Aharonov–Bohm Flux Phase

Worked Example

Let us calculate the relative phase and normalized output intensity for the case where the confined magnetic flux is 0.25 quanta, the visibility of the interference is 1 (perfect coherence), and there is zero extra phase shift along the paths.

Flux quanta f = 0.25
Extra phase shift = 0
Visibility V = 1

Relative Phase (radians):
  phase = 2 * π * f + extra_phase
  phase = 2 * 3.141592653589793 * 0.25 + 0
  phase = 1.5707963267948966

Normalized Output Intensity:
  intensity = (1 + V * cos(phase)) / 2
  intensity = (1 + 1 * cos(1.5707963267948966)) / 2
  intensity = (1 + 0) / 2
  intensity = 0.5

Experiment

Use the Aharonov–Bohm Flux Phase module to interactively explore the quantum interference shift.

1. Set the Flux quanta control to exactly 0.5.
2. Observe that the Relative phase radians becomes approximately 3.14159 (π).
3. Verify that the Normalized output intensity drops to 0, demonstrating completely destructive interference.
4. Change the Visibility control to 0, which removes the interference effect, resulting in an intensity of 0.5 regardless of the flux.

Limitations

This theoretical model and interactive visualization are based on several idealizations. It assumes perfectly coherent one-dimensional particle paths (ideal field-free arms) and a perfectly shielded solenoid, such that the magnetic field is strictly zero everywhere along the electron trajectories. In reality, some finite fringing field might exist and the finite width of the electron beams can cause a spread in the acquired phase. The normalized port intensity calculation relies on these perfect coherence assumptions, which degrades in physical experiments due to inelastic scattering or partial shielding (Visibility < 1).

References

The physics of gauge invariance and the Aharonov-Bohm effect in solid state systems is explored in detail inDavid Tong's lecture notes on Solid State Physics, which supports the concept of phase accumulation purely from the vector potential without any local Lorentz force acting on the particles in the arms.