---
_id: '60959'
abstract:
- lang: eng
  text: 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.
article_number: '34'
article_type: original
author:
- first_name: Maximilian Tim
  full_name: Meyer, Maximilian Tim
  id: '77895'
  last_name: Meyer
  orcid: 0009-0003-4899-0920
- first_name: Arno
  full_name: Schindlmayr, Arno
  id: '458'
  last_name: Schindlmayr
  orcid: 0000-0002-4855-071X
citation:
  ama: Meyer MT, Schindlmayr A. Generalized Miller formulae for quantum anharmonic
    oscillators. <i>Dynamics</i>. 2025;5(3). doi:<a href="https://doi.org/10.3390/dynamics5030034">10.3390/dynamics5030034</a>
  apa: Meyer, M. T., &#38; Schindlmayr, A. (2025). Generalized Miller formulae for
    quantum anharmonic oscillators. <i>Dynamics</i>, <i>5</i>(3), Article 34. <a href="https://doi.org/10.3390/dynamics5030034">https://doi.org/10.3390/dynamics5030034</a>
  bibtex: '@article{Meyer_Schindlmayr_2025, title={Generalized Miller formulae for
    quantum anharmonic oscillators}, volume={5}, DOI={<a href="https://doi.org/10.3390/dynamics5030034">10.3390/dynamics5030034</a>},
    number={334}, journal={Dynamics}, publisher={MDPI}, author={Meyer, Maximilian
    Tim and Schindlmayr, Arno}, year={2025} }'
  chicago: Meyer, Maximilian Tim, and Arno Schindlmayr. “Generalized Miller Formulae
    for Quantum Anharmonic Oscillators.” <i>Dynamics</i> 5, no. 3 (2025). <a href="https://doi.org/10.3390/dynamics5030034">https://doi.org/10.3390/dynamics5030034</a>.
  ieee: 'M. T. Meyer and A. Schindlmayr, “Generalized Miller formulae for quantum
    anharmonic oscillators,” <i>Dynamics</i>, vol. 5, no. 3, Art. no. 34, 2025, doi:
    <a href="https://doi.org/10.3390/dynamics5030034">10.3390/dynamics5030034</a>.'
  mla: Meyer, Maximilian Tim, and Arno Schindlmayr. “Generalized Miller Formulae for
    Quantum Anharmonic Oscillators.” <i>Dynamics</i>, vol. 5, no. 3, 34, MDPI, 2025,
    doi:<a href="https://doi.org/10.3390/dynamics5030034">10.3390/dynamics5030034</a>.
  short: M.T. Meyer, A. Schindlmayr, Dynamics 5 (2025).
date_created: 2025-08-20T09:46:13Z
date_updated: 2025-10-10T07:29:36Z
ddc:
- '530'
department:
- _id: '296'
- _id: '230'
- _id: '15'
- _id: '170'
- _id: '35'
doi: 10.3390/dynamics5030034
external_id:
  isi:
  - '001581270200001'
file:
- access_level: open_access
  content_type: application/pdf
  creator: schindlm
  date_created: 2025-08-28T12:23:26Z
  date_updated: 2025-08-28T12:27:05Z
  description: Creative Commons Attribution 4.0 International Public License (CC BY
    4.0)
  file_id: '61056'
  file_name: dynamics-05-00034.pdf
  file_size: 375897
  relation: main_file
  title: Generalized Miller formulae for quantum anharmonic oscillators
file_date_updated: 2025-08-28T12:27:05Z
has_accepted_license: '1'
intvolume: '         5'
isi: '1'
issue: '3'
language:
- iso: eng
oa: '1'
publication: Dynamics
publication_identifier:
  eissn:
  - 2673-8716
publication_status: published
publisher: MDPI
quality_controlled: '1'
status: public
title: Generalized Miller formulae for quantum anharmonic oscillators
type: journal_article
user_id: '458'
volume: 5
year: '2025'
...