Subject
A study of the geometric phase of multivariate signals

Keywords
Multivariate time series, geometric phase, differential geometry,
quantum
systems.

Location
Grenoble, France.

Supervision
• Nicolas Le Bihan (CNRS, Gipsa-Lab, Grenoble)
• Pierre-Olivier Amblard (CNRS, Gipsa-Lab, Grenoble)
• Julien Flamant (CNRS, CRAN, Nancy)

Application procedure
Send a detailed CV and motivation letter to nicolas.le-bihan@cnrs.fr,
pierre-olivier.amblard@cnrs.fr, and julien.flamant@cnrs.fr.

Context
The concept of geometric phase was discovered by Sir M.V. Berry [1] for
quantum systems. It was since extensively studied and has found numerous
implications [2]. Many classical systems have also been identified to
exhibit
geometric phases, amongst them one can find the Foucault pendulum,
falling
cats or polarized light [3]. Geometric phase can be understood as an
excess
between the dynamical and relative phase of a system during a time
evolution.
This physical quantity is purely geometric as it is inherited from the
curvature of the underlying parameter space of the Hamiltonian. It
arises when
a specific path is followed over time in the parameter space. Geometric
phase
is also gauge-invariant, which makes it desirable for the study of
quantum
systems [2]. Recently, it was showed in [4] that bivariate signals may
exhibit
a geometric phase under mild conditions. Bivariate signals can be
understood
and analyzed with concepts from optics (Poincaré sphere, Stokes
parameters,
etc.). In addition, their geometric phase is one of the
“Pancharatnam-Berry”
type. It is anticipated that multivariate signals should also possess a
geometric phase under some conditions.

Project summary
The proposed work aims at bridging the gap between the study of
multidimensional quantum systems and signal processing theory. It will
consist
in studying and exploring the geometric phase of multivariate signals
(i.e.
vector-valued time series). The candidate will study existence
conditions and
will design algorithms for its practical estimation. Concepts from
differential geometry (fiber bundles, connections, curvature, parallel
transport, etc.) will be used to provide a model for the geometric phase
description of multivariate signals. Simulations will be conducted to
illustrate the cases where multivariate time series possess a geometric
phase
and potential applications in signal processing tasks (denoising,
classification) will be investigated.

Candidate profile
He/she should be enrolled in a M2R or engineer diploma in one or more of
the
following fields: theoretical physics, applied mathematics, signal and
image
processing. The candidate should have good writing and oral
communication
skills.

Environment
Position can be started anytime from February, 2025 and duration is up
to 6
months. The candidate will be based in Gipsa-Lab, Grenoble. This
internship
will be hosted by the RICOCHET ANR project, with regular meetings and
exchanges with researchers from the project. The internship could be
pursued
through a PhD.

References
[1] M.V. Berry, “Quantal phase factors accompanying adiabatic
changes”, Proc.
R. Soc. A 392, 45-57, 1984.
[2] A. Bohm, et al., “The geometric phase in quantum systems”,
Springer, 2003.
[3] M.V. Berry, “The adiabatic phase and Pancharatnam’s phase for
polarized
light”, Journal of Modern Optics, Vol. 34, No. 11, 1987.
[4] N. Le Bihan, J. Flamant, and P.-O. Amblard, ”The geometric phase
of
bivariate signals”, EUSIPCO Conference, Lyon, 2024.