\section{Symmetric pulse}

Figure~\ref{fig_sym} shows the evolution of the $T^8_\text{comp}$ and AFM order parameters in a photo-doped system with $U=8$, $\Omega=3.5$, $a=1$ for two types of excitations: (i) pulse applied only to orbital 3 (solid lines, same as in Fig.~\ref{fig_afm_u8} of the main text), and (ii) pulse applied to all three orbitals (dashed lines). While $T^8_\text{comp}$ is more strongly suppressed by the second pulse, because of the larger density of doublons and holons produced, the result remains qualitatively the same. The photo-doping rapidly melts the AMF order, while the composite order survives for much longer times, resulting in a metastable I,PM,C state. 

Since orbitals $\alpha=1,2$ have little spectral weight near $\omega = \pm \tfrac{\Omega}{2}$ (see Fig.~\ref{fig_spectra} of the main text), most doublons and holons are produced in orbital $\alpha=3$, even if the pulse is applied to all orbitals. 

\begin{figure}[b]
\begin{center}
\includegraphics[angle=0, width=0.49\columnwidth]{plot_all_w35.pdf}
\hfill
\includegraphics[angle=0, width=0.49\columnwidth]{plot_all_w35_log.pdf}
\caption{Time evolution of the $T^8_\text{comp}$ and AMF orders after a photo-excitation, plotted on a linear (left) and logarithmic (right) scale. Solid lines show simulations for a pulse applied to orbital $\alpha=3$ and dashed lines for a pulse applied to all three orbitals. The parameters are the same as in the main text ($U=8$, $\Omega=3.5$, $a=1$). 
}
\label{fig_sym}
\end{center}
\end{figure}

\section{Local density matrix}

It is also instructive to explore the local density matrix and local entropy of the photo-doped system \cite{Werner2020}. We have measured $\rho_{\mu\nu}=|\mu\rangle\langle\nu|$ in the space of half-filled states with one empty, one singly-occupied and one doubly-occupied orbital ($12\times 12$ matrix). The left panel of Fig.~\ref{fig_entropy} plots the evolution of the eigenvalues $e_1,\ldots,e_{12}$ of $\rho$ after the photo-doping pulse and the right panel plots the local entropy defined as $S_\text{loc}=-e_1\ln _1-e_2\ln _2\ldots -e_{12}\ln e_{12}$.   The corresponding results for the thermalized state at $T=1.42$ are indicated by the red dashed lines. 

In the initial I,AFM,C state, a single eigenvalue close to 1 dominates, because of the unique dominant $|dh\rangle_{12}\otimes|s\rangle_{3}$ state, and this results in a low $S_\text{loc}$. In the thermalized state without symmetry breaking, there are two groups of six eigenvalues (dashed lines), associated with ``bonding" and ``anti-bonding" combinations of doublons and holons. The hidden state has a distinctly different eigenvalue structure. Here, the largest eigenvalue becomes doubly degenerate, because of the spin symmetry in $|s\rangle_\gamma$, while the orbital symmetry breaking results in a gap to the eigenvalues of the four sub-dominant states and a reduced $S_\text{loc}$.

\section{Susceptibility for orbital symmetry breaking}

To study the susceptibility for orbital symmetry breaking, we apply slightly different interactions to the three orbitals: $U-dU$, $U$, $U+dU$ for $\alpha=1,2,3$. While the low-temperature PM insulating state has a very large (but non-diverging) susceptibility (see line with circles in Fig.~\ref{fig_susc}) this is not the case at the effective temperature $T_\text{eff}=0.38$ of the photo-doped state (line with crosses). This suggests that the susceptibility for orbital symmetry breaking does not play a dominant role in the stabilization of the hidden state. 

\begin{figure}[t]
\begin{center}
\includegraphics[angle=0, width=0.49\columnwidth]{plot_eigenvalues_b30m0_u8_pumpa1_w35_afm_o3_h001.pdf}
\hfill
\includegraphics[angle=0, width=0.49\columnwidth]{plot_rho_b30m0_u8_pumpa1_w35_afm_o3_h001.pdf}
\caption{Photo-doped I,AFM,C state with $U=8$ and initial $T=0.033$ ($\Omega=3.5$, $a=1$). The left panel shows the evolution of the eigenvalues $e_1$,\dots,$e_{12}$ and the right panel the evolution of the local entropy $S_\text{loc}$. Red dashed lines indicate the values reached in the thermalized state with $T=1.42$. In the right panel, we also show the analogous results for simulations starting from the I,PM state (blue lines, thermalized state with $T=0.92$). 
}
\label{fig_entropy}
\end{center}
\end{figure}

\begin{figure}[b]
\begin{center}
\includegraphics[angle=0, width=0.5\columnwidth]{plot_orb_susc.pdf}
\caption{$T^8_\text{comp}$ in PM equilibrium states at $T=0.033$ (circles) and $T=T_\text{eff}=0.38$ (crosses) for interactions $U-dU,U+dU,U$ in orbitals $\alpha=1,2,3$ ($U=8$). The dashed line shows the value in the initial I,AFM,C state. 
}
\label{fig_susc}
\end{center}
\end{figure}


\end{document}



