Fig. 2: Manipulating matter waves in a high-finesse cavity.

a, Simplified energy-level diagram for ⁸⁷Rb. The empty-cavity resonance used for probing (green) is detuned by δ_c from the \(\left|\uparrow \right\rangle \to \left|e\right\rangle \) transition ω_a. The Raman tones (blue) injected into the cavity drive a spin-changing \(\left|\uparrow \right\rangle \leftrightarrow \left|\downarrow \right\rangle \) transition with two-photon detuning δ defined in a falling reference frame. b, The Raman tones are derived from a laser detuned Δ from ω_a, locked between two adjacent TEM₀₀ modes separated by ω_FSR (grey), and modulated at ω_R ≈ ω_HF/2 for ground-state hyperfine splitting ω_HF, leaving the tones detuned from the cavity resonances by ±23 MHz. c, Atoms are prepared in \(\left|\downarrow \right\rangle \) and allowed to fall for a duration of T_fall = 7.5 ms (orange) or 15 ms (blue). The Raman coupling is applied at a fixed detuning δ, after which the number of atoms in \(\left|\uparrow \right\rangle \) is measured, revealing the axial velocity distribution. The full-width half-maximum of both distributions corresponds to a momentum spread of 5ħk, which is too broad for interferometry. During velocity selection, a group of about 800 atoms with r.m.s. momentum spread Δp = 0.1ħk (red) are kept from the latter distribution whereas the rest are removed with transverse radiation pressure. d, After velocity selection at a two-photon detuning δ_vs, a pair of Raman transitions can be used to place atoms into a superposition of \(\left|0\hbar k,\downarrow \right\rangle \) and \(\left|4\hbar k,\downarrow \right\rangle \). Raman spectroscopy is used to verify the discrete velocity distribution. e, Alternatively, Bragg transitions can be driven by adding amplitude modulation to the Raman tones. Here a Bragg π/2 pulse splits the wave packet, and consecutive π pulses transfer additional momentum to create a superposition \(\left|0\hbar k,\downarrow \right\rangle \) and \(\left|2n\hbar k,\downarrow \right\rangle \) with the momentum difference as large as 10ħk shown here.