Abstract
In classical probability theory, the term cutoff describes the property of some Markov chains to jump from (close to) their initial configuration to (close to) completely mixed in a very narrow window of time. We investigate how coherent quantum evolution affects the mixing properties in two fermionic quantum models (the ``gain/loss'' and ``topological'' models), whose time evolution is governed by a Lindblad equation quadratic in fermionic operators, allowing for a straightforward exact solution. We check that the cutoff phenomenon extends to the quantum case and examine how the mixing properties depend on the initial state. In the topological case, we further show how the mixing properties are affected by the presence of a long-lived edge zero mode when taking open boundary conditions.
Highlights
It seems natural at this stage to ask whether an equivalent phenomenon exists in the quantum context. This was already adressed in [16], where an appropriate distance to equilibrium was defined in the quantum information language and the existence of cutoff established in some specific cases
We consider the evolution of an open quantum system, that is a quantum system coupled to an external environment
Our main interest here will be to investigate both sides of the “Zeno transition”, as well as the effect of the edge modes on the mixing properties, in the case of open boundary conditions
Summary
Coupling to an external environment is one of the many ways to drive a classical or quantum system out of equilibrium. This was already adressed in [16], where an appropriate distance to equilibrium was defined in the quantum information language and the existence of cutoff established in some specific cases These results remain tied to some restrictions on the types of systems considered as well as on the nature of the initial states, and leave several open questions, for instance relating to the initial state dependence of the mixing properties. There are two types of system-bath coupling we shall consider : one corresponds to gain/loss of particles through interaction with the environment, and the other may be considered as a toy-model for Liouvillians with non-trivial topological properties [17] Both these couplings have in common that they reduce in the classical limit to the master equation for the hypercube random walk, and are good candidates for studying the interplay of quantum coherence and classical cutoffs.
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