# Linking the rotation of a rigid body to the Schr\"odinger equation: The quantum tennis racket effect and beyond

## TL;DR

This paper reveals how classical rigid body rotation principles can be applied to quantum control, enabling efficient state transfer, geometric quantum gates, and demonstrating the quantum tennis racket effect experimentally.

## Contribution

It establishes a novel connection between rigid body dynamics and quantum control, providing new control fields and experimental demonstrations for quantum systems.

## Key findings

- Control fields depend on two parameters for efficiency and robustness
- Experimental demonstration of the quantum tennis racket effect using NMR
- Rigid body dynamics can implement one-qubit quantum gates

## Abstract

The design of efficient and robust pulse sequences is a fundamental requirement in quantum control. Numerical methods can be used for this purpose, but with relatively little insight into the control mechanism. Here, we show that the free rotation of a classical rigid body plays a fundamental role in the control of two-level quantum systems by means of external electromagnetic pulses. For a state to state transfer, we derive a family of control fields depending upon two free parameters, which allow us to adjust the efficiency, the time and the robustness of the control process. As an illustrative example, we consider the quantum analog of the tennis racket effect, which is a geometric property of any classical rigid body. This effect is demonstrated experimentally for the control of a spin 1/2 particle by using techniques of Nuclear Magnetic Resonance. We also show that the dynamics of a rigid body can be used to implement one-qubit quantum gates. In particular, non-adiabatic geometric quantum phase gates can be realized based on the Montgomery phase of a rigid body. The robustness issue of the gates is discussed.