Dynamical normal modes for time-dependent Hamiltonians in two dimensions

TL;DR

This paper develops a theoretical framework using time-dependent point transformations to identify independent dynamical normal modes in 2D quantum systems with time-dependent controls, unifying various recent quantum operation techniques.

Contribution

It introduces a condition for defining independent modes and provides a geometric analogy, advancing the understanding of control in 2D quantum systems.

Findings

Unified description of quantum operations like transport and gates

Condition for defining independent dynamical modes

Geometric analogy for control in 2D systems

Abstract

We present the theory of time-dependent point transformations to find independent dynamical normal modes for 2D systems subjected to time-dependent control in the limit of small oscillations. The condition that determines if the independent modes can indeed be defined is identified, and a geometrical analogy is put forward. The results explain and unify recent work to design fast operations on trapped ions, needed to implement a scalable quantum-information architecture: transport, expansions, and the separation of two ions, two-ion phase gates, as well as the rotation of an anisotropic trap for an ion are treated and shown to be analogous to a mechanical system of two masses connected by springs with time dependent stiffness.