Creating quanta with an ‘annihilation’ operator

The asymmetric nature of the boson ‘destruction’ operator â and its ‘creation’ partner ↠is made apparent by applying them to a quantum state |ψ〉 different from the Fock state |n〉. We show that it is possible to increase (by many times or by any quantity) the mean number of quanta in the new ‘photonsubtracted’ state â|ψ〉. Moreover, for certain ‘hyper-Poissonian’ states |ψ〉 the mean number of quanta in the (normalized) state â|ψ〉 can be much greater than in the ‘photon-added’ state â†|ψ〉. The explanation of this ‘paradox’ is given and some examples elucidating the meaning of Mandel’s q-parameter and the exponential phase operators are considered. PACS numbers: 03.65.−w, 42.50.Ar, 42.50.Dv