$\vec{t}_n \equiv (t_1,\ldots,t_n)$. Explicit expressions for the first few Keldysh-ordered cumulants are provided in Eqs. (\[SEq:Kcml\_2\]) and (\[SEq:Kcml\_3\]) of the SM [@SI]. The Keldysh-ordered second cumulant is simply a symmetrized correlation function, whereas the third cumulant corresponds to suppressing time-orderings where the earliest operator appears in the middle of an expectation value.

For stationary noise, the $k$-th order cumulant $C^{(k)}(\vec{t}_{k})$ only depends on the $k-1$ time separations $\tau_j \equiv t_{j+1}-t_1$, $j= 1,\ldots, k-1 $. We thus define the quantum polyspectra as Fourier transforms of the Keldysh-ordered cumulants with respect to $\{ \tau_j \}$: $$\begin{aligned}
    &S_{n} [\vec{\omega}_{n}] \equiv \!\int_{\mathbb{R}^n}\! d\vec{\tau}_{n} \,
    e^{-i \vec{\omega}_{n}\cdot\vec{\tau}_{n}} C^{(n+1)}(\vec{\tau}_{n}), \quad n \geq 1. 
\label{eq::polyspectra}\end{aligned}$$ For more discussion on properties of classical polyspectra, see Refs. [@Elgar1994; @Viola2016; @Ramon2019]. The zero frequency limit $\omega_j \to 0 $ of $S_{n} [\vec{\omega}_{n}]$ characterize fluctuations in $\hat m=\int_0^t dt'  \hat \xi ( t' ) $ in the long-time limit (so-called full counting statistics (FCS)). This is the typical setting where the Keldysh approach has found great utility, largely for studying electronic current fluctuations. Here we extend the method to study non-classical, non-Gaussian noise [*at non-zero frequencies*]{} (see also Ref. [@PekolaPRB2006] for an application to frequency-dependent current noise).

![image](S2_10.eps){width="180mm"}

[*Photon shot noise 