We present a streamlined implementation of a computational framework for constructing and analyzing reduced dynamical maps for complex system--bath models at finite temperature. The methodology is based on three established ingredients of quantum dynamics: the Choi--Jamiołkowski isomorphism for the representation of quantum channels, thermofield (TFD) purification of thermal environments, and tensor-train (TT) propagation of the resulting enlarged pure state. The reduced map is obtained from a single unitary propagation in a thermofield-doubled Hilbert space and represented in matrix form through the Choi--Jamiołkowski isomorphism. The TFD evolution is implemented in the TT representation, enabling efficient propagation of high-dimensional purified thermal states. We illustrate the methodology for exciton transfer in the Fenna--Matthews--Olson complex with site-dependent structured spectral densities represented by discretized bosonic environments. The resulting maps are used to analyze decoherence, relaxation, and finite-memory effects, and to assess the crossover to an effectively time-local description. The proposed approach provides a route to compute reduced propagators and to post-process them into memory kernels, transfer tensors, and effective kinetic rate descriptions for complex molecular systems.
We present a streamlined implementation of a computational framework for constructing and analyzing reduced dynamical maps for complex system--bath models at finite temperature. The methodology is based on three established ingredients of quantum dynamics: the Choi--Jamiołkowski isomorphism for the representation of quantum channels, thermofield (TFD) purification of thermal environments, and tensor-train (TT) propagation of the resulting enlarged pure state. The reduced map is obtained from a single unitary propagation in a thermofield-doubled Hilbert space and represented in matrix form through the Choi--Jamiołkowski isomorphism. The TFD evolution is implemented in the TT representation, enabling efficient propagation of high-dimensional purified thermal states. We illustrate the methodology for exciton transfer in the Fenna--Matthews--Olson complex with site-dependent structured spectral densities represented by discretized bosonic environments. The resulting maps are used to analyze decoherence, relaxation, and finite-memory effects, and to assess the crossover to an effectively time-local description. The proposed approach provides a route to compute reduced propagators and to post-process them into memory kernels, transfer tensors, and effective kinetic rate descriptions for complex molecular systems.
Addressing needs of contemporary nonlinear femtosecond optical spectroscopy, we have developed a fully quantum, numerically accurate wave function-based approach for the calculation of third-order spectroscopic signals of polyatomic molecules and molecular aggregates at finite temperature. The systems are described by multimode nonadiabatic vibronic-coupling Hamiltonians, in which diagonal terms are treated in harmonic approximation, while off-diagonal interstate couplings are assumed to be coordinate independent. The approach is based on the Thermo Field Dynamics (TFD) representation of quantum mechanics and tensor-train (TT) machinery for efficient numerical simulation of quantum evolution of systems with many degrees of freedom. The developed TFD-TT approach is applied to the calculation of time- and frequency-resolved fluorescence spectra of the Fenna-Matthews-Olson (FMO) antenna complex at room temperature taking into account finite time-frequency resolution in fluorescence detection, orientational averaging, and static disorder.
Abstract This review provides the fundamental theoretical tools for the development of a complete wave‐function formalism for the study of time‐evolution of chemico‐physical systems at finite temperature. The methodology is based on the non‐equilibrium thermo‐field dynamics (NE‐TFD) representation of quantum mechanics, which is alternative to the commonly used density matrix representation. TFD concepts are extended and integrated with the tensor‐train (TT) numerical tools leading to a novel and powerful theoretical and computational framework for the study of complex quantum dynamical problems. In addition, NE‐TFD techniques are extended to enable the study of dissipative open systems via a new formulation of the hierarchical equations of motion (HEOM) fully integrated with TT methodologies. We demonstrate that the combination of the TFD machinery with computational advantages of TTs results in a powerful theoretical and computational framework for scrutinizing dynamics of complex multidimensional electron‐vibrational systems. We illustrate the validity and the computational advantages of the developed methodologies by applying them to the study of quantum coherence effects in the energy‐transfer processes in antenna systems, to the analysis of fingerprints of vibrational modes in electron‐transfer and charge‐transfer processes in various model and realistic multidimensional molecular systems, as well as to simulation of other fundamental models of physical chemistry. This article is categorized under: Theoretical and Physical Chemistry > Reaction Dynamics and Kinetics Theoretical and Physical Chemistry > Statistical Mechanics
Accurate quantum dynamics simulations of nonadiabatic processes are important for studies of electron transfer, energy transfer, and photochemical reactions in complex systems. In this comparative study, we benchmark various approximate nonadiabatic dynamics methods with mapping variables against numerically exact calculations based on the tensor-train (TT) representation of high-dimensional arrays, including TT-KSL for zero-temperature dynamics and TT-thermofield dynamics for finite-temperature dynamics. The approximate nonadiabatic dynamics methods investigated include mixed quantum-classical Ehrenfest mean-field and fewest-switches surface hopping, linearized semiclassical mapping dynamics, symmetrized quasiclassical dynamics, the spin-mapping method, and extended classical mapping models. Different model systems were evaluated, including the spin-boson model for nonadiabatic dynamics in the condensed phase, the linear vibronic coupling model for electronic transition through conical intersections, the photoisomerization model of retinal, and Tully's one-dimensional scattering models. Our calculations show that the optimal choice of approximate dynamical method is system-specific, and the accuracy is sensitively dependent on the zero-point-energy parameter and the initial sampling strategy for the mapping variables.
Thermofield dynamics (TFD) is a powerful framework for accounting for thermal effects in a wave function setting and has been extensively used in physics and quantum optics. TFD relies on a duplicated state space and creates a correlated two-mode thermal state via a Bogoliubov transformation acting on the vacuum state. However, a very useful variant of TFD uses the vacuum state as the initial condition and transfers the Bogoliubov transformation into the propagator. This variant, referred to here as the inverse Bogoliubov transformation (iBT) variant, has recently been applied to vibronic coupling problems and coupled-oscillator Hamiltonians in a chemistry context, where the method is combined with efficient tensor network methods for high-dimensional quantum propagation. In the iBT-TFD representation, the mode expectation values are clearly defined and easy to calculate, but the thermalized reduced particle distributions, such as the reduced 1-particle densities or Wigner distributions, are highly non-trivial due to the Bogoliubov back-transformation of the original thermal TFD wave function. Here, we derive formal expressions for the reduced 1-particle density matrix (1-RDM) that use the correlations between the real and tilde modes encoded in the associated reduced 2-particle density matrix. We apply this formalism to define the 1-RDM and the Wigner distributions in the special case of a thermal harmonic oscillator. Moreover, we discuss several approximate schemes that can be extended to higher-dimensional distributions. These methods are demonstrated for the thermal reduced 1-particle density of an anharmonic oscillator.
The most general description of quantum evolution up to a time τ is a completely positive tracing preserving map known as a dynamical mapΛ̂(τ). Here, we consider Λ̂(τ) arising from suddenly coupling a system to one or more thermal baths with a strength that is neither weak nor strong. Given no clear separation of characteristic system/bath time scales, Λ̂(τ) is generically expected to be non-Markovian; however, we do assume the ensuing dynamics has a unique steady state, implying the baths possess a finite memory time τm. By combining several techniques within a tensor network framework, we directly and accurately extract Λ̂(τ) for a small number of interacting fermionic modes coupled to infinite non-interacting Fermi baths. First, we use an orthogonal polynomial mapping and thermofield doubling to arrive at a purified chain representation of the baths whose length directly equates to a time over which the dynamics of the infinite baths is faithfully captured. Second, we employ the Choi-Jamiolkowski isomorphism so that Λ̂(τ) can be fully reconstructed from a single pure state calculation of the unitary dynamics of the system, bath and their replica auxiliary modes up to time τ. From Λ̂(τ), we also compute the time local propagator L̂(τ). By examining the convergence with τ of the instantaneous fixed points of these objects, we establish their respective memory times τmΛ and τmL. Beyond these times, the propagator L̂(τ) and dynamical map Λ̂(τ) accurately describe all the subsequent long-time relaxation dynamics up to stationarity. These timescales form a hierarchy τmL≤τmΛ≤τre, where τre is a characteristic relaxation time of the dynamics. Our numerical examples of interacting spinless Fermi chains and the single impurity Anderson model demonstrate regimes where τre ≫ τm, where our approach can offer a significant speedup in determining the stationary state compared to directly simulating the long-time limit. Our results also show that having access to Λ̂(τ) affords a number of insightful analyses of the open system thus far not commonly exploited.
2024, 161, 151101]. The approach leverages a low-rank decomposition of the Fourier-transform relation connecting the bath correlation function to its spectral density. By capturing the time, frequency, and temperature dependencies encoded in the spectral density-autocorrelation function relation, our method significantly reduces the degrees of freedom required for simulating open quantum system dynamics. We benchmark our approach against existing methods and demonstrate its efficacy through applications to both simple models and a realistic electron transfer process in biological systems. Additionally, we show that this new approach can be effectively combined with the tensor-train formalism to investigate the quantum dynamics of systems interacting with complex non-Markovian environments. Finally, we provide a perspective on the selection and application of various spectral density discretization techniques.