4.2.3 Flux Ring#

Prompts

A charged particle on a ring threads a perpendicular magnetic flux. How does the energy spectrum depend on the flux strength? What makes this periodic?
How does the Aharonov-Bohm effect manifest in the tunneling rate across the ring? Can you calculate it explicitly?
What is a persistent current? Why does it flow in the ground state of a flux ring?
How does the pi-flux case (\(\Phi = \Phi_0/2\)) differ from other flux values? What symmetries protect the degeneracy?
What is a SQUID, and how does the flux ring model explain its operation and sensitivity?

Lecture Notes#

Overview#

A charged particle confined to a ring with a perpendicular magnetic flux provides a concrete, fully solvable model of the Aharonov-Bohm effect. The energy spectrum becomes periodic in flux, tunneling rates depend on the enclosed flux, and the ground state carries a persistent current—a surprising quantum phenomenon with no classical analog and direct applications in superconducting devices.

The Hamiltonian: Particle on a Ring#

Consider a particle of mass \(m\) and charge \(q\) confined to move on a ring of radius \(R\) in the \(xy\)-plane. A uniform magnetic field perpendicular to the ring creates an enclosed magnetic flux \(\Phi\) through the ring’s interior.

Using angular coordinate \(\theta\) (with \(0 \leq \theta < 2\pi\)), the kinetic energy involves the canonical angular momentum:

\[ \hat{L}_{\theta} = -\mathrm{i}\hbar\frac{\partial}{\partial\theta} \]

In the presence of the magnetic field, the canonical momentum is shifted by the vector potential contribution. Via the Peierls substitution, the Hamiltonian becomes:

The Flux Ring Hamiltonian

\[ \hat{H} = \frac{1}{2mR^2}\left(\hat{p}_{\theta} - \frac{q\Phi}{2\pi}\right)^2 \]

where \(\hat{p}_{\theta} = -\mathrm{i}\hbar\frac{\partial}{\partial\theta}\) and \(\Phi\) is the total magnetic flux threading the ring. The term \(q\Phi/(2\pi)\) is the flux-dependent shift in angular momentum, expressed in dimensionless units.

Intuition: The magnetic field creates an effective “boost” to the angular momentum of the particle. A particle moving in one direction sees the flux-induced potential aiding or opposing its motion.

Energy Eigenvalues: Periodic in Flux Quantum#

Since the Hamiltonian commutes with \(\hat{p}_{\theta}\), eigenstates are momentum eigenstates:

\[ \psi_n(\theta) = \frac{1}{\sqrt{2\pi}}\mathrm{e}^{\mathrm{i}n\theta} \]

where \(n \in \mathbb{Z}\) (quantization from periodicity: \(\psi(\theta + 2\pi) = \psi(\theta)\)).

The eigenvalues are:

Flux Ring Energy Spectrum

\[ E_n(\Phi) = \frac{\hbar^2}{2mR^2}\left(n - \frac{\Phi}{\Phi_0}\right)^2 \]

where \(\Phi_0 = \frac{h}{q}\) is the flux quantum (or \(\Phi_0 = h/(2e)\) for electron pairs in superconductivity).

Key feature: The spectrum is periodic in flux with period \(\Phi_0\):

\[ E_n(\Phi + \Phi_0) = E_n(\Phi) \]

Derivation: Setting \(\hat{p}_{\theta}\psi_n = \hbar n\psi_n\) and inserting into the Hamiltonian:

\[ \hat{H}\psi_n = \frac{\hbar^2}{2mR^2}\left(n - \frac{\Phi}{2\pi\hbar}\right)^2\psi_n = \frac{\hbar^2}{2mR^2}\left(n - \frac{\Phi}{\Phi_0}\right)^2\psi_n \]

The periodicity follows because the energy depends only on \(n - \Phi/\Phi_0\), and shifting \(\Phi \to \Phi + \Phi_0\) is equivalent to shifting \(n \to n-1\), which relabels the same set of energy levels.

Ground State and Persistent Current#

The ground state is the lowest-energy level. Since energies are minimized when \(|n - \Phi/\Phi_0|\) is smallest, the ground state quantum number shifts with flux:

For \(0 \leq \Phi/\Phi_0 \leq 1\): ground state is approximately \(n = 0\) if \(\Phi/\Phi_0 < 1/2\), or \(n = 1\) if \(\Phi/\Phi_0 > 1/2\).

The persistent current is the expectation value of the angular momentum current:

\[ I(\Phi) = -\frac{\partial E_0}{\partial\Phi} \]

Persistent Current in the Flux Ring

For \(\Phi/\Phi_0 = \delta\) with \(0 < \delta < 1\) and ground state \(n=0\):

\[ I(\delta) = -\frac{\hbar^2}{mR^2\Phi_0}\left(0 - \delta\right) = \frac{\hbar^2}{mR^2\Phi_0}\delta \]

The current flows continuously in the ground state, even at zero temperature and with no applied voltage. This current reverses sign when the flux crosses \(\Phi_0/2\).

Remarkable consequence: In a superconducting ring (where resistance vanishes), this persistent current is truly permanent and can be detected via the magnetic field it generates.

Discussion: The Persistent Current Puzzle

The persistent current seems to violate the classical intuition that a ground state should be static. Yet here the ground state carries a current that flows indefinitely.

(a) In classical mechanics, a particle in a potential well at rest has zero current. Why does quantum mechanics allow the ground state to have a nonzero current?

(b) The persistent current is proportional to \(\partial E_0/\partial\Phi\). What is the physical meaning of this derivative? Why does increasing the flux change the ground state energy?

(c) Suppose you insert the flux adiabatically (very slowly) into the ring, starting from \(\Phi = 0\). How does the particle respond? Does the persistent current build up gradually, or does it jump suddenly?

Tunneling Across the Ring: Flux Dependence#

Now suppose the ring has a potential barrier that allows tunneling from one side to the other. The tunneling rate depends on the phase accumulated along different paths.

For a path that goes halfway around the ring and acquires a flux-dependent phase:

\[ \phi_{\text{path}} = \frac{q\Phi}{2\hbar c} \]

The tunneling amplitude oscillates with flux, reaching maximum when \(\Phi = m\Phi_0\) (integer multiples) and minimum when \(\Phi = (m+1/2)\Phi_0\).

Aharonov-Bohm Effect in the Flux Ring

The tunneling rate \(\Gamma(\Phi) = \Gamma_0|\cos(\pi\Phi/\Phi_0)|\) oscillates with period \(\Phi_0\). This direct observation of flux modulation is a hallmark AB effect: the particle’s dynamics depend on the enclosed flux even though the particle never enters the field region.

Special Case: Pi-Flux (\(\Phi = \Phi_0/2\))#

At half a flux quantum, a remarkable symmetry emerges.

Degeneracy at Pi-Flux

When \(\Phi = \Phi_0/2\):

\[ E_n\left(\frac{\Phi_0}{2}\right) = \frac{\hbar^2}{2mR^2}\left(n - \frac{1}{2}\right)^2 \]

The ground state is doubly degenerate: both \(n=0\) and \(n=1\) have the same energy.

This degeneracy is protected by a combination of time-reversal symmetry (the Hamiltonian is invariant under \(\theta \to -\theta\) up to a relabeling) and translational symmetry (shifts in \(\theta\) by \(\pi\) interchange the two states).

Physically, at pi-flux, the effective “momentum kick” from the field is exactly half a quantum, and the quantum numbers \(n\) and \(n+1\) experience equal and opposite shifts, leading to a crossing in the spectrum.

Why the Degeneracy Cannot Be Lifted

At \(\Phi = \Phi_0/2\), any perturbation that breaks the time-reversal or translational symmetry of the ring (e.g., a local potential or asymmetry) can lift the degeneracy. However, within the ideal symmetric ring, the degeneracy is protected by symmetry and cannot be removed by continuous deformations of the Hamiltonian that preserve those symmetries.

This is an example of a quantum anomaly: a degeneracy that survives because the symmetries form an anomaly-free representation. Lifting it requires explicitly breaking a fundamental symmetry.

Application: The SQUID (Superconducting Quantum Interference Device)#

A SQUID consists of two Josephson junctions (superconductor-insulator-superconductor tunneling devices) connected to form a loop—essentially a flux ring with adjustable tunneling rates.

Key principle: The interference pattern of Cooper pair tunneling depends on the enclosed magnetic flux. By measuring the voltage across the SQUID as a function of applied flux, one can:

Detect tiny magnetic fields (flux sensitivity down to \(\sim 10^{-15}\) Wb, or \(\sim 10^{-6}\Phi_0\)).
Measure flux quantization directly.
Act as a quantum bit in superconducting quantum computers (fluxonium qubits).

SQUID Operation and Sensitivity

Setup: Two identical Josephson junctions in a loop, each with critical current \(I_c\). A magnetic flux \(\Phi\) threads the loop.

Measurement: The maximum supercurrent \(I_{\text{max}}(\Phi)\) that the loop can carry is:

\[ I_{\text{max}}(\Phi) = 2I_c\left|\cos\left(\frac{\pi\Phi}{\Phi_0}\right)\right| \]

The current oscillates sinusoidally with flux, with period \(\Phi_0\). By measuring the voltage (which is zero below the critical current), one can infer the enclosed flux with extraordinary precision.

Sensitivity: The slope \(\partial I_{\text{max}}/\partial\Phi\) is steepest near \(\Phi = (m+1/2)\Phi_0\), allowing flux changes of order \(10^{-3}\Phi_0\) to be detected with careful lock-in techniques.

Summary#

The flux ring model is a fully solvable system where a charged particle on a ring is threaded by a perpendicular magnetic flux \(\Phi\).
The energy spectrum is periodic in the flux quantum \(\Phi_0 = h/q\) and depends on the quantity \((n - \Phi/\Phi_0)^2\).
A persistent current flows in the ground state, proportional to \(\partial E_0/\partial\Phi\), reflecting the quantum mechanical response to the enclosed flux.
Tunneling rates modulate with flux, directly manifesting the Aharonov-Bohm effect in a calculable system.
At pi-flux (\(\Phi = \Phi_0/2\)), the ground state becomes doubly degenerate, protected by time-reversal and translational symmetry.
SQUIDs exploit flux quantization and interference to detect tiny magnetic fields, achieving sensitivities of order \(10^{-6}\Phi_0\) and enabling quantum computing applications.

Homework#

1. A particle of charge \(q\) and mass \(m\) is confined to a ring of radius \(R\) with perpendicular magnetic flux \(\Phi\). The Hamiltonian is:

\[ \hat{H} = \frac{1}{2mR^2}\left(\hat{p}_{\theta} - \frac{q\Phi}{2\pi}\right)^2 \]

(a) Solve the time-independent Schrödinger equation and write the normalized eigenstates \(\psi_n(\theta)\) and eigenvalues \(E_n(\Phi)\).

(b) Show that the spectrum is periodic in flux: \(E_n(\Phi + \Phi_0) = E_n(\Phi)\) where \(\Phi_0 = h/q\).

(c) For an electron (\(q = -e\)), plot the ground state energy \(E_0(\Phi)\) as a function of \(\Phi/\Phi_0\) for \(0 \leq \Phi/\Phi_0 \leq 2\). Identify the ground state quantum number \(n\) in each region.

2. Persistent Current: The persistent current is defined as \(I(\Phi) = -\frac{\partial E_0}{\partial\Phi}\), the negative gradient of ground state energy with respect to flux.

(a) For an electron on a ring with \(\Phi/\Phi_0 = 0.3\), which quantum number \(n\) is the ground state?

(b) Calculate the persistent current \(I(0.3\Phi_0)\) in terms of \(\hbar\), \(m\), \(R\), and \(e\). Does it flow clockwise or counterclockwise?

(d) Sketch \(I(\Phi)\) as a function of \(\Phi/\Phi_0\) for \(0 \leq \Phi/\Phi_0 \leq 2\).

3. Pi-Flux Degeneracy: At \(\Phi = \Phi_0/2\), the energy spectrum exhibits a special degeneracy.

(a) Show that at pi-flux, the quantum numbers \(n=0\) and \(n=1\) have the same energy. What is that energy (in terms of \(\hbar^2/(2mR^2)\))?

(b) Explain physically why the degeneracy occurs: why do these two angular momentum states become degenerate at precisely this flux value?

(c) Is the degeneracy protected by symmetry? Identify which symmetries of the Hamiltonian prevent the degeneracy from being lifted by small perturbations at \(\Phi = \Phi_0/2\).

4. Flux Quantization and Periodicity: The energy periodicity in flux is a signature of the Aharonov-Bohm effect.

(a) Explain why the energy satisfies \(E_n(\Phi + \Phi_0) = E_n(\Phi)\) in terms of gauge invariance and wavefunction single-valuedness.

(b) Why is the period exactly \(\Phi_0 = h/q\) and not some other value? Derive this from the periodicity condition on the wavefunction.

(c) For a particle with charge \(2e\) (like a Cooper pair), what is the flux quantum \(\Phi_0'\)? How does the energy spectrum change?

5. Tunneling and the AB Effect: A potential barrier of height \(V_0\) at \(\theta = 0\) allows the particle to tunnel across the ring. The tunneling amplitude is approximately:

\[ \mathcal{T}(\Phi) \propto |\langle 0|e^{-\mathrm{i}\pi\Phi/\Phi_0}|1\rangle| = \left|\cos\left(\frac{\pi\Phi}{\Phi_0}\right)\right| \]

(a) At what flux values \(\Phi\) is tunneling maximized? Minimized?

(b) Explain physically why the tunneling amplitude depends on the enclosed flux \(\Phi\) even though the flux does not touch the particle directly (in the region where tunneling occurs).

(c) Sketch the tunneling rate \(\Gamma(\Phi) \propto |\mathcal{T}(\Phi)|^2\) as a function of \(\Phi/\Phi_0\) for \(0 \leq \Phi/\Phi_0 \leq 2\).

6. SQUID Critical Current: A superconducting loop contains two Josephson junctions, each with critical current \(I_c\). The loop encloses a magnetic flux \(\Phi\). The maximum supercurrent the loop can carry is:

\[ I_{\text{max}}(\Phi) = 2I_c\left|\cos\left(\frac{\pi\Phi}{\Phi_0}\right)\right| \]

(a) What is the voltage across the SQUID at zero applied current? What happens if you exceed \(I_{\text{max}}(\Phi)\)?

(b) By what fraction does the critical current change if the flux changes by \(\Delta\Phi = 10^{-2}\Phi_0\)? (Evaluate the sensitivity near \(\Phi = 0\).)

7. Ground State Phase Transition at Pi-Flux: Near \(\Phi = \Phi_0/2\), the ground state is nearly degenerate between \(n=0\) and \(n=1\).

(a) For \(\Phi = \Phi_0/2 + \delta\) with small \(|\delta|\), write the ground state energy difference \(\Delta E = E_1 - E_0\) as a function of \(\delta\).

(b) At what \(\delta\) do the two levels cross and become exactly degenerate?

(c) If you slowly vary the flux through the pi-flux point, starting in the ground state, do you remain in a well-defined ground state, or do you transition between \(n=0\) and \(n=1\)? Explain using adiabatic theorem concepts.

8. Flux Quantization in a Superconductor: A superconducting ring has persistent current \(I(\Phi)\) flowing in its ground state. The self-inductance of the ring is \(L\).

(a) The magnetic flux from the persistent current creates a self-induced flux \(\Phi_{\text{self}} = LI(\Phi)\). Write a self-consistency equation for the total flux \(\Phi_{\text{total}} = \Phi_{\text{applied}} + \Phi_{\text{self}}\).

(b) Show that the only stable values of \(\Phi_{\text{total}}\) are integer multiples of \(\Phi_0\), confirming flux quantization.