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

Nat. Commun. 14, 5441 (2023) · DOI: 10.1038/s41467-023-41129-5

License: CC BY 4.0.

Abstract

We explore, experimentally and theoretically, the emergence of coherent coupled oscillations and synchronization between a pair of non-Hermitian, stochastic, opto-mechanical oscillators, levitated in vacuum. Each oscillator consists of a polystyrene microsphere trapped in a circularly polarized, counter-propagating Gaussian laser beam. Non-conservative, azimuthal forces, deriving from inhomogeneous optical spin, push the micro-particles out of thermodynamic equilibrium. For modest optical powers each particle shows a tendency towards orbital circulation. Initially, their stochastic motion is weakly correlated. As the power is increased, the tendency towards orbital circulation strengthens and the motion of the particles becomes highly correlated. Eventually, centripetal forces overcome optical gradient forces and the oscillators undergo a collective Hopf bifurcation. For laser powers exceeding this threshold, a pair of limit cycles appear, which synchronize due to weak optical and hydrodynamic interactions. In principle, arrays of such Non-Hermitian elements can be arranged, paving the way for opto-mechanical topological materials or, possibly, classical time crystals. In addition, the preparation of synchronized states in levitated optomechanics could lead to new and robust sensors or alternative routes to the entanglement of macroscopic objects.

Figures

38 panels with data across 4 figures. Each panel page shows the plot, its columns and its files; each data.csv begins with a header naming the paper, the panel, the source, the license and the provenance route.

Fig. 1

• panel (d-1): Panel (d), theory: spatial probability density of particle 1 in the $x$-$y$ plane from the stochastic simulation at a laser power of 10 mW, on an 8 nm grid. The print draws the map without axes or colour bar, with a 200 nm scale bar. data.csv
• panel (d-2): Panel (d), theory: spatial probability density of particle 1 in the $x$-$y$ plane from the stochastic simulation at a laser power of 14 mW, on an 8 nm grid. The print draws the map without axes or colour bar, with a 200 nm scale bar. data.csv
• panel (d-3): Panel (d), theory: spatial probability density of particle 1 in the $x$-$y$ plane from the stochastic simulation at a laser power of 18 mW, on an 8 nm grid. The print draws the map without axes or colour bar, with a 200 nm scale bar. data.csv
• panel (d-4): Panel (d), theory: spatial probability density of particle 2 in the $x$-$y$ plane from the stochastic simulation at a laser power of 10 mW, on an 8 nm grid. The print draws the map without axes or colour bar, with a 200 nm scale bar. data.csv
• panel (d-5): Panel (d), theory: spatial probability density of particle 2 in the $x$-$y$ plane from the stochastic simulation at a laser power of 14 mW, on an 8 nm grid. The print draws the map without axes or colour bar, with a 200 nm scale bar. data.csv
• panel (d-6): Panel (d), theory: spatial probability density of particle 2 in the $x$-$y$ plane from the stochastic simulation at a laser power of 18 mW, on an 8 nm grid. The print draws the map without axes or colour bar, with a 200 nm scale bar. data.csv
• panel (d-7): Panel (d), theory: spatial probability density of the breathing quasi-mode coordinates in the $x$-$y$ plane from the stochastic simulation at a laser power of 10 mW, on an 8 nm grid. The print draws the map without axes or colour bar, with a 200 nm scale bar. data.csv
• panel (d-8): Panel (d), theory: spatial probability density of the breathing quasi-mode coordinates in the $x$-$y$ plane from the stochastic simulation at a laser power of 14 mW, on an 8 nm grid. The print draws the map without axes or colour bar, with a 200 nm scale bar. data.csv
• panel (d-9): Panel (d), theory: spatial probability density of the breathing quasi-mode coordinates in the $x$-$y$ plane from the stochastic simulation at a laser power of 18 mW, on an 8 nm grid. The print draws the map without axes or colour bar, with a 200 nm scale bar. data.csv
• panel (d-10): Panel (d), theory: spatial probability density of the centre-of-mass quasi-mode coordinates in the $x$-$y$ plane from the stochastic simulation at a laser power of 10 mW, on an 8 nm grid. The print draws the map without axes or colour bar, with a 200 nm scale bar. data.csv
• panel (d-11): Panel (d), theory: spatial probability density of the centre-of-mass quasi-mode coordinates in the $x$-$y$ plane from the stochastic simulation at a laser power of 14 mW, on an 8 nm grid. The print draws the map without axes or colour bar, with a 200 nm scale bar. data.csv
• panel (d-12): Panel (d), theory: spatial probability density of the centre-of-mass quasi-mode coordinates in the $x$-$y$ plane from the stochastic simulation at a laser power of 18 mW, on an 8 nm grid. The print draws the map without axes or colour bar, with a 200 nm scale bar. data.csv

Fig. 2

• panel (a): Auto-correlation of the breathing-mode coordinate $x_b$, cross covariance [nm$^2$] against delay $\tau$ [ms], for optical powers $P$ = 216, 192, 138 and 76 mW at beam distance $d = 8.6\,\mu$m. Each curve is given at the vertices of the printed line. data.csv
• panel (b): Auto-correlation of the centre-of-mass coordinate $x_c$, cross covariance [nm$^2$] against delay $\tau$ [ms], for optical powers $P$ = 216, 192, 138 and 76 mW at beam distance $d = 8.6\,\mu$m. Each curve is given at the vertices of the printed line. data.csv
• panel (c): Cross-correlation of the centre-of-mass and breathing coordinates $x_c$ and $x_b$, cross covariance [nm$^2$] against delay $\tau$ [ms], for optical powers $P$ = 216, 192, 138 and 76 mW at beam distance $d = 8.6\,\mu$m. Each curve is given at the vertices of the printed line. data.csv
• panel (d): Correlations of the breathing-mode coordinates against delay $\tau$ [ms]: $\langle x_b(t+\tau)\,x_b(\tau)\rangle$ and $\langle x_b(t+\tau)\,y_b(\tau)\rangle$, cross covariance in nm$^2$. Each curve is given at the vertices of the printed line. data.csv
• panel (e): Probability density of the difference of the azimuthal coordinates of the two particles, $\Delta\phi = \phi_1 - \phi_2$ [rad], for beam separation $d = 8.6\,\mu$m at optical powers $P$ = 216, 192, 138 and 76 mW. Each curve is given at the vertices of the printed line; the $\Delta\phi$ axis is in radians where the print labels it in fractions of $\pi$. data.csv
• panel (f): Probability density of the difference of the azimuthal coordinates of the two particles, $\Delta\phi = \phi_1 - \phi_2$ [rad], for beam separation $d = 8.9\,\mu$m at optical powers $P$ = 197, 171, 139 and 104 mW. Each curve is given at the vertices of the printed line; the $\Delta\phi$ axis is in radians where the print labels it in fractions of $\pi$. data.csv
• panel (g): Relative Shannon entropy $S_r$ of the phase-difference distributions against optical power $P$ [mW]: one series for the breathing mode and one for the center of mass mode, points joined by lines. data.csv

Fig. 3

• panel (b-1): Left image of panel (b): camera grey level (8 bit, mean over the 8 pixel rows through the two particles) against time [ms] and position $x$ [$\mu$m] along the line of the particles, one column per frame of the fast CMOS camera (62 frames at 500 kHz). The blue and red curves the print draws over the image are not reproduced. data.csv
• panel (b-2): Right plot of panel (b): displacement $x - x_0$ [nm] of particle 1 and particle 2 along the $x$ axis against time [ms], from the fast CMOS camera record, one point per camera frame. data.csv
• panel (c): Accumulated phase difference $\Delta\Phi = \Phi_1 - \Phi_2$ [rad] of the two particles against time in units of the limit cycle period $T$, one point per camera frame. The $\Delta\Phi$ axis is in radians where the print labels it in multiples of $\pi$. data.csv
• panel (d): Variance of the accumulated phase of a single particle, var($\Phi_1$), and of the phase difference of the two particles, var($\Delta\Phi$), against time [ms], one point per camera frame. data.csv
• panel (e): Probability density of the relative phase $\Delta\phi$ [rad] of the two oscillators on the interval $(-\pi, \pi)$, from the fast CMOS camera record, in 200 bins. The print shades the area under the curve; the $\Delta\phi$ axis is in radians where the print labels it in fractions of $\pi$. data.csv
• panel (f-1): Polar plot 1 of 10 in panel (f), left to right: a short trajectory of particle 1 and of particle 2 in the $x$-$y$ plane, points in the drawn order with the large printed marker last. The print gives no radial scale, so the radius is a fraction of the drawn axis circle; angles are in degrees where the print labels fractions of $\pi$; the shaded sector is not reproduced. data.csv
• panel (f-2): Polar plot 2 of 10 in panel (f), left to right: a short trajectory of particle 1 and of particle 2 in the $x$-$y$ plane, points in the drawn order with the large printed marker last. The print gives no radial scale, so the radius is a fraction of the drawn axis circle; angles are in degrees where the print labels fractions of $\pi$; the shaded sector is not reproduced. data.csv
• panel (f-3): Polar plot 3 of 10 in panel (f), left to right: a short trajectory of particle 1 and of particle 2 in the $x$-$y$ plane, points in the drawn order with the large printed marker last. The print gives no radial scale, so the radius is a fraction of the drawn axis circle; angles are in degrees where the print labels fractions of $\pi$; the shaded sector is not reproduced. data.csv
• panel (f-4): Polar plot 4 of 10 in panel (f), left to right: a short trajectory of particle 1 and of particle 2 in the $x$-$y$ plane, points in the drawn order with the large printed marker last. The print gives no radial scale, so the radius is a fraction of the drawn axis circle; angles are in degrees where the print labels fractions of $\pi$; the shaded sector is not reproduced. data.csv
• panel (f-5): Polar plot 5 of 10 in panel (f), left to right: a short trajectory of particle 1 and of particle 2 in the $x$-$y$ plane, points in the drawn order with the large printed marker last. The print gives no radial scale, so the radius is a fraction of the drawn axis circle; angles are in degrees where the print labels fractions of $\pi$; the shaded sector is not reproduced. data.csv
• panel (f-6): Polar plot 6 of 10 in panel (f), left to right: a short trajectory of particle 1 and of particle 2 in the $x$-$y$ plane, points in the drawn order with the large printed marker last. The print gives no radial scale, so the radius is a fraction of the drawn axis circle; angles are in degrees where the print labels fractions of $\pi$; the shaded sector is not reproduced. data.csv
• panel (f-7): Polar plot 7 of 10 in panel (f), left to right: a short trajectory of particle 1 and of particle 2 in the $x$-$y$ plane, points in the drawn order with the large printed marker last. The print gives no radial scale, so the radius is a fraction of the drawn axis circle; angles are in degrees where the print labels fractions of $\pi$; the shaded sector is not reproduced. data.csv
• panel (f-8): Polar plot 8 of 10 in panel (f), left to right: a short trajectory of particle 1 and of particle 2 in the $x$-$y$ plane, points in the drawn order with the large printed marker last. The print gives no radial scale, so the radius is a fraction of the drawn axis circle; angles are in degrees where the print labels fractions of $\pi$; the shaded sector is not reproduced. data.csv
• panel (f-9): Polar plot 9 of 10 in panel (f), left to right: a short trajectory of particle 1 and of particle 2 in the $x$-$y$ plane, points in the drawn order with the large printed marker last. The print gives no radial scale, so the radius is a fraction of the drawn axis circle; angles are in degrees where the print labels fractions of $\pi$; the shaded sector is not reproduced. data.csv
• panel (f-10): Polar plot 10 of 10 in panel (f), left to right: a short trajectory of particle 1 and of particle 2 in the $x$-$y$ plane, points in the drawn order with the large printed marker last. The print gives no radial scale, so the radius is a fraction of the drawn axis circle; angles are in degrees where the print labels fractions of $\pi$; the shaded sector is not reproduced. data.csv

Fig. 4

• panel (a): Long-time traces of the accumulated phase difference $\Delta\Phi$ [2$\pi$ rads] against time [ms] for oscillators detuned by $\Delta f$ = 41, 117, 664 and 1527 Hz. Each curve is given at the vertices of the printed line. data.csv
• panel (b): Shorter-time traces of the accumulated phase difference $\Delta\Phi$ [2$\pi$ rads] against time [ms] for oscillators detuned by $\Delta f$ = 41, 117, 664 and 1527 Hz. Each curve is given at the vertices of the printed line. The grey horizontal lines the print draws at each integer are not reproduced. data.csv
• panel (c): Probability density of the absolute phase difference $\Delta\phi$ [rad] for oscillators detuned by $\Delta f$ = 41, 117, 664 and 1527 Hz. Each curve is given at the vertices of the printed line. The $\Delta\phi$ axis is in radians where the print labels it in fractions of $\pi$. data.csv
• panel (d): Relative Shannon entropy $S_r$ of the phase-difference distribution against the detuning $\Delta f$ [Hz] of the two oscillators, four points joined by a line. data.csv

Fig. 5

Illustrative figure, no extractable data. Shown in the paper PDF. Page 8 of the PDF.

Cite

Oto Brzobohatý, Martin Duchaň, Petr Jákl, Jan Ježek, Martin Šiler, Pavel Zemánek, Stephen H. Simpson. Synchronization of spin-driven limit cycle oscillators optically levitated in vacuum. Nat. Commun. 14, 5441 (2023). https://doi.org/10.1038/s41467-023-41129-5

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