### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 035, 19 pages      arXiv:2202.02196      https://doi.org/10.3842/SIGMA.2022.035
Contribution to the Special Issue on Non-Commutative Algebra, Probability and Analysis in Action

### The Generalized Fibonacci Oscillator as an Open Quantum System

Franco Fagnola a, Chul Ki Ko b and Hyun Jae Yoo c
a) Mathematics Department, Politecnico di Milano, Piazza L. da Vinci 32, I-20133 Milano, Italy
b) University College, Yonsei University, 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, Korea
c) School of Computer Engineering and Applied Mathematics, Institute for Integrated Mathematical Sciences, Hankyong National University, 327 Jungang-ro, Anseong-si, Gyeonggi-do 17579, Korea

Received February 07, 2022, in final form April 19, 2022; Published online May 11, 2022

Abstract
We consider an open quantum system with Hamiltonian $H_S$ whose spectrum is given by a generalized Fibonacci sequence weakly coupled to a Boson reservoir in equilibrium at inverse temperature $\beta$. We find the generator of the reduced system evolution and explicitly compute the stationary state of the system, that turns out to be unique and faithful, in terms of parameters of the model. If the system Hamiltonian is generic we show that convergence towards the invariant state is exponentially fast and compute explicitly the spectral gap for low temperatures, when quantum features of the system are more significant, under an additional assumption on the spectrum of $H_S$.

Key words: open quantum system; Fibonacci Hamiltonian; deformation of canonical commutation relations; spectral gap; weak-coupling limit; quantum Markov semigroup.

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