Observation of Atom-Wave Beats Using a Kerr Modulator for Atom Waves

A phase modulation puts the atom in a coherent superposition of quantum states with different kinetic energies. We have detected the interference of such modulated waves at the output of our atom interferometer, and we have observed beats at the difference of the modulation frequencies and its harmonics, in good agreement with theory. The phase modulations were produced by a Kerr phase modulator, i.e., by the propagation of the atom wave in a time-dependent electric field. An extension of this technique to electron interferometry should open the way to very high temporal resolution in electron microscopy.

Figure 1

Schematic drawing of our atom interferometer (not to scale). The atomic beam collimated by two slits is diffracted by three quasiresonant laser standing waves produced by reflecting laser beams on three mirrors. Two output atomic beams carry complementary interference signals. A slit selects one of these beams whose intensity is measured by a Langmuir-Taylor surface ionization detector. The interferometer arms labeled $A$ and $B$ are represented by full (green) lines, and the diffraction angle, $\sim 160\mu \mathrm{rad}$, has been grossly exaggerated. Lower part: Detailed view of the interaction region, with the interferometer arms going through two capacitors which share a common grounded electrode. The length of the capacitor is $L\approx 48\mathrm{mm}$, and the distance from their center to the detector is $z_d-z_c=1423\pm 10\mathrm{mm}$.

Figure 2

The interferometer signal in the presence of a phase $\mathrm{\Delta }{\phi }_{\max }\sim 2.7\mathrm{rad}$ oscillating at a frequency $\omega /(2\pi )=21\mathrm{Hz}$. The diffraction phase ${\phi }_d\approx -0.5\mathrm{rad}$ has been chosen to induce a strong non-linearity. Top panel: direct recording of the interferometer signal equal to the number of atoms detected per millisecond. Bottom panel: the modulus of the Fourier transform of a 16.4-s-long record reveals the presence of the first four harmonics.

Figure 3

Measurement of the fringe visibility and the phase shift of the harmonics of the modulation frequency equal to 11 or 21 Hz. Upper panel: $\mathcal{V}(p)/{\mathcal{V}}_0$ as a function of $\mathrm{\Delta }{\phi }_{\max }$ for $p=0$ (squares), 1 (bullets), and 6 (stars); the curves are our theoretical results, with no free parameters. Lower panel: Phase shift ${\psi }_m(p)=p\omega {\tau }_c$ as a function of the harmonic order $p$ for two modulation frequencies $\omega /(2\pi )=11$ and 43 Hz. The total delay ${\tau }_c$ deduced from these experiments is ${\tau }_c\approx 1.65\pm 0.01\mathrm{ms}$: It is the sum of the atom propagation time, estimated near 1.36 ms, and of the delay due to the ionization process. From the measured ${\tau }_c$ value, we deduce an ionization delay near 0.29 ms.

Figure 4

Beats of matter waves: The quantity $\mathcal{V}(p)/{\mathcal{V}}_0$ measured on the modulation at the $p$th harmonics of $({\omega }_A-{\omega }_B)/(2\pi )=13\mathrm{Hz}$ is plotted as a function of the frequency ${\omega }_A/(2\pi )$ in kilohertz. The symbols represent our measurements, while the full lines represent the theoretical results, taking into account the velocity distribution. The value of ${\phi }_{\max }$ is $\approx 6.7\mathrm{rad}$.