# A Unified Stochastic Formulation of Dissipative Quantum Dynamics. II. Beyond Linear Response of Spin Baths

## TL;DR

This paper extends the generalized hierarchical equation of motion to include higher order effects in spin bath models, revealing when linear response approximations are valid or break down due to non-Gaussian and non-Markovian dynamics.

## Contribution

It introduces a systematic method to incorporate anharmonic effects in spin bath decoherence models, analyzing the validity of linear response approximations in different physical scenarios.

## Key findings

- Linear response holds for discretized baths with large N_B due to suppression of non-Gaussian effects.
- Physical spin baths often violate linear response assumptions due to direct spin-spin interactions.
- Symmetries in spin Hamiltonians can lead to non-Markovian dynamics beyond linear response.

## Abstract

We use the "generalized hierarchical equation of motion" proposed in Paper I to study decoherence in a system coupled to a spin bath. The present methodology allows a systematic incorporation of higher order anharmonic effects of the bath in dynamical calculations. We investigate the leading order corrections to the linear response approximations for spin bath models. Two types of spin-based environments are considered: (1) a bath of spins discretized from a continuous spectral density and (2) a bath of physical spins such as nuclear or electron spins. The main difference resides with how the bath frequency and the system-bath coupling parameters are chosen to represent an environment. When discretized from a continuous spectral density, the system-bath coupling typically scales as $\sim 1/\sqrt{N_B}$ where $N_B$ is the number of bath spins. This scaling suppresses the non-Gaussian characteristics of the spin bath and justify the linear response approximations in the thermodynamic limit. For the physical spin bath models, system-bath couplings are directly deduced from spin-spin interactions with no reason to obey the $1/\sqrt{N_B}$ scaling. It is not always possible to justify the linear response approximations. Furthermore, if the spin-spin Hamiltonian and/or the bath parameters are highly symmetrical, these additional constraints generate non-Markovian and persistent dynamics that is beyond the linear response treatments.