# Integrated Source of Path-Entangled Photon Pairs with Efficient Pump Self-Rejection

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## Abstract

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## 1. Introduction

## 2. Scheme

## 3. Results and Discussions

#### 3.1. Operational Regime

#### 3.2. Quantum Perturbation Theory

#### 3.3. Realizations

#### 3.4. Extention to NOON States

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Heisenberg-Picture Operator Perturbation Expansion for the Photon Number

## Appendix B. NOON State Generator

## References

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**Figure 1.**The scheme of the photon-pair generator with in-built pump rejection on the basis of two single-mode waveguides coupled to the common loss reservoir.

**Figure 2.**Average number of twin-photons (solid lines), single-photons (dashed lines), and pump photons (dotted lines) as given by Equations (7), (10) and (12). (

**a**,

**b**) correspond to $\mathsf{\Gamma}=400\gamma $ and $\mathsf{\Gamma}=200\gamma $ respectively and the initial average number of photons of the pump $|{\alpha}_{+}{(0)|}^{2}={10}^{10}$. (

**c**) corresponds to $|{\alpha}_{+}{(0)|}^{2}={10}^{10}$ (thin lines) and $|{\alpha}_{+}{(0)|}^{2}={10}^{11}$ (thick lines) and $\mathsf{\Gamma}=400\gamma $. For all of the figures, the squared nonlinearity is ${U}^{2}={10}^{-20}{\gamma}^{2}$.

**Figure 3.**An example of three-waveguide realization of the basic Scheme (1). The waveguide ${A}_{3}$ is subjected to engineered loss with the rate ${\gamma}_{3}$. Both waveguides ${A}_{1}$ and ${A}_{2}$ are unitary coupled to the waveguide ${A}_{3}$; the coupling constant is g.

**Figure 4.**A minimal waveguide length, ${L}_{min}$, providing for pump rejection with $\delta =0.1$ for typical parameters of the glass waveguides and weak pump.

**Figure 5.**An example of six-waveguide realization of the NOON state generation scheme. The depicted structure shows two mirrored three-waveguide structures of Figure 3 with waveguides labeled as ${A}_{j}$ for the upper, and ${B}_{j}$ for the lower structures. The waveguides ${A}_{3}$ and ${B}_{3}$ are subject to strong engeneered loss. The waveguide ${A}_{1}$ is unitary coupled with the waveguide ${B}_{1}$, the waveguide ${A}_{2}$ is unitary coupled with the waveguide ${B}_{2}$; the coupling constant is v. Other parameters are shown in Figure 3.

**Figure 6.**Average number of photons corresponding to the generation of two photons in one of the antisymmetric modes (blue solid lines) and of two photons being simultaneously in both antisymmetric modes (magenta dashed lines) according to Equation (17) (they are, respectively, $2\times {P}_{2,0}$ and ${P}_{1,1}$). Red dashed-dotted lines show the probability of finding a single photon only in one of the antisymmetric modes, due to photon loss from one of the previous configurations. The coupling constant has been chosen to be $v=30\gamma $. Thick lines correspond to $\mathsf{\Gamma}=400\gamma $, thin lines correspond to $\mathsf{\Gamma}=200\gamma $. Other parameters are the same as in Figure 2.

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**MDPI and ACS Style**

de la Hoz, P.; Sakovich, A.; Mikhalychev, A.; Thornton, M.; Korolkova, N.; Mogilevtsev, D.
Integrated Source of Path-Entangled Photon Pairs with Efficient Pump Self-Rejection. *Nanomaterials* **2020**, *10*, 1952.
https://doi.org/10.3390/nano10101952

**AMA Style**

de la Hoz P, Sakovich A, Mikhalychev A, Thornton M, Korolkova N, Mogilevtsev D.
Integrated Source of Path-Entangled Photon Pairs with Efficient Pump Self-Rejection. *Nanomaterials*. 2020; 10(10):1952.
https://doi.org/10.3390/nano10101952

**Chicago/Turabian Style**

de la Hoz, Pablo, Anton Sakovich, Alexander Mikhalychev, Matthew Thornton, Natalia Korolkova, and Dmitri Mogilevtsev.
2020. "Integrated Source of Path-Entangled Photon Pairs with Efficient Pump Self-Rejection" *Nanomaterials* 10, no. 10: 1952.
https://doi.org/10.3390/nano10101952