+
\frac{\dot{p}}{2\sqrt{Kp}} \sqrt{\tanh \! \frac{p}{K}} \, a^{\dagger} |C_- \rangle.
\label{eq-dotCplus}\end{aligned}$$

Substituting Eq. (\[eq-dotCplus\]) into Eq. (\[eq-Hcounter\]), we obtain $$\begin{aligned}
H_{\mathrm{counter}} 
\approx 
-\frac{i}{2}  \left(
\frac{\dot{p}}{2K} \tanh \! \frac{p}{K} \, 
|C_+ \rangle \langle C_+| 
+ 
\frac{\dot{p}}{2\sqrt{Kp}} \sqrt{\tanh \! \frac{p}{K}} \, 
|C_+ \rangle \langle C_-| a
\right)
+
\mbox{H.c.},
\label{eq-Hcounter2}\end{aligned}$$ where H.c. denotes the Hermitian conjugate.

Using $\displaystyle a^2|C_+ \rangle = \frac{p}{K} |C_+ \rangle$ and $\displaystyle \left( |C_+ \rangle \langle C_-| a \right) |C_+ \rangle 
= \sqrt{\frac{p}{K} \tanh \! \frac{p}{K}} \, |C_+ \rangle
= \sqrt{\frac{K}{p} \tanh \! \frac{p}{K}} \, a^2 |C_+ \rangle$, $H_{\mathrm{counter}}$ acting on $|C_+ \rangle$ is approximated as follows: $$\begin{aligned}
H_{\mathrm{counter}} 
\approx 
-\frac{i}{2}  \left(
\frac{\dot{p}}{2p} \tanh \! \frac{p}{K} \, a^2
+ 
\frac{\dot{p}}{2\sqrt{Kp}} \sqrt{\tanh \! \frac{p}{K}} \, 
\sqrt{\frac{K}{p} \tanh \! \frac{p}{K}} \, a^2
\right)
+
\mbox{H.c.}
=
-\frac{i}{2} \frac{\dot{p}}{p} \tanh \! \frac{p}{K} \, a^2
+
\mbox{H.c.}
\label{eq-Hcounter3}\end{aligned}$$ Thus, we obtain Eqs. (11) and (12) in the main text.

To confirm the validity of the above derivation, we performed numerical simulations of a KPO without the output coupling, where the pump amplitude $p(t)$ is increased linearly from 0 to $2K$ at time $10K^{-1}$. The results are summarized in Fig. \[fig-shortcut\] together with the results in the cases without shortcut-to-adiabaticity technique and with the 