Showing papers on "Master equation published in 2021"

Simulation methods for open quantum many-body systems

Abstract: This article reviews theoretical methods to deal with interacting quantum particles that are in contact with their environment and are thus described by a master equation rather than a Schr\"odinger equation. The similarities and differences are discussed between the pursuit of pure many-body ground states and mixed steady states by different methods, and an outlook is provided on the advances toward simulation of large open many-body system.

Quantum collision models: open system dynamics from repeated interactions

Abstract: We present an extensive introduction to quantum collision models (CMs), also known as repeated interactions schemes: a class of microscopic system-bath models for investigating open quantum systems dynamics whose use is currently spreading in a number of research areas. Through dedicated sections and a pedagogical approach, we discuss the CMs definition and general properties, their use for the derivation of master equations, their connection with quantum trajectories, their application in non-equilibrium quantum thermodynamics, their non-Markovian generalizations, their emergence from conventional system-bath microscopic models and link to the input-output formalism. The state of the art of each involved research area is reviewed through dedicated sections. The article is supported by several complementary appendices, which review standard concepts/tools of open quantum systems used in the main text with the goal of making the material accessible even to readers possessing only a basic background in quantum mechanics. The paper could also be seen itself as a friendly, physically intuitive, introduction to fundamentals of open quantum systems theory since most main concepts of this are treated such as quantum maps, Lindblad master equation, steady states, POVMs, quantum trajectories and stochastic Schrodinger equation.

Few-Mode Field Quantization of Arbitrary Electromagnetic Spectral Densities

Abstract: We develop a framework that provides a few-mode master equation description of the interaction between a single quantum emitter and an arbitrary electromagnetic environment. The field quantization requires only the fitting of the spectral density, obtained through classical electromagnetic simulations, to a model system involving a small number of lossy and interacting modes. We illustrate the power and validity of our approach by describing the population and electric field spatial dynamics in the spontaneous decay of an emitter placed in a complex hybrid plasmonic-photonic structure.

Local master equations bypass the secular approximation

Abstract: Master equations are a vital tool to model heat flow through nanoscale thermodynamic systems. Most practical devices are made up of interacting sub-system, and are often modelled using either local master equations (LMEs) or global master equations (GMEs). While the limiting cases in which either the LME or the GME breaks down are well understood, there exists a 'grey area' in which both equations capture steady-state heat currents reliably, but predict very different transient heat flows. In such cases, which one should we trust? Here, we show that, when it comes to dynamics, the local approach can be more reliable than the global one for weakly interacting open quantum systems. This is due to the fact that the secular approximation, which underpins the GME, can destroy key dynamical features. To illustrate this, we consider a minimal transport setup and show that its LME displays exceptional points (EPs). These singularities have been observed in a superconducting-circuit realisation of the model [1]. However, in stark contrast to experimental evidence, no EPs appear within the global approach. We then show that the EPs are a feature built into the Redfield equation, which is more accurate than the LME and the GME. Finally, we show that the local approach emerges as the weak-interaction limit of the Redfield equation, and that it entirely avoids the secular approximation.

Superconducting-like Heat Current: Effective Cancellation of Current-Dissipation Trade-Off by Quantum Coherence.

Abstract: Quantum coherence is a useful resource for increasing the speed and decreasing the irreversibility of quantum dynamics. Because of this feature, coherence is used to enhance the performance of various quantum information processing devices beyond the limitations set by classical mechanics. However, when we consider thermodynamic processes, such as energy conversion in nanoscale devices, it is still unclear whether coherence provides similar advantages. Here we establish a universal framework, clarifying how coherence affects the speed and irreversibility in thermodynamic processes described by the Lindblad master equation, and give general rules for when coherence enhances or reduces the performance of thermodynamic devices. Our results show that a proper use of coherence enhances the heat current without increasing dissipation; i.e., coherence can reduce friction. In particular, if the amount of coherence is large enough, this friction becomes virtually zero, realizing a superconducting-like "dissipation-less" heat current. Since our framework clarifies a general relation among coherence, energy flow, and dissipation, it can be applied to many branches of science from quantum information theory to biology. As an application to energy science, we construct a quantum heat engine cycle that exceeds the power-efficiency trade-off bound on classical engines and effectively attains the Carnot efficiency with finite power in fast cycles.

Quantum gravitational decoherence from fluctuating minimal length and deformation parameter at the Planck scale.

Abstract: Schemes of gravitationally induced decoherence are being actively investigated as possible mechanisms for the quantum-to-classical transition. Here, we introduce a decoherence process due to quantum gravity effects. We assume a foamy quantum spacetime with a fluctuating minimal length coinciding on average with the Planck scale. Considering deformed canonical commutation relations with a fluctuating deformation parameter, we derive a Lindblad master equation that yields localization in energy space and decoherence times consistent with the currently available observational evidence. Compared to other schemes of gravitational decoherence, we find that the decoherence rate predicted by our model is extremal, being minimal in the deep quantum regime below the Planck scale and maximal in the mesoscopic regime beyond it. We discuss possible experimental tests of our model based on cavity optomechanics setups with ultracold massive molecular oscillators and we provide preliminary estimates on the values of the physical parameters needed for actual laboratory implementations. Current hypotheses towards quantisation of gravity imply the presence of a minimal length scale, which may have a role in explaining quantum-to-classical transition. Here, the authors show how assuming the minimal length scale to be a fluctuating quantity leads to a possible universal decoherence mechanism.

Magnon-assisted photon-phonon conversion in the presence of structured environments

Abstract: Quantum conversion or interface is one of the most prominent protocols in quantum information processing and quantum state engineering. We propose a photon-phonon conversion protocol in a hybrid magnomechanical system comprising a microwave optical mode, a driven magnon mode, and a mechanical-vibrating mode, which has attracted much interest and is expected to become a building block of the future quantum information network due to its controllability in coupling strengths. The microwave photons in the optical cavity are coupled to the magnons by the Zeeman interaction, and the latter are coupled to the mechanical phonons by the magnetostrictive interaction. With a strong photon-magnon interaction and a strong driving on the magnon, an effective Hamiltonian is constructed to describe the conversion between photons and phonons near their resonant point. The cavity-magnon system can then play the role of quantum memory. Moreover, the faithfulness of the photon-phonon conversion is estimated in terms of fidelities for state evolution and state-independent transfer. The former is discussed in the Lindblad master equation, taking account of the leakages of photons, phonons, and magnons into consideration. The latter is derived by the Heisenberg-Langevin equation considering the non-Markovian noise from the structured environments for both optical and mechanical modes. The state-evolution fidelity is found to be robust to the weak leakage. The transfer fidelity can be maintained by the Ohmic and sub-Ohmic environments of the photons and is insensitive to the $1/f$ noise of the phonons. Our work thus provides an interesting application for the magnon system as a photon-phonon converter in the microwave regime.

Quantum thermodynamically consistent local master equations

Abstract: The paper discusses the thermodynamic consistency of quantum master equations revealing an extra contribution to the heat current due to non-compatibility of operators in quantum mechanics.

Quantum Systems Correlated with a Finite Bath: Nonequilibrium Dynamics and Thermodynamics

Abstract: A master equation is derived to describe quantum systems connected to dynamically-evolving mesoscopic baths, progressing towards the control of nanoscale quantum technologies such as engines and refrigerators.

Comparing Relaxation Mechanisms in Quantum and Classical Transverse-Field Annealing

Abstract: Annealing schedule control provides opportunities to better understand the manner and mechanisms by which putative quantum annealers operate. By appropriately modifying the annealing schedule to include a pause (keeping the Hamiltonian fixed) for a period of time, we show that it is possible to more directly probe the dissipative dynamics of the system at intermediate points along the anneal and examine thermal relaxation rates, for example, by observing the repopulation of the ground state after the minimum spectral gap. We provide a detailed comparison of experiments from a D-Wave device, simulations of the quantum adiabatic master equation, and a classical analogue of quantum annealing, spin-vector Monte Carlo, and we observe qualitative agreement, showing that the characteristic increase in success probability when pausing is not a uniquely quantum phenomena. We find that the relaxation in our system is dominated by a single timescale, which allows us to give a simple condition for when we can expect pausing to improve the time to solution, the relevant metric for classical optimization. Finally, we also explore in simulation the role of temperature whilst pausing as a means to better distinguish quantum and classical models of quantum annealers.

Finite Dimensional Approximations of Hamilton--Jacobi--Bellman Equations in Spaces of Probability Measures

Abstract: We prove that viscosity solutions of Hamilton--Jacobi--Bellman (HJB) equations, corresponding either to deterministic optimal control problems for systems of $n$ particles or to stochastic optimal ...

Note on entropy dynamics in the Brownian SYK model

Abstract: We study the time evolution of Renyi entropy in a system of two coupled Brownian SYK clusters evolving from an initial product state. The Renyi entropy of one cluster grows linearly and then saturates to the coarse grained entropy. This Page curve is obtained by two different methods, a path integral saddle point analysis and an operator dynamics analysis. Using the Brownian character of the dynamics, we derive a master equation which controls the operator dynamics and gives the Page curve for purity. Insight into the physics of this complicated master equation is provided by a complementary path integral method: replica diagonal and non-diagonal saddles are responsible for the linear growth and saturation of Ŕenyi entropy, respectively.

Gravitational decoherence: A general nonrelativistic model

Abstract: We derive a general quantum master equation for the dynamics of a scalar bosonic particle interacting with a weak, stochastic and classical external gravitational field. The dynamics predicts decoherence in position, momentum and energy. We show how our master equation reproduces the results present in the literature by taking appropriate limits, thus explaining the apparent contradiction in their dynamical description. Our result is relevant in light of the increasing interest in the low energy quantum-gravity regime.

Origin of Chirality Induced Spin Selectivity in Photoinduced Electron Transfer.

Abstract: Here we propose a mechanism by which spin-polarization can be generated dynamically in chiral molecular systems undergoing photoinduced electron transfer. The proposed mechanism explains how spin-polarization emerges in systems where charge transport is dominated by incoherent hopping, mediated by spin-orbit and electronic exchange couplings through an intermediate charge transfer state. We derive a simple expression for the spin-polarization that predicts a nonmonotonic temperature dependence, consistent with recent experiments, and a maximum spin-polarization that is independent of the magnitude of the spin-orbit coupling. We validate this theory using approximate quantum master equations and the numerically exact hierarchical equations of motion. The proposed mechanism of chirality induced spin selectivity should apply to many chiral systems, and the ideas presented here have implications for the study of spin transport at temperatures relevant to biology and provide simple principles for the molecular control of spins in fluctuating environments.

Open system dynamics from thermodynamic compatibility

Abstract: The paper uses a thermodynamically axiomatic approach to formulate the general form of the Markovian master equation.

A general quantum algorithm for open quantum dynamics demonstrated with the Fenna-Matthews-Olson complex.

Abstract: Using quantum algorithms to simulate complex physical processes and correlations in quantum matter has been a major direction of quantum computing research, towards the promise of a quantum advantage over classical approaches. In this work we develop a generalized quantum algorithm to simulate any dynamical process represented by either the operator sum representation or the Lindblad master equation. We then demonstrate the quantum algorithm by simulating the dynamics of the Fenna-Matthews-Olson (FMO) complex on the IBM QASM quantum simulator. This work represents a first demonstration of a quantum algorithm for open quantum dynamics with a moderately sophisticated dynamical process involving a realistic biological structure. We discuss the complexity of the quantum algorithm relative to the classical method for the same purpose, presenting a decisive query complexity advantage of the quantum approach based on the unique property of quantum measurement. An accurate yet tractable quantum algorithm for the description of complex open quantum systems (like the FMO complex) has a myriad of significant applications from catalytic chemistry and correlated materials physics to descriptions of hybrid quantum systems.

Probabilistic memristive networks: Application of a master equation to networks of binary ReRAM cells

Abstract: The possibility of using non-deterministic circuit components has been gaining significant attention in recent years. The modeling and simulation of their circuits require novel approaches, as now the state of a circuit at an arbitrary moment in time cannot be predicted deterministically. Generally, these circuits should be described in terms of probabilities, the circuit variables should be calculated on average, and correlation functions should be used to explore interrelations among the variables. In this paper, we use, for the first time, a master equation to analyze the networks composed of probabilistic binary memristors. Analytical solutions of the master equation for the case of identical memristors connected in-series and in-parallel are found. Our analytical results are supplemented by results of numerical simulations that extend our findings beyond the case of identical memristors. The approach proposed in this paper facilitates the development of probabilistic/stochastic electronic circuits and advance their real-world applications.

Fermi's Golden Rule for Spontaneous Emission in Absorptive and Amplifying Media.

Abstract: We demonstrate a fundamental breakdown of the photonic spontaneous emission (SE) formula derived from Fermi's golden rule, in absorptive and amplifying media, where one assumes the SE rate scales with the local photon density of states, an approach often used in more complex, semiclassical nanophotonics simulations. Using a rigorous quantization of the macroscopic Maxwell equations in the presence of arbitrary linear media, we derive a corrected Fermi's golden rule and master equation for a quantum two-level system (TLS) that yields a quantum pumping term and a modified decay rate that is net positive. We show rigorous numerical results of the temporal dynamics of the TLS for an example of two coupled microdisk resonators, forming a gain-loss medium, and demonstrate the clear failure of the commonly adopted formulas based solely on the local density of states.

Multiple-qubit controlled unitary quantum gate for Rydberg atoms using shortcut to adiabaticity and optimized geometric quantum operations

Abstract: Multiple-qubit quantum logic gates are an important element in the implementation of quantum computers. The direct construction of multiple-qubit quantum logic gates in an efficient way has important values compared to the construction of multiple-qubit gates using a series of two-qubit and single-qubit gates. We propose a scheme to construct a multiple-qubit ${\mathrm{C}}_{k}\mathrm{U}$ gate ($k$ denotes the number of control qubits and U means the arbitrary universal operation performed on the target qubit) in a neutral atom platform through the Rydberg blockade effect by successively exciting them to Rydberg states. This scheme takes advantage of the shortcut to adiabaticity of inverse engineering, geometric quantum operations, as well as optimized control theory. The geometric quantum computation considered in this manuscript guarantees the robustness to operational errors. Meanwhile, inverse engineering-based shortcut to adiabaticity provides a further advantage in terms of the speed of the system evolution compared to adiabatic processes. An additional feature of our multiple-qubit quantum logic gate is that arbitrary operation on the target atom can be realized by adjusting the amplitude and phase of the laser fields. Numerical simulation of the master equation based on the full Hamiltonian demonstrates the high fidelity of the proposed scheme and its robustness to operational errors and spontaneous emission.

Boundary time crystals in collective d -level systems

Abstract: Boundary time crystals (BTC's) are nonequilibrium phases of matter occurring in quantum systems in contact to an environment, for which a macroscopic fraction of the many-body system breaks time translation symmetry. We study BTC's in collective $d$-level systems, focusing on the cases with $d=2$, 3, and 4. We find that BTC's appear in different forms for the different cases. We first consider the model with collective $d=2$-level systems [Phys. Rev. Lett. 121, 035301 (2018)], whose dynamics is described by a Gorini-Kossakowski-Sudarshan-Lindblad master equation, and perform a throughout analysis of its phase diagram and Jacobian stability for different interacting terms in the coherent Hamiltonian. In particular, using perturbation theory for general (non-Hermitian) matrices, we obtain analytically how a specific ${\mathbb{Z}}_{2}$ symmetry-breaking Hamiltonian term destroys the BTC phase in the model. Based on these results we define a $d=4$ model composed of a pair of collective two-level systems interacting with each other. We show that this model support richer dynamical phases, ranging from limit cycles, period-doubling bifurcations, and a route to chaotic dynamics. The BTC phase is more robust in this case, not annihilated by the former symmetry-breaking Hamiltonian terms. The model with collective $d=3$-level systems is defined similarly, as competing pairs of levels, but sharing a common collective level. The dynamics can deviate significantly from the previous cases, supporting phases with the coexistence of multiple limit cycles, closed orbits and a full degeneracy of zero Lyapunov exponents.

Coordination of gene expression noise with cell size: analytical results for agent-based models of growing cell populations

Abstract: The chemical master equation and the Gillespie algorithm are widely used to model the reaction kinetics inside living cells. It is thereby assumed that cell growth and division can be modelled through effective dilution reactions and extrinsic noise sources. We here re-examine these paradigms through developing an analytical agent-based framework of growing and dividing cells accompanied by an exact simulation algorithm, which allows us to quantify the dynamics of virtually any intracellular reaction network affected by stochastic cell size control and division noise. We find that the solution of the chemical master equation-including static extrinsic noise-exactly agrees with the agent-based formulation when the network under study exhibits stochastic concentration homeostasis, a novel condition that generalizes concentration homeostasis in deterministic systems to higher order moments and distributions. We illustrate stochastic concentration homeostasis for a range of common gene expression networks. When this condition is not met, we demonstrate by extending the linear noise approximation to agent-based models that the dependence of gene expression noise on cell size can qualitatively deviate from the chemical master equation. Surprisingly, the total noise of the agent-based approach can still be well approximated by extrinsic noise models.

Random-matrix theory for the Lindblad master equation.

Abstract: Open quantum systems with Markovian dynamics can be described by the Lindblad equation The quantity governing the dynamics is the Lindblad superoperator We apply random-matrix theory to this superoperator to elucidate its spectral properties The distribution of eigenvalues and the correlations of neighboring eigenvalues are obtained for the cases of purely unitary dynamics, pure dissipation, and the physically realistic combination of unitary and dissipative dynamics

Engineered dissipation induced entanglement transition in quantum spin chains: from logarithmic growth to area law

Abstract: Recent theoretical work has shown that the competition between coherent unitary dynamics and stochastic measurements, performed by the environment, along wavefunction trajectories can give rise to transitions in the entanglement scaling. In this work, complementary to these previous studies, we analyze a situation where the role of Hamiltonian and dissipative dynamics is reversed. We consider an engineered dissipation, which stabilizes an entangled phase of a quantum spin$-1/2$ chain, while competing single-particle or interacting Hamiltonian dynamics induce a disentangled phase. Focusing on the single-particle unitary dynamics, we find that the system undergoes an entanglement transition from a logarithmic growth to an area law when the competition ratio between the unitary evolution and the non-unitary dynamics increases. We evidence that the transition manifests itself in state-dependent observables at a finite competition ratio for Hamiltonian and measurement dynamics. On the other hand, it is absent in trajectory-averaged steady-state dynamics, governed by a Lindblad master equation: although purely dissipative dynamics stabilizes an entangled state, for any non-vanishing Hamiltonian contribution the system ends up irremediably in a disordered phase. In addition, a single trajectory analysis reveals that the distribution of the entanglement entropy constitutes an efficient indicator of the transition. Complementarily, we explore the competition of the dissipation with coherent dynamics generated by an interacting Hamiltonian, and demonstrate that the entanglement transition also occurs in this second model. Our results suggest that this type of transition takes place for a broader class of Hamiltonians, underlining its robustness in monitored open quantum many-body systems.

Fundamental limitations in Lindblad descriptions of systems weakly coupled to baths

Abstract: It is very common in the literature to write down a Markovian quantum master equation in Lindblad form to describe a system with multiple degrees of freedom and weakly connected to multiple thermal baths which can, in general, be at different temperatures and chemical potentials. However, the microscopically derived quantum master equation up to leading order in system-bath coupling is of the so-called Redfield form which is known to not preserve complete positivity in most cases. We analytically show that, in such cases, enforcing complete positivity by imposing any Lindblad form, via any further approximation, necessarily leads to either violation of thermalization, or inaccurate coherences in the energy eigenbasis which then cause a violation of local conservation laws in the non-equilibrium steady state (NESS). In other words, a weak system-bath coupling quantum master equation that is completely positive, shows thermalization and preserves local conservation laws in NESS is fundamentally impossible in generic situations. On the other hand, the Redfield equation, although generically not completely positive, shows thermalization, always preserves local conservation laws and gives correct coherences to leading order. We exemplify our analytical results numerically in an interacting open quantum spin system.

Lindblad equation approach to the determination of the optimal working point in nonequilibrium stationary states of an interacting electronic one-dimensional system: Application to the spinless Hubbard chain in the clean and in the weakly disordered limit

Abstract: Using the Lindblad equation approach, we derive the range of the parameters of an interacting one-dimensional electronic chain connected to two reservoirs in the large bias limit in which an optimal working point (corresponding to a change in the monotonicity of the stationary current as a function of the applied bias) emerges in the nonequilibrium stationary state. In the specific case of the one-dimensional spinless fermionic Hubbard chain, we prove that an optimal working point emerges in the dependence of the stationary current on the coupling between the chain and the reservoirs, both in the interacting and in the noninteracting case. We show that the optimal working point is robust against localized defects of the chain, as well as against a limited amount of quenched disorder. Eventually, we discuss the importance of our results for optimizing the performance of a quantum circuit by tuning its components as close as possible to their optimal working point.

Thermodynamics of Precision in Markovian Open Quantum Dynamics

Abstract: The thermodynamic and kinetic uncertainty relations indicate trade-offs between the relative fluctuation of observables and thermodynamic quantities such as dissipation and dynamical activity. While these relations have been well studied for classical systems, they remain largely unexplored in the quantum regime. In this study, we investigate such trade-off relations for Markovian open quantum dynamics described by the Lindblad master equations. Specifically, we derive finite-time lower bounds on the relative fluctuation of both dynamical observables and the first passage time for arbitrary initial states. The bounds imply that the precision of observables is constrained not only by the thermodynamic quantities but also by quantum coherence. We find that the relative fluctuation is enhanced by quantum coherence in a generic class of thermodynamic processes for systems with non-degenerate energy levels. Our findings provide insights into when and how the classical uncertainty relations survive in the quantum case.

Revealing strong correlations in higher-order transport statistics: A noncrossing approximation approach

Abstract: We present a method for calculating the full counting statistics of a nonequilibrium quantum system based on the propagator noncrossing approximation (NCA). This numerically inexpensive method can provide higher-order cumulants for extended parameter regimes, rendering it attractive for a wide variety of purposes. We compare NCA results to Born-Markov quantum master equations (QME) results to show that they can access different physics, and to numerically exact inchworm quantum Monte Carlo data to assess their validity. As a demonstration of its power, the NCA method is employed to study the impact of correlations on higher-order cumulants in the nonequilibrium Anderson impurity model. The four lowest-order cumulants are examined, allowing us to establish that correlation effects have a profound influence on the underlying transport distributions. Higher-order cumulants are therefore demonstrated to be a proxy for the presence of Kondo correlations in a way that cannot be captured by simple QME methods.

Boundary condition independence of non-Hermitian Hamiltonian dynamics

Abstract: The non-Hermitian skin effect, namely, that the eigenvalues and eigenstates of a non-Hermitian tight-binding Hamiltonian have significant differences under open or periodic boundary conditions, is a remarkable phenomenon of non-Hermitian systems. Inspired by the presence of the non-Hermitian skin effect, we study the evolution of wave packets in non-Hermitian systems, which can be determined using the single-particle Green's function. Surprisingly, we find that in the thermodynamic limit, the Green's function does not depend on boundary conditions, despite the presence of skin effect. We provide a general proof for this statement in arbitrary dimension with finite hopping range, with an explicit illustration in the non-Hermitian Su-Schrieffer-Heeger model. We also explore its applications in noninteracting open quantum systems described by the master equation. We demonstrate that the evolution of the density matrix is independent of the boundary condition.

From nonequilibrium Green's functions to quantum master equations for the density matrix and out-of-time-order correlators: Steady-state and adiabatic dynamics

Abstract: We consider a finite quantum system under slow driving and weakly coupled to thermal reservoirs at different temperatures. We present a systematic derivation of the quantum master equation for the density matrix and the out-of-time-order correlators. We start from the microscopic Hamiltonian and we formulate the equations ruling the dynamics of these quantities by recourse to the Schwinger-Keldysh nonequilibrium Green's function formalism, performing a perturbative expansion in the coupling between the system and the reservoirs. We focus on the adiabatic dynamics, which corresponds to considering the linear response in the ratio between the relaxation time due to the system-reservoir coupling and the time scale associated to the driving. We calculate the particle and energy fluxes. We illustrate the formalism in the case of a qutrit coupled to bosonic reservoirs and of a pair of interacting quantum dots attached to fermionic reservoirs, also discussing the relevance of coherent effects.