normalized by in-cavity drive photon number ${\bar n}_{ \rm{dr} }$. Detunings are: (a) $\delta=0$, and (b) $\delta/\gamma=5$.[]{data-label="fig:timeskn"}](n3K.eps){width="85mm"}

[*Imaginary bispectrum and violations of detailed balance–* ]{} We now turn to another striking feature of the photon shot noise QBS: while in the classical, high-temperature limit it is always real, the quantum correction ${\tilde S}_{ \mathrm{q}}[ {\omega}_{1},{\omega}_{2}]$ has a non-zero imaginary part (see Fig. \[fig:ImS2\]). We stress that this non-trivial imaginary bispectrum can [*only*]{} be probed at finite frequency: by its very definition in Eq. , the imaginary part of the QBS must vanish if either $ \omega_1=0 $ or $\omega_2 =0$. The non-zero imaginary QBS is directly related to the basic symmetries of our quantum noise process, in particular the violation of higher-order Onsager reciprocity relations [@Onsager1931; @Semerjian2004; @Sinitsyn2016]. If a temporal cumulant $C^{(n+1)}(\vec{\tau}_{n})$ is invariant under $\vec{\tau}_{n} \to -\vec{\tau}_{n}$ (i.e. exhibits Onsager reciprocity), then the corresponding polyspectrum must be real [@Brillinger1967]. Further, a classical Markov process obeying detailed balance always respects this symmetry. In our system quantum corrections (as described by ${\tilde S}_{ \mathrm{q}}[ {\omega}_{1},{\omega}_{2}]$) cause a breaking of this symmetry and hence of detailed balance. There is a long history of studying detailed balance in non-equilibrium driven-dissipative quantum systems 