Digital Holography Images Nanometre-to-Micrometre Fluctuations on Superfluid Helium Free Surfaces

Visualising the surface of superfluid helium presents a significant challenge, yet offers a unique window into wave behaviour at extremely low viscosities. Vitor S. Barroso, Patrik Švančara, and Chris Goodwin, all from the School of Mathematical Sciences at the University of Nottingham, alongside colleagues Sreelekshmi C. Ajithkumar, Ilaria Dimina, and Silvia Schiattarella, now demonstrate a breakthrough in this field. The team successfully implements off-axis digital holography to create full-field images of superfluid helium’s free surface, overcoming difficulties posed by the fluid’s low refractive index and the demanding conditions of cryogenic experiments. This innovative technique allows for non-contact measurement of interface fluctuations at the nanometre to micrometre scale, and crucially, enables quantitative studies of surface-wave dynamics with potential applications spanning fluid mechanics, simulation, and optomechanics, establishing a powerful new tool for investigating superfluid behaviour.

The method illuminates the helium surface with a coherent laser beam and records the resulting interference pattern with a camera. Analysis of this pattern reveals the three-dimensional shape of the surface and any changes occurring over time, achieving a temporal resolution of 10 microseconds. This allows detailed observation of surface waves, thermal fluctuations, and other phenomena characteristic of the superfluid state. The team quantifies the amplitude and frequency of surface excitations by employing phase-shifting interferometry to reconstruct the surface profile.

They demonstrate the ability to resolve features with a height resolution of 1. 2 nanometres and a lateral resolution of 2. 5 micrometres. This high resolution enables the study of capillary waves and investigation of the interplay between hydrodynamic and quantum effects. The research introduces a novel experimental approach to studying superfluidity, providing direct visualisation of surface dynamics at the nanoscale and opening new avenues for investigating other quantum fluids and soft matter systems.

Superfluid Helium Surface Imaging via Digital Holography

Visualising the free surface of superfluid helium offers a unique opportunity to explore wave dynamics with negligible viscosity. Such measurements are nonetheless challenging due to helium’s low refractive index contrast, restricted optical access within cryogenic setups, and mechanical vibrations from cooling stages. Overcoming these limitations enables quantitative studies of surface-wave dynamics with applications in fluid mechanics, quantum simulation, and quantum optomechanics. Researchers report the first implementation of off-axis digital holography for full-field imaging of superfluid helium’s free surface.

This technique allows for non-destructive, quantitative measurements of surface topography with high spatial resolution and sensitivity. The experimental setup consists of a cryogenic cell containing superfluid helium, illuminated by a single-mode laser beam. Interference patterns created by reflected light from the helium surface and a reference beam are recorded by a high-resolution camera. These holograms are then digitally reconstructed to obtain a three-dimensional map of the helium surface, revealing details of wave motion and other surface phenomena. The method provides full-field imaging, capturing the entire surface in a single shot, enabling the study of complex wave interactions and dynamics.

Superfluid Helium Surface Imaged with Holography

Researchers have successfully implemented off-axis digital holography to image the free surface of superfluid helium, a challenging feat due to the fluid’s unique properties and the constraints of cryogenic environments. This achievement allows for the non-contact measurement of nanometre- to micrometre-scale fluctuations on the helium surface within both traditional bath cryostats and cryogen-free refrigerators. By employing machine-learning techniques, the team isolated and analysed noise-driven normal modes, reconstructing the dispersion relation for capillary waves and accurately determining film thickness in thicker samples. This work demonstrates the versatility and precision of digital holography as a practical tool for investigating superfluid surfaces, opening new avenues for research in fluid mechanics, simulation, and optomechanics.

The successful deployment of this technique in existing cryogenic setups, without requiring major modifications, highlights its accessibility to a wider range of laboratories. Researchers acknowledge that their implementation prioritised flexibility and broad applicability. Future research directions include exploring surface-mediated quantum turbulence, investigating the coupling between vortices and waves, and developing coherent dynamics studies in superfluid helium-based quantum field theory simulators. The team’s results establish a foundation for next-generation experiments, promising deeper insights into the behaviour of this fascinating quantum fluid.

Detailed Methods, Robustness, and Supplementary Data

This document provides detailed supporting information for the main research paper, explaining the methodology and demonstrating how data was processed, analysed, and interpreted. It also demonstrates the robustness of the results by showing alternative analyses and error estimations, and provides additional data that support the main findings. Researchers detail the process of identifying and characterizing the normal modes of the superfluid helium film, explaining how the spatial profiles of these modes are determined using holographic data and mathematical modelling. They highlight the importance of transforming coordinates to polar form for analysing the modes and discuss the challenges of accurately determining the symmetry origin.

Supplementary figures show the expected Bessel mode patterns, illustrating the experiment’s limited field of view. The document provides a more detailed explanation of how the dispersion relation is reconstructed from the experimental data. Researchers describe how multiple independent reconstructions of the dispersion relation are performed to ensure the robustness of the results and explain how the data is used to estimate the thickness of the helium film. Supplementary figures demonstrate the consistency of the data obtained through this extended reconstruction. Additional visual data supports the main findings, with figures showing interference patterns obtained with and without the superfluid film.

The presence of the superfluid film introduces a divergence in the probe beam, likely due to the curvature of the superfluid interface. Key techniques and analyses highlighted include holographic interferometry, Bessel function fitting, polar coordinate transformation, dispersion relation reconstruction, and statistical analysis. This supplementary information provides a comprehensive and detailed account of the methods and analyses used in the research, allowing readers to fully understand and evaluate the findings.