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Analytical calculation of two-dimensional spectra Joshua D. Bell, Rebecca Conrad, and Mark E. Siemens* Department of Physics and Astronomy, University of Denver, 2112 East Wesley Avenue, Denver, Colorado 80208, USA *Corresponding author: [email protected] Received December 11, 2014; accepted February 9, 2015; posted February 18, 2015 (Doc. ID 230502); published March 17, 2015 We demonstrate an analytical calculation of two-dimensional (2D) coherent spectra of electronic or vibrational resonances. Starting with the solution to the optical Bloch equations for a two-level system in the 2D time domain, we show that a fully analytical 2D Fourier transform can be performed if the projection-slice and Fourier-shift theorems of Fourier transforms are applied. Results can be fit to experimental 2D coherent spectra of resonances with arbitrary inhomogeneity. © 2015 Optical Society of America OCIS codes: (300.3700) Linewidth; (300.6240) Spectroscopy, coherent transient; (300.6290) Spectroscopy, four-wave mixing; (300.6300) Spectroscopy, Fourier transforms; (300.6210) Spectroscopy, atomic; (300.6470) Spectroscopy, semiconductors. http://dx.doi.org/10.1364/OL.40.001157
Two-dimensional coherent spectroscopy (2DCS) has recently emerged as an important tool for direct experimental measurements of nanoscale interactions and quantum transport in biological and solid-state systems. 2DCS using visible light provides deep insight into electronic properties, including quantification of many-body effects in semiconductor quantum wells [1,2] and revealing of unsuspected coherences in photosynthetic proteins [3–5] and conjugated polymers [6]. In the infrared, 2DCS yields unique measurements of structural dynamics in proteins, peptides, and other complex molecules that cannot be observed from linear spectra [7–9]. Once a 2D spectrum has been obtained, retrieving quantitative physical properties related to dephasing dynamics or coupling times requires comparison with a model [9]. Although analytical models are faster and provide intuitive insight, only numerical simulations have been used so far. This is because an analytical calculation of 2D lineshapes cannot be done by direct Fourier transformation of the time domain solution to the optical Bloch equations. Recent work has produced analytical expressions for a 1D slice from a 2D spectrum [10], but complete 2D spectra remain analytically intractable except for homogeneous [11] or inhomogeneous limits. In this report, we present an analytical calculation of the full 2D coherent spectrum for a resonance with arbitrary homogeneous and inhomogeneous broadening. These parameters are critical to quantify for coherent control applications and materials characterization, because the homogeneous linewidth γ is the pure coherence dephasing rate, while the inhomoeneous linewidth σ is due to energy shifts caused by size distributions or changing binding effects from local heterogeneity. We bypass the two analytically unworkable Fourier transforms to a single feasible Fourier transform by applying the projection-slice and Fourier-shift theorems. Finally, we demonstrate the utility of the calculated lineshapes by fitting them to experimental data to extract physical dephasing properties. We model an optical resonance as a two-level system and consider a standard three-pulse transient four-wave mixing (TFWM) setup with three optical pulses (A, B, and C) incident with wavevectors kA , kB , and kC respectively. There is a time delay τ between the first and second pulses, T between the second and third 0146-9592/15/071157-04$15.00/0
pulses, and t from the third pulse to the signal. The optical Bloch equations for this system can be solved with perturbation theory in the rotating wave and Markovian approximations. The T-dependence can be neglected because the signal is independent of T, and we consider scans at a particular value of fixed T. Assuming delta function pulses and selecting only the signal generated in the direction of the phase-matched TFWM signal, ksig kB kC − kA , the solution is st; τ s0 e−γtτiω0 t−τσ
2 t−τ2 ∕2
ΘtΘτ;
(1)
where s0 is the time zero amplitude, ω0 is the resonance frequency, and Θ is the unit step function enforcing causality. The above expression is the “rephasing” signal arising for positive τ when conjugate pulse A arrives first. Equation (1) can be simplified by rotating to pthe more natural 2D time frame along the t0 t τ∕ 2 (homo0 geneous pdecay along the photon echo) and τ t − τ∕ 2 (oscillation and inhomogeneous decay) time directions, yielding p p 0 0 2 02 st0 ; τ0 s0 e−γ 2t i 2ω0 τ σ τ Θt0 τ0 Θt0 − τ0 : (2) p A factor of 2 was omitted from the step functions because their arguments can be scaled arbitrarily. According to the Fourierpshift theorem, the signal oscillation in τ0 at frequency 2ω0 is equivalent to a shift along the ωτ0 axis. To simplify the calculation, we shift to the resonance frequency center p by multiplying the time 0 domain signal st0 ; τ0 by eiω0 2τ . We will shift the calculated signal in the 2D frequency domain to account for this, as described in Eq. (6), yielding p i 2ω 0 τ 0 e sω0 t0 ; τ0 st0 ; τ0 s0 p 0 2 02 e−γ 2t σ τ Θt0 τ0 Θt0 − τ0 : (3)
The rotation of coordinates simplifies the exponential considerably, but direct 2D Fourier transformation of t0 and τ0 remains impossible because of the coordinate mixing in the Θ functions. We will apply the projection-slice and Fourier-shift theorems to enable analytical Fourier transformation. © 2015 Optical Society of America
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The projection-slice theorem states that the Fourier transform of a projection (integral of the signal along the perpendicular direction along an axis) in a particular ˆ x direction in the 2D direction xˆ is equal to a slice in the ω Fourier domain [12]. The projection-slice theorem allows us to calculate a cross-diagonal slice in the 2D spectral domain by Fourier transforming a projection onto a cross-diagonal axis in the 2D time domain. The diagonal shift at which this slice is taken in the 2D frequency domain is determined by the Fourier shift theorem, ˆ x direction which states that a shift of Δωx along the ω in the 2D frequency domain is equivalent to multiplication by e−iΔωx x in the 2D time domain [13]. The calculation produces the same results for either slices along ωt0 and shifts along ωτ0 , or slices along ωτ0 and shifts along ωt0 ; we will proceed with the former. The projection onto the t0 axis, shifted from resonance frequency center by Δω in the ωτ0 direction is written as Z∞ 0 sProj t0 ;ω0 Δω sω0 t0 ;τ0 eiτ Δω dτ0 −∞ Z t0 p 0 0 2 02 e− 2γt eiτ Δω−σ τ dτ0 −t0 p p π − 2γt0 Δω22 4σ Θt0 e 2σ h h Δωi Δωi 0 0 −Erf −σt i ; (4) × Erf σt i 2σ 2σ where Erf is the error function. By the projection-slice theorem, the Fourier transform of sProj t0 ; ω0 ωτ0 (the projection onto the t0 axis) is S slice t0 ; ω0 ωτ0 , so we can calculate the slice in the 2D frequency domain: 0
S slice ωt0 ;ω0 Δω F sProj t ;ω0 Δω p p 2 2 2γ i 2ωt0 −−2γ 2i 2γΔωω2 t0 Δωωt0 4σ e 4σ2γ 2 ω2t0 Δωip2γω 0 p t 2γ iΔω − ωt0 2 σ × e Erfc 2σ p 2γ − iΔω ωt0 ; (5) Erfc 2σ
Fig. 1.
where Erfc is the complementary error function. Figure 1 is a visualization of this mathematical derivation. In order to extend our 1D lineshape slices to a full 2D spectrum calculation, we recognize that Δω in our expressions represents the frequency along the axis perpendicular to ωt0 , that is, Δω ωτ0p . The final steps are to shift the spectrum along ωτ0 by 2ω0 to account for the nonzero resonance frequency and to rotate our 2D spectrum axes to the desired ωt and ωτ axes; the final expression for the complex 2D spectrum is then 1 2σ2γ − iωt ωτ γ−iω −ω 2 t 0 γ − iωt − ω0 p × e 2σ2 Erfc 2σ γ−iωτ ω0 2 γ − iωτ ω0 2 2σ p : (6) Erfc e 2σ Figure 2 shows 2D spectra calculated by plugging values for homogeneous linewidth γ and inhomogeneous linewidth σ into Eq. (6), demonstrating the applicability of this analysis to fitting 2D spectra of arbitrary inhomogeneity. Repeating the above analysis for the nonrephasing signal gives a functionally similar 2D spectrum: S R ωt ; ωτ
S NR ωt ; ωτ
i 2σωt ωτ − 2ω0 γ−iω −ω 2 t 0 γ − iωt − ω0 2 2σ p Erfc × −e 2σ γ−iωτ −ω0 2 γ − iω − ω0 τ 2 p Erfc e 2σ : 2σ
(7)
Correlation spectra can then be calculated by summing Eqs. (6) and (7). We demonstrate the applicability of these analytical lineshapes by fitting to experimental 2DCS data. First, we consider a 2D spectrum of the well-characterized D1 and D2 transitions in potassium (K) vapor. We fit experimental 2D spectra recently published by Li et al. [11]. The K atoms are noninteracting, so the D1 and D2 transitions are highly homogeneous. Because the
Schematic summary of projection-slice and Fourier shift theorems applied to analytically calculate full 2D spectra.
April 1, 2015 / Vol. 40, No. 7 / OPTICS LETTERS
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Table 1. Retrieved Best-Fit Parameters from Fitting 2D Spectra of K-Vapor [11] and GaAs Quantum Well [10] Samplesa Resonance K-Vapor D1 K-Vapor D2 GaAs QW HH GaAs QW LH
γ meV
σ meV
Fit R2
0.137 0.148 0.095 0.269
NA NA 0.441 0.273
0.724 0.934 0.970 0.918
K-vapor D2 resonance fits assumed no inhomogeneity (σ 0) and real and imaginary parts of the data were simultaneously fit to Eq. (8). In the fits to data on the heavy hole exciton in a GaAs quantum well, we fit the absolute value of Eq. (6) to the absolute value of the 2D spectrum. a
1 S homo ωt ; ωτ p : (8) 2π γ iω0 − ωt γ − iω0 ωτ
Fig. 2. Real (first column) and imaginary (second column) 2D spectra are shown, calculated from the analytical expression in Eq. (6). Between rows, γ is held fixed, and σ increases. These 2D spectra have identical axes, with the center frequency ω0 and a frequency spacing of 10γ between ticks.
inhomogeneous linewidth is much smaller than the resolution of our data, analytical 2D lineshapes fail to fit the 2D K-vapor spectra. However, we can simplify the 2D lineshape expression by considering Eq. (6) in the limit σ → 0 (i.e., the homogeneous limit) as follows:
Fig. 3. Experimental 2D data for the D2 transition in K-vapor (top row), 2D fit using Eq. (8) (middle row), and cross-diagonal slices (bottom).
We used the simplified homogeneous 2D lineshape in Eq. (8) to simultaneously fit both real and imaginary parts of the experimental K-vapor 2D spectrum, and results for the D2 resonance are shown in Fig. 3. Fitting parameters for both resonances are summarized in Table 1. Analytical 2D spectra can also be fit to resonances with inhomogeneity, which we demonstrate by fitting to 2DCS data from GaAs quantum wells. The epitaxially grown sample consists of four periods of 10-nm-thick GaAs quantum wells separated by 10-nm Al0.3 Ga0.7 As barriers. The band structure in the valence band leads to two energetically separated resonances corresponding to excitons with heavy and light holes. Many-body effects strongly affect the phase of the signal, so we fit the absolute value 2D spectra to the absolute value of Eq. (6). Data-fit comparisons for the heavy hole exciton are shown in Fig. 4, and best-fit parameters are in Table 1. While the full 2D fit is very strong, it reveals clear differences from the data off of resonance center. These differences are not visible in measurement techniques that use an on-resonance cross-diagonal slice, and
Fig. 4. Left: experimental 2D data for GaAs quantum wells (top left), 2D fit using Eq. (6) (bottom left). Right: crossdiagonal slices 0.4 meV above resonance (top right), on resonance (middle), and 0.4 meV below resonance (bottom right).
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suggest that there could be additional physics involved beyond the assumptions made here. In conclusion, we have derived analytical expressions for calculating full 2D spectra with arbitrary inhomogeneity. These expressions enable direct fitting to experimental data, providing quantitative measurements of both homogeneous and inhomogeneous dephasing rates, and highlighting new areas for study where measured 2D spectra shapes differ from calculations based on dephasing in a simple two-level system. Acknowledgment is made to the donors of the American Chemical Society Petroleum Research Fund for support of this research. The authors thank S. T. Cundiff, R. Singh, and H. Li for very helpful discussions and generously sharing their experimental data for fitting. References 1. X. Li, T. Zhang, C. N. Borca, and S. T. Cundiff, Phys. Rev. Lett. 96, 1 (2006). 2. T. Zhang, I. Kuznetsova, T. Meier, X. Li, R. P. Mirin, P. Thomas, and S. T. Cundiff, Proc. Natl. Acad. Sci. USA 104, 14227 (2007).
3. E. Collini, C. Y. Wong, K. E. Wilk, P. M. G. Curmi, P. Brumer, and G. D. Scholes, Nature 463, 644 (2010). 4. F. D. Fuller, J. Pan, A. Gelzinis, V. Butkus, S. S. Senlik, D. E. Wilcox, C. F. Yocum, L. Valkunas, D. Abramavicius, and J. P. Ogilvie, Nat. Chem. 6, 706 (2014). 5. A. Halpin, P. J. M. Johnson, R. Tempelaar, R. S. Murphy, J. Knoester, T. L. C. Jansen, and M. J. Dwayne, Nat. Chem. 6, 196 (2014). 6. E. Collini and G. D. Scholes, Science 323, 369 (2009). 7. C. T. Middleton, P. Marek, P. Cao, C.-c. Chiu, S. Singh, A. M. Woys, J. J. de Pablo, D. P. Raleigh, and M. T. Zanni, Nat. Chem. 4, 355 (2012). 8. P. Mukherjee, I. Kass, I. T. Arkin, and M. T. Zanni, Proc. Natl. Acad. Sci. USA 103, 3528 (2006). 9. M. T. Zanni, N.-H. Ge, Y. S. Kim, and R. M. Hochstrasser, Proc. Natl. Acad. Sci. 98, 11265 (2001). 10. M. E. Siemens, G. Moody, H. Li, A. D. Bristow, and S. T. Cundiff, Opt. Express 18, 17699 (2010). 11. H. Li, A. D. Bristow, M. E. Siemens, G. Moody, and S. T. Cundiff, Nat. Commun. 4, 1390 (2013). 12. J. W. Goodman, Introduction to Fourier Optics (McGrawHill, 1996). 13. R. R. Ernst, G. Bodenhausen, and A. Wokaun, Principles of Nuclear Magnetic Resonance in One and Two Dimensions (Clarendon, 1988).
1157
Analytical calculation of two-dimensional spectra Joshua D. Bell, Rebecca Conrad, and Mark E. Siemens* Department of Physics and Astronomy, University of Denver, 2112 East Wesley Avenue, Denver, Colorado 80208, USA *Corresponding author: [email protected] Received December 11, 2014; accepted February 9, 2015; posted February 18, 2015 (Doc. ID 230502); published March 17, 2015 We demonstrate an analytical calculation of two-dimensional (2D) coherent spectra of electronic or vibrational resonances. Starting with the solution to the optical Bloch equations for a two-level system in the 2D time domain, we show that a fully analytical 2D Fourier transform can be performed if the projection-slice and Fourier-shift theorems of Fourier transforms are applied. Results can be fit to experimental 2D coherent spectra of resonances with arbitrary inhomogeneity. © 2015 Optical Society of America OCIS codes: (300.3700) Linewidth; (300.6240) Spectroscopy, coherent transient; (300.6290) Spectroscopy, four-wave mixing; (300.6300) Spectroscopy, Fourier transforms; (300.6210) Spectroscopy, atomic; (300.6470) Spectroscopy, semiconductors. http://dx.doi.org/10.1364/OL.40.001157
Two-dimensional coherent spectroscopy (2DCS) has recently emerged as an important tool for direct experimental measurements of nanoscale interactions and quantum transport in biological and solid-state systems. 2DCS using visible light provides deep insight into electronic properties, including quantification of many-body effects in semiconductor quantum wells [1,2] and revealing of unsuspected coherences in photosynthetic proteins [3–5] and conjugated polymers [6]. In the infrared, 2DCS yields unique measurements of structural dynamics in proteins, peptides, and other complex molecules that cannot be observed from linear spectra [7–9]. Once a 2D spectrum has been obtained, retrieving quantitative physical properties related to dephasing dynamics or coupling times requires comparison with a model [9]. Although analytical models are faster and provide intuitive insight, only numerical simulations have been used so far. This is because an analytical calculation of 2D lineshapes cannot be done by direct Fourier transformation of the time domain solution to the optical Bloch equations. Recent work has produced analytical expressions for a 1D slice from a 2D spectrum [10], but complete 2D spectra remain analytically intractable except for homogeneous [11] or inhomogeneous limits. In this report, we present an analytical calculation of the full 2D coherent spectrum for a resonance with arbitrary homogeneous and inhomogeneous broadening. These parameters are critical to quantify for coherent control applications and materials characterization, because the homogeneous linewidth γ is the pure coherence dephasing rate, while the inhomoeneous linewidth σ is due to energy shifts caused by size distributions or changing binding effects from local heterogeneity. We bypass the two analytically unworkable Fourier transforms to a single feasible Fourier transform by applying the projection-slice and Fourier-shift theorems. Finally, we demonstrate the utility of the calculated lineshapes by fitting them to experimental data to extract physical dephasing properties. We model an optical resonance as a two-level system and consider a standard three-pulse transient four-wave mixing (TFWM) setup with three optical pulses (A, B, and C) incident with wavevectors kA , kB , and kC respectively. There is a time delay τ between the first and second pulses, T between the second and third 0146-9592/15/071157-04$15.00/0
pulses, and t from the third pulse to the signal. The optical Bloch equations for this system can be solved with perturbation theory in the rotating wave and Markovian approximations. The T-dependence can be neglected because the signal is independent of T, and we consider scans at a particular value of fixed T. Assuming delta function pulses and selecting only the signal generated in the direction of the phase-matched TFWM signal, ksig kB kC − kA , the solution is st; τ s0 e−γtτiω0 t−τσ
2 t−τ2 ∕2
ΘtΘτ;
(1)
where s0 is the time zero amplitude, ω0 is the resonance frequency, and Θ is the unit step function enforcing causality. The above expression is the “rephasing” signal arising for positive τ when conjugate pulse A arrives first. Equation (1) can be simplified by rotating to pthe more natural 2D time frame along the t0 t τ∕ 2 (homo0 geneous pdecay along the photon echo) and τ t − τ∕ 2 (oscillation and inhomogeneous decay) time directions, yielding p p 0 0 2 02 st0 ; τ0 s0 e−γ 2t i 2ω0 τ σ τ Θt0 τ0 Θt0 − τ0 : (2) p A factor of 2 was omitted from the step functions because their arguments can be scaled arbitrarily. According to the Fourierpshift theorem, the signal oscillation in τ0 at frequency 2ω0 is equivalent to a shift along the ωτ0 axis. To simplify the calculation, we shift to the resonance frequency center p by multiplying the time 0 domain signal st0 ; τ0 by eiω0 2τ . We will shift the calculated signal in the 2D frequency domain to account for this, as described in Eq. (6), yielding p i 2ω 0 τ 0 e sω0 t0 ; τ0 st0 ; τ0 s0 p 0 2 02 e−γ 2t σ τ Θt0 τ0 Θt0 − τ0 : (3)
The rotation of coordinates simplifies the exponential considerably, but direct 2D Fourier transformation of t0 and τ0 remains impossible because of the coordinate mixing in the Θ functions. We will apply the projection-slice and Fourier-shift theorems to enable analytical Fourier transformation. © 2015 Optical Society of America
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The projection-slice theorem states that the Fourier transform of a projection (integral of the signal along the perpendicular direction along an axis) in a particular ˆ x direction in the 2D direction xˆ is equal to a slice in the ω Fourier domain [12]. The projection-slice theorem allows us to calculate a cross-diagonal slice in the 2D spectral domain by Fourier transforming a projection onto a cross-diagonal axis in the 2D time domain. The diagonal shift at which this slice is taken in the 2D frequency domain is determined by the Fourier shift theorem, ˆ x direction which states that a shift of Δωx along the ω in the 2D frequency domain is equivalent to multiplication by e−iΔωx x in the 2D time domain [13]. The calculation produces the same results for either slices along ωt0 and shifts along ωτ0 , or slices along ωτ0 and shifts along ωt0 ; we will proceed with the former. The projection onto the t0 axis, shifted from resonance frequency center by Δω in the ωτ0 direction is written as Z∞ 0 sProj t0 ;ω0 Δω sω0 t0 ;τ0 eiτ Δω dτ0 −∞ Z t0 p 0 0 2 02 e− 2γt eiτ Δω−σ τ dτ0 −t0 p p π − 2γt0 Δω22 4σ Θt0 e 2σ h h Δωi Δωi 0 0 −Erf −σt i ; (4) × Erf σt i 2σ 2σ where Erf is the error function. By the projection-slice theorem, the Fourier transform of sProj t0 ; ω0 ωτ0 (the projection onto the t0 axis) is S slice t0 ; ω0 ωτ0 , so we can calculate the slice in the 2D frequency domain: 0
S slice ωt0 ;ω0 Δω F sProj t ;ω0 Δω p p 2 2 2γ i 2ωt0 −−2γ 2i 2γΔωω2 t0 Δωωt0 4σ e 4σ2γ 2 ω2t0 Δωip2γω 0 p t 2γ iΔω − ωt0 2 σ × e Erfc 2σ p 2γ − iΔω ωt0 ; (5) Erfc 2σ
Fig. 1.
where Erfc is the complementary error function. Figure 1 is a visualization of this mathematical derivation. In order to extend our 1D lineshape slices to a full 2D spectrum calculation, we recognize that Δω in our expressions represents the frequency along the axis perpendicular to ωt0 , that is, Δω ωτ0p . The final steps are to shift the spectrum along ωτ0 by 2ω0 to account for the nonzero resonance frequency and to rotate our 2D spectrum axes to the desired ωt and ωτ axes; the final expression for the complex 2D spectrum is then 1 2σ2γ − iωt ωτ γ−iω −ω 2 t 0 γ − iωt − ω0 p × e 2σ2 Erfc 2σ γ−iωτ ω0 2 γ − iωτ ω0 2 2σ p : (6) Erfc e 2σ Figure 2 shows 2D spectra calculated by plugging values for homogeneous linewidth γ and inhomogeneous linewidth σ into Eq. (6), demonstrating the applicability of this analysis to fitting 2D spectra of arbitrary inhomogeneity. Repeating the above analysis for the nonrephasing signal gives a functionally similar 2D spectrum: S R ωt ; ωτ
S NR ωt ; ωτ
i 2σωt ωτ − 2ω0 γ−iω −ω 2 t 0 γ − iωt − ω0 2 2σ p Erfc × −e 2σ γ−iωτ −ω0 2 γ − iω − ω0 τ 2 p Erfc e 2σ : 2σ
(7)
Correlation spectra can then be calculated by summing Eqs. (6) and (7). We demonstrate the applicability of these analytical lineshapes by fitting to experimental 2DCS data. First, we consider a 2D spectrum of the well-characterized D1 and D2 transitions in potassium (K) vapor. We fit experimental 2D spectra recently published by Li et al. [11]. The K atoms are noninteracting, so the D1 and D2 transitions are highly homogeneous. Because the
Schematic summary of projection-slice and Fourier shift theorems applied to analytically calculate full 2D spectra.
April 1, 2015 / Vol. 40, No. 7 / OPTICS LETTERS
1159
Table 1. Retrieved Best-Fit Parameters from Fitting 2D Spectra of K-Vapor [11] and GaAs Quantum Well [10] Samplesa Resonance K-Vapor D1 K-Vapor D2 GaAs QW HH GaAs QW LH
γ meV
σ meV
Fit R2
0.137 0.148 0.095 0.269
NA NA 0.441 0.273
0.724 0.934 0.970 0.918
K-vapor D2 resonance fits assumed no inhomogeneity (σ 0) and real and imaginary parts of the data were simultaneously fit to Eq. (8). In the fits to data on the heavy hole exciton in a GaAs quantum well, we fit the absolute value of Eq. (6) to the absolute value of the 2D spectrum. a
1 S homo ωt ; ωτ p : (8) 2π γ iω0 − ωt γ − iω0 ωτ
Fig. 2. Real (first column) and imaginary (second column) 2D spectra are shown, calculated from the analytical expression in Eq. (6). Between rows, γ is held fixed, and σ increases. These 2D spectra have identical axes, with the center frequency ω0 and a frequency spacing of 10γ between ticks.
inhomogeneous linewidth is much smaller than the resolution of our data, analytical 2D lineshapes fail to fit the 2D K-vapor spectra. However, we can simplify the 2D lineshape expression by considering Eq. (6) in the limit σ → 0 (i.e., the homogeneous limit) as follows:
Fig. 3. Experimental 2D data for the D2 transition in K-vapor (top row), 2D fit using Eq. (8) (middle row), and cross-diagonal slices (bottom).
We used the simplified homogeneous 2D lineshape in Eq. (8) to simultaneously fit both real and imaginary parts of the experimental K-vapor 2D spectrum, and results for the D2 resonance are shown in Fig. 3. Fitting parameters for both resonances are summarized in Table 1. Analytical 2D spectra can also be fit to resonances with inhomogeneity, which we demonstrate by fitting to 2DCS data from GaAs quantum wells. The epitaxially grown sample consists of four periods of 10-nm-thick GaAs quantum wells separated by 10-nm Al0.3 Ga0.7 As barriers. The band structure in the valence band leads to two energetically separated resonances corresponding to excitons with heavy and light holes. Many-body effects strongly affect the phase of the signal, so we fit the absolute value 2D spectra to the absolute value of Eq. (6). Data-fit comparisons for the heavy hole exciton are shown in Fig. 4, and best-fit parameters are in Table 1. While the full 2D fit is very strong, it reveals clear differences from the data off of resonance center. These differences are not visible in measurement techniques that use an on-resonance cross-diagonal slice, and
Fig. 4. Left: experimental 2D data for GaAs quantum wells (top left), 2D fit using Eq. (6) (bottom left). Right: crossdiagonal slices 0.4 meV above resonance (top right), on resonance (middle), and 0.4 meV below resonance (bottom right).
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suggest that there could be additional physics involved beyond the assumptions made here. In conclusion, we have derived analytical expressions for calculating full 2D spectra with arbitrary inhomogeneity. These expressions enable direct fitting to experimental data, providing quantitative measurements of both homogeneous and inhomogeneous dephasing rates, and highlighting new areas for study where measured 2D spectra shapes differ from calculations based on dephasing in a simple two-level system. Acknowledgment is made to the donors of the American Chemical Society Petroleum Research Fund for support of this research. The authors thank S. T. Cundiff, R. Singh, and H. Li for very helpful discussions and generously sharing their experimental data for fitting. References 1. X. Li, T. Zhang, C. N. Borca, and S. T. Cundiff, Phys. Rev. Lett. 96, 1 (2006). 2. T. Zhang, I. Kuznetsova, T. Meier, X. Li, R. P. Mirin, P. Thomas, and S. T. Cundiff, Proc. Natl. Acad. Sci. USA 104, 14227 (2007).
3. E. Collini, C. Y. Wong, K. E. Wilk, P. M. G. Curmi, P. Brumer, and G. D. Scholes, Nature 463, 644 (2010). 4. F. D. Fuller, J. Pan, A. Gelzinis, V. Butkus, S. S. Senlik, D. E. Wilcox, C. F. Yocum, L. Valkunas, D. Abramavicius, and J. P. Ogilvie, Nat. Chem. 6, 706 (2014). 5. A. Halpin, P. J. M. Johnson, R. Tempelaar, R. S. Murphy, J. Knoester, T. L. C. Jansen, and M. J. Dwayne, Nat. Chem. 6, 196 (2014). 6. E. Collini and G. D. Scholes, Science 323, 369 (2009). 7. C. T. Middleton, P. Marek, P. Cao, C.-c. Chiu, S. Singh, A. M. Woys, J. J. de Pablo, D. P. Raleigh, and M. T. Zanni, Nat. Chem. 4, 355 (2012). 8. P. Mukherjee, I. Kass, I. T. Arkin, and M. T. Zanni, Proc. Natl. Acad. Sci. USA 103, 3528 (2006). 9. M. T. Zanni, N.-H. Ge, Y. S. Kim, and R. M. Hochstrasser, Proc. Natl. Acad. Sci. 98, 11265 (2001). 10. M. E. Siemens, G. Moody, H. Li, A. D. Bristow, and S. T. Cundiff, Opt. Express 18, 17699 (2010). 11. H. Li, A. D. Bristow, M. E. Siemens, G. Moody, and S. T. Cundiff, Nat. Commun. 4, 1390 (2013). 12. J. W. Goodman, Introduction to Fourier Optics (McGrawHill, 1996). 13. R. R. Ernst, G. Bodenhausen, and A. Wokaun, Principles of Nuclear Magnetic Resonance in One and Two Dimensions (Clarendon, 1988).
