2 0 0.2 0.4 0.6 389 389.5 390 390.5 391 391.5 389 389.5 390 390.5 391 391.5 389 389.5 390 390.5 391 391.5 -391.5 -391 -390.5 -390 -389.5 -389 0 0.2 0.4 0.6 0.8 1 389 389.5 390 390.5 391 391.5 -391.5 -391 -390.5 -390 -389.5 -389 -1 0.2 0.6 1 ❄ ❅ ❆ ❇ ❈ ❉ ❊ ❋ ● ❍ ■ ❏ ❑ ▲ ▼ ◆ ❖ GLYPH<a80> ◗ ❘ ❙ ❚ ❯ ❱ ❲ ❳ ❨ ❩ ❬ ❭ ❪ ❫ ❴ ❵ ❛ ❜ ❝ ❞ ❡ ❢ ❣ ❤ ✐ ❥ ❦ ❧ ♠ ♥ ♦ ♣ GLYPH<a113> GLYPH<a114> GLYPH<a115> GLYPH<a116> ✉ ✈ ✇ ①② ③ ④ ⑤ ⑥⑦ ⑧ ⑨ ⑩ ❶ ❷ ❸ ❹ ❺ ❻ ❼ ❽ ❾ ❿ ➀ ➁ ➂ ➃ ➄ ➅ ➆ ➇ ➈ ➉ ➊ ➋ ➌ ➍ ➎ ➏ ➐ ➑ ➒ ➓ ➔ → ➣ ↔ ↕ ➙ ➛ ➜ ➝ ➞ ➟ ➠ ➡ ➢ ➤ ➥ ➦ ➧ ➨ ➩ ➫ ➭ ➯ ➲ ➳ ➵ ➸ ➺ ➻ ➼ ➽ ➾ ➚ ➪ ➶ ➹ ➘ ➴ ➷ ➬ ➮ ➱ ✃ ❐ ❒ ❮ details can be found elsewhere [20, 23]. The TFWM signal, E$_{s}$ ∼ E ∗ $_{A}$E$_{B}$ E$_{C}$ , is generated in the phase-matched direction k$_{s}$ = − k$_{A}$ + k$_{B}$ + k$_{C}$ where E ∗ A is the conjugated pulse. The subscripts, A, B or C only denote direction, not time ordering. If E$_{A}$ arrives first the resulting 2DFT spectrum, S$_{I}$ ( τ, T, t ), is known as “rephasing” because the dephasing due to inhomo- geneous broadening is canceled. In a TFWM experi- ment, this time ordering produces a photon echo. If E$_{A}$ arrives second, the cancelation of dephasing due to in- homogeneous broadening does not occur and the result- ing S$_{II}$ ( τ, T, t ) spectra are referred to as “nonrephasing”. S$_{III}$ ( τ, T, t ) spectra, where E$_{A}$ arrives last, are sensitive to two-quantum resonances [24, 25]. The time resolved spectra S$_{i}$ ( τ, T, t ) are usually Fourier transformed with respect to two time delays, while the third delay is fixed. The time delays τ and T are defined based on time or- dering, not direction, and thus are strictly positive. Reflection cells designed for high temperature vapor experiments [26] can not be used in our multidimensional spectroscopy apparatus because the signal is detected in transmission [20]. Furthermore, in a reflection geometry the phase of the signal depends on the position of the in- terface producing the reflection. Therefore, a thin trans- mission cell for alkali atomic vapor has been designed and manufactured for the current experiment. The cell body is machined from high-grade titanium. Two sapphire windows are separately diffusion bonded to the titanium body. The gap between two windows can be adjusted be- fore the transmission cell is loaded with solid potassium [26]. The vapor temperature is controlled by a heater attached to the transmission cell, up to a maximum of 800 ◦ C. However, for the measurement reported here, the cell temperature is varied from 210 ◦ C to 270 ◦ C, ensur- ing low absorption and number densities in the range 2 . $^{03}$× 10 $^{20}$m − 3 to 1 . $^{87}$× 10 $^{21}$m − 3 [27]. During the mea- surements, the absorption spectrum of the potassium va- por at each temperature is recorded to estimate the gap size between two windows by comparing experimental spectra with calculated absorption spectra. FIG. 2: (color) (a) Excitation pulse spectrum (dotted) and ex- perimental (solid) and calculated (dashed) linear absorption of potassium vapor at 250 ◦ C. α is the absorption coefficient and L is the gap between the two sapphire windows. (b) The relevant energy levels of atomic potassium, with frequencies in THz. (c) Experimental amplitude S$_{I}$ ( ω$_{τ}$ , T, ω$_{t}$ ) (bottom) and S$_{II}$ ( ω$_{τ}$ , T, ω$_{t}$ ) (top) spectra of potassium vapor. (d) Ex- perimental real-part S$_{I}$ ( ω$_{τ}$ , T, ω$_{t}$ ) (bottom) and S$_{II}$ ( ω$_{τ}$ , T, ω$_{t}$ ) (top) spectra.The spectra are normalized to the most intense peak. III. RESULTS AND DISCUSSION Figure 2(c) and (d) show the amplitude and real- part of S$_{I}$ ( ω$_{τ}$, T, ω$_{t}$ ) and S$_{II}$ ( ω$_{τ}$, T, ω$_{t}$ ) 2DFT spectra of potassium vapor at 250 ◦ C. The emission frequency, ω$_{t}$ is defined to be positive, thus for S$_{I}$ ( ω$_{τ}$, T, ω$_{t}$ ), the signal appears at negative ω$_{τ}$ because the first pulse is conju- gated. The S$_{I}$ ( ω$_{τ}$ , T, ω$_{t}$ ) and S$_{II}$ ( ω$_{τ}$ , T, ω$_{t}$ ) spectra can be used to determine homogeneous and inhomogeneous linewidths [30, 31]. The horizontal axis of the spectrum is generated directly from the recorded interferogram be- tween the reference pulse and the TFWM signal [32] at each delay step between the pulses A and B. The vertical axis of the spectrum is generated by fast Fourier trans- form with respect to τ . During the measurements, T was set to 580 fs , which is equal to 1 / ( f$_{D}$$_{2}$ − f$_{D}$$_{1}$ ) where f$_{D}$$_{1}$ and f$_{D}$$_{2}$ are the transitions frequencies corresponding to the transitions D$_{1}$ and D$_{2}$ respectively. The measured linear absorption and the energy level diagram for atomic potassium are shown in Fig. 2(a) and (b) respectively. The laser excites both D$_{1}$ and D$_{2}$ tran- sitions and there are no resonant two-photon transitions. Fig. 2(a) also shows the calculated linear absorption spec- trum at 250 ◦ C. At this temperature the resonance broad- ened linewidths of the D$_{1}$ and D$_{2}$ transitions are below the spectrometer resolution. A ∼ 1500 Torr argon buffer gas is added to the cell to induce collision broadening. The argon reduces the absorbance αL below 1 into the weak absorption regime. In the calculation, the spec- tral linewidth is the sum of resonance from potassium- potassium interactions [28] and collisional broadening with argon [29]. The gap size, L , is estimated to be 19 µ m at 250 ◦ C and varies slightly with temperature over the range explored. Each spectrum has two diagonal peaks, which corre- spond to the D$_{1}$ and D$_{2}$ transitions, and two off-diagonal peaks, which represent coupling between the two reso- nances. However, the relative peak strengths are dif- ferent in S$_{I}$ ( ω$_{τ}$, T, ω$_{t}$ ) and S$_{II}$ ( ω$_{τ}$ , T, ω$_{t}$ ) spectra. This difference can be qualitatively explained by the double- sided Feynman diagrams shown in Fig. 3. Double-sided Feynman diagrams are useful for illustrating all possi-