## Abstract

Multidimensional coherent spectroscopy is a powerful tool for understanding the ultrafast dynamics of complex quantum systems. To fully characterize the nonlinear optical response of a system, multiple pulse sequences must be recorded and quantitatively compared. We present a new single-scan method that enables rapid and parallel acquisition of all unique pulse sequences corresponding to first- and third-order degenerate wave-mixing processes. Signals are recorded with shot-noise limited detection, enabling acquisition times of $\sim 2\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{minutes}$ with $\sim 100\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{zs}$ phase stability and $\sim 8$ orders of dynamic range, in a collinear geometry, on a single-pixel detector. We demonstrate this method using quantum well excitons, and quantitative analysis reveals new insights into the bosonic nature of excitons. This scheme may enable rapid and scalable analysis of unique chemical signatures, metrology of optical susceptibilities, nonperturbative coherent control, and the implementation of quantum information protocols using multidimensional spectroscopy.

## 1. INTRODUCTION

Multidimensional coherent spectroscopy (MDCS) was originally developed in nuclear magnetic resonance spectroscopy to characterize the structure and nature of complex molecules [1]. Since its inception, MDCS has been adapted from nuclear magnetic resonance studies to probe electronic transitions in chemical, physical, and biological systems [2,3]. In ultrafast optical experiments, MDCS records the optical amplitude and phase of degenerate four-wave mixing (FWM) processes [4,5]. MDCS is a powerful tool because of its ability to unfold congested spectra [6,7], measure coupling strengths [8–10], and access nonradiative quantum coherences and recombination dynamics [11–13]. In the infrared and optical regimes, MDCS has been used to study conjugated polymers [14,15], light harvesting complexes [8,16–18], vibrational dynamics in water [19–21], and semiconductors nanostructures [9,10,22–24].

The most powerful motivation for using MDCS is its ability to isolate the different processes involved in generating a nonlinear polarization [25]. On resonance, incident-pulsed radiation will create superpositions of quantum states that temporally interfere. The phase of this temporal interference contains information about the excitation and coupling mechanisms (quantum pathways), which are revealed when emitted signals are Fourier-transformed as a function of interpulse delays. Because all of the system dynamics are encoded in these quantum pathways, it is advantageous to isolate and measure as many multidimensional projections of these pathways as possible. This has been accomplished in three-dimensional (3D) experiments [13,26–30] using a single-pulse sequence. However, because all possible pulse sequences are required to access all quantum pathways, these experiments only measure a subset of all processes [31]. Indeed, two-dimensional coherent spectroscopy (2DCS) has resolved complicated many-body problems in semiconductors by recording spectra from several pulse sequences [9,10]. Therefore, a desirable capability is to characterize the quantum pathways of a system by recording spectra that are linear combinations of pulses, under identical conditions. This measurement, combined with analysis that separates the pathways using the optical phase, would provide a complete picture of the dynamic system response arising from the excitation pulses.

In this work, we demonstrate a novel collinear 2DCS technique that simultaneously measures all quantum pathways up to third-order for pulse sequences arising from linear permutations of the excitation pulses. That is, we record the single-pulse responses, the rephasing, the nonrephasing and the double-quantum pulse sequences simultaneously on a single pixel detector. This is achieved by developing a multichannel superheterodyne optical frequency receiver that has very high dynamic range, short acquisition times, small intrinsic step sizes, temporal dynamic range up to $\sim 1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{ns}$, and the ability to measure in any rotating frame without using common path optics. These attributes and collinear excitation geometry are appropriate for either local or global excitation of individual emitters [32–35], well-suited to probing ensembles, and thus scalable with system size. This method has applications in nonperturbative coherent control, as the different signals can be used for rapid feedback and state characterization. This technique potentially provides a roadmap for using MDCS in quantum information [36–38] or to use ideas from quantum information to analyze the coherent dynamics measured by MDCS [39–42], as we are able to provide a complete picture of the system dynamics due to three excitation pulses.

To demonstrate the utility of recording all third-order quantum pathways, we use this instrument to experimentally characterize heavy-hole excitons in four GaAs quantum wells. Excitons are collective quasi-particles consisting of Coulomb-bound electron-hole pairs that determine the optical properties of semiconductors through their absorption and many-body interactions. The high material quality of GaAs grown by molecular beam epitaxy provides a pristine system to measure and model exciton dynamics. However, to date, it has been challenging to quantitatively measure exciton–exciton interactions and evaluate their role with respect to Pauli blocking because both the transition dipoles and the interaction energies contribute to the amplitude and phase of coherent wave-mixing signals [31,43–49]. However, our measurements record every pulse sequence and thereby provide additional information not present in previous work. To quantitatively elucidate the contribution of Pauli blocking versus many-body interactions on the nonlinear optical response, a Monte Carlo simulation is used to reconstruct the total complex polarization by separating each quantum pathway. In this way, we determine the exciton many-body parameters and find that a rarely used bosonic model [50–52] of exciton dynamics best agrees with the measurements. In stark contrast to atomic systems, where Pauli blocking dominates the nonlinear optical response, these measurements quantitatively show that exciton–exciton interactions are the governing nonlinearity giving rise to the third-order optical response.

## 2. EXPERIMENTAL PLATFORM

A variety of collinear and noncollinear techniques have been developed to perform MDCS, including pulse-shaping approaches [53–61], birefringent methods [62], frequency combs [63–65], feedback-stabilized interferometers [66,67], and diffractive optics [57,68–70]. Each approach has limitations; for instance, wave-vector matching is not appropriate for single nanoemitters due to the lack of a phase-matching condition. Therefore, techniques that are appropriate for long-lived subwavelength structures, such as single quantum dots or single vacancy centers, require collinear approaches with long scan delays. To date, only heterodyne approaches using dynamic phase cycling [71,72] have been found suitable to the FWM of single quantum dots [32–35].

Dynamic phase cycling was first introduced to MDCS by Tekavek *et al.* [72] and differs from static phase cycling [53] in that it relies on continuous pulse-to-pulse phase cycling as set by radio frequency electro-optics. While extremely sensitive, these methods suffer from interpulse stepping errors [32–35,72–74], large step sizes that are set by mechanical delays [32–35,72–74], and slow data acquisition [35,72–74]. Additionally, they have short delay lengths ($\sim 10\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{ps}$) [72,74,75] or suffer from the lack of stable degenerate reference lasers for measuring at different wavelengths (rotating frame synthesis) [35,73]. Our novel approach solves these issues while retaining the advantages, such as arbitrary polarization control of the pulses, collinear excitation, and intrinsically correct phasing of spectra [72].

Our experimental apparatus consisting of two Mach–Zehnder interferometers nested within a larger Mach–Zehnder interferometer [Fig. 1(a)]. This unstable nested interferometer takes a 120 fs pulse centered at 1555 meV from a Ti:Sapphire oscillator ${f}_{\mathrm{rep}}=76\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{MHz}$ and splits it into four replicas: A, B, C, and a local oscillator (LO). Unlike setups where a single pulse acts multiple times, three distinct pulses provide access to all first-, second-, and third-order coherences. The three pulses (A, B, C), denoted alphabetically by time ordering, are recombined collinearly on a beam splitter and focused onto the sample. The time ordering is controlled by mechanical delay lines allowing long $\sim 1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{ns}$ delays. The signal is collected in reflection and coupled with the LO into a fiber-coupled detector that measures the amplitude and phase of the signal through optical heterodyne detection.

Each interferometer arm has an acousto-optical modulator (AOM) driven at a unique radio frequency ${\omega}_{i}$, where $i=A,B,C$, or LO. The AOMs shift the carrier frequency to provide a unique carrier-envelope offset for each pulse. This frequency shift, ${\omega}_{A,B,C,\mathrm{LO}}=(94.408,93.8,93.511,92.111)$ MHz, is much smaller than the optical bandwidth, enabling dynamic phase cycling that manifests as a pulse-to-pulse phase accumulation. Our signal phase cycles $100\times $ faster ($\sim 1\mathrm{\mu s}$) than previous approaches, moving our signals far away from the interferometer, laser, and electronic noise [32,35,71–76]. The interaction of these pulses with the system generates each type of wave-mixing signal corresponding to the linear, rephasing (${S}_{\mathrm{I}}$), nonrephasing (${S}_{\mathrm{II}}$), and double-quantum (${S}_{\mathrm{III}}$) pulse sequences. Due to the wide variety of signals recorded, we introduce the notation (spectrum type-pulse sequence) as tabulated in Table 1 along with the detection frequencies. The signals recorded in this experiment are the 0/1/2 quantum (Q) spectra as well as the linear absorption (LA) signals. Synchronous demodulation of the detector output at these frequencies separates each signal from the others [Fig. 1(b)]. Additionally, demodulation at frequencies ${\omega}_{A}-{\omega}_{B}$ and ${\omega}_{C}-{\omega}_{\mathrm{LO}}$ measures the system’s linear response. As shown in Table 1, by scanning interpulse delay ${t}_{\mathrm{lab}}$ and time-resolving the signal emission during $t$, every process in each pulse sequence can be measured in a single scan. Experimentally, ${t}_{\mathrm{lab}}$ is the delay stage in arm $A$ of Fig. 1(a). By scanning ${t}_{\mathrm{lab}}$ negative, the pulse in interferometer arm $A$ can come after the pulse in interferometer arm $C$. Positive ${t}_{\mathrm{lab}}$ scans the delay between the first and second pulse ($\tau $), while negative ${t}_{\mathrm{lab}}$ scans the delay between the second and third pulse ($T$).

With a few exceptions [72–75,77], most experiments do not record multiple pulse sequences simultaneously. To date, the measurements recorded in this experiment represent the most complete measurement of the wave-mixing processes associated with linear permutations of the exciting pulses. In addition, performing all these measurements at once is not just a convenience; it provides a unique experimental capability in that quantitative comparisons of different signals can be made, as demonstrated in Section 3.C. Finally, measurement of only a single pulse sequence constitutes a conditional measurement in that it is postselecting the system dynamics of interest [36,38]. From a nonperturbative coherent control perspective, the other signals are important as they represent the “error” associated with trying to prepare specific quantum states through specific pulse sequences. In principle, this experiment allows real-time feedback and optimization of multipulse coherent control schemes, and therefore may be of particular interest to quantum information applications of MDCS [36–38].

Measuring the dynamics associated with each pulse sequence is challenging because the interferometer positions must be known to subwavelength precision. Without precise knowledge of the interferometer position, the signal phase can rotate randomly through $2\pi $, removing accurate phase retrieval and the possibility of a Fourier transform. Previous dynamic phase-cycling experiments mitigated this issue by measuring in rotating frames [35,72–75]. However, these experiments suffer from slow scan acquisition rates and unstable interferometers which cause long-term drift errors. In turn, this leads to nonuniform sampling errors that manifest as aliasing and/or limit the useful integration time per time-step. Furthermore, dynamic phase-cycling experiments suitable for scanning long delays currently require stabilized reference lasers that are degenerate with the carrier frequency of the oscillator, thereby limiting the general applicability of these approaches for measuring at arbitrary wavelengths [35,73].

These problems are solved by measuring the interferometer positions in real time using vibrational interferometry. In this experiment, vibrational interferometry consists of time-resolving the vibrational dynamics of the optical table in order to provide an out-of-loop measurement of the interpulse delays. This is achieved by copropagating a far red-detuned (1020 nm) narrow linewidth reference laser through the interferometers. After propagating throughout the setup, the reference laser is separated from the pulses using a dichroic mirror and is measured on a separate detector. The reference detector is synchronously demodulated with the signal detector at frequencies ${\omega}_{A}-{\omega}_{B}$, ${\omega}_{B}-{\omega}_{C}$, and ${\omega}_{C}-{\omega}_{\mathrm{LO}}$. The use of a far red-detuned reference laser avoids background population effects associated with blue-detuned or degenerate reference lasers and enables long interpulse delays and arbitrary rotating frame synthesis.

A major recent innovation in MDCS is the ultra-rapid scanning of interpulse delays through frequency comb-based methods [63–65]. Similarly, rapidly scanning interpulse delays with low repetition rate systems has demonstrated higher signal-to-noise ratios (SNRs) [58,78]. Conventional scan rates using mechanical delay lines are $\sim 30\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{fs}/\mathrm{s}$ [66,72–75,78,79] while noncomb-based rapid-scan methods achieve $\sim 200\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{fs}/\mathrm{s}$ when adjusted for required averaging [57,58,78,80]. We achieve a time-domain $\mathrm{SNR}>1000\text{:}1$ when scanning at $\sim 700\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{fs}/\mathrm{s}$. Additionally, we demonstrate the ability to record at $\sim 3.3\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{ps}/\mathrm{s}$ with a time-domain $\mathrm{SNR}\sim 200\text{:}1$ (Supplement 1, Sec. 1H) acquired within 2.15 min, overall faster than data acquisition on similar samples using frequency combs (3 min) [81]. The data acquisition rate is further enhanced by the five unique wave-mixing signals recorded in parallel. In conventional MDCS approaches, the limiting factors in acquisition speed are usually noise suppression, collection efficiency, and the sample properties, rather than the ability to change an interpulse delay quickly. Similarly, our scan speed is dictated by the sample and the desired SNR, up to the maximum stage velocity of $\sim 66\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{ps}/\mathrm{s}$.

As the delay stage is scanned continuously, the interferometer delay is recorded from the reference laser beat note. The interpulse step size is set by the ratio of the stage velocity to the resampling frequency (28 kHz). Subsequently, intrinsic step sizes $\sim 17\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{as}$ provide a high degree of oversampling (sampling bandwidth 20 eV after binning), which can be changed, if necessary, by adjusting the resampling frequency or stage velocity. Large sampling bandwidth is advantageous, since the sampling bandwidth is at least a partial limitation when measuring higher-order multiquantum coherences [82,83]. The accuracy of the interferometer delay is set by the signal-to-noise of the reference laser beat notes. The interferometer vibrations and stage movement manifest as red–blue relative Doppler shifts (Supplement 1, Sec. 1), span $\sim 500\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{Hz}$ and are dominated by the cryostat motion. In this experiment, the interferometer has an intrinsic resolution of $\sim 100\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{zs}$ based on the noise floor of the amplitude spectrum of the interferometer vibrations [Fig. 1(b)] measured from the reference beat-note phase. Therefore, in an unstable interferometer, we achieve both interferometer step sizes and step-size accuracy comparable to the best common path interferometers [62,84]. Ultimately, the timing resolution and bandwidth are set by the pulse duration/bandwidth ($\sim 10\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{meV}$), and the SNR is determined by the sample and/or integration time. This approach should be extendable to larger optical bandwidths by, for instance, broadening one or more pulses after each AOM.

The detection and signal processing amounts to five superheterodyne optical receivers, as shown in Fig. 1(b). Superheterodyning was invented in the radio-frequency domain for signal processing, and consists of multistage downconversion of high-frequency signals to low frequency for signal processing. Optical heterodyning mixes the signal and LO to create shot-noise limited $\sim \mathrm{MHz}$ beat notes. These beat notes are heterodyned again with an electronically synthesized oscillator at their $\sim \mathrm{MHz}$ carrier frequency and filtered. All electronic frequencies, including those that drive the AOMs, are phase-locked to the same master clock, thus ensuring accurate phase measurements. We use wideband filters to fully sample the interferometer vibrations, with a 3 dB corner of 1 kHz. After heterodyning at $\sim 1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{MHz}$, all eight signals constituting LA, FWM, and reference signals are resampled synchronously at 28 kHz, ensuring no measurement dead time. The phase induced by interferometer motion is removed using the reference laser through a final multistage heterodyning step that moves the measured signals into a rotating frame. This processing step enables physical undersampling, which modifies the Nyquist criterion to be relative to optical, rather than DC frequencies. A final integration equivalent to a moving average filter with bandwidth of $\sim 5\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{THz}$ (the pulse bandwidth) and a subsequent binning step are applied to the data to create a uniform grid for fast-Fourier transformation to the multidimensional frequency domain (Supplement 1, Sec. 1).

## 3. EXPERIMENTAL RESULTS AND ANALYSIS

#### A. Time-Domain Measurements

To demonstrate this technique, we examine the heavy-hole exciton resonance in GaAs/AlGaAs quantum wells. Four 8-nm-wide quantum wells are epitaxially grown above a semiconductor distributed Bragg reflector, which provides a weak twofold cavity enhancement of the optical field at the wells. The sample is held in vacuum at a temperature of 10 K. For three-pulse optical experiments, the relevant energy levels of the heavy-hole exciton consist of the crystal vacuum, the single-exciton state, and a doubly excited two-exciton state [Fig. 2(a)]. The linearly polarized pulses $A$, $B$, and $C$ are focused onto the sample with a spot size of $\sim 9\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}\text{\hspace{0.17em}}({e}^{-2})$ diameter, with 13 μW of average total power. The radiated response is collected in reflection and mixed with a LO on the detector [Fig. 2(a)].

The linear and FWM responses measured at their unique radio frequencies ${\omega}_{1-5}$, are listed in Table 1. These signals are recorded as the interpulse delays $\tau $ (first–second pulse), $T$ (second–third pulse), and $t$ (third-LO pulse) are varied. When the delay line ${t}_{\mathrm{lab}}$ is positive, pulse A arrives first and ${t}_{\mathrm{lab}}=\tau $. For positive delay, the nonlinear signals at ${\omega}_{3-5}$ correspond to the 1Q scans of the ${S}_{\mathrm{I}},{S}_{\mathrm{II}},{S}_{\mathrm{III}}$ pulse sequences as tabulated in the middle column of Table 1. For negative ${t}_{\mathrm{lab}}$, the interpulse delay (T) between pulses C and A (pulse B arrives first) is scanned providing the $0Q/2Q$ signals of the ${S}_{\mathrm{I}},{S}_{\mathrm{II}},{S}_{\mathrm{III}}$ pulse sequences, as tabulated in the last column of Table 1. Thus, by scanning delay line ${t}_{\mathrm{lab}}$ both positive and negative with respect to the global zero delay, the 0/1/2 quantum spectra can be recorded in a single scan.

The normalized amplitude of the time-domain FWM signals is shown in Figs. 2(b)–2(d) as a function of both positive and negative delay ${t}_{\mathrm{lab}}$. A photon-echo is detected [Fig. 2(b), $1\mathrm{Q}-{S}_{\mathrm{I}}$] for positive ${t}_{\mathrm{lab}}=\tau $ and $t$ at signal frequency ${\omega}_{3}$. Free induction decay is observed for the $1Q-{S}_{\mathrm{II}}$ ($1Q-{S}_{\mathrm{III}}$) scans, detected at ${\omega}_{4}$ (${\omega}_{5}$) for positive ${t}_{\mathrm{lab}}=\tau $ and $t$ [Figs. 2(c) and 2(d)]. Recording at frequencies ${\omega}_{3-5}$ for positive ${t}_{\mathrm{lab}}$ captures the absorption processes that are then correlated with the emission pathways during interpulse delay $t$. At negative ${t}_{\mathrm{lab}}$, the signals have a changing zero delay between pulses C and LO (${t}_{0}=t+|{t}_{\mathrm{lab}}|$). As a result, the time-domain data at $-{t}_{\mathrm{lab}}$ appear sloped in Figs. 2(b)–2(d).

Finally, we plot a slice of the data at $\tau =0.5\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{ps}$ for linear and nonlinear wave-mixing processes Fig. 2(e). This plot shows the total polarization during time $t$ up to the third order due to the incident-driving radiation and enables the quantitative comparison of linear and nonlinear optical susceptibilities with respect to each other. In contrast to the linear response, there is a delayed rise in the FWM signal, an indication of many-body interactions. The linear response is more than two orders of magnitude larger than the nonlinear signals and represents a unique tool to evaluate many-body dynamics as other parameters are varied.

#### B. Frequency-Domain Representation

To generate multidimensional spectra from our time-domain data, we perform a numerical two-dimensional fast Fourier transform of the time-domain data. These six spectra [Figs. 3(a)–3(f)] represent every quantum coherence in the relevant Feynman diagrams [31] and correlate the 0/1/2 quantum energies with respect to each other. All presented spectra are phased by removing the phase offset at zero delay following the procedure of Marcus *et al.* [72]. This procedure was checked by using the linear autocorrelation phase offsets and realized similar spectra. In Section 3.C, we present analysis of the two-dimensional spectra discussed here.

In Fig. 3(a), we show the real part of the ($1\mathrm{Q}-{S}_{\mathrm{I}}$) spectrum. A peak on the diagonal dashed line ($-\hslash {\omega}_{\tau}=\hslash {\omega}_{t}$) corresponds to excitation and emission at the exciton resonance energy $\hslash {\omega}_{X}$. This spectrum is slightly elongated compared to the cross-diagonal, indicative of weak inhomogeneous broadening of the exciton resonance due to disordered quantum well/barrier interfaces [85]. The real part of the rephasing spectrum reveals that the exciton line shape has a dispersive component indicative of exciton–exciton many-body interactions [7,12,86–88]. Similar arguments apply for the dispersive line shapes of the real parts of the ($1\mathrm{Q}-{S}_{\mathrm{II}}$, $1\mathrm{Q}-{S}_{\mathrm{III}}$) spectra, shown in Figs. 2(b) and 2(c), and the real part of the ($2\mathrm{Q}-{S}_{\mathrm{III}}$) spectrum [Fig. 2(f)] [7,12,88]. To date, the ($1\mathrm{Q}-{S}_{\mathrm{III}}$) spectrum has not previously been demonstrated, although closely related spectra have recently been reported in cresyl violet [30,89]. This spectrum shows the absorption-emission correlation of the double-quantum pulse sequence during $\tau $ and $t$ (evolving at frequency ${\omega}_{X}$ during both delays) instead of the conventional double-quantum spectrum ($2\mathrm{Q}-{S}_{\mathrm{III}}$) that occurs when scanning the delays $T$ ($2{\omega}_{X}$) and $t$ (${\omega}_{X}$) [Fig. 3(f)]. We show in Section 3.C that the measurement of the of the ($1\mathrm{Q}-{S}_{\mathrm{III}}$) spectrum is crucial to separation of the quantum pathways (Fig. 4, ${S}_{\mathrm{III}}$).

In Figs. 3(d) and 3(e), the real part of the ($0\mathrm{Q}-{S}_{\mathrm{I},\mathrm{II}}$) spectra is shown. The vertical axis corresponds to the low-frequency dynamics of the system, such as the excited state recombination lifetime or coupling between quasi-degenerate states. Because the system is driven to populate the single-exciton manifold during this delay for these pathways, the resonance appears at $\hslash {\omega}_{\tau}\approx 0$. The horizontal axis represents the emission of the system, and thus the resonance appears at $\hslash {\omega}_{t}=\hslash {\omega}_{X}$. In the case of quantum well excitons examined here, both of these spectra provide similar information (i.e., the linewidth along the $\hslash {\omega}_{\tau}$ axis is a measure of the excited state lifetime).

#### C. Exciton Quantum Pathways

Our measurements represent a complete experimental characterization of the quantum pathways for the heavy-hole exciton in GaAs as every first-, second- and third-order quantum coherence is recorded. In this section, the exciton quantum pathways describing these coherences are separated through a Markov chain Monte Carlo (MCMC) simulation of the total nonlinear polarization’s phase and amplitude with a minimal amount of assumptions in that we use only the energy levels provided by first-principles theory and the relevant Feynman diagrams [31]. From this analysis, we determine the exciton many-body parameters and find that a bosonic model [50–52] best agrees with the measurements. Critically, this analysis determines the relative amplitude of different quantum pathways and shows quantitatively that the quantum pathways associated with each pulse sequence destructively interfere in the absence of exciton–exciton interactions. This result thus contrasts the difference between collective quasi-particles such as excitons and atomic systems, where Pauli blocking dominates the nonlinear optical response.

From microscopic theory, a conventional picture of FWM involving excitons in GaAs is a three-level system with a two-body interaction. This system consists of a ground state $g$, a single exciton state $X$, and a two-exciton state $2X$. For these states, there are three pathways for the ${S}_{\mathrm{I}}$ and ${S}_{\mathrm{II}}$ pulse sequences but only two pathways for the ${S}_{\mathrm{III}}$ pulse sequences (Fig. 4) [31]. Coherences that access the $g-X$ transition are different than coherences that access the $X-2X$ transition due to many-body effects. These interactions modify the two-exciton transition energy and dephasing time relative to the single-exciton parameters, as evidenced by the ($2\mathrm{Q}-{S}_{\mathrm{III}}$) spectrum, which provides [Fig. 3(f)] direct evidence for a two-exciton state and many-body interactions [7,12,88]. The other clear signature of these interactions occurs in the real part of the spectrum if the interaction energies are below the exciton homogeneous linewidth. Resonances that are redshifted or blueshifted by an amount less than the homogeneous linewidth will appear dispersive in the real part of the spectra [Figs. 3(a)–3(f)]. Up to the third pulse, the diagrams in the ${S}_{\mathrm{I}}$ and ${S}_{\mathrm{II}}$ pulse sequences (Fig. 4) represent similar physical processes. These diagrams are split with the interaction of the third pulse. Diagrams ${S}_{\mathrm{I}}^{1-2},{S}_{\mathrm{II}}^{1-2}$ are, respectively, the same for their pulse sequence and the diagrams ${S}_{\mathrm{I}}^{3},{S}_{\mathrm{II}}^{3},{S}_{\mathrm{III}}^{1}$ acquire a negative amplitude and sample the doubly excited manifold. The interaction energy and negative amplitude of certain diagrams provide the dispersive profile previously discussed and clearly present in the data [Figs. 3(a)–3(f)].

Previous work on quantum well excitons focused on the collective or single-particle character of excitons [43,52,90–93], while more recent work has focused on interwell or indirect excitons [9,10,52,60,94]. When assigning interaction energies, power-dependent measurements often record shifts in the resonance energy or homogeneous linewidth [47,93,95]; other methods use the amplitude of coupling peaks [9,10,94]. While informative, these approaches can be difficult to interpret due to the complexity of many-body systems. Power-dependent measurements do not directly measure the interaction energy, but rather convolve the excitation density-dependent interactions with the change in the excitation density. This issue has prompted the use of a prepulse before nonlinear experiments [43].

A more direct way to measure many-body dynamics and in the process isolate each quantum pathway is to use the optical phase. In a many-body system, this is a particularly challenging task, as the collective transition dipoles (${\mu}_{g-x},{\mu}_{x-2x}$) are inherently correlated with the interaction energies (${\mathrm{\Delta}}_{x-x},{\gamma}_{x-x}$) in that they both contribute to the amplitude of the FWM signals [31]. Measuring the amplitude of all FWM processes self-consistently enables measurement of the relative value between transition dipoles. Disentangling the interaction energies is further complicated because they are not well resolved in that the interaction energies are often smaller than the exciton homogeneous (${\gamma}_{0}$) and inhomogeneous (${\gamma}_{\text{inhom}}$) linewidths. This makes fitting challenging because both the center frequency ${\omega}_{x}$ and homogeneous linewidth will exhibit correlations in their fit parameters with respect to the many-body interaction energies. To our knowledge, only once has the optical phase of an MDCS spectrum been used to assign interaction energies for semiconductor excitons using model assumptions and different polarizations [52].

We implement a MCMC simulation of the exciton response described by (Fig. 4) and constrain it by using the entire complex third-order polarization. In this simulation, every parameter is free to vary. The MCMC fully samples the parameter space of the Feynman diagrams and evaluates the parameter correlations in a manner that a least-squares fit would not and thereby finds the most likely parameters. Critically, this simulation finds that twice the ground state-exciton transition dipole $2{\mu}_{g-x}^{4}$ and the exciton-two-exciton dipole ${\mu}_{x-2x}^{2}{\mu}_{g-x}^{2}$ have the same value and are perfectly correlated (Supplement 1, Sec. 2).

Due to this measurement, the simulation is re-run with the constraint ${\mu}_{x-2x}=\sqrt{2}{\mu}_{g-x}$. This constrained simulation agrees with the first simulation; we report the most likely exciton parameters in Table 2. A subset of the best-fit slices (Fig. 5) demonstrates excellent ability to describe the entire third-order polarization. The measured parameters are consistent with a previous work [52] that showed a bosonic model could explain the polarization-dependent linewidth of exciton resonances. This result is also consistent with theoretical work that attempts to find low excitation density models of the exciton response that results in a bosonic Hamiltonian [50,51]. The ability to measure the relative value between transition dipoles determines the relative amplitude of the different quantum pathways, subsequently allowing the assignment of a bosonic model. More importantly, a bosonic model indicates that all quantum pathways destructively interfere in the absence of many-body interactions due to the relative amplitudes provided by the relative value between transition dipoles. This indicates that Pauli blocking may only be critical at high excitation density, where the decreasing distance between excitons causes a bosonic approximation to break down.

To provide a more intuitive, albeit more qualitative, illustration of how the present measurements constrain the excitonic parameters, we simulate recently used excitonic models (Fig. 6). The models consider a constant transition dipole ${\mu}_{x-2x}={\mu}_{g-x}$ [9,10], a dipole that changes as ${\mu}_{x-2x}=\sqrt{2}{\mu}_{g-x}$ [52], and a dipole that differs as ${\mu}_{x-2x}=\frac{{\mu}_{g-x}}{\sqrt{2}}$. The first model is often used in the literature, the second arises from a bosonic model of the exciton, and the third is implemented to simulate enhanced Pauli blocking. As Pauli blocking is the only nonlinearity for the commonly used two-level model, its inclusion may mimic these models. We explicitly neglect a two-level model, as the $2\mathrm{Q}-{S}_{\mathrm{III}}$indicates the presence of a two-exciton state as expected from theory [31], and a two-level system cannot give rise to a double-quantum spectrum.

The $1\mathrm{Q}-{S}_{\mathrm{II},\mathrm{III}}$ amplitudes from the simulated models are plotted as a function of interaction energy ${\mathrm{\Delta}}_{x-x}$ relative to the homogeneous linewidth in Fig. 6. We use the exciton–exciton energy shift ${\mathrm{\Delta}}_{x-x}$ rather than the exciton–exciton linewidth shift ${\gamma}_{x-x}$, but both interactions have similar effects on the relative amplitudes of different quantum pathways. Each of these simulated signals is normalized to the amplitude of the exciton resonance in the $1\mathrm{Q}-{S}_{\mathrm{I}}$ spectrum of its own respective model. This normalization choice removes the significant curvature arising from the different models, allowing them to be evaluated on a linear scale. This calculation is meant to be illustrative, as it only compares the relative amplitudes of the different FWM signals as compared to two data points (cross and dot in Fig. 6). We clearly see that, for all interaction energies, the only model that captures the measured relative amplitudes is the model in which the dipole changes as $\sqrt{2}$ consistent with the bosonic model found by the simulation. To illustrate the predictive power of these models based on our parameter uncertainty, the filled bounds in Fig. 6 represent the $\pm 2\sigma $ confidence interval for the model-predicted relative amplitude.

The ability to distinguish between interactions and dipole effects occurs through the quantitative measurement of different pulse sequences with respect to each other. We are thus able to assign all of the relevant exciton parameters by using the entire third-order polarization response and thereby separate the different quantum pathways. Due to the measured relative value between transition dipoles, we find that the quantum pathways that sample the $2X$ state contribute the same amplitude as the sum of those that sample only the single $X$ state. We note that the model elimination problem is not unique to excitonic systems, as excitation ladders and parameter estimation problems are ubiquitous in coherent spectroscopy. Our analysis demonstrates that by measuring all quantum pathways, these ambiguities can be removed and the full coherent response reconstructed.

## 4. CONCLUSION

We have demonstrated a thorough characterization of an excitonic many-body system. Our measurements enable MDCS by measuring all coherences in all quantum pathways up to and including the third order. In principle, our technique is scalable to an arbitrary number of excitation pulses, requiring only that we implement more measurement channels. This instrument has unique characteristics, as it simultaneously offers large sampling bandwidth, arbitrary rotating frame synthesis, arbitrary polarization of pulses, long interpulse delays, and small intrinsic step sizes with collinear excitation while simultaneously measuring all pulse sequences up to the third order. Our analysis determined the parameters of a the heavy-hole exciton, with a minimum number of assumptions, including the relative amplitudes of the Feynman diagrams as determined by the relative value between transition dipoles and determination of the many-body interactions. While not necessary for the analysis presented here, this technique is also compatible with 3D spectroscopy, or, in principle, any arbitrary order of MDCS. Measuring the full 3D response would provide access to all projections and correlations described by quantum pathways.

This technique is broadly applicable to different systems with specific advantages in coherent control gained by simultaneously measuring all pulse sequences up to the third order. Of particular interest are controlled many-body systems such as ion chains that can exhibit long-range interactions and hence have a large number of correlated states. Such correlated states could be directly measured, and the formation of those correlations time-resolved using the multiquantum pulse sequences. Furthermore, since it has been recently shown that multiquantum spectroscopy can be used as an entanglement witness [36–38,96], multiquantum experiments using multichannel measurements may provide a scalable tool to characterize quantum information systems, as the increasing system size renders conventional characterization impractical.

## Funding

National Research Council (NRC); National Institute of Standards and Technology (NIST).

## Acknowledgment

We acknowledge helpful discussions with Jennifer Ellis. We also thank the NIST editorial review board consisting of Jeff Chiles, Ari Feldman, and Norman Sanford. This work is a contribution of the NIST and is not subject to copyright in the United States of America.

See Supplement 1 for supporting content.

## REFERENCES

**1. **R. R. Ernst, G. Bodenhausen, and A. Wokaun, *Principles of Nuclear Magnetic Resonance in One and Two Dimensions* (Clarendon, 1990).

**2. **G. Nardin, T. M. Autry, G. Moody, R. Singh, H. Li, and S. T. Cundiff, “Multi-dimensional coherent optical spectroscopy of semiconductor nanostructures: collinear and non-collinear approaches,” J. Appl. Phys. **117**, 112804 (2015). [CrossRef]

**3. **P. C. Chen, “An introduction to coherent multidimensional spectroscopy,” Appl. Spectrosc. **70**, 1937–1951 (2016). [CrossRef]

**4. **J. D. Hybl, A. W. Albrecht, S. M. G. Faeder, and D. M. Jonas, “Two-dimensional electronic spectroscopy,” Chem. Phys. Lett. **297**, 307–313 (1998). [CrossRef]

**5. **J. D. Hybl, A. A. Ferro, and D. M. Jonas, “Two-dimensional Fourier transform electronic spectroscopy,” J. Chem. Phys. **115**, 6606–6622 (2001). [CrossRef]

**6. **E. L. Read, G. S. Engel, T. R. Calhoun, T. Mancal, T. K. Ahn, R. E. Blankenship, and G. R. Fleming, “Cross-peak-specific two-dimensional electronic spectroscopy,” Proc. Natl. Acad. Sci. USA **104**, 14203–14208 (2007). [CrossRef]

**7. **K. W. Stone, K. Gundogdu, D. B. Turner, X. Li, S. T. Cundiff, and K. A. Nelson, “Two-quantum 2D FT electronic spectroscopy of biexcitons in GaAs quantum wells,” Science **324**, 1169–1173 (2009). [CrossRef]

**8. **V. Tiwari, W. K. Peters, and D. M. Jonas, “Electronic resonance with anticorrelated pigment vibrations drives photosynthetic energy transfer outside the adiabatic framework,” Proc. Natl. Acad. Sci. USA **110**, 1203–1208 (2013). [CrossRef]

**9. **G. Nardin, G. Moody, R. Singh, T. M. Autry, H. Li, F. Morier-Genoud, and S. T. Cundiff, “Coherent excitonic coupling in an asymmetric double InGaAs quantum well arises from many-body effects,” Phys. Rev. Lett. **112**, 046402 (2014). [CrossRef]

**10. **G. Moody, I. A. Akimov, H. Li, R. Singh, D. R. Yakovlev, G. Karczewski, M. Wiater, T. Wojtowicz, M. Bayer, and S. T. Cundiff, “Coherent coupling of excitons and trions in a photoexcited CdTe/CdMgTe quantum well,” Phys. Rev. Lett. **112**, 097401 (2014). [CrossRef]

**11. **N. Takemura, S. Trebaol, M. D. Anderson, V. Kohnle, Y. Léger, D. Y. Oberli, M. T. Portella-Oberli, and B. Deveaud, “Two-dimensional Fourier transform spectroscopy of exciton-polaritons and their interactions,” Phys. Rev. B **92**, 125415 (2015). [CrossRef]

**12. **D. Karaiskaj, A. D. Bristow, L. Yang, X. Dai, R. P. Mirin, S. Mukamel, and S. T. Cundiff, “Two-quantum many-body coherences in two-dimensional Fourier-transform spectra of exciton resonances in semiconductor quantum wells,” Phys. Rev. Lett. **104**, 117401 (2010). [CrossRef]

**13. **D. B. Turner, K. W. Stone, K. Gundogdu, and K. A. Nelson, “Three-dimensional electronic spectroscopy of excitons in GaAs quantum wells,” J. Chem. Phys. **131**, 144510 (2009). [CrossRef]

**14. **E. Collini and G. D. Scholes, “Coherent intrachain energy migration in a conjugated polymer at room temperature,” Science **323**, 369–373 (2009). [CrossRef]

**15. **A. De Sio, F. Troiani, M. Maiuri, J. Réhault, E. Sommer, J. Lim, S. F. Huelga, M. B. Plenio, C. A. Rozzi, G. Cerullo, E. Molinari, and C. Lienau, “Tracking the coherent generation of polaron pairs in conjugated polymers,” Nat. Commun. **7**, 13742 (2016).

**16. **T. Brixner, J. Stenger, H. M. Vaswani, M. Cho, R. E. Blankenship, and G. R. Fleming, “Two-dimensional spectroscopy of electronic couplings in photosynthesis,” Nature **434**, 625–628 (2005). [CrossRef]

**17. **G. S. Engel, T. R. Calhoun, E. L. Read, T.-K. Ahn, T. Mančal, Y.-C. Cheng, R. E. Blankenship, and G. R. Fleming, “Evidence for wavelike energy transfer through quantum coherence in photosynthetic systems,” Nature **446**, 782–786 (2007). [CrossRef]

**18. **H. Lee, Y.-C. Cheng, and G. R. Fleming, “Coherence dynamics in photosynthesis: protein protection of excitonic coherence,” Science **316**, 1462–1465 (2007). [CrossRef]

**19. **J. J. Loparo, S. T. Roberts, and A. Tokmakoff, “Multidimensional infrared spectroscopy of water. II. Hydrogen bond switching dynamics,” J. Chem. Phys. **125**, 194522 (2006). [CrossRef]

**20. **S. Garrett-Roe, F. Perakis, F. Rao, and P. Hamm, “Three-dimensional infrared spectroscopy of isotope-substituted liquid water reveals heterogeneous dynamics,” J. Phys. Chem. B **115**, 6976–6984 (2011). [CrossRef]

**21. **M. Thämer, L. D. Marco, K. Ramasesha, A. Mandal, and A. Tokmakoff, “Ultrafast 2D IR spectroscopy of the excess proton in liquid water,” Science **350**, 78–82 (2015). [CrossRef]

**22. **G. Moody, C. Kavir Dass, K. Hao, C.-H. Chen, L.-J. Li, A. Singh, K. Tran, G. Clark, X. Xu, G. Berghäuser, E. Malic, A. Knorr, and X. Li, “Intrinsic homogeneous linewidth and broadening mechanisms of excitons in monolayer transition metal dichalcogenides,” Nat. Commun. **6**, 8315 (2015). [CrossRef]

**23. **K. Hao, L. Xu, P. Nagler, A. Singh, K. Tran, C. K. Dass, C. Schüller, T. Korn, X. Li, and G. Moody, “Coherent and incoherent coupling dynamics between neutral and charged excitons in monolayer MoSe2,” Nano Lett. **16**, 5109–5113 (2016). [CrossRef]

**24. **K. Hao, L. Xu, F. Wu, P. Nagler, K. Tran, X. Ma, C. Schüller, T. Korn, A. H. MacDonald, G. Moody, and X. Li, “Trion valley coherence in monolayer semiconductors,” 2D Mater. **4**, 025105 (2017). [CrossRef]

**25. **S. T. Cundiff and S. Mukamel, “Optical multidimensional coherent spectroscopy,” Phys. Today **66**(7), 44–49 (2013). [CrossRef]

**26. **D. Hayes and G. S. Engel, “Extracting the excitonic Hamiltonian of the Fenna-Matthews-Olson complex using three-dimensional third-order electronic spectroscopy,” Biophys. J. **100**, 2043–2052 (2011). [CrossRef]

**27. **J. A. Davis, C. R. Hall, L. V. Dao, K. A. Nugent, H. M. Quiney, H. H. Tan, and C. Jagadish, “Three-dimensional electronic spectroscopy of excitons in asymmetric double quantum wells,” J. Chem. Phys. **135**, 044510 (2011). [CrossRef]

**28. **H. Li, A. D. Bristow, M. E. Siemens, G. Moody, and S. T. Cundiff, “Unraveling quantum pathways using optical 3D Fourier-transform spectroscopy,” Nat. Commun. **4**, 1390 (2013). [CrossRef]

**29. **M. Titze and H. Li, “Interpretation of optical three-dimensional coherent spectroscopy,” Phys. Rev. A **96**, 032508 (2017). [CrossRef]

**30. **S. Mueller, S. Draeger, X. Ma, M. Hensen, T. Kenneweg, W. Pfeiffer, and T. Brixner, “Fluorescence-detected two-quantum and one-quantum–two-quantum 2D electronic spectroscopy,” J. Phys. Chem. Lett. **9**, 1964–1969 (2018). [CrossRef]

**31. **L. Yang, I. V. Schweigert, S. T. Cundiff, and S. Mukamel, “Two-dimensional optical spectroscopy of excitons in semiconductor quantum wells: Liouville-space pathway analysis,” Phys. Rev. B **75**, 125302 (2007). [CrossRef]

**32. **W. Langbein and B. Patton, “Heterodyne spectral interferometry for multidimensional nonlinear spectroscopy of individual quantum systems,” Opt. Lett. **31**, 1151–1153 (2006). [CrossRef]

**33. **J. Kasprzak, B. Patton, V. Savona, and W. Langbein, “Coherent coupling between distant excitons revealed by two-dimensional nonlinear hyperspectral imaging,” Nat. Photonics **5**, 57–63 (2011). [CrossRef]

**34. **F. Albert, K. Sivalertporn, J. Kasprzak, M. Strauß, C. Schneider, S. Höfling, M. Kamp, A. Forchel, S. Reitzenstein, E. A. Muljarov, and W. Langbein, “Microcavity controlled coupling of excitonic qubits,” Nat. Commun. **4**, 1747 (2013). [CrossRef]

**35. **E. W. Martin and S. T. Cundiff, “Inducing coherent quantum dot interactions,” Phys. Rev. B **97**, 081301 (2018). [CrossRef]

**36. **M. Gessner, F. Schlawin, and A. Buchleitner, “Probing polariton dynamics in trapped ions with phase-coherent two-dimensional spectroscopy,” J. Chem. Phys. **142**, 212439 (2015). [CrossRef]

**37. **F. Schlawin, M. Gessner, S. Mukamel, and A. Buchleitner, “Nonlinear spectroscopy of trapped ions,” Phys. Rev. A **90**, 023603 (2014). [CrossRef]

**38. **M. Gessner, F. Schlawin, H. Häffner, S. Mukamel, and A. Buchleitner, “Nonlinear spectroscopy of controllable many-body quantum systems,” New J. Phys. **16**, 092001 (2014). [CrossRef]

**39. **J. Yuen-Zhou, J. J. Krich, M. Mohseni, and A. Aspuru-Guzik, “Quantum state and process tomography of energy transfer systems via ultrafast spectroscopy,” Proc. Natl. Acad. Sci. USA **108**, 17615–17620 (2011). [CrossRef]

**40. **J. Yuen-Zhou and A. Aspuru-Guzik, “Quantum process tomography of excitonic dimers from two-dimensional electronic spectroscopy. I. General theory and application to homodimers,” J. Chem. Phys. **134**, 134505 (2011). [CrossRef]

**41. **J. Yuen-Zhou, D. H. Arias, D. M. Eisele, C. P. Steiner, J. J. Krich, M. G. Bawendi, K. A. Nelson, and A. Aspuru-Guzik, “Coherent exciton dynamics in supramolecular light-harvesting nanotubes revealed by ultrafast quantum process tomography,” ACS Nano **8**, 5527–5534 (2014). [CrossRef]

**42. **L. A. Pachon, A. H. Marcus, and A. Aspuru-Guzik, “Quantum process tomography by 2D fluorescence spectroscopy,” J. Chem. Phys. **142**, 212442 (2015). [CrossRef]

**43. **H. Wang, K. Ferrio, D. G. Steel, Y. Z. Hu, R. Binder, and S. W. Koch, “Transient nonlinear optical response from excitation induced dephasing in GaAs,” Phys. Rev. Lett. **71**, 1261–1264 (1993). [CrossRef]

**44. **T. F. Albrecht, K. Bott, T. Meier, A. Schulze, M. Koch, S. T. Cundiff, J. Feldmann, W. Stolz, P. Thomas, S. W. Koch, and E. O. Göbel, “Disorder mediated biexcitonic beats in semiconductor quantum wells,” Phys. Rev. B **54**, 4436–4439 (1996). [CrossRef]

**45. **C. Sieh, T. Meier, F. Jahnke, A. Knorr, S. W. Koch, P. Brick, M. Hübner, C. Ell, J. Prineas, G. Khitrova, and H. M. Gibbs, “Coulomb memory signatures in the excitonic optical Stark effect,” Phys. Rev. Lett. **82**, 3112–3115 (1999). [CrossRef]

**46. **Y. P. Svirko, M. Shirane, H. Suzuura, and M. Kuwata-Gonokami, “Four-wave mixing theory at the excitonic resonance: weakly interacting Boson model,” J. Phys. Soc. Jpn. **68**, 674–682 (1999). [CrossRef]

**47. **T. Meier, S. W. Koch, M. Phillips, and H. Wang, “Strong coupling of heavy- and light-hole excitons induced by many-body correlations,” Phys. Rev. B **62**, 12605–12608 (2000). [CrossRef]

**48. **S. Weiser, T. Meier, J. Möbius, A. Euteneuer, E. J. Mayer, W. Stolz, M. Hofmann, W. W. Rühle, P. Thomas, and S. W. Koch, “Disorder-induced dephasing in semiconductors,” Phys. Rev. B **61**, 13088–13098 (2000). [CrossRef]

**49. **M. Kuwata-Gonokami, T. Aoki, C. Ramkumar, R. Shimano, and Y. Svirko, “Role of exciton–exciton interaction on resonant third-order nonlinear optical responses,” J. Lumin. **87–89**, 162–167 (2000). [CrossRef]

**50. **G. Rochat, C. Ciuti, V. Savona, C. Piermarocchi, A. Quattropani, and P. Schwendimann, “Excitonic Bloch equations for a two-dimensional system of interacting excitons,” Phys. Rev. B **61**, 13856–13862 (2000). [CrossRef]

**51. **S. Rudin and T. L. Reinecke, “Anharmonic oscillator model for driven and vacuum-field Rabi oscillations,” Phys. Rev. B **63**, 075308 (2001). [CrossRef]

**52. **R. Singh, T. Suzuki, T. M. Autry, G. Moody, M. E. Siemens, and S. T. Cundiff, “Polarization-dependent exciton linewidth in semiconductor quantum wells: a consequence of bosonic nature of excitons,” Phys. Rev. B **94**, 081304 (2016). [CrossRef]

**53. **P. Tian, D. Keusters, Y. Suzaki, and W. S. Warren, “Femtosecond phase-coherent two-dimensional spectroscopy,” Science **300**, 1553–1555 (2003). [CrossRef]

**54. **W. Wagner, P. Tian, C. Li, J. Semmlow, and W. S. Warren, “Rapid two-dimensional optical spectroscopy through acousto-optic pulse shaping,” J. Mod. Opt. **51**, 2655–2663 (2004). [CrossRef]

**55. **E. M. Grumstrup, S.-H. Shim, M. A. Montgomery, N. H. Damrauer, and M. T. Zanni, “Facile collection of two-dimensional electronic spectra using femtosecond pulse-shaping technology,” Opt. Express **15**, 16681–16689 (2007). [CrossRef]

**56. **J. A. Myers, K. L. M. Lewis, P. F. Tekavec, and J. P. Ogilvie, “Two-color two-dimensional Fourier transform electronic spectroscopy with a pulse-shaper,” Opt. Express **16**, 17420–17428 (2008). [CrossRef]

**57. **D. B. Turner, K. W. Stone, K. Gundogdu, and K. A. Nelson, “Invited article: the coherent optical laser beam recombination technique (COLBERT) spectrometer: coherent multidimensional spectroscopy made easier,” Rev. Sci. Instrum. **82**, 081301 (2011). [CrossRef]

**58. **D. R. Skoff, J. E. Laaser, S. S. Mukherjee, C. T. Middleton, and M. T. Zanni, “Simplified and economical 2D IR spectrometer design using a dual acousto-optic modulator,” Chem. Phys. **422**, 8–15 (2013). [CrossRef]

**59. **F. D. Fuller, D. E. Wilcox, and J. P. Ogilvie, “Pulse shaping based two-dimensional electronic spectroscopy in a background free geometry,” Opt. Express **22**, 1018–1027 (2014). [CrossRef]

**60. **J. O. Tollerud, C. R. Hall, and J. A. Davis, “Isolating quantum coherence using coherent multi-dimensional spectroscopy with spectrally shaped pulses,” Opt. Express **22**, 6719–6733 (2014). [CrossRef]

**61. **Y. Rodriguez, F. Frei, A. Cannizzo, and T. Feurer, “Pulse-shaping assisted multidimensional coherent electronic spectroscopy,” J. Chem. Phys. **142**, 212451 (2015). [CrossRef]

**62. **D. Brida, C. Manzoni, and G. Cerullo, “Phase-locked pulses for two-dimensional spectroscopy by a birefringent delay line,” Opt. Lett. **37**, 3027–3029 (2012). [CrossRef]

**63. **B. Lomsadze, B. C. Smith, and S. T. Cundiff, “Tri-comb spectroscopy,” Nat. Photonics **12**, 676–680 (2018). [CrossRef]

**64. **B. Lomsadze and S. T. Cundiff, “Frequency-comb based double-quantum two-dimensional spectrum identifies collective hyperfine resonances in atomic vapor induced by dipole-dipole interactions,” Phys. Rev. Lett. **120**, 233401 (2018). [CrossRef]

**65. **B. Lomsadze and S. T. Cundiff, “Frequency combs enable rapid and high-resolution multidimensional coherent spectroscopy,” Science **357**, 1389–1391 (2017). [CrossRef]

**66. **V. Volkov, R. Schanz, and P. Hamm, “Active phase stabilization in Fourier-transform two-dimensional infrared spectroscopy,” Opt. Lett. **30**, 2010–2012 (2005). [CrossRef]

**67. **A. D. Bristow, D. Karaiskaj, X. Dai, T. Zhang, C. Carlsson, K. R. Hagen, R. Jimenez, and S. T. Cundiff, “A versatile ultrastable platform for optical multidimensional Fourier-transform spectroscopy,” Rev. Sci. Instrum. **80**, 073108 (2009). [CrossRef]

**68. **T. Brixner, T. Mančal, I. V. Stiopkin, and G. R. Fleming, “Phase-stabilized two-dimensional electronic spectroscopy,” J. Chem. Phys. **121**, 4221–4236 (2004). [CrossRef]

**69. **T. Brixner, I. V. Stiopkin, and G. R. Fleming, “Tunable two-dimensional femtosecond spectroscopy,” Opt. Lett. **29**, 884–886 (2004). [CrossRef]

**70. **M. L. Cowan, J. P. Ogilvie, and R. J. D. Miller, “Two-dimensional spectroscopy using diffractive optics based phased-locked photon echoes,” Chem. Phys. Lett. **386**, 184–189 (2004). [CrossRef]

**71. **K. L. Hall, G. Lenz, E. P. Ippen, and G. Raybon, “Heterodyne pump-probe technique for time-domain studies of optical nonlinearities in waveguides,” Opt. Lett. **17**, 874–876 (1992). [CrossRef]

**72. **P. F. Tekavec, G. A. Lott, and A. H. Marcus, “Fluorescence-detected two-dimensional electronic coherence spectroscopy by acousto-optic phase modulation,” J. Chem. Phys. **127**, 214307 (2007). [CrossRef]

**73. **G. Nardin, T. M. Autry, K. L. Silverman, and S. T. Cundiff, “Multidimensional coherent photocurrent spectroscopy of a semiconductor nanostructure,” Opt. Express **21**, 28617–28627 (2013). [CrossRef]

**74. **A. A. Bakulin, C. Silva, and E. Vella, “Ultrafast spectroscopy with photocurrent detection: watching excitonic optoelectronic systems at work,” J. Phys. Chem. Lett. **7**, 250–258 (2016). [CrossRef]

**75. **V. Tiwari, Y. A. Matutes, A. T. Gardiner, T. L. C. Jansen, R. J. Cogdell, and J. P. Ogilvie, “Spatially-resolved fluorescence-detected two-dimensional electronic spectroscopy probes varying excitonic structure in photosynthetic bacteria,” Nat. Commun. **9**, 4219 (2018). [CrossRef]

**76. **P. F. Tekavec, T. R. Dyke, and A. H. Marcus, “Wave packet interferometry and quantum state reconstruction by acousto-optic phase modulation,” J. Chem. Phys. **125**, 194303 (2006). [CrossRef]

**77. **K. J. Karki, J. Chen, A. Sakurai, Q. Shi, A. T. Gardiner, O. Kühn, R. J. Cogdell, and T. Pullerits, “Unexpectedly large delocalization of the initial excitation in photosynthetic light harvesting,” arXiv:1804.04840 (2018).

**78. **S. T. Roberts, J. J. Loparo, K. Ramasesha, and A. Tokmakoff, “A fast-scanning Fourier transform 2D IR interferometer,” Opt. Commun. **284**, 1062–1066 (2011). [CrossRef]

**79. **J. Helbing and P. Hamm, “Compact implementation of Fourier transform two-dimensional IR spectroscopy without phase ambiguity,” J. Opt. Soc. Am. B **28**, 171–178 (2011). [CrossRef]

**80. **S. Draeger, S. Roeding, and T. Brixner, “Rapid-scan coherent 2D fluorescence spectroscopy,” Opt. Express **25**, 3259–3267 (2017). [CrossRef]

**81. **B. Lomsadze and S. T. Cundiff, “Multi-heterodyne two dimensional coherent spectroscopy using frequency combs,” Sci. Rep. **7**, 14018 (2017). [CrossRef]

**82. **P. Wen, G. Christmann, J. J. Baumberg, and K. A. Nelson, “Influence of multi-exciton correlations on nonlinear polariton dynamics in semiconductor microcavities,” New J. Phys. **15**, 025005 (2013). [CrossRef]

**83. **S. Yu, M. Titze, Y. Zhu, X. Liu, and H. Li, “Observation of scalable and deterministic multi-atom Dicke states in an atomic vapor,” arXiv:1807.09300 (2018).

**84. **G. S. M. Jansen, D. Rudolf, L. Freisem, K. S. E. Eikema, and S. Witte, “Spatially resolved Fourier transform spectroscopy in the extreme ultraviolet,” Optica **3**, 1122–1125 (2016). [CrossRef]

**85. **M. E. Siemens, G. Moody, H. Li, A. D. Bristow, and S. T. Cundiff, “Resonance lineshapes in two-dimensional Fourier transform spectroscopy,” Opt. Express **18**, 17699–17708 (2010). [CrossRef]

**86. **X. Li, T. Zhang, C. N. Borca, and S. T. Cundiff, “Many-body interactions in semiconductors probed by optical two-dimensional Fourier transform spectroscopy,” Phys. Rev. Lett. **96**, 057406 (2006). [CrossRef]

**87. **T. Zhang, I. Kuznetsova, T. Meier, X. Li, R. P. Mirin, P. Thomas, and S. T. Cundiff, “Polarization-dependent optical 2D Fourier transform spectroscopy of semiconductors,” Proc. Natl. Acad. Sci. USA **104**, 14227–14232 (2007). [CrossRef]

**88. **D. B. Turner and K. A. Nelson, “Coherent measurements of high-order electronic correlations in quantum wells,” Nature **466**, 1089–1092 (2010). [CrossRef]

**89. **H.-S. Tan, “Theory and phase-cycling scheme selection principles of collinear phase coherent multi-dimensional optical spectroscopy,” J. Chem. Phys. **129**, 124501 (2008). [CrossRef]

**90. **D. S. Chemla and D. A. B. Miller, “Room-temperature excitonic nonlinear-optical effects in semiconductor quantum-well structures,” J. Opt. Soc. Am. B **2**, 1155–1173 (1985). [CrossRef]

**91. **B. Deveaud, F. Clérot, N. Roy, K. Satzke, B. Sermage, and D. S. Katzer, “Enhanced radiative recombination of free excitons in GaAs quantum wells,” Phys. Rev. Lett. **67**, 2355–2358 (1991). [CrossRef]

**92. **K. Bott, “Influence of exciton-exciton interactions on the coherent optical response in GaAs quantum wells,” Phys. Rev. B **48**, 17418–17426 (1993). [CrossRef]

**93. **J. M. Shacklette and S. T. Cundiff, “Role of excitation-induced shift in the coherent optical response of semiconductors,” Phys. Rev. B **66**, 045309 (2002). [CrossRef]

**94. **J. O. Tollerud, S. T. Cundiff, and J. A. Davis, “Revealing and characterizing dark excitons through coherent multidimensional spectroscopy,” Phys. Rev. Lett. **117**, 097401 (2016). [CrossRef]

**95. **R. Singh, T. M. Autry, G. Nardin, G. Moody, H. Li, K. Pierz, M. Bieler, and S. T. Cundiff, “Anisotropic homogeneous linewidth of the heavy-hole exciton in (110)-oriented GaAs quantum wells,” Phys. Rev. B **88**, 045304 (2013). [CrossRef]

**96. **M. Gärttner, P. Hauke, and A. M. Rey, “Relating out-of-time-order correlations to entanglement via multiple-quantum coherences,” Phys. Rev. Lett. **120**, 040402 (2018). [CrossRef]