Abstract
Markovian reservoir engineering, in which time evolution of a quantum system is governed by a Lindblad master equation, is a powerful technique in studies of quantum phases of matter and quantum information. It can be used to drive a quantum system to a desired (unique) steady state, which can be an exotic phase of matter difficult to stabilize in nature. It can also be used to drive a system to a unitarily-evolving subspace, which can be used to store, protect, and process quantum information. In this paper, we derive a formula for the map corresponding to asymptotic (infinite-time) Lindbladian evolution and use it to study several important features of the unique state and subspace cases. We quantify how subspaces retain information about initial states and show how to use Lindbladians to simulate any quantum channels. We show that the quantum information in all subspaces can be successfully manipulated by small Hamiltonian perturbations, jump operator perturbations, or adiabatic deformations. We provide a Lindblad-induced notion of distance between adiabatically connected subspaces. We derive a Kubo formula governing linear response of subspaces to time-dependent Hamiltonian perturbations and determine cases in which this formula reduces to a Hamiltonian-based Kubo formula. As an application, we show that (for gapped systems) the zero-frequency Hall conductivity is unaffected by many types of Markovian dissipation. Finally, we show that the energy scale governing leakage out of the subspaces, resulting from either Hamiltonian/jump-operator perturbations or corrections to adiabatic evolution, is different from the conventional Lindbladian dissipative gap and, in certain cases, is equivalent to the excitation gap of a related Hamiltonian.
Highlights
Introduction to LindbladiansLindbladians operate on the space of operators on H, or OpðHÞ ≡ H ⊗ H⋆ [172,173]
It can be used to drive a quantum system to a desired steady state, which can be an exotic phase of matter difficult to stabilize in nature
We show that the quantum information in all subspaces can be successfully manipulated by small Hamiltonian perturbations, jump operator perturbations, or adiabatic deformations
Summary
Consider coupling a quantum mechanical system to a Markovian reservoir which evolves initial states of the system into multiple nonequilibrium (i.e., nonthermal) asymptotic states in the limit of infinite time. The asymptotic subspace attracts all initial states ρin ∈ OpðHÞ, is free from the decoherence effects of L, and any remaining time evolution within AsðHÞ is exclusively unitary. In general Lindbladian-based time evolution, all parts of an initial state ρin that are outside of AsðHÞ will decay as ρin evolves toward an asymptotic state ρ∞ ∈ AsðHÞ. IV C and V C we determine the energy scale governing leakage out of AsðHÞ due to both Hamiltonian perturbations and adiabatic evolution. We show that the zero-frequency Hall conductivity [77] remains quantized under various kinds of Markovian dissipation
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