Shot-noise-dominant regime for ellipsoidal nanoparticles in a linearly polarized beam

Results on the heating and the parametric feedback cooling of an optically trapped anisotropic nanoparticle in the laser-shot-noise-dominant regime are presented. The related dynamical parameters, such as the oscillating frequency and shot noise heating rate, depend on the shape of the trapped particle. For an ellipsoidal particle, the ratio of the axis lengths and the overall size controls the shot noise heating rate relative to the frequency. For a particle with smaller ellipticity or bigger size, the relative heating rate for rotation tends to be smaller than that for translation indicating a better rotational cooling. For one feedback scheme, we also present results on the lowest occupation number that can be achieved as a function of the heating rate and the amount of classical uncertainty in the position measurement.

Figure 1

A symmetric ellipsoidal nanoparticle is trapped in a laser beam (shown by the red line), which is polarized in the $z$ direction and propagating in the positive $y$ direction (shown by the red arrow). Besides the vibrational motion in the center-of-mass degrees of freedom, the ellipsoid also rotationally vibrates with its long axis closely aligned with the laser polarization direction. The angles $\alpha ,\beta ,\gamma$ denote an orientation of the nanoparticle.

Figure 2

The ratio of the occupation number change $\langle {\dot{n}}_R\rangle /\langle {\dot{n}}_T\rangle$ in terms of ellipticity (a) and size (b). (a) The size of particles is fixed at $\sqrt{a^2+b^2}=71\text{nm}$ while the ellipticity increases. (b) The ellipticity is fixed at $e=0.77$ while the particle size increases. The blue curves are for diamonds while the yellow curves are for silica.

Figure 3

The ratio $\mathrm{\Delta }{n}_R/\mathrm{\Delta }{n}_T$ in terms of the particle ellipticity. The blue and yellow curves correspond to diamond and silica, respectively.

Figure 4

The classical simulation results of shot noise heating for nanodiamonds in both the translational and rotational degrees of freedom. Each curve is averaged over 400 individual reheating trajectories. (a) and (b) are for the nanoparticle with half axes $(a=15\text{nm},b=70\text{nm})$, while (c) and (d) with half axes $(a=38\text{nm},b=60\text{nm})$, (e) and (f) with half axes $(a=48\text{nm},b=53\text{nm})$. The dashed lines are the heating curves $T=T_0+\dot{E}t$ with $T_0$ the initial temperature and $\dot{E}$ the corresponding heating rate from Table 1.

Figure 5

The parametric feedback cooling for nanodiamonds in all degrees of freedom, where each curve shows the time evolution of the average occupation number in the corresponding degree of freedom. Data are collected by averaging 30 cooling trajectories. Calculations are for classical parametric feedback cooling; thus results for occupation numbers less than 10 are suggestive. (a) and (b) depict the translational and rotational cooling, respectively, for a nanoparticle with half axes $(a=15\text{nm},b=70\text{nm})$. The cooling parameter ${\mathrm{\Delta }}_1=\{{\eta }_i=1.1\times {10}^{11}{\mathrm{s}/\mathrm{m}}^2,{\zeta }_i={10}^{11}{\mathrm{s}/\mathrm{m}}^2\}$ for $t<100\mathrm{ms}$ and ${\mathrm{\Delta }}_2=10{\mathrm{\Delta }}_1$ for $t>100\mathrm{ms}$. Similarly, (c) and (d) show the cooling for half axes $(a=38\text{nm},b=60\text{nm})$ while (e) and (f) for half axes $(a=48\text{nm},b=53\text{nm})$.

Figure 6

The parametric feedback cooling in only the rotational degrees of freedom for a nanodiamond $(a=48\text{nm},b=53\text{nm})$. All curves are averaged over 400 trajectories. (a) and (b) show the cooling in both ${\beta }_1$ and ${\beta }_2$ with the cooling parameters ${\mathrm{\Delta }}_1=\{{\eta }_{1,2,3}=0,{\zeta }_i={10}^{11}{\mathrm{s}/\mathrm{m}}^2\}$. The black and purple lines show the rotational motion gets cooled as we increase the feedback parameters from ${\mathrm{\Delta }}_1$ to ${\mathrm{\Delta }}_2=10{\mathrm{\Delta }}_1$. The red, green, and blue lines depict the heating trajectories in translational degrees of freedom. (c), (d), and (e) show the result of cooling in only ${\beta }_1$ with parameters ${\mathrm{\Delta }}_1=\{{\eta }_{1,2,3}=0,{\zeta }_2=0,{\zeta }_1={10}^{11}{\mathrm{s}/\mathrm{m}}^2\}$ and ${\mathrm{\Delta }}_2=10{\mathrm{\Delta }}_1$, and heating in ${\beta }_2$ and $x,y,z$, respectively. In (d), resonance heating causes massive heating in the uncooled ${\beta }_2$ degree of freedom. The dashed lines in (b), (d), and (e) are the heating curves $T=T_0+\dot{E}t$ with $T_0$ the initial temperature and $\dot{E}$ the corresponding heating rate from Table 1.

Figure 7

The steady-state occupation in terms of the feedback parameter $\eta$ for the $x$ degree of freedom of the particle ($a=48\text{nm},b=53\text{nm}$) from Table 1. The different curves correspond to several different values of classical uncertainty.

Figure 8

The steady-state occupation for the $x$ degree of freedom of particle ($a=48\text{nm},b=53\text{nm}$) from Table 1 in terms of the feedback parameter $\eta$ with $N=2$. Three different laser powers $P=(40\text{mW},70\text{mW},110\text{mW})$ are used. (a) The quantity $\mathrm{\Delta }n=0.083$. (b) The quantity $\mathrm{\Delta }n=0.026$.

Figure 9

The optimal cooling limit for the $x$ degree of freedom of particle ($a=48\text{nm},b=53\text{nm}$) from Table 1 with respect to $\mathrm{\Delta }n$. The blue and yellow curves correspond to the classical uncertainty measure $N=1$ and $N=2$, respectively. Our data stop at $\mathrm{\Delta }n=0.41$ since the feedback calculation with a larger $\eta$ becomes unstable when we try to reach the optimal cooling limit.