Past exam of the laser science and quantum information processing course of the University of Oxford / 2018 / 5 / a

Take $|0⟩$ to have ${\sigma }_z=+1$ and use angular-frequency gyromagnetic ratios. Neglecting interspin coupling, the Hamiltonian of the Zeeman energy isFor ${\gamma }_1={\gamma }_2=\gamma$, define $ϵ=\hslash \gamma B/(k_{B}T)$. The exact thermal density matrix in the order $|00⟩,|01⟩,|10⟩,|11⟩$ isAt high temperature, $|ϵ|\ll 1$, expand both numerator and normalization to obtainFor positive $\gamma$, $|00⟩$ is the most populated state; a negative gyromagnetic ratio reverses that population preference under this basis convention.