Test of the He-McKellar-Wilkens topological phase by atom interferometry. I. Theoretical discussion

We have recently tested the topological phase predicted by He and McKellar and by Wilkens: this phase appears when an electric dipole propagates in a transverse magnetic field. In the present paper, we first recall the physical origin of this phase and its relations to the Aharonov-Bohm and Aharonov-Casher phases. We then explain possible detection schemes and we briefly describe the lithium atom interferometer we have used for this purpose. Finally, we analyze in great detail the phase shifts induced by electric and magnetic fields acting on such an interferometer, taking into account experimental defects. The experiment and its results are described in the following paper.

Figure 1

Connection between the HMW and AC phases by electric-magnetic duality. The HMW phase arises when an electric dipole moment $\mathbf{d}$ propagates in the radial magnetic field created by a line of magnetic monopoles $q_m$ while the AC phase (between parentheses) appears when a magnetic dipole $\mu$ encircles a line of electric charges $q_e$.

Figure 2

Connection between the HMW and the AB phases following Wei et al. [70]. The interferometer arms (blue full lines) encircle an infinite charged wire (red vertical cylinder) which produces a radial electric field $\mathbf{E}$ and induces an electric dipole $\mathbf{d}$ represented by a positive charge ($+$, large bullet) and a negative charge ($-$, small bullet). Each charge undergoes the AB effect in the uniform magnetic field $\mathbf{B}$. The HMW phase is equal to the sum of the two AB phases and it is proportional to the magnetic flux through the (blue) shaded surface.

Figure 3

Geometry of the electric field for the detection of the HMW phase, following the proposal of Wei et al. [70]. The atom (blue dots) propagates in the interferometer plane, with the homogeneous magnetic field $\mathbf{B}$ perpendicular to this plane. The conductors are shown with their potential (red or black if grounded) and the electric field vector $\mathbf{E}\left(\mathbf{r}\right)$ is represented by red arrows for some sample positions along the interferometer arms. Upper panel (similar to Fig. 2): the original proposal with a charged wire which produces a strongly inhomogeneous electric field. Lower panel: our geometry with homogeneous electric fields produced by plane capacitors.

Figure 4

Schematic top view of our atom interferometer: the HMW interaction region is placed just before the second laser standing wave, at the place where the arm separation is largest. It is thus possible to introduce a septum between the two interferometer arms without any perturbation of the interferometer signal.

Figure 5

Hyperfine Zeeman energies $E(F,m_F)/A$ plotted as a function of $X=-(g_S-g_I){\mu }_{B}B/\left(2A\right)$. It appears clearly that there are four pairs of levels with almost opposite Zeeman energy shifts.

Figure 6

Calculated relative visibility and phase as functions of $J_1$, for a velocity distribution with a typical value of the parallel speed ratio $S_{\parallel }=8$ and for three values of the population imbalance parameter $\chi$: $\chi =0$, full curve; $\chi =0.1$, dash-dotted curve; $\chi =-0.1$, dotted curve. This parameter has a large effect especially when the visibility is very low.

Figure 7

Calculated real and imaginary parts of the complex visibility $\underline{\mathcal{V}}/{\mathcal{V}}_0$ as functions of the coil current $I$. We use the values $J_1/I=0.5$ rad/A and $J_2/I^2=0.01$ rad/A${}^2$, which are close to our experimental values. The $S_{\parallel }$ and $\chi$ values are the same as in Fig. 6.

Figure 8

The population imbalance parameter $\chi$ as a function of laser frequency detuning ${\delta }_L/\left(2\pi \right)$ GHz as given by the simple model [Eq. (A3)]. The $\beta$ parameter has been taken equal to $\beta /\left(2\pi \right)=3.65$ GHz, which is optimum for a detuning ${\delta }_L/\left(2\pi \right)=2$ GHz.