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 analyze the motion of a particle in a linearly nonconservative optical vacuum trap, concentrating on the case where symmetry is broken by optical birefringence, causing nonconservative 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, is discussed.

Cooling the optical-spin driven limit cycle oscillations of a levitated gyroscope

“…As the pressure is decreased the two spectral peaks, discernible at 0.3mbar, merge, leaving a single, dominant frequency, corresponding to the fundamental frequency of an underlying limit cycle [22]. This transition mirrors the behaviour of the decay constants of the correlation functions, providing further evidence that the two processes, resolvable at higher pressure, are replaced by a single, dominant, non-equilibrium process as damping is decreased.…”

“…For isotropic microspheres, this results in striking and characteristic behaviour. With decreasing pressure, the stable trapping point undergoes a Hopf bifurcation [22], giving way to noisy limit cycles (or orbits) whose amplitude increases until the particle is ultimately ejected from the trap. Intriguingly, similar behaviour has not been observed for birefringent, vaterite particles which are known to remain stably trapped even in ultra-high vacuum [4].…”

Gene expression noise accelerates the evolution of a biological oscillator

“…S2G). This well-known phenomenon is called a stochastic Hopf bifurcation (Schenk-Hoppé, 1996; Ma, 2012; Arnold et al, 1999; Li and Zhang, 2019; Simpson et al, 2021). The noise-induced oscillations have a frequency whose value is related to the imaginary part of the eigenvalues of the Jacobian at the fixed point underlying deterministic dynamics.…”

Synchronization of spin-driven limit cycle oscillators optically levitated in vacuum

“…Below a critical, threshold power, we observe biased Brownian motion, featuring correlations which strengthen with increasing optical power. At threshold, a bifurcation occurs [35] and the stable trapping points are replaced with noisy limit cycles that form robust synchronized states with characteristic detuning behaviour [23].…”