Liouville pathways

Figure 2. The Liouville pathway diagrams for third-order phase-matched signals (i.e., ks = signal wavevector = polarization wavevector) into the direction —k +k2+ki. The lower- and uppercase k and K symbols represent one- and two-quantum states of a molecule, respectively.

In this section, we present a rigorous route based on QCL dynamics for deriving quantum lassical expressions for linear and third-order ORFs, which reflect nonequilibrium dynamics on multiple adiabatic surfaces as opposed to equilibrium ground state dynamics.As will be shown, these ORFs consist of contributions from several Liouville pathways that differ with respect to the surfaces on which the dynamics of the photo-inactive DOF takes place between light-matter interactions. 

Figure 10.4 Averaged fundamental transition frequency, coio, along the Liouville pathways 00 10 00 10 (solid line) and 00 10 11 -> 10...

Figure 6.3 Fe5Timan diagrams representing the possible Liouville pathways detected in the kg —ki + 2 + 3 direction, including (a) SE, (b) GSB and (c) ESA. The top row shows rephasing diagrams, where arrives before 2-The bottom row shows non-rephasing diagrams.

Figure 6.24 Simulated 2D spectra for a type-II system, (a) Simulated 2D with small broadening. Feymann diagrams depicting the Liouville pathway that gives rise to each signal are presented, (b) Comparison between experimental 2D spectra (left panel) and the simulated spectra (right panel) with more realistic broadening parameters.

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). 

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.

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).

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... 

---