Lindblad Master Equation Research Articles

Quantum entanglement and non-Hermiticity in free-fermion systems

Abstract This topical review article reports rapid progress on the generalization and application of entanglement in non-Hermitian free-fermion quantum systems. We begin by examining the realization of non-Hermitian quantum systems through the Lindblad master equation, alongside a review of typical non-Hermitian free-fermion systems that exhibit unique features. A pedagogical discussion is provided on the relationship between entanglement quantities and the correlation matrix in Hermitian systems. Building on this foundation, we focus on how entanglement concepts are extended to non-Hermitian systems from their Hermitian free-fermion counterparts, with a review of the general properties that emerge. Finally, we highlight various concrete studies, demonstrating that entanglement entropy remains a powerful diagnostic tool for characterizing non-Hermitian physics. The entanglement spectrum also reflects the topological characteristics of non-Hermitian topological systems, while unique non-Hermitian entanglement behaviors are also discussed. The review is concluded with several future directions. Through this review, we hope to provide a useful guide for researchers who are interested in entanglement in non-Hermitian quantum systems.

Three perspectives on entropy dynamics in a non-Hermitian two-state system

Abstract A comparative study of entropy dynamics as an indicator of physical behavior in an open two-state system with balanced gain and loss is presented. To begin with, we illustrate the phase portrait of this non-Hermitian model on the Bloch sphere, elucidating the changes in behavior as one moves across the phase transition boundary, as well as the emergent feature of unidirectional state evolution in the spontaneously broken PT-symmetry regime. This is followed by an examination of the purity and entropy dynamics. Here we distinguish the perspective taken in utilizing the conventional framework of Hermitian-adjoint states from an approach that is based on biorthogonal-adjoint states and a third case based on an isospectral mapping. In this it is demonstrated that their differences are rooted in the treatment of the environmental coupling mode. For unbroken PT symmetry of the system, a notable characteristic feature of the perspective taken is the presence or absence of purity oscillations, with an associated entropy revival. The description of the system is then continued from its PT-symmetric pseudo-Hermitian phase into the regime of spontaneously broken symmetry, in the latter two approaches through a non-analytic operator-based continuation, yielding a Lindblad master equation based on the PT charge operator C. This phase transition indicates a general connection between the pseudo-Hermitian closed-system and the Lindbladian open-system formalism through a spontaneous breakdown of the underlying physical reflection symmetry.

Hierarchical equations of motion for multiple baths (HEOM-MB) and their application to Carnot cycle.

We have developed a computer code for the thermodynamic hierarchical equations of motion derived from a spin subsystem coupled to multiple Drude baths at different temperatures, which are connected to or disconnected from the subsystem as a function of time. The code can simulate the reduced dynamics of the subsystem under isothermal, isentropic, thermostatic, and entropic conditions. The extensive and intensive thermodynamic variables are calculated as physical observables, and Gibbs and Helmholtz energies are evaluated as intensive and extensive work. The energy contribution of the system-bath interaction is evaluated separately from the subsystem using the hierarchical elements of the hierarchical equations of motion. The accuracy of the calculated results for the equilibrium distribution and the two-body correlation functions is assessed by contrasting the results with those obtained from the time-convolution-less Redfield equation. It is shown that the Lindblad master equation is inappropriate for the thermodynamic description of a spin-boson system. Non-Markovian effects in thermostatic processes are investigated by sequentially turning on and off the baths at different temperatures with different switching times and system-bath coupling. In addition, the Carnot cycle is simulated under quasi-static conditions. To analyze the work performed for the subsystem in the cycle, thermodynamic work diagrams are plotted as functions of intensive and extensive variables. The C++ source codes are provided as supplementary material.

Unconventional magnon blockade in a dissipative photon–magnon coupling system

We investigate unconventional magnon blockade (UMB) in a dissipative photon–magnon coupling cavity, where the coherent and dissipative coupling between photon and magnon can be adjusted by the position of the yttrium iron garnet sphere in the cavity. An approximate analytical solution for the statistical property of magnon is provided, which is in good agreement with the results obtained by numerically solving the Lindblad master equation. Our results indicate that UMB can be achieved in the case of dissipative photon–magnon coupling, but not in the case of coherent photon–magnon coupling. Secondly, by adjusting the amplitude of the external laser field, dynamic control of UMB can be achieved. Finally, by solving the Lindblad master equation containing the occupation number of thermal magnon and photon, the statistical properties of magnons will transition from nonclassical to classical. It is shown that the above phenomena originate from the interference between different quantum paths. Our research may provide theoretical possibilities for quantum manipulation schemes at the single-magnon level and open up a new avenue for designing single-magnon sources in a dissipative photon–magnon coupling system.

Information geometry and parameter sensitivity of non-Hermitian Hamiltonians

Information geometry is the application of differential geometry in statistics, where the Fisher-Rao metric serves as the Riemannian metric on the statistical manifold, providing an intrinsic property for parameter sensitivity. In this paper, we explore the application of information geometry in the realm of non-Hermitian quantum systems, focusing on the Fisher-Rao metric as a measure of parameter sensitivity. We approximate the Lindblad master equation for non-Hermitian Hamiltonians to analyze the temporal evolution of the quantum geometric metric. Utilizing the quantum spin Ising model with an imaginary magnetic field as an exemplar, we investigate the energy spectrum and geometric metric evolution within PT-symmetry Hamiltonians. We demonstrate that the detrimental effects of dissipation can be counteracted by introducing a control Hamiltonian, leading to improved accuracy in parameter estimation. Our work provides insights into the role of quantum control in mitigating dissipative impacts and enhancing the precision of quantum metrological tasks.

CoOMBE: A suite of open-source programs for the integration of the optical Bloch equations and Maxwell-Bloch equations

The programs described in this article and distributed with it aim (1) at integrating the optical Bloch equations governing the time evolution of the density matrix representing the quantum state of an atomic system driven by laser or microwave fields, and (2) at integrating the 1D Maxwell-Bloch equations for one or two laser fields co-propagating in an atomic vapour. The rotating wave approximation is assumed. These programs can also be used for more general quantum dynamical systems governed by the Lindblad master equation. They are written in Fortran 90; however, their use does not require any knowledge of Fortran programming. Methods for solving the optical Bloch equations in the rate equations limit, for calculating the steady-state density matrix and for formulating the optical Bloch equations in the weak probe approximation are also described. Program summaryProgram Title: CoOMBECPC Library link to program files:https://doi.org/10.17632/5wsg9d52dk.1Developers' repository link:https://github.com/durham-qlm/CoOMBELicensing provisions: GPLv3Programming language: Fortran 90Nature of problem: The present programs can be used for the following operations: (1) Integrating the optical-Bloch equations within the rotating wave approximation for a multi-state atomic system. At the choice of the user, the calculation will return either the time-dependent density matrix at given times or the density matrix in the long time limit if the system evolves into a steady state in that limit. The calculation can be done with or without averaging over the thermal velocity distribution of the atoms. The number of atomic states which can be included in the calculation is limited only by the CPU time available and possibly by memory requirements. An arbitrarily large number of laser or microwave fields can be included in the calculation if these fields are all CW. This number is currently limited to one or two for fields that are not all CW. The calculation can be done in the weak probe approximation, or in the rate equations approximation, or without assuming either of these two approximations. Calculating refractive indexes, absorption coefficients and complex susceptibilities is also possible. (2) Integrating the 1D Maxwell-Bloch equations in the slowly varying envelope approximation for one or two fields co-propagating in a single-species atomic vapour. Although geared towards the case of atoms interacting with laser fields, this code can also be used for more general quantum systems with similar equations of motion (e.g., molecular systems, spin systems, etc.).Solution method: The Lindblad master equation is expressed as a system of homogeneous first order linear differential equations, which are transformed as required and solved to obtain the density matrix representing the state of the atomic system. A variety of methods are offered to this end. The same approach is also used in the calculation of the polarisation of the medium when integrating the Maxwell-Bloch equations. The latter are integrated over space using predictor-corrector methods. The library includes a general driving program making it possible to use these codes without additional program development. The distribution also includes examples of the use of a container for running these programs without a pre-installed Fortran compiler.

Satellite-based positioning enhanced by quantum synchronization

This study focuses on the innovative field of quantum synchronization for satellite-based navigation systems including Global Navigation Satellite Systems (GNSSs) and the Non-Terrestrial Network (NTN) component of future 6G networks integrating both communication and navigation services.By combining a four-qubit system with the theoretical approach of the Lindblad master equation, we transcend the inherent limits of standard synchronization techniques. This achievement represents a quantum leap in satellite-based positioning, highlighting the scalability and cost-effectiveness of our technique for smaller satellites. The study demonstrates the possibility of reducing synchronization errors to less than one meter, significantly improving the reliability and precision of satellite-based navigation systems.The results of this study may contribute to the future development of both user-centric localization systems (typically GNSS systems) and network-centric localization systems (typically through the NTN component of 6G networks), leading to better positioning performance, more flexible multi-functional systems with the potential to limit both cost and size of satellites.

Photo-induced dynamics with continuous and discrete quantum baths.

The ultrafast quantum dynamics of photophysical processes in complex molecules is an extremely challenging computational problem with a broad variety of fascinating applications in quantum chemistry and biology. Inspired by recent developments in open quantum systems, we introduce a pure-state unraveled hybrid-bath method that describes a continuous environment via a set of discrete, effective bosonic degrees of freedom using a Markovian embedding. Our method is capable of describing both, a continuous spectral density and sharp peaks embedded into it. Thereby, we overcome the limitations of previous methods, which either capture long-time memory effects using the unitary dynamics of a set of discrete vibrational modes or use memoryless Markovian environments employing a Lindblad or Redfield master equation. We benchmark our method against two paradigmatic problems from quantum chemistry and biology. We demonstrate that compared to unitary descriptions, a significantly smaller number of bosonic modes suffices to describe the excitonic dynamics accurately, yielding a computational speed-up of nearly an order of magnitude. Furthermore, we take into account explicitly the effect of a δ-peak in the spectral density of a light-harvesting complex, demonstrating the strong impact of the long-time memory of the environment on the dynamics.

Exact non-Hermitian mobility edges and robust flat bands in two-dimensional Lieb lattices with imaginary quasiperiodic potentials

The mobility edge (ME) is a critical energy delineates the boundary between extended and localized states within the energy spectrum, and it plays a crucial role in understanding the metal–insulator transition in disordered or quasiperiodic systems. While there have been extensive studies on MEs in one-dimensional non-Hermitian (NH) quasiperiodic lattices recently, the investigation of exact NH MEs in two-dimensional (2D) cases remains rare. In the present study, we introduce a 2D dissipative Lieb lattice (DLL) model with imaginary quasiperiodic potentials applied solely to the vertices of the Lieb lattice. By mapping this DLL model to the 2D NH Aubry–André–Harper model, we analytically derive the exact ME and find it associated with the absolute eigenenergies. We find that the eigenvalues of extended states are purely imaginary when the quasiperiodic potential is strong enough. Additionally, we demonstrate that the introduction of imaginary quasiperiodic potentials does not disrupt the flat bands inherent in the system. Finally, we propose a theoretical framework for realizing our model using the Lindblad master equation. Our results pave the way for further investigation of exact NH MEs and flat bands in 2D dissipative quasiperiodic systems.

Heat currents in qubit systems

There is a current interest in quantum thermodynamics in the context of open quantum systems. An important issue is the consistency of quantum thermodynamics, in particular the second law of thermodynamics, i.e. the flow of heat from a hot reservoir to a cold reservoir. Here recent emphasis has been on composite system and in particular the issue regarding the application of local or global master equations. In order to contribute to this discussion we discuss two cases, namely as an example a single qubit and as a simple composite system two coupled qubits driven by two heat reservoirs at different temperatures, respectively. Applying a global Lindblad master equation approach we present explicit expressions for the heat currents in agreement with the second law of thermodynamics. The analysis is carried out in the Born–Markov approximation. We also discuss issues regarding the possible presence of coherences in the steady state.

Mixed-State Quantum Spin Liquids and Dynamical Anyon Condensations in Kitaev Lindbladians

Quantum spin liquids and anyons, used to be subjects of condensed matter physics, now are realized in various platforms of qubits, offering unprecedented opportunities to investigate fundamental physics of many-body quantum entangled states. Qubits are inevitably exposed to environment effects such as decoherence and dissipation, which are believed to be detrimental to many-body entanglement. Here, we argue that unlike the common belief decoherence and dissipation can give rise to novel topological phenomena in quantum spin liquids. We study open quantum systems of the Kitaev spin liquid and the toric code via the Lindblad master equation approach. By using exact solutions and numerical approaches, we show the dynamical occurrence of anyon condensation by decoherence and dissipation, which results in a topological transition from the initial state spin liquid to the steady state spin liquid. The mechanism of the anyon condensation transition by the Lindblad dynamics is elucidated. We also provide an insight into the relationship between the Kitaev spin liquid and the toric code in the picture of anyon condensation. Our work suggests open quantum systems to be a new venue for topological phenomena of quantum spin liquids and anyons.

Doppler-enhanced quantum magnetometry with thermal Rydberg atoms

We report experimental measurements showing how one can combine quantum interference and thermal Doppler shifts at room temperature to detect weak magnetic fields. We pump 87 Rb atoms to a highly-excited, Rydberg level using a probe and a coupling laser, leading to narrow transmission peaks of the probe due to destructive interference of transition amplitudes, known as Electromagnetically Induced Transparency. While it is customary in such setups to use counterpropagating lasers to minimize the effect of Doppler shifts, here we show, on the contrary, that one can harness Doppler shifts in a copropagating arrangement to produce an enhanced response to a magnetic field. In particular, we demonstrate an order-of-magnitude bigger splitting in the transmission spectrum as compared to the counterpropagating case. We explain and generalize our findings with theoretical modeling and simulations based on a Lindblad master equation. Our results pave the way to using quantum effects for magnetometry in readily deployable room-temperature platforms.

Diagnosing Thermalization Dynamics of Non-Hermitian Quantum Systems via GKSL Master Equations

Abstract The application of the eigenstate thermalization hypothesis to non-Hermitian quantum systems has become one of the most important topics in dissipative quantum chaos, recently giving rise to intense debates. The process of thermalization is intricate, involving many time-evolution trajectories in the reduced Hilbert space of the system. By considering two different expansion forms of the density matrices adopted in the biorthogonal and right-state time evolutions, we derive two versions of the Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) master equations describing the non-Hermitian systems coupled to a bosonic heat bath in thermal equilibrium. By solving the equations, we identify a sufficient condition for thermalization under both time evolutions, resulting in Boltzmann biorthogonal and right-eigenstate statistics, respectively. This finding implies that the recently proposed biorthogonal random matrix theory needs an appropriate revision. Moreover, we exemplify the precise dynamics of thermalization and thermodynamic properties with test models.

Utilizing quantum coherence in Cs Rydberg atoms for high-sensitivity room-temperature terahertz detection: a theoretical exploration

In recent years, terahertz (THz) technology has made significant progress in numerous applications; however, the highly sensitive, room-temperature THz detectors are still rare, which is one of the bottlenecks in THz research. In this paper, we proposed a room-temperature electrometry method for THz detection by laser spectroscopy of cesium (Cs133) Rydberg atoms, and conducted a comprehensive investigation of the five-level system involving electromagnetically induced transparency (EIT), electromagnetically induced absorption (EIA), and Autler–Townes (AT) splitting in Cs133 cascades. By solving the Lindblad master equation, we found that the influence of the THz electric field, probe laser, dressing laser, and Rydberg laser on the ground state atomic population as well as the coherence between the ground state and the Rydberg state, plays a crucial role in the transformation and amplitude of the EIT and EIA signals. Temperature and the atomic vapor cell’s dimensions affect the number of Cs133 atoms involved in the detection, and ultimately determine the sensitivity. We predicted the proposed quantum coherence THz detection method has a remarkable sensitivity of as low as 10−9 V m−1 Hz−1/2. This research offers a valuable theoretical basis for implementing and optimizing quantum coherence effects based on Rydberg atoms for THz wave detection with high sensitivity and room-temperature operation.

Real-time dynamics of the Schwinger model as an open quantum system with Neural Density Operators

Ab-initio simulations of multiple heavy quarks propagating in a Quark-Gluon Plasma are computationally difficult to perform due to the large dimension of the space of density matrices. This work develops machine learning algorithms to overcome this difficulty by approximating exact quantum states with neural network parametrisations, specifically Neural Density Operators. As a proof of principle demonstration in a QCD-like theory, the approach is applied to solve the Lindblad master equation in the 1 + 1d lattice Schwinger Model as an open quantum system. Neural Density Operators enable the study of in-medium dynamics on large lattice volumes, where multiple-string interactions and their effects on string-breaking and recombination phenomena can be studied. Thermal properties of the system at equilibrium can also be probed with these methods by variationally constructing the steady state of the Lindblad master equation. Scaling of this approach with system size is studied, and numerical demonstrations on up to 32 spatial lattice sites and with up to 3 interacting strings are performed.

Stabilization of approximate GHZ state with quasi-local couplings

We propose a reservoir design, composed of fixed dissipation operators acting each on few local subsystems, to stabilize an approximate GHZ state on n qubits. The main idea is to work out how a previously proposed sequence of two stabilization steps can be applied instead in appropriate (probabilistic) superposition. We examine various alternatives to synchronize the superposition using local couplings only, thanks to a chain of ‘clock’ ancillas or to additional levels on the data subsystems. The practical value of these alternatives depends on experimental constraints. They all feature a design tradeoff between approximate stabilization fidelity and protection against perturbations. These proposals illustrate how simple autonomous automata can be implemented in quantum reservoir engineering to replace sequential state preparation procedures. Encoding automaton actions via additional data levels only, appears particularly efficient in this context. Our analysis method, reducing the Lindblad master equation to a Markov chain on virtual output signals, may be of independent interest.

Entangled multiplets, asymmetry, and quantum Mpemba effect in dissipative systems

Recently, the entanglement asymmetry emerged as an informative tool to understand dynamical symmetry restoration in out-of-equilibrium quantum many-body systems after a quantum quench. For integrable systems the asymmetry can be understood in the space-time scaling limit via the quasiparticle picture, as it was pointed out in Ares et al (2023 Nat. Commun. 14 2036) . However, a quasiparticle picture for quantum quenches from generic initial states was still lacking. Here we conjecture a full-fledged quasiparticle picture for the charged moments of the reduced density matrix, which are the main ingredients to construct the asymmetry. Our formula works for quenches producing entangled multiplets of an arbitrary number of excitations. We benchmark our results in the XX spin chain. First, by using an elementary approach based on the multidimensional stationary phase approximation we provide an ab initio rigorous derivation of the dynamics of the charged moments for the quench treated in Ares et al (2023 SciPost Phys. 15 089). Then, we show that the same results can be straightforwardly obtained within our quasiparticle picture. As a byproduct of our analysis, we obtain a general criterion ensuring a vanishing entanglement asymmetry at long times. Next, by using the Lindblad master equation, we study the effect of gain and loss dissipation on the entanglement asymmetry. Specifically, we investigate the fate of the so-called quantum Mpemba effect (QME) in the presence of dissipation. We show that dissipation can induce QME even if unitary dynamics does not show it, and we provide a quasiparticle-based interpretation of the condition for the QME.

Quantum Master Equations: Tips and Tricks for Quantum Optics, Quantum Computing, and Beyond

Quantum master equations are an invaluable tool to model the dynamics of a plethora of microscopic systems, ranging from quantum optics and quantum information processing to energy and charge transport, electronic and nuclear spin resonance, photochemistry, and more. This tutorial offers a concise and pedagogical introduction to quantum master equations, accessible to a broad, cross-disciplinary audience. The reader is guided through the basics of quantum dynamics with hands-on examples that increase in complexity. The tutorial covers essential methods such as the use of the Lindblad master equation, Redfield relaxation, and Floquet theory, as well as techniques such as Suzuki-Trotter expansion and numerical approaches for sparse solvers. These methods are illustrated with code snippets implemented in and other languages, which can be used as a starting point for generalization and more sophisticated implementations. Published by the American Physical Society 2024

Coupling polyatomic molecules to lossy nanocavities: Lindblad vs Schrödinger description.

The use of cavities to impact molecular structure and dynamics has become popular. As cavities, in particular plasmonic nanocavities, are lossy and the lifetime of their modes can be very short, their lossy nature must be incorporated into the calculations. The Lindblad master equation is commonly considered an appropriate tool to describe this lossy nature. This approach requires the dynamics of the density operator and is thus substantially more costly than approaches employing the Schrödinger equation for the quantum wave function when several or many nuclear degrees of freedom are involved. In this work, we compare numerically the Lindblad and Schrödinger descriptions discussed in the literature for a molecular example where the cavity is pumped by a laser. The laser and cavity properties are varied over a range of parameters. It is found that the Schrödinger description adequately describes the dynamics of the polaritons and emission signal as long as the laser intensity is moderate and the pump time is not much longer than the lifetime of the cavity mode. Otherwise, it is demonstrated that the Schrödinger description gradually fails. We also show that the failure of the Schrödinger description can often be remedied by renormalizing the wave function at every step of time propagation. The results are discussed and analyzed.

Normal quantum channels and Markovian correlated two-qubit quantum errors

We study general ‘normally’ (Gaussian) distributed random unitary transformations. These distributions can be defined in terms of a diffusive random walk in the respective group manifold. On the one hand, a Gaussian distribution induces a unital quantum channel, which we will call ‘normal’. On the other hand, the diffusive random walk defines a unital quantum process, generated by a Lindblad master equation. In the single qubit case, we show different distributions may induce the same quantum channel.In the case of two qubits, we study normal quantum channels, induced by Gaussian distributions in SU(2)⊗SU(2) . They provide an appropriate framework for modeling quantum errors with classical correlations. In contrast to correlated Pauli errors, for instance, they conserve their Markovianity, and they lead to very different results in error correcting codes. This is illustrated with an application to entanglement distillation.