q}(t)$. Perturbation theory in $V(t)$ thus necessarily introduces powers of the quantum field. For example, at second order we have:

-   The average $\overline{\xi_{\rm cl}(t) \xi_{\rm q}(0)}$ is directly proportional to the retarded Green function $G^R(t)$, and thus describes linear response to the external field.

-   The average $\overline{\xi_{\rm cl}(t) \xi_{\rm cl}(0)}$ is proportional to $\langle \{ {\hat{\xi}}(t), {\hat{\xi}}(0) \} \rangle$ and thus determines the usual symmetrized noise spectral density.

The same decomposition applies at higher orders. Consider third order correlators. The average of three classical fields $\overline{\xi_{\rm cl}(t_1) \xi_{\rm cl}(t_2) \xi_{\rm cl}(0)}$ is precisely the Keldysh ordered correlator discussed in the main text; it cannot be associated with a response coefficient. The remaining non-zero correlators describe different kinds of response:

-   The average $\overline{\xi_{\rm cl}(t) \xi_{\rm q}(t') \xi_{\rm q}(t'')}$ represents a second-order Kubo response coefficient. It determines to second order how $\langle {\hat{\xi}}(t) \rangle$ is modified by $\hat{H}_{\rm ext}(t')$ at earlier times (i.e. how it depends on $V(t')$ and $V(t'')$).

-   The average $\overline{\xi_{\rm cl}(t) \xi_{\rm cl}(t') \xi_{\rm q}(t'')}$ describes a first order noise-susceptibility [@Reulet2007]. It determines how the symmetrized correlator $\langle \{ {\hat{\xi}}(t), {\hat{\xi}}(t') \} \rangle$ is modified to first order by $\hat{H}_{\rm ext}(t'')$

The arguments sketched here 