Order Diagrams

The product operator formalism is normally based on Cartesian coordinates because that simplifies most of the calculations. However, these operators obscure the coherence order p. For example, we found (Eq. 11.80) that products such as IXSX represent both zero and double quantum coherence. The raising and lowering operators I+ and I are more descriptive in that (as we saw in Eq. 2.8) these operators connect states differing by 1 in quantum number, or coherence order. We can associate I+ and I with p = +1 and — 1, respectively, and (as indicated in Eq. 11.79) the coherences that we normally deal with, Ix and Jy, each include both p = 1. Coherences differing in sign contain partially redundant information, but both are needed to obtain properly phased 2D spectra, in much the same way that both real and imaginary parts of a Fourier transform are needed for phasing. 

It is often helpful to summarize in a diagram the changes in coherence level as a function of the discrete periods within the experiment, as illustrated in Fig. 11.2. We begin at equilibrium, where the density matrix (Iz) has only diagonal terms 

FIGURE 11.2 Coherence pathways for a COSY experiment illustrated in a coherence order diagram, (a) All possible pathways resulting from the two 90° pulses. The width of the second pulse has been exaggerated to clarify the diagram. (b) The two pathways (A and B) selected from a) that lead to observable magnetization (p = —1). Adapted from Gunther.64 

Summation of the results of the four repetitions gives a signal for path A but none for path B. This approach can be used for more complex pulse sequences to devise effective phase cycles. 

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Some of lowest-order diagrams for the temperature GF (A3.6) are shown in Fig. A3.1. The dashed and the solid lines represent the GFs of high-frequency and low-frequency vibrations of a planar lattice in the harmonic approximation ... 

Fig. A3.1. Some lowest-order diagrams for the temperature GF (A3.6). The dashed and solid lines correspond to the GF for high-frequency and resonance low-frequency vibrations of a molecular planar lattice in the harmonic approximation (see Eq. (A3.9) and (A3.10)). Each vertex is associated with the factor -y/N, the integration and summation being performed over each vertex coordinates r, from 0 to / , and over all internal wave vectors K. At ptiClK 1, the main contribution is provided by a-type diagrams.184...

Putting one cross in the second order diagram the two third order localization diagrams, can be obtained (Figure 2.)... 

According to QED an electron continuously emits and absorbs virtual photons (see the leading order diagram in Fig. 2.1) and as a result its electric charge is spread over a finite volume instead of being pointlike ... 

Combining all these, results with the value of the trivial first order diagram we find... 

For low order diagrams the determination of ao is quite simple. With a little experience one can easily identify by inspection those symmetry operations leaving the diagram invariant. For example diagram 5.2a ctd = 1 diagrams 5.3a, b Go = 2 diagram 5.3c = 2 2 = 4. (operations ... 

The first order diagram is reproduced in Fig. 7.1a. Taking the continuous chain limit we from Eq. (4.11) find... 

This equation is an approximation in the sense that high-order diagrams are missing. The first neglected diagrams, in fourth order in the exciton-phonon interaction, are of the form... 

The derivation on a Dyson equation on the Keldysh contour is similar to the derivation presented in Section 4 [54]. The difference is in the interactions. In the present derivation, we also include the interaction with irradiation. For any order diagram with respect to the Coulomb interaction, the semiconductor-quantum dot and quantum dot-light interactions retains the topological structure of a graph in the same manner as for noninteracting case where the ordinary zeroth Green s functions are substituted by the renormalized zeroth Green s functions described by Eqs. (121) and (122). 

Many variants of this scheme are possible and can most readily be devised by considering coherence order diagrams. The requirement for rephasing of a coherence is that... 

Figure 4 Second- and third-order diagrams which arise when the Hartree-Fock model is used to obtain a reference function...

Unlike the second-order and third-order energy diagrams, the fourth-order diagrams can involve intermediate states which are singly-excited, doubly-excited, triply-excited, and quadruply-excited with respect to the Hartree-Fock reference function.8 130... 

Figure 5 Fourth-order diagrams which involve singly-excited intermediate states...

Figure 10 Fifth-order diagrams, using the modified Hugenholtz notation of Paldus and Wong...

We note that iterative schemes have been devised151 to evaluate higher-order diagrams involving double-excitations. These schemes can easily be generalized to handle triple-excitations and quadruple-excitations. 

Figure 14 Third-order diagrams which arise when electron-electron interactions are completely neglected...

Figure 17 Second-order diagrams for electron molecule scattering...

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