Creating Quanta with 'annihilation' Operator

The non-symmetric nature of the boson 'destruction' operatorâ and its 'creation' partnerâ † 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 'photon-subtracted' statê a|ψ. Moreover, in certain cases the mean number of quanta in the (normalized) statê a|ψ can be much greater than in the 'photon-added' statê a † |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. The non-Hermitian bosonic operatorsâ andâ † of the harmonic oscillator, satisfying the canonical com-mutation relation [ˆ a, ˆ a † ] = 1, are usually called 'annihilation' [1] (or 'destruction' [2, 3]) and 'creation' operators (perhaps, only P A M Dirac, in his book [4], did not use these terms, having used instead ofâ andâ † the 'complex dynamical variables' ¯ η and η.) This is due to their action on the Fock (number) state |n (eigenstate of the number operatorˆn = ˆ a † ˆ a) [5] ˆ a |n = √ n |n − 1, (1) ˆ a † |n = √ n + 1 |n + 1. (2) Therefore, there is some belief that the operatorâ may 'destruct' quanta (photons) in an arbitrary state |ψ, which is reflected even in the name 'photon-subtracted state', sometimes used for the statê a|ψ [6, 7, 8]. However, this concept, guided maybe more by intuition than by a sound proof, seems to be misleading when one deals not with single Fock states, but with their superpositions or quantum mixtures. The better known counterexamples are, for instance, the harmonic oscillator coherent states which are eigenstates ofâ, therefore, the statê a|α/|α| preserves the mean number of photons (see a similar remark in [7]). But as we will see below, this example is not the only one. Indeed, if the initial state is (in this article we speak in terms of pure states only for the sake of simplicity of notation) |ψ =