(a) Consider the Hamiltonian $H(t)=H_{0}+\delta H(t)$, where $H_{0}$ is time-independent and non-degenerate. The system is prepared to be in some state $|\psi\rangle=\sum_{r} a_{r}|r\rangle$ at time $t=0$, where $\{|r\rangle\}$ is an orthonormal basis of eigenstates of $H_{0}$. Derive an expression for the state at time $t$, correct to first order in $\delta H(t)$, giving your answer in the interaction picture.

(b) An atom is modelled as a two-state system, where the excited state $|e\rangle$ has energy $\hbar \Omega$ above that of the ground state $|g\rangle$. The atom interacts with an electromagnetic field, modelled as a harmonic oscillator of frequency $\omega$. The Hamiltonian is $H(t)=H_{0}+\delta H(t)$, where

$$H_{0}=\frac{\hbar \Omega}{2}(|e\rangle\langle e|-| g\rangle\langle g|) \otimes 1_{\text {field }}+1_{\text {atom }} \otimes \hbar \omega\left(A^{\dagger} A+\frac{1}{2}\right)$$

is the Hamiltonian in the absence of interactions and

$$\delta H(t)= \begin{cases}0, & t \leqslant 0 \\ \frac{1}{2} \hbar(\Omega-\omega)\left(|e\rangle\langle g|\otimes A+\beta| g\rangle\langle e| \otimes A^{\dagger}\right), & t>0\end{cases}$$

describes the coupling between the atom and the field.

(i) Interpret each of the two terms in $\delta H(t)$. What value must the constant $\beta$ take for time evolution to be unitary?

(ii) At $t=0$ the atom is in state $(|e\rangle+|g\rangle) / \sqrt{2}$ while the field is described by the (normalized) state $e^{-1 / 2} e^{-A^{\dagger}}|0\rangle$ of the oscillator. Calculate the probability that at time $t$ the atom will be in its excited state and the field will be described by the $n^{\text {th }}$excited state of the oscillator. Give your answer to first non-trivial order in perturbation theory. Show that this probability vanishes when $t=\pi /(\Omega-\omega)$.