Stern-Gerlach interferometry in three dimensions: the role of transverse fields

Abstract

We show that superficially similar implementations of Stern-Gerlach Interferometers (SGIs) are expected to differ dramatically in their sensitivity to fields transverse to the primary acceleration direction. These transverse fields unavoidably accompany any static magnetic or electric field gradients, and have been shown by Comparat [Phys. Rev. A 101, 023606 (2020)] to limit the precision application of SGIs. As a concrete example, we consider SGIs with ultracold Rb Rydberg atoms accelerated by spatially-varying electric fields. We find that the deleterious effect of transverse fields imply that only some implementations (sequences of field gradients, internal state swaps, and so-on) may exhibit fringes with high visibility.

Figures (10)

• Figure 1: An SGI as envisioned by early workers illustrating the splitting and recombination of two spin states isbn:9780486659695shortdoi:bdgdmpshortdoi:c34q7x.
• Figure 2: (a) The shifts of the energies of internal states $\ket{r}$ and $\ket{g}$ as a function of electric field magnitude $|\vec{\boldsymbol{\mathbf{E}}}|$ together with an illustration of the origin of the net force on an atom in state $\ket{r}$, and (b) the steps in proposed electric SGI experiment using laser-cooled Rb atoms excited to Rydberg states. After repeating steps (1a) through (1g), with different phases $\phi$ of the final $\pi/2$ pulse, a fringe visibility $\mathcal{V}$ and phase shift $\theta$ may be determined.
• Figure 3: (a) Spatial variations of the electric field in a cylindrically symmetric geometry with: $\partial_z E_z |_0 > 0$ (dotted blue line), and $\partial_z E_z |_0 < 0$ (dashed red line); (b) Corresponding variations in the magnitude of the electric field $E$ along the longitudinal $\hat{\boldsymbol{\mathbf{z}}}$ axis and (c) along the transverse $\hat{\boldsymbol{\mathbf{x}}}$ and $\hat{\boldsymbol{\mathbf{y}}}$ axes. Note that the transverse variation in $E$ is the same for both $\partial_z E_z |_0 > 0$ and $\partial_z E_z |_0 < 0$. Thus, an atom in a high-field seeking state will experience an force pushing it off the $z$-axis; i.e., the inverted harmonic oscillator potential of Eq. \ref{['eq:eff_potential']}.
• Figure 4: An electrode configuration suitable for generating cylindrically symmetric inhomogeneous electric fields. This arrangement gives reasonable optical access for the MOT and Rydberg excitation beams. Voltage differences between the inner and outer concentric electrodes ($\Delta V_{\text{grad}} = V_{t1}-V_{t2} = V_{b1}-V_{b2}$) control the field electric field gradient, $\partial_z E_z|_0 \approx \Delta V_{\text{grad}} / L^2$, whereas voltage differences between the top and bottom electrodes set the electric field at the centre of the cloud of atoms. With the central hole 10mm in diameter and the pairs of electrodes are separated in the $\hat{z}$ direction by $2L = 10mm$, field gradients on the order of $300e3V/m^2$ are generated by electrode voltage differences $\Delta V_{\text{grad}} \approx 10V$.
• Figure 5: Two interferometry sequences that are open in: (a) both final momentum and position (fully open) and (b) just final position (partially open). The dashed red line indicates the internal state $\ket{r}$, which accelerates in an inhomogeneous electric field, whereas the dotted green line indicates the internal state $\ket{g}$ which does not.