Abstract
We show that the dynamics of particles in a one-dimensional harmonic trap with hard-core interactions can be solvable for certain arrangements of unequal masses. For any number of particles, there exist two families of unequal mass particles that have integrable dynamics, and there are additional exceptional cases for three, four, and five particles. The integrable mass families are classified by Coxeter reflection groups and the corresponding solutions are Bethe-ansatz-like superpositions of hyperspherical harmonics in the relative hyperangular coordinates that are then restricted to sectors of fixed particle order. We also provide evidence for superintegrability of these Coxeter mass families and conjecture maximal superintegrability.
- Received 19 April 2017
DOI:https://doi.org/10.1103/PhysRevX.7.041001
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Popular Summary
Experiments with ultracold atoms trapped in pools of light allow physicists to explore quantum dynamics of interacting particles with unprecedented control and precision. The nature of these dynamics runs along a spectrum from simple to complex, from controllable to uncontrollable, and from persistent to transitory. Typically, systems with simple, predictable, and stable dynamics are represented by models with straightforward mathematical properties and with all particles having the same mass. This paper presents a model of interacting particles with different masses, which, depending on the initial conditions, can exhibit the full range of dynamical behaviors.
The model describes a few hard quantum particles trapped and bouncing back and forth in a one-dimensional bowl. The key technical insight is realizing that, for example, four particles in one dimension are equivalent (for most quantum-mechanical purposes) to one particle in four dimensions. By moving to this higher-dimensional space and choosing the right coordinates, some symmetries become more obvious and more useful. These symmetries can be used to map the system we are studying onto a simpler model, namely, a single particle bouncing around within a triangle on the surface of a sphere. The shape of the triangle depends on the masses of the particles and their specific order in the trap.
One consequence is that experimentalists will be able to construct systems that are close to integrable systems, which have as many conserved quantities (like energy and momentum) as they have degrees of freedom. These systems may have practical applications because they do not thermalize as fast and they provide unique opportunities for controlling entanglement and coherence, key resources for quantum information processing.