A Sub-kHz Mechanical Resonator Passively Cooled to 6 mK

TL;DR

This work demonstrates passive cooling of a low-frequency, massive nanomechanical cantilever to sub‑millikelvin temperatures using nuclear demagnetization, achieving 6.1 mK and verifying thermal equilibrium through Boltzmann energy statistics. A lock-in based readout tracks the cantilever energy, with <x>² = k_B T / k linking motion to temperature and a Boltzmann-distributed energy histogram confirming thermodynamic equilibrium. They observe saturation of T_cantilever around 6–8 mK in two runs, attributing it to a fixed thermal load or calibration offsets, and fit T_cantilever = (T_MFFT^n + T₀^n)^{1/n} with n ≈ 4 and T₀ ≈ 6 mK. The results establish a path toward sub-mK operation of low-frequency resonators, promising enhanced coherence and enabling stringent tests of quantum mechanics and collapse models, while outlining concrete improvements to reach lower temperatures and lower force noise.

Abstract

Fundamental tests of quantum mechanics, such as the generation of non-classical states and tests of wavefunction collapse models, are performed on increasingly larger size and mass scales. Highly coherent mechanical resonators, which also prove invaluable in ultrasensitive microscopy methods, are essential tools towards these efforts. Studying these resonators in a thermal equilibrium state at millikelvin temperatures provides a promising path to increase their coherence time. Here, we passively cool a 700 Hz, massive (1.5 ng) mechanical cantilever down to 6.1(4)mK by means of nuclear demagnetization, as confirmed by detecting its thermal motion via a lock-in based detection scheme. At the lowest temperatures the thermal motion of the resonator is still clearly distinguishable from the background noise. Our data analysis confirms that at these temperatures the motion is still thermally distributed. These results pave the way for passiveof cooling low-frequency resonators to the sub-milllikelvin regime, which would enable new tests of quantum mechanics and advances in ultrasensitive force detection.

Figures (14)

• Figure 1: Schematic illustration of the setup used for this work. A thermally isolated silver wire is linked to the nuclear demagnetization stage and connects different parts of the experiment, notably the cantilever, detection chip and input transformer to a temperature below the base temperature of the dilution refrigerator at 20mK. The lower right corner of the schematic shows the circuit that is used for the detection and calibration of the cantilever motion. The position inside the dilution refrigerator of different parts is indicated. In the center part the MFFT is illustrated inside its lead shielding (purple). The lower part shows the cantilever suspended above the detection circuit.
• Figure 2: The cantilever temperature is determined from the readout SQUID signal after post-processing. (a) The resonance frequency of the cantilever is visible as a peak in the amplitude spectral density when the cantilever thermal motion exceeds the (white) detection noise. (b) A digital lock-in amplifier is used at this resonance frequency on the SQUID time signal to obtain the cantilever amplitude as a function of time. (c) After plotting the energy in a histogram, the cantilever temperature is obtained from the mean energy and crosschecked through the slope of the energy distribution. The red shaded area indicates one standard deviation in the number of counts per bin.
• Figure 3: Temperature determined from the cantilever thermal motion versus bath temperature. (a) shows the thermal motion as observed in the power spectral density during measurement run A at various magnetic fields in the nuclear demagnetization stage. In (b) the cantilever temperature $T_{\textrm{cantilever}}$ is plotted against the MFFT temperature $T_{\textrm{MFFT}}$ during run A and B. The dashed lines indicate $T_{\textrm{cantilever}} = c T_{\textrm{MFFT}}$ corresponding to run A and B. The black dashed line indicates $c = 1$.
• Figure 4: The experimental setup used during the experiments presented in this work. The MFFT is visible in the lower right part. The lowest part of the setup contains the piezo motors and cantilever. The silver wire can be seen on the right of the image, it is thermally isolated from the mass-spring system by using LEGO® bricks.
• Figure 5: The sample holder of the instrument. In the center the detection chip can be seen on top of the Macor® plate. The silver wire connected to the nuclear demagnetization stage is also visible.