Bloch Motions and Spinning Tops

TL;DR

This paper explores the dynamics of closed quantum systems using classical rigid body methods, deriving integrability and stability criteria, and constructing solutions that describe oscillating entanglement phenomena.

Contribution

It introduces a novel formalism linking quantum dynamics to classical spinning top models, revealing integrability, stability conditions, and entanglement oscillations.

Findings

Proves Liouville integrability of quantum Bloch dynamics.

Derives stability criteria analogous to classical mechanics.

Constructs solutions showing oscillating entanglement states.

Abstract

This work investigates the dynamics of closed quantum systems in the Bloch vector representation using methods from rigid body dynamics and the theory of integrable systems. To this end, equations of motion for Bloch components are derived from the von Neumann equation, which are mathematically equivalent to equations of motion for a distribution of point masses from classical mechanics. Furthermore, using the Heisenberg equation, another system of Bloch vector equations is derived, which forms an Euler-Poinsot system, as is commonly encountered in the theory of torque-free spinning tops. This is used to prove the Liouville integrability of the corresponding Hamilton equations of motion, whereby formal connections to the Neumann model of classical Hamiltonian dynamics and the Hamiltonian Euler-Poinsot model are drawn to identify the first integrals of motion. Within the same framework, stability criteria for quantum dynamics are then derived which correspond to the Routh-Hurwitz criterion resp. other criteria following from the Energy-Casimir method of classical Newtonian mechanics. Following that, specific solutions to the equations of motion are constructed that encode the complex dynamics of composite quantum systems. Eventually, to show that this formalism provides concrete physical predictions, an analogue of the intermediate axis theorem is derived and the effect of oscillating entanglement is discussed. As a basis for this, special types of solutions to the equations of motion are derived that constitute oscillating entangled states, i.e., dynamical quantum states that change their entanglement structure from maximally entangled to separable and vice versa.