---
_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. Dynamics. 2025;5(3). doi:10.3390/dynamics5030034
apa: Meyer, M. T., & Schindlmayr, A. (2025). Generalized Miller formulae for
quantum anharmonic oscillators. Dynamics, 5(3), Article 34. https://doi.org/10.3390/dynamics5030034
bibtex: '@article{Meyer_Schindlmayr_2025, title={Generalized Miller formulae for
quantum anharmonic oscillators}, volume={5}, DOI={10.3390/dynamics5030034},
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.” Dynamics 5, no. 3 (2025). https://doi.org/10.3390/dynamics5030034.
ieee: 'M. T. Meyer and A. Schindlmayr, “Generalized Miller formulae for quantum
anharmonic oscillators,” Dynamics, vol. 5, no. 3, Art. no. 34, 2025, doi:
10.3390/dynamics5030034.'
mla: Meyer, Maximilian Tim, and Arno Schindlmayr. “Generalized Miller Formulae for
Quantum Anharmonic Oscillators.” Dynamics, vol. 5, no. 3, 34, MDPI, 2025,
doi:10.3390/dynamics5030034.
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'
...
---
_id: '52723'
abstract:
- lang: eng
text: 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.
article_number: '095001'
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. Derivation of Miller’s rule for the nonlinear optical
susceptibility of a quantum anharmonic oscillator. Journal of Physics B: Atomic,
Molecular and Optical Physics. 2024;57(9). doi:10.1088/1361-6455/ad369c'
apa: 'Meyer, M. T., & Schindlmayr, A. (2024). Derivation of Miller’s rule for
the nonlinear optical susceptibility of a quantum anharmonic oscillator. Journal
of Physics B: Atomic, Molecular and Optical Physics, 57(9), Article
095001. https://doi.org/10.1088/1361-6455/ad369c'
bibtex: '@article{Meyer_Schindlmayr_2024, title={Derivation of Miller’s rule for
the nonlinear optical susceptibility of a quantum anharmonic oscillator}, volume={57},
DOI={10.1088/1361-6455/ad369c},
number={9095001}, journal={Journal of Physics B: Atomic, Molecular and Optical
Physics}, publisher={IOP Publishing}, author={Meyer, Maximilian Tim and Schindlmayr,
Arno}, year={2024} }'
chicago: 'Meyer, Maximilian Tim, and Arno Schindlmayr. “Derivation of Miller’s Rule
for the Nonlinear Optical Susceptibility of a Quantum Anharmonic Oscillator.”
Journal of Physics B: Atomic, Molecular and Optical Physics 57, no. 9 (2024).
https://doi.org/10.1088/1361-6455/ad369c.'
ieee: 'M. T. Meyer and A. Schindlmayr, “Derivation of Miller’s rule for the nonlinear
optical susceptibility of a quantum anharmonic oscillator,” Journal of Physics
B: Atomic, Molecular and Optical Physics, vol. 57, no. 9, Art. no. 095001,
2024, doi: 10.1088/1361-6455/ad369c.'
mla: 'Meyer, Maximilian Tim, and Arno Schindlmayr. “Derivation of Miller’s Rule
for the Nonlinear Optical Susceptibility of a Quantum Anharmonic Oscillator.”
Journal of Physics B: Atomic, Molecular and Optical Physics, vol. 57, no.
9, 095001, IOP Publishing, 2024, doi:10.1088/1361-6455/ad369c.'
short: 'M.T. Meyer, A. Schindlmayr, Journal of Physics B: Atomic, Molecular and
Optical Physics 57 (2024).'
date_created: 2024-03-22T08:44:39Z
date_updated: 2024-04-13T11:20:56Z
ddc:
- '530'
department:
- _id: '296'
- _id: '230'
- _id: '15'
- _id: '170'
- _id: '35'
doi: 10.1088/1361-6455/ad369c
external_id:
isi:
- '001196678300001'
file:
- access_level: open_access
content_type: application/pdf
creator: schindlm
date_created: 2024-04-04T09:24:22Z
date_updated: 2024-04-04T09:24:22Z
description: Creative Commons Attribution 4.0 International Public License (CC BY
4.0)
file_id: '53204'
file_name: Meyer_2024_J._Phys._B _At._Mol._Opt._Phys._57_095001.pdf
file_size: 358155
relation: main_file
title: Derivation of Miller's rule for the nonlinear optical susceptibility of a
quantum anharmonic oscillator
file_date_updated: 2024-04-04T09:24:22Z
has_accepted_license: '1'
intvolume: ' 57'
isi: '1'
issue: '9'
language:
- iso: eng
oa: '1'
publication: 'Journal of Physics B: Atomic, Molecular and Optical Physics'
publication_identifier:
eissn:
- 1361-6455
issn:
- 0953-4075
publication_status: published
publisher: IOP Publishing
quality_controlled: '1'
status: public
title: Derivation of Miller's rule for the nonlinear optical susceptibility of a quantum
anharmonic oscillator
type: journal_article
user_id: '458'
volume: 57
year: '2024'
...