Creating quanta with annihilation operator.pdf

a r X i v : q u a n t - p h / 0 2 0 7 0 3 5 v 2 2 5 S e p 2 0 0 2 Creating quanta with ‘annihilation’ operator S.S. Mizrahi ?and V.V. Dodonov ?? Departamento de F??sica, CCT, Universidade Federal de Sa?o Carlos, Rod. Washington Luiz Km 235, Sa?o Carlos, 13565-905, SP, Brazil Abstract An asymmetric nature of the boson ‘destruction’ operator a? and its ‘creation’ partner a?? 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’ state a?|ψ〉. Moreover, for certain ‘hyper-Poissonian’ states |ψ〉 the mean number of quanta in the (normalized) state a?|ψ〉 can be much greater than in the ‘photon-added’ state 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. 1 Introduction The non-Hermitian bosonic operators a? and a?? of the harmonic oscillator, satisfying the canonical commuta- tion relation [a?, 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, introducing instead of a? and a?? the ‘complex dynamical variables’ η? and η.) This is due to their action on the Fock (number) state |n〉 (eigenstate of the number operator n? = a??a?) [5] a? |n〉 = √n |n? 1〉, (1) a??|n〉 = √n+ 1 |n+ 1〉. (2) Therefore, there is some belief that the operator a? may ‘destruct’ quanta (photons) in an arbitrary state |ψ〉. This belief is reflected even in the name ‘photon-subtracted state’, sometimes used for the state 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 counter-example is the harmonic oscillator coherent state