Low-Storage Runge-Kutta Method for Simulating Time-Dependent Quantum Dynamics

Discretized hierarchical equations of motion in mixed Liouville-Wigner space for two-dimensional vibrational spectroscopies of liquid water

A model of a bulk water system describing the vibrational motion of intramolecular and intermolecular modes is constructed, enabling analysis of its linear and nonlinear vibrational spectra as well as the energy transfer processes between the vibrational modes. The model is described as a system of four interacting anharmonic oscillators nonlinearly coupled to their respective heat baths. To perform a rigorous numerical investigation of the non-Markovian and nonperturbative quantum dissipative dynamics of the model, we derive discretized hierarchical equations of motion in mixed Liouville-Wigner space, with Lagrange-Hermite mesh discretization being employed in the Liouville space of the intramolecular modes and Lagrange-Hermite mesh discretization and Hermite discretization in the Wigner space of the intermolecular modes. One-dimensional infrared and Raman spectra and two-dimensional terahertz-infrared-visible and infrared-infrared-Raman spectra are computed as demonstrations of the quantum dissipative description provided by our model.

Modeling Excited-State Proton Transfer Using the Lindblad Equation: Quantification of Time-Resolved Spectroscopy with Mechanistic Insights

The quantum dynamics of excited-state intramolecular proton transfer (ESIPT) is studied using a multilevel vibronic Hamiltonian and the Lindblad master equation. We simulate time-resolved fluorescence spectroscopy of 2-(2'-hydroxyphenyl) benzothiazole (HBT) and 10-hydroxybenzo[h]quinoline (HBQ), which suggests that the underlying mechanism behind the initial ultrafast rise and decay in the spectra is electronic state population that evolves simultaneously with proton wave packet dynamics. The results predict that the initial rise and decay signals at different wavelengths vary significantly with system properties in terms of their shape, the time, and the intensity of the maximum. These findings provide clues for data interpretation, mechanism validation, and control of the dynamics, and the model serves as an attempt toward clarifying ESIPT by direct comparison to time-resolved spectroscopy.

Automatic Feynman diagram generation for nonlinear optical spectroscopies and application to fifth-order spectroscopy with pulse overlaps

Perturbative nonlinear optical spectroscopies are powerful methods to understand the dynamics of excitonic and other condensed phase systems. Feynman diagrams have long provided the essential tool to understand and interpret experimental spectra and to organize the calculation of spectra for model systems. When optical pulses are strictly time ordered, only a small number of diagrams contribute, but in many experiments, pulse-overlap effects are important for interpreting results. When pulses overlap, the number of contributing diagrams can increase rapidly, especially with higher order spectroscopies, and human error is especially likely when attempting to write down all the diagrams. We present an automated Diagram Generator (DG) that generates all the Feynman diagrams needed to calculate any nth-order spectroscopic signal. We characterize all perturbative nonlinear spectroscopies by their associated phase-discrimination condition as well as the time intervals where pulse amplitudes are nonzero. Although the DG can be used to automate impulsive calculations, its greatest strength lies in automating finite pulse calculations where pulse overlaps are important. We consider third-order transient absorption spectroscopy and fifth-order exciton-exciton interaction 2D (EEI2D) spectroscopy, which are described by six or seven diagrams in the impulsive limit, respectively, but 16 or 240 diagrams, respectively, when pulses overlap. The DG allows users to automatically include all relevant diagrams at a relatively low computational cost, since the extra diagrams are only generated for the inter-pulse delays where they are relevant. For EEI2D spectroscopy, we show the important effects of including the overlap diagrams.

Efficient numerical method for predicting nonlinear optical spectroscopies of open systems

Nonlinear optical spectroscopies are powerful tools for probing quantum dynamics in molecular and nanoscale systems. While intuition about ultrafast spectroscopies is often built by considering impulsive optical pulses, actual experiments have finite-duration pulses, which can be important for interpreting and predicting experimental results. We present a new freely available open source method for spectroscopic modeling, called Ultrafast Ultrafast (UF²) spectroscopy, which enables computationally efficient and convenient prediction of nonlinear spectra, such as treatment of arbitrary finite duration pulse shapes. UF² is a Fourier-based method that requires diagonalization of the Liouvillian propagator of the system density matrix. We also present a Runge-Kutta-Euler (RKE) direct propagation method. We include open system dynamics in the secular Redfield, full Redfield, and Lindblad formalisms with Markovian baths. For non-Markovian systems, the degrees of freedom corresponding to memory effects are brought into the system and treated nonperturbatively. We analyze the computational complexity of the algorithms and demonstrate numerically that, including the cost of diagonalizing the propagator, UF² is 20-200 times faster than the direct propagation method for secular Redfield models with arbitrary Hilbert space dimension; it is similarly faster for full Redfield models at least up to system dimensions where the propagator requires more than 20 GB to store; and for Lindblad models, it is faster up to Hilbert space dimension near 100 with speedups for small systems by factors of over 500. UF² and RKE are part of a larger open source Ultrafast Software Suite, which includes tools for automatic generation and calculation of Feynman diagrams.

Numerically "exact" approach to open quantum dynamics: The hierarchical equations of motion (HEOM)

An open quantum system refers to a system that is further coupled to a bath system consisting of surrounding radiation fields, atoms, molecules, or proteins. The bath system is typically modeled by an infinite number of harmonic oscillators. This system-bath model can describe the time-irreversible dynamics through which the system evolves toward a thermal equilibrium state at finite temperature. In nuclear magnetic resonance and atomic spectroscopy, dynamics can be studied easily by using simple quantum master equations under the assumption that the system-bath interaction is weak (perturbative approximation) and the bath fluctuations are very fast (Markovian approximation). However, such approximations cannot be applied in chemical physics and biochemical physics problems, where environmental materials are complex and strongly coupled with environments. The hierarchical equations of motion (HEOM) can describe the numerically "exact" dynamics of a reduced system under nonperturbative and non-Markovian system-bath interactions, which has been verified on the basis of exact analytical solutions (non-Markovian tests) with any desired numerical accuracy. The HEOM theory has been used to treat systems of practical interest, in particular, to account for various linear and nonlinear spectra in molecular and solid state materials, to evaluate charge and exciton transfer rates in biological systems, to simulate resonant tunneling and quantum ratchet processes in nanodevices, and to explore quantum entanglement states in quantum information theories. This article presents an overview of the HEOM theory, focusing on its theoretical background and applications, to help further the development of the study of open quantum dynamics.

Numerically "exact" simulations of entropy production in the fully quantum regime: Boltzmann entropy vs von Neumann entropy

We present a scheme to evaluate thermodynamic variables for a system coupled to a heat bath under a time-dependent external force using the quasi-static Helmholtz energy from the numerically "exact" hierarchical equations of motion (HEOM). We computed the entropy produced by a spin system strongly coupled to a non-Markovian heat bath for various temperatures. We showed that when changes to the external perturbation occurred sufficiently slowly, the system always reached thermal equilibrium. Thus, we calculated the Boltzmann entropy and the von Neumann entropy for an isothermal process, as well as various thermodynamic variables, such as changes in internal energies, heat, and work, for a system in quasi-static equilibrium based on the HEOM. We found that although the characteristic features of the system entropies in the Boltzmann and von Neumann cases as a function of the system-bath coupling strength are similar, those for the total entropy production are completely different. The total entropy production in the Boltzmann case is always positive, whereas that in the von Neumann case becomes negative if we chose a thermal equilibrium state of the total system (an unfactorized thermal equilibrium state) as the initial state. This is because the total entropy production in the von Neumann case does not properly take into account the contribution of the entropy from the system-bath interaction. Thus, the Boltzmann entropy must be used to investigate entropy production in the fully quantum regime. Finally, we examined the applicability of the Jarzynski equality.

A unified view of hierarchy approach and formula of differentiation

The stochastic differential equation is a powerful tool for describing the dynamics of a dissipative system in which noise characterizes the influence of the environment. For the Ornstein-Uhlenbeck noise, both the formula of differentiation and the hierarchy approach provide efficient numerical simulations, with the stochastic differential equation transformed into a set of coupled, linear ordinary differential equations. We show that while these two deterministic schemes result in different sets of equations, they can be regarded as two representations of an underlying linear-dynamics. Moreover, by manipulating the involved Ornstein-Uhlenbeck noise, we propose a unified algorithm that may reduce to the hierarchy approach or the formula of differentiation in different limits. We further analyze the numerical performance of this algorithm and find that the hierarchy approach appears to be more efficient for our numerical model studies.

Dynamical scaling in the Ohmic spin-boson model studied by extended hierarchical equations of motion

Through a decomposition of the bath correlation function, the hierarchical equations of motion are extended to the Ohmic spin-boson model at zero temperature. For two typical cutoff functions of the bath spectral density, the rate kernel of spin dynamics is numerically extracted by a time-convolution equation of the average magnetic moment. A characteristic time is defined accordingly as the inverse of the zeroth-order moment of the rate kernel. For a given Kondo parameter in the incoherent regime, the time evolution of average magnetic moments gradually collapses onto a master curve after rescaling the time variable with the characteristic time. The rescaled spin dynamics is nearly independent of the cutoff frequency and the form of cutoff functions. For a given cutoff frequency, the characteristic time with the change of the Kondo parameter is fitted excellently as a function of the renormalized tunneling amplitude. Despite a significant difference in definition, our result is in good agreement with the characteristic time of the noninteracting blip approximation.