Master Equation For Dynamics Research Articles

Weak second-order quantum state diffusion unraveling of the Lindblad master equation

Simulating mixed-state evolution in open quantum systems is crucial for various chemical physics, quantum optics, and computer science applications. These simulations typically follow the Lindblad master equation dynamics. An alternative approach known as quantum state diffusion unraveling is based on the trajectories of pure states generated by random wave functions, which evolve according to a nonlinear Itô–Schrödinger equation (ISE). This study introduces weak first-order and second-order solvers for the ISE based on directly applying the Itô–Taylor expansion with exact derivatives in the interaction picture. We tested the method on free and driven Morse oscillators coupled to a thermal environment and found that both orders allowed practical estimation with a few dozen iterations. The variance was relatively small compared to the linear unraveling and did not grow with time. The second-order solver delivers a much higher accuracy and stability with bigger time steps than the first-order scheme, with a small additional workload. However, the second-order algorithm has quadratic complexity with the number of Lindblad operators as opposed to the linear complexity of the first-order algorithm.

Intrinsic decoherence dynamics and dense coding in dipolar spin system

The quantum decoherence is an ineluctable process when a quantum system interacts with its surrounding environment. Milburn’s dynamical master equation, as a modified shape of the Schrödinger equation, is typically used to study the effects decoherence even without intervention of environment. We studied the temporal evolution of non-local correlations in a dipolar-coupled spin system under the influence of intrinsic decoherence. Primarily, the generation in correlations of Einstein–Podolsky–Rosen steering, measurement-induced non-locality and entanglement in an uncorrelated initial system state were studied. Also, the effect of the intrinsic decoherence on the quality of the dense coding capacity dynamics for the dipolar-coupled spin system was examined. Our results highlight that the initial system’s role in defining the quantum correlations’ robustness, and we also brought out the impacts of system’s parameters on the non-local correlations and the efficiency of dense coding process.

Master equation emulation and coherence preservation with classical control of a superconducting qubit

Open quantum systems are a topic of intense theoretical research. The use of master equations to model a system's evolution subject to an interaction with an external environment is one of the most successful theoretical paradigms. General experimental tools to study different open system realizations have been limited, and so it is highly desirable to develop experimental tools which emulate diverse master equation dynamics and give a way to test open systems theories. In this paper we demonstrate a systematic method for engineering specific system-environment interactions and emulating master equations of a particular form using classical stochastic noise in a superconducting transmon qubit. We also demonstrate that non-Markovian noise can be used as a resource to extend the coherence of a quantum system and counteract the adversarial effects of Markovian environments.

Intrinsic timing in classical master equation dynamics from an extended quadratic format of the evolution law

Intrinsic timing in classical master equation dynamics from an extended quadratic format of the evolution law

Accelerating the approach of dissipative quantum spin systems towards stationarity through global spin rotations

We consider open quantum systems whose dynamics is governed by a time-independent Markovian Lindblad master equation. Such systems approach their stationary state on a timescale that is determined by the spectral gap of the generator of the master equation dynamics. In a recent paper [Carollo et al., Phys. Rev. Lett. 127, 060401 (2021)] it was shown that under certain circumstances it was possible to exponentially accelerate the approach to stationarity by performing a unitary transformation of the initial state. This phenomenon can be regarded as the quantum version of the so-called Mpemba effect. The transformation of the initial state removes its overlap with the dynamical mode of the open system dynamics that possesses the slowest decay rate and, thus, determines the spectral gap. Whereas this transformation can be exactly constructed in some cases, it is, in practice, challenging to implement. Here we show that even far simpler transformations constructed by a global unitary spin rotation allow to exponentially speed up relaxation. We demonstrate this using simple dissipative quantum spin systems, which are relevant for current quantum simulation and computation platforms based on trapped atoms and ions.

Solving quantum master equations with deep quantum neural networks

Deep quantum neural networks may provide a promising way to achieve a quantum learning advantage with noisy intermediate-scale quantum devices. Here, we use deep quantum feed-forward neural networks capable of universal quantum computation to represent the mixed states for open quantum many-body systems and introduce a variational method with quantum derivatives to solve the master equation for dynamics and stationary states. Owning to the special structure of the quantum networks, this approach enjoys a number of notable features, including an efficient quantum analog of the back-propagation algorithm, resource-saving reuse of hidden qubits, general applicability independent of dimensionality and entanglement properties, as well as the convenient implementation of symmetries. As proof-of-principle demonstrations, we apply this approach to both one-dimensional transverse field Ising and two-dimensional ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ models with dissipation, and show that it can efficiently capture their dynamics and stationary states with a desired accuracy.

Improved estimations of stochastic chemical kinetics by finite-state expansion

Stochastic reaction networks are a fundamental model to describe interactions between species where random fluctuations are relevant. The master equation provides the evolution of the probability distribution across the discrete state space consisting of vectors of population counts for each species. However, since its exact solution is often elusive, several analytical approximations have been proposed. The deterministic rate equation (DRE) gives a macroscopic approximation as a compact system of differential equations that estimate the average populations for each species, but it may be inaccurate in the case of nonlinear interaction dynamics. Here we propose finite-state expansion (FSE), an analytical method mediating between the microscopic and the macroscopic interpretations of a stochastic reaction network by coupling the master equation dynamics of a chosen subset of the discrete state space with the mean population dynamics of the DRE. An algorithm translates a network into an expanded one where each discrete state is represented as a further distinct species. This translation exactly preserves the stochastic dynamics, but the DRE of the expanded network can be interpreted as a correction to the original one. The effectiveness of FSE is demonstrated in models that challenge state-of-the-art techniques due to intrinsic noise, multi-scale populations and multi-stability.

Dynamics of quantum correlations under intrinsic decoherence in a Heisenberg spin chain model with Dzyaloshinskii–Moriya interaction

By taking into account the effect of intrinsic decoherence and by using Milburn’s dynamical master equation, we study the temporal evolution of quantum correlations in a two-qubit XXZ Heisenberg spin chain model with Dzyaloshinskii–Moriya (DM) interaction and an external nonuniform magnetic field both directed along the z-axis. We use the concurrence (C) to detect entanglement and the local quantum uncertainty (LQU) to measure discord-like correlations. We consider three cases of initial quantum states: the mixed state, the Werner state and the pure state. For the mixed initial state and the Werner initial state, our results show that the external magnetic field strongly stimulates the effect of intrinsic decoherence which can highlight the entanglement sudden death phenomenon, while the LQU is resistant to sudden death. In addition, the DM interaction makes the effect of intrinsic decoherence more pronounced. However, a weak DM interaction can markedly improve quantum correlations and thus cause the phenomenon of entanglement sudden revival. On the other hand, and especially for the initial “uncorrelated” state (in terms of entanglement and LQU) with a zero nonuniform magnetic field and no DM interaction, it is easier to generate a strong entanglement, but it is difficult to generate LQU. Finally, we have found that when the system is initially “uncorrelated,” the nonuniform magnetic field can make the system strongly correlated for very remarkable steady-state values (in particular entanglement). Other results will also be discussed.

Spectral theory of fluctuations in time-average statistical mechanics of reversible and driven systems

We present a spectral-theoretic approach to time-average statistical mechanics for general, non-equilibrium initial conditions. We consider the statistics of bounded, local additive functionals of reversible as well as irreversible ergodic stochastic dynamics with continuous or discrete state-space. We derive exact results for the mean, fluctuations and correlations of time average observables from the eigenspectrum of the underlying generator of Fokker-Planck or master equation dynamics, and discuss the results from a physical perspective. Feynman-Kac formulas are re-derived using It\^o calculus and combined with non-Hermitian perturbation theory. The emergence of the universal central limit law in a spectral representation is shown explicitly on large deviation time-scales. For reversible dynamics with equilibrated initial conditions we derive a general upper bound to fluctuations of occupation measures in terms of an integral of the return probability. Simple, exactly solvable examples are analyzed to demonstrate how to apply the theory. As a biophysical example we revisit the Berg-Purcell problem on the precision of concentration measurements by a single receptor. Our results are directly applicable to a diverse range of phenomena underpinned by time-average observables and additive functionals in physical, chemical, biological, and economical systems.

Thermodynamics of optical Bloch equations

Optical Bloch equations (OBE) describe the coherent exchange of energy between a quantum bit (qubit) and a quasi-resonant driving field in the presence of a thermal bath. Despite it being an ubiquitous process in quantum technologies, a sound thermodynamic analysis is still missing. We hereby provide such an analysis, by deriving the relevant framework from first principles. We start from a complete microscopic description of the qubit-bath system where definitions of heat, work and entropy production are unambiguous. We trace out the bath and coarse-grain the resulting expressions in time, using a methodology similar to the derivation of the dynamical master equation, to derive closed expressions for the first and second law in terms of system properties. Long coarse graining times yield the Floquet Master equation and its already known thermodynamic description. Short coarse-graining times yield instead the OBE and a novel thermodynamic framework which explicitly depends on quantum coherences in the qubit’s energy basis which produce quantum signatures in the heat and entropy production flows. This allows us to characterize a genuinely quantum non-equilibrium situation, where the coherences created by the driving field are continuously erased by the bath. Our findings can be readily extended to larger open quantum systems. They carry the seeds for future thermodynamic analyses of quantum gates and the design of quantum engines in the strong coherent driving regime.

A space\u2013time tradeoff for implementing a function with master equation dynamics

Master equations are commonly used to model the dynamics of physical systems, including systems that implement single-valued functions like a computer’s update step. However, many such functions cannot be implemented by any master equation, even approximately, which raises the question of how they can occur in the real world. Here we show how any function over some “visible” states can be implemented with master equation dynamics—if the dynamics exploits additional, “hidden” states at intermediate times. We also show that any master equation implementing a function can be decomposed into a sequence of “hidden” timesteps, demarcated by changes in what state-to-state transitions have nonzero probability. In many real-world situations there is a cost both for more hidden states and for more hidden timesteps. Accordingly, we derive a “space–time” tradeoff between the number of hidden states and the number of hidden timesteps needed to implement any given function.

Exact master equation and general non-Markovian dynamics in open quantum systems

Investigations of quantum and mesoscopic thermodynamics force one to answer two fundamental questions associated with the foundations of statistical mechanics: (i) how does macroscopic irreversibility emerge from microscopic reversibility? (ii) how does the system relax in general to thermal equilibrium with its environment? The answers to these questions rely on a deep understanding of nonequilibrium decoherence dynamics of systems interacting with their environments. Decoherence is also a main concern in developing quantum information technology. In the past two decades, many theoretical and experimental investigations have devoted to this topic, most of these investigations take the Markov (memory-less) approximation. These investigations have provided a partial understanding to several fundamental issues, such as quantum measurement and the quantum-to-classical transition, etc. However, experimental implementations of nanoscale solid-state quantum information processing makes strong non-Markovian memory effects unavoidable, thus rendering their study a pressing and vital issue. Through the rigorous derivation of the exact master equation and a systematical exploration of various non-Markovian processes for a large class of open quantum systems, we find that decoherence manifests unexpected complexities. We demonstrate these general non-Markovian dynamics manifested in different open quantum systems.

Spatial Stochastic Intracellular Kinetics: A Review of Modelling Approaches

Models of chemical kinetics that incorporate both stochasticity and diffusion are an increasingly common tool for studying biology. The variety of competing models is vast, but two stand out by virtue of their popularity: the reaction–diffusion master equation and Brownian dynamics. In this review, we critically address a number of open questions surrounding these models: How can they be justified physically? How do they relate to each other? How do they fit into the wider landscape of chemical models, ranging from the rate equations to molecular dynamics? This review assumes no prior knowledge of modelling chemical kinetics and should be accessible to a wide range of readers.

Exact master equation and non-Markovian decoherence dynamics of Majorana zero modes under gate-induced charge fluctuations

In this paper, we derive the exact master equation to investigate the decoherence dynamics of Majorana zero modes in the Kitaev model, a 1D $p\,$-wave spinless topological superconducting chain (TSC), that is disturbed by charge fluctuations through gate controls. The exact master equation is derived by extending Feynman-Vernon influence functional approach to fermionic open systems involving pairing excitations. We obtain the exact master equations for the zero-energy bogoliubon in the TSC, and then transfer it into the master equation for Majorana zero modes. Within this exact master equation formalism, we can describe in detail the non-Markovian decoherence dynamics of zero-energy bogolibons as well as the Majorana zero modes under local perturbations. We find that at zero temperature, there is a zero-energy localized bound state which is not the original zero-energy bogoliubon or the original Majorana zero mode but a localized bound state of Majorana zero mode after the charge fluctuation is taken into account. It is this zero-energy localized bound state that protects Majorana zero modes from decoherence, as a long-time non-Markovain memory effect. However, for the environment at finite temperature, the zero-energy localized bound state cannot be formed when Majorana zero modes are locally perturbed, and decoherence is inevitable.

Two coupled nonlinear cavities in a driven-dissipative environment

We investigate two coupled nonlinear cavities that are coherently driven in a dissipative environment. We perform semiclassical, numerical and analytical quantum studies of this dimer model when both cavities are symmetrically driven. In the semiclassical analysis, we find steady-state solutions with different photon occupations in two cavities. Such states can be considered analogs of the closed system double well symmetry breaking states. We analyze the occurrence and properties of these localized states in the system parameter space and examine how the symmetry breaking states, in form of a bistable pair, are associated to the single cavity bistable behavior. In a full quantum calculation of the master equation dynamics that includes quantum fluctuations, the symmetry breaking states and bistability disappear due to the quantum fluctuations. In quantum trajectory picture, we observe enhanced quantum jumps and switching which indicate the presence of the underlying semiclassical symmetry breaking states. Finally, we present a set of analytical solutions for the steady state correlation functions using the complex P-representation and discuss its regime of validity.

Minimalistic analytical approach to non-Markovian open quantum systems

The dynamics of finite dimension open quantum systems is studied with the help of the simplest possible form of projection operators, namely the ones which project only onto one-dimensional subspaces. The simplicity of the action of the projection operators always leads to an analytical solution of the dynamical master equation, even in the non-Markovian case, in any perturbative order. The analytical solution correctly reproduces the short-time dynamics, and can be used to recursively recover the dynamics for an arbitrary time interval with arbitrary precision. The necessary number of relevant degrees of freedom to completely characterise an open quantum system is , where n is the dimension of the Hilbert space of the open system. The method is illustrated by two examples, the relaxation of a qubit in a thermal bath and the dynamics of two interacting qubits in a common environment.

Probing the limits of gate-based charge sensing.

Quantum computation requires a qubit-specific measurement capability to readout the final state of individual qubits. Promising solid-state architectures use external readout electrometers but these can be replaced by a more compact readout element, an in situ gate sensor. Gate-sensing couples the qubit to a resonant circuit via a gate and probes the qubit's radiofrequency polarizability. Here we investigate the ultimate performance of such a resonant readout scheme and the noise sources that limit its operation. We find a charge sensitivity of 37 μe Hz(-1/2), the best value reported for this technique, using the example of a gate sensor strongly coupled to a double quantum dot at the corner states of a silicon nanowire transistor. We discuss the experimental factors limiting gate detection and highlight ways to optimize its sensitivity. In total, resonant gate-based readout has advantages over external electrometers both in terms of reduction of circuit elements as well as absolute charge sensitivity.

Nonequilibrium functional renormalization for driven-dissipative Bose-Einstein condensation

We present a comprehensive analysis of critical behavior in the driven-dissipative Bose condensation transition in three spatial dimensions. Starting point is a microscopic description of the system in terms of a many-body quantum master equation, where coherent and driven-dissipative dynamics occur on an equal footing. An equivalent Keldysh real time functional integral reformulation opens up the problem to a practical evaluation using the tools of quantum field theory. In particular, we develop a functional renormalization group approach to quantitatively explore the universality class of this stationary non-equilibrium system. Key results comprise the emergence of an asymptotic thermalization of the distribution function, while manifest non-equilibrium properties are witnessed in the response properties in terms of a new, independent critical exponent. Thus the driven-dissipative microscopic nature is seen to bear observable consequences on the largest length scales. The absence of two symmetries present in closed equilibrium systems - underlying particle number conservation and detailed balance, respectively - is identified as the root of this new non-equilibrium critical behavior. Our results are relevant for broad ranges of open quantum systems on the interface of quantum optics and many-body physics, from exciton-polariton condensates to cold atomic gases.

Dissipative Binding of Lattice Bosons through Distance-Selective Pair Loss

We show that in a gas of ultracold atoms distance selective two-body loss can be engineered via the resonant laser excitation of atom pairs to interacting electronic states. In an optical lattice this leads to a dissipative master equation dynamics with Lindblad jump operators that annihilate atom pairs with a specific interparticle distance. In conjunction with coherent hopping between lattice sites this unusual dissipation mechanism leads to the formation of coherent long-lived complexes that can even exhibit an internal level structure which is strongly coupled to their external motion. We analyze this counterintuitive phenomenon in detail in a system of hard-core bosons. While current research has established that dissipation in general can lead to the emergence of coherent features in many-body systems our work shows that strong nonlocal dissipation can effectuate a binding mechanism for particles.

Modified fluctuation-dissipation and Einstein relation at nonequilibrium steady states

Starting from the pioneering work of Agarwal [G. S. Agarwal, Zeitschrift für Physik 252, 25 (1972)], we present a unified derivation of a number of modified fluctuation-dissipation relations (MFDR) that relate response to small perturbations around nonequilibrium steady states to steady-state correlations. Using this formalism we show the equivalence of velocity forms of MFDR derived using continuum Langevin and discrete master equation dynamics. The resulting additive correction to the Einstein relation is exemplified using a flashing ratchet model of molecular motors.