Variational functionals for the driven quantum harmonic oscillator

Abstract

We study some variational problems related to the controllability of the transition between two quantum states of the driven quantum harmonic oscillator. We focus on the existence of minimum points of the total energy functional which is obtained by adding to the cost of the control a term penalizing the distance from the maximum transition probability between given energy states.

COMMUNICATED BY:

• G. Molica Bisci

KEYWORDS:

• Quantum control
• quantum oscillator
• calculus of variations
• optimal control problem

AMS SUBJECT CLASSIFICATIONS:

• 81Q93 (Quantum control)
• 49K30 (optimalsolutions belonging to restricted classes)

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

GD and FM have been supported by the Research Project of National Relevance “Multiscale Innovative Materials and Structures” granted by the Italian Ministry of Education, University and Research (MIUR Prin 2017, project code 2017J4EAYB), by the Gruppo Nazionale per l'AnalisiMatematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di AltaMatematica (INdAM) GD has been also supported by MIUR – FFABR – 2017 research grant. GF has been supported by the Research Project of National Relevance “Mathematics of active materials: From mechanobiology to smart devices” granted by the Italian Ministry of Education, University and Research (MIUR Prin 2017, project code 2017KL4EF3), by the Gruppo Nazionale per la Fisica Matematica (GNFM) of the Istituto Nazionale di Alta Matematica (INdAM), by the Italian Ministry MISE through the project RAEE SUD-PVP, by INFN through the project “QUANTUM” and by the MIUR – FFABR – 2017 research grant.