Uncertainty Regions for the Vacuum, Number, Coherent, and Squeezed Phonon States

Schematic diagram of the uncertainty areas in the generalized coordinate and momentum (X(q, -q), P(q, -q)) phase space of:

(a) the phonon vacuum state,

(b) a phonon number state,

(c) a phonon coherent state, and

(d) a phonon squeezed state.

Here X(q, -q) and P(q, -q) are the two-mode (±q) coordinate and momentum operators. They are direct generalizations of the single-mode coordinate and momentum operators. Notce that the phonon coherent state has the same uncertainty area as the vacuum state, and that both areas are circular, while the squeezed state has an elliptical uncertainty area. Therefore, in the direction parallel to the q/2 line, the squeezed state has a smaller noise than both the vacuum and coherent states.

 X. Hu and F. Nori, Physical Review Letters 76, 2294 (1996); Physical Review B 53, 2419 (1996); Physical Review Letters 79, 4605 (1997).