Time-convolutionless Master Equation Research Articles

Diagrammatic representation and nonperturbative approximations of the exact time-convolutionless master equation.

The time-convolutionless master equation provides a general framework to model the non-Markovian dynamics of an open quantum system with a time-local generator. A diagrammatic representation is developed and proven for the perturbative expansion of the exact time-local generator for an open quantum system interacting with arbitrary environments. A truncation of the perturbation expansion leads to perturbative time-convolutionless quantum master equations. We further introduce a general iterative approach to construct nonperturbative approximations for the time-local generator as nested time-ordered exponential operators.

Invalidation of the Bloch-Redfield equation in the sub-Ohmic regime via a practical time-convolutionless fourth-order master equation

Invalidation of the Bloch-Redfield equation in the sub-Ohmic regime via a practical time-convolutionless fourth-order master equation

Semiclassical approaches to perturbative time-convolution and time-convolutionless quantum master equations for electronic transitions in multistate systems.

Understanding the dynamics of photoinduced processes in complex systems is crucial for the development of advanced energy-conversion materials. In this study, we investigate the nonadiabatic dynamics using time-convolution (TC) and time-convolutionless (TCL) quantum master equations (QMEs) based on treating electronic couplings as perturbation within the framework of multistate harmonic (MSH) models. The MSH model Hamiltonians are mapped from all-atom simulations such that all pairwise reorganization energies are consistently incorporated, leading to a heterogeneous environment that couples to the multiple electronic states differently. Our exploration encompasses the photoinduced charge transfer dynamics in organic photovoltaic carotenoid-porphyrin-C60 triad dissolved in liquid solution and the excitation energy transfer (EET) dynamics in photosynthetic Fenna-Matthews-Olson complexes. By systematically comparing the perturbative TC and TCL QME approaches with exact quantum-mechanical and various semiclassical approximate kernels, we demonstrate their efficacy and accuracy in capturing the essential features of photoinduced dynamics. Our calculations show that TC QMEs generally yield more accurate results than TCL QMEs, especially in EET, although both methods offer versatile approaches adaptable across different systems. In addition, we investigate various semiclassical approximations featuring the Wigner-transformed and classical nuclear densities as well as the governing dynamics during the quantum coherence period, highlighting the trade-off between accuracy and computational cost. This work provides valuable insights into the applicability and performance of TC and TCL QME approaches via the MSH model, offering guidance for realistic applications to condensed-phase systems on the atomistic level.

Geometric-arithmetic master equation in large and fast open quantum systems

Understanding nonsecular dynamics in open quantum systems is addressed here, with emphasis on systems with large numbers of Bohr frequencies, zero temperature, and fast driving. We employ the master equation, which replaces arithmetic averages of the decay rates in the open system, with their geometric averages, and find that it can improve the second order perturbation theory, known as the Redfield equation, while enforcing complete positivity on quantum dynamics. The characteristic frequency scale that governs the approximation is the minimax frequency: the minimum of the maximum system oscillation frequency and the bath relaxation rate; this needs to be larger than the dissipation rate for it to be valid. The concepts are illustrated on the Heisenberg ferromagnetic spin-chain model. To study the accuracy of the approximation, a Hamiltonian is drawn from the Gaussian unitary ensemble, for which we calculate the fourth order time-convolutionless master equation, in the Ohmic bath at zero temperature. Enforcing the geometric average, decreases the trace distance to the exact solution. Dynamical decoupling of a qubit is examined by applying the Redfield and the geometric-arithmetic master equations, in the interaction picture of the time dependent system Hamiltonian, and the results are compared to the exact path integral solution. The geometric-arithmetic approach is significantly simpler and can be super-exponentially faster compared to the Redfield approach.

Noisy coherent population trapping: applications to noise estimation and qubit state preparation

Coherent population trapping is a well-known quantum phenomenon in a driven Λ system, with many applications across quantum optics. However, when a stochastic bath is present in addition to vacuum noise, the observed trapping is no longer perfect. Here we derive a time-convolutionless master equation describing the equilibration of the Λ system in the presence of additional temporally correlated classical noise, with an unknown decay parameter. Our simulations show a one-to-one correspondence between the decay parameter and the depth of the characteristic dip in the photoluminescence spectrum, thereby enabling the unknown parameter to be estimated from the observed spectra. We apply our analysis to the problem of qubit state initialization in a Λ system via dark states and show how the stochastic bath affects the fidelity of such initialization as a function of the desired dark-state amplitudes. We show that an optimum choice of Rabi frequencies is possible.

Trajectory tracking for non-Markovian quantum systems

We propose a systematic scheme to engineer quantum states of a quantum system governed by a time-convolutionless non-Markovian master equation. According to the idea of reverse engineering, the general algebraic equation to determine the control parameters, such as coherent and incoherent control fields, is presented. Without artificially engineering the time-dependent decay rates and retaining the environment-induced Lamb shifts, the quantum state can still be transferred into the target state in a finite period of time along an arbitrary designed trajectory strictly in Hilbert space. As an application, we apply our scheme to a driven two-level non-Markovian system and realize instantaneous-steady-state tracking and a complete population inversion with control parameters which are available in experimental settings.

Phonon signatures in spectra of exciton polaritons in transition metal dichalcogenides

Embedding a monolayer of a transition metal dichalcogenide in a high-$Q$ optical cavity results in the formation of distinct exciton polariton modes. The polaritons are affected by the strong exciton-phonon interaction in the monolayer. We use a time convolutionless master equation to calculate the phonon influence on the spectra of the polaritons. We discuss the nontrivial dependence of the line shapes of both branches on temperature and detuning. The peculiar polariton dispersion relation results in a linewidth of the lower polariton being largely independent of the coupling to acoustic phonons. For the upper polariton, acoustic phonons lead to a low-energy shoulder of the resonance in the linear response. Furthermore, we analyze the influence of inhomogeneous broadening being the dominant contribution to the lower polariton linewidth at low temperatures. Our results point towards interesting phonon features in polariton spectra in transition metal dichalcogenides.

How Quantum Evolution with Memory is Generated in a Time-Local Way

Two widely used but distinct approaches to the dynamics of open quantum systems are the Nakajima-Zwanzig and time-convolutionless quantum master equation, respectively. Although both describe identical quantum evolutions with strong memory effects, the first uses a time-nonlocal memory kernel $\mathcal{K}$, whereas the second achieves the same using a time-local generator $\mathcal{G}$. Here we show that the two are connected by a simple yet general fixed-point relation: $\mathcal{G} = \hat{\mathcal{K}}[\mathcal{G}]$. This allows one to extract nontrivial relations between the two completely different ways of computing the time-evolution and combine their strengths. We first discuss the stationary generator, which enables a Markov approximation that is both nonperturbative and completely positive for a large class of evolutions. We show that this generator is not equal to the low-frequency limit of the memory kernel, but additionally "samples" it at nonzero characteristic frequencies. This clarifies the subtle roles of frequency dependence and semigroup factorization in existing Markov approximation strategies. Second, we prove that the fixed-point equation sums up the time-domain gradient / Moyal expansion for the time-nonlocal quantum master equation, providing nonperturbative insight into the generation of memory effects. Finally, we show that the fixed-point relation enables a direct iterative numerical computation of both the stationary and the transient generator from a given memory kernel. For the transient generator this produces non-semigroup approximations which are constrained to be both initially and asymptotically accurate at each iteration step.

Open quantum systems beyond Fermi's golden rule: Diagrammatic expansion of the steady-state time-convolutionless master equations

Steady-state observables, such as occupation numbers and currents, are crucial experimental signatures in open quantum systems. The time-convolutionless (TCL) master equation, which is both exact and time-local, is an ideal candidate for the perturbative computation of such observables. We develop a diagrammatic approach to evaluate the steady-state TCL generator based on operators rather than superoperators. We obtain the steady-state occupation numbers, extend our formulation to the calculation of currents, and provide a simple physical interpretation of the diagrams. We further benchmark our method on a single non-interacting level coupled to Fermi reservoirs, where we recover the exact expansion to next-to-leading order. The low number of diagrams appearing in our formulation makes the extension to higher orders accessible. Combined, these properties make the steady-state time-convolutionless master equation an effective tool for the calculation of steady-state properties in open quantum systems.

Time-Convolutionless Master Equation for Multi-Level Open Quantum Systems with Initial System-Environment Correlations

Time-Convolutionless Master Equation for Multi-Level Open Quantum Systems with Initial System-Environment Correlations

Theoretical Study of Non-Markov Effects on Singlet Fission Dynamics of Model Pentacene Dimers Using the Second-Order Time-Convolutionless Quantum Master Equation Method

Non-Markov effects on singlet fission (SF) dynamics of model pentacene dimers are investigated using the second-order time-convolutionless quantum master equation method with the state-independent (SI)/dependent (SD) spectral density of vibronic coupling for each diabatic exciton state. It is found on the early time scale (∼80 fs) that, as compared to Markov dynamics, non-Markov dynamics using the SD spectral density causes a significant increase of the double-triplet (TT) population by 15%, though the SI one just causes a negligible variation (∼3%) of the TT population. The difference in the non-Markov effects between the cases of using SI and SD spectral densities is found to be related to the frequency giving the peak of spectral density for the charge-transfer (CT) state (which is concerned with amplification of the non-Markov population relaxation) and the amplitude of the spectral density for the TT state at the energy gap between high-lying CT-like (CT′) and TT-like (TT′) adiabatic states (which is concerned with acceleration of the non-Markov population relaxation from the initial (Frenkel exciton (FE)-like; FE′) to the final (TT′) adiabatic states). We also examine the dependence of non-Markov effects on the energy gap between FE′ and TT′ states using SD spectral densities. It turns out (i) that non-Markov effects depend on the relationship between the SF time scale and the detuning degree between the peak frequency of the primary contributing spectral density and the energy gap between the related adiabatic exciton states, and (ii) that they appear more significantly in the case of the detuning degree closer to the SF time scale. The obtained results reveal the conditions for inducing the non-Markov effects in realistic SF materials and thereby pave the way toward constructing design guidelines for highly efficient SF systems based on non-Markov dynamics.

Theory of the absorption line shape in monolayers of transition metal dichalcogenides

The linear absorption spectra in monolayers of transition metal dichalcogenides show pronounced signatures of the exceptionally strong exciton-phonon interaction in these materials. To account for both exciton and phonon physics in such optical signals, we compare different theoretical methods to calculate the absorption spectra using the example of $\mathrm{MoSe_2}$. In this paper, we derive the equations of motion for the polarization either using a correlation expansion up to 4th Born approximation or a time convolutionless master equation. We show that the Born approximation might become problematic when not treated in high enough order, especially at high temperatures. In contrast, the time convolutionless formulation gives surprisingly good results despite its simplicity when compared to higher-order corrrelation expansion and therefore provides a powerful tool to calculate the lineshape of linear absorption spectra in the very popular monolayer materials.

Dynamical and thermodynamical approaches to open quantum systems

The non-Markovian dynamics of open quantum systems is studied from two different points of view. The first one coincides with the traditional tracing out of the environmental degrees of freedom, presented in classical textbooks on open quantum systems. The second one is an approximation of the exact density operator with the knowledge of only several dynamical variables in the spirit of non-equilibrium thermodynamics. The approximation is based on the principle of maximal entropy. We discuss the information and the Renyi entropies, which lead to different approximations. The time-convolutionless master equation governs the dynamics of both traditional and approximated reduced density operator with a particular projection operator. Considering the example of two interacting qubits in a thermal environment, we compare the traditional and thermodynamical approaches.

Quantum algorithm for the simulation of open-system dynamics and thermalization

The quantum open-system simulation is an important category of quantum simulation. By simulating the thermalisation process at the zero temperature, we can solve the ground-state problem of quantum systems. To realise the open-system evolution on the quantum computer, we need to encode the environment using qubits. However, usually the environment is much larger than the system, i.e. numerous qubits are required if the environment is directly encoded. In this paper, we propose a way to simulate open-system dynamics by reproducing reservoir correlation functions using a minimised Hilbert space. In this way, we only need a small number of qubits to represent the environment. To simulate the $n$-th-order expansion of the time-convolutionless master equation by reproducing up to $n$-time correlation functions, the number of qubits representing the environment is $\sim \lfloor \frac{n}{2} \rfloor \log_2(N_\omega N_\beta)$. Here, $N_\omega$ is the number of frequencies in the discretised environment spectrum, and $N_\beta$ is the number of terms in the system-environment interaction. By reproducing two-time correlation functions, i.e. taking $n = 2$, we can simulate the Markovian quantum master equation. In our algorithm, the environment on the quantum computer could be even smaller than the system.

Quantum Dynamics in a Fluctuating Environment

We theoretically investigate the dynamics of a quantum system which is coupled to a fluctuating environment based on the framework of Kubo-Anderson spectral diffusion. By employing the projection operator technique, we derive two types of dynamical equations, namely, time-convolution and time-convolutionless quantum master equations, respectively. We derive the exact quantum master equations of a qubit system with both diagonal splitting and tunneling coupling when the environmental noise is subject to a random telegraph process and a Ornstein-Uhlenbeck process, respectively. For the pure decoherence case with no tunneling coupling, the expressions of the decoherence factor we obtained are consistent with the well-known existing ones. The results are significant to quantum information processing and helpful for further understanding the quantum dynamics of open quantum systems.

Time-Convolutionless Master Equation for Multi-Level Open Quantum Systems with Initial System-Environment Correlations

Time-Convolutionless Master Equation for Multi-Level Open Quantum Systems with Initial System-Environment Correlations

Excitation energy transfer with initial system-bath correlations for coherent initial conditions in a toy donor-acceptor model

A theory of coherent resonant energy transfer with initial system-bath correlations [Qin et al., Phys. Rev. A 99, 032111 (2019)] is extended for a coherent initial system condition, which is a linear superposition of donor and acceptor excitations, and is more general and actual in many respects. The nonequilibrium bath state, containing initial system-bath correlations, is expanded in powers of coupling strength within the polaron formalism of second-order time-convolutionless quantum master equation. Detailed expressions for both homogeneous and inhomogeneous terms are derived and calculations are performed under the assumption of super-Ohmic spectral densities where the case of common bath modes is also included. Numerical results indicate that, for certain initial system conditions, the nonequilibrium due to the initial system-bath correlations brings about larger amplitude of population oscillation. And the initial system condition is of crucial importance in determining whether the initial system-bath correlations would play a significant role in the transfer dynamics. We identify a sensitive window in which the transfer dynamics is especially vulnerable to the bath such that there exists a greatly obvious difference between the dynamics with and without initial correlations. Besides, the system-bath coupling strength, the evolution time, and the nature of common bath modes are also shown to contribute to these effects. Rate equations based on F\orster-Dexter energy transfer theory are derived for comparison.

Simulating conical intersection dynamics in the condensed phase with hybrid quantum master equations

We present a framework for simulating relaxation dynamics through a conical intersection of an open quantum system that combines methods to approximate the motion of degrees of freedom with disparate time and energy scales. In the vicinity of a conical intersection, a few degrees of freedom render the nuclear dynamics nonadiabatic with respect to the electronic degrees of freedom. We treat these strongly coupled modes by evolving their wavepacket dynamics in the absence of additional coupling exactly. The remaining weakly coupled nuclear degrees of freedom are partitioned into modes that are fast relative to the nonadiabatic coupling and those that are slow. The fast degrees of freedom can be traced out and treated with second-order perturbation theory in the form of the time-convolutionless master equation. The slow degrees of freedom are assumed to be frozen over the ultrafast relaxation and treated as sources of static disorder. In this way, we adopt the recently developed frozen-mode extension to second-order quantum master equations. We benchmark this approach to numerically exact results in models of pyrazine internal conversion and rhodopsin photoisomerization. We use this framework to study the dependence of the quantum yield on the reorganization energy and the characteristic time scale of the bath in a two-mode model of photoisomerization. We find that the yield is monotonically increasing with reorganization energy for a Markovian bath but monotonically decreasing with reorganization energy for a non-Markovian bath. This reflects the subtle interplay between dissipation and decoherence in conical intersection dynamics in the condensed phase.

Excitation-energy transfer from weak to strong coupling: Role of initial system-bath correlations

Excitation-energy transfer problems in molecular aggregates have attracted intensive research interests for their fundamental importance. It is usually supposed that the total density operator at time $t=0$ is a factorized product of single-excitation system state and thermal equilibrium bath state. The Franck-Condon principle ensures that it is a good approximation for a rapid photoexcitation. For natural photosynthesis, however, the transfer process in light-harvesting complexes mostly is initialized from an excited molecule, not direct absorption of a photon. Such intermolecular excitations span the same time scale as the transfer dynamics, and longer than the characteristic time of bath relaxation. Therefore, the initial system-bath correlations should be reconsidered. This work extends the coherent resonant energy transfer theory by including initial system-bath correlations. Within the approach of a second-order time-convolutionless quantum master equation in the polaron frame, a general time evolution equation for the reduced system density operator is obtained, including detailed expressions for both homogeneous and inhomogeneous terms. Two essentially distinct kinds of nonequilibrium are identified: one stems from subsistent initial system-bath correlations, while the other is from the theory of polaron transformation itself. The two kinds of nonequilibrium can accelerate or slow down the dynamical evolution for different energetic situations. Rate equations based on F\orster-Dexter theory with and without initial correlations are also derived for comparison. Besides, our conclusions provide a positive perspective accounting for the quantum coherence induced by initial system-bath correlations, which may help to clarify the long-lasting issue of whether quantum phenomenons might be observed under natural conditions of excitation by incoherent solar light, and deepen understandings of the physical mechanisms underlying the natural photosynthesis process.

Non-Markovian quantum interference in multilevel quantum systems: exact master equation approach

We study the non-Markovian dynamics of multilevel quantum systems coupled with a bosonic dissipative environment. Based on the known exact quantum-state diffusion (QSD) equations, we propose a systematic approach to derive exact time-convolutionless master equations for multilevel quantum systems. Through a combination of analytical and numerical approaches, we extract the non-Markovian dynamics of quantum interference in different time scales. Also, we demonstrate the evolution of quantum interference in a four-level system controlled by an external electromagnetic field. Our findings are extended to few-body quantum networks, with a universal formalism established.