# Dynamical Decoupling of Unbounded Hamiltonians

## TL;DR

This paper explores the feasibility of dynamical decoupling in quantum systems with unbounded Hamiltonians, providing criteria for success and illustrating cases where it works or fails.

## Contribution

It develops decoupling criteria for unbounded Hamiltonians and analyzes their effectiveness through explicit examples and convergence analysis.

## Key findings

- Decoupling can be achieved for certain unbounded Hamiltonians.
- Explicit examples demonstrate when decoupling succeeds or fails.
- Convergence speed and limiting evolution can be explicitly computed.

## Abstract

We investigate the possibility to suppress interactions between a finite dimensional system and an infinite dimensional environment through a fast sequence of unitary kicks on the finite dimensional system. This method, called dynamical decoupling, is known to work for bounded interactions, but physical environments such as bosonic heat baths are usually modelled with unbounded interactions, whence here we initiate a systematic study of dynamical decoupling for unbounded operators. We develop a sufficient decoupling criterion for arbitrary Hamiltonians and a necessary decoupling criterion for semibounded Hamiltonians. We give examples for unbounded Hamiltonians where decoupling works and the limiting evolution as well as the convergence speed can be explicitly computed. We show that decoupling does not always work for unbounded interactions and provide both physically and mathematically motivated examples.