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_nleft\(v_n1(x\)q_n^k+v_n2(x)p_n^lright) 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_n2m_n^-2ω_n^-2f(x) where C_n1,2 are the coupling constants with the nth 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 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.