## Abstract

We derive an analytical form for resonance lineshapes in two-dimensional (2D) Fourier transform spectroscopy. Our starting point is the solution of the optical Bloch equations for a two-level system in the 2D time domain. Application of the projection-slice theorem of 2D Fourier transforms reveals the form of diagonal and cross-diagonal slices in the 2D frequency data for arbitrary inhomogeneity. The results are applied in quantitative measurements of homogeneous and inhomogeneous broadening of multiple resonances in experimental data.

©2010 Optical Society of America

## 1. Introduction

Two-dimensional Fourier transform spectroscopy (2DFTS) has many advantages over one-dimensional techniques, including isolation of quantum interaction pathways and clear distinction of many-body and biexciton effects. Based on the same concept as nuclear magnetic resonance (NMR) experiments that use radio frequency (RF) radiation to reveal the atomic structure of complex proteins [1,2], 2DFTS has recently been applied at much shorter wavelengths [3,4]. Using pulsed lasers as light sources, these optical analogues of NMR enable access to sub-picosecond resonance dynamics in systems insensitive to RF radiation. For example, molecular vibrations have been extensively studied using 2DFTS in the infrared [5,6], while visible 2DFTS has been used to investigate electronic transitions in dye molecules [7], photosynthetic processes [8], and semiconductor nanostructures [9]. 2DFTS provides clear insight into coherent and incoherent coupling processes in each of these systems.

2DFTS can also clearly separate the homogenous broadening of individual oscillators from inhomogeneous broadening due to sample irregularities in semiconductors [10] or Doppler broadening in atomic vapors [11]. This ability is in contrast to linear spectroscopies such as absorption and photoluminescence, which yield a linewidth that is the combination of the homogeneous and inhomogeneous broadenings [12]. Nonlinear one-dimensional techniques such as four-wave mixing (FWM) can isolate the homogeneous broadening of a single resonance in the homogeneous or inhomogeneous limit [13], but in more complicated systems containing both homogeneous and inhomogeneous broadening or multiple resonances, 2DFTS gives clearer insight.

A glance at a 2DFT spectrum can give a qualitative sense of the inhomogeneity in a system: for a given resonance, the linewidth in the cross-diagonal direction is associated with homogeneous broadening, while the diagonal linewidth is related to inhomogeneous broadening, as shown in Fig. 1
. However, acquiring quantitative information about the homogeneous and inhomogeneous broadenings is more difficult because they are coupled along the diagonal and cross-diagonal directions of the spectrum. Consider the 2DFT spectrum of a purely homogeneously broadened resonance, which has the classic star shape familiar from NMR [1]. In this case, the diagonal and cross-diagonal slices of an absolute value spectrum are Lorentzians with identical widths, and although there is no inhomogeneous broadening here, the diagonal width is not zero. If inhomogeneous broadening is added, the diagonal will broaden as expected, but the cross-diagonal will also widen slightly and change shape, as shown in the inset to Fig. 1. Clearly there is coupling between the diagonal and cross-diagonal widths, and additional insight is needed in order to obtain *quantitative* information about the broadening in a two-dimensional (2D) spectrum.

Past work on 2D lineshapes has focused on managing the coupling between inhomogeneous and homogeneous broadening rather than understanding and isolating the individual contributions [1,14]. The coupling degraded frequency resolution in NMR experiments; windowing functions were used to improve the resolution of resonance peaks, but provided no insight into the connection between the lineshapes and resonance dephasings. A different approach for molecular systems considered both rephasing and nonrephasing signals together, which reduced the coupling [15]. In theoretical work, Tokmakoff derived envelope lineshapes in the homogeneous and inhomogeneous limits from the Fourier transform of an absolute-value 2D time-domain solution of the optical Bloch equations [16]. Phenomenological fitting to simulations was used to obtain correlation information [17,18], as well as ratios of dephasing parameters in the presence of many-body effects [19,20], but a method for determining absolute (quantitative and physically meaningful) homogeneous and inhomogeneous broadening parameters from 2D lineshapes has not yet been presented, to the best of our knowledge.

In this paper, we derive an analytical form for complex resonance lineshapes in 2DFTS signals for arbitrary homogeneous and inhomogeneous linewidths. We begin in the 2D time domain with the solution of the optical Bloch equations for a two-level system. Instead of Fourier transforming this 2D time signal to get the full 2D frequency signal as is usually done, we apply the projection-slice theorem of 2D Fourier transforms. This approach allows us to determine an analytical form of diagonal and cross-diagonal slices in the 2D frequency data. This result provides a method of extracting the absolute homogeneous and inhomogeneous linewidths from a 2D Fourier-transform spectrum with arbitrary amounts of homogeneous and inhomogeneous broadening. We fit the resulting lineshapes to experimental data from semiconductor quantum wells to obtain quantitative homogeneous and inhomogeneous linewidths of multiple resonances.

## 2. 2D time domain

In order to calculate 2D frequency domain lineshapes, we first consider the signal in the 2D time domain. We will see that the expected signal is strongly modified by the requirements of causality and pathway selection, which will also significantly affect the 2D frequency lineshape.

We consider a two-pulse excitation scheme in which *τ* is the time between pulse 1 (incident with wavevector k_{1}) and pulse 2 (k_{2}), and *t* is the time of signal emission after the arrival of the second pulse. This is equivalent to three-pulse excitation with zero delay between the second and third pulses. We begin with the optical Bloch equations (OBEs) for a two-level system, apply perturbation theory and the rotating wave approximation, assume delta-function pulses, and select only the signal emitted in the phase-matched FWM direction 2k_{2}-k_{1} [13]. We work within the Markovian approximation, which results in monotonic exponential decays in time [12]. Including Gaussian inhomogeneous broadening, we find the signal in the 2D time domain

*s*is the amplitude at time zero,

_{0,0}*ω*is the center resonance frequency,

_{0}*γ*is the homogeneous linewidth,

*σ*is the inhomogeneous linewidth, and the

*Θ*’s are unit step functions establishing that a signal cannot be emitted before the pulses are arrive. We consider only the rephasing pulse sequence where the conjugate pulse comes first and a photon echo is emitted at a time

*t*=

*τ*after the arrival of the final pulse. This photon echo can be clearly seen as a sharp ridge along the diagonal in Figs. 2b and c , which shows the real part of the signal field from Eq. (1) in the 2D time domain for various values of inhomogeneous broadening.

The signal can be decomposed into a homogeneous decay along the photon echo direction *t’ = t + τ* and an oscillation multiplied by a Gaussian envelope along the anti-echo *τ’ = t-τ*:

*Θ*functions enforcing causality (or time ordering) involve

*t’*and

*τ’*in an inseparable way. It is this time-ordering that causes the mixing between homogeneous and inhomogeneous broadenings along

*ω*and

_{τ’}*ω*; without it, the homogeneous 2D lineshapes would be simply given by the Fourier transform of the inhomogeneous decay along

_{t’}*ω*and the Fourier transform of the homogeneous decay along

_{τ’}*ω*.

_{t’}## 3. Analytical lineshapes in the 2D frequency domain

#### 3.1 Projection-slice theorem

In order to extend these results to the 2D frequency domain, we apply the projection-slice theorem, a fundamental property of 2D Fourier transforms [21]. A projection onto a line in a particular direction is performed by integrating the signal perpendicular to the line at each point. The projection-slice theorem states that a Fourier transform of this projection is equivalent to a slice in the 2D Fourier pair plane. In our case, Fourier transforming the 2D time domain data projected onto a line at an angle θ with respect to the *t* axis yields a slice in the 2D frequency domain, at the same angle θ from the ω_{t} axis.

A signal oscillating at frequency *ω _{0}* will be shifted along the

*ω*axis by

_{τ’}*ω*on a 2D frequency spectrum, as illustrated in Fig. 3b [22]. This is a result of the

_{0}*e*term in the 2D time signal in Eq. (2); according to the Fourier shift theorem, this oscillation in time translates to a shift along the

^{-iω0τ’}*ω*axis. In order to obtain a slice in the 2D frequency domain that cuts through the resonance peak, we apply a shift from the origin of

_{τ’}*ω*along the

_{0}*ω*axis. Accounting for the shift and normalizing by

_{τ’}*s*, the signal in the 2D time domain will be

_{0,0}*ω*, the resulting oscillations will zero the signal when the projection operation is applied. This demonstrates how 2D spectroscopy is capable of determining properties of multiple resonances or along an inhomogeneous distribution.

_{i}≉ ω_{0}Homogeneous and inhomogeneous broadenings are most clearly separated along the diagonal and cross-diagonal in 2D frequency space, which corresponds to slices along the *ω _{τ’}* and

*ω*directions shown in Fig. 3b. We therefore evaluate projections onto the

_{t’}*τ’*(-π/4 from

*t*) and

*t’*(π /4 from

*t*) axes in the time domain shown, in Fig. 3a, in order to determine slices along the

*ω*and

_{τ’}*ω*axes in the 2D frequency domain, shown in Fig. 3b. We project onto a given axis by integrating the signal perpendicular to the axis and adjusting the limits of integration to account for the time-ordering limits. The projection onto the

_{t’}*t’*axis and centered at

*ω*is illustrated in Fig. 3c and written as

_{0}The projection onto *τ’* is illustrated in Fig. 3d and written as

*t’*and

*τ’*axes are depicted in Fig. 4 for homogeneous, inhomogeneous, and moderate inhomogeneous broadening.

#### 3.2 Inhomogeneous and homogeneous limits

First we consider the inhomogeneous and homogeneous limits to confirm the suitability of the projection-slice theorem. In the case of dominant inhomogeneity, where the distribution of resonance frequencies is much larger than the dephasing rate of a particular resonance (*σ*≫γ), the Gaussian term in Eq. (4) is narrow enough to be treated as a delta-function, and the photon echo signal is restored along the diagonal (*t’ = t + τ*). The projected signal onto *t’ s*
_{ProjIn}
*(t’)* is purely homogeneous dephasing, and the Fourier transform is straightforward to perform:

*τ’*axis will yield the lineshape of a diagonal slice in 2D frequency space. In the inhomogeneous limit, the projection is a very narrow Gaussian, so the slice is Gaussian as well:

*σ*≪γ), enforcing time-ordering (zero signal before

*t*= 0 and

*τ*= 0, as shown in Fig. 3c as the gray triangles) strongly affects the projections. Looking first at the projection onto the

*t’*axis, we return to Eq. (4) and consider the limit of

*σ*➔

*0*. In this case, the Gaussian can be treated as a constant, the projection integral is trivial, and the Fourier transform yields a complex Lorentzian:

*τ’*axis in the homogeneous limit is given by Eq. (5) with the Gaussian term approaching a constant:

Equations (6-9) provide the lineshapes in the inhomogeneous and homogeneous limits. Table 1 summarizes the complex shapes and widths, Fig. 1 shows the amplitude, Fig. 5 the imaginary part, and Fig. 6 the real part of the lineshapes in these limits.

#### 3.3 Arbitrary inhomogeneity

Finally, we investigate the case of moderate inhomogeneity (*σ~γ*), where the homogeneous and inhomogeneous broadenings will each contribute to both diagonal and cross-diagonal lineshapes. The integrals in Eqs. (4) and 5 can be evaluated and Fourier transformed analytically without any restriction on homogeneous or inhomogeneous broadenings.

First we consider projection onto the *t’* axis. Evaluating Eq. (4), we find

## 4. Comparison with experimental data

The analytical lineshape expressions in Eq. (11) and Eq. (13) can be fit to experimental data to obtain quantitative measurement of the homogeneous and inhomogeneous linewidths. As a demonstration, we apply this analysis to 2D spectra obtained from exciton resonances in a GaAs quantum well. The sample and technique are described in detail elsewhere [9,23,24]. Many-body effects (MBE) are known to strongly modify the coherent response of semiconductor nanostructures [10,25,26], and our lineshape analysis has neglected these interactions. We therefore focus our attention on the lineshape amplitude and acknowledge that the homogeneous linewidths reported here include significant excitation-induced many-body effects. The radiative limit of exciton resonances is narrower than the widths reported here, and a quantitative study of linewidth dependence on excitation is an area of future study [27].

A 2D amplitude spectrum obtained from cocircular-polarized excitation is shown in Fig. 7a
. The two resonances along the diagonal correspond to the light-hole exciton (X_{LH}~1556 meV) and heavy-hole exciton (X_{HH}~1547 meV) in a quantum well. The peaks above and below the diagonal indicate coupling from the LH to the HH excitons. Centered at the peak of each resonance, we take diagonal and cross-diagonal slices of the data, shown by the dots in Fig. 7c and d. We then fit Eq. (11) to the cross-diagonal and Eq. (13) to the diagonal slices, using *γ* and *σ* as fitting parameters. The fits and extracted homogeneous and inhomogeneous values are shown in Fig. 7c and d. These values are plugged in to Eq. (1) and Fourier transformed to model the expected 2D signal; the results of the model are shown in Fig. 7b. We see excellent agreement with the experimental lineshapes for both LH and HH excitons. All diagonal and cross-diagonal slices of the model match exactly with the slice fits (not shown), confirming that this is an absolute measurement of homogeneous and inhomogeneous linewidths for both resonances.

Close examination of the cross-diagonal X_{HH} data reveals wings on the experimental data that deviate significantly from the fitted analytical lineshape. These sidebands indicate the presence of non-Markovian behavior in these quantum wells, as observed previously [28]. Clearly, 2DFTS is a powerful tool that may provide insight into the physics governing dynamics in molecular and solid systems, including non-Markovian processes.

## 5. Conclusion

2D Fourier transform spectroscopy is a powerful tool for separating inhomogeneous and homogeneous broadening in coherent signals, but a means for determining quantitative linewidth information from experimental spectra has been missing. We derived analytical expressions for slices of 2D lineshapes from rephasing signals. These lineshapes were determined in the limits of homogeneous and inhomogeneous broadening, as well as arbitrary inhomogeneity. The results can be applied to extract quantitative values of homogeneous and inhomogeneous broadening from experimental 2D signals.

## Acknowledgements

The authors would like to thank David Jonas and Warren Warren for helpful discussions and Richard Mirin for providing samples. This work was supported by National Science Foundation and the Chemical Sciences, Geosciences, and Biosciences Division Office of Basic Energy Sciences, U.S. Department of Energy. MES acknowledges funding from the National Academy of Sciences/National Research Council postdoctoral fellows program.

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