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Journal of the Chinese Chemical Society

Diagram Technique for Nonlinear Optical Spectroscopy in the Fast Electronic Dephasing Limit

Boris D. Fainberg

Published Aug 1, 2000• 4 citations
Publisher

Abstract

Abstract We show that applying the double‐sided Feynman diagrams to systems with fast electronic dephasing opens up new possibilities for using this technique in resonance nonlinear optical spectroscopy. The novel potentials are including non‐perturbative system‐bath interactions (non‐Markovian relaxation) and the summation of diagrams. The last procedure enables us to obtain new equations for a nonperturbative description of the light‐matter interaction.

DOI
10.1002/jccs.200000079

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Abstract We show that applying the double‐sided Feynman diagrams to systems with fast electronic dephasing opens up new possibilities for using this technique in resonance nonlinear optical spectroscopy. The novel potentials are including non‐perturbative system‐bath interactions (non‐Markovian relaxation) and the summation of diagrams. The last procedure enables us to obtain new equations for a nonperturbative description of the light‐matter interaction.
The analysis of nonlinear spectroscopy, widely used to study the dynamics and structures of condensed-phase matter, typically employs a perturbative approach noticing the weak interaction between the laser and the matter of interest. However, such perturbative approach is no longer applicable once the interaction between the laser and the matter is strong. We adapt the method of quantum dynamical evolution into the calculation of signal and present the response formalism of the nonlinear spectroscopy in a nonperturbative approach. In this new approach, we demonstrate that in addition to the third-order term in the perturbative method, the higher-order terms have essential contributions to the nonlinear signal of the two-pulse and three-pulse photon echo (2PPE and 3PPE). The detailed calculations are demonstrated with the example of a three-level V-type system, which is widely used in the studies of quantum optics. We consider the effect of the environment via a pure dephasing mechanism with both the localized modes of each molecule and the shared modes between molecules.
A diagrammatic density matrix technique is developed and applied to study nonlinear optical processes involving responses from weak probe fields interacting with a system under strong excitation. These diagrams consist of lowest-order double-Feynman diagrams which represent the density matrix perturbation terms describing the system response to the weak probe fields1,2 combined with dressed propagators which represent the density matrix elements of the system under strong excitation. The dressed propagators are described by a set of linear equations which, in most cases, may be derived in inspection. The methodology is similar to the bare-atom approach.2 The advantages of using this diagrammatic technique are that (1) it allows one to select and express analytically the dominant and appropriate density matrix elements (diagrams and dressed propagators); (2) analytical expressions may be constructed directly from the diagrams and thus lengthy calculations needed to obtain closed-form solutions are greatly reduced; (3) it provides a physical interpretation for the density matrix solutions and nonlinear optical processes; and (4) it may be generalized to more complicated, multilevel systems. We demonstrate the application of this technique to describe gain saturation, Stark splitting, and Raman scattering.
We show that dephasing-induced nonlinear-optical effects whose amplitudes are proportional to trinomials of dephasing rates, ${\ensuremath{\Gamma}}_{\mathrm{ij}}$+${\ensuremath{\Gamma}}_{\mathrm{jk}}$-${\ensuremath{\Gamma}}_{\mathrm{ik}}$, exist in all nonlinear orders of perturbation theory. To do this, we derive a new form for the perturbation expansion of the susceptibility, equivalent to the currently used expansion, but which writes each order as a sum of (1) terms due to single-sided diagrams, and (2) correction terms, all of which are proportional to trinomials of dephasing rates. The proof of this result utilizes a diagrammatic approach involving a new type of double-sided diagram for the latter terms. These diagrams immediately yield the terms proportional to trinomials of dephasing rates in all orders without the need for algebraic manipulation. This approach also reveals some general properties of the Nth-order susceptibility.
Diagrammatic methods for the calculation of the nonlinear response of a system that is dressed by resonant external fields are developed. The double-sided diagrams used earlier by Prior [IEEE J. Quantum Electron. QE-20, 37 (1984)] for the calculation of nonlinear susceptibilities in the presence of weak fields are generalized. Additional types of double-sided diagrams are given that incorporate explicitly (a) the relaxation-induced population transfers among the dressed levels and (b) the effects of the nonvanishing of the diagonal elements of the dipole operator in the dressed-state basis. The rules for the diagrams are worked out from the analytical structure of the nonlinear response. Applications of the diagrammatic techniques to several intense-field phenomena are discussed.
The analysis of nonlinear spectroscopy, widely used to study the dynamics and structures of condensed-phase matter, typically employs a perturbative approach noticing the weak interaction between the laser and the matter of interest. However, such a perturbative approach is no longer applicable once the interaction between the laser and the matter is strong. We adapt the method of quantum dynamical evolution into the calculation of signal and present the response formalism of the nonlinear spectroscopy in a nonperturbative approach. In this approach, we demonstrate that, in addition to the third-order term in the perturbative method, the higher-order terms have essential contributions to the nonlinear signal of the two-pulse and three-pulse photon echo (2PPE and 3PPE). The detailed calculations are demonstrated with the example of a three-level V-type system, which is widely used in the studies of quantum optics. We consider the effect of the environment via a pure dephasing mechanism with both the localized modes of each molecule and the shared modes between molecules.
The nonperturbative approach to the calculation of nonlinear optical spectra of Seidner et al. [J. Chem. Phys. 103, 3998 (1995)] is extended to describe four-wave mixing experiments. The system-field interaction is treated nonperturbatively in the semiclassical dipole approximation, enabling a calculation of third order nonlinear spectroscopic signals directly from molecular dynamics and an efficient modeling of multilevel systems exhibiting relaxation and transfer phenomena. The method, coupled with the treatment of dynamics within the Bloch model, is illustrated by calculations of the two-dimensional three-pulse photon echo spectra of a simple model system-a two-electronic-level molecule. The nonperturbative calculations reproduce well-known results obtained by perturbative methods. Technical limitations of the nonperturbative approach in dealing with a dynamic inhomogeneity are discussed, and possible solutions are suggested. An application of the approach to an excitonically coupled dimer system with emphasis on the manifestation of complex exciton dynamics in two-dimensional optical spectra is presented in paper II Pisliakov et al. [J. Chem. Phys. 124, 234505 (2006), following paper].