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arXiv:2601.00431

Multistep quantum master equation theory for response functions in four wave mixing electronic spectroscopy of multichromophoric macromolecules

Seogjoo J. Jang

quant-phphysics.chem-phphysics.opticsSubmitted Jan 1, 2026• Announced Jan 2, 2026
arXivHTMLPDF

Abstract

This work provides an alternative derivation of third order response functions in four wave mixing spectroscopy of multichromophoric macromolecular systems considering only single exciton states. For the case of harmonic oscillator bath linearly and diagonally coupled to exciton states, closed form expressions showing all the explicit time dependences are derived. These expressions can provide more solid physical basis for understanding 2-dimensional electronic spectroscopy signals. For more general cases of system-bath coupling, the quantum master equation (QME) approach is employed for the derivation of multistep time evolution equations for Green function-like operators. Solution of these equations is feasible at the level of 2nd order non-Markovian QME, and the new approach can account for inter-exciton coupling, dephasing, relaxation, and non-Markovian effects in a consistent manner.

Journal reference
Bulletin of the Korean Chemical Society 33, 997-1008 (2012)
DOI
10.5012/bkcs.2012.33.3.997
Comments
14 pages, 0 figure

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This work provides an alternative derivation of third order response functions in four wave mixing spectroscopy of multichromophoric macromolecular systems considering only single exciton states. For the case of harmonic oscillator bath linearly and diagonally coupled to exciton states, closed form expressions showing all the explicit time dependences are derived. These expressions can provide more solid physical basis for understanding 2-dimensional electronic spectroscopy signals. For more general cases of system-bath coupling, the quantum master equation (QME) approach is employed for the derivation of multistep time evolution equations for Green function-like operators. Solution of these equations is feasible at the level of 2nd order non-Markovian QME, and the new approach can account for inter-exciton coupling, dephasing, relaxation, and non-Markovian effects in a consistent manner.
Fourth-order quantum master equations (FQMEs) are derived in both time nonlocal and local forms for a general system Hamiltonian, with new detailed expressions for the fourth-order kernel, where the bath correlation functions are explicitly decoupled from the system superoperators. Further simplifications can be made for the model of linearly coupled harmonic oscillator bath. Consideration of the high temperature Ohmic bath limit leads to a general Markovian FQME with compact forms of time independent superoperators. Two examples of this equation are then considered. For the system of a quantum particle in a continuous potential field, the equation reduces to a known form of the quantum Fokker–Planck equation, except for a fourth-order potential renormalization term that can be neglected only in the weak system-bath interaction regime. For a two-level system with off-diagonal coupling to the bath, fourth-order corrections do not alter the relaxation characteristics of the second-order equation and introduce additional coherence terms in the equations for the off-diagonal elements.
The simulation of electronically nonadiabatic dynamics and spectroscopy remains as a central challenge in computational chemistry. Amongst the reduced dynamic methods, various forms of the perturbative quantum master equation (QME) including Fermi's golden rule (FGR), Redfield equation, and Marcus theory enjoy widespread use for their efficiency in the perturbative regime, and provides important physical intuition for the underlying dynamics. In the first chapter of this dissertation, the accuracy of various QMEs that treat off-diagonal electronic coupling perturbatively (OD-QME) will be examined on a spin-boson model. OD-QME works in the interaction picture, limiting its application to population dynamics. A novel way of retrieving Schrodinger picture coherences from interaction picture dynamics will be introduced, and the resulting dynamics will be compared against the numerically exact multi-configuration time dependent Hartree method. It will be shown that OD-QME can generate accurate population and coherence in appropriate parameter ranges, and that the time-nonlocal QME is more accurate than time-local QME, especially if the QME uses a population only projection. In the second chapter, the aforementioned method of calculating electronic coherence via OD-QME is applied to the simulation of absorption spectra. The result will be compared against the numerically exact hierarchical equations of motion, as well as the Redfield time-local QME which is widely applied in the context of spectroscopy. It will be shown that OD-QME exhibits comparable accuracy to Redfield QME over a wide parameter range. Both QMEs fail to predict the spectra accurately in the case of a high-frequency bath at a low temperature, and this error is likely to be linked to the quantum behavior of the bath. In the third chapter, the FGR rates of electronically nonadiabatic reactions enhanced by a microcavity is calculated. It will be shown that the cavity-enhanced reaction rates can be evaluated from the cavity-free FGR correlation function with minimum additional effort, and this method will be applied to a model CT reaction. While a high-frequency cavity enhances reactions in the Marcus inverted region selectively, a low-frequency cavity enhances reactions in all regions uniformly. In the fourth chapter, the FGR charge transfer rate constants for a molecular triad system solvated in explicit solvent is calculated via linear semiclassical (LSC) approach. A Marcus-like expression can be derived after applying a hierarchy of approximations. The validity of such approximations will be confirmed for the triad model. It will also be shown that the charge transfer rate depends deeply on the conformation of the molecule, the fluctuation between which is the rate-limiting step of the charge transfer.
An extremely useful evolution equation that allows systematically calculating the two-time correlation functions (CF's) of system operators for non-Markovian open (dissipative) quantum systems is derived. The derivation is based on perturbative quantum master equation approach, so non-Markovian open quantum system models that are not exactly solvable can use our derived evolution equation to easily obtain their two-time CF's of system operators, valid to second order in the system-environment interaction. Since the form and nature of the Hamiltonian are not specified in our derived evolution equation, our evolution equation is applicable for bosonic and/or fermionic environments and can be applied to a wide range of system-environment models with any factorized (separable) system-environment initial states (pure or mixed). When applied to a general model of a system coupled to a finite-temperature bosonic environment with a system coupling operator L in the system-environment interaction Hamiltonian, the resultant evolution equation is valid for both L = L(†) and L ≠ L(†) cases, in contrast to those evolution equations valid only for L = L(†) case in the literature. The derived equation that generalizes the quantum regression theorem (QRT) to the non-Markovian case will have broad applications in many different branches of physics. We then give conditions on which the QRT holds in the weak system-environment coupling case and apply the derived evolution equation to a problem of a two-level system (atom) coupled to the finite-temperature bosonic environment (electromagnetic fields) with L ≠ L(†).
Considering an electronic two-level system coupled to vibrational degrees of freedom and driven by short and intense non-overlapping laser pulses, we introduce the concept of nonperturbative response functions. These response functions are expressed in terms of effective electronic transition dipole moments which depend on the strength of the field-matter coupling and on the pulse durations. It is shown that the nonlinear polarization representing four-wave-mixing signals can elegantly be expressed in terms of these nonperturbative response functions to all orders in the field-matter coupling. The nonperturbative response functions generalize the interpretative power of the familiar third-order response functions to four-wave-mixing experiments with intense laser pulses.
Quantum coherence and its destruction (dephasing) by coupling to a dissipative environment plays an important role in time-resolved nonlinear optical response as well as nonadiabatic transitions and tunneling processes in condensed phases. Generating functions of density-matrix elements and multitime coordinate and momentum correlation functions related to these phenomena are calculated using a pathintegral approach by performing functional integration. The dissipative environment is assumed to be an ensemble of harmonic oscillators and is taken into account by using Feynman-Vernon influence functional. Closed-form expressions for generating functions in terms of the bath spectral density are derived. The present theory generalizes earlier calculations of these quantities to arbitrary temperatures, any dependence of the transition coupling on coordinates (non-Condon effects), and arbitrary order in the interstate coupling. Conditions for factorizing the Liouville-space generating functions that allow a reduced description based on the classical Langevin equation are established. Possible applications to four-wave-mixing spectroscopy and nonadiabatic rate processes are discussed.
The Casida equations of linear response TDDFT are extended to calculate linear spectra of open quantum systems evolving according to a Markovian master equation. By mapping a many-body open quantum system onto an open, non-interacting Kohn-Sham system, extrinsic line broadening due to electron-bath coupling can be described exactly within TDDFT. The structure of the resulting matrix equations are analyzed for the generic case of electrons linearly coupled to a harmonic bath within Redfield theory. An approximate form of the exchange-correlation kernel based on first-order G\\"orling-Levy perturbation theory is derived.