Liouville space pathways

S. Tanaka, V. Chernyak, and S. Mukamel, Time-resolved X-ray spectroscopies nonlinear response functions and Liouville space pathways. Phys. Rev. A 63(6), 063405 (2001). 

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

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... 

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.

Figure 5. Pictoral representation of the Liouville space pathways that contribute to the nonlinear response function [Eqs. (49) and (53)]. Solid lines denote radiative coupling V, horizontal (vertical) lines represent action of V from the right (left). Starting at aa, after three perturbations, the system finds itself along the dashed line. The dotted lines represent the last V, which acts from the left. At the end of four perturbations, the system is in a diagonal state (aa, bb, cc, or dd). The number of three-bond pathways leading to ad, ba, dc, and cb is 1, 1,3, and 3, respectively. Altogether, there are, therefore, eight pathways, which are shown in Fig. 6. In each pathway, each of the three incoming fields acts once.

Figure 6. The eight Liouville space pathways that contribute to the nonlinear response function [Eq. (49) or (53)]. The eight terms in Eqs. (49), (53), (57), (60), and (63) correspond, respectively, to pathways (i)-(viii).

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 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.)...

Figure 7. Pictorial representation of the pathways in Liouville space that contribute to SRF spectra. Solid lines denote radiative coupling V. Horizontal (vertical) lines represent action of V from the right (left). The SRF process is obtained by all pathways that start at aa and end at cc in fourth order (four bonds). There are six pathways that contribute. However, owing to symmetry, we need consider only the three pathways shown in Fig. 8. The other three are obtained by a complex conjugation and permutation of b and d.

Fig. 11.1 Pathways in Liouville space. The circles labeled a,a and b,b represent the diagonal elements of the density matrix (populations) for a two-state system those labeled a,b and b,a represent off-diagonal elements (coherences). Lines represent individual interactions with a radiation field, with vertical lines denoting interactions that change the left-hand (bra) index of the density matrix and horizontal lines those that change the right-hand (ket) index. In the convention used here, the zero-order density matrix (p ° ) is at the lower left, and time increases upwards and to the right downward or leftward steps are not allowed. The coherences in the shaded circles are endpoints of the two one-step pathways [pa,a — pb (B) and pa — Paj, (C)] that contribute to the first-order density matrix (p< )) and the first-order optical polarization (P ). A second interaction with the radiation field (dotted line) can convert a coherence to the excited state (Pbb) Of the ground (paa) state. The pathways in (B) and (C) are described as complex conjugates because one can be generated fi om the other by interchanging the two indices at each step...

Pab> pha) can be formed by a pathway in Liouville space that involves a single interaction with the radiation field paa Pab or Paa Pbcd, a second interaction then is required to generate Take the pathway through p b- Rewriting... 

Fig. 11.4 Generation of the third-order polarization by pathways in Liouville space. (A) An extension of Fig. 11.1 A, with the coherences that contribute to p and denoted by shaded circles. There are eight three-step pathways that start at a/i lower-left corner) and end at one of these ovals. (B) Four of these pathways are shown the other four are the complex conjugates of these. A fourth interaction with the field vertical or horizontal dotted line in A) generates either the excited state bjy) or the ground state (a,a) (not shown). Pathways Ri, Rz, Rs and Rn correspond to the four individual response functions /fj to R4 (Eq. 11.37) that combine with their complex conjugates to make the third-rader nonlinear response function (Eq. 11.36)...

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... 

Kramers and Heisenberg [2], who predicted the phenomenon of Raman scattering several years before Raman discovered it experimentally, advanced a semiclas-sical theory in which they treated the scattering molecule quantum mechanically and the radiation field classically. Dirac [3] soon extended the theory to include quantization of the radiatiOTi field, and Placzec, Albrecht and others explored the selection rules for molecules with various symmetries [4, 5]. A theory of the resonance Raman effect based on vibratiOTial wavepackets was developed by Heller, Mathies, Meyers and their colleagues [6-11]. Mukamel [1, 12] presented a comprehensive theory that considered the nonlinear response functions for pathways in LiouvUle space. Having briefly described the pertinent pathways in Liouville space above, we will first develop the Kramers-Heisenberg-Dirac theory by a second-order perturbation approach, and then turn to the wavepacket picture. 

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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