Relationship between the overall permutation and time-reversal symmetries in the nonlinear optical hyperpolarizabilities

The overall-permutation-symmetry property of the nonlinear optical hyperpolarizabilities in a nondissipative medium is examined to find that the time-reversal symmetry is already assumed in the Manley-Rowe power relation, a basic relation adopted in the derivation of the overall permutation symmetry. This indicates that the overall permutation symmetry is a consequence of the time-reversal symmetry. The quantum-mechanical expression of hyperpolarizability is shown to satisfy both the overall permutation and the time-reversal symmetry explicitly. From the extended symmetry relation, it is predicted that second-harmonic generation and difference-frequency generation, two different nonlinear optical processes, are governed by the same parameter.