  Understanding the nature and behaviour of excitations in quantum spin
liquids, and in topological phases of matter in general, is of fundamental
importance and has proven crucial for experimental detection and
characterisation of candidate materials. Current theoretical and numerical
techniques, however, have limited capabilities, especially when it comes to
studying gapped excitations. Here, we propose a semiclassical numerical method
to study systems whose spin liquid behaviour is underpinned by perturbative
ring-exchange Hamiltonians. Our method can readily access both thermodynamic
and spectral properties. We focus in particular on quantum spin ice and its
photon and vison excitations. After benchmarking the method against existing
results on photons, we use it to characterise visons and their thermodynamic
behaviour, which remained hitherto largely unexplored. We find that visons, in
contrast to spinons in classical spin ice, form a weak electrolyte: vison pairs
are the dominant population at low temperatures. This is reflected in the
behaviour of thermodynamic quantities, such as pinch point motifs in the
relevant correlators. Visons also appear to strongly hybridise with the photon
background, a phenomenon that affects the way these quasiparticles may show up
in inelastic response measurements. Our results demonstrate that the method,
and generalisations thereof, can substantially help our understanding of
quasiparticles and their interplay in quantum spin ice and other quantum spin
liquids, quantum dimer models, and lattice gauge theories in general.
