Office of Science, Basic Energy Sciences.

**Supplementary Material: Spectral characterization of non-Gaussian quantum noise**

Distinguishing fluctuations from response properties
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As discussed in the main text, a general $n$-point quantum correlation function describes both the intrinsic fluctuation properties of the system of interest (i.e. quantities that play the role of classical noise), as well as the response properties of the system to external applied fields. The situation is very clear at second order, where the product ${\hat{\xi}}(t) {\hat{\xi}}(t')$ can be decomposed as the sum of a commutator and an anti-commutator. The commutator determines the retarded Green function $$G^R(t) \equiv -i \Theta(t) \left \langle [ {\hat{\xi}}(t), {\hat{\xi}}(0) ] \right \rangle.$$ This describes how the average value $\langle {\hat{\xi}}(t) \rangle$ changes to first order in response to an external perturbing field $V(t)$ entering the Hamiltonian as $$\hat{H}_{\rm ext}(t) = V(t) {\hat{\xi}}.$$ The relevant Kubo formula is: $$\delta \langle {\hat{\xi}}(t) \rangle = 
        \int_{-\infty}^{\infty} dt' G^R(t-t') V(t').$$ In contrast, the anti-commutator describes the symmetrized noise spectral density: $$S[\omega] \equiv \frac{1}{2} \int dt e^{i \omega t} \langle \{ {\hat{\xi}}(t), {\hat{\xi}}(0) \} \rangle .$$ As has been discussed in many places (see e.g. Ref. [@ClerkRMP]), this spectral density plays the role of a classical noise spectral 