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arXiv:1512.08079 · Physical Review X

Geometry and Response of Lindbladians

Victor V. Albert, Barry Bradlyn, Martin Fraas, Liang Jiang

Published Nov 16, 2016• 162 citations

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 zerofrequency 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 or 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.

DOI: 10.1103/physrevx.6.041031

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Markovian master equations, often called Liouvillians or Lindbladians, are used to describe decay and decoherence of a quantum system induced by that system's environment. While a natural environment is detrimental to fragile quantum properties, an engineered environment can drive the system toward exotic phases of matter or toward subspaces protected from noise. These cases often require the Lindbladian to have more than one steady state, and such Lindbladians are dissipative analogues of Hamiltonians with multiple ground states. This thesis studies Lindbladian extensions of topics commonplace in degenerate Hamiltonian systems, providing examples and historical context along the way.

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