Decay of the survival probability of a local excitation in multi-qubit platforms

TL;DR

This paper provides an analytical study of the survival probability decay of a local excitation in multi-qubit systems, highlighting the role of extended states and their impact on equilibration.

Contribution

It introduces a theoretical framework using random matrix theory to analyze survival probability decay, independent of the specific interaction ensemble.

Findings

Decay properties depend on the presence of extended states in the Hamiltonian.

Survival probability decay is insensitive to whether the interaction is chaotic or solvable.

Extended eigenstates influence the quantum return time estimates.

Abstract

We present a theoretical study of the survival probability of a state initially prepared in the one-particle sector of a multi-qubit system. The motivation for our work is the ongoing laboratory development of multi-qubit platforms based on superconducting circuits. Using elementary concepts of random matrix theory, we obtain analytic expressions for the survival probability in mathematical models of platforms which, albeit stylized, have been previously shown to provide relevant benchmarks for experimental data. In particular, we show that the decay properties are sensitive to the property of the Hamilton operator to have extended states. The survival probability does not appear instead to depend on whether the interaction between qubits is described by a Gaussian orthogonal ensemble (often interpreted as a model of ''chaotic'' dynamics) or is modeled by an analytically solvable chain. We interpret this phenomenon as a manifestation of a general mechanism for the emergence of equilibration in purely unitary dynamics. Finally, under the same hypothesis of an initial preparation with projection on a large fraction of the extended eigenstates of the Hamilton operator, we show how to extend the classical Kac-Mazur-Montroll estimate of the return time to the quantum survival probability.