^{1}, Geoffrey A. Lott

^{1}and Andrew H. Marcus

^{2,a)}

### Abstract

Two-dimensional electronic coherence spectroscopy (ECS) is an important method to study the coupling between distinct optical modes of a material system. Such studies often involve excitation using a sequence of phased ultrashort laser pulses. In conventional approaches, the delays between pulse temporal envelopes must be precisely monitored or maintained. Here, we introduce a new experimental scheme for phase-selective nonlinear ECS, which combines acousto-optic phase modulation with ultrashort laser excitation to produce intensity modulated nonlinear fluorescence signals. We isolate specific nonlinear signal contributions by synchronous detection, with respect to appropriately constructed references. Our method effectively decouples the relative temporal phases from the pulse envelopes of a collinear train of four sequential pulses. We thus achieve a robust and high signal-to-noise scheme for phase-selective ECS to investigate the resonant nonlinear optical response of photoluminescent systems. We demonstrate the validity of our method using a model quantum three-level system—atomic Rb vapor. Moreover, we show how our measurements determine the resonant complex-valued third-order susceptibility.

We thank Professor Jeff Cina and Professor Tom Dyke for useful discussions and Cliff Dax for his assistance with the design and implementation of custom electronics. This research is supported by the National Science Foundation CHE-0303715 and the National Institutes of Health 1 R01 GM67891.

I. INTRODUCTION

II. THEORETICAL BACKGROUND

A. Two-dimensional ECS

III. EXPERIMENTAL METHODS

A. Two-dimensional ECS with AO phase modulation

B. Phase-sensitive signal detection

C. Determination of the third-order susceptibility

D. Two-dimensional PM-ECS instrumentation

IV. DISCUSSION OF RESULTS

A. Time-domain interferograms

B. Third-order susceptibilities

V. CONCLUSIONS

### Key Topics

- Monochromators
- 32.0
- Excited states
- 23.0
- Fluorescence
- 16.0
- Fourier transforms
- 15.0
- Phase modulation
- 15.0

## Figures

Energy level diagram for the D line transitions (described in text and Table I). For the purposes of our experiments, the system behaves as a three-level atom with ground state , first excited state , and second excited state .

Energy level diagram for the D line transitions (described in text and Table I). For the purposes of our experiments, the system behaves as a three-level atom with ground state , first excited state , and second excited state .

Illustration of a train of four sequential phase-modulated pulses. Each pulse sequence is labeled by the superscript ; the individual pulses are labeled by subscripts 1–4. A pulse sequence is characterized by the interpulse delays , , and and the relative temporal phases and .

Illustration of a train of four sequential phase-modulated pulses. Each pulse sequence is labeled by the superscript ; the individual pulses are labeled by subscripts 1–4. A pulse sequence is characterized by the interpulse delays , , and and the relative temporal phases and .

Wave packet pathway diagrams for the four nonlinear population terms isolated by PM-ECS (described in text). Separate panels illustrate the pathways for (a) , (b) , (c) , and (d) . The overlaps described in panels (a) and (d) contribute to our “sum signal” and the overlaps described in panels (b) and (c) contribute to our “difference signal.”

Wave packet pathway diagrams for the four nonlinear population terms isolated by PM-ECS (described in text). Separate panels illustrate the pathways for (a) , (b) , (c) , and (d) . The overlaps described in panels (a) and (d) contribute to our “sum signal” and the overlaps described in panels (b) and (c) contribute to our “difference signal.”

Diagrams illustrating the four distinct wave packet pathways that contribute to the overlap (described in text). The pathways depicted in panels (a) and (b) are “single-mode” pathways, while those depicted in panels (c) and (d) are “coupling” pathways.

Diagrams illustrating the four distinct wave packet pathways that contribute to the overlap (described in text). The pathways depicted in panels (a) and (b) are “single-mode” pathways, while those depicted in panels (c) and (d) are “coupling” pathways.

(Color online) Schematic diagram of the experimental setup for phase modulation (PM)-ECS (described in text). Abbreviations have the following meanings. APD, amplified photodiode; PD, photodiode; AO, acousto-optic Bragg cell; BS, beam splitter.

(Color online) Schematic diagram of the experimental setup for phase modulation (PM)-ECS (described in text). Abbreviations have the following meanings. APD, amplified photodiode; PD, photodiode; AO, acousto-optic Bragg cell; BS, beam splitter.

Grayscale contour diagrams of the calculated difference [panels (g)–(l)] and sum signals for the Rb system, obtained from PM-ECS (described in text). Real and imaginary parts of each signal are given in the top and bottom rows, respectively. In the left column [panels (a), (d), (g), and (j)] are plots of the undersampled two-dimensional interferograms. In the center column [panels (b), (e), (h), and (k)] are plots of the fully sampled interferograms. In the right column [panels (c), (f), (i), and (l)] are plots of the instrumental phase functions.

Grayscale contour diagrams of the calculated difference [panels (g)–(l)] and sum signals for the Rb system, obtained from PM-ECS (described in text). Real and imaginary parts of each signal are given in the top and bottom rows, respectively. In the left column [panels (a), (d), (g), and (j)] are plots of the undersampled two-dimensional interferograms. In the center column [panels (b), (e), (h), and (k)] are plots of the fully sampled interferograms. In the right column [panels (c), (f), (i), and (l)] are plots of the instrumental phase functions.

Comparison between theoretical calculations [panels (a)–(d)] and experimental data [panels (e)–(h)] for the difference signal interferograms obtained by PM-ECS (described in text).

Comparison between theoretical calculations [panels (a)–(d)] and experimental data [panels (e)–(h)] for the difference signal interferograms obtained by PM-ECS (described in text).

Comparison between theoretical calculations [panels (a)–(d)] and experimental data [panels (e)–(h)] for the sum signal interferograms obtained by PM-ECS (described in text).

Comparison between theoretical calculations [panels (a)–(d)] and experimental data [panels (e)–(h)] for the sum signal interferograms obtained by PM-ECS (described in text).

Comparison between theoretical calculations [panels (a)–(d)] and experimental data [panels (e)–(h)] for the rephasing susceptibility obtained by PM-ECS (described in text).

Comparison between theoretical calculations [panels (a)–(d)] and experimental data [panels (e)–(h)] for the rephasing susceptibility obtained by PM-ECS (described in text).

Comparison between theoretical calculations [panels (a)–(d)] and experimental data [panels (e)–(h)] for the nonrephasing susceptibility obtained by PM-ECS (described in text).

Comparison between theoretical calculations [panels (a)–(d)] and experimental data [panels (e)–(h)] for the nonrephasing susceptibility obtained by PM-ECS (described in text).

Comparison between theoretical calculations [panels (a)–(d)] and experimental data [panels (e)–(h)] for the two-dimensional correlation spectra obtained by PM-ECS (described in text).

Comparison between theoretical calculations [panels (a)–(d)] and experimental data [panels (e)–(h)] for the two-dimensional correlation spectra obtained by PM-ECS (described in text).

## Tables

Physical constants associated with optical line transitions. is the transition index, is the transition frequency, is the transition wavelength, is the transition dipole moment (in units of Debye), is the lifetime, and is the natural line width (Ref. 41).

Physical constants associated with optical line transitions. is the transition index, is the transition frequency, is the transition wavelength, is the transition dipole moment (in units of Debye), is the lifetime, and is the natural line width (Ref. 41).

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