  As the energy relaxation time of superconducting qubits steadily improves,
non-equilibrium quasiparticle excitations above the superconducting gap emerge
as an increasingly relevant limit for qubit coherence. We measure fluctuations
in the number of quasiparticle excitations by continuously monitoring the
spontaneous quantum jumps between the states of a fluxonium qubit, in
conditions where relaxation is dominated by quasiparticle loss. Resolution on
the scale of a single quasiparticle is obtained by performing quantum
non-demolition projective measurements within a time interval much shorter than
$T_1$, using a quantum limited amplifier (Josephson Parametric Converter). The
quantum jumps statistics switches between the expected Poisson distribution and
a non-Poissonian one, indicating large relative fluctuations in the
quasiparticle population, on time scales varying from seconds to hours. This
dynamics can be modified controllably by injecting quasiparticles or by seeding
quasiparticle-trapping vortices by cooling down in magnetic field.
