# Uncertainty Regions for the Vacuum, Number, Coherent, and Squeezed States for Superconducting Josephson Junctions

Schematic diagram of the uncertainty areas in the phase difference and
charge number (f, *n*) phase space of:

**(a)** the vacuum state,

**(b)** a "number" state,

**(c)** a coherent state, and

**(d)** a squeezed state.

In these diagrams, the phases are measured in units of (*EC*/*EJ*)^0.25,
while the charge numbers in units of (*EJ*/*EC*)^0.25.
In addition, the "number" state here is not an eigenstate of the charge
number operator *n*. It is actually and eigenstate of the harmonic
part of the Hamiltonians. Notice that the 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 areas. Therefore, in the director parallel
to the q/2 line, the squeezed state has a smaller
noise than both the vacuum and coherent states.

Image Source: X. Hu, *Quantum Fluctuations In Condensed Matter Systems*,
UM Ph.D. thesis 1997, Page 103.

X. Hu and F. Nori, UM Preprint, 1995;.