Laser-Manipulating Macroscopic Quantum States of a Bose–Einstein Condensate Held in a Kronig–Penny Potential

Abstract

We investigate the boundary value problem (BVP) of a quasi-one-dimensional Gross–Pitaevskii equation with the Kronig–Penney potential (KPP) of period d, which governs a repulsive Bose–Einstein condensate. Under the zero and periodic boundary conditions, we show how to determine n exact stationary eigenstates {Rn} corresponding to different chemical potentials {μn} from the known solutions of the system. The n-th eigenstate Rn is the Jacobian elliptic function with period 2d/n for n = 1,2,…, and with zero points containing the potential barrier positions. So Rn is differentiable at any spatial point and Rn2 describes n complete wave-packets in each period of the KPP. It is revealed that one can use a laser pulse modeled by a δ potential at site xi to manipulate the transitions from the states of {Rn} with zero point x ≠ xi to the states of {Rn} with zero point x = xi. The results suggest an experimental scheme for applying BEC to test the BVP and to observe the macroscopic quantum transitions.