effective optical path lengths. These paths are then recombined at the second beamsplitter, where the signal waves interfere and exit the interferometer through the two output ports. An intensity measurement at either output port exhibits interference fringes depending on the relative path length.

Here, we instead utilize LZS transitions at level crossings as beamsplitters for the atomic state [@Shytov03]. We drive the persistent-current qubit with a harmonic driving field of the form $$\varepsilon(t) = \varepsilon_0 + A \sin \omega t$$ with $\omega = 2\pi \nu $ the driving frequency and $A>\varepsilon_0$ the field amplitude (in units of energy), which is proportional to the RF voltage at the source. As illustrated in Fig. \[fig:2\]D, the qubit state undergoes two LZS transitions during one period of the driving field. The first LZS transition at time $t_1$ splits the qubit into a superposition of excited and ground states. A relative phase $\Delta \theta_{12}$ accumulates $$\Delta\theta_{12}=\frac1{\hbar}\int_{t_1}^{t_2} \epsilon(t)dt ,\quad \epsilon(t)=\epsilon_{|R\rangle}(t)-\epsilon_{|L\rangle}(t)$$ until the second LZS transition at time $t_2$, at which point the qubit state collides with itself and interferes. Interference fringes appear at half-integer and integer values of $\Delta\theta_{12}/2 \pi$, which is tunable via the drive amplitude.

It is clear, however, that a second phase 