Pancharatnam phase: A tool for atom optics

The Pancharatnam phase belongs to the family of geometric Berry phases. We use this optical phase to control the phase of our atom interferometer, which involves diffraction of the atom wave by laser standing waves in the Bragg regime. The Pancharatnam phase of the reflected beam of one standing wave controls the phase imprinted on the atom wave by the diffraction process. In addition to the expected phase shift, the experimental data exhibits the signature of several defects which are described and quantified. From this analysis, we estimate that a Pancharatnam phase shifter can be reliably used to control the phase of an atom interferometer in the sub-mrad regime. Moreover, as the geometric nature of the Pancharatnam phase renders this phase achromatic, its use in multispecies atom interferometers may be of great interest.

Figure 1

(Left part) The optical arrangement of Chyba et al. [24]. A linearly polarized laser beam crosses two quarter-wave plates QWP1 and QWP2 before being reflected by a mirror and crossing back QWP2 and QWP1. The optical axis of QWP1 makes an angle $\pi /4$ with the polarization vector while the optical axis of QWP2 makes an angle $\theta$ with the optical axis of QWP1. (Right part) Circuit followed on the Poincaré sphere by the light beam polarization. The initial linear polarization (point A) becomes right circular after QWP1 (pole B) then linear after QWP2 (point C). The polarization, not modified by reflection on the mirror, becomes left circular after QWP2 (pole D) and back linear after QWP1 (point A). The angle between the planes BAD and BCD being equal to $2\theta$, the solid angle spanned by the ABCDA circuit is $4\theta$ and the Pancharatnam phase is equal to $2\theta$.

Figure 2

Schematic drawing of our atom interferometer. A lithium atomic beam (full green lines) is diffracted by three laser standing waves produced by reflecting three laser beams on the mirrors $M1,M2$, and $M3$. The (violet) box below $M3$ represents the piezo-electric device which displaces this mirror in the $\mathbf{x}$ direction and modifies the diffraction phase ${\phi }_d$ given by Eq. (3). In the present experiment, this displacement is replaced by the Pancharatnam phase produced by the quarter-wave plates QWP1 and QWP2.

Figure 3

Atom interference fringes obtained by the rotation of QWP2. The (red) triangles correspond to the atom count each $100\text{ms}$, normalized to the atom flux [i.e., $\mathcal{I}/\left(2{\mathcal{I}}_0\right)]$. Four (blue) circles indicate typical error bars. The (black) dotted line results from a fit assuming an ideal Pancharatnam phase ${\phi }_d=2\left(\theta -{\theta }_0\right)$: There are systematic deviations due to the defects described by Eq. (8). The (green) solid line results from a second fit using Eq. (8) with the following fitted values $u_1=0.38\left(3\right)$ rad, $u_2=0.03\left(1\right)\text{rad}$, and $u_3=0.000\left(2\right)\text{rad}$ (see text).

Figure 4

Fractional part $\varepsilon$ of the quantity $p_0=2ne/\lambda$ where $e$ is the thickness of the plate QWP2 as a function of the laser spot's position on the plate.

Figure 5

$\mathbf{L}$ is the incoming laser beam and $\mathbf{N}$ the normal of the QWP2 faces. The rotation axe is along $\mathbf{Z}$ and the rotated frame $\left({\mathbf{X}}^{{}^{\prime }},{\mathbf{Y}}^{{}^{\prime }},{\mathbf{Z}}^{{}^{\prime }}\right)$ with $\omega t+\theta$ the rotation angle.

Figure 6

Trajectory on the Poincaré sphere when the QWP retardations is not perfect. To emphasize the deviation from the perfect trajectory, the QWP's defects for this figure are ${\varepsilon }_1=0.1$ and ${\varepsilon }_2=-0.1$.

Figure 7

Trajectory on the Poincaré sphere when the initial polarization is not perfectly linear. For this trajectory, the Stokes parameters for the incident light were $S_1=99.5\%,S_2=8.7\%$, and $S_1=4.8\%$ and the QWP were considered perfect. This corresponds to ${\epsilon }_P=0.05\text{rad}$ and a very small amount of ellipticity.