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 governed by a Lindblad equation ddtρ(t)=Lγρ(t) where Lγρ=-i[H,ρ]+γDρ. Here, H is a Hamiltonian on HAB while D is a dissipator DA⊗1 acting non-trivially only on part A of the system, which can be thought of as the boundary, and γ is a parameter. It is known that the dynamics may simplify as the Zeno limit, γ→∞, is approached, so that after a initial time of order γ-1, ρ(t) is well approximated by πA⊗R(t) where πA is a density matrix on HA such that DAπA=0, and R(t) is an approximate solution of ddtR(t)=LP,γR(t) where LP,γR:=-i[HP,R]+γ-1DPR with HP being a Hamiltonian on HB and DP being a Lindblad generator acting on density matrices on HB. We give a rigorous proof of this holding in greater generality than in previous work; we assume only that DA is ergodic and gapped. Moreover, we precisely control the error terms, and use this to show that the mixing times of Lγ and LP,γ are tightly related near the Zeno limit. Despite this connection, the errors in the approximate description of the evolution accumulate on times of order γ2, so it is difficult to directly access steady states ρ¯γ of Lγ through study of LP,γ. In order to better control the long time behavior, and in particular the steady states ρ¯γ, we introduce a third Lindblad generator DP♯ that does not involve γ, but is still closely related to Lγ and LP,γ. We show that if DP♯ is ergodic and gapped, then so are Lγ and LP,γ for all large γ, and in this case, if ρ¯γ denotes the unique steady state for Lγ, then limγ→∞ρ¯γ=πA⊗R¯ where R¯ is the unique steady state for DP♯. We further show that there is a trace norm convergent expansion ρ¯γ=πA⊗R¯+γ-1∑k=0∞γ-kn¯k where, defining n¯-1:=πA⊗R¯, Dn¯k=-i[H,n¯k-1] for all k⩾0. Using properties of DP and DP♯, we show that this system of equations has a unique solution. This is illustrated in a simple example for which one can compute ρ¯γ, and can carry out the expansion explicitly.