Generalized Hamiltonian for a two-mode fermionic model and asymptotic equilibria

Abstract

In some recent papers, the so called (H, ρ)-induced dynamics of a system S whose time evolution is deduced adopting an operatorial approach, has been introduced. According to the formal mathematical apparatus of quantum mechanics, H denotes the Hamiltonian for S, while ρ is a certain rule applied periodically on S. In this approach the rule acts at specific times kτ , with k integer and τ fixed, by modifying some of the parameters entering H according to the state variation of the system. As a result, a dynamics admitting an asymptotic equilibrium state can be obtained. Here, we consider the limit for τ → 0, so that we introduce a generalized model leading to asymptotic equilibria. Moreover, in the case of a two-mode fermionic model, we are able to derive a relation linking the parameters involved in the Hamiltonian to the asymptotic equilibrium states.

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“…This approach allows us to modify some of the parameters entering the Hamiltonian according to the evolution of the system. In [8], we showed that this approach can provide similar results to the ones derived when considering open quantum systems or in the case where the Hamiltonian is time-dependent (see also [12]). The remarkable effect of the approach, besides keeping low the computational complexity, is that of producing time evolutions usually approaching some equilibrium state, even for finite-dimensional systems.…”

“…The choice of τ plays a role in the dynamics too. Indeed, if τ is very large, τ T, it is as if no rule ρ is truly acting on S; vice versa, if τ 0, it is as if ρ is acting continuously on S [12], producing a form of Zeno effect.…”