Amplitude Real Ab sorption Frequency, ω τ /2 π (THz) Emission Frequency, ω$_{t}$ /2 π (THz) 3 ❰ GLYPH<a207> GLYPH<a208> GLYPH<a209> GLYPH<a210> GLYPH<a211> GLYPH<a212> GLYPH<a213> GLYPH<a214>GLYPH<a215>GLYPH<a216> GLYPH<a217> GLYPH<a218> GLYPH<a219> GLYPH<a220> GLYPH<a221> GLYPH<a222> GLYPH<a223>GLYPH<a224>GLYPH<a225> GLYPH<a226> GLYPH<a227> GLYPH<a228> GLYPH<a229> GLYPH<a230> GLYPH<a231> GLYPH<a232>GLYPH<a233> GLYPH<a234> GLYPH<a235> GLYPH<a236>GLYPH<a237>GLYPH<a238> GLYPH<a239> GLYPH<a240>GLYPH<a241> GLYPH<a242> GLYPH<a243> GLYPH<a244> GLYPH<a245> GLYPH<a246> GLYPH<a247> GLYPH<a248>GLYPH<a249>GLYPH<a250> GLYPH<a251> GLYPH<a252> GLYPH<a253> GLYPH<a254> GLYPH<a255> ✂ ☎ ✄ GLYPH<a0> ✁✆ ✝ ✞ ✟ ✠ ✡ ☛ ☞✌✍ ✎ ✏✑ ✒ ✓ ✔ ✕ ✖ ✗ ✘✙✚ ✛ ✜ ✢ ✣ ✤ ✥ ✦ ✧ ★ ✩ ✪ ✫ ✬✭✮ ✯ ✰✱ ✲ ✶ ✷ ✸✹✺ ✻ ✼ ✽ ✾ ✿ ❀ ❁ ❂ ❃ ❄ ❅ ❆ ❇❈❉ ❊ ❋● ❍ ■ ❏ ❑ ▲ ▼ ◆❖GLYPH<a80> ◗ ❘ ❙ ❚ ❯ ❱ ❲ ❳ ❨ ❩ ❬ ❭ ❪❫❴ ❵ ❛❜ ❝ ❞ ❡ ❢ ❣ ❤ ✐❥❦ ❧ ♠ ♥ ♦ ♣ GLYPH<a113> GLYPH<a114> GLYPH<a115> GLYPH<a116> ✉ ✈ ✇ ①② ③ ④ ⑤⑥ ⑦ ⑧ ⑨ ⑩ ❶ ❷ ❸❹ ❺ ❻ ❼ ❽ ❾ ❿ ➀ ➁ ➂ ➃ ➄ ➅ ➆ ➇ ➈➉ ➊ ➋➌ ➍ ➑ ➒ ➓➔→ ➣ ↔ ↕ ➙ ➛ ➜ ➞ ➠ ➡ ➢ ➤ ➥➦ ➧ ➨➩ ➫ ➭ ➯ ➲ ➳ ➵ ➸➺➻ ➼ ➽ ➾ ➚ ➪ ➶ ➹ ➘ ➴ ➷ ➬ ➮ ➱✃❐ ❒ ❮❰ GLYPH<a207> GLYPH<a208> GLYPH<a209> GLYPH<a210> GLYPH<a211> GLYPH<a212> GLYPH<a213>GLYPH<a214>GLYPH<a215> GLYPH<a216> GLYPH<a217> GLYPH<a218> GLYPH<a219> GLYPH<a220> GLYPH<a221> GLYPH<a222> GLYPH<a223> GLYPH<a224> GLYPH<a225> GLYPH<a226> GLYPH<a227> GLYPH<a228> GLYPH<a229> GLYPH<a230> GLYPH<a231> GLYPH<a232> GLYPH<a233> GLYPH<a234>GLYPH<a235>GLYPH<a236> GLYPH<a237> GLYPH<a238> GLYPH<a239> GLYPH<a240> GLYPH<a241> GLYPH<a242> GLYPH<a243>GLYPH<a244>GLYPH<a245> GLYPH<a246> GLYPH<a247> GLYPH<a248> GLYPH<a249> GLYPH<a250> GLYPH<a251> GLYPH<a252>GLYPH<a253> GLYPH<a254> GLYPH<a255> ✁GLYPH<a0>✆ ✂ ✄✝ ☎ ✡ ☛ ☞✌✍ ✎ ✏ ✑ ✒ ✓ ✔ ✖ ✗ ✘✙ ✚ ✛ ✜ ✢ ✣ ✤ ✥ ✦ ✧★✩ ✪ ✫ ✬ ✭ ✮ ✯ ✰✱✲ ✳ ✴ ✵ ✶ ✷ ✸ ✹✺ ✻ ✼ ✽✾✿ ❀ ❁❂ ❃ ❄ ❅ ❆ ❇ ❈ ❉❊❋ ● ❍ ■ ❏ ❑ ▲ ▼ ◆ ❖ GLYPH<a80>◗ ❘ ❙ ❚ ❯ ❱ ❲ ❳ ❨ ❩❬❭ ❪ ❫ ❴ ❵ ❛ ❜ ❝❞❡ ❢ ❣ ❤ ✐ ❥ ❦ ❧♠ ♥ ♦ ♣GLYPH<a113>GLYPH<a114> GLYPH<a115> GLYPH<a116>✉ ✈ ③ ④ ⑤⑥⑦ ⑧ ⑨ ⑩ ❶ ❷ ❸ ❺ ❻ ❼❽ ❾ ❿ ➀ ➁ ➂ ➃ ➄ ➅ ➆ ➇ ➈ ➉ ➊ ➋ ➌ ➍ ➎ ➏ ➐ ➑ F IG. 4: (colo r) Calculated amplitude (left) and real-part ( right) 2DFT spectra of potassium vapor for S$_{I}$ ( ω$_{τ}$ , T, ω$_{t}$ ) ( bottom) and S$_{II}$ ( ω$_{τ}$ , T, ω$_{t}$ ) (top) measurements. The spec- t ra are normali zed to the most intense peaks. FIG. 3: (color online) Double-sided Feynman diagrams for all possible Liouville-space pathways contributing to the third- order optical response of a V-type system. The pathways for the S$_{I}$ ( ω$_{τ}$ , T, ω$_{t}$ ) and S$_{I}$ ( τ, ω$_{T}$ , ω$_{t}$ ) measurements are listed in top two rows; while the pathways for the S$_{II}$ ( ω$_{τ}$ , T, ω$_{t}$ ) measurement are listed in bottom two rows. The state num- bers, 0, 1 and 2, correspond to 4 $^{2}$S 1 / 2 , 4 $^{2}$P 1 / 2 and 4 $^{2}$P 3 / 2 of potassium atoms. 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]: 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]: 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$_{2}$ 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 E ( τ, T, ω$_{t}$ ) = 2 πL n ( ω$_{t}$ ) c iω$_{t}$P $^{(3)}$( τ, T, ω$_{t}$ ) (4) ˙ ρ$_{00}$ = − ˙ ρ$_{1}$$_{1}$ (1) ˙ ρ$_{11}$ = − γ s$^{p}$ρ$_{11}$ + i $_{ℏ}$µ$_{01}$ E ( ρ$_{01}$ − ρ$_{10}$ ) (2) i ˙ ρ$_{01}$ = − γ p$^{h}$ρ$_{01}$ + iω$_{01}$ρ$_{01}$ + $_{ℏ}$µ$_{01}$ E ( ρ$_{11}$ − ρ$_{00}$ ) (3)