Parametric feedback cooling of rigid body nanodumbbells in levitated optomechanics

Abstract

We theoretically investigate the rigid body dynamics of an optically levitated nanodumbbell under parametric feedback cooling and provide a simplified model for describing the motion. Differing from previous studies, the spin of the nanoparticle about its symmetry axis is considered non-negligible. Simulations reveal that standard parametric feedback cooling can extract energy from two of the five rotational degrees of freedom when the nanoparticle is levitated using a linearly polarized laser beam. The dynamics after feedback cooling are characterized by a normal mode describing precession about the laser polarization axis together with spin about the nanoparticle's symmetry axis. Cooling the remaining mode requires an asymmetry in the two librational frequencies associated with motion about the polarization axis as well as information about the two frequencies of rotation about the polarization axis. Introducing an asymmetric potential allows full cooling of the librational coordinates if the frequencies of both are used in the feedback modulation and is an avenue for entering the librational quantum regime. The asymmetry in the potential needs to be large enough for practical cooling times as the cooling rate of the system depends non-linearly on the degree of asymmetry, a condition that is easily achieved experimentally.

Stability and dynamics of optically levitated dielectric disks in a Gaussian standing wave beyond the harmonic approximation

“…As was shown in Ref. [5], for a symmetric top-like rigid body the rotational kinetic energy naturally involves coupling between the α,α, β, andβ degrees of freedom. Whether these terms are significant depends on the geometry.…”

“…The disk remains stable in the trap after several thousand oscillations for all of the beam waists explored w 0 = 2, 2.5, 3, 4 μm. The a = 200 nm disk was found to be stable at all frequencies, even with inclusion of the nonlinear coupling terms in the rotational kinetic energy [5]. Recall from Sec.…”

“…The principal moments of inertia are I z = ma 2 /2 and I x = I y = m(3a 2 + T 2 )/12. The disk's center of mass is located at r 0 = x 0 , y 0 , z 0 and rotations are described in terms of the Euler angles (α, β, γ ) in the z-y -z convention [5,25,26].…”

“…One noteworthy feature not mentioned in the main text is that the ω θ z rotational frequency depends on χ ⊥ rather than χ . Rotational frequencies in the Rayleigh approximation depend on χ solely [5] as the particle's long axis tries to align with the electric field. In the Rayleigh approximation, ω θ z = 0 and is only nonzero here due to the electric field gradient across the finite extension of the disk.…”

“…A nanorod has large differences in moments of inertia and polarizability which allows rotations to be described as decoupled librations about the laser polarization axis. The motion of nanodumbells or generally anisotropic materials requires rigid-body dynamics since these particles have moments of inertia of similar magnitude [5,6]. Increasing the size of the particle relative to the wavelength of the laser further complicates the motion for any particle shape [7].…”