Past exam of the laser science and quantum information processing course of the University of Oxford / 2005 / 5 / 1 / Solution

Treat $|a⟩=|0⟩$ and $|b⟩=|1⟩$ as an isolated ground-state hyperfine qubit. Their transition angular frequency is ${\omega }_0=2\pi \times 320.7\mathrm{MHz}$. The radiofrequency magnetic field couples them through an allowed magnetic dipole transition. After choosing its phase reference, take the two-level laboratory Hamiltonian operator to bewhere $X,Z$ are Pauli operators. Here $\Omega$ is defined by the on-resonance rotating-frame off-diagonal matrix element $\hslash \Omega /2$.

Use a quantum rotating frame and the rotating-wave approximation, valid for $\Omega ,|\delta |\ll {\omega }_0$, with Rabi detuning $\delta ={\omega }_0-\omega$. Ignoring off-resonant coupling to other levels givesAt resonance $\delta =0$, the unitary time evolution isThus the Born rule gives coherent Rabi oscillations,The stated amplitudes are in the rotating frame; the laboratory carrier phases are tracked by the drive reference.

These logical states are orthogonal, individually resolvable hyperfine levels in the same electronic ground state. They avoid fast optical decay, can be coherently coupled by radiofrequency control, and have a state-selective optical readout through $|c⟩$. The excited level $|c⟩$ is useful for fluorescence detection rather than for the storage superposition in this model. Long lifetimes alone do not guarantee a useful qubit: control isolation and adequate quantum coherence are also required.