Adler, R. A study of locking phenomena in oscillators. Proc. IRE 34, 351–357 (1946).
Kuramoto, Y. Self-entrainment of a population of coupled non-linear oscillators. in International Symposium on Mathematical Problems in Theoretical Physics (Springer Berlin Heidelberg, Berlin, Heidelberg, 1975).
Acebrón, J. A., Bonilla, L. L., Pérez Vicente, C. J., Ritort, F. & Spigler, R. The kuramoto model: A simple paradigm for synchronization phenomena. Rev. Mod. Phys. 77, 137–185 (2005).
Kiselev, S. I. et al. Microwave oscillations of a nanomagnet driven by a spin-polarized current. Nature 425, 380–383 (2003).
Kaka, S. et al. Mutual phase-locking of microwave spin torque nano-oscillators. Nature 437, 389–392 (2005).
Mancoff, F. B., Rizzo, N. D., Engel, B. N. & Tehrani, S. Phase-locking in double-point-contact spin-transfer devices. Nature 437, 393–395 (2005).
Houssameddine, D. et al. Spin-torque oscillator using a perpendicular polarizer and a planar free layer. Nat. Mater. 6, 447 (2007).
Pribiag, V. S. et al. Magnetic vortex oscillator driven by d.c. spin-polarized current. Nat. Phys. 3, 498–503 (2007).
Slavin, A. & Tiberkevich, V. Nonlinear auto-oscillator theory of microwave generation by spin-polarized current. IEEE Trans. Magn. 45, 1875–1918 (2009).
Safranski, C. et al. Spin caloritronic nano-oscillator. Nat. Commun. 8, 117 (2017).
Zhu, J.-G., Zhu, X. & Tang, Y. Microwave assisted magnetic recording. IEEE Trans. Magn. 44, 125–131 (2008).
Dieny, B. et al. Opportunities and challenges for spintronics in the microelectronics industry. Nat. Electron. 3, 446 (2020).
Sharma, R. et al. Electrically connected spin-torque oscillators array for 2.4-ghz wifi band transmission and energy harvesting. Nat. Commun. 12, 2924 (2021).
Torrejon, J. et al. Neuromorphic computing with nanoscale spintronic oscillators. Nature 547, 428–431 (2017).
Zahedinejad, M. et al. Two-dimensional mutually synchronized spin hall nano-oscillator arrays for neuromorphic computing. Nat. Nanotechnol. 15, 47 (2020).
Romera, M. et al. Vowel recognition with four coupled spin-torque nano-oscillators. Nature 563, 230 (2018).
Albertsson, D. I. et al. Ultrafast ising machines using spin torque nano-oscillators. Appl. Phys. Lett. 118, 112404 (2021).
Houshang, A. et al. Phase-binarized spin hall nano-oscillator arrays: Towards spin hall ising machines. Phys. Rev. Appl. 17, 014003 (2022).
Demokritov, S. O. et al. Bose-einstein condensation of quasi-equilibrium magnons at room temperature under pumping. Nature 443, 430–433 (2006).
Bozhko, D. A. et al. Supercurrent in a room-temperature bose-einstein magnon condensate. Nat. Phys. 12, 1057–1062 (2016).
Schneider, M. et al. Bose-einstein condensation of quasiparticles by rapid cooling. Nat. Nano. 15, 457–461 (2020).
L’vov, V. S., Pomyalov, A., Bozhko, D. A., Hillebrands, B. & Serga, A. A. Correlation-enhanced interaction of a bose-einstein condensate with parametric magnon pairs and virtual magnons. Phys. Rev. Lett. 131, 156705 (2023).
Bunkov, Y. M. & Volovik, G. E. Spin superfluidity and magnon BEC. arXiv:1003.4889. https://doi.org/10.48550/arXiv.1003.4889 (2010).
Bao, M. et al. Coplanar waveguide radio frequency ferromagnetic parametric amplifier. Appl. Phys. Lett. 93, 072509 (2008).
Brächer, T. et al. Parallel parametric amplification of coherently excited propagating spin waves in a microscopic ni81fe19 waveguide. Appl. Phys. Lett. 104, 202408 (2014).
Kamimaki, A., Iihama, S., Suzuki, K., Yoshinaga, N. & Mizukami, S. Parametric amplification of magnons in synthetic antiferromagnets. Phys. Rev. Appl. 13, 044036 (2020).
Bejarano, M. et al. Parametric magnon transduction to spin qubits. Sci. Adv. 10, eadi2042 (2024).
Ordóñez Romero, C. L. et al. Three-magnon splitting and confluence processes for spin-wave excitations in yttrium iron garnet films: Wave vector selective brillouin light scattering measurements and analysis. Phys. Rev. B 79, 144428 (2009).
Qu, T., Hamill, A., Victora, R. H. & Crowell, P. A. Oscillations and confluence in three-magnon scattering of ferromagnetic resonance. Phys. Rev. B 107, L060401 (2023).
Makiuchi, T. et al. Persistent magnetic coherence in magnets. Nat. Mater. 23, 627–632 (2024).
Qu, T., Xiong, Y., Zhang, X., Li, Y. & Zhang, W. Pump-induced magnon anticrossing due to three-magnon splitting and confluence. Phys. Rev. B 111, L180410 (2025).
Anderson, P. W. & Suhl, H. Instability in the motion of ferromagnets at high microwave power levels. Phys. Rev. 100, 1788–1789 (1955).
Suhl, H. The theory of ferromagnetic resonance at high signal powers. J. Phys. Chem. Solids 1, 209–227 (1957).
Kurebayashi, H. et al. Controlled enhancement of spin-current emission by three-magnon splitting. Nat. Mater. 10, 660–664 (2011).
Lendinez, S. et al. Nonlinear multi-magnon scattering in artificial spin ice. Nat. Commun. 14, 3419 (2023).
Körber, L. et al. Pattern recognition in reciprocal space with a magnon-scattering reservoir. Nat. Commun. 14, 3954 (2023).
Nikolaev, K. O., Lake, S. R., Schmidt, G., Demokritov, S. O. & Demidov, V. E. Resonant generation of propagating second-harmonic spin waves in nano-waveguides. Nat. Commun. 15, 1827 (2024).
Schultheiss, H., Vogt, K. & Hillebrands, B. Direct observation of nonlinear four-magnon scattering in spin-wave microconduits. Phys. Rev. B 86, 054414 (2012).
Pirro, P. et al. Non-gilbert-damping mechanism in a ferromagnetic heusler compound probed by nonlinear spin dynamics. Phys. Rev. Lett. 113, 227601 (2014).
An, K. et al. Emergent coherent modes in nonlinear magnonic waveguides detected at ultrahigh frequency resolution. Nat. Commun. 15, 7302 (2024).
Geilen, M. et al. Parametric excitation and instabilities of spin waves driven by surface acoustic waves. Adv. Phys. Res. 4, 2400086 (2025).
Wang, B. PT-symmetric magnon laser in cavity optomagnonics. Phys. Rev. A 105, 053705 (2022).
Wang, Z. Q. et al. Single-mode magnon-polariton lasing and amplification controlled by dissipative coupling. Phys. Rev. Lett. 135, 186704 (2025).
Li, Y. et al. Unidirectional microwave transduction with chirality selected short-wavelength magnon excitations. Appl. Phys. Lett. 123, 022406 (2023).
See the Supplemental Information for details.
Hula, T. et al. Spin-wave frequency combs. Appl. Phys. Lett. 121, 112404 (2022).
Yamamoto, T. et al. Flux-driven josephson parametric amplifier. Appl. Phys. Lett. 93, 042510 (2008).
Castellanos-Beltran, M. A., Irwin, K. D., Hilton, G. C., Vale, L. R. & Lehnert, K. W. Amplification and squeezing of quantum noise with a tunable josephson metamaterial. Nat. Phys. 4, 929–931 (2008).
Macklin, C. et al. A near-quantum-limited josephson traveling-wave parametric amplifier. Science 350, 307–310 (2015).
Anferov, A., Suleymanzade, A., Oriani, A., Simon, J. & Schuster, D. I. Millimeter-wave four-wave mixing via kinetic inductance for quantum devices. Phys. Rev. Appl. 13, 024056 (2020).
Xu, M., Cheng, R., Wu, Y., Liu, G. & Tang, H. X. Magnetic field-resilient quantum-limited parametric amplifier. PRX Quantum 4, 010322 (2023).
Khalifa, M. & Salfi, J. Nonlinearity and parametric amplification of superconducting nanowire resonators in magnetic field. Phys. Rev. Appl. 19, 034024 (2023).
Yurke, B. & Buks, E. Performance of cavity-parametric amplifiers, employing kerr nonlinearites, in the presence of two-photon loss. J. Lightwave Tech. 24, 5054 (2006).
Clerk, A. A., Devoret, M. H., Girvin, S. M., Marquardt, F. & Schoelkopf, R. J. Introduction to quantum noise, measurement, and amplification. Rev. Mod. Phys. 82, 1155–1208 (2010).
Brächer, T. et al. Time- and power-dependent operation of a parametric spin-wave amplifier. Appl. Phys. Lett. 105, 232409 (2014).
Verba, R., Carpentieri, M., Finocchio, G., Tiberkevich, V. & Slavin, A. Amplification and stabilization of large-amplitude propagating spin waves by parametric pumping. Appl. Phys. Lett. 112, 042402 (2018).
Brächer, T., Pirro, P. & Hillebrands, B. Parallel pumping for magnon spintronics: Amplification and manipulation of magnon spin currents on the micron-scale. Phys. Rep. 699, 1–34 (2017).
Padrón-Hernández, E., Azevedo, A. & Rezende, S. M. Amplification of spin waves in yttrium iron garnet films through the spin hall effect. Appl. Phys. Lett. 99, 192511 (2011).
Navabi, A. et al. Control of spin-wave damping in yig using spin currents from topological insulators. Phys. Rev. Appl. 11, 034046 (2019).
Breitbach, D. et al. Stimulated amplification of propagating spin waves. Phys. Rev. Lett. 131, 156701 (2023).
Merbouche, H. et al. True amplification of spin waves in magnonic nano-waveguides. Nat. Commun. 15, 1560 (2024).
Wang, Q., Csaba, G., Verba, R., Chumak, A. V. & Pirro, P. Nanoscale magnonic networks. Phys. Rev. Appl. 21, 040503 (2024).
Mohseni, M. et al. Controlling the nonlinear relaxation of quantized propagating magnons in nanodevices. Phys. Rev. Lett. 126, 097202 (2021).
Schultheiss, K. et al. Excitation of whispering gallery magnons in a magnetic vortex. Phys. Rev. Lett. 122, 097202 (2019).
Körber, L. et al. Nonlocal stimulation of three-magnon splitting in a magnetic vortex. Phys. Rev. Lett. 125, 207203 (2020).