Quantum dynamics and relaxation in comb turbulent diffusion

TL;DR

This paper investigates quantum dynamics in comb geometries, revealing two scenarios where quantum and classical relaxation interplay, leading to phenomena like quantum swimming upstream, with detailed analytical solutions for wave and Green's functions.

Contribution

It introduces two novel models of continuous time quantum walks in comb structures involving non-Hermitian operators, highlighting unique quantum relaxation and transport phenomena.

Findings

Quantum swimming upstream observed due to dilatation operator effects.

Analytical solutions derived for wave and Green's functions in both scenarios.

Distinct roles of unitary and non-Hermitian dynamics in quantum transport.

Abstract

Continuous time quantum walks in the form of quantum counterparts of turbulent diffusion in comb geometry are considered. The interplay between the backbone inhomogeneous advection $\delta(y)x\partial_x$ along the $x$ axis, which takes place only at the $y=0$, and normal diffusion inside fingers $\partial_y^2$ along the $y$ axis leads to turbulent diffusion. This geometrical constraint of transport coefficients due to comb geometry and properties of a dilatation operator lead to consideration of two possible scenarios of quantum mechanics. These two variants of continuous time quantum walks are described by non-Hermitian operators of the form $\hat{\cal H}=\hat{A}+i\hat{B}$. Operator $\hat{A}$ is responsible for the unitary transformation, while operator $i\hat{B}$ is responsible for quantum/classical relaxation. At the first quantum scenario, the initial wave packet can move against the classical streaming. This quantum swimming upstream is due to the dilatation operator, which is responsible for the quantum (not unitary) dynamics along the backbone, while the classical relaxation takes place in fingers. In the second scenario, the dilatation operator is responsible for the quantum relaxation in the form of an imaginary optical potential, while the quantum unitary dynamics takes place in fingers. Rigorous analytical analysis is performed for both wave and Green's functions.