If the radius modification is not smoothly performed, sound waves will not be able to adiabatically follow the change of the transverse profile of the initially excited supermode along the propagation direction, exciting other supermodes of the system and we would lose the control of sound propagation. It is important to remark that spatial adiabatic passage processes have the advantage that they are robust in front of variations of the parameter values. In our case, fluctuations in the size of the defects are not critical to perform a spatial adiabatic passage process, only a smooth enough variation of the radii size is needed to adiabatically follow the considered supermode.

Two-supermode case {#sc_twosmode}
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The propagation of sound along two defects for frequencies in which both S and A supermodes exist (light gray shadow in Figs. \[sc\_fig2\](a) and (b), or yellow in the online version) can be approximately described by the coupled-mode equations, as it can be done for TIR waveguides [@chen_foundations_2007] or photonic crystal linear defects.[@chien_tight-binding_2007] For the most general case of two coupled waveguides, coupled-mode equations can be written as: $$i\frac{d}{dz}\left(\begin{array}{c}C_{U}(z)\\C_{L}(z)\end{array}\right)= \frac{1}{2}\left( \begin{array}{cc}
2k_U (z) & -\Omega_{LU} (z)\\
-\Omega_{UL} (z) &2k_L (z)
\end{array} \right)\left(\begin{array}{c}C_{U} (z)\\C_{L} (z)\end{array}\right),
\label{sc_coupled}$$ where $k_{U}$ 