Well-posedness and convergence of the Lindblad master equation for a quantum harmonic oscillator with multi-photon drive and damping

Rémi Azouit 1, 2 Alain Sarlette 1 Pierre Rouchon 1, 2
1 QUANTIC - QUANTum Information Circuits
Inria de Paris, MINES ParisTech - École nationale supérieure des mines de Paris, ENS Paris - École normale supérieure - Paris, UPMC - Université Pierre et Marie Curie - Paris 6
Abstract : 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 2k with integer k > 1 for a k-photon process. We prove that the corresponding PDE makes the state converge, for large time, to an invariant subspace spanned by a set of k 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.
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Rémi Azouit, Alain Sarlette, Pierre Rouchon. 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, EDP Sciences, 2016, Special Issue in honor of Jean-Michel Coron for his 60th birthday, 22 (4), pp.1353-1369. ⟨10.1051/cocv/2016050⟩. ⟨hal-01395585⟩

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