While an axion-clouded black hole (BH) encounters a pulsar (PSR) or has a PSR companion, a gravitational molecule can be formed. In such a system, the axion cloud evolves at the binary hybrid orbitals, as it happens at microscopic level to electron cloud in a chemical molecule. To demonstrate this picture, we develop a semi-analytical formalism using the method of linear combination of atomic orbitals with an adiabatic approximation. An oscillating axion-cloud profile and a perturbed binary rotation, together with unique and novel detection signals, are then predicted. Remarkably, the proposed PSR timing and polarization observables, namely the oscillation of periastron time shift and the birefringence with multiple modulations, correlate in pattern, and thus can be properly combined to strengthen the detection.
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