  The theory of Gaussian quantum fluctuations around classical steady states in
nonlinear quantum-optical systems (also known as standard linearization) is a
cornerstone for the analysis of such systems. Its simplicity, together with its
accuracy far from critical points or situations where the nonlinearity reaches
the strong coupling regime, has turned it into a widespread technique, which is
the first method of choice in most works on the subject. However, such a
technique finds strong practical and conceptual complications when one tries to
apply it to situations in which the classical long-time solution is time
dependent, a most prominent example being spontaneous limit-cycle formation.
Here we introduce a linearization scheme adapted to such situations, using the
driven Van der Pol oscillator as a testbed for the method, which allows us to
compare it with full numerical simulations. On a conceptual level, the scheme
relies on the connection between the emergence of limit cycles and the
spontaneous breaking of the symmetry under temporal translations. On the
practical side, the method keeps the simplicity and linear scaling with the
size of the problem (number of modes) characteristic of standard linearization,
making it applicable to large (many-body) systems.
