Uncertainty relations for the photon number and phase of electromagnetic field operators for quantum phase superpositions of coherent states

We examine various uncertainty relations: (electromagnetic-field phase (the so-called “Pegg–Barnett operator”))–(number of photons), (trigonometric functions of the phase)–(number of photons), and (trigonometric functions of the phase)–(field phase). An electromagnetic field in quantum states of phase superpositions of coherent states, which are general superpositions of two coherent states with identical moduli but arbitrary phases, is considered. The rigorous uncertainty relation (Cauchy inequality) and the soft uncertainty relation (Heisenberg inequality) are examined and compared.