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ble Liouville-space pathways in the systems [33]. Two pathways contribute to each peak in the S$_{I}$ ( ω$_{τ}$ , T, ω$_{t}$ ) spectra; whereas for S$_{II}$ ( ω$_{τ}$ , T, ω$_{t}$ ) three pathways con- tribute to each diagonal peak (NA or NB) and only one pathway to each cross peak (NC or ND). Generally, the peak strengths are proportional to the sum over partic- ipating diagrams while the strength of individual dia- gram is proportional to ∏ µ$_{i}$ , where index i runs from 1 to 4 and µ$_{i}$ is the transition dipole moment for each excitation step in each double-sided Feynman diagram. The ratio between the transition dipole moment for D$_{2}$ 

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t ransition, µ$_{D}$$_{2}$ , to the dipole moment for D$_{1}$ transition , µ$_{D}$$_{1}$ , is appro ximately √$_{2 : 1. Assuming each quantum}$ p athway has t he same contribution to the peak strength, t he peak stre ngth ratio RA:RC:RD:RB is proportional t o 2 µ 4 D 1 : 2 µ 2 D $_{1}$µ 2 D 2 : 2 µ 2 D $_{2}$µ 2 D 1 : 2 µ 4 D $_{2}$, or relative val- ues of 1:2:2:4. Similarly, the peak strength ratio in the S$_{II}$ ( ω$_{τ}$ , T, ω$_{t}$ ) measurement NA:NC:ND:NB is equal to 2:1:1:5, which agrees well with the experimental result. 

The interac tion between light and a closed two-level system can b e described by the optical Bloch equations [34]: 

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where ρ$_{00}$ an d ρ$_{11}$ are density matrix elements for the ground and ex cited state population respectively, and ρ$_{01}$ is the off-diag onal term of the density matrix. γ sp and γ ph are the population relaxation and dephasing rates respectively, µ$_{01}$ is the transition dipole moment, ω$_{01}$ is the resonant transition frequency and E is the electric field amplitude. We extended the equations to a multi- level system and numerically solved them [34, 35]. In the calculation, γ ph is the total linewidth due to resonance and collision broadening. The induced polarization of the system is P = N T r ( µρ ). Then the electric field of the emitted signal is determined from the third order polarization [34]: 

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where L is the sample thickness, n ( ω$_{t}$ ) is the index of refraction of the sample, and c is the speed of light in vacuum. S$_{I}$ ( ω$_{τ}$ , T, ω$_{t}$ ) and S$_{II}$ ( ω$_{τ}$ , T, ω$_{t}$ ) 2DFT spectra 

