Anharmonic coupling matrix elements

Equation (5), VER involves a higher-order anharmonic coupling matrix element, which gives rise to decay via simultaneous emission of several phonons nftjph (multiphonon emission). In the ACN case, three phonons must be emitted simultaneously via quartic anharmonic coupling (or four phonons via fifth-order coupling, etc.). 

Finally, in t-stilbene there may be large off-diagonal anharmonic coupling matrix elements that can couple widely separated zero-order states. This can lead to less than ideal modulation depths. 

Fig. 5.20. The two experimental Q-branches assigned to Uj — 0 can only be quantitatively reproduced by adding an anharmonic coupling matrix element (/ ). The low-frequency band with more intensity (band b, while la 0.54 0.46) is assumed to be the bright state in the fitting. As increases, the dark state borrows more intensity from the bright state, and the separation between the two shifted band origins increases (as the dotted lines shown between the different panels).

The matrix elements of C13 (and C23) are precisely of the type (5.24) (Amat-Nielsen couplings), except that they include anharmonic cutoff effects. This can be seen in Table 5.1. 

In this last expression we take into account that the excitation of n quanta of the j mode will be an anharmonic process, and hence the transferred energy may not be simply proportional to the mode frequency hflj. The anharmonic coupling becomes readily apparent if we simplify the matrix element in Eq. (11) ... 

There is a formal similarity in the mathematics used to describe vibrational transitions pumped by a resonant radiation field [148] and vibrational transitions pumped by phonons in a crystal lattice. In the lowest-order approximations, the radiation field and the vibrational transition are coupled by a transition dipole matrix element that is a linear function of a coordinate. The transition dipole describes charge displacement that occurs during the transition. Some of the cubic anharmonic coupling terms described by Eq. (10) result in a similar coupling between vibrational transitions and a phonon coordinate. These generally have the form / vibVph, so that the energy of the vibration with normal coordinate /vib is linearly proportional to the phonon coordinate /ph. Thus either an incoherent photon field or an incoherent phonon field can result in incoherent... 

These matrix elements are equivalent to those of Eq. (3.124), apart from anharmonic contributions of the order of v/N. So we see that the extended Majorana operator has the required effect on the states involved in the resonance mechanism. At the same time > SB does not preserve the coupled 65 (2) symmetry in other words, + Vg is not conserved anymore. Consequently, the block-diagonal structure of the Hamiltonian operator is destroyed and the numerical diagonalization of... 

Besides Fermi interactions, another class of anharmonic couplings among vibrational levels exist and they are often referred to as Darling-Dennison interaction [77]. They typically arise in conjunction with a normal-mode molecular picture, since the associated matrix elements can be written as... 

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