# Equilibration properties of small quantum systems: further examples

Abstract : It has been proposed to investigate the equilibration properties of a small isolated quantum system by means of the matrix of asymptotic transition probabilities in some preferential basis. The trace $T$ of this matrix measures the degree of equilibration of the system prepared in a typical state of the preferential basis. This quantity may vary between unity (ideal equilibration) and the dimension $N$ of the Hilbert space (no equilibration at all). Here we analyze several examples of simple systems where the behavior of $T$ can be investigated by analytical means. We first study the statistics of $T$ when the Hamiltonian governing the dynamics is random and drawn from a distribution invariant under the group U$(N)$ or O$(N)$. We then investigate a quantum spin $S$ in a tilted magnetic field making an arbitrary angle with the preferred quantization axis, as well as a tight-binding particle on a finite electrified chain. The last two cases provide examples of the interesting situation where varying a system parameter -- such as the tilt angle or the electric field -- through some scaling regime induces a continuous transition from good to bad equilibration properties.
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https://hal-cea.archives-ouvertes.fr/cea-01484658
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Submitted on : Tuesday, March 7, 2017 - 3:32:39 PM
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• HAL Id : cea-01484658, version 1
• ARXIV : 1702.05909

### Citation

J. M. Luck. Equilibration properties of small quantum systems: further examples. 2017. ⟨cea-01484658⟩

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