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Lieb-Robinson Bounds and Existence of the Thermodynamic Limit for a Class of\n Irreversible Quantum Dynamics

Bruno Nachtergaele, Anna Vershynina, Valentin Anatol'evich Zagrebnov

Published Mar 6, 2011• 49 citations
OpenAlex

Abstract

We prove Lieb-Robinson bounds and the existence of the thermodynamic limit for a\n general class of irreversible dynamics for quantum lattice systems with time-dependent\n generators that satisfy a suitable decay condition in space.

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Existence of the thermodynamic limit and asymptotic behavior of some irreversible quantum dynamical systems

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We discuss the properties of two open quantum systems with a general class of irreversible quantum dynamics. First we study Lieb-Robinson bounds in a quantum lattice systems. The time-dependent generator of the dynamics of the system is of the Lindblad-Kossakowski type. This generator satisfies some suitable decay condition in space. We show that the dynamics with a such generator on a finite system is a well-defined quantum dynamics in a sense of a norm-continuous cocycle of unit preserving completely positive maps. After proving Lieb-Robinson bounds for a finite lattice systems we also show the existence of the thermodynamic limit of the dynamics. We show that in a strong limit there exits a strongly continuous cocycle of unit preserving completely positive maps. Which means that the dynamics exists in an infinite system, where Lieb-Robinson bounds also holds. In the second part of the paper we consider a system that consists of a beam of two-level atoms that pass one by one through the microwave cavity. The atoms are randomly excited and there is exactly one atom present in the cavity at any given moment. We consider both the ideal and leaky cavity and study the time asymptotic behavior of the state of the cavity. We show that the number of photons increases indefinitely in the case of the ideal cavity. In case of leaky cavity we prove that the mean photon number in the cavity stabilizes in time. The limiting state of the cavity in this case exists and it is independent of the initial state. We calculate the characteristic functional of this non-quasi-free non-equilibrium state. We also calculate the energy flux in both the ideal and leaky and the entropy production for the ideal cavity.
In this note, we study a broad class of time-dependent and irreversible dynamics for quantum spin systems in both finite and infinite volume. We develop a simple and flexible strategy for expanding commutators of the form [A,τs,t B][A,τ_{s,t} \, B][A,τs,t​B] in terms of self-avoiding paths of sets, where AAA and BBB are local operators and τs,tτ_{s,t}τs,t​ is the Heisenberg time evolution. This adaptive Dyson expansion requires no backwards time evolution and is thus also applicable to irreversible dynamics. Furthermore, we discuss how such an expansion directly implies Lieb-Robinson bounds. A central feature of our result is that the obtained Lieb-Robinson bounds are independent of any Hamiltonian and dissipative on-site terms of the generator.