Nanoscale real-time detection of quantum vortices at millikelvin temperatures

Abstract

Since we still lack a theory of classical turbulence, attention has focused on the conceptually simpler turbulence in quantum fluids. Reaching a better understanding of the quantum case may provide additional insight into the classical counterpart. That said, we have hitherto lacked detectors capable of the real-time, non-invasive probing of the wide range of length scales involved in quantum turbulence. Here we demonstrate the real-time detection of quantum vortices by a nanoscale resonant beam in superfluid ⁴He at 10 mK. Essentially, we trap a single vortex along the length of a nanobeam and observe the transitions as a vortex is either trapped or released, detected through the shift in the beam resonant frequency. By exciting a tuning fork, we control the ambient vortex density and follow its influence on the vortex capture and release rates demonstrating that these devices are capable of probing turbulence on the micron scale.

Fig. 1. Schematic of the experimental setup.

A tuning fork generates quantum turbulence, whilst a 70-μm-long nanomechanical beam, suspended 1 μm above the substrate, acts as the detector. The beam and fork are driven by vector network analysers or signal generators through several stages of attenuation at various temperatures. The beam and fork signals are amplified at room temperature by an 80-dB amplifier and an I/V converter. For a detailed description see the Supplementary Information.

Fig. 2. The magnitude of the nanobeam response at each excitation frequency against time taken from the start of the first event in heat-map format.

Before point α₁ the beam is in the default vortex-free state. Between α₁ and β₁ a vortex interacting with the beam gradually raises the beam frequency by 3 kHz, finally becoming captured along the entire length of the beam at β₁. From β₁ to γ₁ the resonance is stable for 20 ms. The captured vortex interacts with a nearby vortex and at point γ₁/δ₁ the system suddenly resets via reconnection of the trapped and attracted vortices and the beam resonance jumps back to the vortex-free state. After 14.35 s a second event at α₂ occurs with similar features. The cartoons along the top of the figure sketch the broad processes involved, although the precise details of the capture and release mechanisms are not completely understood.

Fig. 3. The capture process.

a The tuning fork velocity as a function of the applied force on the left axis and the rate of detected events by the beam on the right axis. The blue circles correspond to the tuning fork force-velocity dependence, while the symbols on the right show the beam detection rate at various fork forces. The dotted blue line corresponds to the onset of turbulence production by the tuning fork. b A probability density function of the wait time between events t_δα at the same fork velocities. The solid lines correspond to exponential fits, of the form $\propto \exp \left(-t/\tau \right)$. Note symbol colour matching between panels a and b, black points in panel a represent data from other runs not used for panel b. For details, see text.

Fig. 4. The release process.

A probability density function (PDF) of captured vortex lifetimes t_αδ at selected fork velocities. The discrete data at long lifetimes are the result of single observed events. The data point colours reflect the same data as in Fig. 3.