Case Study 2: The Aharonov-Bohm Effect

Overview

In 1959, Yakir Aharonov and David Bohm proposed one of the most counterintuitive predictions in quantum mechanics: a charged particle can be physically affected by an electromagnetic potential even in a region where the electromagnetic fields are identically zero. This prediction — the Aharonov-Bohm (AB) effect — struck at the heart of the classical worldview, in which only fields (not potentials) are physically real. After decades of increasingly refined experiments, the effect is now established beyond doubt, and it serves as one of the clearest demonstrations that quantum mechanics assigns physical reality to gauge potentials.

In this case study, we trace the AB effect from its theoretical prediction through its experimental confirmation and connect it to the Berry phase framework developed in Chapter 32.

Part 1: The Theoretical Prediction

The Classical View of Potentials

In classical electrodynamics, the electric field $\mathbf{E}$ and magnetic field $\mathbf{B}$ are the fundamental physical quantities. The scalar potential $\phi$ and vector potential $\mathbf{A}$ are introduced as mathematical conveniences:

$$\mathbf{B} = \nabla \times \mathbf{A}, \quad \mathbf{E} = -\nabla\phi - \frac{\partial\mathbf{A}}{\partial t}.$$

Because $\mathbf{A}$ is not unique — any gauge transformation $\mathbf{A} \to \mathbf{A} + \nabla\chi$ leaves $\mathbf{B}$ unchanged — classical physicists reasonably concluded that $\mathbf{A}$ has no independent physical meaning. Only the gauge-invariant fields $\mathbf{E}$ and $\mathbf{B}$ are "real."

The Lorentz force law reinforces this view:

$$\mathbf{F} = q(\mathbf{E} + \mathbf{v} \times \mathbf{B}).$$

If $\mathbf{E} = 0$ and $\mathbf{B} = 0$ in the region where the particle moves, there is no force — and classically, there is no effect.

Aharonov and Bohm's Gedankenexperiment

Aharonov and Bohm proposed the following experiment:

1. An infinitely long solenoid carries a steady current, producing a uniform magnetic field $\mathbf{B}$ inside and $\mathbf{B} = 0$ outside.

2. The vector potential outside the solenoid is not zero. In cylindrical coordinates, for a solenoid of radius $R$ and flux $\Phi$:

$$\mathbf{A} = \frac{\Phi}{2\pi r}\hat{\boldsymbol{\phi}}, \quad r > R.$$

Note: $\nabla \times \mathbf{A} = 0$ for $r > R$, confirming $\mathbf{B} = 0$ outside. But $\oint \mathbf{A} \cdot d\mathbf{l} = \Phi \neq 0$ for any loop encircling the solenoid.

3. An electron beam is split into two paths that pass on opposite sides of the solenoid and recombine at a detector screen.

4. The two paths enclose the solenoid, so the enclosed flux is $\Phi$. The difference in phase between the two paths is:

$$\Delta\phi = \frac{e}{\hbar}\oint \mathbf{A} \cdot d\mathbf{l} = \frac{e\Phi}{\hbar}.$$

This phase difference shifts the interference pattern on the detector screen. The shift depends on the flux $\Phi$ inside the solenoid, which the electrons never visit.

Why This Is Shocking

The AB effect says: - The electrons are influenced by a region of space they never enter. - The influence is mediated by the vector potential $\mathbf{A}$, which is nonzero where the electrons are, even though $\mathbf{B} = 0$. - The effect is intrinsically quantum — it involves interference and has no classical analogue.

This challenges the locality principle: how can a particle be affected by a field configuration in a region it does not visit? The resolution is that quantum mechanics is non-local in a specific sense: the phase of the wave function is affected by the topology of the accessible region (the first homotopy group of the space with the solenoid removed), and this topology "knows" about the enclosed flux.

The Phase as a Berry Phase

As shown in Section 32.5 of the main chapter, the AB phase is a Berry phase. The parameter space is the position $\mathbf{R}$ of the electron, the Berry connection is $\mathcal{A}_i = (e/\hbar)A_i$, and the Berry phase around the loop is:

$$\gamma = \frac{e}{\hbar}\oint \mathbf{A} \cdot d\mathbf{R} = \frac{e\Phi}{\hbar}.$$

This identification shows that the AB effect is not an isolated curiosity but part of a universal geometric structure in quantum mechanics.

Part 2: The Experimental Confirmation

Early Experiments (1960s)

The first experiments testing the AB effect were performed by Chambers (1960) and Mollenstedt and Bayh (1962) using electron biprism interferometers and tiny magnetized iron whiskers. These experiments observed interference fringe shifts consistent with the AB prediction, but they were not conclusive because the magnetic field was not perfectly confined — some leakage field could account for the shifts.

The Tonomura Experiments (1980s)

The definitive experiments were performed by Akira Tonomura and collaborators at Hitachi, using electron holography and a revolutionary trick to confine the magnetic flux.

The key innovation: Tonomura used tiny toroidal magnets, covered by a superconducting niobium layer. The superconductor (exploiting the Meissner effect) perfectly confined the magnetic field to the interior of the toroid, with $\mathbf{B} = 0$ everywhere outside — rigorously, not just approximately. Furthermore, the quantization of magnetic flux in a superconductor meant that $\Phi$ was known exactly: $\Phi = n\Phi_0/2$, where $\Phi_0 = h/e$ is the flux quantum.

The experiment (1986): 1. An electron beam from a field-emission source was directed at the toroidal magnet. 2. The beam split into components passing through the hole and around the outside of the toroid. 3. The components recombined and produced an interference pattern recorded by electron holography. 4. For $\Phi = 0$ (zero flux), the interference pattern was symmetric. 5. For $\Phi = \Phi_0/2$ (half a flux quantum), the interference pattern shifted by exactly half a fringe.

The results were unambiguous. The fringe shift was exactly $\Delta\phi = e\Phi/\hbar = \pi$ for half a flux quantum, in perfect agreement with the AB prediction. No magnetic field leakage could explain the result because the superconductor guaranteed $\mathbf{B} = 0$ outside.

Mesoscopic Ring Experiments

In 1985, Webb, Washburn, Umbach, and Laibowitz observed Aharonov-Bohm oscillations in the electrical resistance of a gold ring (diameter $\sim 800$ nm) at low temperatures. As an external magnetic field was swept, the resistance oscillated with a period corresponding to one flux quantum $\Phi_0 = h/e$ threading the ring. This was the first observation of the AB effect in a solid-state system and opened the field of mesoscopic quantum transport.

Precision and Modern Experiments

Modern experiments have confirmed the AB effect to extraordinary precision:

• Bachtold et al. (1999): AB oscillations in carbon nanotubes, with period $h/e$ as predicted.
• Caprez et al. (2007): Demonstrated the AB effect with a new geometry (electron beams passing through a gap in a charged capacitor), confirming the electric AB effect.
• Electrostatic AB effect: There is also an electric analog: a time-dependent scalar potential $\phi(t)$, applied only during a period when the particle is in a field-free region, shifts the phase. This has been observed but is less famous than the magnetic version.

Part 3: Theoretical Implications

The Physical Reality of Gauge Potentials

The AB effect demonstrated that gauge potentials cannot be dismissed as mere mathematical bookkeeping. In quantum mechanics, the potential $\mathbf{A}$ enters the Hamiltonian directly through the minimal coupling prescription $\hat{\mathbf{p}} \to \hat{\mathbf{p}} - e\mathbf{A}$. The particle "feels" $\mathbf{A}$, not just $\mathbf{B}$.

However, this does not mean $\mathbf{A}$ at a single point is observable. Only the gauge-invariant quantities — the circulation $\oint \mathbf{A} \cdot d\mathbf{l}$ around closed loops — are physically meaningful. This is exactly the Berry phase, which is gauge-invariant for closed loops but gauge-dependent for open paths.

Non-Locality and Topology

The AB effect is often described as "non-local" because the particle is affected by flux it never encounters. A more precise statement is that the AB effect is topological: it depends on the topology of the region accessible to the particle. The region outside a solenoid is not simply connected (it has a "hole"), and the fundamental group $\pi_1$ of this region classifies the possible AB phases.

If the region were simply connected (no holes), the AB phase would be zero regardless of the vector potential (because $\oint \mathbf{A} \cdot d\mathbf{l} = \iint (\nabla \times \mathbf{A}) \cdot d\mathbf{S} = \iint \mathbf{B} \cdot d\mathbf{S} = 0$ by Stokes' theorem, since $\mathbf{B} = 0$ everywhere in the region). The non-trivial topology is essential.

Quantization of Flux

In a superconducting ring, the magnetic flux is quantized: $\Phi = n\Phi_0/2$, where $\Phi_0 = h/e = 4.136 \times 10^{-15}$ Wb. This can be understood as a requirement that the AB phase for a Cooper pair (charge $2e$) encircling the ring be a multiple of $2\pi$:

$$\frac{2e\Phi}{\hbar} = 2\pi n \quad \Rightarrow \quad \Phi = n\frac{h}{2e} = n\frac{\Phi_0}{2}.$$

The flux quantum $\Phi_0/2$ is one of the most precisely measured quantities in physics and is used to define voltage standards.

Part 4: The Electric Aharonov-Bohm Effect

Prediction

Aharonov and Bohm also predicted an electric analogue: if a charged particle passes through a region where there is a time-dependent scalar potential $\phi(t)$ but zero electric field ($\mathbf{E} = -\nabla\phi = 0$ requires $\phi$ to be spatially uniform), the particle acquires a phase:

$$\Delta\phi = -\frac{e}{\hbar}\int \phi(t)\,dt.$$

If two paths experience different potentials $\phi_1(t)$ and $\phi_2(t)$ (applied during the time the particle is inside a Faraday cage where $\mathbf{E} = 0$), the interference pattern shifts by:

$$\Delta\phi = \frac{e}{\hbar}\int [\phi_2(t) - \phi_1(t)]\,dt.$$

The Conceptual Significance

The electric AB effect is arguably even more fundamental than the magnetic one. It shows that the scalar potential $\phi$ — not just $\mathbf{E}$ — has physical consequences in quantum mechanics, even in regions where $\mathbf{E} = 0$.

Together, the magnetic and electric AB effects show that the full four-potential $A^\mu = (\phi, \mathbf{A})$ is the fundamental quantity in quantum electrodynamics, not the field strength tensor $F^{\mu\nu}$. The gauge potential is physically real; only its gauge-dependent part is unphysical.

Part 5: The AB Effect in Modern Technology

Electron Holography

Tonomura's electron holography technique, developed to test the AB effect, has become a powerful tool in materials science. Electron holography can image magnetic domain structures, electric potential distributions in semiconductors, and charge distributions in biological molecules with nanometer resolution. The AB phase shift is the signal that is measured.

SQUID Magnetometers

The Superconducting Quantum Interference Device (SQUID) is essentially an AB interferometer for Cooper pairs. It consists of a superconducting loop interrupted by one or two Josephson junctions. The critical current of the device oscillates as a function of the enclosed flux with period $\Phi_0/2$, providing the most sensitive magnetometers ever built (sensitivity $\sim 10^{-15}$ T).

SQUIDs are used in: - Magnetoencephalography (imaging brain magnetic fields) - Geological surveying - Materials characterization - Fundamental physics experiments (gravitational wave detection, axion searches)

Quantum Computing

The AB effect is used in flux qubits — superconducting loops where the quantum states correspond to clockwise and counterclockwise persistent currents. The energy level splitting depends on the AB phase (the flux through the loop), which can be tuned by an external magnetic field. This tunability is essential for implementing quantum gates.

Discussion Questions

1. The AB effect shows that $\mathbf{B} = 0$ in a region does not mean "nothing quantum-mechanical is happening." What does this tell us about the concept of "local physical reality"? Is the AB effect truly non-local, or is it better described as topological?

2. The AB effect is sometimes used to argue that the vector potential $\mathbf{A}$ is "more fundamental" than the magnetic field $\mathbf{B}$ in quantum mechanics. Others argue that only gauge-invariant quantities (like the enclosed flux $\Phi$) are physical. Who is right? Can both be right?

3. If you could place a perfect detector inside the solenoid to determine which side of the solenoid each electron passes, would the AB effect still be observed? Why or why not? (Hint: think about which-path information and interference.)

4. The AB effect requires a multiply-connected region (space with a hole). In a simply connected region, the phase can always be gauged away. What does this suggest about the relationship between topology and quantum observability?

5. Compare the AB effect with the Berry phase for a spin-1/2 particle in a rotating magnetic field. What are the parameter spaces, Berry connections, and Berry curvatures in each case? How are they similar and different?

Quantitative Summary

Further Exploration

• The original paper: Y. Aharonov and D. Bohm, "Significance of Electromagnetic Potentials in the Quantum Theory," Physical Review 115, 485 (1959).

• Tonomura's definitive experiment: A. Tonomura et al., "Evidence for Aharonov-Bohm effect with magnetic field completely shielded from electron wave," Phys. Rev. Lett. 56, 792 (1986).

• For the connection to topology: M. Peshkin and A. Tonomura, The Aharonov-Bohm Effect, Springer (1989).

• For the AB effect in mesoscopic physics: S. Washburn and R. A. Webb, "Aharonov-Bohm effect in normal metal quantum coherence and transport," Adv. Phys. 35, 375 (1986).