Markovian Open Quantum Systems Research Articles

Accelerated Decay due to Operator Spreading in Bulk-Dissipated Quantum Systems.

Markovian open many-body quantum systems display complicated relaxation dynamics. The spectral gap of the Liouvillian characterizes the asymptotic decay rate towards the stationary state, but it has recently been pointed out that the spectral gap does not necessarily determine the overall relaxation time. Our understanding on the relaxation process before the asymptotically long-time regime is still limited. We here present a collective relaxation dynamics of autocorrelation functions in the stationary state. As a key quantity in the analysis, we introduce the instantaneous decay rate, which characterizes the transient relaxation and converges to the conventional asymptotic decay rate in the long-time limit. Our theory predicts that a bulk-dissipated system generically shows an accelerated decay before the asymptotic regime due to the scrambling of quantum information associated with the operator spreading.

Quantum thermodynamics of boundary time-crystals

Time-translation symmetry breaking is a mechanism for the emergence of non-stationary many-body phases, so-called time-crystals, in Markovian open quantum systems. Dynamical aspects of time-crystals have been extensively explored over the recent years. However, much less is known about their thermodynamic properties, also due to the intrinsic nonequilibrium nature of these phases. Here, we consider the paradigmatic boundary time-crystal system, in a finite-temperature environment, and demonstrate the persistence of the time-crystalline phase at any temperature. Furthermore, we analyze thermodynamic aspects of the model investigating, in particular, heat currents, power exchange and irreversible entropy production. Our work sheds light on the thermodynamic cost of sustaining nonequilibrium time-crystalline phases and provides a framework for characterizing time-crystals as possible resources for, e.g. quantum sensing. Our results may be verified in experiments, for example with trapped ions or superconducting circuits, since we connect thermodynamic quantities with mean value and covariance of collective (magnetization) operators.

Universal and nonuniversal probability laws in Markovian open quantum dynamics subject to generalized reset processes.

We consider quantum-jump trajectories of Markovian open quantum systems subject to stochastic in time resets of their state to an initial configuration. The reset events provide a partitioning of quantum trajectories into consecutive time intervals, defining sequences of random variables from the values of a trajectory observable within each of the intervals. For observables related to functions of the quantum state, we show that the probability of certain orderings in the sequences obeys a universal law. This law does not depend on the chosen observable and, in the case of Poissonian reset processes, not even on the details of the dynamics. When considering (discrete) observables associated with the counting of quantum jumps, the probabilities in general lose their universal character. Universality is only recovered in cases when the probability of observing equal outcomes in the same sequence is vanishingly small, which we can achieve in a weak-reset-rate limit. Our results extend previous findings on classical stochastic processes [N. R. Smith et al., Europhys. Lett. 142, 51002 (2023)0295-507510.1209/0295-5075/acd79e] to the quantum domain and to state-dependent reset processes, shedding light on relevant aspects for the emergence of universal probability laws.

Criteria for Davies irreducibility of Markovian quantum dynamics

The dynamics of Markovian open quantum systems are described by Lindblad master equations, generating a quantum dynamical semigroup. An important concept for such systems is (Davies) irreducibility, i.e. the question whether there exist non-trivial invariant subspaces. Steady states of irreducible systems are unique and faithful, i.e. they have full rank. In the 1970s, Frigerio showed that a system is irreducible if the Lindblad operators span a self-adjoint set with trivial commutant. We discuss a more general and powerful algebraic criterion, showing that a system is irreducible if and only if the multiplicative algebra generated by the Lindblad operators La and the operator iH+∑aLa†La , involving the Hamiltonian H, is the entire operator space. Examples for two-level systems, show that a change of Hamiltonian terms as well as the addition or removal of dissipators can render a reducible system irreducible and vice versa. Examples for many-body systems show that a large class of spin chains can be rendered irreducible by dissipators on just one or two sites. Additionally, we discuss the decisive differences between (Davies) reducibility and Evans reducibility for quantum channels and dynamical semigroups which has lead to some confusion in the recent physics literature, especially, in the context of boundary-driven systems. We give a criterion for quantum reducibility in terms of associated classical Markov processes and, lastly, discuss the relation of the main result to the stabilization of pure states and argue that systems with local Lindblad operators cannot stabilize pure Fermi-sea states.

Variational quantum algorithms for simulation of Lindblad dynamics

We introduce variational hybrid classical-quantum algorithms to simulate the Lindblad master equation and its adjoint for time-evolving Markovian open quantum systems and quantum observables. Our methods are based on a direct representation of density matrices and quantum observables as quantum superstates. We design and optimize low-depth variational quantum circuits that efficiently capture the unitary and non-unitary dynamics of the solutions. We benchmark and test the algorithms on different models and system sizes, showing their potential for utility with near-future hardware.

Influence phase of a dS observer. Part I. Scalar exchange

Inspired by real-time computations in AdS black holes, we propose a method to obtain the influence phase of a cosmological observer by calculating the on-shell action on a doubled spacetime geometry. The influence phase is the effective action for an open system: for a dS static patch observer coupled to a scalar field it incorporates the radiation reaction due to the bulk fields and their dS Hawking radiation. For a general extended source in dS, we describe how to account for finite size effects. In the long-time limit, we get a Markovian open quantum system susceptible to cosmological fluctuations, whereas the short-time limit reproduces the worldline theory of flat-space radiation reaction. We also present a fully covariantised form for the cubic corrections to the radiation reaction in even spacetime dimensions, including Hubble contributions, and find an intriguing recursive structure across dimensions.

Metastable dynamics of Rydberg atomic system under electromagnetically induced transparency

Open quantum system, coupled with an external bath is a key research field of quantum physics. Steady state is the state in which any initial state converges after a long time and is usually of the most interest. However, relatively speaking, the nonequilibrium dynamical processes of quantum many-body systems have been rarely studied. This is mainly due to the fact that quantum many-body systems generally have interactions, and the Hilbert space required for a complete description of their dynamical processes will grow exponentially with the number of particles increasing, and the computational difficulty will increase dramatically as well. Hence, it is a difficult problem to completely describe their dynamical processes completely. With the development of quantum technologies, the interest in the nonequilibrium dynamics of open quantum many-body systems is aroused. A common phenomenon is the metastable state, where the system initially relaxes into a long-lived state and then converges to the final stationary state for a longer time. In this paper, we establish a low-dimensional approximation to describe the metastability dynamics in Markovian open quantum system, based on the spectra of the Liouvillian super-operator. The separation of time scales implies a splitting in the spectrum, and this spectral division allows us to eliminate the fast decay modes by using the perturbation method, and then we establish the effective description in the low-lying eigenmodes subspace. Furthermore, we study the dynamics process of the Rydberg atomic system under electromagnetically induced transparency (EIT) conditions and find that the system can handle metastable dynamics if the atomic interactions are considered. We compare the effective dynamics in the subspace with the actual dynamics in the full space, and the results show that the effective dynamics works well on condition that the perturbation approximation holds. Our work provides a feasible idea and method for establishing an effective and simplified description of the dynamical process of open quantum many-body systems.

Digital Simulation of Single Qubit Markovian Open Quantum Systems: A Tutorial

One of the first proposals for the use of quantum computers was the simulation of quantum systems. Over the past three decades, great strides have been made in the development of algorithms for simulating closed quantum systems and the more complex open quantum systems. In this tutorial, we introduce the methods used in the simulation of single qubit Markovian open quantum systems. It combines various existing notations into a common framework that can be extended to more complex open system simulation problems. The only currently available algorithm for the digital simulation of single qubit open quantum systems is discussed in detail. A modification to the implementation of the simpler channels is made that removes the need for classical random sampling, thus making the modified algorithm a strictly quantum algorithm. The modified algorithm makes use of quantum forking to implement the simpler channels that approximate the total channel. This circumvents the need for quantum circuits with a large number of CNOT gates.Quanta 2023; 12: 131–163.

Attraction domain analysis for steady states of Markovian open quantum systems

This article concerns the attraction domain analysis for steady states in Markovian open quantum systems, which are mathematically described by Lindblad master equations. The central question is proposed as: given a steady state, which part of the state space of density operators does it attract and which part does it not attract? We answer this question by presenting necessary and sufficient conditions that determine, for any steady state and initial state, whether the latter belongs to the attraction domain of the former. Furthermore, it is found that the attraction domain of a steady state is the intersection between the set of density operators and an affine space which contains that steady state. Moreover, we show that steady states without uniqueness in the set of density operators have attraction domains with measure zero under some translation invariant and locally finite measures. Finally, an example regarding an open Heisenberg XXZ spin chain is presented. We pick two of the system’s steady states with different magnetization profiles and analyse their attraction domains.

Thermodynamics of Markovian Open Quantum Systems with Application to Lasers

Göran Lindblad was one of the pioneers of what is now called quantum thermodynamics. From this vast and rapidly developing field, we have selected a sample of results concerning quantum open systems described by Markovian master equations of the Lindblad (Gorini–Kossakowski–Sudarshan) type, which are applied to models of lasers. One can study their thermodynamics using the properties of quantum relative entropy, also introduced by Lindblad.

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.

Entanglement Phase Transition Induced by the Non-Hermitian Skin Effect

Recent years have seen remarkable development in open quantum systems effectively described by non-Hermitian Hamiltonians. A unique feature of non-Hermitian topological systems is the skin effect, anomalous localization of an extensive number of eigenstates driven by nonreciprocal dissipation. Despite its significance for non-Hermitian topological phases, the relevance of the skin effect to quantum entanglement and critical phenomena has remained unclear. Here, we find that the skin effect induces a nonequilibrium quantum phase transition in the entanglement dynamics. We show that the skin effect gives rise to a macroscopic flow of particles and suppresses the entanglement propagation and thermalization, leading to the area law of the entanglement entropy in the nonequilibrium steady state. Moreover, we reveal an entanglement phase transition induced by the competition between the unitary dynamics and the skin effect even without disorder or interactions. This entanglement phase transition accompanies nonequilibrium quantum criticality characterized by a nonunitary conformal field theory whose effective central charge is extremely sensitive to the boundary conditions. We also demonstrate that it originates from an exceptional point of the non-Hermitian Hamiltonian and the concomitant scale invariance of the skin modes localized according to the power law. Furthermore, we show that the skin effect leads to the purification and the reduction of von Neumann entropy even in Markovian open quantum systems described by the Lindblad master equation. Our work opens a way to control the entanglement growth and establishes a fundamental understanding of phase transitions and critical phenomena in open quantum systems far from thermal equilibrium.

Lyapunov control of finite-dimensional quantum systems based on bi-objective quantum-behaved particle swarm optimization algorithm

This paper presents a Lyapunov control scheme to drive finite-dimensional closed and Markovian open quantum systems into any target pure state with as high fidelity and as little time as possible. The control law is established by a Lyapunov function with an unknown Hermitian operator, where the eigenvectors of the operator are constructed based on the condition that the LaSalle invariant set contains the desired target state and a set of optimal eigenvalues of the Hermitian operator is searched by the quantum-behaved particle swarm optimization (QPSO) algorithm. In particular, for open systems, when the denominator in the control law is quite small, we propose an improved QPSO algorithm with constraints to enhance the flexibility in implementation. Finally, numerical simulations are carried out on a five-level and a three-qubit closed quantum systems, and a five-level and a ten-level open quantum systems. Simulation results demonstrate that the proposed control scheme can achieve high-fidelity transfer to desired target states for high-dimensional closed and open quantum systems.

Solving quasi-free and quadratic Lindblad master equations for open fermionic and bosonic systems

The dynamics of Markovian open quantum systems are described by Lindblad master equations. For fermionic and bosonic systems that are quasi-free, i.e. with Hamiltonians that are quadratic in the ladder operators and Lindblad operators that are linear in the ladder operators, we derive the equation of motion for the covariance matrix. This determines the evolution of Gaussian initial states and the steady states, which are also Gaussian. Using ladder super-operators (a.k.a. third quantization), we show how the Liouvillian can be transformed to a many-body Jordan normal form which also reveals the full many-body spectrum. Extending previous work by Prosen and Seligman, we treat fermionic and bosonic systems on equal footing with Majorana operators, shorten and complete some derivations, also address the odd-parity sector for fermions, give a criterion for the existence of bosonic steady states, cover non-diagonalizable Liouvillians also for bosons, and include quadratic systems. In extension of the quasi-free open systems, quadratic open systems comprise additional Hermitian Lindblad operators that are quadratic in the ladder operators. While Gaussian states may then evolve into non-Gaussian states, the Liouvillian can still be transformed to a useful block-triangular form, and the equations of motion for k-point Green’s functions form a closed hierarchy. Based on this formalism, results on criticality and dissipative phase transitions in such models are discussed in a companion paper.

Thermodynamics of quantum-jump trajectories of open quantum systems subject to stochastic resetting

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.

Exact solution of a boundary time-crystal phase transition: Time-translation symmetry breaking and non-Markovian dynamics of correlations

The breaking of the continuous time-translation symmetry manifests, in Markovian open quantum systems, through the emergence of non-stationary dynamical phases. Systems that display nonequilibrium transitions into these phases are referred to as time-crystals, and they can be realized, for example, in many-body systems governed by collective dissipation and long-ranged interactions. Here, we provide a complete analytical characterization of a boundary time-crystal phase transition. This involves exact expressions for the order parameter and for the dynamics of quantum fluctuations, which, in the time-crystalline phase, remains asymptotically non-Markovian as a consequence of the time-translation symmetry breaking. We demonstrate that boundary time-crystals are intrinsically critical phases, where fluctuations exhibit a power-law divergence with time. Our results show that a dissipative time-crystal phase is far more than merely a classical non-linear and non-stationary (limit cycle) dynamics of a macroscopic order parameter. It is rather a genuine many-body phase where the properties of correlations distinctly differs from that of stationary ones.

Reachable-set characterization of an open quantum system by the quantum speed limit

In recent years, the idea of reachable-set characterization based on the quantum speed limit (QSL) was proposed by Arenz et al. [New J. Phys. 19, 103015 (2017)]; that is, the reachable set of the target unitary gate in a closed qubit system can be characterized by considering the QSL as the necessary condition that the control setup must satisfy in order to achieve the goal. Inspired by this idea, in this paper we characterize a general Markovian open quantum system based on the QSL derived by Kobayashi and Yamamoto [Phys. Rev. A 102, 042606 (2020)]. Note that this bound is not only explicitly computable with respect to system parameters but also tighter than the other bounds. Some examples for demonstrating this analysis will be given.

Thermodynamics of Precision in Markovian Open Quantum Dynamics.

The thermodynamic and kinetic uncertainty relations indicate trade-offs between the relative fluctuation of observables and thermodynamic quantities such as dissipation and dynamical activity. Although these relations have been well studied for classical systems, they remain largely unexplored in the quantum regime. In this Letter, we investigate such trade-off relations for Markovian open quantum systems whose underlying dynamics are quantum jumps, such as thermal processes and quantum measurement processes. Specifically, we derive finite-time lower bounds on the relative fluctuation of both dynamical observables and their first passage times for arbitrary initial states. The bounds imply that the precision of observables is constrained not only by thermodynamic quantities but also by quantum coherence. We find that the product of the relative fluctuation and entropy production or dynamical activity is enhanced by quantum coherence in a generic class of dissipative processes of systems with nondegenerate energy levels. Our findings provide insights into the survival of the classical uncertainty relations in quantum cases.

Deriving a kinetic uncertainty relation for piecewise deterministic processes: from classical to quantum

From the perspective of Markovian piecewise deterministic processes (PDPs), we investigate the derivation of a kinetic uncertainty relation (KUR), which was originally proposed in Markovian open quantum systems. First, stationary distributions of classical PDPs are explicitly constructed. Then, a tilting method is used to derive a rate functional of large deviations. Finally, based on an improved approximation scheme, we recover the KUR. These classical results are directly extended to the open quantum systems. We use a driven two-level quantum system to exemplify the quantum results.

Exponentially Accelerated Approach to Stationarity in Markovian Open Quantum Systems through the Mpemba Effect.

Ergodicity breaking and slow relaxation are intriguing aspects of nonequilibrium dynamics both in classical and quantum settings. These phenomena are typically associated with phase transitions, e.g., the emergence of metastable regimes near a first-order transition or scaling dynamics in the vicinity of critical points. Despite being of fundamental interest the associated divergent timescales are a hindrance when trying to explore steady-state properties. Here we show that the relaxation dynamics of Markovian open quantum systems can be accelerated exponentially by devising an optimal unitary transformation that is applied to the quantum system immediately before the actual dynamics. This initial "rotation" is engineered in such a way that the state of the quantum system no longer excites the slowest decaying dynamical mode. We illustrate our idea-which is inspired by the so-called Mpemba effect, i.e., water freezing faster when initially heated up-by showing how to achieve an exponential speeding-up in the convergence to stationarity in Dicke models, and how to avoid metastable regimes in an all-to-all interacting spin system.