in Eq. \[Eq:P\_LZS\] such that $|t|^2= 1-P_{\textrm{D-LZS}}$, $|r|^2= P_{\textrm{D-LZS}}$, and $U^{\dagger}_{\textrm{D-LZS}} U_{\textrm{D-LZS}} = I$. Note that we have defined $|t|^2$ and $|r|^2$ from the perspective of a beamsplitter, which “transmits” (“reflects”) an input state to the same (opposite) diabatic state, respectively. After the transition, a relative phase $\Delta \theta_{12}$ accrues due to the energy difference between the states $| L \rangle$ and $| R \rangle$. If the drive $\varepsilon(t)$ then returns the system through the avoided crossing a second time, the atomic state collides and quantum mechanically interferes with itself during the second LZS transition (Fig. \[fig:2\]D). The cumulative result is an atom-state interferometer whose output state depends on the LZS transition amplitudes and the interference phase.

An analogy can be made to an optical Mach-Zehnder interferometer: the atomic states play the role of the photon modes, the LZS transitions play the role of the photon beamsplitters, and the energy-level splitting, which determines the interference phase, plays the role of optical path length difference. The quantum interference is robust provided the evolution time of the state through the interferometer is short compared with the atom’s coherence times. In addition to superconducting artificial atoms [@Oliver05a; @Berns06a; @Nakamura01; @Sillanpaa06a; @Wilson07a; @Saito06a], this 