  Connecting quantum dots through Andreev bound states in a
semiconductor-superconductor hybrid provides a platform to create a Kitaev
chain. Interestingly, in a double quantum dot, a pair of poor man's Majorana
zero modes can emerge when the system is fine-tuned to a sweet spot, where
superconducting and normal couplings are equal in magnitude. Control of the
Andreev bound states is crucial for achieving this, usually implemented by
varying its chemical potential. In this work, we propose using Andreev bound
states in a short Josephson junction to mediate both types of couplings, with
the ratio tunable by the phase difference across the junction. Now a minimal
Kitaev chain can be easily tuned into the strong coupling regime by varying the
phase and junction asymmetry, even without changing the dot-hybrid coupling
strength. Furthermore, we identify an optimal sweet spot at $\pi$ phase,
enhancing the excitation gap and robustness against phase fluctuations. Our
proposal introduces a new device platform and a new tuning method for realizing
quantum-dot-based Kitaev chains.
