Interpretation of optical three-dimensional coherent spectroscopy

As an extension to powerful two-dimensional coherent spectroscopy, optical three-dimensional (3D) coherent spectroscopy has been experimentally implemented and found beneficial in studying various systems in physics and chemistry. A critical challenge is how to interpret 3D spectra and extract useful quantitative information, given the richness and complexity of 3D data. Here, we demonstrate how the information of a system's optical response is manifested in 3D spectra by theoretical simulations of a few representative examples including a homogeneous three-level V system, an inhomogeneous three-level V system, and an inhomogeneous three-level ladder system. These examples show that important parameters of the system can be extracted from the spectral pattern, peak positions, amplitudes, and line shapes. The method developed here can be used to analyze 3D spectra of more sophisticated systems which might be a generalization or combination of the three examples, contributing to develop a general approach for the interpretation of 3D spectra.

Figure 1

(a) The box geometry in which three excitation pulses are aligned to three corners of a box and the signal emits at the fourth corner. (b) The time ordering of the excitation pulses in the so-called rephasing excitation sequence.

Figure 2

(a) The energy-level diagram for a three-level V system. (b) Double-sided Feynman diagrams representing eight possible excitation quantum pathways in the experiment.

Figure 3

Simulated 3D spectra of a three-level V system. (a) The amplitude of the spectrum, where the solid (semitransparent) red isosurface has a value of 0.3 (0.1) with the maximum amplitude normalized to 1. (b) The real part of the spectrum, where the red (blue) isosurface has a value of 0.1 ($-0.1$) with the spectrum normalized to a range from $-1$ to 1.

Figure 4

A 3D spectrum with its 2D projections onto three different 2D planes. Various 2D spectra can be retrieved by projecting a 3D spectrum onto a proper plane.

Figure 5

(a) An isolated 3D peak with 2D projections. Slices through the peak center along the directions of (b) the absorption frequency ${\omega }_{\tau }$, (c) the emission frequency ${\omega }_t$, and (d) the mixing frequency ${\omega }_T$.

Figure 6

Simulated 3D spectra of a three-level V system with different inhomogeneous broadening parameters: (a) $R=1$, $\delta {\omega }_{10}=\delta {\omega }_{20}=0.3$ THz; (b) $R=0.7$, $\delta {\omega }_{10}=\delta {\omega }_{20}=0.3$ THz; and (c) $R=1$, $\delta {\omega }_{10}=0.4$ THz, $\delta {\omega }_{20}=0.2$ THz. The solid (semitransparent) red isosurface represents a magnitude of 0.1 (0.07) with the maximum normalized to 1. The corresponding 2D projections on the bottom and back planes are also shown for each case.

Figure 7

(a) A three-level ladder system with a singly excited state $|1\rangle$ and a doubly excited state $|2\rangle$. The dashed line denotes the double energy of the singly excited state. (b) Double-sided Feynman diagrams representing three possible excitation quantum pathways in the experiment.

Figure 8

Simulated 3D spectra of a three-level ladder system with different inhomogeneous broadening parameters: (a) $R=1$, $\delta {\omega }_{10}=\delta {\omega }_{21}=0.3$ THz; (b) $R=0.7$, $\delta {\omega }_{10}=\delta {\omega }_{21}=0.3$ THz; and (c) $R=1$, $\delta {\omega }_{10}=0.4$ THz, $\delta {\omega }_{21}=0.2$ THz. The solid red isosurface represents a magnitude of 0.2 with the maximum normalized to 1. The corresponding 2D projections on the bottom and back planes are also shown for each case.

Figure 9

Four possible types of vertices in double-sided Feynman diagrams.