Liouville space diagram

Figure 1- Liouville space diagram corresponding to the only term that contributes to the spontaneous light emission from a two-level system within the rotating-wave approximation [Eq. (2.7)]. Here ]g) and e) denote the ground and the excited states, respectively.

Figure 4. TW° ways to describe the time-evolution of the dipole operators single (a) and double (b) Liouville space diagrams.

The Liouville-space diagrams in Fig. 12.3 help to clarify the main physical distinction between Raman scattering and ordinary fluorescence. Both processes require four interactions with a radiation field, and therefore four steps in Liouville space [1]. There are six possible pathways with four steps between the initial state whose population is indicated by a,a at the lower-left comer of Fig. 12.3 A and the final state b,b) at the upper right the three paths shown in Fig. 12.3B-D and their complex conjugates. Ordinary fluorescence occurs by paths B and C, whereas... 

Fig. 12.3 Liouville-space diagrams for spontaneous fluorescence and Raman scattering. (A) Liouville-space pathways connecting an initial state (a), intermediate state (k) and a flnal state (h). (See Sect. 11.1, Figs. 11.1 and 11.4 for an explanation of these diagrams.) (B-D) Three of the six possible paths from atoh with four steps (four interactions with a radiation held). The other three paths are the complex conjugates of the ones shown. All six paths contribute to spontaneous fluorescence Raman scattering involves only path (D) (and its complex conjugate), in which the intermediate state is never populated. (E) A double-sided Feynman diagram for path (D)...

Figure Al.6.18. Liouville space lattice representation in one-to-one correspondence with the diagrams in figure A1.6.17. Interactions of the density matrix with the field from the left (right) is signified by a vertical (liorizontal) step. The advantage to the Liouville lattice representation is that populations are clearly identified as diagonal lattice points, while coherences are off-diagonal points. This allows innnediate identification of the processes subject to population decay processes (adapted from [37]).

Figure 14. Liouville space coupling schemes and their respective double-sided Feynman diagrams for three of the six pathways in Liouville space which contribute to p 2. The complex conjugates are not shown. All pathways proceed only via coherences, created by the interactions with the two fields shown as incoming arrows. Solid curves pertain to e( 11 and dashed curves to r/2T (Reproduced with permission from Ref. 47, Copyright 2005 American Institute of Physics.)...

Fig. 1. Two showing the fifth-order Raman pulse sequence. (A) Definition of fields, (B) Energy level diagram showing one possible Liouville space pathway.

Figure 3. Diagrams showing how to divide the triple integral in (2.7) to get the six terms of (2.10). Domains (a), (b), and (c) correspond to the three Liouville space paths given in Fig. 2 and domains (d), (e), and (/) to the complex-conjugate paths.

In resonant infrared multidimensional spectroscopies the excitation pulses couple directly to the transition dipoles. The lowest order possible technique in noncentrosymmetrical media involves three-pulses, and is, in general, three dimensional (Fig. 1A). Simulating the signal requires calculation of the third-order response function. In a small molecule this can be done by applying the sum-over-states expressions (see Appendix A), taking into account all possible Liouville space pathways described by the Feynman diagrams shown in Fig. IB. The third-order response of coupled anharmonic vibrations depends on the complete set of one- and two-exciton states coupled to thermal bath (18), and the sum-over-states approach rapidly becomes computationally more expensive as the molecule size is increased. 

Figure 5 Double-sided Feynman diagrams representing the two Liouville space pathways contributing to photon echo representing (1) correlations between one-exciton states, and (2) correlations between one- and two-exciton states.

In this appendix we present the sum-over-one- and two-exciton state expressions for the third-order response function. Double-sided Feynman diagrams representing the Liouville space pathways contributing to the four wave mixing in the RWA are given in Fig. IB. The response function is... 

The present study uses a quantum statistical method to show how the time-profile of CARS from molecules in liquids is associated with the inteimolecular dephasing mechanisms. Liouville space Feynman diagrams are used for the development of relevant transitions associated with pmrs of molecules. - The structure of the inteimolecular dephasing constant is clarified to identify the difference between the inteimolecular dephasing constant and intramolecular dephasing constant in Sec.ll. Section III presents the rovibrational interference mechanism. The sub-picosecond decay observed in CARS profile of neat benzene is explained in terms of this mechanism. 

Fig. 18.16 A Liouville space pathway diagram describing the Ujj, -> ffout,out transition. The state a represents either p or s (one can think of two diagrams like this, one for p the other for s, which are connected only at the ap,p o as,s junction.

Fig. 11.5 Double-sided Feynman diagrams for third-order polarization in a two-state system. The symbols have the same meanings as in Fig. 11.2. The times between the interactions with the radiation field are indicated by tj, and T3. Diagrams Ri-R4 correspond to the Liouville-space paths in Fig. 11.4B and to third-order nonlinear response functions/fj to/f4 (Eq. 11.37). However, only stimulated emission that repopulates the ground state is shown fin the fourth interaction with the field. This interaction also could convert the last coherence or its ctnnplex conjugate to...

Fig. 12.9 Liouville-space and double-sided Feynman diagrams for two-photon absOTptirai (A, B) and a representative pathway resulting in ordinary excited-state absorption (C, D). The ground state and the final excited state are labeled a and b. Excited-state absorption requires populating an intermediate state (i), whereas two-photon absorption proceeds entirely through coherences. Both processes also occur by the complex conjugates of the pathways shown. Excited-state absorption also can occur by the pathway shown in Fig. 12.3B and its complex conjugate...

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