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  • Open Access

Tangent equations of motion for nonlinear response functions

Atsushi Ono*

  • Department of Physics, Graduate School of Science, Tohoku University, Sendai 980-8578, Japan

  • *Contact author: a.ono@tohoku.ac.jp
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Phys. Rev. Research 8, 033131 – Published 3 August, 2026

DOI: https://doi.org/10.1103/hbjl-f5cw

Abstract

Nonlinear response functions, formulated as multipoint correlation functions or Volterra kernels, encode the dynamical and spectroscopic properties of physical systems and underpin a wide range of nonlinear transport and optical phenomena. However, their evaluation rapidly becomes prohibitive at high orders because of combinatorial (often factorial) scaling or severe numerical errors. Here, we establish a systematic and efficient framework to compute nonlinear response functions directly from real-time dynamics, without explicitly constructing multipoint correlators or relying on numerically unstable finite-difference methods for order-resolved extraction. Our approach is based on the Gateaux derivative with respect to the external field in function space, which yields a closed hierarchy of tangent equations of motion (TEOM). Propagating the TEOM alongside the original dynamics isolates each perturbative order with high accuracy, providing a term-by-term decomposition of physical contributions. The computational cost scales exponentially with response order in the fully general setting and reduces to polynomial complexity when all perturbation directions are identical; both regimes avoid the factorial scaling of explicit multipoint-correlator evaluations. We demonstrate the power of TEOM by computing frequency-resolved fifth-order response functions for a solid-state electron model and by obtaining nonlinear response functions up to the 49th order with controlled accuracy in a classical Duffing oscillator. We further show that our time-evolution formulation allows optical conductivities to be evaluated directly while remaining numerically stable even near zero frequency. TEOM can be incorporated seamlessly into existing real-time evolution methods, yielding a general framework for computing nonlinear response functions in quantum and classical dynamical systems.

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References (168)

  1. H. B. Callen and T. A. Welton, Irreversibility and generalized noise, Phys. Rev. 83, 34 (1951).
  2. R. Kubo, Statistical-mechanical theory of irreversible processes. I. General theory and simple applications to magnetic and conduction problems, J. Phys. Soc. Jpn. 12, 570 (1957).
  3. R. Kubo, The fluctuation-dissipation theorem, Rep. Prog. Phys. 29, 255 (1966).
  4. R. L. Peterson, Formal theory of nonlinear response, Rev. Mod. Phys. 39, 69 (1967).
  5. S. Mukamel, Principles of Nonlinear Optical Spectroscopy (Oxford University Press, New York, NY, 1995).
  6. R. W. Boyd, Nonlinear Optics (Academic Press, San Diego, CA, 2020).
  7. J.-M. Liu, Nonlinear Photonics (Cambridge University Press, Cambridge, 2022).
  8. Y. R. Shen, Recent advances in nonlinear optics, Rev. Mod. Phys. 48, 1 (1976).
  9. S. Mukamel, Multidimensional femtosecond correlation spectroscopies of electronic and vibrational excitations, Annu. Rev. Phys. Chem. 51, 691 (2000).
  10. R. Matsunaga, Y. I. Hamada, K. Makise, Y. Uzawa, H. Terai, Z. Wang, and R. Shimano, Higgs amplitude mode in the BCS superconductors Nb1−xTixN induced by terahertz pulse excitation, Phys. Rev. Lett. 111, 057002 (2013).
  11. R. Matsunaga, N. Tsuji, H. Fujita, A. Sugioka, K. Makise, Y. Uzawa, H. Terai, Z. Wang, H. Aoki, and R. Shimano, Light-induced collective pseudospin precession resonating with Higgs mode in a superconductor, Science 345, 1145 (2014).
  12. N. Tsuji and H. Aoki, Theory of Anderson pseudospin resonance with Higgs mode in superconductors, Phys. Rev. B 92, 064508 (2015).
  13. T. Cea, C. Castellani, and L. Benfatto, Nonlinear optical effects and third-harmonic generation in superconductors: Cooper pairs versus Higgs mode contribution, Phys. Rev. B 93, 180507(R) (2016).
  14. D. Pekker and C. Varma, Amplitude/Higgs modes in condensed matter physics, Annu. Rev. Condens. Matter Phys. 6, 269 (2015).
  15. R. Shimano and N. Tsuji, Higgs mode in superconductors, Annu. Rev. Condens. Matter Phys. 11, 103 (2020).
  16. L. Luo, M. Mootz, J. H. Kang, C. Huang, K. Eom, J. W. Lee, C. Vaswani, Y. G. Collantes, E. E. Hellstrom, I. E. Perakis, C. B. Eom, and J. Wang, Quantum coherence tomography of light-controlled superconductivity, Nat. Phys. 19, 201 (2023).
  17. C. Huang, M. Mootz, L. Luo, D. Cheng, A. Khatri, J.-M. Park, R. H. J. Kim, Y. Qiang, V. L. Quito, Y. Yao, P. P. Orth, I. E. Perakis, and J. Wang, Discovery of an unconventional quantum echo by interference of Higgs coherence, Sci. Adv. 11, eads8740 (2025).
  18. Y. Wan and N. P. Armitage, Resolving continua of fractional excitations by spinon echo in THz 2D coherent spectroscopy, Phys. Rev. Lett. 122, 257401 (2019).
  19. Z.-L. Li, M. Oshikawa, and Y. Wan, Photon echo from lensing of fractional excitations in Tomonaga-Luttinger spin liquid, Phys. Rev. X 11, 031035 (2021).
  20. M. Potts, R. Moessner, and O. Benton, Signatures of spinon dynamics and phase structure of dipolar-octupolar quantum spin ices in two-dimensional coherent spectroscopy, Phys. Rev. Lett. 133, 226701 (2024).
  21. F. Mahmood, D. Chaudhuri, S. Gopalakrishnan, R. Nandkishore, and N. P. Armitage, Observation of a marginal Fermi glass, Nat. Phys. 17, 627 (2021).
  22. D. Xiao, M.-C. Chang, and Q. Niu, Berry phase effects on electronic properties, Rev. Mod. Phys. 82, 1959 (2010).
  23. I. Sodemann and L. Fu, Quantum nonlinear Hall effect induced by Berry curvature dipole in time-reversal invariant materials, Phys. Rev. Lett. 115, 216806 (2015).
  24. K. Kang, T. Li, E. Sohn, J. Shan, and K. F. Mak, Nonlinear anomalous Hall effect in few-layer WTe2, Nat. Mater. 18, 324 (2019).
  25. Q. Ma, S.-Y. Xu, H. Shen, D. MacNeill, V. Fatemi, T.-R. Chang, A. M. Mier Valdivia, S. Wu, Z. Du, C.-H. Hsu, et al., Observation of the nonlinear Hall effect under time-reversal-symmetric conditions, Nature (London) 565, 337 (2019).
  26. T. Morimoto and N. Nagaosa, Topological nature of nonlinear optical effects in solids, Sci. Adv. 2, e1501524 (2016).
  27. F. de Juan, A. G. Grushin, T. Morimoto, and J. E. Moore, Quantized circular photogalvanic effect in Weyl semimetals, Nat. Commun. 8, 15995 (2017).
  28. J. Ahn, G.-Y. Guo, and N. Nagaosa, Low-frequency divergence and quantum geometry of the bulk photovoltaic effect in topological semimetals, Phys. Rev. X 10, 041041 (2020).
  29. J. Ahn, G.-Y. Guo, N. Nagaosa, and A. Vishwanath, Riemannian geometry of resonant optical responses, Nat. Phys. 18, 290 (2022).
  30. A. Gao, Y.-F. Liu, J.-X. Qiu, B. Ghosh, T. V. Trevisan, Y. Onishi, C. Hu, T. Qian, H.-J. Tien, S.-W. Chen, et al., Quantum metric nonlinear Hall effect in a topological antiferromagnetic heterostructure, Science 381, 181 (2023).
  31. J. Orenstein, J. Moore, T. Morimoto, D. Torchinsky, J. Harter, and D. Hsieh, Topology and symmetry of quantum materials via nonlinear optical responses, Annu. Rev. Condens. Matter Phys. 12, 247 (2021).
  32. P. Bhalla, K. Das, D. Culcer, and A. Agarwal, Resonant second-harmonic generation as a probe of quantum geometry, Phys. Rev. Lett. 129, 227401 (2022).
  33. T. Morimoto, S. Kitamura, and N. Nagaosa, Geometric aspects of nonlinear and nonequilibrium phenomena, J. Phys. Soc. Jpn. 92, 072001 (2023).
  34. M. Schetzen, The Volterra and Wiener Theories of  Nonlinear Systems (Wiley, New York, NY, 1980).
  35. H. K. Khalil, Nonlinear Systems (Prentice-Hall, Upper Saddle River, NJ, 2002).
  36. C. Cheng, Z. Peng, W. Zhang, and G. Meng, Volterra-series-based nonlinear system modeling and its engineering applications: A state-of-the-art review, Mech. Syst. Sig. Process. 87, 340 (2017).
  37. B. Orr and J. Ward, Perturbation theory of the non-linear optical polarization of an isolated system, Mol. Phys. 20, 513 (1971).
  38. J. F. Ward, Calculation of nonlinear optical susceptibilities using diagrammatic perturbation theory, Rev. Mod. Phys. 37, 1 (1965).
  39. W. Wu, Nonlinear optical susceptibilities of a one-dimensional semiconductor, Phys. Rev. Lett. 61, 1119 (1988).
  40. S. A. Jafari, T. Tohyama, and S. Maekawa, Dynamical mean field theory of optical third harmonic generation, J. Phys. Soc. Jpn. 75, 083706 (2006).
  41. D. E. Parker, T. Morimoto, J. Orenstein, and J. E. Moore, Diagrammatic approach to nonlinear optical response with application to Weyl semimetals, Phys. Rev. B 99, 045121 (2019).
  42. T. Holder, D. Kaplan, and B. Yan, Consequences of time-reversal-symmetry breaking in the light-matter interaction: Berry curvature, quantum metric, and diabatic motion, Phys. Rev. Res. 2, 033100 (2020).
  43. S. M. João and J. M. Viana Parente Lopes, Basis-independent spectral methods for non-linear optical response in arbitrary tight-binding models, J. Phys.: Condens. Matter 32, 125901 (2020).
  44. N. Tsuji, Y. Murakami, and H. Aoki, Nonlinear light–Higgs coupling in superconductors beyond BCS: Effects of the retarded phonon-mediated interaction, Phys. Rev. B 94, 224519 (2016).
  45. N. Tsuji and Y. Nomura, Higgs-mode resonance in third harmonic generation in NbN superconductors: Multiband electron-phonon coupling, impurity scattering, and polarization-angle dependence, Phys. Rev. Res. 2, 043029 (2020).
  46. M. Silaev, Nonlinear electromagnetic response and Higgs-mode excitation in BCS superconductors with impurities, Phys. Rev. B 99, 224511 (2019).
  47. Y.-T. Chang and Y.-H. Chan, Diagrammatic approach to excitonic effects on nonlinear optical response, Phys. Rev. B 109, 155437 (2024).
  48. T. Tanabe, T. Kaneko, and Y. Ohta, Third-harmonic generation in excitonic insulators, Phys. Rev. B 104, 245103 (2021).
  49. H. Tanaka and Y. Yanase, Vertex correction for the linear and nonlinear optical responses in superconductors: Multiband effect and topological superconductivity, Phys. Rev. B 112, 094503 (2025).
  50. Y. Michishita and R. Peters, Effects of renormalization and non-Hermiticity on nonlinear responses in strongly correlated electron systems, Phys. Rev. B 103, 195133 (2021).
  51. X. Wan, J. Dong, M. Qian, and W. Zhang, Nonlinear optical properties of perovskite YMnO3 studied by real-space recursion method, Phys. Rev. B 61, 10664 (2000).
  52. U. R. Meza, B. S. Mendoza, and W. L. Mochán, Second-harmonic generation in nanostructured metamaterials, Phys. Rev. B 99, 125408 (2019).
  53. K. A. Hallberg, Density-matrix algorithm for the calculation of dynamical properties of low-dimensional systems, Phys. Rev. B 52, R9827 (1995).
  54. S. Sota, S. Yunoki, and T. Tohyama, Density-matrix renormalization group study of third harmonic generation in one-dimensional Mott insulator coupled with phonon, J. Phys. Soc. Jpn. 84, 054403 (2015).
  55. H. Mori, A continued-fraction representation of the time-correlation functions, Prog. Theor. Phys. 34, 399 (1965).
  56. A. Weiße, G. Wellein, A. Alvermann, and H. Fehske, The kernel polynomial method, Rev. Mod. Phys. 78, 275 (2006).
  57. J. E. Sipe and E. Ghahramani, Nonlinear optical response of semiconductors in the independent-particle approximation, Phys. Rev. B 48, 11705 (1993).
  58. J. E. Sipe and A. I. Shkrebtii, Second-order optical response in semiconductors, Phys. Rev. B 61, 5337 (2000).
  59. G. B. Ventura, D. J. Passos, J. M. B. Lopes dos Santos, J. M. Viana Parente Lopes, and N. M. R. Peres, Gauge covariances and nonlinear optical responses, Phys. Rev. B 96, 035431 (2017).
  60. D. J. Passos, G. B. Ventura, J. M. Viana Parente Lopes, J. M. B. Lopes dos Santos, and N. M. R. Peres, Nonlinear optical responses of crystalline systems: Results from a velocity gauge analysis, Phys. Rev. B 97, 235446 (2018).
  61. H. Watanabe and Y. Yanase, Chiral photocurrent in parity-violating magnet and enhanced response in topological antiferromagnet, Phys. Rev. X 11, 011001 (2021).
  62. T. Kaneko, Z. Sun, Y. Murakami, D. Golež, and A. J. Millis, Bulk photovoltaic effect driven by collective excitations in a correlated insulator, Phys. Rev. Lett. 127, 127402 (2021).
  63. A. Kofuji and R. Peters, Two-particle correlation effects on nonlinear optical responses in the one-dimensional interacting Rice-Mele model, Phys. Rev. B 109, 155111 (2024).
  64. S. R. White and A. E. Feiguin, Real-time evolution using the density matrix renormalization group, Phys. Rev. Lett. 93, 076401 (2004).
  65. G. Vidal, Efficient simulation of one-dimensional quantum many-body systems, Phys. Rev. Lett. 93, 040502 (2004).
  66. G. Vidal, Classical simulation of infinite-size quantum lattice systems in one spatial dimension, Phys. Rev. Lett. 98, 070201 (2007).
  67. J. Haegeman, J. I. Cirac, T. J. Osborne, I. Pižorn, H. Verschelde, and F. Verstraete, Time-dependent variational principle for quantum lattices, Phys. Rev. Lett. 107, 070601 (2011).
  68. A. Holzner, A. Weichselbaum, I. P. McCulloch, U. Schollwöck, and J. von Delft, Chebyshev matrix product state approach for spectral functions, Phys. Rev. B 83, 195115 (2011).
  69. H. Aoki, N. Tsuji, M. Eckstein, M. Kollar, T. Oka, and P. Werner, Nonequilibrium dynamical mean-field theory and its applications, Rev. Mod. Phys. 86, 779 (2014).
  70. Y. Murakami, D. Golež, M. Eckstein, and P. Werner, Photoinduced nonequilibrium states in Mott insulators, Rev. Mod. Phys. 97, 035001 (2025).
  71. E. Runge and E. K. U. Gross, Density-functional theory for time-dependent systems, Phys. Rev. Lett. 52, 997 (1984).
  72. K. Yabana and G. F. Bertsch, Time-dependent local-density approximation in real time, Phys. Rev. B 54, 4484 (1996).
  73. K. Yabana, T. Sugiyama, Y. Shinohara, T. Otobe, and G. F. Bertsch, Time-dependent density functional theory for strong electromagnetic fields in crystalline solids, Phys. Rev. B 85, 045134 (2012).
  74. M. Marques and E. Gross, Time-dependent density functional theory, Annu. Rev. Phys. Chem. 55, 427 (2004).
  75. C. Attaccalite, M. Grüning, and A. Marini, Real-time approach to the optical properties of solids and nanostructures: Time-dependent Bethe-Salpeter equation, Phys. Rev. B 84, 245110 (2011).
  76. S. A. Sato, H. Hübener, U. De Giovannini, and A. Rubio, Technical review: Time-dependent density functional theory for attosecond physics ranging from gas-phase to solids, npj Comput. Mater. 11, 233 (2025).
  77. G. Lindblad, On the generators of quantum dynamical semigroups, Commun. Math. Phys. 48, 119 (1976).
  78. H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, Oxford, 2002).
  79. I. de Vega and D. Alonso, Dynamics of non-Markovian open quantum systems, Rev. Mod. Phys. 89, 015001 (2017).
  80. Y. Tanimura and R. Kubo, Time evolution of a quantum system in contact with a nearly Gaussian-Markoffian noise bath, J. Phys. Soc. Jpn. 58, 101 (1989).
  81. Y. Tanimura, Numerically “exact” approach to open quantum dynamics: The hierarchical equations of motion (HEOM), J. Chem. Phys. 153, 020901 (2020).
  82. S. Ghimire, A. D. DiChiara, E. Sistrunk, P. Agostini, L. F. DiMauro, and D. A. Reis, Observation of high-order harmonic generation in a bulk crystal, Nat. Phys. 7, 138 (2011).
  83. S. Ghimire and D. A. Reis, High-harmonic generation from solids, Nat. Phys. 15, 10 (2019).
  84. E. Goulielmakis and T. Brabec, High harmonic generation in condensed matter, Nat. Photon. 16, 411 (2022).
  85. G. Vampa, C. R. McDonald, G. Orlando, D. D. Klug, P. B. Corkum, and T. Brabec, Theoretical analysis of high-harmonic generation in solids, Phys. Rev. Lett. 113, 073901 (2014).
  86. G. F. Bertsch, J.-I. Iwata, A. Rubio, and K. Yabana, Real-space, real-time method for the dielectric function, Phys. Rev. B 62, 7998 (2000).
  87. R. G. Pereira, S. R. White, and I. Affleck, Spectral function of spinless fermions on a one-dimensional lattice, Phys. Rev. B 79, 165113 (2009).
  88. H. N. Phien, G. Vidal, and I. P. McCulloch, Infinite boundary conditions for matrix product state calculations, Phys. Rev. B 86, 245107 (2012).
  89. C. Shao, T. Tohyama, H.-G. Luo, and H. Lu, Numerical method to compute optical conductivity based on pump-probe simulations, Phys. Rev. B 93, 195144 (2016).
  90. C. Attaccalite and M. Grüning, Nonlinear optics from an ab initio approach by means of the dynamical Berry phase: Application to second- and third-harmonic generation in semiconductors, Phys. Rev. B 88, 235113 (2013).
  91. M. Uemoto, Y. Kuwabara, S. A. Sato, and K. Yabana, Nonlinear polarization evolution using time-dependent density functional theory, J. Chem. Phys. 150, 094101 (2019).
  92. D. Skachkov, D. R. Englund, and M. N. Leuenberger, Linear and nonlinear optical response based on many-body GW-Bethe–Salpeter and Kadanoff–Baym approaches for two-dimensional layered semiconductors, npj 2D Mater. Appl. 10, 15 (2026).
  93. J. Iguchi, H. Watanabe, Y. Murakami, T. Nomoto, and R. Arita, Bulk photovoltaic effect in antiferromagnet: Role of collective spin dynamics, Phys. Rev. B 109, 064407 (2024).
  94. K. Hattori, H. Watanabe, and R. Arita, Nonlinear Hall effect driven by spin-charge-coupled motive force, Phys. Rev. B 111, 174416 (2025).
  95. Y. Takimoto, F. D. Vila, and J. J. Rehr, Real-time time-dependent density functional theory approach for frequency-dependent nonlinear optical response in photonic molecules, J. Chem. Phys. 127, 154114 (2007).
  96. F. Wang, C. Y. Yam, and G. Chen, Time-dependent density-functional theory/localized density matrix method for dynamic hyperpolarizability, J. Chem. Phys. 126, 244102 (2007).
  97. F. Ding, B. E. Van Kuiken, B. E. Eichinger, and X. Li, An efficient method for calculating dynamical hyperpolarizabilities using real-time time-dependent density functional theory, J. Chem. Phys. 138, 064104 (2013).
  98. S. Boyd, Y. Tang, and L. Chua, Measuring Volterra kernels, IEEE Trans. Circuits Syst. 30, 571 (1983).
  99. G. Bicken, G. F. Carey, and R. O. Stearman, Frequency domain kernel estimation for 2nd‐order volterra models using random multi‐tone excitation, VLSI Design 15, 701 (2002).
  100. A. Guandalini, C. Cocchi, S. Pittalis, A. Ruini, and C. A. Rozzi, Nonlinear light absorption in many-electron systems excited by an instantaneous electric field: A non-perturbative approach, Phys. Chem. Chem. Phys. 23, 10059 (2021).
  101. M. N. Pionteck, M. Grüning, S. Sanna, and C. Attaccalite, A real-time approach to frequency-mixing spectroscopies: Application to sum and difference frequency generation in two-dimensional crystals, SciPost Phys. 19, 129 (2025).
  102. Y. Kim, H. Kim, H. Jang, J.-H. Ahn, and J. Lee, Dual resonant sum frequency generations from two-dimensional materials, Nano Lett. 20, 4530 (2020).
  103. A. Liu, Multidimensional terahertz probes of quantum materials, npj Quantum Mater. 10, 18 (2025).
  104. C. Huang, M. Mootz, L. Luo, I. E. Perakis, and J. Wang, Terahertz 2D coherent spectroscopy for probing and controlling multicorrelations in quantum matter, Nat. Rev. Phys. 8, 171 (2026).
  105. D. M. Jonas, Two-dimensional femtosecond spectroscopy, Annu. Rev. Phys. Chem. 54, 425 (2003).
  106. M. Cho, Coherent two-dimensional optical spectroscopy, Chem. Rev. 108, 1331 (2008).
  107. T. Brixner, T. Mančal, I. V. Stiopkin, and G. R. Fleming, Phase-stabilized two-dimensional electronic spectroscopy, J. Chem. Phys. 121, 4221 (2004).
  108. P. F. Tekavec, G. A. Lott, and A. H. Marcus, Fluorescence-detected two-dimensional electronic coherence spectroscopy by acousto-optic phase modulation, J. Chem. Phys. 127, 214307 (2007).
  109. J. Lüttig and P. Malý, A simple method to extract nonlinear signals in time-resolved spectroscopy, J. Phys. B: At. Mol. Opt. Phys. 59, 063001 (2026).
  110. Y. Tanimura and S. Mukamel, Two-dimensional femtosecond vibrational spectroscopy of liquids, J. Chem. Phys. 99, 9496 (1993).
  111. K. Tominaga and K. Yoshihara, Fifth order optical response of liquid CS2 observed by ultrafast nonresonant six-wave mixing, Phys. Rev. Lett. 74, 3061 (1995).
  112. A. Tokmakoff, M. J. Lang, D. S. Larsen, G. R. Fleming, V. Chernyak, and S. Mukamel, Two-dimensional Raman spectroscopy of vibrational interactions in liquids, Phys. Rev. Lett. 79, 2702 (1997).
  113. D. A. Blank, L. J. Kaufman, and G. R. Fleming, Fifth-order two-dimensional Raman spectra of CS2 are dominated by third-order cascades, J. Chem. Phys. 111, 3105 (1999).
  114. D. A. Blank, L. J. Kaufman, and G. R. Fleming, Direct fifth-order electronically nonresonant Raman scattering from CS2 at room temperature, J. Chem. Phys. 113, 771 (2000).
  115. A. Ono, Extracting nonlinear dynamical response functions from time evolution, Phys. Rev. Lett. 135, 026401 (2025).
  116. G. Benettin, L. Galgani, A. Giorgilli, and J.-M. Strelcyn, Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; a method for computing all of them. Part 1: Theory, Meccanica 15, 9 (1980).
  117. F.-X. L. Dimet and O. Talagrand, Variational algorithms for analysis and assimilation of meteorological observations: Theoretical aspects, Tellus A 38, 97 (1986).
  118. O. Talagrand and P. Courtier, Variational assimilation of meteorological observations with the adjoint vorticity equation. I: Theory, Q. J. R. Meteorol. Soc. 113, 1311 (1987).
  119. M. B. Giles and N. A. Pierce, An introduction to the adjoint approach to design, Flow, Turbul. Combust. 65, 393 (2000).
  120. M. Ghil and P. Malanotte-Rizzoli, Data assimilation in meteorology and oceanography, Adv. Geophys. 33, 141 (1991).
  121. R. M. Errico, What is an adjoint model? Bull. Am. Meteorol. Soc. 78, 2577 (1997).
  122. Y. Cao, S. Li, L. Petzold, and R. Serban, Adjoint sensitivity analysis for differential-algebraic equations: The adjoint DAE system and its numerical solution, SIAM J. Sci. Comput. 24, 1076 (2003).
  123. D. G. Luenberger, Optimization by Vector Space Methods (Wiley-Interscience, New York, NY, 1997).
  124. A. Ono, TEOM-RiceMele: Julia implementation of tangent equations of motion for the Rice–Mele model [software] Zenodo, 2026, https://doi.org/10.5281/zenodo.20658978.
  125. C. Aversa and J. E. Sipe, Nonlinear optical susceptibilities of semiconductors: Results with a length-gauge analysis, Phys. Rev. B 52, 14636 (1995).
  126. A. Taghizadeh, F. Hipolito, and T. G. Pedersen, Linear and nonlinear optical response of crystals using length and velocity gauges: Effect of basis truncation, Phys. Rev. B 96, 195413 (2017).
  127. M. J. Rice and E. J. Mele, Elementary excitations of a linearly conjugated diatomic polymer, Phys. Rev. Lett. 49, 1455 (1982).
  128. J. Wilhelm, P. Grössing, A. Seith, J. Crewse, M. Nitsch, L. Weigl, C. Schmid, and F. Evers, Semiconductor Bloch-equations formalism: Derivation and application to high-harmonic generation from Dirac fermions, Phys. Rev. B 103, 125419 (2021).
  129. Y. Murakami and M. Schüler, Doping and gap size dependence of high-harmonic generation in graphene: Importance of consistent formulation of light-matter coupling, Phys. Rev. B 106, 035204 (2022).
  130. M. D. Tokman, Gauge invariance of phenomenological models of the interaction of quantum dissipative systems with electromagnetic fields, Phys. Rev. A 79, 053415 (2009).
  131. M. S. Wismer and V. S. Yakovlev, Gauge-independent decoherence models for solids in external fields, Phys. Rev. B 97, 144302 (2018).
  132. V. I. Belinicher and B. I. Sturman, The photogalvanic effect in media lacking a center of symmetry, Sov. Phys. Usp. 23, 199 (1980).
  133. R. von Baltz and W. Kraut, Theory of the bulk photovoltaic effect in pure crystals, Phys. Rev. B 23, 5590 (1981).
  134. S. M. Young and A. M. Rappe, First principles calculation of the shift current photovoltaic effect in ferroelectrics, Phys. Rev. Lett. 109, 116601 (2012).
  135. S. A. Sato and A. Rubio, Limitations of mean-field approximations in describing shift-current and injection-current in materials, Phys. Rev. B 109, 195205 (2024).
  136. I. Martin and C. D. Batista, Itinerant electron-driven chiral magnetic ordering and spontaneous quantum Hall effect in triangular lattice models, Phys. Rev. Lett. 101, 156402 (2008).
  137. Y. Akagi and Y. Motome, Spin chirality ordering and anomalous Hall effect in the ferromagnetic Kondo lattice model on a triangular lattice, J. Phys. Soc. Jpn. 79, 083711 (2010).
  138. H. Takagi, R. Takagi, S. Minami, T. Nomoto, K. Ohishi, M.-T. Suzuki, Y. Yanagi, M. Hirayama, N. D. Khanh, K. Karube, H. Saito, D. Hashizume, R. Kiyanagi, Y. Tokura, R. Arita, T. Nakajima, and S. Seki, Spontaneous topological Hall effect induced by non-coplanar antiferromagnetic order in intercalated van der Waals materials, Nat. Phys. 19, 961 (2023).
  139. P. Park, W. Cho, C. Kim, Y. An, Y.-g. Kang, M. Avdeev, R. Sibille, K. Iida, R. Kajimoto, K. H. Lee, W. Ju, E.-J. Cho, H.-J. Noh, M. J. Han, S.-S. Zhang, C. D. Batista, and J.-G. Park, Tetrahedral triple-Q magnetic ordering and large spontaneous Hall conductivity in the metallic triangular antiferromagnet Co1/3TaS2, Nat. Commun. 14, 8346 (2023).
  140. A. Ono, S. Okumura, S. Imai, and Y. Akagi, High harmonic generation from electrons moving in topological spin textures, Phys. Rev. B 110, 125111 (2024).
  141. J. S. Aldridge and A. N. Cleland, Noise-enabled precision measurements of a Duffing nanomechanical resonator, Phys. Rev. Lett. 94, 156403 (2005).
  142. J. F. Rhoads, S. W. Shaw, and K. L. Turner, Nonlinear dynamics and its applications in micro- and nanoresonators, J. Dyn. Syst. Meas. Control 132, 034001 (2010).
  143. R. Vijay, M. H. Devoret, and I. Siddiqi, Invited review article: The Josephson bifurcation amplifier, Rev. Sci. Instrum. 80, 111101 (2009).
  144. G. K. Yu, Y. D. Zhang, and Y. Shen, Nonlinear amplitude-frequency response of a Helmholtz resonator, J. Vib. Acoust. 133, 024502 (2011).
  145. X. Fang, W. Lacarbonara, and L. Cheng, Advances in nonlinear acoustic/elastic metamaterials and metastructures, Nonlinear Dyn. 113, 23787 (2025).
  146. A. H. Nayfeh and D. T. Mook, Nonlinear Oscillations (Wiley-VCH, Weinheim, 1995).
  147. P. Krantz, M. Kjaergaard, F. Yan, T. P. Orlando, S. Gustavsson, and W. D. Oliver, A quantum engineer's guide to superconducting qubits, Appl. Phys. Rev. 6, 021318 (2019).
  148. Y. Ueda, Survey of regular and chaotic phenomena in the forced duffing oscillator, Chaos, Solitons Fractals 1, 199 (1991).
  149. A. Griewank and A. Walther, Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation (Society for Industrial and Applied Mathematics, Philadelphia, PA, 2008).
  150. M. Berz, Differential algebraic description of beam dynamics to very high orders, Part. Accel. 24, 109 (1989).
  151. M. Eckstein and M. Kollar, Theory of time-resolved optical spectroscopy on correlated electron systems, Phys. Rev. B 78, 205119 (2008).
  152. G. De Filippis, V. Cataudella, E. A. Nowadnick, T. P. Devereaux, A. S. Mishchenko, and N. Nagaosa, Quantum dynamics of the Hubbard-Holstein model in equilibrium and nonequilibrium: Application to pump-probe phenomena, Phys. Rev. Lett. 109, 176402 (2012).
  153. Z. Lenarčič, D. Golež, J. Bonča, and P. Prelovšek, Optical response of highly excited particles in a strongly correlated system, Phys. Rev. B 89, 125123 (2014).
  154. K. Shinjo and T. Tohyama, Ultrafast transient interference in pump-probe spectroscopy of band and Mott insulators, Phys. Rev. B 98, 165103 (2018).
  155. A. Ono and S. Ishihara, Photocontrol of magnetic structure in an itinerant magnet, Phys. Rev. B 98, 214408 (2018).
  156. A. Ono and S. Ishihara, Nonequilibrium susceptibility in photoinduced Floquet states, Phys. Rev. B 100, 075127 (2019).
  157. A. Kumar and A. F. Kemper, Higgs oscillations in time-resolved optical conductivity, Phys. Rev. B 100, 174515 (2019).
  158. A. Eskandari-asl and A. Avella, Generalized linear response theory for pumped systems and its application to transient optical properties, Phys. Rev. A 110, 043520 (2024).
  159. S. Ejima, F. Lange, and H. Fehske, Nonequilibrium dynamics in pumped Mott insulators, Phys. Rev. Res. 4, L012012 (2022).
  160. M. Udono, T. Kaneko, and K. Sugimoto, Wannier-Stark ladders and Stark shifts of excitons in Mott insulators, Phys. Rev. B 108, L081304 (2023).
  161. T. Tohyama, K. Shinjo, S. Sota, and S. Yunoki, Anomalous suppression of photoinduced in-gap weight in the optical conductivity of a two-leg Hubbard ladder, Phys. Rev. B 108, 035113 (2023).
  162. K. Takubo, A. Ono, S. Ueno, S. Banu, H. Yu, K. En-ya, R. Nishimori, M. Kuwahara, T. Asaka, K. Maeda, D. Ono, K. Ozawa, T. Itoh, K. Shigematsu, M. Azuma, T. Ishikawa, Y. Okimoto, M. Hada, and S.-Y. Koshihara, Photocontrol of ferroelectricity in multiferroic BiFeO3 via structural modification coupled with photocarrier, Commun. Mater. 5, 261 (2024).
  163. K. Shinjo and T. Tohyama, Keldysh crossover in one-dimensional Mott insulators, APL Mater. 12, 041109 (2024).
  164. K. Shinjo, S. Sota, S. Yunoki, and T. Tohyama, Optical absorption activated by an ultrashort half-cycle pulse in metallic and superconducting states of the Hubbard model, Phys. Rev. B 111, 235125 (2025).
  165. M. Volkov, S. A. Sato, A. Niedermayr, A. Rubio, L. Gallmann, and U. Keller, Floquet-Bloch resonances in near-petahertz electroabsorption spectroscopy of SiO2, Phys. Rev. B 107, 184304 (2023).
  166. Y. Kim, Pseudospins revealed through the giant dynamical Franz-Keldysh effect in massless Dirac materials, npj Quantum Mater. 9, 86 (2024).
  167. A. Osterkorn, Y. Murakami, T. Kaneko, Z. Sun, A. J. Millis, and D. Golež, Optical signatures of dynamical excitonic condensates, Phys. Rev. Lett. 135, 106902 (2025).
  168. A. Ono, Temporal modulation of second harmonic generation in ferroelectrics by a pulsed electric field, Phys. Rev. B 112, 115146 (2025).

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Phys. Rev. Research 8, 033131– Published 3 August, 2026
  • Received 3 March 2026
  • Accepted 29 June 2026
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