Generalized Quantum Master Equation Research Articles

From Liouville to Landauer: Electron transport and the bath assumptions made along the way

A generalized quantum master equation approach is introduced to describe electron transfer in molecular junctions that spans both the off-resonant (tunneling) and resonant (hopping) transport regimes. The model builds on prior insights from scattering theory but is not limited to a certain parameter range with regard to the strength of the molecule–electrode coupling. The framework is used to study the simplest case of energy and charge transfer between the molecule and the electrodes for a single site noninteracting Anderson model in the limit of symmetric and asymmetric coupling between the molecule and the electrodes. In the limit of elastic transport, the Landauer result is recovered for the current by invoking a single active electron Ansatz and a binary collision approximation for the memory kernel. Inelastic transport is considered by allowing the excitation of electron–hole pairs in the electrodes in tandem with charge transport. In the case of low bias voltages where the Fermi levels of the electrodes remain below the molecular state, it is shown that the current arises from tunneling and the molecule remains neutral. However, once the threshold is reached for aligning the fermi level of one electrode with the molecular orbital, a small amount of charge transfer occurs with a negligible amount of hopping current. While inelasticity in the current has a minimal impact on the shape of the current–voltage curve in the case of symmetric electrode coupling, the results for a slight asymmetry in coupling demonstrate complete charge transfer and a significant drop in current. These results provide encouraging confirmation that the framework can describe charge transport across a wide range of electrode–molecule coupling and provide a unique perspective for developing new master equation treatments for energy and charge transport in molecular junctions. An extension of this work to account for inelastic scattering from electron–vibrational coupling at the molecule is straightforward and will be the subject of subsequent work.

Efficient formulation of multitime generalized quantum master equations: Taming the cost of simulating 2D spectra.

Modern 4-wave mixing spectroscopies are expensive to obtain experimentally and computationally. In certain cases, the unfavorable scaling of quantum dynamics problems can be improved using a generalized quantum master equation (GQME) approach. However, the inclusion of multiple (light-matter) interactions complicates the equation of motion and leads to seemingly unavoidable cubic scaling in time. In this paper, we present a formulation that greatly simplifies and reduces the computational cost of previous work that extended the GQME framework to treat arbitrary numbers of quantum measurements. Specifically, we remove the time derivatives of quantum correlation functions from the modified Mori-Nakajima-Zwanzig framework by switching to a discrete-convolution implementation inspired by the transfer tensor approach. We then demonstrate the method's capabilities by simulating 2D electronic spectra for the excitation-energy-transfer dimer model. In our method, the resolution of data can be arbitrarily coarsened, especially along the t2 axis, which mirrors how the data are obtained experimentally. Even in a modest case, this demands O(103) fewer data points. We are further able to decompose the spectra into one-, two-, and three-time correlations, showing how and when the system enters a Markovian regime where further measurements are unnecessary to predict future spectra and the scaling becomes quadratic. This offers the ability to generate long-time spectra using only short-time data, enabling access to timescales previously beyond the reach of standard methodologies.

Simulating Open Quantum System Dynamics on NISQ Computers with Generalized Quantum Master Equations.

We present a quantum algorithm based on the generalized quantum master equation (GQME) approach to simulate open quantum system dynamics on noisy intermediate-scale quantum (NISQ) computers. This approach overcomes the limitations of the Lindblad equation, which assumes weak system-bath coupling and Markovity, by providing a rigorous derivation of the equations of motion for any subset of elements of the reduced density matrix. The memory kernel resulting from the effect of the remaining degrees of freedom is used as input to calculate the corresponding non-unitary propagator. We demonstrate how the Sz.-Nagy dilation theorem can be employed to transform the non-unitary propagator into a unitary one in a higher-dimensional Hilbert space, which can then be implemented on quantum circuits of NISQ computers. We validate our quantum algorithm as applied to the spin-boson benchmark model by analyzing the impact of the quantum circuit depth on the accuracy of the results when the subset is limited to the diagonal elements of the reduced density matrix. Our findings demonstrate that our approach yields reliable results on NISQ IBM computers.

Tensor-Train Thermo-Field Memory Kernels for Generalized Quantum Master Equations.

The generalized quantum master equation (GQME) approach provides a rigorous framework for deriving the exact equation of motion for any subset of electronic reduced density matrix elements (e.g., the diagonal elements). In the context of electronic dynamics, the memory kernel and inhomogeneous term of the GQME introduce the implicit coupling to nuclear motion and dynamics of electronic density matrix elements that are projected out (e.g., the off-diagonal elements), allowing for efficient quantum dynamics simulations. Here, we focus on benchmark quantum simulations of electronic dynamics in a spin-boson model system described by various types of GQMEs. Exact memory kernels and inhomogeneous terms are obtained from short-time quantum-mechanically exact tensor-train thermo-field dynamics (TT-TFD) simulations and are compared with those obtained from an approximate linearized semiclassical method, allowing for assessment of the accuracy of these approximate memory kernels and inhomogeneous terms. Moreover, we have analyzed the computational cost of the full and reduced-dimensionality GQMEs. The scaling of the computational cost is dependent on several factors, sometimes with opposite scaling trends. The TT-TFD memory kernels can provide insights on the main sources of inaccuracies of GQME approaches when combined with approximate input methods and pave the road for the development of quantum circuits that implement GQMEs on digital quantum computers.

Generalized Quantum Master Equations for Simulating Electronic Dynamics in Molecular Systems

Generalized Quantum Master Equations for Simulating Electronic Dynamics in Molecular Systems

Quasiclassical approaches to the generalized quantum master equation.

The formalism of the generalized quantum master equation (GQME) is an effective tool to simultaneously increase the accuracy and the efficiency of quasiclassical trajectory methods in the simulation of nonadiabatic quantum dynamics. The GQME expresses correlation functions in terms of a non-Markovian equation of motion, involving memory kernels that are typically fast-decaying and can therefore be computed by short-time quasiclassical trajectories. In this paper, we study the approximate solution of the GQME, obtained by calculating the kernels with two methods: Ehrenfest mean-field theory and spin-mapping. We test the approaches on a range of spin-boson models with increasing energy bias between the two electronic levels and place a particular focus on the long-time limits of the populations. We find that the accuracy of the predictions of the GQME depends strongly on the specific technique used to calculate the kernels. In particular, spin-mapping outperforms Ehrenfest for all the systems studied. The problem of unphysical negative electronic populations affecting spin-mapping is resolved by coupling the method with the master equation. Conversely, Ehrenfest in conjunction with the GQME can predict negative populations, despite the fact that the populations calculated from direct dynamics are positive definite.

Simulating the dynamics of electronic observables via reduced-dimensionality generalized quantum master equations.

We describe a general-purpose framework for formulating the dynamics of any subset of electronic reduced density matrix elements in terms of a formally exact generalized quantum master equation (GQME). Within this framework, the effect of coupling to the nuclear degrees of freedom, as well as to any projected-out electronic reduced density matrix elements, is captured by a memory kernel and an inhomogeneous term, whose dimensionalities are dictated by the number of electronic reduced density matrix elements included in the subset of interest. We show that the memory kernel and inhomogeneous term within such GQMEs can be calculated from projection-free inputs of the same dimensionality, which can be cast in terms of the corresponding subsets of overall system two-time correlation functions. The applicability and feasibility of such reduced-dimensionality GQMEs is demonstrated on the two-state spin-boson benchmark model. To this end, we compare and contrast the following four types of GQMEs: (1) a full density matrix GQME, (2) a single-population scalar GQME, (3) a populations-only GQME, and (4) a subset GQME for any combination of populations and coherences. Using a method based on the mapping Hamiltonian approach and linearized semiclassical approximation to calculate the projection-free inputs, we find that while single-population GQMEs and subset GQMEs containing only one population are less accurate, they can still produce reasonable results and that the accuracy of the results obtained via the populations-only GQME and a subset GQME containing both populations is comparable to that obtained via the full density matrix GQMEs.

On simulating the dynamics of electronic populations and coherences via quantum master equations based on treating off-diagonal electronic coupling terms as a small perturbation.

Quantum master equations provide a general framework for describing the dynamics of electronic observables within a complex molecular system. One particular family of such equations is based on treating the off-diagonal coupling terms between electronic states as a small perturbation within the framework of second-order perturbation theory. In this paper, we show how different choices of projection operators, as well as whether one starts out with the time-convolution or the time-convolutionless forms of the generalized quantum master equation, give rise to four different types of such off-diagonal quantum master equations (OD-QMEs), namely, time-convolution and time-convolutionless versions of a Pauli-type OD-QME for only the electronic populations and an OD-QME for the full electronic density matrix (including both electronic populations and coherences). The fact that those OD-QMEs are given in terms of the interaction picture makes it non-trivial to obtain Schrödinger picture electronic coherences from them. To address this, we also extend a procedure for extracting Schrödinger picture electronic coherences from interaction picture populations recently introduced by Trushechkin in the context of time-convolutionless Pauli-type OD-QME to the other three types of OD-QMEs. The performance of the aforementioned four types of OD-QMEs is explored in the context of the Garg-Onuchic-Ambegaokar benchmark model for charge transfer in the condensed phase across a relatively wide parameter range. The results show that time-convolution OD-QMEs can be significantly more accurate than their time-convolutionless counterparts, particularly in the case of Pauli-type OD-QMEs, and that rather accurate Schrödinger picture coherences can be obtained from interaction picture electronic inputs.

Nonuniqueness of generalized quantum master equations for a single observable

When deriving exact generalized master equations for the evolution of a reduced set of degrees of freedom, one is free to choose what quantities are relevant by specifying projection operators. However, obtaining a reduced description does not always need to be achieved through projections-one can also use conservation laws for this purpose. Such an operation should be considered as distinct from any kind of projection; that is, projection onto a single observable yields a different form of master equation compared to that resulting from a projection followed by the application of a constraint. We give a simple example to show this pointand give relationships that the different memory kernels must satisfy to yield the same dynamics.

Generalized quantum master equation: A tutorial review and recent advances

The generalized quantum master equation (GQME) provides a general and exact approach for simulating the reduced dynamics in open quantum systems where a quantum system is embedded in a quantum environment. Dynamics of open quantum systems is important in excitation energy, charge, and quantum coherence transfer as well as reactive photochemistry. The system is usually chosen to be the interested degrees of freedom such as the electronic states in light-harvesting molecules or tagged vibrational modes in a condensed-phase system. The environment is also called the bath, whose influence on the system has to be considered, and for instance can be described by the GQME formalisms using the projection operator technique. In this review, we provide a heuristic description of the development of two canonical forms of GQME, namely the time-convoluted Nakajima-Zwanzig form (NZ-GQME) and the time-convolutionless form (TCL-GQME). In the more popular NZ-GQME form, the memory kernel serves as the essential part that reflects the non-Markovian and non-perturbative effects, which gives formally exact dynamics of the reduced density matrix. We summarize several schemes to express the projection-based memory kernel of NZ-GQME in terms of projection-free time correlation function inputs that contain molecular information. In particular, the recently proposed modified GQME approach based on NZ-GQME partitions the Hamiltonian into a more general diagonal and off-diagonal parts. The projection-free inputs in the above-mentioned schemes expressed in terms of different system-dependent time correlation functions can be calculated via numerically exact or approximate dynamical methods. We hope this contribution would help lower the barrier of understanding the theoretical pillars for GQME-based quantum dynamics methods and also envisage that their combination with the quantum computing techniques will pave the way for solving complex problems related to quantum dynamics and quantum information that are currently intractable even with today’s state-of-the-art classical supercomputers.

A Road Map to Various Pathways for Calculating the Memory Kernel of the Generalized Quantum Master Equation.

The generalized quantum master equation (GQME) provides a powerful framework for simulating electronic energy, charge, and coherence transfer dynamics in molecular systems. Within this framework, the effect of the nuclear degrees of freedom on the time evolution of the electronic reduced density matrix is fully captured by a memory kernel superoperator. However, the actual memory kernel depends on the choice of projection operator and is therefore not unique. Furthermore, calculating the memory kernel can be done in multiple ways that use different forms of projection-free inputs. Although the electronic dynamics is invariant to those choices when quantum-mechanically exact projection-free inputs are used, this is not the case when they are obtained via more feasible semiclassical or mixed quantum-classical approximate methods. Furthermore, the accuracy and numerical stability of the resulting electronic dynamics has been observed to be sensitive to the above-mentioned choices when approximate methods are used to calculate the projection-free inputs. In this article, we provide a systematic road map to 30 possible pathways for calculating the memory kernel and highlight how they are related as well as the ways in which they differ. We also compare the performance of different pathways in the context of the spin-boson benchmark model, with the projection-free inputs obtained via a mapping Hamiltonian linearized semiclassical method. In this case, we find that expressing the memory kernel with an exponential operator where the projection operator precedes the Liouvillian yields the most accurate and most numerically stable results.

Simulating energy transfer dynamics in the Fenna-Matthews-Olson complex via the modified generalized quantum master equation.

The generalized quantum master equation (GQME) provides a general and formally exact framework for simulating the reduced dynamics of open quantum systems. The recently introduced modified approach to the GQME (M-GQME) corresponds to a specific implementation of the GQME that is geared toward simulating the dynamics of the electronic reduced density matrix in systems governed by an excitonic Hamiltonian. Such a Hamiltonian, which is often used for describing energy and charge transfer dynamics in complex molecular systems, is given in terms of diabatic electronic states that are coupled to each other and correspond to different nuclear Hamiltonians. Within the M-GQME approach, the effect of the nuclear degrees of freedom on the time evolution of the electronic density matrix is fully captured by a memory kernel superoperator, which can be obtained from short-lived (compared to the time scale of energy/charge transfer) projection-free inputs. In this paper, we test the ability of the M-GQME to predict the energy transfer dynamics within a seven-state benchmark model of the Fenna-Matthews-Olson (FMO) complex, with the short-lived projection-free inputs obtained via the Ehrenfest method. The M-GQME with Ehrenfest-based inputs is shown to yield accurate results across a wide parameter range. It is also found to dramatically outperform the direct application of the Ehrenfest method and to provide better-behaved convergence with respect to memory time in comparison to an alternative implementation of the GQME approach previously applied to the same FMO model.

Markovian embedding procedures for non-Markovian stochastic Schrödinger equations

We present embedding procedures for the non-Markovian stochastic Schrödinger equations, arising from studies of quantum systems coupled with bath environments. By introducing auxiliary wave functions, it is demonstrated that the non-Markovian dynamics can be embedded in extended, but Markovian, stochastic models. Two embedding procedures are presented. The first method leads to nonlinear stochastic equations, the implementation of which is much more efficient than the non-Markovian stochastic Schrödinger equations. The stochastic Schrödinger equations obtained from the second procedure involve more auxiliary wave functions, but the equations are linear, and a closed-form generalized quantum master equation for the density-matrix can be obtained. The accuracy of the embedded models is ensured by fitting to the power spectrum. The stochastic force is represented using a linear superposition of Ornstein-Uhlenbeck processes, which are incorporated as multiplicative noise in the auxiliary Schrödinger equations. It is shown that the asymptotic behavior of the spectral density in the low frequency regime, which is responsible for the long-time behavior of the quantum dynamics, can be preserved by using correlated stochastic processes. The approximations are verified by using a spin-boson system as a test example.

Protecting quantum coherences from static noise and disorder

Quantum coherences are paramount resources for applications, such as quantum-enhanced light-harvesting or quantum computing, which are fragile against environmental noise. We here derive generalized quantum master equations using perturbation theory in order to describe the effective ensemble-averaged time-evolution of finite-size quantum systems subject to static noise on all time scales. We then analyse the time-evolution of the coherences under energy broadening noise in a variety of systems characterized by both short and long-range interactions, by strongly correlated and fully uncorrelated noise -- a single qubit, a lattice model with on-site disorder and a potential ladder, and bosons in a double-well potential with random interaction strength -- and show that couplings can partially protect the system from the ensemble-averaging induced loss of coherence. Our work suggests that suitably tuned couplings could be employed to counteract part of the dephasing detrimental to quantum applications. Conversely, tailored noise distributions can be utilized to reach target non-equilibrium quantum states.

Comparative study on model parameter evaluations for the energy transfer dynamics in Fenna–Matthews–Olson complex

Fenna–Matthews–Olson (FMO) light-harvesting protein that connects the outer antenna system with the reaction center in green sulfur bacteria provides a paradigmatic model system in terms of the observed quantum coherence possibly associated with highly efficient excitation energy transfer (EET) in the photosynthetic system. There have been many computational studies on the EET dynamics of FMO complex, in which two sets of model parameters, site energies and excitonic coupling energies for eight bacteriochlorophyll (BChl) a pigments, play pivotal roles. In this study, we examine the optimality of these energy parameters evaluated by various research groups on the basis of generalized quantum master equations describing the EET dynamics among eight BChls. We thus find that most energy parameters are nearly optimal for high efficiency in the cases that BChl 1 and/or BChl 6 are initially excited, with some exceptions due to putative inaccuracies in the parameter evaluations.

Stochastic thermodynamics of an electron spin resonance quantum dot system

We present a stochastic thermodynamics analysis of an electron-spin-resonance pumped quantum dot device in the Coulomb-blocked regime, where a pure spin current is generated without an accompanying net charge current. Based on a generalized quantum master equation beyond secular approximation, quantum coherences are accounted for in terms of an effective average spin in the Floquet basis. Elegantly, this effective spin undergoes a precession about an effective magnetic field, which originates from the non-secular treatment and energy renormalization. It is shown that the interaction between effective spin and effective magnetic field may have the dominant roles to play in both energy transport and irreversible entropy production. In the stationary limit, the energy and entropy balance relations are also established based on the theory of counting statistics.

Simulating Electronically Nonadiabatic Dynamics via the Generalized Quantum Master Equation

Simulating Electronically Nonadiabatic Dynamics via the Generalized Quantum Master Equation

Real-Time Path Integral Methods, Quantum Master Equations, and Classical vs Quantum Memory.

We investigate the use of accurate path integral methods, namely the quasi-adiabatic propagator path integral (QuAPI) and the quantum-classical path integral (QCPI), for generating the memory kernel entering generalized quantum master equations (GQME). Our calculations indicate that the length of the memory kernel in system-bath models is equal to the full length of time nonlocality encoded in the Feynman-Vernon influence functional and that the solution of the GQME with a QuAPI kernel is identical to that obtained through an iterative QuAPI calculation with the same memory length. Further, we show that the memory length in iterative QCPI calculations is always shorter than the GQME kernel memory length. This stems from the ability of the QCPI methodology to pretreat all memory effects of a classical nature (i.e., those associated with phonon absorption and stimulated emission), as well as some of the quantum memory contributions (arising from spontaneous phonon emission). Furthermore, trajectory-based iterative QCPI simulations can fully account for important structural/conformational changes that may occur on very long time scales and that cannot be captured via master equation treatments.

Combining the mapping Hamiltonian linearized semiclassical approach with the generalized quantum master equation to simulate electronically nonadiabatic molecular dynamics.

The generalized quantum master equation (GQME) provides a powerful framework for simulating electronically nonadiabatic molecular dynamics. Within this framework, the effect of the nuclear degrees of freedom on the time evolution of the electronic reduced density matrix is fully captured by a memory kernel superoperator. In this paper, we consider two different procedures for calculating the memory kernel of the GQME from projection-free inputs obtained via the combination of the mapping Hamiltonian (MH) approach and the linearized semiclassical (LSC) approximation. The accuracy and feasibility of the two procedures are demonstrated on the spin-boson model. We find that although simulating the electronic dynamics by direct application of the two LSC-based procedures leads to qualitatively different results that become increasingly less accurate with increasing time, restricting their use to calculating the memory kernel leads to an accurate description of the electronic dynamics. Comparison with a previously proposed procedure for calculating the memory kernel via the Ehrenfest method reveals that MH/LSC methods produce memory kernels that are better behaved at long times and lead to more accurate electronic dynamics.

Efficient construction of generalized master equation memory kernels for multi-state systems from nonadiabatic quantum-classical dynamics

Methods derived from the generalized quantum master equation (GQME) framework have provided the basis for elucidating energy and charge transfer in systems ranging from molecular solids to photosynthetic complexes. Recently, the nonperturbative combination of the GQME with quantum-classical methods has resulted in approaches whose accuracy and efficiency exceed those of the original quantum-classical schemes while offering significant accuracy improvements over perturbative expansions of the GQME. Here, we show that, while the non-Markovian memory kernel required to propagate the GQME scales quartically with the number of subsystem states, the number of trajectories required scales at most quadratically when using quantum-classical methods to construct the kernel. We then present an algorithm that allows further acceleration of the quantum-classical GQME by providing a way to selectively sample the kernel matrix elements that are most important to the process of interest. We demonstrate the utility of these advances by applying the combination of Ehrenfest mean field theory with the GQME (MF-GQME) to models of the Fenna-Matthews-Olson (FMO) complex and the light harvesting complex II (LHCII), with 7 and 14 states, respectively. This allows us to show that the MF-GQME is able to accurately capture all the relevant dynamical time scales in LHCII: the initial nonequilibrium population transfer on the femtosecond time scale, the steady state-type trapping on the picosecond time scale, and the long time population relaxation. Remarkably, all of these physical effects spanning tens of picoseconds can be encoded in a memory kernel that decays only after ∼65 fs.