  Dissipative Kerr solitons arising from parametric gain in ring
microresonators are usually described within a classical mean-field framework.
Here, we develop a quantum-mechanical model of dissipative Kerr solitons in
terms of the truncated Wigner method, which accounts for quantum effects to
lowest order. We show that the soliton experiences a finite lifetime due to
quantum fluctuations originating from losses. Reading the results in terms of
the theory of open quantum systems, allows to estimate the Liouvillian spectrum
of the system. It is characterized by a set of eigenvalues with finite
imaginary part and vanishing real part in the limit of vanishing quantum
fluctuations. This feature shows that dissipative Kerr solitons are a specific
class of dissipative time crystals.
