Quantum Brownian motion with non-Gaussian noises: Fluctuation-Dissipation Relation and nonlinear Langevin equation

Building upon the work of Hu, Paz, and Zhang [1,2] on open quantum systems we consider the quantum Brownian motion (QBM) model with one oscillator (position variable $x$) as the system, {\it nonlinearly} coupled to an environment of $N$ harmonic oscillators (with mass $m_n$, natural frequency $ω_n$, position $q_n$ and momentum $p_n$ variables) in the form $\sum_{n}\left(v_{n1}(x)q_{n}^{k}+v_{n2}(x)p_{n}^{l}\right)$ 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 $v_{n1}=λC_{n1} f(x), v_{n2}(x)=-λ,C_{n2}m_{n}^{-2}ω_{n}^{-2}f(x)$ where $C_{n1,2}$ are the coupling constants with the $n$th oscillator, $f(x)$ is any arbitrary function of $x$, and $λ$ is a dimensionless constant. Employing the closed-time-path formalism the influence action $S_{IF}$ is calculated using a perturbative expansion in $λ$. It is possible to identify the terms in $S_{IF}$ quadratic or higher in $Δ(s)\equiv f(x_{+}(s))-f(x_{-}(s))$ to constitute the noise kernel, while terms linear in $Δ$ to that of the dissipation kernel. The non-Gaussian noise kernel gives rise to non-zero three-point correlation function of the corresponding stochastic force. The pathway presented here should be useful for the exploration of \textit{non-Gaussian properties of systems nonlinearly coupled with their environments}; examples in early universe cosmology and in quantum optomechanics (QOM) are mentioned. A modified fluctuation-dissipation relation (FDR) 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.

💡 Research Summary

The paper extends the standard quantum Brownian motion (QBM) framework to incorporate nonlinear system‑environment couplings of the form (v_{n1}(x)q_n^2+v_{n2}(x)p_n^2). Here the vertex functions are taken as (v_{n1}= \lambda C_{n1} f(x)) and (v_{n2}= -\lambda C_{n2} m_n^{-2}\omega_n^{-2} f(x)), where (f(x)) is an arbitrary system function, (C_{n1,2}) are coupling constants, and (\lambda) is a dimensionless parameter controlling the strength of the nonlinearity. The authors employ the closed‑time‑path (CTP) or Schwinger‑Keldysh formalism to integrate out the bath of (N) harmonic oscillators and obtain the influence functional (S_{\rm IF}