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arXiv:0906.2934

The rolling sphere and the quantum spin

Alberto G. Rojo, Anthony M. Bloch

Published Jun 16, 2009• 0 citations
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Abstract

We consider the problem of a sphere rolling of a curved surface and solve it by mapping it to the precession of a spin 1/2 in a magnetic field of variable magnitude and direction. The mapping can be of pedagogical use in discussing both rolling and spin precession, and in particular in understanding the emergence of geometrical phases in classical problems.

DOI
10.48550/arxiv.0906.2934

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We consider the problem of a sphere rolling of a curved surface and solve it by mapping it to the precession of a spin 1/2 in a magnetic field of variable magnitude and direction. The mapping can be of pedagogical use in discussing both rolling and spin precession, and in particular in understanding the emergence of geometrical phases in classical problems.
We solve the problem of a sphere rolling on a curved surface by mapping it to the precession of a spin 1/2 in a magnetic field of variable magnitude and direction. The mapping can be instructive for discussing both rolling and spin precession. As an example, we show that the Landau–Zener problem corresponds to the rolling of a sphere on a Cornu spiral and derive the probability of a nonadiabatic transition using this correspondence. We also discuss the adiabatic limit and the vanishing of geometric phases for rolling on curved surfaces.
This work explores how geometry shapes the rules of quantum mechanics. Just as planets move along curved paths in space, quantum states evolve on geometric “phase spaces” with unique shapes. We show how these spaces like the famous Bloch sphere for qubits help us understand quantum dynamics, speed limits, and even entanglement. A key focus is the “geometric phase,” a memory-like effect acquired by quantum systems during cyclic journeys, akin to a hiker gaining elevation on a mountain trail. We solve the “quantum brachistochrone” problem finding the fastest path between quantum states using geometric tools. For two interacting spins, we reveal how entanglement warps the geometry of their shared state space, linking stronger correlations to flatter curvature. Finally, we connect these ideas to quantum computing, showing how geometric phases enable robust logic gates in experiments like nuclear magnetic resonance (NMR). This bridges abstract math to real-world quantum technologies.
To quantitatively characterize the behaviors of a spin-12 particle moving on a rotating curved surface and in a time-varying field, we derived the Schrödinger equation of the particle on the surface with the induced Pauli matrices based on the local curvilinear coordinate. To consider the relativistic effects, Minkowski space M is employed. Based on the solution of the Schrödinger equation, we found that the evolution of phase is influenced by the coupling of external fields and the geometry of surfaces as well as the relativistic effect originating from rotation. As examples, the solutions on a rotating cylinder/sphere in a damping magnetic field are calculated. The spin-flip behavior of the particle and the effects of external field and rotation are demonstrated. The results may help stimulate the ideas of modulating and controlling the spin on nano-surfaces.
The entanglement dependence of the noncyclic geometric phase is analyzed. A pair of noninteracting spin-12\frac{1}{2}21​ particles prepared in an arbitrarily entangled state and precessing in an external time-independent uniform magnetic field is considered. It is shown that the geometric phase reduces to a sum of one-particle geometric phases for product states and takes on the two values corresponding to the phase factors $\ifmmode\pm\else\textpm\fi{}1$ for maximally entangled states. If only one of the particles is affected by the magnetic field it is demonstrated that the influence of entanglement on the geometric phase may be interpreted as an effective reduction of the degree of polarization of the affected particle. The generalization to more than two precessing spin-12\frac{1}{2}21​ particles, in the particular case where Schmidt decompositions exists, is briefly outlined. The geometric phase for a pair of spin-12\frac{1}{2}21​ particles with a spin-spin interaction is calculated. In this model we show that the noncyclic geometric phase for a certain class of states may be interpreted solely in terms of the solid angle enclosed by the geodesically closed curve on a two-sphere parametrized by the evolving Schmidt coefficients. This suggests a geometric interpretation of Schmidt decompositions for spin-12\frac{1}{2}21​ pairs analogous to that of the Poincar\'e sphere for a single spin 12.\frac{1}{2}.21​.