The importance of trapping and cooling techniques is evidenced by the series of Nobel Prizes awarded to this theme (1989 ion trapping, 1997 atomic trapping, 2001 Bose-Einstein condensation, 2012 trapping of photons and ions). The trapping and cooling of mesoscopic objects, such as nanoparticles, is a new territory and lies at the boundary of classical and quantum physics. The objective of this project is to control the dynamics of a nanoscale object with high precision and to study interactions on the mesoscopic length scale, - the grey zone between the discrete atomistic world and the continuous world of macroscopic systems.

A dielectric nanoparticle is captured by the gradient force of a focused laser beam in ultrahigh vacuum and its center-of-mass motion is controlled by optical back-action. To cool the nanoparticle close to its quantum ground state we explore active parametric feedback in combination with passive cavity-based cooling. The figure on the right shows a photograph of light scattered from a 85 nm fused silica particle (arrow). The object to the right of the particle is the outline of the objective lens. Also shown is a time trace of the particle’s Χ coordinate (transverse to the optical axis) at 2mbar pressure.

The classical equation of motion of the particle’s Χ coordinate can be written as the following function

where Ffluct is a random Langevin force that satisfies <Ffluct (t) Ffluct(t')> = 2mΓ0kBTδ(t-t'). Γ0 and Ω0 are the particle’s natural damping constant and oscillation frequency, respectively. Optical back-action gives rise to a modification of the particle's response, which is accounted for by ΔΩ and ΔΓ, respectively. The nature of the back-action force is in principle not relevant: it can be a force due to radiation pressure, a thermal force, or an optical force due to the modulation of laser power.

The heating of the center-of-mass motion is due to the random impact of air molecules inside the vacuum chamber, that is Γ0 ∞ Pgas, where Pgas is the gas pressure. In high vacuum at Pgas = 10^{-6}mbar we measure Γ0 **≈ **3 μHz. This corresponds to a quality factor of Q = 10^{8}!

To increase δΓ and hence to slow down the center-of-mass motion we employ parametric feedback cooling (see figure above). The feedback hinders the particle’s motion by increasing the trap stiffness whenever the particle moves away from the trap center and reducing it when the particle falls back toward the trap. In the frequency domain, this corresponds to a modulation at twice the trap frequency with an appropriate phase shift. Frequency doubling and phase shifting is done independently for each of the photodetector signals x, y and z. Since the three directions are spectrally separated (see figure below), there is no cross-coupling between the three signals, that is, modulating one of the signals does not affect the other signals. Therefore, it is possible to sum up all three feedback signals and use the result to drive a single Pockels cell that modulates the power of the trapping laser. Thus, using a single beam we are able to effectively cool all spatial degrees of freedom.

According to the equipartition principle, the center-of-mass temperature Tcm follows from

kBTcm = m(Ω0 +δΩ)2 <χ2>. Considering that δΩ≪ 0 we obtain

where T is the equilibirum temperature in the absence of parametric feedback (δΓ = 0). Thus, the temperature of the oscillator can be raised or lowered, depending on the sign of δΓ. With parametric feedback cooling we reach temperatures of a few milliKelvin.

A laser-trapped nanoparticle is physically decoupled from its environment, which guarantees extremely long coherence times and quality factors as high as 10^{8} at a pressure of 10^{-6}mbar. Force sensitivities of 10^{-20} N/√Hz can be achieved even at room temperature, which outperforms many other measurement techniques by orders of magnitude. Thus, a laser-trapped nanoparticle can be used as a local probe for measuring mesoscopic interactions, such as Casimir forces, vacuum friction, non-equilibrium dynamics and phase transitions, with ultrahigh accuracy. A nanoparticle cooled to its quantum ground state also opens up the possibility to measure the collapse of quantum superposition states under the influence of noise and gravity-induced quantum state reduction.

Related Publications:

J. Gieseler, R. Quidant, C. Dellago, and L. Novotny, ”Dynamic relaxation of a levitated nanoparticle form a non-equilibrium steady state,” Nat.Nanotechnol. 9, 358 (2014).

J. Gieseler, M. Spasenovic, L. Novotny, and R. Quidant, ”Nonlinear mode coupling and synchronization of vacuum-trapped nanoparticles,” Phys. Rev. Lett. 112, 103603 (2014).

J. Gieseler, L. Novotny, and R. Quidant, ”Thermal nonlinearities in a nanomechanical oscillator” Nature Physic 9, 806-810 (2013).

J. Gieseler, B. Deutsch, R. Quidant, and L. Novotny, ”Sub-Kelvin parametric feedback cooling of a laser-trapped nanoparticle,” Phys. Rev. Lett. 109, 103603 (2012).