\delta \hat \xi (t_{\pi _1}) \delta \hat \xi (t_{\pi _2}) \delta \hat \xi (t_{\pi _3})  \rangle,$$ where $\mathcal{P}_3$ denotes the set of all possible permutations of $(123)$ indices. Such ordering is given by an average over all permutations of the three displaced operators $\delta \hat \xi (t_j)$, except for the terms where the earliest time appears in the middle position (as implied by the step functions), in agreement with expansion of the operator in Eq.  in powers of coupling $F(t')$. A similar expression of Keldysh-ordered third cumulant has also been derived for current operators in Ref. [@PekolaPRB2006].

Phase space method for computing Keldysh-ordered cumulants of a driven damped cavity
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In this section, we outline the phase space method to calculate Keldysh-ordered cumulants. However, we remark that once we have defined the unique Keldysh ordering for each higher cumulant using Eqs.  and , standard techniques for computing multi-point correlation functions (e.g. Langevin equations of motion, and quantum regression theorem) work equally well for the Keldysh-ordered cumulants.

In the phase space method, we need to solve the time evolution of qubit coherence $\hat \rho _{ \uparrow  \downarrow } (t) \equiv \langle\uparrow \! |\hat \rho (t)|\! \downarrow \rangle$ $$\dot {\hat \rho}_{ \uparrow  