-plus-reservoir approach to investigate how strongly inhomogeneous environments and frustration compete in the relaxation of finite open chains. We find a sufficient condition for our inhomogeneous environments to break ergodicity. We use the microscopic model to derive a Markovian quantum master equation for a generic chain with ultrastrong intrachain couplings. We show that this microscopic model avoids a spurious broken ergodicity we find in the phenomenological model. We work out an explicit example of broken ergodicity due to the inhomogeneous environment of an unfrustrated spin chain as far as simulating a recent experiment on protein denaturation (where environment inhomogeneity is especially relevant). We finally show that an inhomogeneous environment can mitigate the effects of frustration-induced degeneracies.

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.

Noise-Driven Universal Dynamics towards an Infinite Temperature State

Dynamical universality is the observation that the dynamical properties of different systems might exhibit universal behavior that are independent of the system details. In this Letter, we study the longtime dynamics of a one-dimensional noisy quantum magnetic model, and find that even though the system is inevitably driven to an infinite temperature state, the relaxation dynamics towards such a featureless state can be highly nontrivial and universal. The effect of various mode-coupling mechanisms (external potential, disorder, interaction, and the interplay between them) as well as the conservation law on the longtime dynamics of the systems have been studied, and their relevance with current ultracold atomic experiments has been discussed.

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.

Universal Dynamics of Stochastically Driven Quantum Impurities

We show that the dynamics of a quantum impurity subject to a stochastic drive on one side and coupled to a quantum critical system on the other display a universal behavior inherited from the quantum critical scaling. Using boundary conformal field theory, we formulate a generic ansatz for the dynamical scaling form of the typical Loschmidt echo and corroborate it with exact numerical calculations in the case of a spin impurity driven by shot noise in a quantum Ising chain. We find that due to rare events the dynamics of the mean echo can follow very different dynamical scaling than the typical echo for certain classes of drives. Our results are insensitive to irrelevant perturbations of the bulk critical model and apply to all the microscopic models in the same universality class.

Heating-Induced Long-Range η Pairing in the Hubbard Model

We show how, upon heating the spin degrees of freedom of the Hubbard model to infinite temperature, the symmetries of the system allow the creation of steady states with long-range correlations between η pairs. We induce this heating with either dissipation or periodic driving and evolve the system towards a nonequilibrium steady state, a process which melts all spin order in the system. The steady state is identical in both cases and displays distance-invariant off-diagonal η correlations. These correlations were first recognized in the superconducting eigenstates described in Yang's seminal Letter [Phys. Rev. Lett. 63, 2144 (1989)PRLTAO0031-900710.1103/PhysRevLett.63.2144], which are a subset of our steady states. We show that our results are a consequence of symmetry properties and entirely independent of the microscopic details of the model and the heating mechanism.

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.

Locality Bound for Dissipative Quantum Transport

We prove an upper bound on the diffusivity of a dissipative, local, and translation invariant quantum Markovian spin system: D≤D_{0}+(αv_{LR}τ+βξ)v_{C}. Here v_{LR} is the Lieb-Robinson velocity, v_{C} is a velocity defined by the current operator, τ is the decoherence time, ξ is the range of interactions, D_{0} is a decoherence-induced microscopic diffusivity, and α and β are precisely defined dimensionless coefficients. The bound constrains quantum transport by quantities that can either be obtained from the microscopic interactions (D_{0}, v_{LR}, v_{C}, ξ) or else determined from independent local nontransport measurements (τ, α, β). We illustrate the general result with the case of a spin-half XXZ chain with on-site dephasing. Our result generalizes the Lieb-Robinson bound to constrain the sub-ballistic diffusion of conserved densities in a dissipative setting.

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.

Fluctuating hydrodynamics, current fluctuations, and hyperuniformity in boundary-driven open quantum chains

We consider a class of either fermionic or bosonic noninteracting open quantum chains driven by dissipative interactions at the boundaries and study the interplay of coherent transport and dissipative processes, such as bulk dephasing and diffusion. Starting from the microscopic formulation, we show that the dynamics on large scales can be described in terms of fluctuating hydrodynamics. This is an important simplification as it allows us to apply the methods of macroscopic fluctuation theory to compute the large deviation (LD) statistics of time-integrated currents. In particular, this permits us to show that fermionic open chains display a third-order dynamical phase transition in LD functions. We show that this transition is manifested in a singular change in the structure of trajectories: while typical trajectories are diffusive, rare trajectories associated with atypical currents are ballistic and hyperuniform in their spatial structure. We confirm these results by numerically simulating ensembles of rare trajectories via the cloning method, and by exact numerical diagonalization of the microscopic quantum generator.

Microscopic theory of a nonequilibrium open bosonic chain

Quantum master equations form an important tool in the description of transport problems in open quantum systems. However, they suffer from the difficulty that the shape of the Lindblad dissipator depends sensibly on the system Hamiltonian. Consequently, most of the work done in this field has focused on phenomenological dissipators which act locally on different parts of the system. In this paper we show how to construct Lindblad dissipators to model a one-dimensional bosonic tight-binding chain connected to two baths at the first and last site, kept at different temperatures and chemical potentials. We show that even though the bath coupling is local, the effective Lindblad dissipator stemming from this interaction is inherently nonlocal, affecting all normal modes of the system. We then use this formalism to study the current of particles and energy through the system and find that they have the structure of Landauer's formula, with the bath spectral density playing the role of the transfer integral. Finally, we consider infinitesimal temperature and chemical potential gradients and show that the currents satisfy Onsager's reciprocal relations, which is a consequence of the fact that the microscopic quantum dynamics obeys detailed balance.

Nonequilibrium quantum chains under multisite Lindblad baths

We study a quantum XX chain coupled to two heat reservoirs that act on multiple sites and are kept at different temperatures and chemical potentials. The baths are described by Lindblad dissipators, which are constructed by direct coupling to the fermionic normal modes of the chain. Using a perturbative method, we are able to find analytical formulas for all steady-state properties of the system. We compute both the particle or magnetization current and the energy current, both of which are found to have the structure of Landauer's formula. We also obtain exact formulas for the Onsager coefficients. All properties are found to differ substantially from those of a single-site bath. In particular, we find a strong dependence on the intensity of the bath couplings. In the weak-coupling regime, we show that the Onsager reciprocal relations are satisfied.

Diffusive and Subdiffusive Spin Transport in the Ergodic Phase of a Many-Body Localizable System

We study high temperature spin transport in a disordered Heisenberg chain in the ergodic regime. By employing a density matrix renormalization group technique for the study of the stationary states of the boundary-driven Lindblad equation we are able to study extremely large systems (400 spins). We find both a diffusive and a subdiffusive phase depending on the strength of the disorder and on the anisotropy parameter of the Heisenberg chain. Studying finite-size effects, we show numerically and theoretically that a very large crossover length exists that controls the passage of a clean-system dominated dynamics to one observed in the thermodynamic limit. Such a large length scale, being larger than the sizes studied before, explains previous conflicting results. We also predict spatial profiles of magnetization in steady states of generic nondiffusive systems.

Dynamical signatures of molecular symmetries in nonequilibrium quantum transport

Symmetries play a crucial role in ubiquitous systems found in Nature. In this work, we propose an elegant approach to detect symmetries by measuring quantum currents. Our detection scheme relies on initiating the system in an anti-symmetric initial condition, with respect to the symmetric sites, and using a probe that acts like a local noise. Depending on the position of the probe the currents exhibit unique signatures such as a quasi-stationary plateau indicating the presence of metastability and multi-exponential decays in case of multiple symmetries. The signatures are sensitive to the characteristics of the probe and vanish completely when the timescale of the coherent system dynamics is much longer than the timescale of the probe. These results are demonstrated using a 4-site model and an archetypal example of the para-benzene ring and are shown to be robust under a weak disorder.

Towards a Theory of Metastability in Open Quantum Dynamics

By generalizing concepts from classical stochastic dynamics, we establish the basis for a theory of metastability in Markovian open quantum systems. Partial relaxation into long-lived metastable states-distinct from the asymptotic stationary state-is a manifestation of a separation of time scales due to a splitting in the spectrum of the generator of the dynamics. We show here how to exploit this spectral structure to obtain a low dimensional approximation to the dynamics in terms of motion in a manifold of metastable states constructed from the low-lying eigenmatrices of the generator. We argue that the metastable manifold is in general composed of disjoint states, noiseless subsystems, and decoherence-free subspaces.

Dissipative Remote-State Preparation in an Interacting Medium

Standard quantum state preparation methods work by preparing a required state locally and then distributing it to a distant location by a free-space propagation. We instead study procedures of preparing a target state at a remote location in the presence of an interacting background medium on which no control is required, manipulating only local dissipation. In mathematical terms, we characterize a set of reduced steady states stabilizable by local dissipation. An explicit local method is proposed by which one can construct a wanted one-site reduced steady state at an arbitrary remote site in a lattice of any size and geometry. In the chain geometry we also prove uniqueness of such a steady state. We demonstrate that the convergence time to fixed precision is smaller than the inverse gap, and we study robustness of the scheme in different medium interactions.