Probing the spatial distribution of k-vectors in situ with Bose-Einstein condensates

Abstract

We present a novel method for mapping \textit{in situ} the spatial distribution of photon momentum across a laser beam using a Bose-Einstein condensate (BEC) as a moving probe. By displacing the BEC, we measure the photon recoil by atom interferometry at different positions in the laser beam and thus reconstruct a two-dimensional map of the local intensity and effective dispersion of the $k$ wave vector. Applied to a beam diffracted by a diaphragm, this method reveals a local \textit{extra recoil} effect, which exceeds the magnitude $hν/c$ of the individual plane-waves over which the beam can be decomposed. This method offers a new way to precisely characterize wavefront distortions and to evaluate one of the major systematic bias sources in quantum sensors based on atom interferometry.

Figures (4)

• Figure 1: (a) Schematic of the experimental setup showing the cold atoms chamber, the detection and the atom interferometry areas, as well as the propagation of the Raman and Bloch beams along the vertical axis ($z$-axis). Raman beams are in a orthogonal linear polarizations arrangement ($\bigodot$ and $\leftrightarrow$) while both Bloch beams have the same linear polarization ($\bigodot$). (b) Experimental setup used to generate the laser beams forming the dipole trap for the BEC production. We employ two acousto-optic modulators (AOMs) to control the position of the center of the dipole trap and to impart a transverse velocity to the Bose-Einstein condensate before being released. The inset illustrates how a change of the trap position is used to impart a velocity to the BEC (c) Atom trajectories (for simplicity in the free falling reference frame) and temporal sequence used to measure recoil velocity using a Ramsey-Bordé interferometer. It consists of two pairs of $\pi/2$ pulses with $T_R=20$ ms, separated by a duration $T_D=35$ ms. These pulses induce stimulated Raman transitions between the two hyperfine states $|F=1\rangle$ and $|F=2\rangle$ of 87 rubidium atom. They separate and recombine the atomic wave packets that interfere at the end of the time sequence. Between the two pairs of pulses, an accelerated optical lattice is switched on for 6 ms. Atoms perform $N_B=500$ Bloch oscillations and acquire a momentum of $2 N_\mathrm{B} \hbar \vec{k}_\mathrm{B}$.
• Figure 2: Typical interference fringes shown for different configurations : $(\alpha)$ BEC positioned at the beam center, $(\beta)$ BEC with transverse velocity displaced by $\sim$ 2 mm from the beam center and $(\gamma)$ measurements performed with optical molasses. Right) Values of $\kappa_\mathrm{rel}$ extracted from Eq. \ref{['Kappa']} using measurements of the Doppler shifted frequency $f_D$ and as reference frequency $f_D^\mathrm{ref}$ obtained with optical molasses. The data were acquired over 117 hours, using BEC and optical molasses alternately.
• Figure 3: One-dimensional profile of $\kappa_{\mathrm{rel}}$ measured with a BEC along the $y$-axis are depicted by blue dots, using a beam clipped by a diaphragm. The $\kappa_{\mathrm{rel}}$ correction calculated from the beam intensity measurement is illustrated with red dots. These measurements are compared to Monte-Carlo simulations of the experiment, represented by an orange dashed line.
• Figure 4: Reconstruction of beam intensity profile within the vacuum chamber using the light shift method with the BEC. Each data point corresponds to the phase shift value in radian for a position in the transverse plan, which is directly proportional to the intensity. (a) Spatial profile of the upward-propagating Bloch beam (b) spatial profile of the upward-propagating Bloch beam diffracted by an iris with an aperture of 4 mm diameter.