(see e.g. [@Kubo1966; @Agarwal1973; @Carmichael1976; @Tomita1973; @Tomita1974; @Carmichael2002]); the QBS provides yet another tool for exploring this physics. In the SM, we discuss another related quantum system which exhibits an apparent breaking of detailed balance, namely a cavity driven by squeezed noise [@SI].

For a heuristic understanding of this symmetry breaking, we consider a simpler object, the temporal (Keldysh-ordered) third cumulant $C^{(3)}(\tau_1,\tau_2)$ at $\tau_1 = \tau_2 = t$. The non-zero imaginary QBS implies that this correlator differs for $t$ and $-t$ (see Fig. \[fig:timeskn\]). Using the definition of Keldysh ordering in Eqs. (\[Eq:MGFK\]) and (\[eq:ncdef\]) we find: $$\begin{aligned}
     &  \langle \delta \hat n(0) \delta \hat n(t ) \delta \hat n(t )\rangle_\mathcal{K} =  
    \label{eq:KeldyshOrderedSkewness}  \\
    & \frac{1 }{2} \langle \{\delta \hat n(0) ,[\delta \hat n(t )]^2 \} \rangle
           -\frac{ \Theta(-t)}{4}   \langle [ \delta \hat n(t), [ \delta \hat n(t ),\delta \hat n(0)] ] \rangle
  \nonumber ,\end{aligned}$$ where $\delta \hat{n}(t) = \hat{n}(t) - \langle \hat{n}(t) \rangle$, and $\Theta(t)$ is the Heavisde step-function. One finds that any imaginary quantum correction $ \mathrm{Im}  \tilde{S}_q[\omega_1,\omega_2]$ is entirely due to the second term on the RHS; it is thus completely responsible for the the lack of time-symmetry. What does this mean physically? As we have emphasized, 