On physicality of electromagnetic potential from causal structure of flux quantization

TL;DR

The paper addresses whether electromagnetic potentials are physically real or merely mathematical tools, revisiting the Aharonov-Bohm debate and Vaidman’s loophole. It proposes a gedanken experiment with a superconducting ring encircling a distant solenoid, leveraging flux quantization and locality via time-dependent Ginzburg-Landau dynamics. Through a carefully staged sequence of flux initialization, dissipation, and rapid cooling, the authors argue that a detectable field signal in a zero-field region would be mediated by the vector potential, not by nonlocal fields. This framework highlights flux quantization as a platform to test potential physicality and outlines future work for quantitative predictions and AB-like extensions.

Abstract

Recent work by Vaidman [Phys. Rev. A 86,040101 (2012)] showed that Aharonov-Bohm effect can be explained in terms of local fields, thus effectively restating an old problem of physicality of potentials. In this work, we propose an argument demonstrating the physicality of electromagnetic potential (upon the assumption of locality) based on the causal structure in flux quantization setup. Crucially, we discuss the fundamental difference between the considered setup and the Aharonov-Bohm experiment that allows for avoiding Vaidman's loophole in our scenario.

Figures (5)

• Figure 1: a) Scheme of the setup for the gedanken experiment. After introducing the flux not equal to the integer multiple of the flux quantum to the solenoid, the superconducting ring (SR), initially prepared in the normal (non-superconducting) phase, dissipates the information about the field appearing during the initial flux generation into the environment. Then SR is coupled to a cooling system to induce phase-transition which results in appearance of supercurrent in response to magnetic potential. This current generates non-zero magnetic field which is then measured at anticipated moment in time determined by the time necessary for magnetic field to travel the distance from the SR to the detector. b) Aharonov-Bohm experiment with single electron. The imposed superposition of trajectories ($e_L,e_R$) of electron results in its entanglement with the magnetic field generated by the electron, which finally leads to the entanglement with solenoid: $({|{e_L,B\odot, S_L}\rangle}+{|{e_R,B\otimes,S_R}\rangle})/\sqrt{2}$. Here $B\odot,B\otimes$ represent states of the magnetic field generated by the moving electron and $S_{L,R}$ -- the states of the solenoid. This entanglement allows for formulating Vaidman's loophole.
• Figure 2: A space-time diagram of the crucial steps of our gedanken experiment. $r$ denotes the radial dimension of the system. Blue rectangle represents the cross-section of the solenoid, whereas orange rectangles represent cross-section of the ring. Blue solid lines represent light trajectories, which determine causal structure of the experiment. The experiment starts with the process of flux generation (purple rectangle G. F.), which includes turning on the current in the solenoid and is followed by dissipation of any fields induced in the ring by the initial impulse. The initial flux is not equal to the multiple of the flux quantum. The dissipation stage lasts sufficiently long in order to achieve a state of constant flux within the ring and lack of any electromagnetic fields propagating in between the ring and the solenoid, which is represented by the dots "..." . After the dissipation stage a phase transition to the superconducting state (P. T.) is induced in the ring. Since a superconducting ring can surround only a quantized flux, and the information about non-quantized flux is accessible to the ring solely via vector potential $\vec{A}$, a supercurrent is induced in the (now) superconducting ring, which forces flux quantization within the ring. Finally just after the phase transition in the ring, the signal due to the field induced by the supercurrent reaches the magnetic field detector (placed anywhere within the hatched green triangles F.D.) where it is measured. Detector's time window is chosen such that the only source of the signal after phase transition could be from the ring side. Then induction of the field in the system must have been caused by vector potential "informing" the superconducting ring that the flux surrounded by the ring is not properly quantized.
• Figure 3: Causal diagram representing proposed experiment. Blue rectangles represent events taking place within the solenoid, orange ones -- within the ring, and white ones - in between the ring and the solenoid. Green arrows represent the desired causal structure explaining observation of a field between ring and the solenoid, in which the existence of a vector potential is a necessary factor for inducing the super current compensating lack of flux quantization within the superconducting ring after the phase transition. Purple rectangles and red arrows represent possible loopholes in the experiment, namely causal explanations for observing magnetic field in between the ring and the solenoid, which do not demand the vector potential as a part of a causal explanation of the final effect. The first loophole due to initial generation of the flux is suppressed by dissipation process, leading to the stable flux (this is denoted by a dotted red arrow). Second loophole is due to possible field generation during the process of cooling which leads to phase transition. This could then provoke generation of response field from the solenoid to which one could try to attribute generation of the supercurrent. This loophole is rejected by appropriately designed temporal structure of the experiment, namely by performing field measurement precisely in the green-hatched spacetime region as shown in Fig. \ref{['fig:spacetime']}.
• Figure 4: (a) Average time interval $t$ needed to achieve the asymptotic regime for different radii $R$ of the ring while keeping constant ratio between the initial flux $\Phi_0$ and radius $R$. Radius is expressed in the units of penetration depth $\lambda$ whereas time is presented in units $\xi^2/D$, in which $\xi$ stands for coherence length and $D$ for diffusion coefficient. Results are obtained from 50 runs of simulation. Bars in the plot represent sample standard deviation. Clearly the time interval of achieving equilibration appears to be independent of the radius. Time interval needed for light to traverse one penetration depth is given by $D\lambda/c\xi^2$, with $c$ being the speed of light. This time can vary depending on the material and it's purity. For Niobium $\xi\approx\lambda\approx 4\cdot 10^{-8}$m and $D\sim10^{-4}$ for impure samples Diff and for pure samples $D$ can reach $D\sim10^{-1}$ (based on estimation from mean free path $810$nm diff2), and thus this time is of the order $10^{-5}-10^{-2}\xi^2/D$ depending on the purity. Thus, in this example, to obtain the effect before signal reaches solenoid one would need to choose a ring of radius of the order of $10^7-10^4\lambda$, which is even in the worst case scenario in the reasonable range of decimeters. (b) Time dependence of average dimensionless current density $\langle J\rangle$ integrated over the ring (see \ref{['app:simulation']} for details). Average was calculated over 50 runs of simulations. Blue curve was obtained for flux $\Phi_0=1000.2\, \Phi_Q$, whereas red curve for $\Phi_0=0$, and $R=1500\lambda$. Clearly nonzero flux results in organized current in the ring which however does not appear when there is no flux through the solenoid. While for this example $R$ is too small to guarantee that currents are big enough to be measurable before signal passes to solenoid, one should expect that this behavior extends also to wider rings. In \ref{['app:simulation']} we present such behavior using simulations which utilize additional approximations.
• Figure 5: a) Time interval $t$ needed to achieve the asymptotic regime for different radii $R$ of the ring while keeping constant ratio between the initial flux $\Phi_0$ and radius $\tilde{R}$ for calculations with no enough dense grid to capture full solution. Bars in the plot represent sample standard deviation. For this case the time interval of achieving equilibration is also independent of the radius as in the main text. b) Time dependence of average dimensionless current $\tilde{j}_s$ integrated over the ring. Blue curve stands for flux $\Phi_0=(10^7+0.2)\,\Phi_Q$, whereas red curve for $\Phi_0=0$ and $\tilde{R}=1.5*10^7$. Clearly nonzero flux results in organized current in the ring also for high value of $\tilde{R}$. The behavior for zero flux is also analogous to the simulation presented in the main text. Note that for this radius the time needed for light to pass to solenoid is of the order $10^2-10^5\xi^2/D$, and thus it is longer then our bound for the time needed for first non-negligible current to appear.