Thermodynamic free-energy spectrum theory for open quantum systems.

In this work, we develop the free-energy spectrum theory for thermodynamics of open quantum impurity systems that can be either fermionic or bosonic or combined. We identify two types of thermodynamic free-energy spectral functions for open quantum systems and further consider the thermodynamic limit, which supports the Gaussian-Wick description of hybrid environments. We can then relate the thermodynamic spectral functions to the local impurity properties. These could be experimentally measurable quantities, especially for the cases of quantum dots embedded in solid surfaces. Another type of input is the bare-bath coupling spectral densities, which could be accurately determined with various methods. For illustration, we consider the simplest noninteracting systems, with focus on the strikingly different characteristics between the bosonic and fermionic scenarios.

Equilibrium and transient thermodynamics: A unified dissipaton-space approach.

This work presents a unified dissipaton-equation-of-motion (DEOM) theory and its evaluations on the Helmholtz free energy change due to the isotherm mixing of two isolated subsystems. One is a local impurity, and the other is a nonlocal Gaussian bath. DEOM constitutes a fundamental theory for such open quantum mixtures. To complete the theory, we also construct the imaginary-time DEOM formalism via an analytical continuation of dissipaton algebra, which would be limited to equilibrium thermodynamics. On the other hand, the real-time DEOM deals with both equilibrium structural and nonequilibrium dynamic properties. Its combination with the thermodynamic integral formalism would be a viable and accurate means to both equilibrium and transient thermodynamics. As illustrations, we report the numerical results on a spin-boson system, with elaborations on the underlying anharmonic features, the thermodynamic entropy vs the von Neumann entropy, and an indication of "solvent-cage" formation. Beside the required asymptotic equilibrium properties, the proposed transient thermodynamics also supports the basic spontaneity criterion.

Hierarchical equations of motion method based on Fano spectrum decomposition for low temperature environments.

The hierarchical equations of motion (HEOM) method has become one of the most popular methods for the studies of the open quantum system. However, its applicability to systems at ultra-low temperatures is largely restrained by the enormous computational cost, which is caused by the numerous exponential functions required to accurately characterize the non-Markovian memory of the reservoir environment. To overcome this problem, a Fano spectrum decomposition (FSD) scheme has been proposed recently [Cui et al., J. Chem. Phys. 151, 024110 (2019)], which expands the reservoir correlation functions using polynomial-exponential functions and hence greatly reduces the size of the memory basis set. In this work, we explicitly establish the FSD-based HEOM formalisms for both bosonic and fermionic environments. The accuracy and efficiency of the FSD-based HEOM are exemplified by the calculated low-temperature dissipative dynamics of a spin-boson model and the dynamic and static properties of a single-orbital Anderson impurity model in the Kondo regime. The encouraging numerical results highlight the practicality and usefulness of the FSD-based HEOM method for general open systems at ultra-low temperatures.

System-bath entanglement theorem with Gaussian environments.

In this work, we establish a so-called "system-bath entanglement theorem," for arbitrary systems coupled with Gaussian environments. This theorem connects the entangled system-bath response functions in the total composite space to those of local systems, as long as the interacting bath spectral densities are given. We validate the theorem with direct evaluation via the exact dissipaton-equation-of-motion approach. Therefore, this work enables various quantum dissipation theories, which originally describe only the reduced system dynamics, for their evaluations on the system-bath entanglement properties. Numerical demonstrations are carried out on the Fano interference spectroscopies of spin-boson systems.

Highly efficient and accurate sum-over-poles expansion of Fermi and Bose functions at near zero temperatures: Fano spectrum decomposition scheme.

The Fano spectrum decomposition (FSD) scheme is proposed as an efficient and accurate sum-over-poles expansion of Fermi and Bose functions at cryogenic temperatures. The new method practically overcomes the discontinuity of Fermi and Bose functions near zero temperature, which causes slow convergence in conventional schemes such as the state-of-the-art Padé spectrum decomposition (PSD). The FSD scheme fragments Fermi or Bose function into a high-temperature reference and a low-temperature correction. While the former is efficiently decomposed via the standard PSD, the latter can be accurately described by several modified Fano functions. The resulting FSD scheme is found to converge overwhelmingly faster than the standard PSD method. Remarkably, the low-temperature correction supports further a recursive and scalable extension to access the near-zero temperature regime. Thus, the proposed FSD scheme, which obeys rather simple recursive relations, has a great value in efficient numerical evaluations of Fermi or Bose function-involved integrals for various low-temperature condensed physics formulations and problems. For numerical demonstrations, we exemplify FSD for the efficient unraveling of fermionic reservoir correlation functions and the exact hierarchical equations of motion simulations of spin-boson dynamics, both at extremely low temperatures.

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

The hierarchical equations of motion (HEOM) theory is in principle exact for describing the dissipative dynamics of quantum systems linearly coupled to Gaussian environments. In practice, the hierarchy needs to be truncated at a finite tier. We demonstrate that, for general systems described by the fermionic HEOM, the (n+L̃)th-tier truncation with L̃=2NσNν yields the exact density operators up to the nth tier. Here, Nσ = 2 for fermionic systems and Nν is the system degrees of freedom. For noninteracting systems, L̃ is further reduced by half. Such an exact termination pattern originates from the Pauli exclusion principle for fermions, and it holds true regardless of the system-environment coupling strength, the number of coupling reservoirs, or the specific scheme employed to unravel the environment memory contents. The relatively small L̃ emphasizes the nonperturbative nature of the HEOM theory. We also propose a simplified HEOM approach to further reduce the memory cost for practical calculations.

Theoretical insights into the reactivity of Fe-based catalysts for water oxidation: the role of electron-withdrawing groups.

Recent experiments have shown that complex (1), [Fe(OTf)2(Pytacn)] (OTf = CF3SO3-, Pytacn = 1-(2'-pyridylmethyl)-4,7-dimethyl-1,4,7-triazacyclononane), is a promising artificial photosynthetic catalyst because of its distinct capability in water oxidation. Experimentalists have also synthesized several derivatives, e.g., [Fe(OTf)2(E,HPytacn)] (E = -Cl (2), -CO2Et (3) and -NO2 (4)) and [Fe(OTf)2 (E,RPytacn)] (R = -F (5) and R = -Me (6)), and proposed that the E-substituted electron-withdrawing groups could improve the catalytic efficiency. However, the mechanism remains somewhat unclear, especially on the relative catalytic efficiency of these complexes. In this work, we propose an oxygen radical mechanism based on density functional theory (DFT) calculations for the six complexes. The crucial O-O bond-formation step is elucidated. Our calculations reveal that the FeIV-oxyl radical is the active species during the reaction, and the catalytic activities follow the sequence of (4) > (3) > (2) > (1) > (5) > (6), which agrees consistently with the experimental findings. Furthermore, we propose a simple charge-pair interaction model to characterize the effect of electron-withdrawing groups on the catalytic efficiency. It is clearly demonstrated that an electron-withdrawing group with a higher electronegativity is associated with a lower Gibbs free energy barrier for the O-O bond formation, which then leads to a more active catalyst. We also emphasize that the accurate description of dispersive interactions in DFT calculations is crucially important to retrieve the correct sequence of the catalytic efficiency. The theoretical insights provided in this work could be useful for the design of highly efficient Fe-based water oxidation catalysts.

Theories of quantum dissipation and nonlinear coupling bath descriptors.

The quest of an exact and nonperturbative treatment of quantum dissipation in nonlinear coupling environments remains in general an intractable task. In this work, we address the key issues toward the solutions to the lowest nonlinear environment, a harmonic bath coupled both linearly and quadratically with an arbitrary system. To determine the bath coupling descriptors, we propose a physical mapping scheme, together with the prescription reference invariance requirement. We then adopt a recently developed dissipaton equation of motion theory [R. X. Xu et al., Chin. J. Chem. Phys. 30, 395 (2017)], with the underlying statistical quasi-particle ("dissipaton") algebra being extended to the quadratic bath coupling. We report the numerical results on a two-level system dynamics and absorption and emission line shapes.

Efficient steady-state solver for hierarchical quantum master equations.

Steady states play pivotal roles in many equilibrium and non-equilibrium open system studies. Their accurate evaluations call for exact theories with rigorous treatment of system-bath interactions. Therein, the hierarchical equations-of-motion (HEOM) formalism is a nonperturbative and non-Markovian quantum dissipation theory, which can faithfully describe the dissipative dynamics and nonlinear response of open systems. Nevertheless, solving the steady states of open quantum systems via HEOM is often a challenging task, due to the vast number of dynamical quantities involved. In this work, we propose a self-consistent iteration approach that quickly solves the HEOM steady states. We demonstrate its high efficiency with accurate and fast evaluations of low-temperature thermal equilibrium of a model Fenna-Matthews-Olson pigment-protein complex. Numerically exact evaluation of thermal equilibrium Rényi entropies and stationary emission line shapes is presented with detailed discussion.

Low-frequency logarithmic discretization of the reservoir spectrum for improving the efficiency of hierarchical equations of motion approach.

An efficient low-frequency logarithmic discretization (LFLD) scheme for the decomposition of fermionic reservoir spectrum is proposed for the investigation of quantum impurity systems. The scheme combines the Padé spectrum decomposition (PSD) and a logarithmic discretization of the residual part in which the parameters are determined based on an extension of the recently developed minimum-dissipaton ansatz [J. J. Ding et al., J. Chem. Phys. 145, 204110 (2016)]. A hierarchical equations of motion (HEOM) approach is then employed to validate the proposed scheme by examining the static and dynamic system properties in both the Kondo and noninteracting regimes. The LFLD scheme requires a much smaller number of exponential functions than the conventional PSD scheme to reproduce the reservoir correlation function and thus facilitates the efficient implementation of the HEOM approach in extremely low temperature regimes.

Fokker-Planck quantum master equation for mixed quantum-semiclassical dynamics.

We revisit Caldeira-Leggett's quantum master equation representing mixed quantum-classical theory, but with limited applications. Proposed is a Fokker-Planck quantum master equation theory, with a generic bi-exponential correlation function description on semiclassical Brownian oscillators' environments. The new theory has caustic terms that bridge between the quantum description on primary systems and the semiclassical or quasi-classical description on environments. Various parametrization schemes, both analytical and numerical, for the generic bi-exponential environment bath correlation functions are proposed and scrutinized. The Fokker-Planck quantum master equation theory is of the same numerical cost as the original Caldeira-Leggett's approach but acquires a significantly broadened validity and accuracy range, as illustrated against the exact dynamics on model systems in quantum Brownian oscillators' environments, at moderately low temperatures.

Minimum-exponents ansatz for molecular dynamics and quantum dissipation.

A unified theory for minimum exponential-term ansatzes on bath correlation functions is proposed for numerically efficient and physically insightful treatments of non-Markovian environment influence on quantum systems. For a general Brownian oscillator bath of frequency Ω and friction ζ, the minimum ansatz results in the correlation function a bi-exponential form, with the effective Ω¯ and friction ζ¯ being temperature dependent and satisfying Ω¯/Ω=(ζ¯/ζ)1/2=r¯BO/rBO≤ 1, where r¯BO=ζ¯/(2Ω¯) and rBO=ζ/(2Ω). The maximum value of r¯BO=rBO can effectively be reached when kBT≥ 0.8Ω. The bi-exponential correlation function can further reduce to single-exponential form, in both the diffusion (rBO≫1) limit and the pre-diffusion region that could occur when rBO≥ 2. These are remarkable results that could be tested experimentally. Moreover, the impact of the present work on the efficient and accuracy controllable evaluation of non-Markovian quantum dissipation dynamics is also demonstrated.

Effects of Herzberg-Teller vibronic coupling on coherent excitation energy transfer.

In this work, we study the effects of non-Condon vibronic coupling on the quantum coherence of excitation energy transfer, via the exact dissipaton-equation-of-motion evaluations on excitonic model systems. Field-triggered excitation energy transfer dynamics and two dimensional coherent spectroscopy are simulated for both Condon and non-Condon vibronic couplings. Our results clearly demonstrate that the non-Condon vibronic coupling intensifies the dynamical electronic-vibrational energy transfer and enhances the total system-and-bath quantum coherence. Moreover, the hybrid bath dynamics for non-Condon effects enriches the theoretical calculation, and further sheds light on the interpretation of the experimental nonlinear spectroscopy.

Enhancing pairwise state-transition weights: A new weighting scheme in simulated tempering that can minimize transition time between a pair of conformational states.

Simulated tempering (ST) is a widely used enhancing sampling method for Molecular Dynamics simulations. As one expanded ensemble method, ST is a combination of canonical ensembles at different temperatures and the acceptance probability of cross-temperature transitions is determined by both the temperature difference and the weights of each temperature. One popular way to obtain the weights is to adopt the free energy of each canonical ensemble, which achieves uniform sampling among temperature space. However, this uniform distribution in temperature space may not be optimal since high temperatures do not always speed up the conformational transitions of interest, as anti-Arrhenius kinetics are prevalent in protein and RNA folding. Here, we propose a new method: Enhancing Pairwise State-transition Weights (EPSW), to obtain the optimal weights by minimizing the round-trip time for transitions among different metastable states at the temperature of interest in ST. The novelty of the EPSW algorithm lies in explicitly considering the kinetics of conformation transitions when optimizing the weights of different temperatures. We further demonstrate the power of EPSW in three different systems: a simple two-temperature model, a two-dimensional model for protein folding with anti-Arrhenius kinetics, and the alanine dipeptide. The results from these three systems showed that the new algorithm can substantially accelerate the transitions between conformational states of interest in the ST expanded ensemble and further facilitate the convergence of thermodynamics compared to the widely used free energy weights. We anticipate that this algorithm is particularly useful for studying functional conformational changes of biological systems where the initial and final states are often known from structural biology experiments.

Kinetic Rate Kernels via Hierarchical Liouville-Space Projection Operator Approach.

Kinetic rate kernels in general multisite systems are formulated on the basis of a nonperturbative quantum dissipation theory, the hierarchical equations of motion (HEOM) formalism, together with the Nakajima-Zwanzig projection operator technique. The present approach exploits the HEOM-space linear algebra. The quantum non-Markovian site-to-site transfer rate can be faithfully evaluated via projected HEOM dynamics. The developed method is exact, as evident by the comparison to the direct HEOM evaluation results on the population evolution.

Onsets of hierarchy truncation and self-consistent Born approximation with quantum mechanics prescriptions invariance.

The issue of efficient hierarchy truncation is related to many approximate theories. In this paper, we revisit this issue from both the numerical efficiency and quantum mechanics prescription invariance aspects. The latter requires that the truncation approximation made in Schrödinger picture, such as the quantum master equations and their self-consistent-Born-approximation improvements, should be transferable to their Heisenberg-picture correspondences, without further approximations. We address this issue with the dissipaton equation of motion (DEOM), which is a unique theory for the dynamics of not only reduced systems but also hybrid bath environments. We also highlight the DEOM theory is not only about how its dynamical variables evolve in time, but also the underlying dissipaton algebra. We demonstrate this unique feature of DEOM with model systems and report some intriguing nonlinear Fano interferences characteristics that are experimentally measurable.

Nonperturbative spin-boson and spin-spin dynamics and nonlinear Fano interferences: a unified dissipaton theory based study.

We consider the hybrid system-bath dynamics, based on the Yan's dissipaton formalism [Y. J. Yan, J. Chem. Phys. 140, 054105 (2014)]. This theory provides a unified quasi-particle treatment on three distinct classes of quantum bath, coupled nonperturbatively to arbitrary quantum systems. In this work, to study the entangled system and bath polarization and nonlinear Fano interference, we incorporate further the time-dependent light field, which interacts with both the molecular system and the collective bath dipoles directly. Numerical demonstrations are carried out on a two-level system, with comparison between phonon and exciton baths, in both linear and nonlinear Fano interference regimes.

Correlated driving and dissipation in two-dimensional spectroscopy.

The correlation between coherent driving and non-Markovian dissipation plays a vital role in optical processes. To exhibit its effect on the simulation of optical spectroscopy, we explore the correlated driving-dissipation equation (CODDE) [R. X. Xu and Y. J. Yan, J. Chem. Phys. 116, 9196 (2002)], which modifies the conventional Redfield theory with the inclusion of correlated driving-dissipation effect at the second-order system-bath coupling level. With an exciton model mimicking the Fenna-Matthews-Olson pigment-protein complex, we compare between the Redfield theory, CODDE, and exact hierarchical dynamics, for their results on linear absorption and coherent two-dimensional spectroscopy. We clarify that the failure of Redfield approach originates mainly from the neglect of driving-dissipation correlation, rather than its second-order nature. We further propose a dynamical inhomogeneity parameter to quantify the applicable range of CODDE. Our results indicate that CODDE is an efficient and quantifiable theory for many light-harvesting complexes of interest. To facilitate the evaluation of multi-dimensional spectroscopy, we also develop the mixed Heisenberg-Schrödinger picture scheme that is valid for any dynamics implementation on nonlinear response functions.