Unidimensional time-domain quantum optics

Choosing the right first-quantization basis in quantum optics is critical for the interpretation of experimental results. The usual frequency basis is, for instance, inappropriate for short, subcycle waveforms. Deriving first quantization in the time domain shows that the electromagnetic field is not directly proportional, nor even causally related, to the photonic field (the amplitude probability of a photon detection). We derive the relation between the two and calculate the statistics of the electromagnetic field for specific states in the time domain, such as the single photon Fock state. We introduce the dual of the Hamiltonian in the time domain and extend the concept of quadratures to all first quantization bases.