Research ArticleQUANTUM PHYSICS

Coherent oscillations of a levitated birefringent microsphere in vacuum driven by nonconservative rotation-translation coupling

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Science Advances  03 Jun 2020:
Vol. 6, no. 23, eaaz9858
DOI: 10.1126/sciadv.aaz9858
  • Fig. 1 Experimental coordinates and oscillatory motion of a vaterite microsphere at the gas pressure of 1.2 mbar.

    (A) Trapping geometry and coordinate axes used throughout. The beam is polarized in the x direction and propagates in the positive z direction. The symmetry axis of the vaterite particle, û, is specified in terms of the polar and azimuthal angles, θ and φ. (B) Time-lapse images of a vaterite microparticle oscillating along the E-field (x direction), where yellow crosses indicate the CoM of the particle. (C) Time traces of the particle’s CoM position both in x and y directions (transverse to the beam axis in the z direction).

  • Fig. 2 Experimentally measured dynamics of a vaterite microsphere compared with a silica microsphere trapped with a linearly polarized beam.

    (A) CoM position distributions in the x-y plane (transverse to the beam axis, z) and (B) their histograms in the x direction (along E-field). (C) Power spectra of x(t) showing the trap frequency at 530 Hz. (D) Autocorrelation of x(t) normalized by 〈x2〉, where τc denotes the correlation time. Rows (1) to (4) represent data at different gas pressures.

  • Fig. 3 Experimentally measured position variance and its reciprocal of a vaterite microsphere.

    (A) Position variance in both x and y directions (left axis) and the relative energy in the x motion (right axis) at different gas pressures P. 〈x2〉 increases with decreasing P toward 1 mbar, whereas 〈y2〉 remains at room temperature. (B) The same data as (A), replotted as the reciprocal of the variance versus pressure to show the scaling behavior as the critical viscosity and pressure are approached.

  • Fig. 4 Simulations describing the correlation between the x and θ coordinates and their coupling forces and torques.

    (A and B) The correlation between x and θ for three separate cases: a birefringent sphere (Δn = 0.1) at low air viscosity (μ = 1 × 10−6 Pa s) (red), the same birefringent sphere at a higher viscosity (μ = 3.25 × 10−6 Pa s) (green), and an isotropic sphere (Δn = 0) at low viscosity (μ = 1 × 10−6 Pa s) (blue). (A) Scatter plot of the simulated coordinates. The insets at each corner show a complete cycle of successive rotations about the y axis and translations along the x axis (i.e., the electric polarization direction). (B) Cross-correlations, 〈x(t)θ(t + τ)〉, evaluated from the simulation data. (C and D) Calculations of the coupled forces Fx and torques Ty for x displacements (with θ = 0) and θ rotations (with x = 0), respectively.

  • Fig. 5 PSD with different oscillation modes.

    Experimentally measured PSD (orange dots) fitted with a Lorentzian (blue curves) of the oscillating particle at the oscillation frequency in the range from 530 to 840 Hz (centered at 0 Hz): (A) Self-sustained oscillations showing a linewidth Δf0 of 0.90 Hz, yielding Q = 924. (B) Phase-locked oscillations showing Δf0 = 10.5 μHz and Q = 5.53 × 107. (C) Frequency-locked oscillations showing Δf0 = 2.20 μHz and Q = 2.45 × 108.

  • Fig. 6 Pressure measurement by a phase-locked oscillator.

    The oscillation frequency (orange dots) fitted with a linear function (blue line) is monitored, while the pressure is changed in the range of 4 to 7 mbar.

Supplementary Materials

  • Supplementary Materials

    Coherent oscillations of a levitated birefringent microsphere in vacuum driven by nonconservative rotation-translation coupling

    Yoshihiko Arita, Stephen H. Simpson, Pavel Zemánek, Kishan Dholakia

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    • Sections S1 to S5
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