Creating quanta with "annihilation" operator

An asymmetric 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 $|ψ>$ different from the Fock state $|n>$. We show that it is possible to {\em increase} (by many times or by any quantity) the mean number of quanta in the new `photon-subtracted' state $\hat{a}|ψ>$. Moreover, for certain `hyper-Poissonian' states $|ψ>$ the mean number of quanta in the (normalized) state $\hat{a}|ψ>$ can be much greater than in the `photon-added' state $\hat{a}^{\dagger}|ψ>$. 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.

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