Electrodynamic Aharonov-Bohm effect

We propose an electrodynamic Aharonov-Bohm (AB) scheme where a nonzero AB phase difference appears even if the interferometer paths do not enclose a magnetic flux and are subjected to negligible scalar potential differences during the propagation of the quantum charged particle. In the proposal, the current in a solenoid outside the interferometer varies in time while the quantum particle is in a superposition state inside two Faraday cages, such that it is always subjected to negligible electromagnetic fields. At first glance, this result could challenge the topological nature of the AB effect. However, by considering the topology of the electromagnetic field configuration and the possible particle trajectories in spacetime, we demonstrate the topological nature of this situation.

Figure 1

Magnetic AB effect. A quantum charged particle has two possible paths in an interferometer that encloses an infinite solenoid. The AB phase difference between the paths depends on the solenoid magnetic flux, with the particle only propagating in regions with negligible electromagnetic fields.

Figure 2

Electric AB effect. (a) A quantum charged particle has two possible paths in an interferometer, each one with a Faraday cage. When the particle is in a superposition state inside both cages, their potentials $V_a$ and $V_b$ vary and come back to zero before each component of the wave function leaves the corresponding cage. The AB phase difference depends on the potential difference between the paths during the particle propagation. (b) Possible spatiotemporal trajectories ${\mathbf{x}}_a\left(t\right)$ and ${\mathbf{x}}_b\left(t\right)$ of the particle in the interferometer, in black. We are only showing the spatial $xy$ plane in the figure, assuming that the particle trajectories are in this plane. The time coordinate is represented in the vertical direction. The spacetime region with a nonzero electric field is represented in green. One of the possible spacetime surfaces whose boundaries are the trajectories ${\mathbf{x}}_a\left(t\right)$ and ${\mathbf{x}}_b\left(t\right)$ is represented in blue. (c) Construction of the spacetime surface $\mathcal{S}$, whose boundaries are the trajectories ${\mathbf{x}}_a\left(t\right)$ and ${\mathbf{x}}_b\left(t\right)$. This figure is in three-dimensional (3D) spatial coordinates. For each time $t$, we define a continuous curve $C_t$ in space that goes from the position ${\mathbf{x}}_a\left(t\right)$ to the position ${\mathbf{x}}_b\left(t\right)$ through infinitesimal displacements $d\mathbf{r}\left(t\right)$. The spacetime surface $\mathcal{S}$ is formed by connecting consecutive curves, as illustrated here between times $t$ and $t+dt$. The spatial component of the formed infinitesimal spacetime surface is also defined in the figure.

Figure 3

Proposed electrodynamic AB scheme. (a) A charged particle propagates through the interferometer, with two possible paths. While it is in a superposition state inside both Faraday cages, the solenoid current linearly changes. The AB phase difference between the paths in general is nonzero. (b) Possible spatiotemporal trajectories ${\mathbf{x}}_a\left(t\right)$ and ${\mathbf{x}}_b\left(t\right)$ of the particle in the interferometer, with the same representation as in Fig. 2. The spacetime region with a nonzero electric field is represented in green and the regions with nonzero magnetic field are in yellow and orange. One of the possible spacetime surfaces whose boundaries are the trajectories ${\mathbf{x}}_a\left(t\right)$ and ${\mathbf{x}}_b\left(t\right)$ is represented in blue. (c) The same as in (b), but with a different spacetime surface formed by a different deformation of one trajectory into the other.