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arXiv:2512.12584

Extended dissipaton theory for higher-order bath couplings and application to non-Condon spectroscopy with anharmonicity

Zi-Fan Zhu, Yu Su, Yao Wang, Rui-Xue Xu, YiJing Yan

physics.chem-phquant-phSubmitted Dec 14, 2025• Announced Dec 16, 2025
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Abstract

In this work, we develop an extended dissipaton theory that generalizes the environmental couplings beyond the conventional linear and quadratic forms, enabling the treatment of arbitrary order of bath couplings. Applying this theoretical framework to the condensed-phase non-Condon spectroscopy, we demonstrate the interplay of anharmonicity, non-Condon and solvent effects on optical spectra. Precise simulations are carried out with high efficiency on linear absorption spectra involving the above mentioned correlated effects. We exhibit how an anharmonic potential modulates the vibronic feature, offering insights into the role of nonlinear environmental couplings in spectroscopic signatures and exemplifying the success of the extended dissipaton formalism as an exact and efficient method for higher-order bath couplings.

Comments
Part of Special Issue dedicated to Professor Qing-shi Zhu on the occasion of his 80th birthday

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In this work, we develop an extended dissipaton theory that generalizes the environmental couplings beyond the conventional linear and quadratic forms, enabling the treatment of arbitrary order of bath couplings. Applying this theoretical framework to the condensed-phase non-Condon spectroscopy, we demonstrate the interplay of anharmonicity, non-Condon and solvent effects on optical spectra. Precise simulations are carried out with high efficiency on linear absorption spectra involving the above mentioned correlated effects. We exhibit how an anharmonic potential modulates the vibronic feature, offering insights into the role of nonlinear environmental couplings in spectroscopic signatures and exemplifying the success of the extended dissipaton formalism as an exact and efficient method for higher-order bath couplings.
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.
Coherent multidimensional spectroscopy enables detailed investigations of vibronic effects in molecular and solid-state systems. We present explicit analytical expressions for multidimensional nonlinear response functions in the presence of Herzberg-Teller (non-Condon) coupling, within the displaced harmonic oscillator model. The formulation applies to electronic systems with an arbitrary number N of electronic states and to response functions of arbitrary order M in the light-matter interaction. We show that Herzberg-Teller coupling introduces additional oscillatory factors in the time-domain response functions, leading, upon Fourier transformation, to replicas of the Franck-Condon multidimensional spectra shifted by integer multiples of the vibrational frequencies. The present results provide a general analytical framework for the interpretation of non-Condon effects in coherent multidimensional spectroscopies. A Python code that implements the present approach and simulates multidimensional spectra for N-level systems is available on GitHub.
We have obtained Nth-order response functions for a two-level system described by displaced anharmonic potential surfaces coupled to a heat bath. The anharmonicity of the potentials has been taken into account as a perturbation of harmonic potentials. The heat-bath was assumed to be an ensemble of harmonic oscillators. Coupling between the two-level system and the bath was assumed to be bilinear. The calculations were done analytically using the Liouville-space generating functional, which had been obtained by way of the path-integral approach. The response functions have been defined in terms of line-shape functions with these line-shape functions being expressed in terms of the bath spectral density and the temperature. We have carried out model calculations of the first-, third-, and fifth-order optical processes for various parameters of anharmonicity to show that anharmonicity plays a minor role in linear absorption, impulsive pump–probe, and photon echo experiments, but plays a major role, in some cases, in fifth-order two-dimensional resonant spectroscopy which is proposed in this paper.
We develop an operator formulation of statistical quasiparticle theory for open quantum systems with nonlinear system-bath coupling, extending beyond the linear coupling setting of the standard Gaussian influence functional formulation. For Gaussian baths, the statistical quasiparticles, termed dissipatons, are explicitly constructed as collective operators in the thermofield bath space. Our construction provides a microscopic operator basis for the dissipaton algebra and generates hierarchical equations of motion directly from the Liouville equation. Nonlinear interactions are incorporated through a recursive generalized Wick's theorem, enabling automatic generation of hierarchical equations for polynomial couplings. The resulting framework connects explicit bath operator representations to nonperturbative, non-Markovian dynamics and unifies the treatment of linear and nonlinear system-bath interactions.
The quantum Fokker-Planck equation is derived for a system nonlinearly coupled to a harmonic oscillator bath. The system-bath interaction is assumed to be linear in the bath coordinates but quadratic in the system coordinate. The relaxation induced dynamics of a harmonic system are investigated by simulating the higher-order correlation functions of the Raman polarizability and the dipole moment, which represent the nonlinear optical responses of Raman or infrared spectroscopy. The 5th-order Raman response shows that, in addition to the frequency fluctuations induced by the bath, higher-order energy transfer between the system and bath plays a role. The nonlinearity of the system-bath interaction yields also an interesting feature in the 7th-order Raman echo or the 3rd-order infrared photon echo response: The calculations predict a finite signal for the case of a harmonic potential and a linear coordinate dependence of the polarizability or dipole while for linear system-bath coupling this response vanishes completely due to destructive interference of different Liouville space pathways.
This work establishes a strongly correlated system-and-bath dynamics theory, the many-dissipaton density operators formalism. It puts forward a quasi-particle picture for environmental influences. This picture unifies the physical descriptions and algebraic treatments on three distinct classes of quantum environments, electron bath, phonon bath, and two-level spin or exciton bath, as their participating in quantum dissipation processes. Dynamical variables for theoretical description are no longer just the reduced density matrix for system, but remarkably also those for quasi-particles of bath. The present theoretical formalism offers efficient and accurate means for the study of steady-state (nonequilibrium and equilibrium) and real-time dynamical properties of both systems and hybridizing environments. It further provides universal evaluations, exact in principle, on various correlation functions, including even those of environmental degrees of freedom in coupling with systems. Induced environmental dynamics could be reflected directly in experimentally measurable quantities, such as Fano resonances and quantum transport current shot noise statistics.