Quantum bifurcations in modulated vibrational systems

In classical physical systems, bifurcations are associated with the change of the topology of the phase portrait that results from merging of stationary or periodic states. Such merging is incompatible with quantum mechanics, because of the uncertainty principle. Still, quantum dynamics in the vicinity of classical bifurcation points should display certain universal features. This is a consequence of the slowing down of the system near bifurcation parameter values, reminiscent of the onset of soft modes near phase transition points. We study quantum dynamics in the vicinity of several most frequently encountered bifurcation points, including different types of saddle-node and pitchfork bifurcations. They describe the onset of different types of branches of periodic states in modulated vibrational systems. We consider the generic forms of the Hamiltonian near such bifurcation points and the scaling of the energy spectrum and the expectation values of the dynamical variables with the distance to the bifurcation point. We also consider the role of dissipation and the associated quantum noise, and the scaling of the dynamical variables and quantum fluctuations in the presence of dissipation. The results refer primarily to Floquet systems, but are not limited to such systems. The analysis of quantum effects near bifurcation points is of particular interest for quantum sensing, as it imposes constraints on the sensitivity, but also suggests new approaches to sensing.