  In ref. [1] we analyzed the properties of a Degenerate Optical Parametric
Oscillator (DOPO) tuned to the first transverse mode family at the signal
frequency. Above threshold, a Hermite-Gauss mode with an arbitrary orientation
in the transverse plane is emitted, and thus the rotational invariance of the
system is broken. When quantum effects were taken into account, it was found on
the one hand, that quantum noise is able to induce a random rotation on this
classically emitted mode. On the other hand, the analysis of a balanced
homodyne detection in which the local oscillator (LO) was orthogonal to the
excited mode at any time, showed that squeezing in the quadrature selected by
the LO was found for every phase of this one, squeezing being perfect for a
pi/2 phase. This last fact revealed an apparent paradox: If all quadratures are
below shot noise level, the uncertainty principle seems to be violated. In [1]
we stated that the explanation behind this paradox is that the quadratures of
the rotating orthogonal mode do not form a canonical pair, and the extra noise
is transferred to the diffusing orientation. Thes notes are devoted to prove
this claim.
