Past exam of the laser science and quantum information processing course of the University of Oxford / 2005 / 5 / 2 / Solution

A resonant Rabi pi pulse satisfies $\Omega T_{\pi }=\pi$. The supplied $T_{\pi }=0.256\mathrm{s}$ therefore givesThe ideal rotating-frame operation is $-iX$, a Pauli X gate up to global phase.

Both logical states belong to the electronic ground level. A fast electric dipole transition between them is forbidden, and spontaneous radiative relaxation at a hyperfine radiofrequency is weak. The optical readout field is off during the coherent pulse. Neglecting relaxation is therefore consistent with a population lifetime much longer than $0.256\mathrm{s}$. Magnetic noise, collisions and technical dephasing are additional limitations; the ideal coherent calculation assumes their effects are small over this interval.

At detuning $\delta$, define ${\Omega }_{\mathrm{eff}}=\sqrt{{\Omega }^2+{\delta }^2}$. The same two-level Hamiltonian operator givesFor an accurate fixed-duration $\pi$ rotation, the detuning must be small compared with the Rabi angular frequency. Writing $\delta \nu =\delta /(2\pi )$,a fractional mismatch much smaller than $6.1\times 10^{-9}$ relative to the $320.7\mathrm{MHz}$ carrier. This is an accuracy scale, not an assertion that a particular nonzero detuning gives a perfect gate.

The small-detuning error of a Rabi pi pulse quantifies a chosen tolerance. For an initial $|a⟩$ and the resonant pulse duration,A population-transfer error at most $ϵ$ consequently requires approximately $|\delta \nu |\lesssim 1.95\sqrt{ϵ}\mathrm{Hz}$; for example, about $0.20\mathrm{Hz}$ for one percent error. Full gate calibration also includes pulse area and drive phase, beyond this population-transfer criterion.