The Exercise

An atom with two levels sits in a cavity and exchanges energy with a single mode of the field. Two real numbers fix the model: the detuning $\delta$ between the atom and the field, and the coupling strength $g$. On the three states $|00\rangle$, $|01\rangle$, $|10\rangle$ the Hamiltonian $H$ of Jaynes and Cummings is a $3 \times 3$ matrix, and the exercise asks for its time evolution operator $e^{-iHt}$ at time $t$, in units with $\hbar = 1$, as an explicit matrix whose entries oscillate at the Rabi frequency, and for the phase it puts on $|00\rangle$.

An Atom in a Cavity

The Rabi frequency $\Omega$ of a detuning $\delta$ and a coupling $g$ is the real number $$\Omega = \sqrt{g^2 + \delta^2}.$$(1)^✦

The Hamiltonian $H$ of Jaynes and Cummings, with detuning $\delta$ and coupling $g$, is the operator on the span of $|00\rangle$, $|01\rangle$, $|10\rangle$ whose matrix in that basis is $$H = -\begin{pmatrix} \delta & 0 & 0 \\ 0 & \delta & g \\ 0 & g & -\delta \end{pmatrix}.$$(2)^✦

The Rabi unitary $U(t)$ at time $t$, for detuning $\delta$ and coupling $g$, is the operator on the same three states with matrix $$U(t) = \begin{pmatrix} e^{it(\delta - \Omega)}\left(\cos \Omega t + i \sin \Omega t\right) & 0 & 0 \\ 0 & \cos \Omega t + i \tfrac{\delta}{\Omega} \sin \Omega t & i \tfrac{g}{\Omega} \sin \Omega t \\ 0 & i \tfrac{g}{\Omega} \sin \Omega t & \cos \Omega t - i \tfrac{\delta}{\Omega} \sin \Omega t \end{pmatrix}.$$(3)^✦

The Claim

The entries of (3) divide by $\Omega$, which vanishes only when $g$ and $\delta$ are both zero.

What Is Checked

The claim is stated over the Hamiltonian $H$ of (2), given by its matrix on the three states, and over the Rabi unitary $U(t)$ of (3), whose entries are built from the Rabi frequency (1).