Classical and quantum field in beam-splitter

Question

Why is the annihilation operator $ \hat a $ the quantum analog of a classical field amplitude $ \mathcal{E} $? Is the quantized electromagnetic field not a sum of the annihilation and creator operator? Does this work for the beam splitter only, or in general this analogy is accepted? Any broad explanation would be helpful.

Answer 1

It is just a matter of complex vs real notation of the field amplitude : Let your classical field is $E(t)=\mathcal E \cos(ωt -φ)$, of classical (real) amplitude $\mathcal E$ and dephasing $φ$. The corresponding complex amplitude is $\mathcal E_c=\mathcal Ee^{iφ}$, and $E(t)=\frac12({\mathcal E_ce^{-iωt}}+\mathcal E_c^*e^{iωt})=\mathrm{Re}({\mathcal E_ce^{-iωt}})$.

So, when one writes the quantum field as $\hat E(t)∝\hat ae^{-iωt}+\hat a^\dagger e^{-iωt}$, $\hat a$ and $\hat a^\dagger$ are the quantum analogue of the complex field amplitudes $\mathcal E_c$ and $\mathcal E_c^*$.

Notes:

1. I wrote all the above in the Heisenberg representation (where states are fixed and observables change over time). If you’re more familiar with the Schrödinger representation (where states change over time and observable are fixed), just remove all the $e^{±iωt}$ in the equations.

2. I probably did irrelevant sign mistakes in the answer above. Check the signs in your textbook if they’re relevant for you.