An infinite-temperature limit for a quantum scattering process

TL;DR

This paper investigates the limit of a quantum scattering process driven by a Lindblad generator with a Poisson-timed scattering component, showing convergence to a limiting dynamics as the mass ratio tends to zero and collisions become more frequent.

Contribution

It introduces a new limiting dynamics for a quantum scattering process in the infinite-temperature regime, with precise operator norm bounds on the convergence.

Findings

The dynamics approaches a limiting process with second order error as the mass ratio tends to zero.

The difference between the original and limiting dynamics is bounded by a term proportional to  in operator norm.

The results apply to test functions in the space of bounded operators on L^2().

Abstract

We study a quantum dynamical semigroup driven by a Lindblad generator with a deterministic Schr\"odinger part and a noisy Poission-timed scattering part. The dynamics describes the evolution of a test particle in $\R^{n}$, $n=1,2,3$, immersed in a gas, and the noisy scattering part is defined by the reduced effect of an individual interaction, where the interaction between the test particle and a single gas particle is via a repulsive point potential. In the limit that the mass ratio $\lambda=\frac{m}{M}$ tends to zero and the collisions become more frequent as $\frac{1}{\lambda}$, we show that our dynamics $\Phi_{t,\lambda}$ approaches a limiting dynamics $\Phi_{t,\lambda}^{\diamond}$ with second order error. Working in the Heisenberg representation, for $G\in \Bi(L^{2}(\R^{n}))$ $n=1,3$ we bound the difference between $\Phi_{t,\lambda}(G)$ and $\Phi_{t,\lambda}^{\diamond}(G)$ in operator norm proportional to $\lambda^{2}$.