Two-photon matrix element

Our present focus is on correlated electronic structure methods for describing molecular systems interacting with a structured environment where the electronic wavefunction for the molecule is given by a multiconfigurational self-consistent field wavefunction. Using the MCSCF structured environment response method it is possible to determine molecular properties such as (i) frequency-dependent polarizabilities, (ii) excitation and deexcitation energies, (iii) transition moments, (iv) two-photon matrix elements, (v) frequency-dependent first hyperpolarizability tensors, (vi) frequency-dependent polarizabilities of excited states, (vii) frequency-dependent second hyperpolarizabilities (y), (viii) three-photon absorptions, and (ix) two-photon absorption between excited states. 

Since the ground state 0 > and the excited state 1 > are both to be coupled via the continuum by dipole-allowed transitions, it follows that they must have the same parity, and that they are coupled to each other by a two-photon matrix element which is called the two-photon Rabi frequency (for a discussion of the Rabi frequency, see section 9.10). 

The residue analysis of the CRF yields different types of excited-state quantities such as three-photon transition matrix elements (three-photon absorption) [27], the two-photon matrix elements between excited states (the cross section for second-order transitions), and the excited-state polarizability (dynamic second-order property). 

For convenience consider the case of diatomic dissociation. Examination of the selection rules shows that when the transition-dipole operators deg and dLje are parallel to the nuclear axis, the two-photon amplitude is nonzero only if Jj—Jj — 2,0. By contrast, in that case the one-photon matrix element (Eu Jh M, dg Ej, Jj, Mj) is nonzero only if Jj —Jf = 1. Since these two conditions are contradictory, Pql2 E) is zero. Hence coherent control over integral cross sections is not possible using the one- vs. two-photon scenario. 

When the states P1 and P2 are described as linear combinations of CSFs as introduced earlier ( Fi = Zk CiKK), these matrix elements can be expressed in terms of CSF-based matrix elements < K I eri IOl >. The fact that the electric dipole operator is a one-electron operator, in combination with the SC rules, guarantees that only states for which the dominant determinants differ by at most a single spin-orbital (i.e., those which are "singly excited") can be connected via electric dipole transitions through first order (i.e., in a one-photon transition to which the < Fi Ii eri F2 > matrix elements pertain). It is for this reason that light with energy adequate to ionize or excite deep core electrons in atoms or molecules usually causes such ionization or excitation rather than double ionization or excitation of valence-level electrons the latter are two-electron events. 

Here, Ri f and Rf i are the rates (per moleeule) of transitions for the i ==> f and f ==> i transitions respeetively. As noted above, these rates are proportional to the intensity of the light souree (i.e., the photon intensity) at the resonant frequeney and to the square of a matrix element eonneeting the respeetive states. This matrix element square is oti fp in the former ease and otf ip in the latter. Beeause the perturbation operator whose matrix elements are ai f and af i is Hermitian (this is true through all orders of perturbation theory and for all terms in the long-wavelength expansion), these two quantities are eomplex eonjugates of one another, and, henee ai fp = af ip, from whieh it follows that Ri f = Rf i. This means that the state-to-state absorption and stimulated emission rate eoeffieients (i.e., the rate per moleeule undergoing the transition) are identieal. This result is referred to as the prineiple of microscopic reversibility. 

If we restrict ourselves to the case of a hermitian U(ia), the vanishing of this commutator implies that the /S-matrix element between any two states characterized by two different eigenvalues of the (hermitian) operator U(ia) must vanish. Thus, for example, positronium in a triplet 8 state cannot decay into two photons. (Note that since U(it) anticommutes with P, the total momentum of the states under consideration must vanish.) Equation (11-294) when written in the form... 

Let us emphasize that the total contribution of the double transverse exchange is given by the matrix element of the two-photon exchanges between... 

That the observed resonances for m 0 become narrower and more symmetric as the power is increased can be understood in the following way, using the one-photon process as an example. The coupling matrix element of Eq. (15.2) has two... 

The fact that the single photon transitions require so little power prompts us to consider two photon transitions. Consider the Na 16d — 16g transition via the virtual intermediate 16f state which is detuned from the real intermediate 16f state as shown by the inset of Fig. 16.3. If the detuning between the real and virtual intermediate states is A and the matrix elements between the real states arefxx and fi2, the expression analogous to Eq. (16.5) for a two photon transition is... 

Another possible means of distinguishing electronic from vibronic lines is two-photon spectroscopy. The relevant matrix element for a single beam two-photon transition is... 

Figure 6.19. Relaxation rate tor Raman transitions (R) and two-photon emission (T) in a two-level system as a function of temperature. The value of asymmetry is 0, 4, and 32 cm 1 as indicated. The tunneling matrix element is 0.25 cm 1 and the Debye frequency is 80 cm 1. (From Silbey and Trommsdorf [1990].)...

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