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The projection operator proof that the transition dipole product must be real in Ref. 25 does not apply to four-wave mixing when all four polarizations are different. The time reversal invariance proof applies even in this case.
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35
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note
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The rectangular 2D box transition dipoles are not equal, but this inequality has no effect on the pump-probe signal to first order when the box size simultaneously changes by +ε( -ε) along x(y). In contrast, the level splitting is first order in ε. Use of a common transition dipole magnitude is justified as equivalent to a first-order expansion in the deformation, which is accurate only for small deformations.
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When two components of a single degenerate level are involved, the vibrational wave functions for each component must be the same (or else the degeneracy would be lifted) so that the product of overlap integrals has the form «AB»«BC»«CB»«BA», which is always real and positive.
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If the nondegenerate ground state is not totally symmetric, this statement must be modified: the Jahn-Teller distortion symmetry must match the change in symmetry between the ground state and the two doubly excited states with common symmetry to cause anisotropy decay.
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44
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It is assumed that coupling between the singly and doubly excited states can be neglected
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It is assumed that coupling between the singly and doubly excited states can be neglected.
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