the slope of $\Delta E$ for $P < 2~P_{{\mathrm{th}}}$ is predominantly caused by an electronic disorder potential, which starts to saturate for $P = 2~P_{{\mathrm{th}}}$, and that for $P \geq 4~P_{{\mathrm{th}}}$ the blueshift is governed by condensate-condensate interactions. Further discussions are presented in Appendix \[sec:experiment\_disorderorigin\] and references therein.

For the comparison between the theoretical model and the experimental data, the parameters of the eGPE  are chosen according to the experiment, see Tab. \[tab:parameters\]. [^2] We note that a quantitative determination of the disorder parameter from experiment is very challenging, cf. discussion in Appendix \[sec:experiment\_disorderorigin\], and we chose $\xi_{{\mathrm{V}}} V_0 \approx 0.15 \: \mu{\mathrm{m}} \, {{\mathrm{m}} {\mathrm{e}} {\mathrm{V}}}$ for simulations.

                          $m$                                  $\tau$            $\xi_{{\mathrm{P}}}$                $P_{{\mathrm{th}}}$                $P/P_{{\mathrm{th}}}$   $d \Delta \! E / d(P \! / \! {P_{{\mathrm{th}}}})$        
  --------------------------------------------------- ------------------------ ------------------------ ---------------------------------------------- ----------------------- ---------------------------------------------------- -- -- --
   $4.4 \cdot 10^{-5}\, {\mathrm{m}}_{{\mathrm{e}}}$   $0.4 \, {\mathrm{ps}}$   $2 \, \mu{\mathrm{m}}$   $79 \, {\mathrm{W}} \! / \! {\mathrm{cm}}^2$       $2,\ldots,20$                    $0.7 \: {\mathrm{meV}}$                      

  : Parameters extracted from experiment and corresponding parameters used for simulations as well as relevant length scales. For definitions see Appendix \[sec:theory\_detailsmodel\].[]{data-label="tab:parameters"}

   $\alpha$   $P/P_{{\mathrm{th}}}$   ${\xi_{{\mathrm{V}}} V_0}/{l_c \, \hbar \gamma_c}$   ${\xi_{{\mathrm{P}}}}/{l_{{\mathrm{c}}}}$   ${\mathcal{L}_{\mathrm{n}}}/ {L_{c}}$   ${\mathcal{L}_{\phi}}/ {L_{c}}$   ${\mathcal{L}_{s}}/ {L_{c}}$
  ---------- ----------------------- ---------------------------------------------------- ------------------------------------------- --------------------------------------- --------------------------------- ------------------------------
     $7$          $5,11,15,20$                             $0.125$                                            