Developing a simulation tool to investigate a novel trapped two-state Bose-Einstein condensate Ramsey interferometer driven by dipole oscillations and gravitational sag

We propose and explore the feasibility of a novel Ramsey interferometer created by a trapped two-state Bose-Einstein condensate (BEC) driven by dipole oscillations and gravitational sag. The BEC is formed in a pure cigar shaped compressed magnetic trap (CMT) via a dilute atom cloud of $^{87}Rb$ atoms in state $\vert F=2, m_F=+2 \rangle$ $(\vert +2 \rangle)$ of the $5 ^2S_{\frac{1}{2}}$ ground state. Here, Rmasey interferometry is performed with states $\vert F=2, m_F=+1 \rangle$ $(\vert +1 \rangle)$ and $\vert +2 \rangle$. The proposed interferometer utilises the response of atoms to the harmonic oscillator trapping potential and the gravitational sag due to the variation in the $m_F$ state. Briefly, the state $\vert +1 \rangle$ experiences a shallower radial trap with a larger gravitational sag; whereas, state $\vert +2 \rangle$ experiences a tighter radial trap with a gravitational sag which is half of state $\vert +1 \rangle$. Due to this, a superposition between the states $\vert +1 \rangle$ and $\vert +2 \rangle$ experiences multipath propagation resulting in an interference pattern. This may be utilised to measure local gravitational fields and measure inter-sate scattering lengths. Here, a theoretical framework is reported which is developed via the two-level system in combination with the Gross-Pitaevskii equation (GPE). Further, the development of a simulation tool via GPELabs in MATLAB that explores the prosed interferometer is reported along with key insights and findings.