# Two-Colour Spectrally Multimode Integrated SU(1,1) Interferometer

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Model

## 3. Dispersion Suppression and Phase Sensitivity

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Giovannetti, V.; Lloyd, S.; Maccone, L. Quantum Metrology. Phys. Rev. Lett.
**2006**, 96, 010401. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Polino, E.; Valeri, M.; Spagnolo, N.; Sciarrino, F. Photonic quantum metrology. AVS Quantum Sci.
**2020**, 2, 024703. [Google Scholar] [CrossRef] - Tóth, G.; Apellaniz, I. Quantum metrology from a quantum information science perspective. J. Phys. A Math. Theor.
**2014**, 47, 424006. [Google Scholar] [CrossRef] [Green Version] - Giovannetti, V.; Lloyd, S.; Maccone, L. Advances in quantum metrology. Nat. Photonics
**2011**, 5, 222–229. [Google Scholar] [CrossRef] - Řehaček, J.; Hradil, Z.; Stoklasa, B.; Paúr, M.; Grover, J.; Krzic, A.; Sánchez-Soto, L.L. Multiparameter quantum metrology of incoherent point sources: Towards realistic superresolution. Phys. Rev. A
**2017**, 96, 062107. [Google Scholar] [CrossRef] [Green Version] - Abbott, B.P. Observation of Gravitational Waves from a Binary Black Hole Merger. Phys. Rev. Lett.
**2016**, 116, 061102. [Google Scholar] [CrossRef] - Luo, K.H.; Santandrea, M.; Stefszky, M.; Sperling, J.; Massaro, M.; Ferreri, A.; Sharapova, P.R.; Herrmann, H.; Silberhorn, C. Quantum optical coherence: From linear to nonlinear interferometers. arXiv
**2021**, arXiv:2104.02641. [Google Scholar] - Chekhova, M.V.; Ou, Z.Y. Nonlinear interferometers in quantum optics. Adv. Opt. Photonics
**2016**, 8, 104–155. [Google Scholar] [CrossRef] - Giovannetti, V.; Lloyd, S.; Maccone, L. Quantum-Enhanced Measurements: Beating the Standard Quantum Limit. Science
**2004**, 306, 1330–1336. [Google Scholar] [CrossRef] [Green Version] - Dowling, J.P. Quantum optical metrology—the lowdown on high-N00N states. Contemp. Phys.
**2008**, 49, 125–143. [Google Scholar] [CrossRef] - Slussarenko, S.; Weston, M.M.; Chrzanowski, H.M.; Shalm, L.K.; Verma, V.B.; Nam, S.W.; Pryde, G.J. Unconditional violation of the shot-noise limit in photonic quantum metrology. Nat. Photonics
**2017**, 11, 700. [Google Scholar] [CrossRef] [Green Version] - Demkowicz-Dobrzański, R.; Jarzyna, M.; Kołodyński, J. Chapter Four—Quantum Limits in Optical Interferometry. In Progress in Optics; Wolf, E., Ed.; Elsevier: Amsterdam, The Netherlands, 2015; Volume 60, pp. 345–435. [Google Scholar]
- Yurke, B.; McCall, S.L.; Klauder, J.R. SU(2) and SU(1,1) interferometers. Phys. Rev. A
**1986**, 33, 4033–4054. [Google Scholar] [CrossRef] [PubMed] - Ou, Z.Y. Fundamental quantum limit in precision phase measurement. Phys. Rev. A
**1997**, 55, 2598–2609. [Google Scholar] [CrossRef] - Shih, Y.H.; Sergienko, A.V.; Rubin, M.H.; Kiess, T.E.; Alley, C.O. Two-photon entanglement in type-II parametric down-conversion. Phys. Rev. A
**1994**, 50, 23–28. [Google Scholar] [CrossRef] [PubMed] - Rubin, M.H.; Klyshko, D.N.; Shih, Y.H.; Sergienko, A.V. Theory of two-photon entanglement in type-II optical parametric down-conversion. Phys. Rev. A
**1994**, 50, 5122–5133. [Google Scholar] [CrossRef] - Keller, T.E.; Rubin, M.H. Theory of two-photon entanglement for spontaneous parametric down-conversion driven by a narrow pump pulse. Phys. Rev. A
**1997**, 56, 1534–1541. [Google Scholar] [CrossRef] - Marino, G.; Solntsev, A.S.; Xu, L.; Gili, V.F.; Carletti, L.; Poddubny, A.N.; Rahmani, M.; Smirnova, D.A.; Chen, H.; Lemaître, A.; et al. Spontaneous photon-pair generation from a dielectric nanoantenna. Optica
**2019**, 6, 1416–1422. [Google Scholar] [CrossRef] [Green Version] - Santiago-Cruz, T.; Fedotova, A.; Sultanov, V.; Weissflog, M.A.; Arslan, D.; Younesi, M.; Pertsch, T.; Staude, I.; Setzpfandt, F.; Chekhova, M. Photon Pairs from Resonant Metasurfaces. Nano Lett.
**2021**, 21, 4423–4429. [Google Scholar] [CrossRef] - Alibart, O.; Fulconis, J.; Wong, G.K.L.; Murdoch, S.G.; Wadsworth, W.J.; Rarity, J.G. Photon pair generation using four-wave mixing in a microstructured fibre: Theory versus experiment. New J. Phys.
**2006**, 8, 67. [Google Scholar] [CrossRef] - Zhang, D.; Ahmed, I.; Hao, L.; Cai, Y.; Li, C.; Zhang, Y.; Zhang, Y. Temporally Correlated Narrowband Biphoton Interference Based on Mach-Zehnder Interferometer. Ann. Der Phys.
**2019**, 531, 1900300. [Google Scholar] [CrossRef] - Lemieux, S.; Manceau, M.; Sharapova, P.R.; Tikhonova, O.V.; Boyd, R.W.; Leuchs, G.; Chekhova, M.V. Engineering the Frequency Spectrum of Bright Squeezed Vacuum via Group Velocity Dispersion in an SU(1,1) Interferometer. Phys. Rev. Lett.
**2016**, 117, 183601. [Google Scholar] [CrossRef] [Green Version] - Triginer, G.; Vidrighin, M.D.; Quesada, N.; Eckstein, A.; Moore, M.; Kolthammer, W.S.; Sipe, J.E.; Walmsley, I.A. Understanding High-Gain Twin-Beam Sources Using Cascaded Stimulated Emission. Phys. Rev. X
**2020**, 10, 031063. [Google Scholar] [CrossRef] - Roux, F.S. Stimulated parametric down-conversion for spatiotemporal metrology. Phys. Rev. A
**2021**, 104, 043514. [Google Scholar] [CrossRef] - Ou, Z.Y.; Li, X. Quantum SU(1,1) interferometers: Basic principles and applications. APL Photonics
**2020**, 5, 080902. [Google Scholar] [CrossRef] - Frascella, G.; Mikhailov, E.E.; Takanashi, N.; Zakharov, R.V.; Tikhonova, O.V.; Chekhova, M.V. Wide-field SU(1,1) interferometer. Optica
**2019**, 6, 1233–1236. [Google Scholar] [CrossRef] [Green Version] - Ferreri, A.; Ansari, V.; Silberhorn, C.; Sharapova, P.R. Temporally multimode four-photon Hong–Ou–Mandel interference. Phys. Rev. A
**2019**, 100, 053829. [Google Scholar] [CrossRef] [Green Version] - Ferreri, A.; Santandrea, M.; Stefszky, M.; Luo, K.H.; Herrmann, H.; Silberhorn, C.; Sharapova, P.R. Spectrally multimode integrated SU(1,1) interferometer. Quantum
**2021**, 5, 461. [Google Scholar] [CrossRef] - Paterova, A.V.; Krivitsky, L.A. Nonlinear interference in crystal superlattices. Light. Sci. Appl.
**2020**, 9, 82. [Google Scholar] [CrossRef] - Kitaeva, G.K.; Kornienko, V.V.; Leontyev, A.A.; Shepelev, A.V. Generation of optical signal and terahertz idler photons by spontaneous parametric down-conversion. Phys. Rev. A
**2018**, 98, 063844. [Google Scholar] [CrossRef] - Dvernik, L.S.; Prudkovskii, P.A. Azimuthal eigenmodes at strongly non-degenerate parametric down-conversion. Appl. Phys. B
**2021**, 127, 1–10. [Google Scholar] [CrossRef] - Luo, K.H.; Herrmann, H.; Krapick, S.; Brecht, B.; Ricken, R.; Quiring, V.; Suche, H.; Sohler, W.; Silberhorn, C. Direct generation of genuine single-longitudinal-mode narrowband photon pairs. New J. Phys.
**2015**, 17, 073039. [Google Scholar] [CrossRef] - Kuznetsov, K.A.; Malkova, E.I.; Zakharov, R.V.; Tikhonova, O.V.; Kitaeva, G.K. Nonlinear interference in the strongly nondegenerate regime and Schmidt mode analysis. Phys. Rev. A
**2020**, 101, 053843. [Google Scholar] [CrossRef] - Riedmatten, H.d.; Marcikic, I.; Tittel, W.; Zbinden, H.; Gisin, N. Quantum interference with photon pairs created in spatially separated sources. Phys. Rev. A
**2003**, 67, 022301. [Google Scholar] [CrossRef] [Green Version] - Tanzilli, S.; Martin, A.; Kaiser, F.; De Micheli, M.; Alibart, O.; Ostrowsky, D. On the genesis and evolution of Integrated Quantum Optics. Laser Photonics Rev.
**2012**, 6, 115–143. [Google Scholar] [CrossRef] [Green Version] - Caspani, L.; Xiong, C.; Eggleton, B.J.; Bajoni, D.; Liscidini, M.; Galli, M.; Morandotti, R.; Moss, D.J. Integrated sources of photon quantum states based on nonlinear optics. Light. Sci. Appl.
**2017**, 6, e17100. [Google Scholar] [CrossRef] - O’Brien, J.; Patton, B.; Sasaki, M.; Vučković, J. Focus on integrated quantum optics. New J. Phys.
**2013**, 15, 035016. [Google Scholar] [CrossRef] [Green Version] - Sharapova, P.R.; Luo, K.H.; Herrmann, H.; Reichelt, M.; Meier, T.; Silberhorn, C. Toolbox for the design of LiNbO3-based passive and active integrated quantum circuits. New J. Phys.
**2017**, 19, 123009. [Google Scholar] [CrossRef] - Ono, T.; Sinclair, G.F.; Bonneau, D.; Thompson, M.G.; Matthews, J.C.F.; Rarity, J.G. Observation of nonlinear interference on a silicon photonic chip. Opt. Lett.
**2019**, 44, 1277–1280. [Google Scholar] [CrossRef] - Krapick, S.; Herrmann, H.; Quiring, V.; Brecht, B.; Suche, H.; Silberhorn, C. An efficient integrated two-color source for heralded single photons. New J. Phys.
**2013**, 15, 033010. [Google Scholar] [CrossRef] [Green Version] - Herrmann, H.; Yang, X.; Thomas, A.; Poppe, A.; Sohler, W.; Silberhorn, C. Post-selection free, integrated optical source of non-degenerate, polarization entangled photon pairs. Opt. Express
**2013**, 21, 27981–27991. [Google Scholar] [CrossRef] [Green Version] - Sharapova, P.R.; Tikhonova, O.V.; Lemieux, S.; Boyd, R.W.; Chekhova, M.V. Bright squeezed vacuum in a nonlinear interferometer: Frequency and temporal Schmidt-mode description. Phys. Rev. A
**2018**, 97, 053827. [Google Scholar] [CrossRef] [Green Version] - Klyshko, D. Ramsey interference in two-photon parametric scattering. J. Exp. Theor. Phys.
**1993**, 104, 2676–2684. [Google Scholar] - Klyshko, D. Parametric generation of two-photon light in anisotropic layered media. J. Exp. Theor. Phys.
**1994**, 105, 1574–1582. [Google Scholar] - Santandrea, M.; Stefszky, M.; Ansari, V.; Silberhorn, C. Fabrication limits of waveguides in nonlinear crystals and their impact on quantum optics applications. New J. Phys.
**2019**, 21, 033038. [Google Scholar] [CrossRef] - Helmfrid, S.; Arvidsson, G.; Webjörn, J. Influence of various imperfections on the conversion efficiency of second-harmonic generation in quasi-phase-matching lithium niobate waveguides. J. Opt. Soc. Am. B
**1993**, 10, 222–229. [Google Scholar] [CrossRef] - Law, C.K.; Walmsley, I.A.; Eberly, J.H. Continuous Frequency Entanglement: Effective Finite Hilbert Space and Entropy Control. Phys. Rev. Lett.
**2000**, 84, 5304–5307. [Google Scholar] [CrossRef] - Manceau, M.; Khalili, F.; Chekhova, M. Improving the phase super-sensitivity of squeezing-assisted interferometers by squeeze factor unbalancing. New J. Phys.
**2017**, 19, 013014. [Google Scholar] [CrossRef] - Anderson, B.E.; Schmittberger, B.L.; Gupta, P.; Jones, K.M.; Lett, P.D. Optimal phase measurements with bright- and vacuum-seeded SU(1,1) interferometers. Phys. Rev. A
**2017**, 95, 063843. [Google Scholar] [CrossRef] [Green Version] - Li, D.; Yuan, C.H.; Ou, Z.Y.; Zhang, W. The phase sensitivity of an SU(1,1) interferometer with coherent and squeezed-vacuum light. New J. Phys.
**2014**, 16, 073020. [Google Scholar] [CrossRef] - Plick, W.N.; Dowling, J.P.; Agarwal, G.S. Coherent-light-boosted, sub-shot noise, quantum interferometry. New J. Phys.
**2010**, 12, 083014. [Google Scholar] [CrossRef] - Li, D.; Yuan, C.H.; Yao, Y.; Jiang, W.; Li, M.; Zhang, W. Effects of loss on the phase sensitivity with parity detection in an SU(1,1) interferometer. J. Opt. Soc. Am. B
**2018**, 35, 1080–1092. [Google Scholar] [CrossRef] [Green Version] - Marino, A.M.; Corzo Trejo, N.V.; Lett, P.D. Effect of losses on the performance of an SU(1,1) interferometer. Phys. Rev. A
**2012**, 86, 023844. [Google Scholar] [CrossRef] - Xin, J.; Wang, H.; Jing, J. The effect of losses on the quantum-noise cancellation in the SU(1,1) interferometer. Appl. Phys. Lett.
**2016**, 109, 051107. [Google Scholar] [CrossRef] - Hu, X.L.; Li, D.; Chen, L.Q.; Zhang, K.; Zhang, W.; Yuan, C.H. Phase estimation for an SU(1,1) interferometer in the presence of phase diffusion and photon losses. Phys. Rev. A
**2018**, 98, 023803. [Google Scholar] [CrossRef] [Green Version] - Tiedau, J.; Schapeler, T.; Anant, V.; Fedder, H.; Silberhorn, C.; Bartley, T.J. Single-channel electronic readout of a multipixel superconducting nanowire single photon detector. Opt. Express
**2020**, 28, 5528–5537. [Google Scholar] [CrossRef] - Ferrari, S.; Kahl, O.; Kovalyuk, V.; Goltsman, G.N.; Korneev, A.; Pernice, W.H.P. Waveguide-integrated single- and multi-photon detection at telecom wavelengths using superconducting nanowires. Appl. Phys. Lett.
**2015**, 106, 151101. [Google Scholar] [CrossRef] - Luo, K.H.; Brauner, S.; Eigner, C.; Sharapova, P.R.; Ricken, R.; Meier, T.; Herrmann, H.; Silberhorn, C. Nonlinear integrated quantum electro-optic circuits. Sci. Adv.
**2019**, 5, eaat1451. [Google Scholar] [CrossRef] [Green Version] - Quesada, N.; Sipe, J.E. Effects of time ordering in quantum nonlinear optics. Phys. Rev. A
**2014**, 90, 063840. [Google Scholar] [CrossRef] [Green Version] - Christ, A.; Brecht, B.; Mauerer, W.; Silberhorn, C. Theory of quantum frequency conversion and type-II parametric down-conversion in the high-gain regime. New J. Phys.
**2013**, 15, 053038. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**Schematic model of the two-colour SU(1,1) interferometer. To produce spectrally non-degenerate and orthogonally polarised photons, the pump laser interacts with two PDC sections consisting of periodically poled KTP waveguides of length L. Different polarisations are depicted by the red and blue colours. The PDC sections are separated by the non-poled KTP section of length l. Two polarisation converters, located at the ${x}_{1}$ and ${x}_{2}$ positions, switch the polarisations of signal and idler photons in order to ensure a proper compensation of their group velocities when coming to the second PDC section. An additional phase for the idler photon is provided and controlled by the phase modulator PM. The signal photon is finally detected.

**Figure 2.**Normalised spectra of (

**a**) signal and (

**b**) idler photons. The FWHM of both is $\Delta \omega $ = 2 × ${10}^{12}$ rad/s, and the spectral detuning of both is $\delta \omega $ = 10 × ${10}^{12}$ rad/s. The following parameters were used for the calculations: L = 8 mm, ${x}_{1}\simeq 1.038$ mm, ${x}_{2}\simeq 8.962$ mm, $\mathsf{\Lambda}$ = 133 $\mathsf{\mu}$m, pump wavelength ${\lambda}_{p}$ = 766 nm, $\varphi =\phi +\mathsf{\Phi}=0$; see the discussion around Equation (12) for more details.

**Figure 3.**Normalised spectra of signal photons at (

**a**) $\varphi =0$, (

**b**) $\varphi =\pi /2$, and (

**c**) $\varphi =\pi $. The normalisation is performed with respect to the maximum intensity in the constructive interference case, when $\varphi =0$. The following parameters were chosen and fixed for all further calculations: ${\lambda}_{p}$ = 766 nm, L = 8 mm, ${x}_{1}\simeq 1.038$ mm, ${x}_{2}\simeq 8.962$ mm, $\mathsf{\Lambda}$ = 133 $\mathsf{\mu}$m. The choice of $\delta \omega ={10}^{13}$ rad/s ensures the fully spectral distinguishability of signal and idler photons.

**Figure 4.**The phase sensitivity normalised to the SNL versus the phase at different gains. The SNL is shown by the black line.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Ferreri, A.; Sharapova, P.R.
Two-Colour Spectrally Multimode Integrated SU(1,1) Interferometer. *Symmetry* **2022**, *14*, 552.
https://doi.org/10.3390/sym14030552

**AMA Style**

Ferreri A, Sharapova PR.
Two-Colour Spectrally Multimode Integrated SU(1,1) Interferometer. *Symmetry*. 2022; 14(3):552.
https://doi.org/10.3390/sym14030552

**Chicago/Turabian Style**

Ferreri, Alessandro, and Polina R. Sharapova.
2022. "Two-Colour Spectrally Multimode Integrated SU(1,1) Interferometer" *Symmetry* 14, no. 3: 552.
https://doi.org/10.3390/sym14030552