# Nanoscale Waveguide Beam Splitter in Quantum Technologies

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

## 3. Results

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Bromberg, Y.; Lahini, Y.; Morandotti, R.; Silberberg, Y. Quantum and Classical Correlations in Waveguide Lattices. Phys. Rev. Lett.
**2009**, 102, 253904. [Google Scholar] [CrossRef] [PubMed] - Politi, A.; Cryan, M.J.; Rarity, J.G.; Yu, S.; O’Brien, J.L. Silica-on-Silicon Waveguide Quantum Circuits. Sience
**2008**, 320, 646–649. [Google Scholar] [CrossRef] [PubMed][Green Version] - Pan, J.W.; Chen, Z.; Lu, C.Y.; Weinfurter, H.; Zeilinger, A.; Zukowski, M. Multiphoton entanglement and interferometry. Rev. Mod. Phys.
**2012**, 84, 777. [Google Scholar] [CrossRef] - Knill, E.; Laflamme, R.; Milburn, G.J. A Scheme for Efficient Quantum Computation With Linear Optics. Nature
**2001**, 409, 46–52. [Google Scholar] [CrossRef] [PubMed] - Kok, P.; Munro, W.J.; Nemoto, K.; Ralph, T.C.; Dowling, J.P.; Milburn, G.J. Linear optical quantum computing with photonic qubits. Rev. Mod. Phys.
**2007**, 79, 135. [Google Scholar] [CrossRef][Green Version] - Ladd, T.D.; Jelezko, F.; Laflamme, R.; Nakamura, Y.; Monroe, C.; O’Brien, J.L. Quantum Computers. Nature
**2010**, 464, 45–53. [Google Scholar] [CrossRef] [PubMed][Green Version] - Tan, S.H.; Rohde, P.P. The resurgence of the linear optics quantum interferometer–recent advances and applications. Rev. Phys.
**2019**, 4, 100030. [Google Scholar] [CrossRef] - Pezze, L.; Smerzi, A.; Oberthaler, M.K.; Schmied, R.; Treutlein, P. Quantum metrology with nonclassical states of atomic ensembles. Rev. Mod. Phys.
**2018**, 90, 035005. [Google Scholar] [CrossRef][Green Version] - Weedbrook, C.; Pirandola, S.; Garcia-Patron, R.; Cerf, N.J.; Ralph, T.C.; Shapiro, J.H.; Lloyd, S. Gaussian quantum information. Rev. Mod. Phys.
**2012**, 84, 621. [Google Scholar] [CrossRef] - Hong, C.K.; Ou, Z.Y.; Mandel, L. Measurement of subpicosecond time intervals between two photons by interference. Phys. Rev. Lett.
**1987**, 59, 2044–2046. [Google Scholar] [CrossRef] - Sangouard, N.; Simon, C.; de Riedmatten, H.; Gisin, N. Quantum repeaters based on atomic ensembles and linear optics. Rev. Mod. Phys.
**2011**, 83, 33. [Google Scholar] [CrossRef][Green Version] - Harris, N.C.; Steinbrecher, G.R.; Prabhu, M.; Lahini, Y.; Mower, J.; Bunandar, D.; Chen, C.; Wong, F.N.C.; Baehr-Jones, T.; Hochberg, M.; et al. Quantum transport simulations in a programmable nanophotonic processor. Nat. Photonics
**2017**, 11, 447–452. [Google Scholar] [CrossRef][Green Version] - Tambasco, J.L.; Corrielli, G.; Chapman, R.J.; Crespi, A.; Zilberberg, O.; Osellame, R.; Peruzzo, A. Quantum interference of topological states of light. Sci Adv.
**2018**, 4, eaat3187. [Google Scholar] [CrossRef] [PubMed][Green Version] - Campos, R.A.; Saleh, B.E.A.; Teich, M.C. Quantum-mechanical lossless beam splitter: SU(2) symmetry and photon statistics. Phys. Rev. A
**1989**, 40, 1371. [Google Scholar] [CrossRef] [PubMed] - Kim, M.S.; Son, W.; Buzek, V.; Knight, P.L. Entanglement by a beam splitter: Nonclassicality as a prerequisite for entanglement. Phys. Rev. A
**2002**, 65, 032323. [Google Scholar] [CrossRef][Green Version] - Makarov, D. Quantum entanglement and reflection coefficient for coupled harmonic oscillators. Phys. Rev. E
**2020**, 102, 052213. [Google Scholar] [CrossRef] - Chen, Y.F.; Hsieh, M.X.; Ke, H.T.; Yu, Y.T.; Liang, H.C.; Huang, K.F. Quantum entanglement by a beam splitter analogous to laser mode transformation by a cylindrical lens. Opt. Lett.
**2021**, 46, 5129–5132. [Google Scholar] [CrossRef] - Makarov, D.N. Theory of HOM interference on coupled waveguides. Opt. Lett.
**2020**, 45, 6322–6325. [Google Scholar] [CrossRef] - Makarov, D.N. Fluctuations in the detection of the HOM effect. Sci. Rep.
**2020**, 10, 20124. [Google Scholar] [CrossRef] - Makarov, D.N. Theory of a frequency-dependent beam splitter in the form of coupled waveguides. Sci. Rep.
**2021**, 11, 5014. [Google Scholar] [CrossRef] - Makarov, D.; Gusarevich, E.; Goshev, A.; Makarova, K.; Kapustin, S.; Kharlamova, A.; Tsykareva, Y.V. Quantum entanglement and statistics of photons on a beam splitter in the form of coupled waveguides. Sci. Rep.
**2021**, 11, 10274. [Google Scholar] [CrossRef] - Makarov, D.; Tsykareva, Y. Quantum Entanglement of Monochromatic and Non-Monochromatic Photons on a Waveguide Beam Splitter. Entropy
**2022**, 24, 49. [Google Scholar] [CrossRef] [PubMed] - Makarov, D. Theory for the beam splitter in quantum optics: Quantum entanglement of photons and their statistics, HOM effect. arXiv
**2022**. [Google Scholar] [CrossRef] - Holland, M.; Burnett, K. Interferometric detection of optical phase shifts at the heisenberg limit. Phys. Rev. Lett.
**1993**, 71, 1355. [Google Scholar] [CrossRef] [PubMed] - Polino, E.; Valeri, M.; Spagnolo, N.; Sciarrino, F. Photonic Quantum Metrology. AVS Quantum Sci.
**2020**, 2, 024703. [Google Scholar] [CrossRef] - Phoenix, S.; Knight, P. Fluctuations and entropy in models of quantum optical resonance. Ann. Phys.
**1988**, 186, 381–407. [Google Scholar] [CrossRef] - Christodoulides, D.N.; Lederer, F.; Silberberg, Y. Discretizing light behaviour in linear and nonlinear waveguide lattices. Nature
**2003**, 424, 817–823. [Google Scholar] [CrossRef] - Regensburger, A.; Bersch, C.; Miri, M.A.; Onishchukov, G.; Christodoulides, D.; Peschel, U. Parity–time synthetic photonic lattices. Nature
**2012**, 488, 167–171. [Google Scholar] [CrossRef] - Guo, Z.; Sun, Y.; Jiang, H.; Ding, Y.Q.; Li, Y.; Zhang, Y.; Chen, H. Experimental demonstration of an anomalous Floquet topological insulator based on negative-index media. arXiv
**2020**. [Google Scholar] [CrossRef] - Guo, Z.; Song, J.; Jiang, H.; Chen, H. Miniaturized Backward Coupler Realized by the Circuit-Based Planar Hyperbolic Waveguide. Adv. Photonics Res.
**2021**, 2, 2100035. [Google Scholar] [CrossRef] - Guo, Z.; Jian, Y.; Wu, X.; Deng, F.; Dong, L.; Chen, H. Multiple linear-crossing metamaterials for directional refraction. Front. Mater.
**2022**, 9, 1001233. [Google Scholar] [CrossRef]

**Figure 1.**3D representation of the waveguide BS. Photons (in the general case nonmonochromatic) fall on the input ports BS. At the output ports of the BS are detectors ${D}_{1},{D}_{2}$ registering photons. The figure highlights the coupling region of the waveguide, where the electromagnetic fields from ports 1 and 2 overlap.

**Figure 2.**The calculation is presented: (

**a**) probabilities ${P}_{0,2}$ of detecting 2 photons at the second detector and 0 of photons at 1 detector (with ${P}_{0,2}={P}_{2,0}$ ) at different parameters $\sigma /\Omega $={0,1/2,3/2,3,5,10} (respectively, the color of the graphs in the figure: {black, yellow, green, red, blue, brown}) depending on the dimensionless BS length $L/{L}_{BS}$; (

**b**) the same but with larger dimensions of the beamsplitter, i.e., at $L/{L}_{BS}\to \infty $; (

**c**) the same as in (

**a**) but only for the probability ${P}_{1,1}$ of one photon detected at each detector; (

**d**) the same as in (

**b**) but only for the probability ${P}_{1,1}$.

**Figure 3.**The calculation is presented: (

**a**) quantum entanglement ${S}_{N}$ at different parameters $\sigma /\mathrm{\Omega}$={0,1/2,3/2,3,5,10} (respectively, the color of the graphs in the figure: {black, yellow, green, red, blue, brown}) depending on the dimensionless BS length $L/{L}_{BS}$; (

**b**) the same but with larger dimensions of the BS, i.e., at $L/{L}_{BS}\to \infty $.

**Figure 4.**The calculation for the initial state $|4,4\rangle $ is presented: (

**a**) quantum entanglement ${S}_{N}$ at different parameters $\sigma /\mathrm{\Omega}$={0,1/2,3/2,3,5,10} (respectively, the color of the graphs in the figure: {black, yellow, green, red, blue, brown}) depending on the dimensionless BS length $L/{L}_{BS}$; (

**b**) same but at larger BS size, i.e., at $L/{L}_{BS}\to \infty $.

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## Share and Cite

**MDPI and ACS Style**

Makarov, D.; Makarova, K.; Tsykareva, Y.; Kapustin, S.; Kharlamova, A.; Gusarevich, E.; Goshev, A.
Nanoscale Waveguide Beam Splitter in Quantum Technologies. *Nanomaterials* **2022**, *12*, 4030.
https://doi.org/10.3390/nano12224030

**AMA Style**

Makarov D, Makarova K, Tsykareva Y, Kapustin S, Kharlamova A, Gusarevich E, Goshev A.
Nanoscale Waveguide Beam Splitter in Quantum Technologies. *Nanomaterials*. 2022; 12(22):4030.
https://doi.org/10.3390/nano12224030

**Chicago/Turabian Style**

Makarov, Dmitry, Ksenia Makarova, Yuliana Tsykareva, Sergey Kapustin, Anastasia Kharlamova, Eugeny Gusarevich, and Andrey Goshev.
2022. "Nanoscale Waveguide Beam Splitter in Quantum Technologies" *Nanomaterials* 12, no. 22: 4030.
https://doi.org/10.3390/nano12224030