3. Physics

A lot of laboratory physics is a phase measurement. A spin sits in a magnetic field and precesses. An atomic clock lets that precession run for a known time. A neutron or atom interferometer splits a path, lets one arm pick up a phase, and recombines the paths. The circuit is the same shape every time.

The question

A phase $\theta$ was written onto a qubit. What is $\theta$?

You cannot read a phase by measuring the qubit in the state the phase was applied to. The interference lesson is why: a second Hadamard turns the hidden phase into a probability.

The quantum idea

Start from $|0\rangle$. The first Hadamard splits the qubit into two amplitudes. $\mathrm{RZ}(\theta)$ multiplies one of them by a phase and leaves the other alone, which is the precession, or the extra path length, or the extra time in the clock. The second Hadamard recombines the amplitudes. The probability of outcome 1 is

So $\theta = 0$ returns 0 for certain, $\theta = \pi / 2$ is an even coin, and $\theta = \pi$ returns 1 for certain.

A toy you can run

import math
import numpy as np
from fqkit import QuantumCircuit, Hadamard, RZ, run
 
np.set_printoptions(precision=4, suppress=True)
 
def ramsey(theta):
    qc = QuantumCircuit(1)
    qc.add_gate(Hadamard(), [0])
    qc.add_gate(RZ(theta), [0])
    qc.add_gate(Hadamard(), [0])
    probabilities = np.abs(run(qc)) ** 2
    print(f"theta = {theta:.4f}   P = {np.round(probabilities, 4)}")
 
for theta in (0.0, math.pi / 2, math.pi):
    ramsey(theta)

theta = 0.0000   P = [1. 0.]
theta = 1.5708   P = [0.5 0.5]
theta = 3.1416   P = [0. 1.]

The right-hand number is $P(1)$. It matches $\sin^2(\theta/2)$.

Follow up

1. In the physics notebook, set the angle to $3\pi/2$ before you run it. Predict $P(1)$ from the formula, then check the printout.
2. The interference lesson is this circuit with the angle fixed at $\pi$, plus the explanation of why the second Hadamard is required.
3. Two detectors and a phase shifter in an optics lab are this circuit with a different story attached. The arithmetic does not change.

The real scale

Atomic clocks, atom interferometers, and spin-resonance spectrometers are this sequence with a better phase reference and a longer wait between the two Hadamards. Quantum sensing, on the further sectors page, is the same circuit pointed at a magnetic field or a gravitational gradient. Noise, drift, and how many repetitions you can afford are what the laboratory spends its time on. The simulator here is noiseless, so the formula is exact.