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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. 

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. 

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## 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. 

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- 

