Creating quanta with an  annihilation  operator

Abstract

The non-symmetric nature of the boson "destruction" operator $\\hat{a} and its "creation" partner $\\hat{a}^{\\dagger}$ is made apparent by applying them to a quantum state $|\\psi>$ 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 "photon-subtracted" state $\\hat{a}|\\psi >$. Moreover, in certain cases the mean number of quanta in the (normalized) state $\\hat{a}|\\psi>$ can be much greater than in the "photon-added" state $\\hat{a}^{\\dagger}|\\psi>$. 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.