Cooling Mechanical Oscillators by Coherent Control

In optomechanics, electromagnetic fields are harnessed to control a single mode of a mechanically compliant system, while other mechanical degrees of freedom remain unaffected due to the modes’ mutual orthogonality and high quality factor. Extension of the optical control beyond the directly addressed mode would require a controlled coupling between mechanical modes. Here, we introduce an optically controlled coupling between two oscillation modes of an optically levitated nanoparticle. We sympathetically cool one oscillation mode by coupling it coherently to the second mode, which is feedback cooled. Furthermore, we demonstrate coherent energy transfer between mechanical modes and discuss its application for ground-state cooling.

Figure 1

(a) Experimental setup. The trapping laser (1064 nm) passes two modulators for intensity and polarization modulation, before it is combined with a measurement laser (780 nm) at a dichroic beam splitter. Both lasers are focused by an objective inside a vacuum chamber and recollimated by a lens. The trapping laser is split off by a dichroic beam splitter. The measurement laser is guided to the detection system to measure the particle position $x$, $y$, $z$, which is used to derive a feedback signal to heat or cool the particle’s motion. (b) Power spectral densities of particle motion along $x$ and $y$. The solid lines are Lorentzian fits. (c) Illustration of the optical potential in the focal plane. The potential is stiffer along the $y$ direction due to the polarization of the trapping laser. (d) Illustration of the optical potential rotated by an angle $\varphi$ around the optical axis.

Figure 2

(a) Rabi oscillations of mode energy. The particle is initialized with high energy in the $x$ mode by parametric heating, while the $y$ mode is feedback cooled. At time $t=0$ the feedback is turned off and remains off during the entire measurement. At time $t=5.5\mathrm{ms}$ [denoted as (1)], we turn on the modulation of the polarization angle of the trapping beam and observe periodic energy exchange between the modes. The symbols denote measurements, the solid lines theory. (b) Bloch sphere representation of the measurement in (a). At $t=0$ the system is initialized to a point on the green dashed line on the upper hemisphere, with the precise location determined by the phase between the oscillator modes. When the modulation is switched on, the system in (a) is at the point denoted as (1), and the Bloch vector of the system rotates along the solid green line.

Figure 3

Sympathetic cooling of the $x$ mode. The system is started with the coupling off and the $y$ mode cooled to $0.01k{T}_0$. The $y$ mode is feedback cooled the entire time. No feedback cooling is applied to the $x$ mode throughout the entire measurement. The initial energy of the $x$ mode is $0.8k{T}_0$. At time $t=0$, the coupling between the $x$ and $y$ modes is turned on and energy is transferred from the hot $x$ mode to the $y$ mode, from where it is removed by feedback cooling.

Figure 4

(a) Protocol for cooling by energy transfer. The system is started at an arbitrary point on the Bloch sphere, here chosen to be the point on the equator denoted by (1). Coupling between the modes rotates the Bloch vector around the ${\mathit{e}}_1$ axis. Observing the Rabi oscillations allows prediction of the passage through point (2), where the phase of the modulation signal is switched, such that the Bloch vector rotates around the ${\mathit{e}}_2$ axis. Again, the Rabi oscillations are monitored to predict passage through the pole of the Bloch sphere, denoted by (3), where the modulation is switched off. (b) Experimental implementation of the energy-transfer protocol. The system is brought from point (1), with equal energy of $0.6k_B{T}_0$ in both modes, to a state where the $y$ mode carries $0.01k_B{T}_0$. The data points are measured; the solid lines serve as guides to the eye.