##
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).