Quantum Multibaker Maps: Extreme Quantum Regime

TL;DR

This paper introduces quantum multibaker maps as models for quantum random walks, analyzing their eigenstates, steady states, and relaxation properties in the deep quantum regime, with analytical and numerical results.

Contribution

It presents a new family of quantum multibaker maps, providing analytical solutions for uniform cases and exploring their quantum behavior in the deep quantum regime.

Findings

Eigenstates are extended in uniform QMB and localized in random QMB.

Analytical solutions for time-dependent states in the deep quantum regime.

Steady state properties and relaxation dynamics are analytically described.

Abstract

We introduce a family of models for quantum mechanical, one-dimensional random walks, called quantum multibaker maps (QMB). These are Weyl quantizations of the classical multibaker models previously considered by Gaspard, Tasaki and others. Depending on the properties of the phases parametrizing the quantization, we consider only two classes of the QMB maps: uniform and random. Uniform QMB maps are characterized by phases which are the same in every unit cell of the multibaker chain. Random QMB maps have phases that vary randomly from unit cell to unit cell. The eigenstates in the former case are extended while in the latter they are localized. In the uniform case and for large $\hbar$, analytic solutions can be obtained for the time dependent quantum states for periodic chains and for open chains with absorbing boundary conditions. Steady state solutions and the properties of the relaxation to a steady state for a uniform QMB chain in contact with ``particle'' reservoirs can also be described analytically. The analytical results are consistent with, and confirmed by, results obtained from numerical methods. We report here results for the deep quantum regime (large $\hbar$) of the uniform QMB, as well as some results for the random QMB. We leave the moderate and small $\hbar$ results as well as further consideration of the other versions of the QMB for further publications.