  Quantum dots exhibit a variety of strongly correlated effects, e.g., when
tuned to emulate localized magnetic impurities that form a Kondo singlet with
their surrounding environment. Interestingly, in double-dots setups, the
magnetic impurities couple to each other by direct
Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction, which wins over the Kondo
physics. In this work, we investigate a double-dot device where the dots are
coupled via off-resonant ballistic whispering gallery modes, dubbed electronic
cavity modes. Within this cavity-double-dot system, we study, using variational
matrix product state techniques, the competition between Kondo formation and
the coherent RKKY-like interaction that the cavity facilitates. Specifically,
we find that (i) Kondo can win and form on each dot individually, or (ii) the
cavity can win and mediate between the two dots either a singlet or a novel
nonlocal Kondo-like effect phase, which we call "cat Kondo". We systematically
study the quantum phase transitions between the different many-body states. Our
discoveries lay the foundation for the experimental observation of
unconventional nonlocal magnetic impurities.
