Monopole oscillation of Bose-Einstein condensate in a harmonic potential: Two-mode theory

The low-lying monopole oscillation is an important collective excitation in a Bose–Einstein condensate (BEC). In this paper, we develop a two-mode theory to describe the monopole oscillation of a BEC in a harmonic potential. We demonstrate that the mean-square radius, a key indicator of the monopole oscillation, is directly linked to the population imbalance between the two lowest even-parity modes. This relation shows that the mean-square radius contains both the fundamental population-oscillation frequency and a weaker second-harmonic component. The model also reveals a dynamical crossover from expanding to contracting oscillations, corresponding to a reversal of the net population transfer between the two modes. Including additional even-parity modes systematically extends the agreement with Gross–Pitaevskii simulations to stronger interactions and recovers the decreasing and saturating behavior of the oscillation frequency.