limit ${\bar n}_{ \rm{th} }=0$, one finds that real part of the QBS only becomes negative above a critical value of the detuning $|\delta|$. Moreover, the initial onset of negativity occurs at $\omega_1 = \omega_2 = 0$. In the large-detuning regime $|\delta| \gg \gamma$, the negative region of the QBS is peaked near a polygon whose shape is defined by the resonance conditions $\omega_j = \pm \delta$ ($j=1,2,3$).

![Frequency dependence of the imaginary parts of the photon-shot noise bispectrum $\mathrm{Im} S [ {\omega}_{1} ,{\omega}_{2} ]$ in the extreme quantum limit ${\bar n}_{ \rm{th} }=0$ at different detunings. Parameters: (a) $\delta / \gamma = 0$, (b) $\delta / \gamma = 1$. Note that for ${\bar n}_{ \rm{th} }=0$ the full quantum bispectrum coincides with the drive-dependent contribution $S_{\rm dr}   [ {\omega}_{1},{\omega}_{2}  ]$.[]{data-label="fig:ImS2"}](ImS2.eps){width="85mm"}

![Time-dependent Keldysh-ordered photon-shot noise third cumulant $C_{\mathrm{dr} }^{(3)}(| t|,|t|)$ in the quantum limit ${\bar n}_{ \rm{th} } =0$ for $t>0$ (red solid lines) and $t<0$ (orange dashed lines). Difference between curves highlights asymmetry under time reversal $t \to -t$. In contrast, the thin blue curves correspond to the same correlator $C_{\mathrm{dr} }^{(3)}(| t|,|t|)$ in the classical limit normalized by thermal photon number, which is symmetric. All correlation functions are 