@article{60959,
  abstract     = {{Miller's rule originated as an empirical relation between the nonlinear and linear optical coefficients of materials. It is now accepted as a useful tool for guiding experiments and computational materials discovery, but its theoretical foundation had long been limited to a derivation for the classical Lorentz model with a weak anharmonic perturbation. Recently, we developed a mathematical framework which enabled us to prove that Miller's rule is equally valid for quantum anharmonic oscillators, despite different dynamics due to zero-point fluctuations and further quantum-mechanical effects. However, our previous derivation applied only to one-dimensional oscillators and to the special case of second- and third-harmonic generation in a monochromatic electric field. Here we extend the proof to three-dimensional quantum anharmonic oscillators and also treat all orders of the nonlinear response to an arbitrary multi-frequency field. This makes the results applicable to a much larger range of physical systems and nonlinear optical processes. The obtained generalized Miller formulae rigorously express all tensor elements of the frequency-dependent nonlinear susceptibilities in terms of the linear susceptibility and thus allow a computationally inexpensive quantitative prediction of arbitrary parametric frequency-mixing processes from a small initial dataset.}},
  author       = {{Meyer, Maximilian Tim and Schindlmayr, Arno}},
  issn         = {{2673-8716}},
  journal      = {{Dynamics}},
  number       = {{3}},
  publisher    = {{MDPI}},
  title        = {{{Generalized Miller formulae for quantum anharmonic oscillators}}},
  doi          = {{10.3390/dynamics5030034}},
  volume       = {{5}},
  year         = {{2025}},
}

@article{52723,
  abstract     = {{Miller's rule is an empirical relation between the nonlinear and linear optical coefficients that applies to a large class of materials but has only been rigorously derived for the classical Lorentz model with a weak anharmonic perturbation. In this work, we extend the proof and present a detailed derivation of Miller's rule for an equivalent quantum-mechanical anharmonic oscillator. For this purpose, the classical concept of velocity-dependent damping inherent to the Lorentz model is replaced by an adiabatic switch-on of the external electric field, which allows a unified treatment of the classical and quantum-mechanical systems using identical potentials and fields. Although the dynamics of the resulting charge oscillations, and hence the induced polarizations, deviate due to the finite zero-point motion in the quantum-mechanical framework, we find that Miller's rule is nevertheless identical in both cases up to terms of first order in the anharmonicity. With a view to practical applications, especially in the context of ab initio calculations for the optical response where adiabatically switched-on fields are widely assumed, we demonstrate that a correct treatment of finite broadening parameters is essential to avoid spurious errors that may falsely suggest a violation of Miller's rule, and we illustrate this point by means of a numerical example.}},
  author       = {{Meyer, Maximilian Tim and Schindlmayr, Arno}},
  issn         = {{1361-6455}},
  journal      = {{Journal of Physics B: Atomic, Molecular and Optical Physics}},
  number       = {{9}},
  publisher    = {{IOP Publishing}},
  title        = {{{Derivation of Miller's rule for the nonlinear optical susceptibility of a quantum anharmonic oscillator}}},
  doi          = {{10.1088/1361-6455/ad369c}},
  volume       = {{57}},
  year         = {{2024}},
}