  The pair coherent state (PCS) is a theoretical extension of the Glauber
coherent state to two harmonic oscillators. It is an interesting class of
non-Gaussian continuous-variable entangled state and is also at the heart of a
promising quantum error correction code: the pair cat code. Here we report an
experimental demonstration of the pair coherent state of microwave photons in
two superconducting cavities. We implement a cross-cavity pair-photon driven
dissipation process, which conserves the photon number difference between
cavities and stabilizes the state to a specific complex amplitude. We further
introduce a technique of quantum subspace tomography, which enables direct
measurements of individual coherence elements of a high-dimensional quantum
state without global tomographic reconstruction. We characterize our two-mode
quantum state with up to 4 photons in each cavity using this subspace
tomography together with direct measurements of the photon number difference
and the joint Wigner function. We identify the spurious cross-Kerr interaction
between the cavities and our dissipative reservoir mode as a prominent
dephasing channel that limits the steady-state coherence in our current scheme.
Our experiment provides a set of reservoir engineering and state
characterization tools to study quantum optics and implement multi-mode bosonic
codes in superconducting circuits.
