0 0.5 1 1.5 2 0.3 0.4 0.5 M ixing Frequency, ω T  /2 π (THz) TA TB TC 389 390 391 392 -2.5 -2 -1.5 -1 -0.5 0 0.1 0.2 Emission Frequency, ω$_{t}$ /2 π (THz) TD 389 389.5 390 390.5 391 391.5 389 389.5 390 390.5 391 391.5 389 389.5 390 390.5 391 391.5 -1 -0.6 -0.2 0.2 0.6 1 ➒ ➓ ➔ → ➣ ↔ ↕ ➙ ➛ ➜ ➝ ➞ ➟ ➠ ➡ FIG. 6: (color) Experimental amplitude S$_{I}$ ( τ, ω$_{T}$ , ω$_{t}$ ) spec- t r um. F IG. 5: (color) Experimental (left) and theoretical (right) purely absorptive spectra obtained by the addition of the S$_{I}$ ( ω$_{τ}$ , T, ω$_{t}$ ) and S$_{II}$ ( ω$_{τ}$ , T, ω$_{t}$ ) spectra. are defined as the Fourier transform with respect to delay τ : ton energies corresponding to the transitions D$_{1}$ and D$_{2}$ separately. The amplitude of the spectrum is normalized according to the strength of the strongest peak, however, the spectrum is displayed with a color scale of 0 to 0.5 to emphasize two side peaks. It is clear that the two quan- tum pathways for Raman coherences (TC and TD) can be isolated as two side peaks in the S$_{I}$ ( τ, ω$_{T}$ , ω$_{t}$ ) spec- trum. Weak asymmetry between TC and TD features is observed, where the stronger peak has the same emission energy as the stronger zero mixing energy feature. This asymmetry is also observed in semiconductors [21]. The numerical simulations of the S$_{I}$ ( ω$_{τ}$ , T, ω$_{t}$ ) and S$_{II}$ ( ω$_{τ}$ , T, ω$_{t}$ ) spectra shown in Fig. 4 are in very good agreement with the experiment; correct peak strengths and symmetric star-like line shapes are reproduced in the amplitude spectra, indicating the dominance of homoge- neous broadening [30]. As expected the lineshapes are absorptive, which in contrast to semiconductors where many-body interactions result in dispersive lineshapes [19, 23, 34]. The dephasing rate of the Raman coherence is related to the dephasing rates of the optical transitions by [37] There are both absorptive and dispersive contributions in the S$_{I}$ ( ω$_{τ}$ , T, ω$_{t}$ ) and S$_{II}$ ( ω$_{τ}$ , T, ω$_{t}$ ) spectra. Purely absorptive contributions can be isolated by adding the S$_{I}$ ( ω$_{τ}$ , T, ω$_{t}$ ) and S$_{II}$ ( ω$_{τ}$ , T, ω$_{t}$ ) signals together[12, 36]. The experimental and theoretical purely absorptive spec- tra are shown in Fig. 5. Compared with the S$_{I}$ ( ω$_{τ}$ , T, ω$_{t}$ ) and S$_{II}$ ( ω$_{τ}$ , T, ω$_{t}$ ) spectra, the disappearance of disper- sive wings in the purely absorptive spectra results in a higher resolution of spectral features. where γ$_{D}$$_{2}$ − D $_{1}$, γ$_{D}$$_{2}$ and γ$_{D}$$_{1}$ are the dephasing rates for the Raman coherences, the D$_{2}$ and D$_{1}$ transitions re- spectively. The coefficient R$_{ph}$ describes the degree of correlation between fluctuations of the two transitions. The dephasing rates of D$_{2}$ and D$_{1}$ extracted from the S$_{I}$ ( ω$_{τ}$ , T, ω$_{t}$ ) measurement are 0.128 THz and 0.118 THz. The dephasing rates for the Raman coherences at the en- ergy position of TC and TD are 0.127 THz and 0.140 THz respectively so the correlation coefficients R$_{ph}$ are 0.45 and 0.43 respectively. The positive correlation coef- ficients means that the energies of the D$_{2}$ and D$_{1}$ lines simultaneously shift in the same direction relative to each other during the collisional scattering, as expected. Cross peaks in S$_{I}$ ( ω$_{τ}$ , T, ω$_{t}$ ) have contributions from ground-state bleaching and excited-state emission path- ways. These two pathways differ during the delay time T . After the second excitation pulse, the ground-state bleaching pathway leaves the system in the ground state, while the excited-state emission or Raman-like pathway (labeled TC or TD in Fig. 3) leaves the system in a co- herent superposition of 4 $^{2}$P$_{1}$$_{/}$$_{2}$ and 4 $^{2}$P$_{3}$$_{/}$$_{2}$ . S$_{I}$ ( τ, ω$_{T}$ , ω$_{t}$ ) spectra can be used to isolate Raman coherences terms, projecting the 2DFT signal along ω$_{T}$ and ω$_{t}$ instead of ω$_{τ}$ and ω$_{t}$ [21]. Technically, the measurement is achieved by scanning the third pulse C while keeping the time delay τ between first two pulses constant. The spectrum is then obtained by fast Fourier transform of time domain sig- nal with respect to T . Figure 6 shows an experimental spectrum of S$_{I}$ ( τ, ω$_{T}$ , ω$_{t}$ ), with τ = 0 in the measure- ment. In the spectrum, two strong peaks are observed along the zero mixing energy and two side peaks show up at the energy position of TC ( E$_{D}$$_{2}$, E$_{D}$$_{2}$ − E$_{D}$$_{1}$ ) and TD ( E$_{D}$$_{1}$, E$_{D}$$_{1}$ − E$_{D}$$_{2}$ ), where E$_{D}$$_{1}$ and E$_{D}$$_{2}$ are the pho- As in other types of spectroscopy, the lineshapes in 2DFT spectroscopy can be affected by the propagation of the excitation laser pulses and signals in the reso- nant medium. Propagation effects in TFWM experi- ments of alkali vapors have been discussed [5], as have distortions of 2DFT spectra due to propagation effects [38–41]. When the number density of atomic potassium is increased by raising the temperature, the absorption also strengthens. Distortion of 2DFT spectra peaks due γ$_{D}$$_{2}$ − D 1 = γ$_{D}$$_{2}$ + γ$_{D}$$_{1}$ − 2 R$_{ph}$ ( γ$_{D}$$_{1}$γ$_{D}$$_{2}$ ) 1 / 2 (6) S$_{I,II}$ ( ω$_{τ}$ , T, ω$_{t}$ ) = ∫ ∞ −∞ E ( τ, T, ω$_{t}$ ) e iω τ $^{τ}$d τ (5)