The Exercise

An atom in an optical cavity is coupled to two of its modes, $a$ and $b$. It has a ground state $|g\rangle$ and two excited states: absorbing a photon of mode $a$ takes it to $|e_a\rangle$ with coupling strength $g_a$, and absorbing a photon of mode $b$ takes it to $|e_b\rangle$ with coupling strength $g_b$. Each transition is detuned from its mode by $2\delta$. Absorption and emission conserve the number of excitations, photons plus the excitation of the atom, so the Hamiltonian splits into blocks, one for each excitation number.

Write a state as $|s,n_an_b\rangle$, with $s$ the state of the atom and $n_a$, $n_b$ the photon numbers of the two modes. Start with the atom in its ground state and one photon in each mode, the state $|g,11\rangle$. The exercise asks for the phase $\phi_{ab}$ this state acquires over a time $t$, relative to the vacuum $|g,00\rangle$: the Kerr phase the atom gives a pair of photons.

Two Blocks of the Hamiltonian

The vacuum block $H_0$ is the $1\times1$ matrix of the Hamiltonian on the single state $|g,00\rangle$, whose one entry is the energy of the ground state: $$H_0 = \begin{pmatrix} -\delta \end{pmatrix}.$$(1)^✦

The block $H_2$ with two excitations is the $3\times3$ matrix of the Hamiltonian on the states $|g,11\rangle$, $|e_a,01\rangle$ and $|e_b,10\rangle$, in that order, for real couplings $g_a$ and $g_b$: $$H_2 = \begin{pmatrix} -\delta & g_a & g_b \\ g_a & \delta & 0 \\ g_b & 0 & \delta \end{pmatrix}.$$(2)^✦

Within this block the state $|g,11\rangle$ oscillates between two levels at energies $\pm\Omega'$, where the Rabi frequency $\Omega'$ of the block $H_2$ is the nonnegative real number $$\Omega' = \sqrt{\delta^2 + g_a^2 + g_b^2}.$$(3)^✦

The Claim

What Is Checked

The claim is stated over the vacuum block $H_0$ of (1), the block $H_2$ with two excitations of (2) and its Rabi frequency $\Omega'$ of (3). What the library proves about them is the closed form (4) of the phase.