The Exercise

An atom sits in an optical cavity with two modes, $a$ and $b$. It has a ground level $G$ and two excited levels $E_a$ and $E_b$: a photon of mode $a$ can lift it from $G$ to $E_a$, with coupling strength $g_a$, and a photon of mode $b$ from $G$ to $E_b$, with coupling strength $g_b$. Both transitions are detuned from their modes by the same $\delta$. A state of the atom and the field is written $|n_a n_b, X\rangle$, with $n_a$ and $n_b$ the numbers of photons in the two modes and $X$ the level of the atom.

The atom starts in its ground level, with one photon in mode $a$, the state $|10,G\rangle$, or with one photon in each mode, the state $|11,G\rangle$. The exercise asks for the probability that by time $t$ the atom has absorbed a photon.

The Hamiltonian

For real $\delta$ and $g$, the detuned block $M(\delta, g)$ is the Hamiltonian of one mode and the transition it drives, on the state with a photon in the mode and the atom in its ground level and the state in which the atom has absorbed the photon; in this basis it is the $2 \times 2$ matrix $$M(\delta, g) = \begin{pmatrix} -\delta & g \\ g & \delta \end{pmatrix}.$$(1)^✦

The single-excitation block $H_1$, for a detuning $\delta$ and couplings $g_a$ and $g_b$, is the Hamiltonian on the four states with one excitation, taken in the order $|10,G\rangle$, $|00,E_a\rangle$, $|01,G\rangle$, $|00,E_b\rangle$. It is block diagonal, with one detuned block for each mode: $$H_1 = \begin{pmatrix} M(\delta, g_a) & 0 \\ 0 & M(\delta, g_b) \end{pmatrix}.$$(2)^✦

The two-excitation block $H_2$, for the same $\delta$, $g_a$ and $g_b$, is the Hamiltonian on $|11,G\rangle$ and the two states it is coupled to, $|01,E_a\rangle$, where the atom has absorbed the photon of mode $a$, and $|10,E_b\rangle$, where it has absorbed that of mode $b$: $$H_2 = \begin{pmatrix} -\delta & g_a & g_b \\ g_a & \delta & 0 \\ g_b & 0 & \delta \end{pmatrix}.$$(3)^✦

The Claim

For a real detuning $\delta$, coupling $g$ and time $t$, the detuned absorption profile $L(\delta, g, t)$ is the Rabi formula for a transition driven off resonance, $$L(\delta, g, t) = \frac{g^2}{\delta^2 + g^2}\,\sin^2\!\bigl(\sqrt{\delta^2 + g^2}\,t\bigr).$$(4)^✦

With $\hbar = 1$, a state evolves under a Hamiltonian $H$ by $e^{-iHt}$.

What Is Checked

The claim is stated over the single-excitation block $H_1$ of (2), built from two detuned blocks (1), over the two-excitation block $H_2$ of (3), and over the profile $L$ of (4).