Building upon the work of Hu, Paz, and Zhang [Phys. Rev. D 45, 2843 (1992) and Phys. Rev. D 47, 1576 (1993)] on open quantum systems we consider the quantum Brownian motion model with one oscillator (position variable $x$) as the system, nonlinearly coupled to an environment of $N$ harmonic oscillators (with mass $m_n$, natural frequency ${\omega }_n$, position $q_n$, and momentum $p_n$ variables) in the form ${\sum }_n(v_{n1}(x)q_n^k+v_{n2}(x)p_n^l)$ where $k$, $l$ are integers (the present work only considers the $k=l=2$ cases). The vertex functions $v_{n1},v_{n2}$ are of the form $v_{n1}=\lambda C_{n1}f(x),v_{n2}(x)=-\lambda C_{n2}{m}_n^{-2}{\omega }_n^{-2}f(x)$ where $C_{n1,2}$ are the coupling constants with the $n$th oscillator, $f(x)$ is any arbitrary function of $x$, and $\lambda$ is a dimensionless constant. Employing the closed-time-path formalism the influence action $S_{IF}$ is calculated using a perturbative expansion in $\lambda$. It is possible to identify the terms in $S_{IF}$ quadratic or higher in $\mathrm{\Delta }(s)\equiv f(x_{+}(s))-f(x_{-}(s))$ to constitute the noise kernel, while terms linear in $\mathrm{\Delta }$ to that of the dissipation kernel. The non-Gaussian noise kernel gives rise to nonzero three-point correlation function of the corresponding stochastic force. The pathway presented here should be useful for the exploration of non-Gaussian properties of systems nonlinearly coupled with their environments; examples in early universe cosmology and in quantum optomechanics are mentioned. A fluctuation-dissipation relation is also established, which ensures the consistency of the model and the accuracy of results even at higher perturbative orders. Another result of significance is the derivation of a nonlinear Langevin equation which is expected to be useful for many open quantum system applications.

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• Brownian motion
• Dissipative dynamics
• Fluctuation-dissipation theorem
• Open quantum systems
• Langevin equation