THE JOURNAL OF CHEMICAL PHYSICS 142, 212420 (2015)

Lineshape analysis of coherent multidimensional optical spectroscopy using incoherent light Darin J. Ulness1 and Daniel B. Turner2,a) 1 2

Department of Chemistry, Concordia College, Moorhead, Minnesota 56562, USA Department of Chemistry, New York University, New York, New York 10003, USA

(Received 10 December 2014; accepted 31 March 2015; published online 13 April 2015) Coherent two-dimensional electronic spectroscopy using incoherent (noisy) light, I(4) 2D ES, holds intriguing challenges and opportunities. One challenge is to determine how I(4) 2D ES compares to femtosecond 2D ES. Here, we merge the sophisticated energy-gap Hamiltonian formalism that is often used to model femtosecond 2D ES with the factorized time-correlation formalism that is needed to describe I(4) 2D ES. The analysis reveals that in certain cases the energy-gap Hamiltonian is insufficient to model the spectroscopic technique correctly. The results using a modified energy-gap Hamiltonian show that I(4) 2D ES can reveal detailed lineshape information, but, contrary to prior reports, does not reveal dynamics during the waiting time. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4917320] I. INTRODUCTION

The interpretation of femtosecond 2D ES measurements on biological light-absorbing proteins motivated the development of coherent two-dimensional electronic spectroscopy using incoherent (noisy) light, I(4) 2D ES.1,2 I(4) 2D ES is an extension of a now 30-year old field called noisy-light spectroscopy.3–9 Noisy-light spectroscopy provides a natural apparatus (both theoretical and experimental) for investigating the effects of spectrally incoherent light as an excitation source. Noisy light differs from pulsed light in important ways. Both are broadband and spatially coherent, but the similarities end there. In a femtosecond laser, the frequencies are phase locked to produce a pulse. In a noisy-light source, the phase of each frequency is random. Noisy light is said to be color locked because each frequency is coherent only with itself. This produces a beam with a temporally stochastic electric field. A few simplifications are made regarding noisy light. The light is taken to be cross-spectrally pure,10 which allows for separation of the time and spatial behaviors. In particular, the light is taken to be spatially coherent. Further, the stochastic function of time is taken to obey complex circular Gaussian statistics,11 thus allowing high-order timecorrelation functions (which appear in analyzing nonlinear optical processes) to be expressed in terms of two-point timecorrelation functions. Finally, the stochastic process is taken to be ergodic; this allows time averages to be represented by ensemble averages. Following the ergodic assumption, the Wiener-Khintchine theorem relates the time-correlation function to the spectrum.12 In this paper, we use the term “noisy light” to refer to light that is cross-spectrally pure, circular Gaussian, and ergodic and the term “incoherent light” to refer more generally to all incoherent light. To date, the majority of theoretical treatments of noisylight spectroscopy have employed the Bloch model, resulting a)Electronic mail: [email protected]

0021-9606/2015/142(21)/212420/12/$30.00

in Lorentzian lineshapes. The treatments that investigated other lineshapes—Voigt,13 two versions of the Brownian oscillator,14 and non-Ohmic (Drude) dissipation15—focused only on noisy-light coherent Raman scattering. This paper extends the existing work by developing I(4) 2D ES under the overdamped Brownian oscillator and displaced harmonic oscillator models, which are commonly used to describe the lineshapes of molecules in the condensed phase. The former models solvation dynamics while the latter models underdamped intramolecular vibrations. The 2D electronic spectra therefore yield insight into both the electronic and vibrational degrees of freedom of a molecule. To introduce the analysis method, we quickly describe the zerothorder detection process and linear (first-order) absorption spectroscopy. We then present I(4) 2D ES using frequencydomain heterodyne detection. Numeric simulations are used to elucidate details of the models when analytic solutions become unmanageable.

II. ZEROTH-ORDER INTERFEROGRAM

We present the mathematics of noisy-light interferometry at zeroth order by considering a Michelson interferometer without a sample present. The goal is to determine the stochastically averaged intensity at the detector as a function of the interferometric delay time, η. We depict this in Fig. 1. The presentation here is in direct analogy with the higher-order calculations in the rest of the paper. The total scalar electric field at the detector at time t is Etot(t) = EB(t) + EB′(t). Detection occurs at the intensity level, so I(η) = |EB(t) + EB′(t)|2

= EB(t) + EB′(t) EB∗ (s) + EB∗ ′(s) .

(1)

There is no need for distinct timelines, t and s, but they are required when material is involved. More broadly, in the 142, 212420-1

© 2015 AIP Publishing LLC

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field. Because the ω terms will cancel, we do not include them in the argument of I(η). We use the ergodicity of the field to express the integral over t as a stochastic (ensemble) average T ( T1 2T dt ⇒ ⟨. . .⟩), and we anticipate that the integrand is −2 a function of t − s so that we can express the s (Riemann) integral as a Stieltjes integral with differential d(t − s), hence  +∞  +∞ I(η) ∝ dω S D J (ω S ) d(t − s) e+iω S (t−s) −∞ −∞ ( × ⟨p(t)p∗(s)⟩ e−iω(t−s) + ⟨p(t)p∗(s − η)⟩ e−iω(t−s+η) + ⟨p(t − η)p∗(s)⟩ e−iω(t−s−η) ) + ⟨p(t − η)p∗(s − η)⟩ e−iω(t−s) .

FIG. 1. Depictions of the time-correlation process and the experimental apparatus for zeroth-order detection using a Michelson interferometer without a sample (left) and linear absorption spectroscopy of a sample using a CCD (right). The ∗ indicates a common time point on stochastic electric fields.

mathematical treatment of all noisy-light experiments— including this work—one develops the appropriate polarization, usually via perturbation theory. The conjugate polarization must be developed independently on a second generic chromophore (on a second timeline); the polarizations are then multiplied and stochastically averaged to describe the observed noisy-light signal intensity.16 The stochastic average is essential and nontrivial because one must account for intensity-level signals arising from cross terms between the two polarizations at any time. Even though incorporating two timelines complicates the present problem in which there is no sample, it permits this calculation to serve as a model. For modelling many femtosecond experiments, one also finds the total intensity by considering polarizations developed from two chromophores. This is most relevant for some homodyne-detected measurements such as those performed to distinguish between quantum beats and polarization interference.17,18 The distinction between noisy-light and femtosecond measurements is that the two polarizations are time coincident (or near so) in femtosecond measurements, but in noisy light one must consider polarizations from any time. We work with the complex analytic fields rather than the real fields and take the real part of the final answer. The interferogram—with detector response D J (ω S )—is  T  +∞  +∞ 2 1 I(η) ∝ dt dω S D J (ω S ) ds e+iω S (t−s) T − T2 −∞ −∞ ( × p(t)p∗(s)e−iω(t−s) + p(t)p∗(s − η)e−iω(t−s+η) + p(t − η)p∗(s)e−iω(t−s−η) ) + p(t − η)p∗(s − η)e−iω(t−s) ,

(2)

where ω is the carrier frequency and p(t) is a complex stochastic function representing the spectrally incoherent light

(3)

Strictly speaking, this is only true in the limit T → ∞, but since T is much greater than any dynamical variable, this limit is effectively realized in all spectroscopy applications. At this point, a spectral density for the light, J(ω), must be postulated. Although a spectral density such as Lorentzian or Gaussian could be used,16 we use the “white” spectrum J(ω) = C in this work for computational convenience. This implies ⟨p(ζ)p∗(0)⟩ = δ(ζ).

(4)

Direct calculation involves integration of the time variables. Using Eq. (4) and the stationarity principle, Eq. (3) becomes  +∞  +∞ I(η) ∝ dω S D J (ω S ) d(t − s) e+iω S (t−s) −∞ −∞ ( × 2δ(t − s)e−iω(t−s) + δ(t − s + η)e−iω(t−s+η) ) + δ(t − s − η)e−iω(t−s−η) , (5) where the first term in the parentheses originates from the first and last terms in parentheses of Eq. (3)—they are identical due to stationarity. A change of variables,19 x = t − s, yields  +∞  +∞ ( I(η) ∝ dω S D J (ω S ) dx e+iω S x 2δ(x)e−iω x −∞ −∞ ) −iω(x+η) + δ(x + η)e + δ(x − η)e−iω(x−η) . (6) Upon x integration via the becomes  +∞ I(η) ∝ dω S D J (ω S ) −∞  +∞ ∝ dω S D J (ω S )

δ-functions, the expression 

2 + e+iω S η + e−iω S η

[2 + 2 cos(ω S η)] .



(7) (8)

−∞

In the limit of spectrally integrated (achromatic) detection, D J (ω S ) = 1, and Eq. (8) becomes I(η) = 2 + 2δ(η).

(9)

The achromatic limit represents what a power meter would detect. In the limit of monochromatic detection, D J (ω S ) = δ(ω S − ω D ), and Eq. (8) becomes I0 I0 + cos(ω D η), (10) 2 2 where I0 is a constant of proportionality. There is no decay in η in this limit. The monochromatic limit approximates what I(η) =

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a single pixel on a CCD would detect. In a real instrument, each pixel has finite resolution given by the optics of the spectrometer.

III. LINEAR ABSORPTION SPECTROSCOPY USING NOISY LIGHT A. Frequency-domain detection

In this work, we study heterodyne detection in the frequency domain. As depicted in Fig. 1, the grating in the spectrometer spectrally disperses the signal and residual input light before the CCD detects the total intensity of the fields. ˜ Thus, the signal I(ω) is  2 (1) ˜ I(ω) = E˜out(ω) + E˜sig (ω)  (1)  (1) ∗ = I˜out(ω) + I˜sig (ω) + 2 Re E˜sig (ω) E˜out (ω) . (11) An initial measurement of the input field yields I˜out(ω) for (1) future removal. The homodyne signal contribution, I˜sig (ω), will be negligible in the low-density limit. This leaves the cross terms. Analysis of  (1)  ∗ ˜ ∆ I(ω) = 2 Re E˜sig (ω) E˜out (ω) (12) begins with the signal field  ∞ (1) Esig (t) = i A dτ R(1)(τ)Ein(t − τ),

(13)

0

where A is an amplitude factor, R(1)(τ) is the first-order response function defined below, and the input field is Ein(t) = E0 p(t)e−iωt ,

We then19 make a change of variables (t, s) → (t, x = t − s),  +∞  +∞ 1 (1) ∗ ˜ ∆ I(ω) = Re dt dx Esig (t)Eout (t − x) π −∞ −∞  × e+iωtt e−iωs(t−x) . (19) This procedure resembles a transformation22 using a Stieltjes integral with difference differential d(t − s). The heterodyne substitution (ω ≡ ω t = ω s ) yields  +∞  +∞  1 (1) ∗ +iω x ˜ dx dt Esig (t)Eout(t − x)e ∆ I(ω) = Re . (20) π −∞ −∞ The conjugate variable of ω is x. Recognizing that the integral over t is a time-correlation function yields  +∞    1 (1) ∗ +iω x ˜ dx Esig (t)Eout(t − x) e , (21) ∆ I(ω) = Re t π −∞ where the subscript reminds us that the correlation function occurs over time variable t. This expression does not assume a particular response function and thus retains generality.

(14)

where ω is the carrier frequency, E0 represents the field strength, and p(t) is a complex stochastic function representing the spectrally incoherent light field. The heterodyning reference field is given by Eout(s) = E0 p(s)e−iωs ,

The conjugate variable pairs are (s,ωs) and (t,ωt). Later, meaning after Eq. (19), we set ω ≡ ωt = ωs, which is heterodyne detection in the frequency domain. Anticipating this variable change, we let ∆ I˜ be a function purely of ω. The cross term becomes  (1)  ∗ ˜ ∆ I(ω) = 2 Re E˜sig (ωt) E˜out (ωs)  +∞  +∞  1 (1) ∗ +iω tt−iω s s = Re dt ds Esig (t)Eout(s)e . π −∞ −∞ (18)

(15)

where we assume a negligible amount of sample absorption. The residual input light will act as a local oscillator and therefore must have its own timeline, s. Briefly, the bichromophoric model, first introduced by Hanamura and Mukamel20,21 in the context of short pulse experiments, demands a second timeline. As a consequence, homodyne-detected noisy light experiments require two distinct timelines, one for each chromophore.16 Heterodyne detection is different, but it is productive to treat the local oscillator as if it originated from the second chromophore in the bichromophoric model. Also, the treatment is then consistent with factorized time correlation diagram analysis, which is a powerful tool for analyzing noisylight spectroscopies.9 The spectrometer Fourier transforms the residual input light and the linear signal to the frequency domain,  +∞ 1 E˜out(ωs) = √ ds Eout(s)e+iωss (16) 2π −∞ and  +∞ 1 (1) (1) E˜sig (ωt) = √ dt Esig (t)e+iωtt . (17) 2π −∞

B. Response functions

In general, the first-order response function is   2  2i (22) R(1)(τ) = Θ(τ) µeg Im e−iω eg τ e−g (τ) , ~ where Θ(. . .) represents the Heaviside step function and g(τ) is the lineshape function23 that describes fluctuations of an electronic transition due to its interactions with the environment. The lineshape function for the displaced harmonic oscillator model, describing interactions of a two-level electronic system with an underdamped vibration is, following Ref. 24, ( ) g(τ) = −D e−i(ω0−iγ0)τ − 1 , (23) where D is the Huang-Rhys factor and ω0 is the natural frequency of the harmonic oscillator that decays with damping parameter γ0. The functional form of the model conserves vibrational energy. The Brownian oscillator model describes the interaction of a two-level electronic system with a bath of harmonic oscillators. In the overdamped, high-temperature limit,23 the lineshape is given by ( )
λ 2λk BT −i g(τ) = e−Λτ + Λτ − 1 , (24) Λ ~Λ2 where k B is the Boltzmann constant, T is the temperature, and λ and Λ parametrize the oscillator-bath couplings.

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The displaced harmonic oscillator and overdamped Brownian oscillator have been used extensively in the literature of femtosecond 2D ES25–29 as well as many other third-order nonlinear optical spectroscopy techniques. C. Predicted spectrum for general lineshape

For the general lineshape derived from the energy-gap Hamiltonian, the first-order field, using Eqs. (13) and (22), is  2AE0  2 −iωt ∞ (1) µeg e dτ Θ(τ) Esig (t) = − ~ 0   (25) × Im e−iω eg τ e−g (τ) e+iωτ p(t − τ). Directly inserting the signal and output fields into Eq. (21) yields  2  +∞  ∞ 2AI0 µeg −i(ω−ω)x dτ Re dx e π~ 0 −∞   × Θ(τ)Im e−iω eg τ e−g (τ) e+iωτ 

 × p(t − τ)p∗(t − x) t , (26)

˜ ∆ I(ω) =−

where I0 = |E0|2. Due to the Wiener-Khintchine theorem, the time-correlation function becomes ⟨p(t − τ)p∗(t − x)⟩t = δ(τ − x).

(27)

The δ-function assists the τ integration, leading to  2  +∞ 2AI0 µeg Re dx Θ(x)e+iω x π~ −∞   −iω eg x −g (x) × Im e e .

˜ ∆ I(ω) =−

(28)

To proceed, we must choose a model and insert the appropriate lineshape function. D. Predicted spectrum for displaced harmonic oscillator

∞  1 n −in(ω0−iγ0)x D e . n! n=0

Fig. 2 shows linear absorption spectra under two values of D and constant dephasing and frequency values. We observe the expected Franck–Condon vibronic progression that becomes more pronounced as the Huang-Rhys factor increases. The spacing between peaks depends on the natural frequency of the oscillator, ω0. For the third-order spectra, we focus specifically on the case where D = 1.

E. Predicted spectrum for Brownian oscillator

We can rewrite the expression for the overdamped, hightemperature limit for the Brownian oscillator as
g(x) = (a − ib) e−Λx + Λx − 1 , (31) where a ≡ 2λk BT/(~Λ2) and b ≡ λ/Λ. The signal becomes  2  +∞ 2AI0 µeg ˜ Re dx Θ(x)e+iω x ∆ I(ω) = − π~ −∞   −iω eg x −(a−ib)(e −Λx −Λx−1) , (32) × Im e e which yields an analytic solution using a gamma function and an incomplete gamma function,  2  2AI0 µeg −(a−ib) (a + ib)a+i(b+c) ˜ ∆ I(ω) = − e Re π~ Λ   × Γ(a − i (b + c)) − Γ(a − i (b + c) , a − ib) , (33) where c ≡ ωeg − ω /Λ. Fig. 3 shows a linear absorption spectrum for a temperature of 300 K, λ = 3 THz, andΛ = 5 THz. As expected,


2

For the displaced harmonic oscillator, we follow the usual approach and expand the Huang-Rhys term as e−g (x) = e−D

Including electronic dephasing characterized by γeg , this yields  2 2AI0 µeg −D ˜ e ∆ I(ω) = π~ ∞  1 Dn × . (30) n! (ωeg − iγeg ) + n(ω0 − iγ0) − ω n=0

(29)

~Λ for an intermediate case where 2λk ∼ 1, we observe a BT lineshape that is neither Gaussian nor Lorentzian. In this intermediate modulation limit, the model describes a coupling of the fluctuations that is on the same timescale as nuclear dynamics. By also evaluating the fluorescence spectrum (not shown), we find a Stokes shift of about 2 THz. At third order, we also evaluate this case.

FIG. 2. Linear absorption spectra for the displaced harmonic oscillator under two levels of coupling. The HuangRhys factors are indicated; other parameters (in THz) are ω 0/(2π) = 18, γ 0 = 1, ω eg /(2π) = 500, γ eg = 10.

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EC(t, τ, σ) = E0 p(t + τ − σ)e−iω(t+τ−σ), ∗ (s, κ) ELO

FIG. 3. Linear absorption spectra for the overdamped Brownian oscillator for temperature of 300 K and parameters (in THz) of ω eg /(2π) = 500, λ = 3, and Λ = 5. The Stokes shift is about 2 THz.

IV. THIRD-ORDER NONLINEAR OPTICAL SPECTROSCOPY USING NOISY LIGHT A. Frequency-domain detection

The two examples above reveal how to treat heterodyne detection in the frequency domain and interferometric delays. For I(4) 2D ES, we will incorporate multiple interferometric delays as well as heterodyne detection in the frequency domain. The reference field is a local oscillator instead of residual input, and the signal field is a third-order field. We begin analysis of   ∗ ˜ σ, τ, κ) = 2 Re E˜ (3)(ω t , σ, τ) E˜LO ∆ I(ω, (ω s , κ) (34) sig in the time domain, where  t  (3) Esig (σ, τ,t) ∝ dt 3 −∞



t3

t2

dt 2

−∞

−∞

dt 1 EA(t)EB∗ (t, σ)

× EC(t, σ, τ)S (3)(t 1,t 2,t 3).

(35)

The function S (t 1,t 2,t 3) is the third-order response function. Working with the complex analytic signal, the temporal characteristics of the electric fields of the four input beams that are arranged in the BOX geometry, see Fig. 4, are (3)

EA(t) = E0 p(t)e−iωt , EB∗ (t, σ)

=

E0∗ p∗(t

− σ)e

(36a) +iω(t−σ)

,

(36b)

=

E0∗ p∗(s

− κ)e

(36c)

+iω(s−κ)

,

(36d)

where E0 represents the field strength, p(t) is a complex stochastic function describing the random envelope of the noisy field, and ω is the carrier frequency. The local oscillator has its own timeline, s. The experimentally controlled interferometric delays are σ, τ, ζ, and κ. Interferometric variable ζ is equivalent to the LO delay in femtosecond 2D ES and it is removed by spectral interferometry.30 In numeric simulations, we set ζ = 0 for convenience. Variable κ is redundant, κ = ζ + σ − τ, but we define it in this manner for symmetry in the final expressions. The interferometric variables, which are illustrated in Fig. 4, can be positive or negative, as in the case of the motion of a translation stage. The CCD detects  +∞ 1 ˜ dx e+iω x ∆ I(ω, σ, τ, ζ) = Re π −∞  (3)  ∗ × Esig (σ, τ,t)ELO(κ,t − x) . (37) t

We will eventually Fourier transform over the interferometric ˜ σ , τ,ω). If we delay σ to yield a 2D spectrum given by ∆ I(ω let t I be the interaction time for field EI and suppress the time and frequency arguments of ∆ I˜ for brevity, the signal becomes   +∞ I02 −iω(τ+κ) ˜ ∆ I = Re e dx e−i(ω−ω)x π −∞  t  t3  t2 × dt 3 dt 2 dt 1 −∞ −∞  −∞  × p(t A)p∗(t B − σ)p(t C + τ − σ)p∗(t − x − κ) t  (38) × e−iω(tA−tB+tC−t) S (3)(t 1,t 2,t 3) . Several complications occur because in noisy-light spectroscopy, the experimenter cannot control the time at which a particular noisy field acts on the material. Thus, t A, t B, t C does not imply t A < t B < t C, which is the case in femtosecond 2D ES using δ-function excitation.31 We must select interaction

FIG. 4. The fields and their factorization. (Left) The BOX geometry and heterodyne detection in the frequency domain. (Right) Factorization of the fourpoint time-correlation function leads to a sum of a pair of two-point timecorrelation functions. Subscripts i and ii distinguish the pairs.

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times for the fields, meaning to replace the t I with the t i . This must be done for all possible time orderings of t A, t B, and t C. We must also choose a response function. In noisylight spectroscopy, one treats the four-point time-correlation function by assuming circular complex Gaussian statistics for the stochastic functions. This assumption—used successfully in noisy-light spectroscopy for decades—allows us to express the four-point time-correlation function in terms of products of two-point time-correlation functions11  







∗ ∗ ∗ EC EB∗ . + EA ELO = EA EB∗ EC ELO EA EB∗ EC ELO

(3) SNR (τ3, τ2, τ1) =

( )3 i Θ(τ3)Θ(τ2)Θ(τ1) ~   × Im R1(τ3, τ2, τ1) + R4(τ3, τ2, τ1) ,

(43)

where the response functions are  4 ∗ ∗ R1(τ3, τ2, τ1) = µeg e−iω eg (τ3+τ1)e−g (τ1)e−g (τ2)e−g (τ3) × e+g (τ1+τ2)e+g (τ2+τ3)e−g (τ1+τ2+τ3),  4 R4(τ3, τ2, τ1) = µeg e−iω eg (τ3+τ1)e−g (τ1)e−g (τ2)e−g (τ3) ∗

(44)

× e+g (τ1+τ2)e+g (τ2+τ3)e−g (τ1+τ2+τ3).

(45)

(39) Decomposing the four-point time-correlation function makes the mathematics more tractable and captures the essential features of noisy-light

spectroscopy. 

 Below, the subscript ∗ ∗ i denotes the pair E E E E A C

 B ∗  LO and the subscript ii ∗ denotes the pair EA ELO EC EB . Fig. 4 provides a schematic depiction of the fields and the factorization. We note that Zhang and coworkers have investigated other treatments of the fourpoint time-correlation function including chaotic field, phasediffusion, and Gaussian-amplitude models of noisy light.32–37 B. Model

At this point in the analysis, we must choose a model for the system. In this work, we continue with analysis of the energy-gap Hamiltonian describing time-dependent fluctuations of a two-level electronic system. It is well known that this system has four terms, excluding conjugates. The third-order rephasing signal, SR(3), is ( )3 i SR(3)(τ3, τ2, τ1) = Θ(τ3)Θ(τ2)Θ(τ1) ~   × Im R2(τ3, τ2, τ1)+ R3(τ3, τ2, τ1) . (40) The time arguments, τi , represent interaction time intervals in the system response. This notation is different from Mukamel, see Fig. 5.1 in Ref. 23, but we use it to stay consistent with the noisy-light literature. Moreover, unlike femtosecond methods assuming δ-function excitation, the interaction time intervals inherent to the response functions (τi ) are distinct from the interferometric time delay intervals of the fields (σ, τ, ζ, κ). Unlike femtosecond 2D ES, in I(4) 2D ES, signals from improperly time-ordered pathways can contribute well beyond the overlap duration given by the coherence time. The response functions  4 ∗ ∗ R2(τ3, τ2, τ1) = µeg e−iω eg (τ3−τ1)e−g (τ1)e+g (τ2)e−g (τ3) × e−g (τ1+τ2)e−g (τ2+τ3)e+g (τ1+τ2+τ3),  4 −iω eg (τ −τ ) −g ∗(τ ) +g ∗(τ ) −g (τ ) 3 1 e 1 e 2 e 3 R3(τ3, τ2, τ1) = µeg e ∗

× e−g

∗

∗(τ +τ ) −g ∗(τ +τ ) +g ∗(τ +τ +τ ) 1 2 2 3 1 2 3

e

e

The response function R1 is the stimulated emission term while R4 is the ground-state bleach term. In previous works,1,2 R1 and R4 were labeled PI I and G I I , respectively. Fig. 5 depicts the double-sided Feynman diagrams that represent the pathways. C. Predicted spectrum for general lineshape

Decomposing the four-point time-correlation function and accounting for the interchange of nonconjugate field interactions yields a total of sixteen terms, four for each response function. The treatment follows previous works.1,2 Briefly, we convert the pair correlators to δ-functions, switch to time-interval variables using Tn = t n+1 − t n , identify that τi → Ti , and then use the δ-functions to solve two of the Ti integrals. Below, we list the expressions to this point for all terms, suppressing the transform from σ to ωσ for brevity. The expressions are the first result of this work:  4 2  µeg I0 Θ(σ) Re ie−iω eg (ζ+σ)e−g (σ) PI I αi = ~3π +∞ ∗ × dx Θ(x + ζ)e−i(ω eg −ω)x e−g (x+ζ) −∞ ∞ ∗ ∗ × dT2 Θ(T2)e−g (T2)e+g (T2+x+ζ) 0  +g (σ+T2) −g (σ+T2+x+ζ) ×e e , (46)  4 2  µeg I0 ∗ Θ(−τ) Re ie−iω eg (τ+κ)e−g (−τ) PI I αii = ~3π  ∞ +∞ −i(ω eg −ω)x −g (x+κ) × dx e e dT3 −∞

× Θ(x + κ + τ − T3)Θ(T3)e−g ×e

0

∗(T ) 3

+g (x+κ−T3) +g ∗(T3−τ) −g (x+κ+τ−T3)

e

e

 ,

(47)

(41) (42)

are given by the lineshape function, g(t), using the cumulant expansion to break down the four-point dipole-correlation function.23 R2 is the stimulated emission term while R3 is the ground-state bleach term. In previous works,1,2 R2 and R3 were labeled PI and G I , respectively. The third-order nonrephasing (3) signal SNR is

FIG. 5. As a model, we consider the energy-gap Hamiltonian describing fluctuations of a two-level system. The response functions of the third-order signal are depicted by double-sided Feynman diagrams. Both femtosecond and noisy-light labels are given.

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4  µeg I02 G I I αii = Θ(−τ) Re ie−iω eg (τ+κ)e−g (−τ) ~3π  +∞  ∞ × dx e−i(ω eg −ω)x e−g (x+κ) dT3

4  µeg I02 ∗ Θ(−σ) Re ie−iω eg (ζ+σ)e−g (−σ) = ~3π 





PI I βi

+∞

×

∞

dx e−i(ω eg −ω)x e−g (x+ζ)

−∞

× Θ(x + σ + ζ − T3)Θ(T3)e

dT3 0

−∞

−g ∗(T3)

+g ∗(T3−σ) +g (x+ζ−T3) −g (x+ζ+σ−T3)



×e e e ,  4 2  µeg I0 PI I βii = Θ(τ) Re ie−iω eg (κ+τ)e−g (τ) ~3π +∞

×

PI αi

dx Θ(x + κ)e−i(ω eg −ω)x e−g

+∞

−∞ ∞

(48) G I I βi

∗(x+κ)

−∞ ∞ ∗ ∗ × dT2 Θ(T2)e−g (T2)e+g (T2+x+κ) 0  +g (τ+T2) −g (τ+T2+x+κ) ×e e ,  4 2  µeg I0 ∗ = Θ(−σ) Re ie−iω eg (ζ+σ)e−g (−σ) 3 ~ π ×

−∞

(49) G I I βii

∗(x+ζ)

dx Θ(x + ζ)e−i(ω eg −ω)x e−g

dT2 Θ(T2)e+g (T2)e−g (T2+x+ζ)  −g ∗(T2−σ) +g ∗(T2+x+ζ−σ) ×e e ,  4 2  µeg I0 ∗ Θ(−τ) Re ie−iω eg (τ+κ)e−g (−τ) = ~3π  0

+∞

× −∞

× Θ(−T2 − τ)Θ(x + κ − T2)e

dT2 Θ(T2) 0

× −∞

PI βii

+∞

×

−∞ ∞

G I αii

dx e−i(ω eg −ω)x e−g (x+ζ)

dT2 Θ(T2) 0 −g ∗(−T2−σ) +g (T2)

−∞

(52)

0

(53) G I βii

0

(54)



×e e e . (59)  4 2  µeg I0 ∗ = Θ(−σ) Re ie−iω eg (ζ+σ)e−g (−σ) ~3π  +∞  ∞ −i(ω eg −ω)x −g ∗(x+ζ) × dx e e dT2 Θ(T2) −∞

×

0

∗(−T −τ) 2

+g ∗(T2) −g ∗(x+κ−T2) +g ∗(x+κ−τ−T2)

G I βi

dx Θ(x + ζ)e−i(ω eg −ω)x e−g (x+ζ)

dT2 Θ(T2)e−g (T2)e+g (T2+x+ζ)  × e+g (σ+T2)e−g (σ+T2+x+ζ) ,

(55)

 × e+g (T3−σ)e+g (x+ζ−T3)e−g (x+ζ+σ−T3) , (56)  4 2  µeg I0 Θ(τ) Re ie−iω eg (κ+τ)e−g (τ) = ~3π  +∞ × dx Θ(x + κ)e−i(ω eg −ω)x e−g (x+κ) −∞  ∞ × dT2 Θ(T2)e−g (T2)e+g (T2+x+κ) 0  +g (τ+T2) −g (τ+T2+x+κ) ×e e , (57)  4 2  µeg I0 ∗ = Θ(−σ) Re ie−iω eg (ζ+σ)e−g (−σ) ~3π  +∞ × dx Θ(x + ζ)e−i(ω eg −ω)x e−g (x+ζ) −∞  ∞ ∗ ∗ × dT2 Θ(T2)e+g (T2)e−g (T2+x+ζ) 0  −g ∗(T2−σ) +g ∗(T2+x+ζ−σ) ×e e , (58)  4 2  µeg I0 ∗ ie−iω eg (τ+κ)e−g (−τ) = Θ(−τ) Re ~3π  +∞  ∞ −i(ω eg −ω)x −g ∗(x+κ) × dx e e dT2 Θ(T2) × Θ(−T2 − τ)Θ(x + κ − T2)e−g

−i(ω eg −ω)x −g ∗(x+κ)

dx Θ(x + κ)e e −∞ ∞ ∗ × dT2 Θ(T2)e+g (T2)e−g (T2−τ) 0  −g (T2+x+κ) +g ∗(T2+x+κ−τ) ×e e ,  4 2  µeg I0 = Θ(σ) Re ie−iω eg (ζ+σ)e−g (σ) ~3π ×

(51)

∞

× Θ(x + ζ − T2)Θ(−T2 − σ)e e  ∗ ∗ × e−g (x+ζ−T2)e+g (x+ζ−σ−T2) ,  4 2  µeg I0 ∗ Θ(−τ) Re ie−iω eg (τ+κ)e−g (−τ) = ~3π +∞

G I I αi



×e e e ,  4 2  µeg I0 ∗ Θ(−σ) Re ie−iω eg (ζ+σ)e−g (−σ) = 3 ~ π  +∞

G I αi

−g ∗(−T2−τ)

+g (T2) −g (x+κ−T2) +g ∗(x+κ−τ−T2)

PI βi

(50)

∞

dx e−i(ω eg −ω)x e−g (x+κ)

 × e+g (x+κ−T3)e+g (T3−τ)e−g (x+κ+τ−T3) ,  4 2  µeg I0 = Θ(−σ) Re ie−iω eg (ζ+σ)e−g (−σ) ~3π  +∞  ∞ × dx e−i(ω eg −ω)x e−g (x+ζ) dT3 × Θ(x + σ + ζ − T3)Θ(T3)e−g (T3)

×

PI αii

0

× Θ(x + κ + τ − T3)Θ(T3)e−g (T3)

0

× Θ(x + ζ − T2)Θ(−T2 − σ)e−g (−T2−σ)e+g  −g (x+ζ−T2) +g ∗(x+ζ−σ−T2) ×e e ,  4 2  µeg I0 ∗ = Θ(−τ) Re ie−iω eg (τ+κ)e−g (−τ) ~3π  +∞ × dx Θ(x + κ)e−i(ω eg −ω)x e−g (x+κ) −∞ ∞ ∗ ∗ × dT2 Θ(T2)e+g (T2)e−g (T2−τ) 0  −g ∗(T2+x+κ) +g ∗(T2+x+κ−τ) ×e e . ∗

∗(T ) 2

(60)

(61)

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FIG. 6. For a nonrephasing-like scan, only two terms contribute to the total signal. The top spectra are the magnitude of P I I αi and G I I αi and the bottom row are the real parts. The peaks in the stimulated-emission term are located in a quadrant that reflects the Stokes shift while the peaks from ground-state bleach term are not shifted. The right-hand column is the total signal. We do not normalize the spectra because the relative amplitudes are important for comparison.

The sixteen terms can be categorized by topological descriptions of the associated integration variables.1,2 Eight terms—those with αi and βii subscripts—fall into the unrestricted class. These terms involve full integration over time interval T2. Four other terms—PI I αii , PI I βi , G I I αii , and G I I βi —are in the singly restricted class; these are the only four terms where it was convenient to evaluate the T2 integral rather than the T3 integral. The final four terms—PI αii , PI βi , G I αii , and G I βi —are part of the doubly restricted class. Insertion of the Bloch model lineshape (g(τ) = Γτ) produces expressions that are equivalent to those found previously.2 To perform the final three integrations (the remaining Ti integral and the Fourier transform integrals), we must choose a model for the lineshape and integrate numerically. This is similar to femtosecond treatments. D. Predicted spectrum for linear lineshape

Before we consider the numeric solutions, we can investigate an analytic but approximate solution. In femtosecond spectroscopy, researchers occasionally invoke the approximation38,39 that the lineshape function is linear, meaning g(a + b) = g(a) + g(b). This is equivalent to the Bloch model. Using the term PI I αi as an example, this approximation reduces the expression to  4 2  µeg I0 PI I αi = Θ(σ) Re ie−iω eg (ζ+σ)e−g (σ) ~3π +∞ × dx Θ(x + ζ)e−i(ω eg −ω)x e−g (x+ζ)  −∞ ∞ × dT2 Θ(T2) . (62) 0

The integral over T2 does not converge. In femtosecond measurements, this would not occur because there would be a δ(T2) from the field interactions. Noisy-light spectroscopy reveals that populations must decay, and consequently the traditional energy-gap Hamiltonian is insufficient to model the technique correctly. This discontinuity also occurs in the nonlinearized analyses. We must add a population decay term to the model to enforce convergence of the integral, 

∞

dT2 Θ(T2)e−ΓeeT2 = 0

1 , Γee

(63)

and thereby make the signal finite. This has been worked out for the Bloch model.40 Because the lineshape models used below are nonlinear, we do not use this approximation. Nevertheless, the linear approximation succinctly shows the energy-gap Hamiltonian must be modified to include population decay during T2. The adjustment must be used for both the Brownian oscillator and displaced harmonic oscillator models in this paper, and it may apply more generally. In experiments, the excitation is pulsed on the order of tens of nanoseconds. In such case, the signal will converge due to the finite duration of the excitation, even in the case of very long-lived chemical species. E. Predicted spectrum for displaced harmonic oscillator

For the displaced harmonic oscillator model, we expect a series of peaks throughout the 2D spectrum, spaced by the natural vibrational frequency, ω0, similar to femtosecond 2D ES predictions and measurements. In I(4) 2D ES, the

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FIG. 7. For a rephasing-like scan, six terms contribute to the total signal. The top spectra are the magnitude and real parts of P I αi and G I αi . The peaks in the stimulated-emission term are located in a quadrant that reflects the Stokes shift while the peaks from ground-state bleach term are not shifted. The bottom row γ eg contains spectra from the four weak terms. The terms are weaker than P I αi and G I αi by a factor of about Γee because integration over the population lifetime is restricted. The right-hand column is the sum of all six terms.

spectra can contain a complicated pattern of overlapping peaks. The challenge is to identify a set of interferometricdelay parameters that yields clean signals. The cleanest measurement may be a nonrephasing-like scan where σ varies over positive values while τ is much greater than the optical dephasing. Under these parameters, fourteen terms vanish. The terms with Θ(−σ) and Θ(−τ) immediately vanish, and the terms with Θ(τ) also vanish because they dephase at the optical dephasing rate. The only nonzero terms therefore are PI I αi and G I I αi because they have Θ(σ) and are independent of τ. These terms originate from nonrephasing pathways. In Fig. 6, we display the two terms individually as well 4 as the sum. The simulation parameters were ζ = 0, µeg I02 = ~3π, 0 ≤ x ≤ 1.5 ps in steps of 3 fs, 0 ≤ σ ≤ 1.5 ps in steps of 3 fs, 0 ≤ T2 ≤ 3.5 ps in steps of 1 fs, ω0/(2π) = 18 THz, γ0 = 1 THz, ωeg /(2π) = 500 THz, γeg = 5 THz, D = 1, Γee = 0.1 THz, 400 ≤ ω/(2π) ≤ 600 THz in steps of 1 THz, and 400 ≤ ωσ /(2π) ≤ 600 THz in steps of 1 THz. Under these parameters, the amplitude of the brightest

feature is about 4 × 104. The computation takes about 5 min on a typical workstation using M 2013a. For the nonrephasing-like scan, we predict that the stimulated-emission term produces a series of peaks located to the upper left of the diagonal, where excitation energy is greater than emission energy. The peaks form a twodimensional Franck-Condon progression akin to those predicted for femtosecond 2D ES.25,41 The cross peaks indicate the coupling between the vibrational mode and the electronic transition, and the quadrant in which the peaks are located reflects that the Stokes shift causes emission at red-shifted frequencies. For comparison, the linearized version of the model does not yield a Stokes shift for peaks originating from the stimulated-emission term. The ground-state bleach term leads to a series of peaks that is centered about the diagonal. In a ground-state bleach pathway, there should not be a Stokes shift because only the ground electronic state is probed during time period T2, and peaks’ frequencies should be symmetric with respect to the diagonal. The features result from the

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FIG. 8. The sum of the rephasing-like and nonrephasing-like spectra yields a total spectrum that has sharper lineshapes than either the rephasing-like or nonrephasing-like spectra.

full integration over T2 and are analogous to a fully relaxed femtosecond 2D ES such as Fig. 7 of Ref. 25. Again, the cross peaks report on the electronic-vibrational coupling. The total signal therefore has a series of peaks in two quadrants, the quadrants where excitation energies are greater than the fundamental transition. A second clean scan is a rephasing-like scan where σ varies over negative values while τ is very large. Under these scan parameters, the six terms with Θ(σ) will contribute. Two of those terms, PI αi and G I αi , will be the rephasing terms analogous to those in the nonrephasing-like scan. They will be strong in amplitude under the typical condition where the optical dephasing is faster than the population lifetime. The γ eg other four terms will be weaker by a factor of Γee , which is 50 here. Similar to the nonrephasing-like scan, the series of peaks from the stimulated-emission term is located in the upper left quadrant while the series of peaks from the groundstate bleach term are symmetric about the diagonal. Of the four weak terms, the two that fall into the singly restricted class, PI I αii and G I I αii , originate from nonrephasing pathways and the two that fall into the doubly restricted class, PI βi and G I βi , originate from rephasing pathways. The unusual lineshapes for the weak terms are reminiscent of simulations using the Bloch model.1 Fig. 7 displays the individual terms and the sum of all six terms.

In femtosecond 2D spectroscopy, researchers often sum the rephasing and nonrephasing spectra to produce a total spectrum that has multiple advantages for information extraction.25,42 Phase twist is removed, and the real and imaginary components more cleanly separate absorptive and dispersive components. We find similar advantages here, and Fig. 8 displays the total spectrum in real, imaginary, and magnitude components. The Franck-Condon progression is more sharply resolved. The real part has absorptive lineshapes. We can compare the predicted spectra to those measured in experiments for prototypical dye molecules in solution.2 The spectra predicted here compare favorably to the spectra measured in IR144, where the brightest cross peak involved high-energy excitation and low-energy emission. This is consistent with the spectra in Fig. 8. The I(4) 2D ES of DTP, on the other hand, is less consistent. In particular, the relative intensities of the cross peaks are not reproduced well by the simulation. In the experiment, the cross peak involving highenergy emission and low-energy excitation was the brightest feature. This is likely because DTP photoisomerizes while IR144 does not. The simulations here highlight the anomalous nature of the bright cross peak in the measurement of DTP. A surprising result of the displaced harmonic oscillator model is that there are no dynamics during time period τ. The lack of dynamics can be seen in the analytic expressions

FIG. 9. The total spectrum for the overdamped Brownian oscillator model in real, imaginary, and magnitude components. Due to the Stokes shift, the peak is located above the diagonal by about 2 THz.

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and is not a result of the parameters chosen for the numeric simulation. This contrasts the previous theoretical analysis using the Bloch model1 that included explicit sublevels and yielded coherent oscillations during time period τ. Measurements on the conjugated molecule IR144 observed what appeared to be vibrational coherences,2 supporting the Bloch multilevel analysis.

provides more general insights about the underlying physics of noisy-light spectroscopy. However, such results lack analytic form. The current work thus complements previous FTC diagram analysis for I(4) 2D ES.1,2 To perform analogous experiments will require the development of a light source meeting the properties of noisy light that has bandwidth several times greater than any used previously.

F. Predicted spectrum for Brownian oscillator

ACKNOWLEDGMENTS

Using the methodology developed for the displaced harmonic oscillator model, we can change to the overdamped Brownian oscillator lineshape. The computation parameters were identical to the linear absorption case, where λ = 3 THz and Λ = 5 THz. In Fig. 9, we display the total spectrum in real, imaginary, and magnitude components. The peak is located about 2 THz above the diagonal due to the Stokes shift induced by the coupling. As in the displaced harmonic oscillator model, the shift arises from the stimulated-emission terms, not the ground-state bleach terms. Overall, the spectra for this model have fewer distinct features than the spectra for the underdamped oscillator. This is analogous to femtosecond 2D ES.

D.B.T. thanks New York University for start-up funding. D.J.U. thanks the Concordia College Research Fund. We thank Tobias Gellen for useful conversations and for reading the manuscript carefully.

V. CONCLUSIONS

This work presented analytic and numeric results for coherent multidimensional spectroscopy using noisy light, I(4) 2D ES, and a general lineshape function. Noisy light has a few specific properties but it is nonetheless a good model for incoherent light. The overdamped Brownian oscillator and the displaced harmonic oscillator models were used as examples of theoretical treatments that go beyond the Bloch model. The main result is that I(4) 2D ES is sensitive to lineshapes in many of the same ways as femtosecond 2D ES with the selfevident caveat that dynamics are not directly probed. Instead, dynamics information is encoded in the width and brightness of peaks. In contrast to theoretical results using the multilevel Bloch model1 and experiments,2 the present analysis suggests that vibronic transitions will not induce oscillations as a function of the waiting time. The discrepancy needs further examination, but could be due to the consideration of only a single electronic transition in this work. The analysis parallels an initial theoretical study of femtosecond 2D ES performed by Jonas and coworkers.25 The computations provide insight in that they supply explicit solutions for important lineshape models that are analytically formidable. While expected features in the 2D spectra were present and matched those observed using femtosecond excitation, new features also emerged. In particular, the relative amplitude of the cross peaks points to the sensitivity of I(4) 2D ES to photoisomers. The analysis presented here reveals that I(4) 2D ES has a striking specificity compared to noisy-light methods developed in the 1980s and 1990s. I(4) 2D ES has specificity equivalent to that of femtosecond 2D ES; in many cases, individual Liouville pathways can be isolated. Finally, we note that noisy-light spectroscopy has long been described diagrammatically by factorized time correlation (FTC) diagrams.9 FTC diagram analysis transcends the specifics of the lineshape model and thereby

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