Quantum speed limit in the thermal spin-boson system with and without tunneling term

In this paper, we study the spin-bosonic model, with and without tunneling terms, in detail. The spin-bosonic model without tunneling is studied by using the thermofield dynamics approach. Indeed, by considering temperature, we show that environmental states, while they become entangled with system, approach thermal coherent states with different phases. In addition, by considering the tunneling term, we study the interplay of the environmental cut-off frequency as well as the impacts of environmental temperature on the quantum speed limit in both cases, i.e., spin-boson system with and without tunneling term. In these studies, we indicate temperature play more important role in compare with cut-off frequency to control the quantumness of a spin system.