Spectroscopy of collective modes in a Bose–Einstein condensate: From single to double excitation periods

Collective modes are coherent excitations in Bose–Einstein condensates (BECs), and their study provides insight into the macroscopic quantum phenomena that govern these systems. Collective mode frequencies can be used to probe the properties of BECs, such as the trap geometry, the interatomic interactions, and the presence of defects; hence, it is essential to develop methods for high-resolution determination of collective mode frequencies. A standard technique consists of a single pulse of an external oscillatory field, which the authors denote Rabi-like due to the analogy with the field of nuclear magnetic resonances. In this work, the authors propose a method to achieve a better resolution than the Rabi-like protocol, which consists of two oscillating fields separated in time, which the authors call Ramsey-like. The authors focus on BECs in harmonic traps, considering mainly the quadrupole and breathing modes for different trap anisotropy values and comparing the results using single and double excitation periods. First, the authors employ a variational approach with a Thomas–Fermi ansatz, which gives rise to dynamical equations that are numerically solved. Then, the authors modeled the problem as a three-level system, offering a concise and alternative description of the same dynamics, enabling the idea of coherent control of these collective modes. Both approaches show that the Ramsey-like protocol provides a better resolution than the Rabi-like. This offers the possibility of measuring collective mode frequencies with higher precision, which is important in experiments where they are used as indirect measurements of properties that are difficult to probe.