An analysis is given of the correlation in detection times of two successive radiations emitted spontaneously in a sequential decay. By a resolvent operator method, the time evolution of the coupled particle-field state is derived for a system consisting of a three-level atom coupled to the photon field. Photons A and B, emitted in the first and second stages of the cascade, are detected at distances rA and rB from the atom. When one of the photons is detected, the state vector for the system is assumed to be reduced, by projection onto the observed state. After this measurement, the system again evolves continuously, until the second photon is detected. The probability that a retarded time interval overlineT = (t B - r B/C) - (t A - r A/C) will elapse between the detection of photon A, at time tA, and the detection of photon B, at time tB, is found to be proportional to exp ( -Γ overlineT) if overlineT > 0 and to be zero if overlineT < 0 , where 1/Γ is the mean lifetime of the intermediate state. This result is consistent with a prequantum view of successive photon emission, but is obtained by a method which avoids the conceptual difficulties of a point-particle photon model.