The possible existence of deeply bound $\bar K$-nucleus bound states (kaonic nuclei) has been widely discussed as a consequence of the strongly attractive $\bar K N$ interaction in I = 0 channels.
For the simplest kaonic nucleus, $\bar KNN$, we observed a significant peak structure that can be interpreted as the ``$K^-pp$’’ bound state using the $^3$He$(K^-,N)$ reactions at J-PARC.
To further understand the kaonic nuclei, we have proposed and prepared an experiment to precisely measure the $\bar K NNN$ system using the $^4$He$(K^-,N)$ reactions as a first step towards a comprehensive study of the light kaonic nuclei from ``$\bar K N$'' (=$\Lambda$(1405)) to ``$\bar K NNNN$''.
Through the experiments and detailed theoretical calculations, we will unravel the nature of the kaonic nuclei from the changes in their properties with increasing mass number~$A$.