Dirac formalism: quantum states and two-level systems

Equipment

• Rotatable 3D Bloch sphere with the state vector starting at |0⟩
• “Coupling Ω” slider — the drive's Rabi frequency
• “Detuning Δ” slider off resonance
• “Time t” slider and the P₁(t) plot in solid/dashed lines

Procedure

1. Set resonance and follow the Rabi oscillation

   Set “Detuning Δ” to 0 and sweep “Time t”: the Bloch vector rotates about the x-axis and the state flips completely |0⟩ → |1⟩ at $\Omega t = \pi$ — a π-pulse. Check that the dashed P₁(t) curve (resonant case) reaches 1.

2. Increase the detuning and measure the transition amplitude

   Raise “Detuning Δ”: the rotation axis tilts toward $(\Omega, 0, \Delta)$, the Bloch vector no longer reaches the south pole, and P₁ oscillates with smaller amplitude at the larger frequency $\Omega_{\text{eff}} = \sqrt{\Omega^2+\Delta^2}$. Compare the amplitude with $P_{\max} = \Omega^2/(\Omega^2+\Delta^2)$.

3. Vary Ω and read the bra–ket bookkeeping

   Raise “Coupling Ω” at fixed Δ: both the frequency and the amplitude increase. Write the state $|\psi(t)\rangle = c_0(t)|0\rangle + c_1(t)|1\rangle$ and the probability $P_1 = |c_1|^2 = |\langle 1|\psi\rangle|^2$ — illustrating how Dirac notation encodes states and measurements.

Simulation

Experiment history

Related physicists

Related library topics