Correction of dephasing oscillations in matter-wave interferometry

Vibrations, electromagnetic oscillations, and temperature drifts are among the main reasons for dephasing in matter-wave interferometry. Sophisticated interferometry experiments, e.g., with ions or heavy molecules, often require integration times of several minutes due to the low source intensity or the high velocity selection. Here we present a scheme to suppress the influence of such dephasing mechanisms—especially in the low-frequency regime—by analyzing temporal and spatial particle correlations available in modern detectors. Such correlations can reveal interference properties that would otherwise be washed out due to dephasing by external oscillating signals. The method is shown experimentally in a biprism electron interferometer where a perturbing oscillation is artificially introduced by a periodically varying magnetic field. We provide a full theoretical description of the particle correlations where the perturbing frequency and amplitude can be revealed from the disturbed interferogram. The original spatial fringe pattern without the perturbation can thereby be restored. The technique can be applied to lower the general noise requirements in matter-wave interferometers. It allows for the optimization of electromagnetic shielding and decreases the efforts for vibrational or temperature stabilization.

Figure 1

In-vacuum setup of the electron biprism interferometer, which is a modified version of the one described in [4, 11, 12] (not to scale). An electron beam is field emitted from a single-atom tip [18, 19] and adjusted by deflector electrodes toward the optical axis. The electron matter waves are coherently separated and recombined by an electrostatic biprism [1]. The resulting interference pattern is magnified by quadrupole lenses and detected in a delay line detector [17]. It can be artificially disturbed by a time-varying field originating from two magnetic coils, which are placed outside the vacuum chamber. A mu-metal shield (not shown) is placed between the in-vacuum setup and the magnetic coils.

Figure 2

(a) Electron biprism interference pattern and corresponding integrated line profile recorded with the setup of Fig. 1. (b) The same interference pattern after the introduction of a periodic 50 Hz magnetic-field oscillation with a dc amplitude of 2$\pi$. The fringes are completely washed out. (c) Histogram and integrated line profile of spatial distances $\Delta x$ and $\Delta y$ between temporal adjacent events. They clearly show correlations in the position of two consecutive events, revealing the existence of an interference pattern even after perturbation. The integrated line profile has been corrected by a factor ${(1-|\Delta y|/Y)}^{-1}$ to correct for the finite pattern width of $Y=20$  mm.

Figure 3

Two-dimensional correlation functions $g^{\left(2\right)}(u,\tau )$ of disturbed interference patterns as a function of the spatial ($u$) and temporal ($\tau$) distance between two detection events. The axes are normalized to the spatial periodicity ($\lambda$) and the perturbation period (${\nu }^{-1}$). The plots visualize the results of Eq. (5) for different peak phase deviations ${\phi }_0$ of (a) $0\pi$, (b) $1/3\pi$, (c) $2/3\pi$, and (d) $1\pi$. The absolute value of $A\left(\tau \right)$ normalized to the contrast $K^2$ is shown below each plot.

Figure 4

(a) Experimentally determined two-dimensional correlation function $g^{\left(2\right)}(u,\tau )$, extracted according to Eq. (8) from the detector signal of the disturbed interference pattern in Fig. 2. (b) Fit of the pattern in (a) with the theoretical expression in Eqs. (5, 6, 7). (c) The residual of theoretical and experimental data. The remaining periodic structure might be related to diffraction at the biprism. (d) Reconstruction of the original undisturbed interference pattern from the strongly disturbed data of Fig. 2 using the extracted fitting parameters.