# Coherent Superpositions of Photon Creation Operations and Their Application to Multimode States of Light

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Heralded Single-Photon Addition by PDC

#### 2.2. Multimode Superposition of Heralded Single-Photon Additions

#### 2.3. Two-Mode Homodyne Detection and Tomographic State Reconstruction

#### 2.4. Entanglement Measurements

## 3. Results

#### 3.1. Delocalized Single Photon

#### 3.2. Hybrid Entanglement

#### 3.3. Entanglement of Macroscopic Coherent States

#### 3.4. Discorrelation

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Experimental scheme for heralded single-photon addition. Symbols and abbreviations are defined in the text.

**Figure 2.**Experimental scheme for heralded multimode superposition of photon additions on consecutive temporal wavepacket modes by means of an unbalanced interferometer in the idler path.

**Figure 3.**Scheme of the ultrafast modulation of the laser pulse train for the phase control of the local oscillator (LO) pulses in two-mode homodyne detection.

**Figure 5.**Experimental values (blue dots) of the entanglement witness $S\left(R\right)$ in Equation (6), as a function of the superposition weight R, measured on the states of Equation (4). The black dashed line is the theoretical threshold above which $S\left(R\right)$ detects entanglement. Experimental values larger than this threshold certify the presence of entanglement in the state, whereas values below it cannot reveal any information about this property. The red area represents the theoretical behavior of the witness calculated on the states of Equation (4), considering fluctuations in the detection efficiency during the measurements.

**Figure 6.**(

**a**) Experimental scheme for hybrid entangled state generation. (

**b**) Measured values and theoretical behavior of the NPT as a function of the amplitude $\alpha $ of the input coherent state.

**Figure 7.**(

**a**) Experimental scheme for the generation of macro-macro entanglement. (

**b**) Experimental NPT (green dots) of the generated states as a function of the mean number of photons $\overline{n}$ in the input coherent states. Red and blue solid curves are the ideal NPT for $\phi =\pi $ and $\phi =0$, respectively. The orange curve is calculated for $\phi =\pi $ when all the experimental imperfections are considered.

**Figure 8.**(

**Top**) Raw correlations between the quadratures in the two modes as acquired from homodyne detection. Dots represent experimental points for different values of $\overline{n}$, while dashed lines are the theoretical predictions considering the measured experimental inefficiencies. (

**Bottom**) Quadrature correlations for different values of $\overline{n}$ after removal of the mean field from the measured quadrature distributions. Dots are the experimental points, while the black dashed line is the theoretical prediction for a balanced single-photon path-entangled state, with a detection efficiency of ${\eta}_{p}=0.92$.

**Figure 9.**Raw joint photon number probability distributions as a function of the mean number of photons $\overline{n}$ of the input coherent states, clearly showing the effect of discorrelation.

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**MDPI and ACS Style**

Biagi, N.; Francesconi, S.; Zavatta, A.; Bellini, M.
Coherent Superpositions of Photon Creation Operations and Their Application to Multimode States of Light. *Entropy* **2021**, *23*, 999.
https://doi.org/10.3390/e23080999

**AMA Style**

Biagi N, Francesconi S, Zavatta A, Bellini M.
Coherent Superpositions of Photon Creation Operations and Their Application to Multimode States of Light. *Entropy*. 2021; 23(8):999.
https://doi.org/10.3390/e23080999

**Chicago/Turabian Style**

Biagi, Nicola, Saverio Francesconi, Alessandro Zavatta, and Marco Bellini.
2021. "Coherent Superpositions of Photon Creation Operations and Their Application to Multimode States of Light" *Entropy* 23, no. 8: 999.
https://doi.org/10.3390/e23080999