We consider the model of a quantum harmonic oscillator governed by a Lindblad master equation where the typical drive and loss channels are multi-photon processes instead of single-photon ones; this implies a dissipation operator of order with integer for a -photon process. We prove that the corresponding PDE makes the state converge, for large time, to an invariant subspace spanned by a set of selected basis vectors; the latter physically correspond to so-called coherent states with the same amplitude and uniformly distributed phases. We also show that this convergence features a finite set of bounded invariant functionals of the state (physical observables), such that the final state in the invariant subspace can be directly predicted from the initial state. The proof includes the full arguments towards the well-posedness of the corresponding dynamics in proper Banach spaces of Hermitian trace-class operators equipped with adapted nuclear norms. It relies on the Hille−Yosida theorem and Lyapunov convergence analysis.
Accepté le :
DOI : 10.1051/cocv/2016050
Mots-clés : Infinite-dimensional dissipative dynamical systems, Lyapunov functions and stability, accretive operators, Lindblad master equation, decoherence, quantum control, quantum electrodynamics and circuits.
@article{COCV_2016__22_4_1353_0,
author = {Azouit, R\'emi and Sarlette, Alain and Rouchon, Pierre},
title = {Well-posedness and convergence of the {Lindblad} master equation for a quantum harmonic oscillator with multi-photon drive and damping},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {1353--1369},
publisher = {EDP-Sciences},
volume = {22},
number = {4},
year = {2016},
doi = {10.1051/cocv/2016050},
zbl = {1354.37088},
mrnumber = {3570505},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2016050/}
}
TY - JOUR AU - Azouit, Rémi AU - Sarlette, Alain AU - Rouchon, Pierre TI - Well-posedness and convergence of the Lindblad master equation for a quantum harmonic oscillator with multi-photon drive and damping JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2016 SP - 1353 EP - 1369 VL - 22 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2016050/ DO - 10.1051/cocv/2016050 LA - en ID - COCV_2016__22_4_1353_0 ER -
%0 Journal Article %A Azouit, Rémi %A Sarlette, Alain %A Rouchon, Pierre %T Well-posedness and convergence of the Lindblad master equation for a quantum harmonic oscillator with multi-photon drive and damping %J ESAIM: Control, Optimisation and Calculus of Variations %D 2016 %P 1353-1369 %V 22 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2016050/ %R 10.1051/cocv/2016050 %G en %F COCV_2016__22_4_1353_0
Azouit, Rémi; Sarlette, Alain; Rouchon, Pierre. Well-posedness and convergence of the Lindblad master equation for a quantum harmonic oscillator with multi-photon drive and damping. ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 4, pp. 1353-1369. doi : 10.1051/cocv/2016050. http://www.numdam.org/articles/10.1051/cocv/2016050/
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