Quasi-adiabatic Propagator Path Integral Research Articles

Improved memory truncation scheme for quasi-adiabatic propagator path integral via influence functional renormalization.

Accurately simulating non-Markovian quantum dynamics in system-bath coupled problems remains challenging. In this work, we present a novel memory truncation scheme for the iterative quasi-adiabatic propagator path integral (iQuAPI) method to improve accuracy. Conventional memory truncation in iQuAPI discards all influence functional beyond a certain time interval, which is not effective for problems with a long memory time. Our proposed scheme selectively retains the most significant parts of the influence functional using the density matrix renormalization group algorithm. We validate the effectiveness of our scheme through simulations of the spin-boson model across various parameter sets, demonstrating faster convergence and improved accuracy compared to the conventional scheme. Our findings suggest that the new memory truncation scheme significantly advances the capabilities of iQuAPI for problems with a long memory time.

PathSum: A C++ and Fortran suite of fully quantum mechanical real-time path integral methods for (multi-)system + bath dynamics.

This paper reports the release of PathSum, a new software suite of state-of-the-art path integral methods for studying the dynamics of single or extended systems coupled to harmonic environments. The package includes two modules, suitable for system-bath problems and extended systems comprising many coupled system-bath units, and is offered in C++ and Fortran implementations. The system-bath module offers the recently developed small matrix path integral (SMatPI) and the well-established iterative quasi-adiabatic propagator path integral (i-QuAPI) method for iteration of the reduced density matrix of the system. In the SMatPI module, the dynamics within the entanglement interval can be computed using QuAPI, the blip sum, time evolving matrix product operators, or the quantum-classical path integral method. These methods have distinct convergence characteristics and their combination allows a user to access a variety of regimes. The extended system module provides the user with two algorithms of the modular path integral method, applicable to quantum spin chains or excitonic molecular aggregates. An overview of the methods and code structure is provided, along with guidance on method selection and representative examples.

B800-to-B850 relaxation of excitation energy in bacterial light harvesting: All-state, all-mode path integral simulations.

We report fully quantum mechanical simulations of excitation energy transfer within the peripheral light harvesting complex (LH2) of Rhodopseudomonas molischianum at room temperature. The exciton-vibration Hamiltonian comprises the 16 singly excited bacteriochlorophyll states of the B850 (inner) ring and the 8 states of the B800 (outer) ring with all available electronic couplings. The electronic states of each chromophore couple to 50 intramolecular vibrational modes with spectroscopically determined Huang-Rhys factors and to a weakly dissipative bath that models the biomolecular environment. Simulations of the excitation energy transfer following photoexcitation of various electronic eigenstates are performed using the numerically exact small matrix decomposition of the quasiadiabatic propagator path integral. We find that the energy relaxation process in the 24-state system is highly nontrivial. When the photoexcited state comprises primarily B800 pigments, a rapid intra-band redistribution of the energy sharply transitions to a significantly slower relaxation component that transfers 90% of the excitation energy to the B850 ring. The mixed character B850* state lacks the slow component and equilibrates very rapidly, providing an alternative energy transfer channel. This (and also another partially mixed) state has an anomalously large equilibrium population, suggesting a shift to lower energy by virtue of exciton-vibration coupling. The spread of the vibrationally dressed states is smaller than that of the eigenstates of the bare electronic Hamiltonian. The total population of the B800 band is found to decay exponentially with a 1/e time of 0.5ps, which is in good agreement with experimental results.

Differential Equation Based Path Integral for Open Quantum Systems

We propose the differential equation based path integral (DEBPI) method to simulate the real-time evolution of open quantum systems. In this method, a system of partial differential equations is derived based on the continuation of a classical numerical method called iterative quasi-adiabatic propagator path integral (i-QuAPI). While the resulting system has infinite equations, we introduce a reasonable closure to obtain a series of finite systems. New numerical schemes can be derived by discretizing these differential equations. It is numerically verified that in certain cases, by selecting appropriate systems and applying suitable numerical schemes, the memory cost required in the i-QuAPI method can be significantly reduced.

Pairwise connected tensor network representation of path integrals

It has been recently shown how the tensorial nature of real-time path integrals involving the Feynman-Vernon influence functional can be utilized using matrix product states, taking advantage of the finite length of the non-Markovian memory. Tensor networks promise to provide a new, unified language to express the structure of path integral. Here, a generalized tensor network is derived and implemented specifically incorporating the pairwise interaction structure of the influence functional, allowing for a compact representation and efficient evaluation. This pairwise connected tensor network path integral (PCTNPI) is illustrated through applications to typical spin-boson problems and explorations of the differences caused by the exact form of the spectral density. The storage requirements and performance are compared with iterative quasi-adiabatic propagator path integral and iterative blip-summed path integral. Finally, the viability of using PCTNPI for simulating multistate problems is demonstrated taking advantage of the compressed representation.

Small Matrix Path Integral for Driven Dissipative Dynamics.

The small matrix decomposition of the quasi-adiabatic propagator path integral (SMatPI) for a system coupled to a harmonic bath, which accounts for multitime memory correlations in the influence functional without the use of tensors, is extended to include a time-dependent term that drives the system. In the case of a periodic field, the algorithm requires the construction of SMatPI matrices initialized over a short time interval. The SMatPI algorithm circumvents the large array storage of tensor-based iterative path integral decompositions and, in the case of a periodic field, also eliminates the demanding tensor multiplication at each time step, leading to dramatic savings which allow the fully quantum mechanical treatment of multistate systems and long-memory environments.

Simulated quantum annealing as a simulator of nonequilibrium quantum dynamics

Simulated quantum annealing based on the path-integral Monte Carlo is one of the most common tools to simulate quantum annealing on classical hardware. Nevertheless, it is in principle highly non-trivial whether or not this classical algorithm can correctly reproduce the quantum dynamics of quantum annealing, particularly in the diabatic regime. We study this problem numerically through the generalized Kibble-Zurek mechanism of defect distribution in the simplest ferromagnetic one-dimensional transverse-field Ising model with and without coupling to the environment. We find that,in the absence of coupling to the environment, simulated quantum annealing correctly describes the annealing-time dependence of the average number of defects, but a detailed analysis of the defect distribution shows clear deviations from the theoretical prediction. When the system is open (coupled to the environment), the average number of defects does not follow the theoretical prediction but is qualitatively compatible with the numerical result by the infinite time-evolving block decimation combined with the quasi-adiabatic propagator path integral, which is valid in a very short time region. The distribution of defects in the open system turns out to be not far from the theoretical prediction. It is surprising that the classical stochastic dynamics of simulated quantum annealing ostensibly reproduce some aspects of the quantum dynamics. However, a serious problem is that it is hard to predict for which physical quantities in which system it is reliable. Those results suggest the necessity to exert a good amount of caution in using simulated quantum annealing to study the detailed quantitative aspects of the dynamics of quantum annealing.

Asymmetry in Charge Transfer Pathways Caused by Pigment–Protein Interactions in the Photosystem II Reaction Center Complex

This article discusses the photoinduced charge transfer (CT) kinetics within the reaction center complex of photosystem II (PSII RC). The PSII RC exhibits a structural symmetry in its arrangement of pigments forming two prominent branches, D1 and D2. Despite this symmetry, the CT has been observed to occur exclusively in the D1 branch. The mechanism to realize such functional asymmetry is yet to be understood. To approach this matter, we applied the theoretical tight-binding model of pigment excitations and simulated CT dynamics based upon the framework of an open quantum system. This simulation used a recently developed method of computation based on the quasi-adiabatic propagator path integral. A quantum CT state is found to be dynamically active when its site energy is resonant with the exciton energies of the PSII RC, regardless of the excitonic landscape we utilized. Through our investigation, it was found that the relative displacement between the local molecular energy levels of pigments can play a crucial role in realizing this resonance and therefore greatly affects the CT asymmetry in the PSII RC. Using this mechanism phenomenologically, we demonstrate that a near 100-to-1 ratio of reduction between the pheophytins in the D1 and D2 branches can be realized at both 77 K and 300 K. Our results indicate that the chlorophyll Chl D 1 is the most active precursor of the primary charge separation in the D1 branch and that the reduction of the pheophytins can occur within pico-seconds. Additionally, a broad resonance of the active CT state implies that a large static disorder observed in the CT state originates in the fluctuations of the relative displacements between the local molecular energy levels of the pigments in the PSII RC.

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.

Coarse-grained representation of the quasi adiabatic propagator path integral for the treatment of non-Markovian long-time bath memory.

The description of non-Markovian effects imposed by low frequency bath modes poses a persistent challenge for path integral based approaches like the iterative quasi-adiabatic propagator path integral (iQUAPI) method. We present a novel approximate method, termed mask assisted coarse graining of influence coefficients (MACGIC)-iQUAPI, that offers appealing computational savings due to substantial reduction of considered path segments for propagation. The method relies on an efficient path segment merging procedure via an intermediate coarse grained representation of Feynman-Vernon influence coefficients that exploits physical properties of system decoherence. The MACGIC-iQUAPI method allows us to access the regime of biological significant long-time bath memory on the order of hundred propagation time steps while retaining convergence to iQUAPI results. Numerical performance is demonstrated for a set of benchmark problems that cover bath assisted long range electron transfer, the transition from coherent to incoherent dynamics in a prototypical molecular dimer and excitation energy transfer in a 24-state model of the Fenna-Matthews-Olson trimer complex where in all cases excellent agreement with numerically exact reference data is obtained.

Finite-temperature time-dependent variation with multiple Davydov states.

The Dirac-Frenkel time-dependent variational approach with Davydov Ansätze is a sophisticated, yet efficient technique to obtain an accurate solution to many-body Schrödinger equations for energy and charge transfer dynamics in molecular aggregates and light-harvesting complexes. We extend this variational approach to finite temperature dynamics of the spin-boson model by adopting a Monte Carlo importance sampling method. In order to demonstrate the applicability of this approach, we compare calculated real-time quantum dynamics of the spin-boson model with that from numerically exact iterative quasiadiabatic propagator path integral (QUAPI) technique. The comparison shows that our variational approach with the single Davydov Ansätze is in excellent agreement with the QUAPI method at high temperatures, while the two differ at low temperatures. Accuracy in dynamics calculations employing a multitude of Davydov trial states is found to improve substantially over the single Davydov Ansatz, especially at low temperatures. At a moderate computational cost, our variational approach with the multiple Davydov Ansatz is shown to provide accurate spin-boson dynamics over a wide range of temperatures and bath spectral densities.

Influences of different environmental factors to 2D non-linear spectra of bio-molecular aggregates

In this paper, we investigate the influences of different environmental factors to rephasing spectra of 2D, third-order, non-linear (2D3ONL), photon echo signals for bio-molecular aggregates by using an exact numerical dynamics approach—the quasi-adiabatic propagator path integral (QUAPI) method. It is shown that the increase of temperature and coupling strength between the system and its environment results in not only the inhomogeneous but also the homogeneous broadening of the 2D3ONL rephasing spectra of photon echo signals. However, it is interested that the increase of dynamical memory times, namely, the increase of the non-Markovian effects will mainly lead to the homogeneous broadening of the spectra. The evolutions of the 2D3ONL rephasing spectra with population times are also investigated.

On iterative path integral calculations for a system interacting with a shifted dissipative bath.

Real-time path integral calculations for the propagation of a system in contact with a harmonic dissipative environment often employ the iterative quasi-adiabatic propagator path integral (i-QuAPI) methodology. We compare two simple ways of applying this methodology to a bath initially in equilibrium with the localized state of the system (e.g., the donor in the case of charge transfer). The first way involves modifying the phase of the system via a time-local phase given in terms of integrals of the spectral density or in terms of the coefficients entering the QuAPI-discretized influence functional. In the iterative decomposition of the path integral, this approach requires consistent memory truncation to avoid extremely slow convergence. The second, alternative approach involves shifting the coordinate of the system, to bring the donor state in equilibrium with the bath, and requires no further modification of the i-QuAPI algorithm.

Exploiting classical decoherence in dissipative quantum dynamics: Memory, phonon emission, and the blip sum

The mechanisms that lead to the quenching of the coherence of a quantum system interacting with a dissipative environment are analyzed within the path integral/influence functional framework. Classical and quantum contributions are identified and linked to stimulated and spontaneous phonon emission processes. Three ways of exploiting decoherence in numerical calculations are discussed. The first is based on the decay of memory effects with time separation and leads to well-known iterative quasi-adiabatic propagator path integral (i-QuAPI) schemes. The other two mechanisms exploit the prominent role of classical decoherence. Removal of classical memory is possible through the introduction of auxiliary variables, leading to efficient and accurate quantum–classical path integral (QCPI) methods. A third way is proposed, which involves systematic corrections to the fully incoherent limit. An efficient procedure for summing all paths associated with ‘blips’ (short off-diagonal path pair segments) is described. The ideas are illustrated with numerical calculations of tunneling dynamics.

Decoherence dynamics of coherent electronic excited states in the photosynthetic purple bacteriumRhodobacter sphaeroides

In this paper, we present a theoretical description to the quantum coherence and decoherence phenomena of energy transfer in photosynthesis observed in a recent experiment [Science 316, 1462 (2007)]. As a successive two-color laser pulses with selected frequencies cast on a sample of the photosynthetic purple bacterium Rb. sphaeroides two resonant excitations of electrons in chromophores can be generated. However, this effective two-level subsystem will interact with its protein environment and decoherence is inevitable. We describe this subsystem coupled with its environment as a dynamical spin-boson model. The non-Markovian decoherence dynamics is described using a quasiadiabatic propagator path integral (QUAPI) approach. With the photon-induced effective time-dependent level splitting energy and level flip coupling coefficient between the two excited states and the environment-induced non-Markovian decoherence dynamics, our theoretical result is in good agreement with the experimental data.

Coherent control of an effective two-level system in a non-Markovian biomolecular environment

We investigate the quantum coherent dynamics of an externally driven effective two-level system (TLS) subjected to a slow Ohmic environment characteristic of biomolecular protein–solvent reservoirs in photosynthetic light-harvesting complexes. By means of the numerically exact quasi-adiabatic propagator path integral (QUAPI) method we are able to include non-Markovian features of the environment and show the dependence of the quantum coherence on the characteristic bath cut-off frequency ωc as well as on the driving frequency ωl and the field amplitude A. Our calculations extend from the weak-coupling regime to the incoherent strong-coupling regime. In the latter case, we find evidence for a resonant behaviour, beyond the expected behaviour, when the reorganization energy Er coincides with the driving frequency. Moreover, we investigate how the coherent destruction of tunnelling within the TLS is influenced by the non-Markovian environment.

Decoherence of Josephson charge qubit

In this paper we investigate decoherence time of superconducting Josephson charge qubit (JCQ). Two kinds of methods, iterative tensor multiplication (ITM) method derived from the quasiadiabatic propagator path integral (QUAPI) and Bloch equations method are used. Using the non-Markovian ITM method we correct the decoherence time predicted by Bloch equations method. By comparing the exact theoretical result with the experimental data we suggest that the Ohmic noise plays the central role to the decoherence of the JCQ.

Quantum‐mechanical reaction rate constants from centroid molecular dynamics simulations: Barrier crossing in an asymmetrical double‐well

AbstractQuantum‐mechanical reaction rate constants were calculated from centroid molecular dynamics (CMD) simulations, for the case of barrier crossing in an asymmetrical double‐well potential bilinearly coupled to a harmonic bath. The calculation is based on a recently proposed formulation of the reaction rate constant in terms of the position—flux correlation function, which can be approximated via CMD in a well‐defined manner. The predictions of CMD and various simplified versions of it are compared to exact results, which were obtained via the quasi‐adiabatic propagator path integral (QUAPI) method, and/or path integral quantum transition state theory (PI‐QTST). The predictions based on CMD are found to be in good agreement with both.

Numerical path integral techniques for long time dynamics of quantum dissipative systems

Recent progress in numerical methods for evaluating the real-time path integral in dissipative harmonic environments is reviewed. Quasi-adiabatic propagators constructed numerically allow convergence of the path integral with large time increments. Integration of the harmonic bath leads to path integral expressions that incorporate the exact dynamics of the quantum particle along the adiabatic path, with an influence functional that describes nonadiabatic corrections. The resulting quasi-adiabatic propagator path integral is evaluated by efficient system-specific quadratures in most regimes of parameter space, although some cases are handled by grid Monte Carlo sampling. Exploiting the finite span of nonlocal influence functional interactions characteristic of broad condensed phase spectra leads to an iterative scheme for calculating the path integral over arbitrary time lengths. No uncontrolled approximations are introduced, and the resulting methodology converges to the exact quantum result with modest amounts of computational power. Applications to tunneling dynamics in the condensed phase are described.

Quantum rates for a double well coupled to a dissipative bath: Accurate path integral results and comparison with approximate theories

We present accurate fully quantum calculations of thermal rate constants for a symmetric double well system coupled to a dissipative bath. The calculations are performed using the quasiadiabatic propagator path integral (QUAPI) methodology to evaluate the flux–flux correlation function whose time integral determines the rate coefficient. The discretized path integral converges very rapidly in the QUAPI representation, allowing efficient calculation of quantum correlation functions for sufficiently long times. No ad hoc assumption is introduced and thus these calculations yield the true quantum mechanical rate constants. The results presented in the paper demonstrate the applicability of the QUAPI methodology to practically all regimes of chemical interest, from thermal activation to deep tunneling, and the quantum transmission factor exhibits a Kramers turnover. Our calculations reveal an unusual step structure of the integrated reactive flux in the weak friction regime as well as quantum dynamical enhancement of the rate above the quantum transition state theory value at low temperatures, which is largely due to vibrational coherence effects. The quantum rates are compared to those obtained from classical trajectory simulations. We also use the numerically exact classical and quantum results to establish the degree of accuracy of several analytic and numerical approximations, including classical and quantum Grote–Hynes theories, semiclassical transition state theory (periodic orbit) estimates, classical and quantum turnover theories, and the centroid density approximation.