Sideband cooling of nearly degenerate micromechanical oscillators in a multimode optomechanical system

Multimode optomechanical systems are an emerging platform for studying fundamental aspects of matter near the quantum ground state and are useful in sensitive sensing and measurement applications. We study optomechanical cooling in a system where two nearly degenerate mechanical oscillators are coupled to a single microwave cavity. Due to an optically mediated coupling the two oscillators hybridize into a bright mode with a strong optomechanical cooling rate and a dark mode nearly decoupled from the system. We find that at high coupling, sideband cooling of the dark mode is strongly suppressed. Our results are relevant to novel optomechanical systems where multiple closely spaced modes are intrinsically present.

Figure 1

Setup. (a) Our multimode optomechanical system is analogous to an optical cavity with two compliant end mirrors. (b) Pumping scheme. (c) Cryogenic reflection measurement setup. The pump tone ${\omega }_P$ is combined with the probe tone ${\omega }_{\text{in}}$ from a vector network analyzer (VNA) and coupled to the sample. The reflected signal is amplified and recorded by either the VNA or a signal analyzer (SA).

Figure 2

Linear response. (a) Cavity linear response for pump powers increasing in steps of 5 dB from bottom to top, corresponding to $G_1=2\pi \times 357\mathrm{Hz}$ (bottom curve) and to $G_1=2\pi \times 113\mathrm{kHz}$ (top curve). Each curve is vertically offset by 10 dB. Black curves are a theory prediction (see text). At high pump powers, splitting of the spectrum into hybrid normal modes is visible. (b) Detail of panel (a), showing anti-Stokes peaks of the individual mechanical modes at low power, and the formation of a single dark mode at high power. (c) Response as function of pump detuning for $G_1\approx 2\pi \times 94\mathrm{kHz}$. Inset: Detail showing dark mode resonance.

Figure 3

Noise spectrum. (a) Cavity output spectrum for varying pump power ranging from $G_1=2\pi \times 1.6\mathrm{kHz}$ (lower curve) to $G_1=2\pi \times 101\mathrm{kHz}$ (upper curve) in 3 dB steps. Black lines show a theory fit (see text). For each curve the background level is subtracted, and subsequently vertically offset by ten units for clarity. (b) Detail of panel (a). (c) Frequency and (d) linewidth of the eigenmodes. Symbols correspond to Lorentzian fits to the data, lines show the predicted eigenmodes from the three-mode model. At small $G_1$ these correspond to the mechanical (red circles, blue squares) and cavity (green dashed line) modes, respectively; at large $G_1$ the mechanical modes mix to dark (purple downward triangles) and bright (yellow upward triangles) modes. Dotted line shows the expected linewidth for a system with a single mechanical oscillator. Error bars show statistical 95% confidence levels.

Figure 4

Sideband cooling. (a) Theory prediction of the mode occupation of the bare mechanical modes (dashed lines) and the mechanic-like eigenmodes of the three-mode system (solid lines), in the absence of technical heating effects. Modes are colored as in Figs. 3 and 3 and labeled in the figure. (b) Fitted effective environment temperatures in the experimental data, expressed in quanta. (c) Occupation of the eigenmodes taking the fitted environment temperatures into account (solid lines). Symbols show the temperatures inferred based on Lorentzian fits to the data according to Eq. (3). The data were normalized to correspond to the fridge temperature at low pumping power. Error bars show statistical 95% confidence levels. Colors and symbols as in Figs. 3 and 3.