Stochastic Hopf bifurcations in vacuum optical tweezers

Abstract

The forces acting on an isotropic microsphere in optical tweezers are effectively conservative. However, reductions in the symmetry of the particle or trapping field can break this condition. Here we theoretically analyse the motion of a particle in a linearly non-conservative optical vacuum trap, concentrating on the case where symmetry is broken by optical birefringence, causing non-conservative coupling between rotational and translational degrees of freedom. Neglecting thermal fluctuations, we first show that the underlying deterministic motion can exhibit a Hopf bifurcation in which the trapping point destabilizes and limit cycles emerge whose amplitude grows with decreasing viscosity. When fluctuations are included, the bifurcation of the underlying deterministic system is expressed as a transition in the statistical description of the motion. For high viscosities, the probability distribution is normal, with a kurtosis of three, and persistent probability currents swirl around the stable trapping point. As the bifurcation is approached the distribution and currents spread out in phase space. Following the bifurcation the probability distribution function hollows out, reflecting the underlying limit cycle, and the kurtosis halves abruptly. The system is seen to be a noisy self sustained oscillator featuring a highly uneven limit cycle. A variety of applications, from autonomous stochastic resonance to synchronization, are discussed.