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Our work here presents the analytical expressions for a typical Ramseyinterferometric sequence for a three- and a five-level system. The analyticalexpressions are derived starting from the first principals of unitary timeevolution operators. We focus on the three- and five-level systems because wepropose a novel Ramsey interferometer created by a trapped two-stateBose-Einstein Condensate driven by dipole oscillations and gravitational sag.It involves the $^{87}Rb$ atoms in states $\vert F=2, m_F=+2 \rangle$ $(\vert+2 \rangle)$ and $\vert F=2, m_F=+1 \rangle$ $(\vert +1 \rangle)$ of the $5^2S_{\frac{1}{2}}$ ground state. Though the interferometer focusses on thetwo-levels, the experimental readouts involve all the five states in $F = 2$hyperfine manifold. Therefore, the analytical derivation was first tested forthree-levels and then expanded to five-levels. We developed the expressions forfive-levels for greater analytical accuracy of the experimental scenario. Thiswork provides a step-by-step outline for the derivation and methodology for theanalytical expressions. These analytical formulae denote the populationvariation during Rabi and Ramsey oscillations for each state as well as theoverall average for both the three- and five-level cases. The expressions arederived within the rotating wave approximation (RWA) under the equal Rabicondition. Further, by following the derivation methodology, these analyticalexpressions can be easily expanded for Ramsey sequences with unequal pulses,and Ramsey sequences with spin echo techniques.

Ramsey interferometry in three-level and five-level systems of $^{87}Rb$ Bose-Einstein condensates