Cavity cooling of an optically trapped nanoparticle

We study the cooling of a dielectric nanoscale particle trapped in an optical cavity. We derive the frictional force for motion in the cavity field and show that the cooling rate is proportional to the square of oscillation amplitude and frequency. Both the radial and axial components of the center-of-mass motion of the trapped particle, which are coupled by the cavity field, are cooled. This motion is analogous to two coupled but damped pendulums. Our simulations show that the nanosphere can be cooled to $e^{-1}$ of its initial momentum over time scales of hundredths of milliseconds.

Figure 1

Schematic diagram for cooling a small dielectric nanoscale particle in a high-finesse cavity.

Figure 2

A plot of the particle (a) position, (b) velocity, (c) square of field amplitude [Eq. (5)], and (d) the friction force $\frac{2\pi m\beta }{\kappa }{\Omega }^2{x}^{2}v$ derived from Eq. (6). These values are derived from the numerical solution of Eq. (6) for optimal detuning and initial particle position of 100 nm and velocity of 3 cm/s over a time of scale 40 $\mu$s.

Figure 3

A three-dimensional plot showing the variation of the energy damping rate as a function of cavity detuning $\Delta$ and oscillation amplitude of nanosphere position $x_0$. Maximal cooling occurs at negative detuning.

Figure 4

A phase-space plot showing damped motion in position over 1 ms. Note that the cooling rate decreases as the particle becomes localized in the lattice potential.

Figure 5

Plot of the amplitude of the oscillatory energy as a function of time to compare of cooling times for a constant input intensity into the cavity of 1 mW (solid line) and an exponentially decreasing input intensity with a $e^{-1}$ decay constant of 50 ms (dashed).

Figure 6

Plots illustrating the evolution in the axial and radial velocity of the trapped sphere. The axial velocity has a period of 8.2 microseconds and the radial velocity a period of 2.2 ms. The reduction in the amplitude of the axial oscillation indicates cooling over this small time period.

Figure 7

A plot of intracavity intensity calculated for the first 3 ms of cooling. The fast modulation of intensity with a period of 4.1 microseconds (not resolved) is due to the axial motion of the particle in the cavity field. The slower modulation with a period of 1.1 ms is due to the radial motion of the particle. The axial and radial motion as shown in Fig. 6. oscillates at the half the frequency of the intensity modulation.

Figure 8

A plot of the amplitude of the oscillatory velocity of the particle for both the axial ($v_x$) and radial ($v_r$) oscillatory motion as a function of time for an input field of constant intensity. The lower curve which more rapidly decays is the axial velocity while the upper curve is the radial velocity.

Figure 9

Plots of the decay of the amplitude of oscillating total kinetic energy in the cavity as a function of time, including both axial and radial motions. The upper curve, which slowly decays, is for a fixed input intensity while the lower curve is for an exponential decaying input intensity with a time constant of 50 ms.