Past exam of the laser science and quantum information processing course of the University of Oxford / 2021 / 3 / d / Solution

Interpret the unspecified Gaussian width $\Delta$ as the standard deviation, soThe characteristic function of this Gaussian distribution isAverage the free-evolution Bloch vectors from part (c):It starts at $-y$ and spirals towards the origin in the equatorial plane, with period $2\pi /{\delta }_0$ and shrinking length $e^{-{\Delta }^2{T}^2/2}$. Individual pure states retain unit-length Bloch vectors; only ensemble averaging produces the apparent loss of quantum coherence.

After the final pulse, the corrected return population isThe oscillations lie between $[1\pm e^{-{\Delta }^2{T}^2/2}]/2$, start at zero, and tend to $1/2$. The upper envelope has no implication of exponential energy relaxation: this is static inhomogeneous dephasing. If instead the intended Gaussian convention was $p\propto e^{-(\delta -{\delta }_0{)}^2/{\Delta }^2}$, every envelope above becomes $e^{-{\Delta }^2{T}^2/4}$. The printed-plus-sign convention would give the complementary fringe.