  Gaussian quantum Markov semigroups are the natural non-commutative extension
of classical Ornstein-Uhlenbeck semigroups. They arise in open quantum systems
of bosons where canonical non-commuting random variables of positions and
momenta come into play. If there exits a faithful invariant density we
explicitly compute the optimal exponential convergence rate, namely the
spectral gap of the generator, in non-commutative $L^2$ spaces determined by
the invariant density showing that the exact value is the lowest eigenvalue of
a certain matrix determined by the diffusion and drift matrices. The spectral
gap turns out to depend on the non-commutative $L^2$ space considered, whether
the one determined by the so-called GNS or KMS multiplication by the square
root of the invariant density. In the first case, it is strictly positive if
and only if there is the maximum number of linearly independent noises. While,
we exhibit explicit examples in which it is strictly positive only with KMS
multiplication. We do not assume any symmetry or quantum detailed balance
condition with respect to the invariant density.
