Stability Analysis of Three Coupled Kerr Oscillators: Implications for Quantum Computing
Chmielewski Kuba 1, Grygiel Krzysztof 2*, Bartkiewicz Karol 2
1 Adam Mickiewicz University
Faculty of Physics and Astronomy
61-614 Poznań, Poland
2 Adam Mickiewicz University
Department of Quantum Information, Faculty of Physics and Astronomy
61-614 Poznań, Poland
∗E-mail: grygielk@amu.edu.pl
Received:
Received: 27 June 2025; accepted: 22 July 2025; published online: 18 August 2025
DOI: 10.12921/cmst.2025.0000012
Abstract:
We investigate the classical dynamics of optical nonlinear Kerr couplers, focusing on their potential relevance to quantum computing applications. The system consists of three Kerr-type nonlinear oscillators arranged in two configurations: a triangular arrangement, where each oscillator is coupled to the others, and a sandwich arrangement, where only the middle oscillator interacts with the two outer ones. The system is driven by an external periodic field and subject to dissipative processes. Its evolution is governed by six non-autonomous differential equations derived from a Kerr Hamiltonian with nonlinear coupling terms. We demonstrate that even for identical Kerr media, the interplay between nonlinear couplings and mismatched fundamental and pump frequencies gives rise to rich and complex dynamics, including the emergence of multiple stable attractors. These attractors are highly sensitive to both the coupling configuration and initial conditions. A key contribution of this work is a detailed stability analysis based on numerical calculation of Lyapunov exponents, revealing transitions from regular to chaotic dynamics as damping is reduced. We identify critical damping thresholds for the onset of chaos and characterize phenomena such as chaotic beats. These findings offer insights for potential experimental realizations and are directly relevant to emerging quantum technologies, where Kerr parametric oscillators play a central role in quantum gates, error correction protocols, and quantum neural network architectures.
Key words:
chaotic beats, Kerr oscillators, Lyapunov exponents, quantum technology
References:
[1] R.W. Boyd, Nonlinear Optics, Academic Press, Amsterdam (2008).
[2] Y.S. Kivshar, G.P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals, Academic Press, San Diego (2003).
[3] M.O. Scully, M.S. Zubairy, Quantum Optics, Cambridge University Press, Cambridge, UK (1997).
[4] C. Gerry, P. Knight, Introductory Quantum Optics, Cambridge University Press, Cambridge (2006).
[5] D.F. Walls, G.J. Milburn, Quantum Optics, Springer, Berlin (2006).
[6] X. Gu, A.F. Kockum, A. Miranowicz, Y. Liu, F. Nori, Microwave photonics with superconducting quantum circuits, Phys. Rep. 718–719, 1–102 (2017).
[7] C. Cui, L. Zhang, L. Fan, In situ control of effective Kerr nonlinearity with Pockels integrated photonics, Nat. Phys. 18, 497–501 (2022).
[8] M. Perez, G. Moille, X. Lu, J. Stone, F. Zhou, K. Srinivasan, High-performance Kerr microresonator optical parametric oscillation on a silicon chip, Nature Commun. 14, 242 (2023).
[9] Y. Yin, H. Wang, M. Mariantoni, R.C. Bialczak, R. Barends, Y. Chen, M. Lenander, E. Lucero, M. Neely et al., Dynamic quantum Kerr effect in circuit quantum electrodynamics, Phys. Rev. A 85, 023826 (2021).
[10] IBM, IBM Debuts Next-Generation Quantum Processor & IBM Quantum System Two, IBM Newsroom (2023).
[11] T. Kanao, S. Masuda, S. Kawabata, H. Goto, Quantum Gate for a Kerr Nonlinear Parametric Oscillator Using Effective Excited States, Phys. Rev. Applied 18, 014019 (2022).
[12] G. Margiani, O. Ameye, O. Zilberberg, A. Eichler, Three strongly coupled Kerr parametric oscillators forming a Boltzmann machine, arXiv:2504.04254 (2025).
[13] I. Śliwa, K. Grygiel, Periodic orbits, basins of attraction and chaotic beats in two coupled Kerr oscillators, Nonlin. Dynam. 67, 755–765 (2012).
[14] J.K. Kalaga, A. Kowalewska-Kudłaszyk, W. Leoński, A. Barasiński, Quantum correlations and entanglement in a model comprised of a short chain of nonlinear oscillators, Rev. A 94, 032304 (2016).
[15] M.S.M. Hanapi, A-B.M.A. Ibrahim, R. Julius, P.K. Choudhury, H. Eleuch, Nonclassical light in a three-waveguide coupler with second-order nonlinearity, EPJ Quant. Techn. 11, 51 (2024).
[16] G.D. Van Wiggeren, R. Roy, Communication with Chaotic Lasers, Science 279, 1198–1200 (1998).
[17] A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, P. Davis, Fast physical random bit generation with chaotic semiconductor lasers, Nat. Photonics 2, 728–732 (2008).
[18] D. Brunner, M.C. Soriano, C.R. Mirasso, I. Fischer, Parallel photonic information processing at gigabyte per second data rates using transient states, Nat. Commun. 4, 1364 (2013).
[19] S. Haroche, J.M. Raimond, Exploring the Quantum: Atoms, Cavities and Photons, Oxford University Press, Oxford (2006).
[20] S. Aldana, C. Bruder, A. Nunnenkamp, Equivalence between an optomechanical system and a Kerr medium, Phys. Rev. A 88, 043826 (2013).
[21] X. Wang, W. Qin, A. Miranowicz, S. Savasta, F. Nori, Unconventional Cavity Optomechanics: Nonlinear Control of Phonons in the Acoustic Quantum Vacuum, Phys. Rev. A 100, 063827 (2019).
[22] A.F. Kockum, A. Miranowicz, S. Liberato, S. Savasta, F. Nori, Ultrastrong coupling between light and matter, Nat. Rev. Phys. 1, 19 (2019).
[23] W. Qin, A.F. Kockum, C.S. Muñoz, A. Miranowicz, F. Nori, Quantum amplification and simulation of strong and ultrastrong coupling of light and matter, Phys. Rep. 1078, 1 (2024).
[24] R. Tanaś, A. Miranowicz, Ts. Gantsog, Quantum phase properties of nonlinear optical phenomena, [In:] Progress in Optics 35, Eds. E. Wolf, Elsevier, Amsterdam (1996), 355–446.
[25] W. Leoński, R. Tanaś, Possibility of producing the onephoton state in a kicked cavity with a nonlinear Kerr medium, Phys. Rev. A 49, R20(R) (1994).
[26] A. Imamogˇlu, H. Schmidt, G. Woods, M. Deutsch, Strongly Interacting Photons in a Nonlinear Cavity, Phys. Rev. Lett. 79, 1467 (1997).
[27] W. Leoński, A. Kowalewska-Kudłaszyk, Quantum Scissors: Finite-Dimensional States Engineering, Prog. Opt. 56, 131 (2011).
[28] A. Miranowicz, M. Paprzycka, Y. Liu, J. Bajer, F. Nori, Two-photon and three-photon blockades in driven nonlinear systems, Phys. Rev. A 87, 023809 (2013).
[29] C. Hamsen, K.N. Tolazzi, T. Wilk, G. Rempe, Two-Photon Blockade in an Atom-Driven Cavity QED System, Phys. Rev. Lett. 118, 133604 (2017).
[30] A. Kowalewska-Kudłaszyk, S.I. Abo, G. Chimczak, J. Perina Jr., F. Nori, A. Miranowicz, Two-photon blockade and photon-induced tunneling generated by squeezing, Phys. Rev. A 100, 053857 (2019).
[31] R. Huang, A. Miranowicz, J.-Q. Liao, F. Nori, H. Jing, Nonreciprocal Photon Blockade, Phys. Rev. Lett. 121, 153601 (2018).
[32] Y. Zuo, Y.-F. Jiao, X.-W. Xu, A. Miranowicz, L.-M. Kuang, H. Jing, Chiral photon blockade, Opt. Express 32, 22020– 22030 (2024).
[33] Y.-X. Liu, A. Miranowicz, Y.B. Gao, J. Bajer, C.P. Sun, F. Nori, Qubit-induced phonon blockade as a signature of quantum behavior in nanomechanical resonators, Phys. Rev. A 82, 032101 (2010).
[34] S. Abo, G. Chimczak, A. Kowalewska-Kudłaszyk, J. Perina Jr., R. Chhajlany, A. Miranowicz, Hybrid photon-phonon blockade, Sci. Rep. 12, 17655 (2022).
[35] W. Leoński, A. Miranowicz, Kerr nonlinear coupler and entanglement, J. Opt. B 6, S37 (2004).
[36] A. Miranowicz, W. Leoński, Two-mode optical state truncation and generation of maximally entangled states in pumped nonlinear couplers, J. Phys. B 39, 1683 (2006).
[37] A. Miranowicz, J. Bajer, N. Lambert, Y. Liu, F. Nori, Tunable multiphonon blockade in coupled nanomechanical resonators, Phys. Rev. A 93, 013808 (2016).
[38] T.C.H. Liew, V. Savona, Single Photons from Coupled Quantum Modes, Phys. Rev. Lett. 104, 183601 (2010).
[39] B. Li, R. Huang, X.-W. Xu, A. Miranowicz, H. Jing, Nonreciprocal unconventional photon blockade in a spinning optomechanical system, Photonics Res. 7, 630–641 (2019).
[40] R. Tanaś, Theory of Non-Classical States of Light, [In:] V. Dodonov, V.I. Man’ko (Eds.), Taylor & Francis, London (2003).
[41] R. Tanaś, S. Kielich, Self-squeezing of light propagating through nonlinear optically isotropic media, Opt. Commun. 45, 351 (1983).
[42] Y. Yamamoto, N. Imoto, S. Machida, Amplitude squeezing in a semiconductor laser using quantum nondemolition measurement and negative feedback, Phys. Rev. A 33, 3243 (1986).
[43] R. Tanaś, A. Miranowicz, S. Kielich, Squeezing and its graphical representations in the anharmonic oscillator model, Phys. Rev. A 43, 4014 (1991).
[44] J. Bajer, A. Miranowicz, R. Tanaś, Limits of noise squeezing in Kerr effect, Czech. J. Phys. 52, 1313 (2002).
[45] B. Yurke, D. Stoler, Generating quantum mechanical superpositions of macroscopically distinguishable states via amplitude dispersion, Phys. Rev. Lett. 57, 13 (1986).
[46] P. Tombesi, A. Mecozzi, Generation of macroscopically distinguishable quantum states and detection by the squeezed-vacuum technique, J. Opt. Soc. Am. B 4, 1700 (1987).
[47] A. Miranowicz, R. Tanaś, S. Kielich, Generation of discrete superpositions of coherent states in the anharmonic oscillator model, Quantum Opt. 2, 253 (1990).
[48] G. Kirchmair, B. Vlastakis, Z. Leghtas, S.E. Nigg, H. Paik, E. Ginossar, M. Mirrahimi, L. Frunzio, S.M. Girvin, R.J. Schoelkopf, Observation of quantum state collapse and revival due to the single-photon Kerr effect, Nature 495, 205 (2013).
[49] X.L. He, Y. Lu, D.Q. Bao, H. Xue, W.B. Jiang, Z. Wang, A.F. Roudsari, P. Delsing, J.S. Tsai, Z.R. Lin, Fast generation of Schrödinger cat states using a Kerr-tunable superconducting resonator, Nat. Commun. 14, 6358 (2023).
[50] D. Iyama, T. Kamiya, S. Fujii, H. Mukai, Y. Zhou, T. Nagase, A. Tomonaga, R. Wang, J.-J. Xue, S. Watabe, S. Kwon, J.-S. Tsai, Observation and manipulation of quantum interference in a superconducting Kerr parametric oscillator, Nat. Commun. 15, 86 (2024).
[51] G.J. Milburn, D.F. Walls, Quantum nondemolition measurements via quadratic coupling, Phys. Rev. A 28, 2065 (1983).
[52] N. Imoto, H.A. Haus, Y. Yamamoto, Quantum nondemolition measurement of the photon number via the optical Kerr effect, Phys. Rev. A 32, 2287 (1985).
[53] H. Chono, T. Kanao, H. Goto, Two-qubit gate using conditional driving for highly detuned Kerr-nonlinear parametric oscillators, Phys. Rev. Research 4, 043054 (2022).
[54] F.-F. Du, G. Fan, X.-M. Ren, Kerr-effect-based quantum logical gates in decoherence-free subspace, Quantum 8, 1342 (2024).
[55] Y.-H. Chen, R. Stassi, W. Qin, A. Miranowicz, F. Nori, Fault-tolerant multiqubit geometric entangling gates using photonic cat-state qubits, Phys. Rev. Appl. 18, 024076 (2022).
[56] M. Bartkowiak, L.-A. Wu, A. Miranowicz, Quantum circuits for amplification of Kerr nonlinearity via quadrature squeezing, J. Phys. B 47, 145501 (2014).
[57] A. Wolf, J.B. Swift, H.L. Swinney, J.A. Vastano, Determining Lyapunov exponents from a time series, Physica D 16, 285–317 (1985).
[58] K. Grygiel, P. Szlachetka, Generation of chaotic beats, Int. J. Bifurcation Chaos 12, 635–644 (2002).
[59] D. Cafagna, G. Grassi, A new phenomenon in nonautonomous Chua’s circuits: Generation of chaotic beats, Int. J. Bifurcation Chaos 14, 1773–1788 (2004).
[60] I. Śliwa, P. Szlachetka, K. Grygiel, Chaotic beats in a nonautonomous system governing second-harmonic generation of light, Int. J. Bifurcation Chaos 17, 3253–3257 (2007).
[61] A.I. Ahamed, K. Srinivasan, K. Murali, M. Lakshmanan, Observation of chaotic beats in a driven memristive Chua’s circuit, Int. J. Bifurcation Chaos 21, 737–757 (2011).
[62] M.P. Asir, A. Jeevarekha, P. Philominathan, Experimental observation of chaotic beats in oscillators sharing nonlinearity, Int. J. Bifurcation Chaos 26, 1630027 (2016).
[63] J. Maldacena, S.H. Shenker, D. Stanford, A bound on chaos, J. High Energy Phys. 2016, 106 (2016).
We investigate the classical dynamics of optical nonlinear Kerr couplers, focusing on their potential relevance to quantum computing applications. The system consists of three Kerr-type nonlinear oscillators arranged in two configurations: a triangular arrangement, where each oscillator is coupled to the others, and a sandwich arrangement, where only the middle oscillator interacts with the two outer ones. The system is driven by an external periodic field and subject to dissipative processes. Its evolution is governed by six non-autonomous differential equations derived from a Kerr Hamiltonian with nonlinear coupling terms. We demonstrate that even for identical Kerr media, the interplay between nonlinear couplings and mismatched fundamental and pump frequencies gives rise to rich and complex dynamics, including the emergence of multiple stable attractors. These attractors are highly sensitive to both the coupling configuration and initial conditions. A key contribution of this work is a detailed stability analysis based on numerical calculation of Lyapunov exponents, revealing transitions from regular to chaotic dynamics as damping is reduced. We identify critical damping thresholds for the onset of chaos and characterize phenomena such as chaotic beats. These findings offer insights for potential experimental realizations and are directly relevant to emerging quantum technologies, where Kerr parametric oscillators play a central role in quantum gates, error correction protocols, and quantum neural network architectures.
Key words:
chaotic beats, Kerr oscillators, Lyapunov exponents, quantum technology
References:
[1] R.W. Boyd, Nonlinear Optics, Academic Press, Amsterdam (2008).
[2] Y.S. Kivshar, G.P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals, Academic Press, San Diego (2003).
[3] M.O. Scully, M.S. Zubairy, Quantum Optics, Cambridge University Press, Cambridge, UK (1997).
[4] C. Gerry, P. Knight, Introductory Quantum Optics, Cambridge University Press, Cambridge (2006).
[5] D.F. Walls, G.J. Milburn, Quantum Optics, Springer, Berlin (2006).
[6] X. Gu, A.F. Kockum, A. Miranowicz, Y. Liu, F. Nori, Microwave photonics with superconducting quantum circuits, Phys. Rep. 718–719, 1–102 (2017).
[7] C. Cui, L. Zhang, L. Fan, In situ control of effective Kerr nonlinearity with Pockels integrated photonics, Nat. Phys. 18, 497–501 (2022).
[8] M. Perez, G. Moille, X. Lu, J. Stone, F. Zhou, K. Srinivasan, High-performance Kerr microresonator optical parametric oscillation on a silicon chip, Nature Commun. 14, 242 (2023).
[9] Y. Yin, H. Wang, M. Mariantoni, R.C. Bialczak, R. Barends, Y. Chen, M. Lenander, E. Lucero, M. Neely et al., Dynamic quantum Kerr effect in circuit quantum electrodynamics, Phys. Rev. A 85, 023826 (2021).
[10] IBM, IBM Debuts Next-Generation Quantum Processor & IBM Quantum System Two, IBM Newsroom (2023).
[11] T. Kanao, S. Masuda, S. Kawabata, H. Goto, Quantum Gate for a Kerr Nonlinear Parametric Oscillator Using Effective Excited States, Phys. Rev. Applied 18, 014019 (2022).
[12] G. Margiani, O. Ameye, O. Zilberberg, A. Eichler, Three strongly coupled Kerr parametric oscillators forming a Boltzmann machine, arXiv:2504.04254 (2025).
[13] I. Śliwa, K. Grygiel, Periodic orbits, basins of attraction and chaotic beats in two coupled Kerr oscillators, Nonlin. Dynam. 67, 755–765 (2012).
[14] J.K. Kalaga, A. Kowalewska-Kudłaszyk, W. Leoński, A. Barasiński, Quantum correlations and entanglement in a model comprised of a short chain of nonlinear oscillators, Rev. A 94, 032304 (2016).
[15] M.S.M. Hanapi, A-B.M.A. Ibrahim, R. Julius, P.K. Choudhury, H. Eleuch, Nonclassical light in a three-waveguide coupler with second-order nonlinearity, EPJ Quant. Techn. 11, 51 (2024).
[16] G.D. Van Wiggeren, R. Roy, Communication with Chaotic Lasers, Science 279, 1198–1200 (1998).
[17] A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, P. Davis, Fast physical random bit generation with chaotic semiconductor lasers, Nat. Photonics 2, 728–732 (2008).
[18] D. Brunner, M.C. Soriano, C.R. Mirasso, I. Fischer, Parallel photonic information processing at gigabyte per second data rates using transient states, Nat. Commun. 4, 1364 (2013).
[19] S. Haroche, J.M. Raimond, Exploring the Quantum: Atoms, Cavities and Photons, Oxford University Press, Oxford (2006).
[20] S. Aldana, C. Bruder, A. Nunnenkamp, Equivalence between an optomechanical system and a Kerr medium, Phys. Rev. A 88, 043826 (2013).
[21] X. Wang, W. Qin, A. Miranowicz, S. Savasta, F. Nori, Unconventional Cavity Optomechanics: Nonlinear Control of Phonons in the Acoustic Quantum Vacuum, Phys. Rev. A 100, 063827 (2019).
[22] A.F. Kockum, A. Miranowicz, S. Liberato, S. Savasta, F. Nori, Ultrastrong coupling between light and matter, Nat. Rev. Phys. 1, 19 (2019).
[23] W. Qin, A.F. Kockum, C.S. Muñoz, A. Miranowicz, F. Nori, Quantum amplification and simulation of strong and ultrastrong coupling of light and matter, Phys. Rep. 1078, 1 (2024).
[24] R. Tanaś, A. Miranowicz, Ts. Gantsog, Quantum phase properties of nonlinear optical phenomena, [In:] Progress in Optics 35, Eds. E. Wolf, Elsevier, Amsterdam (1996), 355–446.
[25] W. Leoński, R. Tanaś, Possibility of producing the onephoton state in a kicked cavity with a nonlinear Kerr medium, Phys. Rev. A 49, R20(R) (1994).
[26] A. Imamogˇlu, H. Schmidt, G. Woods, M. Deutsch, Strongly Interacting Photons in a Nonlinear Cavity, Phys. Rev. Lett. 79, 1467 (1997).
[27] W. Leoński, A. Kowalewska-Kudłaszyk, Quantum Scissors: Finite-Dimensional States Engineering, Prog. Opt. 56, 131 (2011).
[28] A. Miranowicz, M. Paprzycka, Y. Liu, J. Bajer, F. Nori, Two-photon and three-photon blockades in driven nonlinear systems, Phys. Rev. A 87, 023809 (2013).
[29] C. Hamsen, K.N. Tolazzi, T. Wilk, G. Rempe, Two-Photon Blockade in an Atom-Driven Cavity QED System, Phys. Rev. Lett. 118, 133604 (2017).
[30] A. Kowalewska-Kudłaszyk, S.I. Abo, G. Chimczak, J. Perina Jr., F. Nori, A. Miranowicz, Two-photon blockade and photon-induced tunneling generated by squeezing, Phys. Rev. A 100, 053857 (2019).
[31] R. Huang, A. Miranowicz, J.-Q. Liao, F. Nori, H. Jing, Nonreciprocal Photon Blockade, Phys. Rev. Lett. 121, 153601 (2018).
[32] Y. Zuo, Y.-F. Jiao, X.-W. Xu, A. Miranowicz, L.-M. Kuang, H. Jing, Chiral photon blockade, Opt. Express 32, 22020– 22030 (2024).
[33] Y.-X. Liu, A. Miranowicz, Y.B. Gao, J. Bajer, C.P. Sun, F. Nori, Qubit-induced phonon blockade as a signature of quantum behavior in nanomechanical resonators, Phys. Rev. A 82, 032101 (2010).
[34] S. Abo, G. Chimczak, A. Kowalewska-Kudłaszyk, J. Perina Jr., R. Chhajlany, A. Miranowicz, Hybrid photon-phonon blockade, Sci. Rep. 12, 17655 (2022).
[35] W. Leoński, A. Miranowicz, Kerr nonlinear coupler and entanglement, J. Opt. B 6, S37 (2004).
[36] A. Miranowicz, W. Leoński, Two-mode optical state truncation and generation of maximally entangled states in pumped nonlinear couplers, J. Phys. B 39, 1683 (2006).
[37] A. Miranowicz, J. Bajer, N. Lambert, Y. Liu, F. Nori, Tunable multiphonon blockade in coupled nanomechanical resonators, Phys. Rev. A 93, 013808 (2016).
[38] T.C.H. Liew, V. Savona, Single Photons from Coupled Quantum Modes, Phys. Rev. Lett. 104, 183601 (2010).
[39] B. Li, R. Huang, X.-W. Xu, A. Miranowicz, H. Jing, Nonreciprocal unconventional photon blockade in a spinning optomechanical system, Photonics Res. 7, 630–641 (2019).
[40] R. Tanaś, Theory of Non-Classical States of Light, [In:] V. Dodonov, V.I. Man’ko (Eds.), Taylor & Francis, London (2003).
[41] R. Tanaś, S. Kielich, Self-squeezing of light propagating through nonlinear optically isotropic media, Opt. Commun. 45, 351 (1983).
[42] Y. Yamamoto, N. Imoto, S. Machida, Amplitude squeezing in a semiconductor laser using quantum nondemolition measurement and negative feedback, Phys. Rev. A 33, 3243 (1986).
[43] R. Tanaś, A. Miranowicz, S. Kielich, Squeezing and its graphical representations in the anharmonic oscillator model, Phys. Rev. A 43, 4014 (1991).
[44] J. Bajer, A. Miranowicz, R. Tanaś, Limits of noise squeezing in Kerr effect, Czech. J. Phys. 52, 1313 (2002).
[45] B. Yurke, D. Stoler, Generating quantum mechanical superpositions of macroscopically distinguishable states via amplitude dispersion, Phys. Rev. Lett. 57, 13 (1986).
[46] P. Tombesi, A. Mecozzi, Generation of macroscopically distinguishable quantum states and detection by the squeezed-vacuum technique, J. Opt. Soc. Am. B 4, 1700 (1987).
[47] A. Miranowicz, R. Tanaś, S. Kielich, Generation of discrete superpositions of coherent states in the anharmonic oscillator model, Quantum Opt. 2, 253 (1990).
[48] G. Kirchmair, B. Vlastakis, Z. Leghtas, S.E. Nigg, H. Paik, E. Ginossar, M. Mirrahimi, L. Frunzio, S.M. Girvin, R.J. Schoelkopf, Observation of quantum state collapse and revival due to the single-photon Kerr effect, Nature 495, 205 (2013).
[49] X.L. He, Y. Lu, D.Q. Bao, H. Xue, W.B. Jiang, Z. Wang, A.F. Roudsari, P. Delsing, J.S. Tsai, Z.R. Lin, Fast generation of Schrödinger cat states using a Kerr-tunable superconducting resonator, Nat. Commun. 14, 6358 (2023).
[50] D. Iyama, T. Kamiya, S. Fujii, H. Mukai, Y. Zhou, T. Nagase, A. Tomonaga, R. Wang, J.-J. Xue, S. Watabe, S. Kwon, J.-S. Tsai, Observation and manipulation of quantum interference in a superconducting Kerr parametric oscillator, Nat. Commun. 15, 86 (2024).
[51] G.J. Milburn, D.F. Walls, Quantum nondemolition measurements via quadratic coupling, Phys. Rev. A 28, 2065 (1983).
[52] N. Imoto, H.A. Haus, Y. Yamamoto, Quantum nondemolition measurement of the photon number via the optical Kerr effect, Phys. Rev. A 32, 2287 (1985).
[53] H. Chono, T. Kanao, H. Goto, Two-qubit gate using conditional driving for highly detuned Kerr-nonlinear parametric oscillators, Phys. Rev. Research 4, 043054 (2022).
[54] F.-F. Du, G. Fan, X.-M. Ren, Kerr-effect-based quantum logical gates in decoherence-free subspace, Quantum 8, 1342 (2024).
[55] Y.-H. Chen, R. Stassi, W. Qin, A. Miranowicz, F. Nori, Fault-tolerant multiqubit geometric entangling gates using photonic cat-state qubits, Phys. Rev. Appl. 18, 024076 (2022).
[56] M. Bartkowiak, L.-A. Wu, A. Miranowicz, Quantum circuits for amplification of Kerr nonlinearity via quadrature squeezing, J. Phys. B 47, 145501 (2014).
[57] A. Wolf, J.B. Swift, H.L. Swinney, J.A. Vastano, Determining Lyapunov exponents from a time series, Physica D 16, 285–317 (1985).
[58] K. Grygiel, P. Szlachetka, Generation of chaotic beats, Int. J. Bifurcation Chaos 12, 635–644 (2002).
[59] D. Cafagna, G. Grassi, A new phenomenon in nonautonomous Chua’s circuits: Generation of chaotic beats, Int. J. Bifurcation Chaos 14, 1773–1788 (2004).
[60] I. Śliwa, P. Szlachetka, K. Grygiel, Chaotic beats in a nonautonomous system governing second-harmonic generation of light, Int. J. Bifurcation Chaos 17, 3253–3257 (2007).
[61] A.I. Ahamed, K. Srinivasan, K. Murali, M. Lakshmanan, Observation of chaotic beats in a driven memristive Chua’s circuit, Int. J. Bifurcation Chaos 21, 737–757 (2011).
[62] M.P. Asir, A. Jeevarekha, P. Philominathan, Experimental observation of chaotic beats in oscillators sharing nonlinearity, Int. J. Bifurcation Chaos 26, 1630027 (2016).
[63] J. Maldacena, S.H. Shenker, D. Stanford, A bound on chaos, J. High Energy Phys. 2016, 106 (2016).
