  We study simultaneous parametric oscillations in a system composed of two
distributed-element-circuit Josephson parametric oscillators in the
single-photon Kerr regime coupled via a static capacitance. The energy of the
system is described by a two-bit Ising Hamiltonian with an effective coupling
whose amplitude and sign depend on the relative phase between parametric pumps.
We demonstrate that the binary phases of the parametric oscillations are
correlated with each other, and that the parity and strength of the correlation
can be controlled by adjusting the pump phase. The observed correlation is
reproduced in our simulation taking pure dephasing into account. The present
result demonstrates the tunability of the Hamiltonian parameters by the phase
of external microwave, which can be used in the Ising machine hardware composed
of the KPO network.
