Abstract: We show that quantum particles constrained to move along curves undergoing cyclic deformations acquire, in general, geometric phases. We treat explicitly an example, involving particular deformations of a circle, and ponder on potential applications. Wires with Quantum Memory 2 1
We consider particles constrained to move in the close vicinity of a space curve through a steep quadratic potential in the plane normal to the curve. As is known, the effective 1D Hamiltonian that governs the motion along the curve involves both its curvature and torsion. Thus, the adiabatic cyclic deformations of the curve might give rise to geometric phases being accumulated by the particle. We report that this is indeed, in general, the case, and analyze in detail the origin of the phases, as described in both an inertial Cartesian frame and a noninertial one, adapted to the curve. We extend previous work to include non-locally arclength preserving deformations, and derive general formulas for Berry’s curvature in this case.
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.