Digital holographic imaging for free surfaces of superfluid helium

Visualising the free surface of superfluid helium offers a rare opportunity to explore wave dynamics in the limit of vanishing viscosity. Such measurements are nonetheless challenging due to helium's low refractive index contrast, restricted optical access to the cryogenic setups required to maintain helium in its superfluid phase, and mechanical vibrations from the various cooling stages. Overcoming these limitations will enable quantitative studies of surface-wave dynamics with applications in fluid mechanics, quantum simulation, and quantum optomechanics. Here we report an implementation of off-axis digital holography for full-field imaging of the free surface of superfluid $^\text{4}$He. We perform non-contact measurements of nanometre- to micrometre-scale interface fluctuations in two cryogenic systems: a traditional helium bath cryostat and a cryogen-free refrigerator. We employ machine-learning-based analysis to isolate noise-driven normal modes and their spatial structure in both systems. This enables reconstruction of the dispersion relation for gravity-capillary waves in macroscopic samples and, for thick films, determination of the film thickness from the measured dispersion, providing a quantitative benchmark for our approach. These proof-of-concept experiments show that digital holography is a powerful and versatile tool for high-resolution, minimally invasive studies of superfluid surfaces, with strong potential for integration into diverse experimental platforms.