Quantum Hamiltonian and dipole moment identification in presence of large control perturbations
ESAIM: Control, Optimisation and Calculus of Variations, Volume 23 (2017) no. 3, pp. 1129-1143.

The problem of recovering the Hamiltonian and dipole moment is considered in a bilinear quantum control framework. The process uses as inputs some measurable quantities (observables) for each admissible control. If the implementation of the control is noisy the data available is only in the form of probability laws of the measured observable. Nevertheless it is proved that the inversion process still has unique solutions (up to phase factors). Both additive and multiplicative noises are considered. Numerical illustrations support the theoretical results.

Received:
Accepted:
DOI: 10.1051/cocv/2016026
Classification: 93-XX, 49-XX, 81Q93
Keywords: Quantum control, quantum identification
Fu, Ying 1; Turinici, Gabriel 1, 2

1 UniversitéParis-Dauphine, PSL Research University, CNRS, Ceremade, 75016 Paris, France.
2 Institut Universitaire de France, 1 Rue Descartes, 75231 Paris cedex 05, France.
@article{COCV_2017__23_3_1129_0,
     author = {Fu, Ying and Turinici, Gabriel},
     title = {Quantum {Hamiltonian} and dipole moment identification in presence of large control perturbations},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {1129--1143},
     publisher = {EDP-Sciences},
     volume = {23},
     number = {3},
     year = {2017},
     doi = {10.1051/cocv/2016026},
     mrnumber = {3660462},
     zbl = {1364.93156},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv/2016026/}
}
TY  - JOUR
AU  - Fu, Ying
AU  - Turinici, Gabriel
TI  - Quantum Hamiltonian and dipole moment identification in presence of large control perturbations
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2017
SP  - 1129
EP  - 1143
VL  - 23
IS  - 3
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv/2016026/
DO  - 10.1051/cocv/2016026
LA  - en
ID  - COCV_2017__23_3_1129_0
ER  - 
%0 Journal Article
%A Fu, Ying
%A Turinici, Gabriel
%T Quantum Hamiltonian and dipole moment identification in presence of large control perturbations
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2017
%P 1129-1143
%V 23
%N 3
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/cocv/2016026/
%R 10.1051/cocv/2016026
%G en
%F COCV_2017__23_3_1129_0
Fu, Ying; Turinici, Gabriel. Quantum Hamiltonian and dipole moment identification in presence of large control perturbations. ESAIM: Control, Optimisation and Calculus of Variations, Volume 23 (2017) no. 3, pp. 1129-1143. doi : 10.1051/cocv/2016026. http://www.numdam.org/articles/10.1051/cocv/2016026/

L. Baudouin and A. Mercado, An inverse problem for Schrodinger equations with discontinuous main coefficient. Applicable Analysis 87 (2008) 1145–1165. | DOI | MR | Zbl

K. Beauchard, J.-M. Coron, and P. Rouchon, Controllability issues for continuous-spectrum systems and ensemble controllability of Bloch equations. Comm. Math. Phys. 296 (2010) 525–557. | DOI | MR | Zbl

M. Belhadj, J. Salomon, and G. Turinici, Ensemble controllability and discrimination of perturbed bilinear control systems on connected, simple, compact Lie groups. Eur. J. Control 22 (2015) 23–29. | DOI | MR | Zbl

S. Bonnabel, M. Mirrahimi, and P. Rouchon, Observer-based Hamiltonian identification for quantum systems. Automatica 45 (2009) 1144–1155. | MR | Zbl

C. Brif, R. Chakrabarti, and H. Rabitz, Control of quantum phenomena: past, present and future. New J. Phys. 12 (2010) 075008. | DOI | Zbl

C. Cohen-Tannoudji, B. Diu and F. Laloë, Quantum Mechanics, Vol 1. Wiley, New-York (1977).

D. D’Alessandro, Introduction to quantum control and dynamics. Chapman & Hall/CRC Applied Mathematics and Nonlinear Science Series. Chapman & Hall/CRC, Boca Raton, FL (2008). | MR | Zbl

A. Donovan and H. Rabitz, Exploring the Hamiltonian inversion landscape. Phys. Chem. Chem. Phys. 16 (2014) 15615–15622. | DOI

J.W. Eaton, D. Bateman, S. Hauberg and R. Wehbring, GNU Octave version 4.0.0 manual: a high-level interactive language for numerical computations. Available at http://www.gnu.org/software/octave/doc/interpreter (2015).

J.W. Eaton et al., GNU Octave 4.0.0. Available at http://www.octave.org (2015).

J.M. Geremia and H. Rabitz, Optimal Hamiltonian identification: The synthesis of quantum optimal control and quantum inversion. J. Chem. Phys. 118 (2003) 5369–5382. | DOI

D. Hocker, Co. Brif, M.D. Grace, A. Donovan and T.-S. Ho, K. Moore Tibbetts, R. Wu and H. Rabitz, Characterization of control noise effects in optimal quantum unitary dynamics. Phys. Rev. A 90 (2014) 062309. | DOI

V. Jurdjevic and H.J. Sussmann. Control systems on Lie groups. J. Differ. Eq. 12 (1972) 313–329. | DOI | MR | Zbl

K. Khodjasteh and L. Viola, Dynamical quantum error correction of unitary operations with bounded controls. Phys. Rev. A 80 (2009) 032314. | DOI

K. Khodjasteh and L. Viola, Dynamically error-corrected gates for universal quantum computation. Phys. Rev. Lett. 102 (2009) 080501. | DOI

C. Le Bris, M. Mirrahimi, H. Rabitz and G. Turinici, Hamiltonian identification for quantum systems: well-posedness and numerical approaches. ESAIM: COCV 13 (2007) 378–395. | Numdam | MR | Zbl

J.-S. Li and N. Khaneja, Control of inhomogeneous quantum ensembles. Phys. Rev. A 73 (2006) 030302. | DOI

Y. Maday and J. Salomon, A greedy algorithm for the identification of quantum systems. In Decision and Control, 2009 held jointly with the 2009 28th Chinese Control Conference. Proc. of the 48th IEEE Conference on CDC/CCC 2009 (2009) 375–379.

A.M. Souza, G.A. Álvarez and D. Suter, Experimental protection of quantum gates against decoherence and control errors. Phys. Rev. A 86 (2012) 050301. | DOI

G. Turinici, Beyond bilinear controllability: applications to quantum control. In Control of coupled partial differential equations, Vol. 155 of Internat. Ser. Numer. Math. Oberwolfach, Allemagne. Birkhauser (2007) 293–309. | MR | Zbl

G. Turinici, V. Ramakhrishna, B. Li and H. Rabitz, Optimal discrimination of multiple quantum systems: controllability analysis. J. Phys. A 37 (2004) 273. | DOI | MR | Zbl

C. Villani, Topics in optimal transportation. Graduate Studies in Mathematics. American Mathematical Society, cop., Providence R.I. (2003). | MR | Zbl

W. Zhu and H. Rabitz, Potential surfaces from the inversion of time dependent probability density data. J. Chem. Phys. 111 (1999) 472–480. | DOI

I.R. Zola and H. Rabitz, The influence of laser field noise on controlled quantum dynamics. J. Chem. Phys. 120 (2004) 9009–9016. | DOI

Cited by Sources: