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

We study composite open quantum systems with a finite-dimensional state space HAB = HA⊗ HB H AB = H A ⊗ H B governed by a Lindblad equation d/dtρ (t) = L_γ ρ (t) d d t ρ ( t ) = L γ ρ ( t ) where L_γ ρ = -i[H,ρ ] + γ Dρ L γ ρ = - i [ H , ρ ] + γ D ρ . Here, H is a Hamiltonian on HAB H AB while D D is a dissipator DA⊗ 1 D A ⊗ 1 acting non-trivially only on part A of the system, which can be thought of as the boundary, and γ γ is a parameter. It is known that the dynamics may simplify as the Zeno limit, γ → ∞ γ → ∞ , is approached, so that after a initial time of order γ ⁻¹ γ - 1 , ρ (t) ρ ( t ) is well approximated by π A⊗ R(t) π A ⊗ R ( t )