NUMBER-PHASE UNCERTAINTY PRODUCT FOR DISPLACED NUMBER STATES

The number-phase uncertainty product for displaced number states of th e harmonic oscillator is examined within the framework of three differ ent approaches: (i) the Carruthers-Nieto number-phase uncertainty rela tion in terms of the Susskind-Glogower sine and cosine phase operators , (ii) a similar relation to this calculated with the Pegg-Barnett uni tary phase operator, and (iii) the number-phase uncertainty relation a rising from the Pegg-Barnett Hermitian phase operator. The correspondi ng number-phase uncertainty product is calculated extending from avera ge photon numbers 30 down to m for the first few classes of displaced number states (m = 0,...,5). It is found that, for a displaced number state with a reasonable average number of excited quanta, all three ri val phase formalisms yield similar number-phase uncertainty products, tending, for increasingly large magnitude of the displacement paramete r, to the constant value m + 1/2. On the other hand, for a small numbe r of excited quanta it is found that, according to the first two forma lisms, the number-phase uncertainty product for a given class of displ aced number states tends to the maximum value square-root 2m + 1/(squa re-root m + 1 - square-root m), while the third phase formalism predic ts an entirely different behavior; with decreasing magnitude of displa cement parameter, after passing through a maximum, the number-phase un certainty product falls off eventually to zero making, in particular, the search for minimum number-phase uncertainty states futile. It is a rgued in favor of this last result, and the possibility of experimenta l verification, in the realm of quantum optics, is briefly considered.