The operator $|e\rangle\langle g|\otimes A$ annihilates one field quantum and excites the atom. Its adjoint $|g\rangle\langle e|\otimes A^\dagger$ de-excites the atom and creates one field quantum. These are the two exchange processes in the <Jaynes-Cummings model>.

<Unitary time evolution> requires the Hamiltonian to be Hermitian. Since the common coefficient $\tfrac12\hbar(\Omega-\omega)$ is real, the two terms must be adjoints with equal coefficients. Therefore
$$
\boxed{\beta=1}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 2 / 34A / b / i / Solution

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The Jaynes-Cummings model couples a two-level atom to one quantized oscillator mode through
$$
H_{\rm int}=\hbar g(\sigma_+A+\sigma_-A^\dagger).
$$
The first term excites the atom while annihilating one field quantum; the second performs the Hermitian-conjugate process.


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