Boundary-driven quantum systems near the Zeno limit: steady states and long-time behavior

Abstract We study composite open quantum systems with a finite-dimensional state space $$\mathcal {H}_{AB} = \mathcal {H}_A\otimes \mathcal {H}_B$$ H AB = H A ⊗ H B governed by a Lindblad equation $$\displaystyle {\frac{\textrm{d}}{\textrm{d}t}\rho (t) = \mathcal {L}_\gamma \rho (t)}$$ d d t ρ ( t ) = L γ ρ ( t ) where $$\mathcal {L}_\gamma \rho = -i[H,\rho ] + \gamma \mathcal {D}\rho $$ L γ ρ = - i [ H , ρ ] + γ D ρ . Here, H is a Hamiltonian on $$\mathcal {H}_{AB}$$ H AB while $$\mathcal {D}$$ D is a dissipator $$\mathcal {D}_A\otimes \mathbbm {1}$$ D A ⊗ 1 acting non-trivially only on part A of the system, which can be thought of as the boundary, and $$\gamma $$ γ is a parameter. It is known that the dynamics may simplify as the Zeno limit, $$\gamma \rightarrow \infty $$ γ → ∞ , is approached, so that after a initial time of order $$\gamma ^{-1}$$ γ - 1 , $$\rho (t)$$ ρ ( t ) is well approximated by $$\pi _A\otimes R(t)$$ π A ⊗ R ( t )