Kerr Enhanced Backaction Cooling in Magnetomechanics

Optomechanics is a prime example of light matter interaction, where photons directly couple to phonons, allowing the precise control and measurement of the state of a mechanical object. This makes it a very appealing platform for testing fundamental physics or for sensing applications. Usually, such mechanical oscillators are in highly excited thermal states and require cooling to the mechanical ground state for quantum applications, which is often accomplished by using optomechanical backaction. However, while massive mechanical oscillators are desirable for many tasks, their frequency usually decreases below the cavity linewidth, significantly limiting the methods that can be used to efficiently cool. Here, we demonstrate a novel approach relying on an intrinsically nonlinear cavity to backaction-cool a low frequency mechanical oscillator. We experimentally demonstrate outperforming an identical, but linear, system by more than 1 order of magnitude. Furthermore, our theory predicts that with this approach we can also surpass the standard cooling limit of a linear system. By exploiting a nonlinear cavity, our approach enables efficient cooling of a wider range of optomechanical systems, opening new opportunities for fundamental tests and sensing.

Figure 1

Nonlinear cooling illustration. (a) Optomechanical interaction between a mechanical oscillator and a nonlinear cavity. (b) To illustrate the enhanced cooling, we compare the response of the cavity photon number, $n_c$, of a linear to an identical nonlinear cavity. The cavity frequency changes due to the optomechanical interaction ($g_0⟨\widehat{x}⟩$), changing the probe-cavity detuning. For the linear case we recover a symmetric response, while for the nonlinear one the typical nonlinear response. This is identical to sweeping the probe tone itself through a fixed frequency cavity. The shaded area indicates the cooling work done on a mechanical system within a cycle by the cavity for parameters from the experiment. $X_{PP}$ denotes the peak-to-peak amplitude of the mechanical resonator. The cooling enhancement provided by the nonlinearity is clearly visible.

Figure 2

Setup and characterization. (a) Schematic depiction of our setup. $L$ and $L_J$ are the cavity and the junction inductance, $C$, $C_c$, and $C_G$ denote the self capacitance of the cavity, the coupling, and the ground capacitance. (b) Picture of the microstrip cavity (white) on top of a silicon substrate (golden). The cantilever chip (gray) is glued to the silicon substrate. An optical microscope picture (enlargement) shows the cantilever with false colored magnet above the SQUID loop. (c) Change of cavity frequency when applying external magnetic flux. Indicated are the measured coupling values $g_0$ as the flux sensitivity changes. White crosses symbolize measurements presented in the main text, gray crosses in [26]. (d) Characterization of the cavity nonlinearity. Close to bistability, the response is in good agreement with the model for a Kerr nonlinear cavity [26]. For low drive powers we effectively recover a linear response.

Figure 3

Nonlinear cooling measurement at $g_0/2\pi =201\pm 3\mathrm{Hz}$. (a) Phonon number $⟨n_m⟩$ against probe-cavity detuning, $\mathrm{\Delta }$, for two powers. (b),(c) Mechanical linewidth and frequency against probe-cavity detuning for the high power case. The suppressed heating prevents us from entering the instable regime ($\mathrm{\Gamma }/2\pi <0$). (d) Lowest phonon number, $⟨n_m{⟩}_{\min }$, for different input power. Right before bistability (shaded region), the cooling is limited by flux noise, which is captured in a simple model [26]. The errors shown are the standard errors [26].

Figure 4

Best cooling using higher $g_0$. (a) Mechanical noise spectral densities and corresponding fits at 40 mK and $g_0/2\pi =2136\pm 26\mathrm{Hz}$ for increasing backaction by changing the input power and the probe-cavity detuning. The noise floor decreases with increasing backaction, which is also a result of our cavity acting as a parametric amplifier [44]. $⟨n_{\mathrm{c}}⟩$ are the average number of photons circulating in the cavity for the measurement; the uncertainties arise from uncertainties in the input photon number, linewidth, and detuning. The uncertainties for the phonon numbers are fit errors. (b) Spectral density for one of the lowest mechanical occupation numbers we measured. We fit this trace with a model including flux noise (dashed, [26]). The uncertainty on the phonon number comes from the uncertainty of the offset determination for the numeric integration [26]). (c) Theory prediction for the lowest phonon occupation reachable when increasing $g_0$ always using the optimal input power for the nonlinear case ($\mathcal{K}/2\pi =-12\mathrm{kHz}/n_c$) up to bistability (orange, the dotted line is the input power). For the comparison to linear theory we either use the same power (blue line) or the ideal power for the linear case (red dashed). All other parameters remain constant. Nonlinear cooling allows cooling to $⟨n_m⟩=2.97$, while in the linear case cooling is limited to 3.18 phonons.