Initial State
Upper Arm Rotation
Lower Arm Rotation
Recombination
Detection Probabilities
Interference Visibility
V = 0.00
V = 1: perfect interference
V = 0: no interference
V = 0: no interference
Send spin-½ particles through Stern-Gerlach magnets—rotate and recombine beams to see quantum interference
The spin interferometry experiment combines the famous Stern-Gerlach experiment with quantum interference. A spin-1/2 particle enters a Stern-Gerlach apparatus that splits it into two spatial paths based on spin orientation. We then apply controlled rotations to the spin in each arm before recombining the paths. This reveals the profound connection between spin rotation, quantum phase, and interference.
The visualization shows a spin particle traveling from left to right. The first Stern-Gerlach magnet (SG1) splits the beam into upper and lower paths corresponding to spin-up |↑⟩ and spin-down |↓⟩ states. In each arm, magnetic fields can rotate the spin by controllable angles θ₁ (upper) and θ₂ (lower). When recombination is enabled, a second Stern-Gerlach magnet (SG2) merges the paths and measures in the original basis.
The Bloch sphere on the right shows the quantum state in each arm. The spin-1/2 system is a qubit: |↑⟩ is the north pole, |↓⟩ is the south pole. Rotations appear as trajectories on the sphere surface. The color of each path indicates the relative quantum phase accumulated during rotation.
When recombination is enabled, the two paths interfere quantum mechanically. The final detection probabilities depend on the relative phase difference (θ₁ - θ₂). This isn't classical probability—the amplitudes add, and their squares give probabilities. You can get constructive or destructive interference depending on the rotation angles.
Spin-1/2 particles (like electrons, neutrons, or silver atoms) are two-level quantum systems. Rotations around the z-axis are generated by σ_z, around x by σ_x, and around y by σ_y. A rotation by angle θ around axis n̂ is R(θ, n̂) = exp(-iθσ·n̂/2). The factor of 1/2 means you need to rotate by 4π (not 2π) to return to the starting state— this is the famous spinor property of spin-1/2.
A Stern-Gerlach apparatus measures spin in a chosen direction (here, the z-direction). It physically separates |↑⟩ and |↓⟩ components into distinct spatial paths. Crucially, if you don't observe which path the particle took, the quantum coherence is preserved. The particle is in a spatial superposition, and the two paths can later interfere.
The interference visibility V quantifies how much the probabilities oscillate with changing phase. With no recombination, V = 0 (you simply see independent paths). With recombination of a pure superposition, V can reach 1 (perfect interference fringes). The visibility is fundamentally limited by the coherence of the initial state and any which-path information.
No Rotation, No Recombination: Start with |+⟩ = (|↑⟩ + |↓⟩)/√2. With both rotations at zero and recombination off, you see 50% probability in each path—the Stern-Gerlach apparatus has separated the superposition into spatial components.
Enable Recombination: Now enable recombination with θ₁ = θ₂ = 0. The paths merge coherently, and you should see the original state recovered. If you started with |+⟩, you get back to a symmetric superposition.
Rotate Only Upper Arm: Set θ₁ = π and θ₂ = 0, with recombination on. The upper arm flips its spin: |↑⟩ → |↓⟩. This introduces a relative phase shift. Watch how the interference pattern changes. The visibility oscillates as you change θ₁.
Equal and Opposite Rotations: Set θ₁ = π and θ₂ = -π (or 5.28 rad). Both arms rotate, but in opposite directions. The relative phase difference is 2π. What happens? You should see the same result as no rotation—2π is a full cycle. But try θ₁ = π, θ₂ = 0 and compare.
Scan for Interference Fringes: Start with |+⟩, enable recombination, fix θ₂ = 0, and slowly sweep θ₁ from 0 to 2π. Watch the detection probabilities oscillate. This is quantum interference—the hallmark of superposition and coherence. The visibility V tells you how "quantum" the interference is.
Spinor Behavior: Start with |↑⟩. Set θ₁ = 2π (a full rotation) and θ₂ = 0. The upper arm returns to |↑⟩ geometrically, but with a quantum phase of -1 (spinors pick up a minus sign under 2π rotation). With recombination on, this minus sign creates destructive interference. This is not a classical effect!
Build a Spin Echo: Start with |+⟩. Apply θ₁ = π/2 to the upper arm only. Then apply θ₂ = -π/2 to the lower arm. The rotations partially cancel when recombined. This is the principle behind spin echo techniques used in NMR and quantum computing to fight decoherence.
Spin interferometry is more than an abstract demonstration—it's the foundation of technologies like:
Every time you see interference fringes in this toy, you're witnessing the same quantum mechanics that powers these technologies. The mathematics of spin-1/2 is identical to the mathematics of qubits, photon polarization, and any two-level quantum system. Master this, and you understand the building blocks of the quantum world.