^{1,a)}

### Abstract

Three-dimensional-IR spectroscopy is proposed as a new spectroscopic technique that is sensitive to three-point frequency fluctuation correlation functions. This will be important when the statistics of the underlying stochastic process is non-Gaussian, and hence when the system does not follow the linear response hypothesis. Furthermore, a very general classification of nonlinear spectroscopy in terms of higher order frequency fluctuation correlation functions is introduced, according to which certain moments of a multidimensional spectrum are related to certain frequency fluctuation correlation functions. The classification is rigorous in the so-called inhomogenous limit, but remains valid approximately also when motional narrowing becomes important. The work also puts a recent paper [J. Bredenbeck *et al.*, Phys. Rev. Lett.95, 083201 (2005)] onto solid theoretical grounds, where we have shown for the first time that fifth-order spectroscopy—in this case transient two-dimensional spectroscopy—is indeed sensitive to the three-point frequency fluctuation correlation function.

I am grateful to Gerhard Stock, Jens Bredenbeck, and Jan Helbing for many insightful discussions and careful reading of the manuscript. The work was supported by the Swiss Science Foundation (SNF) under Grant No. 200020–107492/1.

I. INTRODUCTION

II. THEORY

A. Response functions

1. 1D spectroscopy

2. 2D spectroscopy

3. 3D spectroscopy

B. Connection to frequency fluctuation correlation functions (FFCF’s)

C. Projection slice theorem for 3D spectroscopy

III. MODEL CALCULATIONS

A. Stochastic model

B. Results

IV. DISCUSSION

V. CONCLUSION

### Key Topics

- Nonlinear spectroscopy
- 19.0
- Absorption spectra
- 11.0
- Stochastic processes
- 11.0
- Correlation functions
- 10.0
- Solvents
- 10.0

## Figures

Pulse sequences and possible Feynman diagrams in (a) a first-order, (b) a third-order, and (c) a fifth-order experiment of a two-level system. one, three, or five pulses hit the sample, and the emitted first-order , third-order , and fifth-order polarizations- is phase sensitively detected by heterodyning it with a local oscillator (LO). We restrict ourselves to Feynman pathways which are in a population states during time periods and (c). Only diagrams that are in the ground state ∣0⟩⟨0∣ during these times are exemplified here. For each diagram, additional diagrams exist which go through ∣1⟩⟨1∣ states, but which lead to the same response functions as long as Kubo’s stochastic theory of line shapes (Ref. 47) is assumed.

Pulse sequences and possible Feynman diagrams in (a) a first-order, (b) a third-order, and (c) a fifth-order experiment of a two-level system. one, three, or five pulses hit the sample, and the emitted first-order , third-order , and fifth-order polarizations- is phase sensitively detected by heterodyning it with a local oscillator (LO). We restrict ourselves to Feynman pathways which are in a population states during time periods and (c). Only diagrams that are in the ground state ∣0⟩⟨0∣ during these times are exemplified here. For each diagram, additional diagrams exist which go through ∣1⟩⟨1∣ states, but which lead to the same response functions as long as Kubo’s stochastic theory of line shapes (Ref. 47) is assumed.

(a) Contour plots of the potentials of mean force in units of of model 1 (left column), model 2 (middle), and model 3 (right column). Short Langevin trajectories are exemplified. The resulting two-point FFCF’s and three-point FFCF’s are shown in (b) and (c). Note that is plotted here, while we had used in Ref. 42. (d) shows an example of a three-time point, fourth-order FFCF .

(a) Contour plots of the potentials of mean force in units of of model 1 (left column), model 2 (middle), and model 3 (right column). Short Langevin trajectories are exemplified. The resulting two-point FFCF’s and three-point FFCF’s are shown in (b) and (c). Note that is plotted here, while we had used in Ref. 42. (d) shows an example of a three-time point, fourth-order FFCF .

1D (a), 2D (b), and 3D (c) spectra for model 1 (left column), model 2 (middle column), and model 3 (right column). The contour lines in (b) are on a linear scale between zero and the peak maximum, while 20% contour surfaces are shown in (c). The faces in (c) show the contour plots of projections of the 3D spectra onto the corresponding planes, thereby regaining 2D spectra [Eq. (19)]. 2D and 3D spectra are for and and , respectively.

1D (a), 2D (b), and 3D (c) spectra for model 1 (left column), model 2 (middle column), and model 3 (right column). The contour lines in (b) are on a linear scale between zero and the peak maximum, while 20% contour surfaces are shown in (c). The faces in (c) show the contour plots of projections of the 3D spectra onto the corresponding planes, thereby regaining 2D spectra [Eq. (19)]. 2D and 3D spectra are for and and , respectively.

The covariance (a) and generalized skewness (b) of the 2D and 3D spectra of model 1 (left column), model 2 (middle column), and model 3 (right column) as a function of times and . (d) shows an example of a higher moment [see Eq. (18)]. Although the inhomogeneous limit is not strictly valid, covariance , generalized skewness , and nicely resemble the corresponding FFCF’s [Figs. 2(b)–2(d)].

The covariance (a) and generalized skewness (b) of the 2D and 3D spectra of model 1 (left column), model 2 (middle column), and model 3 (right column) as a function of times and . (d) shows an example of a higher moment [see Eq. (18)]. Although the inhomogeneous limit is not strictly valid, covariance , generalized skewness , and nicely resemble the corresponding FFCF’s [Figs. 2(b)–2(d)].

Multilevel diagrams for third-order and fifth-order spectroscopies, exemplified for and . The number above the diagram indicates the multiplicity of the corresponding diagram, considering the number of possibilities to reach a particular coherence states (i.e., the number of possibilities to go through ∣0⟩⟨0∣, ∣1⟩⟨1∣, etc., population states and still reach the same coherence states, see, e.g., the two first diagrams in (a) which are embraced since they yield the same response function), as well as the corresponding transition dipoles. Harmonic values have been used for the transition dipoles: and . The other diagrams of Fig. 1 lead to the same frequencies during the coherence times (with different signs) and the same multiplicities.

Multilevel diagrams for third-order and fifth-order spectroscopies, exemplified for and . The number above the diagram indicates the multiplicity of the corresponding diagram, considering the number of possibilities to reach a particular coherence states (i.e., the number of possibilities to go through ∣0⟩⟨0∣, ∣1⟩⟨1∣, etc., population states and still reach the same coherence states, see, e.g., the two first diagrams in (a) which are embraced since they yield the same response function), as well as the corresponding transition dipoles. Harmonic values have been used for the transition dipoles: and . The other diagrams of Fig. 1 lead to the same frequencies during the coherence times (with different signs) and the same multiplicities.

## Tables

Parameters used for models (1)–(3).

Parameters used for models (1)–(3).

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