Fig. 1: Schematics of the vortex matter-wave interferometer.

a The interferometer is composed of two BSs (BS1 and BS2). The input vortex state \(\left|{l}_{1}\right\rangle\) is in the spin state \(\left|\downarrow \right\rangle\). The BS1 (a two-photon Raman pulse) splits the input state \(\left|\downarrow \right\rangle \left|{l}_{1}\right\rangle\) into two paths \(\left|\downarrow \right\rangle \left|{l}_{1}\right\rangle\) and \(\left|\uparrow \right\rangle \left|{l}_{2}\right\rangle\). The BS2 (an RF pulse) combines two vortex states \(\left|{l}_{1}\right\rangle\) and \(\left|{l}_{2}\right\rangle\) in each spin state. ϕ_↑ and ϕ_↓ are the relative phases between the two vortex modes in the two spin states, respectively. b is the pulse sequence. τ_R and τ_RF are the periods of the Raman and RF pulses, respectively. T is the evolution time. c An exemplary interferometer between two vortex states \(\left|{l}_{1}=0\right\rangle\) and \(\left|{l}_{2}=-2\right\rangle\). The Raman pulse contains two optical fields with OAM numbers L₁ =−2 and L₂ = 0. After BS1, there is a vortex state \(\left|{l}_{1}=0\right\rangle\) (\(\left|{l}_{2}=-2\right\rangle\)) in the spin state \(\left|\downarrow \right\rangle\) (\(\left|\uparrow \right\rangle\)). After BS2, vortex states \(\left|{l}_{1}=0\right\rangle\) and \(\left|{l}_{2}=-2\right\rangle\) interfere in each spin state.