Optimal control of quantum systems: Applications to the robust control of Bose-Einstein Condensates and to quantum speed limit with piecewise constant control. – (Etienne Dionis / LCAR / Seminar). – 29/06/2023, 9H

Etienne Dionis (Laboratoire Interdisciplinaire Carnot de Bourgogne, Université de Bourgogne-Franche Comté)

Séminaire LCAR Jeudi 29 juin, 9H00. – Nouvelle Salle de conférence, 3R4

Résumé :
This presentation will focus on two recent applications of optimal control techniques to quantum
systems.

In the first part, the goal is to manipulate the motional states of an atomic Bose-Einstein
condensate (BEC) in a one-dimensional optical lattice. This study is a joint work wit the experi-
mental group of Pr. D. Guéry-Odelin in Toulouse University (France). The protocols operate on
the momentum comb associated with the lattice through its amplitude and phase [3]. A precise and
versatile control for a wide variety of targets has been demonstrated. However, in order to improve
the agreement between theory and experiment, it is important to design control processes that are
robust with respect to experimental uncertainties (Figure 1). One limitation of the experimental
setup under study is the value of the quasi-momentum which is not exactly zero as assumed in
[3]. Due to the large dimension of the Hilbert space, numerical algorithms such as GRAPE [1]
have to be used. In the case of the simultaneous control of an ensemble of systems, we propose
a new formulation of GRAPE with a sequential-update of the control. We show numerically the
superiority of this sequential approach with respect to the standard one [2].
In the second part, we apply a recent extension of the Pontryagin Maximum Principle (PMP) [4]
to piecewise constant control. PMP, on which optimal control theory is based, was originally for-
mulated for continuous functions. In practice, experimental implementations resort to piecewise
constant signals. In [5], the authors have mathematically proved that for general non-linear dy-
namics, the PMP should be modified in a non-trivial way. We have transposed this result to
quantum systems and applied it to the time-optimal control of simple quantum systems with two
levels (Figure 2). In these systems, we derive exact quantum speed limits accounting for this key
experimental constraint [6]. This should be an important step toward understanding the role of
technological limitations in the manipulation of quantum systems, a key issue in quantum control.

References
[1] Glaser S. J., Boscain U., Calarco T., Koch C., Kockenberger W., Kosloff R., Kuprov I., Luy
B., Schirmer S., Schulte-Herbrüggen T., Sugny D. and Wilhelm F. 2015 Eur. Phys. J. D 69,
279
[2] Dionis E. and Sugny D. 2022 J. Phys. B: At. Mol. Opt. Phys. 55, 184006
[3] Dupont N., Chatelain G., Gabardos L., Arnal M., Billy J., Peaudecerf B., Sugny D. and
Guéry-Odelin D. 2021 PRX Quantum 2, 040303
[4] Boscain U., Sigalotti M. and Sugny D. 2021 PRX Quantum 2, 030203
[5] Bourdin L. and Trélat E. 2017 Automatica 79, 273
[6] Dionis E. and Sugny D. 2023 Physical Review A 107, 032613