Exact solutions for a spin-orbit coupled ultracold atom held in a driven double well

Exact solutions for spin-orbit (SO)-coupled cold atomic systems are very important and rare in physics. In this paper, we propose a simple method of combined modulations to generate the exactly analytic solutions for a single SO-coupled ultracold atom held in a driven double well. For the cases of synchronous combined modulations and the spin-conserving tunneling, we obtain the generally accurate solutions of this system respectively. For the case of spin-flipping tunneling under asynchronous combined modulations, we get the specially exact solutions in simple form when the driving parameters are appropriately chosen. Based on these obtained exact solutions, we reveal some intriguing quantum spin dynamical phenomena, for instance, the arbitrary coherent population transfer with and/or without spin-flipping, the controlled coherent population conservation, and the controlled coherent population inversion. The results may provide a possibility for generating the accurate quantum entangled states and the exact control of spin dynamics for a SO-coupled ultracold atomic system.