From two Algebraic Bethe Ansätze to the dynamics of Dicke-Jaynes-Cummings-Gaudin quantum integrable models through eigenvalue-based determinants

Abstract

The work presented in this thesis was inspired by precedent results on the Gaudin models (which are integrable) for spins-12 only which, by a change of variables in the algebraic Bethe equations, manage to considerably simplify the numerical treatment of such models. This numerical optimisation is carried out by the construction of determinants, only depending on the previously mentioned variables, for every scalar products appearing in the expression of the mean value of an observable of interest at a given time. By showing it is possible to use the Quantum Inverse ScatteringMethod (QISM), even when the vacuum state is not eigenstate of the transfer matrix, the previous results concerning spins-12 only are generalised tomodels including an additional spin-boson interaction. De facto, this generalisation opened different possible paths of research. First of all, we show that it is possible to further generalise the use of determinants for spin models describing the interaction of one spin of arbitrary normwith many spins-12. We give the method leading to the explicit construction of determinants’ expressions. Moreover, we can extend this work to other Gaudin models where the vacuumstate is not an eigenstate of the transfer matrix. We did this work for spins-12interacting with an arbitrarily oriented magnetic field. Finally, a numerical treatment of systems describing the interaction ofmany spins-12 with a single bosonic mode is presented. We study the time evolution of bosonic occupation and of local magnetisation for two different Hamiltonians, the Tavis-Cummings Hamiltonian and a central spin Hamiltonian. We learn that the dynamics of these systems, relaxing from an initial state to a stationary state, leads to a superradiant-like state for certain initial states.

Date of Award	2017
Original language	English
Awarding Institution	Coventry University Université de Lorraine
Supervisor	Thierry Platini (Supervisor) & Alexandre Faribault (Supervisor)