Fig. 1. Two-dimensional spectroscopy in phase matching and collinear arrangements. In the phase matching arrangement shown on the top plot, multiple pulses are spatially crossed in a time delayed sequence through an ensemble of dipole oscillators. The resulting nonlinear polarization coherently constructs in a phase matched direction for background free signal detection. In the collinear arrangement, shown on the bottom plot with a blue reference wave used to emphasize the relevance of the relative pulse carrier phases, the first pulse causes all dipole oscillators within the laser bandwidth to oscillate at their natural frequencies. The second pulse can intensify, suppress, or map the oscillations into other coherences depending on its phase. The third pulse serves as a local oscillator. Here, the first and last pulses are 1800 out of carrier phase with respect to the center pulse. τ_1,2 are the time delays, and k⃗ is the propagation vector.

Fig. 2. Level structure and linear/nonlinear spectra of rubidium.

Fig. 3. Experimental setup. 50fs, broadband pulses enter the shaper at 20KHz. They are spectrally dispersed by an 1800 line/millimeter diffraction grating and focussed to a zero-dispersion line in an acousto-optic modulator by a 37.5 centimeter focal length spherical mirror. The modulator deflects the pulses as discussed in the text. The pulses are recombined through a second identical lens and grating pair. The three pulses created by the three RF waveforms in the acousto-optic modulator are focussed on a rubidium cell by 20cm lenses. Isotropic fluorescence is collected by a large f-number lens and focussed on a photomultiplier tube. The transmission-echo signal is retrieved by isolating light just from the two transitions shown in Fig. 2(a). The transmitted light is filtered by a 4F notch-pass filter which allows only light within 0.3nm of the 5S _1/2 →5P _1/2 and 5S _1/2 →5P _3/2 transitions through. The back-reflected light is picked off and sent into a photomultiplier.

Fig. 4. Rapid phase cycling setup electronics. For proper synchronization of the RF drive pulse in the AOM, all RF frequencies must be derived from the 80MHz repetition rate of our oscillator laser. The RF unit for the laser amplifier provides a 20KHz synchronization signal for triggering the boxcar. A frequency divider provides a 10MHz signal to externally oscillate both the 120MHz RF source and the AWG. The Ch2 marker from the AWG is mixed up by the 120MHz to create a 200MHz RF pulse for the AOM. This creates the central of the three sampling pulses for the experiment. The Ch1 and Ch2 outputs are mixed with the 120MHz to create RF pulses which vary from 200MHz. Ch1 derived pulses increase in frequency to create more delayed pulses from the shaper. Ch2 RF pulses create pulses which are less delayed then the center 200MHz pulse. When all three RF pulses are mixed, bandpass (BP) filtered, amplified (AMP), and sent to the AOM, three optical pulses with arbitrary relative delays and phases are created. The 20KHz signal is shown with yellow lines, the 10MHz with green, and the sampling pulse from the Ch1 marker, necessary to allow the computer oscilloscope to sample the data from the boxcar with red. The red arrow from indicates data flow from the boxcar to the scope.

Fig. 5. Rubidium photon echo data taken in fluorescence and transmission mode. This un-averaged data is retrieved from 16 time-scan experiments combined according to Eq. (3).