# Bayesian Analysis of Femtosecond Pump-Probe Photoelectron-Photoion Coincidence Spectra with Fluctuating Laser Intensities

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## Abstract

**:**

## 1. Introduction

## 2. Experiment

## 3. Bayesian Data Analysis

#### 3.1. Preliminary Considerations

#### 3.2. The Posterior PDF

#### 3.3. The Prior PDF

#### 3.4. The Likelihood

#### 3.5. Remarks on the Posterior Sampling

## 4. Mock Data Analysis

#### 4.1. False Coincidences

#### 4.2. Background Subtraction

## 5. Application to Experimental Data

## 6. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Solution of the 〈λ^{n} 〉-Integral

## Appendix B. Channel-Resolved Single Coincidences

## Appendix C. Transformation of the Dirac Distribution

## Appendix D. The Jacobian Determinant

## Appendix E. Probabilities of the Count-Pairs (N_{e}, N_{i})

## References

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**Figure 1.**Utterly simplified sketch of a time-resolved photoionization study carried out with a pump-probe setup and a time-of-flight spectrometer. A commercial Ti:sapphire laser system delivers pulses of $800$ nm in center wavelength and $25$ fs in temporal length at a repetition rate of $3$ kHz. The delay stage is used to control the length of the optical path, and hence the time delay. The energy level diagram shows how the electron kinetic energy, given the energy of the states and the photons, identifies the state the system was in at the moment of ionization. A detailed description of the setup can be found in our previous publications [7,28].

**Figure 3.**Simulation with mock data for studying the influence of $\lambda $-fluctuations on false coincidences. The black lines are the spectra used to generate the data; the green (blue) lines including $\pm \sigma $ error bands are the reconstructed spectra (not) including $\lambda $-fluctuations in the reconstruction. The parameters are ${\xi}_{i}={\xi}_{e}=0.5$ and ${\mathcal{N}}_{p}={10}^{7}$. For ${\underline{\lambda}}_{2}=1.5$, differences between the algorithms are negligible even at relatively high $\lambda $-fluctuations with ${\sigma}_{2}=0.5$; see spectra (

**a**,

**b**). When choosing ${\underline{\lambda}}_{2}=0.5$ (

**c**,

**d**), the algorithm not including $\lambda $-fluctuations produces small deviations, e.g., underestimation of the false coincidences at the first Gaussian in the fragment spectrum.

**Figure 4.**Simulated test spectra for studying the influence of $\lambda $-fluctuations on the background subtraction. The parameters are ${\underline{\lambda}}_{1}={\underline{\lambda}}_{2}=0.5$, ${\xi}_{i}={\xi}_{e}=0.5$, and ${\mathcal{N}}_{p}={10}^{7}$. ${\sigma}_{1}$ and ${\sigma}_{2}$ are different for every sub-figure. If ${\sigma}_{1}={\sigma}_{2}=0.1$ (

**a**) or ${\sigma}_{1}={\sigma}_{2}=0.5$ (

**b**), both algorithms (with (green line) and without (blue line) including $\lambda $-fluctuations) reconstruct the spectra correctly. ${\sigma}_{1}=0.1$ and ${\sigma}_{2}=0.5$ lead to an underestimation of the background when neglecting $\lambda $-fluctuations (

**c**). Overestimation of the background happens in the case of ${\sigma}_{1}=0.5$ and ${\sigma}_{2}=0.1$ (

**d**).

**Table 1.**Estimated parameters ${\underline{\lambda}}_{2}$, ${\sigma}_{2}$, ${\xi}_{i}$, and ${\xi}_{e}$. In the lines showing the results of the algorithm presented in [22], ${\lambda}_{2}$ is shown instead of ${\underline{\lambda}}_{2}$. Each value denotes the mean and standard deviation of the parameter’s distribution.

λ${}_{2}$ | ${\mathit{\sigma}}_{2}$ | ${\mathit{\xi}}_{\mathit{i}}$ | ${\mathit{\xi}}_{\mathit{e}}$ | |
---|---|---|---|---|

Parameters (Figure 3a,b) | $1.5$ | $0.5$ | $0.5$ | $0.5$ |

Algorithm in [22] | $1.4177\pm 0.0008$ | - | $0.5235\pm 0.0003$ | $0.5235\pm 0.0003$ |

Algorithm with $\lambda $-fluctuations | $1.499\pm 0.001$ | $0.501\pm 0.002$ | $0.4998\pm 0.0003$ | $0.5000\pm 0.0003$ |

Parameters (Figure 3c,d) | $0.5$ | $0.5$ | $0.5$ | $0.5$ |

Algorithm in [22] | $0.4365\pm 0.0003$ | - | $0.5618\pm 0.0004$ | $0.5621\pm 0.0004$ |

Algorithm with $\lambda $-fluctuations | $0.4992\pm 0.0005$ | $0.499\pm 0.001$ | $0.5000\pm 0.0004$ | $0.5004\pm 0.0004$ |

**Table 2.**Estimated parameters ${\underline{\lambda}}_{1}$, ${\underline{\lambda}}_{2}$, ${\sigma}_{1}$, ${\sigma}_{2}$, ${\xi}_{i}$, and ${\xi}_{e}$. The parameter regimes denoted by the identifications (a–d) are according to Figure 4. For each parameter set, the first line denotes the true value, while Line 2 (3) contains the parameter estimation performed with the algorithm without (with) $\lambda $-fluctuations, respectively. In the lines showing the results of the algorithm ignoring $\lambda $-fluctuations, ${\lambda}_{j}$ is shown instead of ${\underline{\lambda}}_{j}$. Each value denotes the mean and standard deviation of the parameter’s distribution.

λ${}_{1}$ | λ${}_{2}$ | ${\mathit{\sigma}}_{1}$ | ${\mathit{\sigma}}_{2}$ | ${\mathit{\xi}}_{\mathit{i}}$ | ${\mathit{\xi}}_{\mathit{e}}$ | |
---|---|---|---|---|---|---|

(a) | $0.5$ | $0.5$ | $0.1$ | $0.1$ | $0.5$ | $0.5$ |

$0.4970\pm 0.0003$ | $0.4972\pm 0.0005$ | - | - | $0.5032\pm 0.0002$ | $0.5029\pm 0.0002$ | |

$0.5000\pm 0.0003$ | $0.5007\pm 0.0005$ | $0.098\pm 0.002$ | $0.101\pm 0.005$ | $0.5003\pm 0.0003$ | $0.5000\pm 0.0003$ | |

(b) | $0.5$ | $0.5$ | $0.5$ | $0.5$ | $0.5$ | $0.5$ |

$0.4367\pm 0.0003$ | $0.4317\pm 0.0004$ | - | - | $0.5618\pm 0.0002$ | $0.5620\pm 0.0002$ | |

$0.5002\pm 0.0004$ | $0.5009\pm 0.0006$ | $0.5026\pm 0.0008$ | $0.496\pm 0.002$ | $0.4996\pm 0.0003$ | $0.4998\pm 0.0003$ | |

(c) | $0.5$ | $0.5$ | $0.1$ | $0.5$ | $0.5$ | $0.5$ |

$0.4840\pm 0.0003$ | $0.4608\pm 0.0005$ | - | - | $0.5201\pm 0.0002$ | $0.5201\pm 0.0002$ | |

$0.4991\pm 0.0003$ | $0.5013\pm 0.0006$ | $0.098\pm 0.002$ | $0.500\pm 0.002$ | $0.5001\pm 0.0002$ | $0.5001\pm 0.0003$ | |

(d) | $0.5$ | $0.5$ | $0.5$ | $0.1$ | $0.5$ | $0.5$ |

$0.4452\pm 0.0003$ | $0.4675\pm 0.0004$ | - | - | $0.5441\pm 0.0002$ | $0.5440\pm 0.0002$ | |

$0.4998\pm 0.0004$ | $0.5005\pm 0.0006$ | $0.5002\pm 0.0009$ | $0.102\pm 0.004$ | $0.4998\pm 0.0003$ | $0.4997\pm 0.0003$ |

**Table 3.**Estimated parameters ${\underline{\lambda}}_{1}$, ${\underline{\lambda}}_{2}$, ${\sigma}_{1}$, ${\sigma}_{2}$, ${\xi}_{i}$, and ${\xi}_{e}$. Line 1 (2) contains the parameter estimations performed with the algorithm without (with) $\lambda $-fluctuations, respectively. In the line showing the results of the algorithm ignoring $\lambda $-fluctuations, ${\lambda}_{j}$ is shown instead of ${\underline{\lambda}}_{j}$. Each value denotes the mean and standard deviation of the parameter’s distribution.

λ${}_{1}$ | λ${}_{2}$ | ${\mathit{\sigma}}_{1}$ | ${\mathit{\sigma}}_{2}$ | ${\mathit{\xi}}_{\mathit{i}}$ | ${\mathit{\xi}}_{\mathit{e}}$ |
---|---|---|---|---|---|

$0.3328\pm 0.0009$ | $0.132\pm 0.001$ | - | - | $0.3247\pm 0.0008$ | $0.542\pm 0.001$ |

$0.3328\pm 0.0009$ | $0.1335\pm 0.0003$ | $0.0004\pm 0.0010$ | $0.0004\pm 0.0012$ | $0.3245\pm 0.0008$ | $0.541\pm 0.001$ |

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**MDPI and ACS Style**

Heim, P.; Rumetshofer, M.; Ranftl, S.; Thaler, B.; Ernst, W.E.; Koch, M.; von der Linden, W.
Bayesian Analysis of Femtosecond Pump-Probe Photoelectron-Photoion Coincidence Spectra with Fluctuating Laser Intensities. *Entropy* **2019**, *21*, 93.
https://doi.org/10.3390/e21010093

**AMA Style**

Heim P, Rumetshofer M, Ranftl S, Thaler B, Ernst WE, Koch M, von der Linden W.
Bayesian Analysis of Femtosecond Pump-Probe Photoelectron-Photoion Coincidence Spectra with Fluctuating Laser Intensities. *Entropy*. 2019; 21(1):93.
https://doi.org/10.3390/e21010093

**Chicago/Turabian Style**

Heim, Pascal, Michael Rumetshofer, Sascha Ranftl, Bernhard Thaler, Wolfgang E. Ernst, Markus Koch, and Wolfgang von der Linden.
2019. "Bayesian Analysis of Femtosecond Pump-Probe Photoelectron-Photoion Coincidence Spectra with Fluctuating Laser Intensities" *Entropy* 21, no. 1: 93.
https://doi.org/10.3390/e21010093