Hierarchical Equations Of Motion Research Articles

High accuracy exponential decomposition of bath correlation functions for arbitrary and structured spectral densities: Emerging methodologies and new approaches.

This study investigates the decomposition of bath correlation functions (BCFs) in terms of complex exponential functions, with an eye on the realistic modeling of open quantum systems based on the hierarchical equations of motion. We introduce the theoretical background of various decomposition schemes in both time and frequency domains and assess their efficiency and accuracy by demonstrating the decomposition of various BCFs. We further develop a new procedure for the decomposition of BCFs originating from highly structured spectral densities with a high accuracy and compare it with existing fitting techniques. Advantages and disadvantages of each methodology are discussed in detail with special attention to their application to the corresponding quantum dynamical problem. This work provides fundamental tools for choosing and using a variety of decomposition techniques of BCFs for the study of open quantum systems in structured environments.

Effects of Soliton Creation on Transient Transport through a Polymer Chain.

By using a nonadiabatic molecular dynamics method combined with the hierarchical equations of motion, we have investigated the nonequilibrium transient transport through a conjugated polymer chain. The polymer chain is described by the Su-Schrieffer-Heeger model, and its two ends are coupled with metal electrodes of different chemical potentials. In order to present the evolutions of the electronic injection and transport in the real-time domain, we have mainly discussed the dynamic relaxation processes of the excited states and transient transport currents. It is found that due to the existence of electron-phonon couplings in the conjugated polymers, creation of solitons not only affects the time of the system achieving the steady state but also leads to periodical oscillations of the steady-state transport currents with time in our simulations. Furthermore, with increasing applied bias voltage, the steady-state transport electronic current increases, which proved that the creation of the solitons can assist the electronic transport. These results have shown that the creation of the excited states is important in understanding the transport properties in organic nanostructures.

Simulating the operation of a quantum computer in a dissipative environment.

The operations of current quantum computers are still significantly affected by decoherence caused by interaction with the environment. In this work, we employ the non-perturbative hierarchical equations of motion (HEOM) method to simulate the operation of model quantum computers and reveal the effects of dissipation on the entangled quantum states and on the performance of well-known quantum algorithms. Multi-qubit entangled states in Shor's factorizing algorithm are first generated and propagated using the HEOM. It is found that the failure of factorization is accompanied by a loss of fidelity and mutual information. An important challenge in using the HEOM to simulate quantum computers in a dissipative environment is how to efficiently treat systems with many qubits. We propose a two-dimensional tensor network scheme for this problem and demonstrate its capability by simulating a one-dimensional random circuit model with 21 qubits.

Investigating the dissipation of heat and quantum information from DNA-scaffolded chromophore networks.

Scaffolded molecular networks are important building blocks in biological pigment-protein complexes, and DNA nanotechnology allows analogous systems to be designed and synthesized. System-environment interactions in these systems are responsible for important processes, such as the dissipation of heat and quantum information. This study investigates the role of nanoscale molecular parameters in tuning these vibronic system-environment dynamics. Here, genetic algorithm methods are used to obtain nanoscale parameters for a DNA-scaffolded chromophore network based on comparisons between its calculated and measured optical spectra. These parameters include the positions, orientations, and energy level characteristics within the network. This information is then used to compute the dynamics, including the vibronic population dynamics and system-environment heat currents, using the hierarchical equations of motion. The dissipation of quantum information is identified by the system's transient change in entropy, which is proportional to the heat currents according to the second law of thermodynamics. These results indicate that the dissipation of quantum information is highly dependent on the particular nanoscale characteristics of the molecular network, which is a necessary first step before gleaning the systematic optimization rules. Subsequently, the I-concurrence dynamics are calculated to understand the evolution of the vibronic system's quantum entanglement, which are found to be long-lived compared to these system-bath dissipation processes.

An efficient Julia framework for hierarchical equations of motion in open quantum systems

The hierarchical equations of motion (HEOM) approach can describe the reduced dynamics of a system simultaneously coupled to multiple bosonic and fermionic environments. The complexity of exactly describing the system-environment interaction with the HEOM method usually results in time-consuming calculations and a large memory cost. Here, we introduce an open-source software package called HierarchicalEOM.jl: a Julia framework integrating the HEOM approach. HierarchicalEOM.jl features a collection of methods to compute bosonic and fermionic spectra, stationary states, and the full dynamics in the extended space of all auxiliary density operators (ADOs). The required handling of the ADOs multi-indexes is achieved through a user-friendly interface. We exemplify the functionalities of the package by analyzing a single impurity Anderson model, and an ultra-strongly coupled charge-cavity system interacting with bosonic and fermionic reservoirs. HierarchicalEOM.jl achieves a significant speedup with respect to the corresponding method in the Quantum Toolbox in Python (QuTiP), upon which this package is founded.

Holstein polaron transport from numerically "exact" real-time quantum dynamics simulations.

Numerically "exact" methods addressing the dynamics of coupled electron-phonon systems have been intensively developed. Nevertheless, the corresponding results for the electron mobility μdc are scarce, even for the one-dimensional (1d) Holstein model. Building on our recent progress on single-particle properties, here we develop the momentum-space hierarchical equations of motion (HEOM) method to evaluate real-time two-particle correlation functions of the 1d Holstein model at a finite temperature. We compute numerically "exact" dynamics of the current-current correlation function up to real times sufficiently long to capture the electron's diffusive motion and provide reliable results for μdc in a wide range of model parameters. In contrast to the smooth ballistic-to-diffusive crossover in the weak-coupling regime, we observe a temporally limited slow-down of the electron on intermediate time scales already in the intermediate-coupling regime, which translates to a finite-frequency peak in the optical response. Our momentum-space formulation lowers the numerical effort with respect to existing HEOM-method implementations, while we remove the numerical instabilities inherent to the undamped-mode HEOM by devising an appropriate hierarchy closing scheme. Still, our HEOM remains unstable at too low temperatures, for too strong electron-phonon coupling, and for too fast phonons.

On stability issues of the HEOM method

AbstractThe Hierarchical Equations of Motion (HEOM) method has become one of the cornerstones in the simulation of open quantum systems and their dynamics. It is commonly referred to as a non-perturbative method. Yet, there are certain instances, where the necessary truncation of the hierarchy of auxiliary density operators seems to introduce errors which are not fully controllable. We investigate the nature and causes of this type of critical error both in the case of pure decoherence, where exact results are available for comparison, and in the spin-boson system, a full system-reservoir model. We find that truncating the hierarchy to any finite size can be problematic for strong coupling to a dissipative reservoir, in particular when combined with an appreciable reservoir memory time.

Tree tensor network state approach for solving hierarchical equations of motion.

The hierarchical equations of motion (HEOM) method is a numerically exact open quantum system dynamics approach. The method is rooted in an exponential expansion of the bath correlation function, which in essence strategically reshapes a continuous environment into a set of effective bath modes that allow for more efficient cutoff at finite temperatures. Based on this understanding, one can map the HEOM method into a Schrödinger-like equation, with a non-Hermitian super-Hamiltonian for an extended wave function being the tensor product of the central system wave function and the Fock state of these effective bath modes. In this work, we explore the possibility of representing the extended wave function as a tree tensor network state (TTNS) and the super-Hamiltonian as a tree tensor network operator of the same structure as the TTNS, as well as the application of a time propagation algorithm using the time-dependent variational principle. Our benchmark calculations based on the spin-boson model with a slow-relaxing bath show that the proposed HEOM+TTNS approach yields consistent results with those of the conventional HEOM method, while the computation is considerably sped up. In addition, the simulation with a genuine TTNS is four times faster than a one-dimensional matrix product state decomposition scheme.

Efficient low-temperature simulations for fermionic reservoirs with the hierarchical equations of motion method: Application to the Anderson impurity model

The hierarchical equations of motion (HEOM) approach is an accurate method to simulate open system quantum dynamics, which allows for systematic convergence to numerically exact results. To represent effects of the bath, the reservoir correlation functions are usually decomposed into summation of multiple exponential terms in the HEOM method. Since the reservoir correlation functions become highly non-Markovian at low temperatures or when the bath has complex band structures, a present challenge is to obtain accurate exponential decompositions that allow efficient simulation with the HEOM. In this work, we employ the barycentric representation to approximate the Fermi function and hybridization functions in the frequency domain. This method, by approximating these functions with optimized rational decomposition, greatly reduces the number of basis functions in decomposing the reservoir correlation functions, which further allows the HEOM method to be applied to ultralow temperature and general band structures. We demonstrate the efficiency, accuracy, and long-time stability of this decomposition scheme by applying it to the Anderson impurity model in the low-temperature regime with the Lorentzian and tight-binding hybridization functions.

Chiral current regulation and detection of Berry phase in triangular triple quantum dots

Based on the hierarchical equations of motion (HEOM) calculation, we theoretically investigate the corresponding control of a triangular triple-quantum-dots (TTQD) ring which is connected to two reservoirs. We initially demonstrate by adding bias voltage and further adjusting the coupling strength between quantum dots, the chiral current induced by bias will go through a transformation of clockwise to counterclockwise direction and an unprecedented effective Hall angle will be triggered. The transformation is very rapid, with a corresponding characteristic time of 80–200 ps. In addition, by adding a magnetic flux to compensate for the chiral current in the original system, we elucidate the relationship between the applied magnetic flux and the Berry phase, which can realize direct measurement of the chiral current and reveal the magnetoelectric coupling relationship.

Hierarchical equations of motion analog for systems with delay: Application to intercavity photon propagation

Over the last two decades, the hierarchical equations of motion (HEOM) of Tanimura and Kubo have become the equation of motion-based tool for numerically exact calculations of system-bath problems. The HEOM is today generalized to many cases of dissipation and transfer processes through an external bath. In spatially extended photonic systems, the propagation of photons through the bath leads to retardation/delays in the coupling of quantum emitters. Here, the idea behind the HEOM derivation is generalized to the case of photon retardation and applied to the simple example of two dielectric slabs. The derived equations provide a simple reliable framework for describing retardation and may provide an alternative to path-integral treatments.

QuTiP-BoFiN: A bosonic and fermionic numerical hierarchical-equations-of-motion library with applications in light-harvesting, quantum control, and single-molecule electronics

The "hierarchical equations of motion" (HEOM) method is a powerful exact numerical approach to solve the dynamics and find the steady-state of a quantum system coupled to a non-Markovian and non-perturbative environment. Originally developed in the context of physical chemistry, it has also been extended and applied to problems in solid-state physics, optics, single-molecule electronics, and biological physics. Here we present a numerical library in Python, integrated with the powerful QuTiP platform, which implements the HEOM for both bosonic and fermionic environments. We demonstrate its utility with a series of examples. For the bosonic case, we include demonstrations of fitting arbitrary spectral densities, and an example of the dynamics of energy transfer in the Fenna-Matthews-Olson photosynthetic complex, showing how a suitable non-Markovian environment can protect against pure dephasing. We also demonstrate how the HEOM can be used to benchmark different strategies for dynamical decoupling of a spin from its environment, and show that the Uhrig pulse-spacing scheme is less optimal than equally spaced pulses when the environment's spectral density is very broad. For the fermionic case, we present an integrable single-impurity example, used as a benchmark of the code, and a more complex example of an impurity strongly coupled to a single vibronic mode, with applications to single-molecule electronics.

Hierarchical equations of motion approach for accurate characterization of spin excitations in quantum impurity systems.

Recent technological advancement in scanning tunneling microscopes has enabled the measurement of spin-field and spin-spin interactions in single atomic or molecular junctions with an unprecedentedly high resolution. Theoretically, although the fermionic hierarchical equations of motion (HEOM) method has been widely applied to investigate the strongly correlated Kondo states in these junctions, the existence of low-energy spin excitations presents new challenges to numerical simulations. These include the quest for a more accurate and efficient decomposition for the non-Markovian memory of low-temperature environments and a more careful handling of errors caused by the truncation of the hierarchy. In this work, we propose several new algorithms, which significantly enhance the performance of the HEOM method, as exemplified by the calculations on systems involving various types of low-energy spin excitations. Being able to characterize both the Kondo effect and spin excitation accurately, the HEOM method offers a sophisticated and versatile theoretical tool, which is valuable for the understanding and even prediction of the fascinating quantum phenomena explored in cutting-edge experiments.

Recent advances in fermionic hierarchical equations of motion method for strongly correlated quantum impurity systems

Investigations of strongly correlated quantum impurity systems (QIS), which exhibit diversified novel and intriguing quantum phenomena, have become a highly concerning subject in recent years. The hierarchical equations of motion (HEOM) method is one of the most popular numerical methods to characterize QIS linearly coupled to the environment. This review provides a comprehensive account of a formally rigorous and numerical convergent HEOM method, including a modeling description of the QIS and an overview of the fermionic HEOM formalism. Moreover, a variety of spectrum decomposition schemes and hierarchal terminators have been proposed and developed, which significantly improve the accuracy and efficiency of the HEOM method, especially in cryogenic temperature regimes. The practicality and usefulness of the HEOM method to tackle strongly correlated issues are exemplified by numerical simulations for the characterization of nonequilibrium quantum transport and strongly correlated Kondo states as well as the investigation of nonequilibrium quantum thermodynamics.

Tweezer-like magnetic tip control of the local spin state in the FeOEP/Pb(111) adsorption system: a preliminary exploration based on first-principles calculations.

The magnetic interactions between the spin-polarized scanning tunnelling microscopy (SP-STM) tip and the localized spin impurities lead to various forms of the Kondo effect. Although these intriguing phenomena enrich Kondo physics, detailed theoretical simulations and explanations are still lacking due to the rather complex formation mechanisms. Here, by combining density functional theory (DFT), complete active space self-consistent field (CASSCF) theory, and hierarchical equations of motion (HEOM) methods, we perform first-principles-based simulation to elaborate the regulation process of the magnetic Co-tip on the spin state and transport behaviour of FeOEP/Pb(111) system. Compared with the non-magnetic tip, the stronger interaction between the magnetic tip and FeOEP molecule results in a more drastic deformation of the molecular structure with more electron transfer from the local environment to Fe-3d orbitals. The magnetic anisotropy of FeOEP changes very drastically from positive values in the tunnelling region to negative values in the contact region. The ferromagnetic electron correlation between the magnetic tip and the molecule induces an asymmetric Kondo line-shape near the Fermi level. This work highlights that the DFT + CASSCF + HEOM approach can not only predict complex quantum phenomena and explain underlying physical mechanisms, but also facilitate the design of more fascinating quantum control experiments.

Excitation energy equilibration in a trimeric LHCII complex involves unusual pathways.

We explore the energy equilibration within the LHCII trimer using various approaches, including the Redfield-Förster method (with different compartmentalization schemes) and the exact hierarchical equation of motion (HEOM). We demonstrate that the inter-monomeric migration in the trimeric LHCII complex is not limited to direct transfers between quasi-equilibrated chlorophylls (Chls) a, but also involves additional pathways with uphill transfers from Chls a to the stromal-side Chls b (connecting the Chls a clusters from different monomeric subunits). Although these uphill transfers are slow they still can increase the total rate of inter-monomeric transfers by a factor of 1.5. The same stromal-side Chls b also promote a depopulation of the Chl a604 long-lived state (blue-shifted and mixed with the lumenal-side Chls b). Due to the connection between the stromal- and lumenal-side Chls b clusters the intra- and inter-monomeric transfers from a604 to the main Chls a become faster by a factor of 1.6 and 1.75, respectively.

On the practical truncation tier of fermionic hierarchical equations of motion.

The fermionic hierarchical equations of motion (HEOM) approach has found wide application in the exploration of open quantum systems, and extensive efforts have been committed to improving its efficiency and accuracy in practical calculations. In this work, by scrutinizing the stationary-state and dynamic properties of Kondo-correlated quantum impurity systems, we show that the strength of Kondo correlation induced by the system-environment entanglement primarily determines the converged hierarchical truncation tier of the HEOM method. This complements the rule of thumb regarding the positive correlation between the height of hierarchy and system-environment coupling strength. These insights will provide useful guidelines for developing a more sophisticated fermionic HEOM method for the investigation of many-body open quantum systems.

Taming Quantum Noise for Efficient Low Temperature Simulations of Open Quantum Systems.

The hierarchical equations of motion (HEOM), derived from the exact Feynman-Vernon path integral, is one of the most powerful numerical methods to simulate the dynamics of open quantum systems. Its applicability has so far been limited to specific forms of spectral reservoir distributions and relatively elevated temperatures. Here we solve this problem and introduce an effective treatment of quantum noise in frequency space by systematically clustering higher order Matsubara poles, equivalent to an optimized rational decomposition. This leads to an elegant extension of the HEOM to arbitrary temperatures and very general reservoirs in combination with efficiency, high accuracy, and long-time stability. Moreover, the technique can directly be implemented in other approaches such as Green's function, stochastic, and pseudomode formulations. As one highly nontrivial application, for the subohmic spin-boson model at vanishing temperature the Shiba relation is quantitatively verified which predicts the long-time decay of correlation functions.

Laser-controlled electronic symmetry breaking in a phenylene ethynylene dimer: Simulation by the hierarchical equations of motion and optimal control

The ability to prepare a specific superposition of electronic excited states leading to a transitory symmetry breaking of the electronic density in complex systems remains a challenging concern. We investigate how an initial coherence can be controlled by laser fields. The selected molecular system is a symmetric dimer of phenylene ethynylene presenting different interesting properties: Two bright nearly degenerate excited states coupled through a conical intersection are addressable by orthogonal transition dipole moments and are well separated from neighboring states. Creating a superposed state with equal weights corresponds to a right or left electronic localization as in the double-well system followed by a transitory oscillation between the two wells. To ensure a spectral bandwidth typically smaller than 0.25 eV, the pulse duration is in the tens of femtoseconds range, so nuclear motion cannot be neglected. Optimal control theory (OCT) is applied with guess fields that effectively create the target coherence in the absence of dephasing due to the vibrational baths. We analyze the field reshaping proposed by the control and we further fit a sequence of pulses on the optimal field. The overall result is efficient and robust disymmetry control over reasonable timescales of few tens of femtoseconds, exceeding the pulse duration. The monotonically convergent algorithm is combined with the hierarchical equations of motion (HEOM) able to treat strongly coupled non-Markovian dynamics. We also check the implementation of the combined OCT-HEOM approach in the tensor-train representation with propagation using the time-dependent variational method.

Machine learning for excitation energy transfer dynamics

A well-known approach to describe the dynamics of an open quantum system is to compute the master equation evolving the reduced density matrix of the system. This approach plays an important role in describing excitation transfer through photosynthetic light harvesting complexes (LHCs). The hierarchical equations of motion (HEOM) was adapted by Ishizaki and Fleming (J. Chem. Phys., 2009) to simulate open quantum dynamics in the biological regime. We generate a set of time dependent observables that depict the coherent propagation of electronic excitations through the LHCs by solving the HEOM. We solve the inverse problem using classical machine learning (ML) models as this is a computationally intractable problem. The objective here is to determine whether a trained ML model can perform Hamiltonian tomography by using the time dependence of the observables as inputs. We demonstrate the capability of convolutional neural networks to tackle this research problem. The models developed here can predict Hamiltonian parameters such as excited state energies and inter-site couplings of a system up to 99.28\% accuracy and mean-squared error as low as 0.65.