One of the relevant tasks in modern hydrodynamics is the study of quantum turbulence, as well as the mechanisms of formation and decay of quantum vortices in quantum liquids and gases. Quantum vortices in superfluid helium‑4 are topological defects that must begin and end on the surface of the liquid or on the walls of the container [1], or they can form closed loops in the bulk [2]. The principal difference from classical vortices is the quantization of the flow circulation around the vortex core, with the circulation quantum being equal to $h/m_{He}$ , where  is the Planck constant and $m_{He}$ is the mass of a helium‑4 atom.

Recent theoretical and experimental studies of the nonlinear interaction between waves propagating on the surface of a classical viscous fluid (for example, water) have demonstrated that the surface waves’ interaction can lead to the generation of vortex flows on the surface of a viscous fluid [3]. The features of vortex flow generation by waves on the surface of a quantum fluid, where normal and superfluid helium coexist simultaneously, had not been previously investigated theoretically. As our experiments [4] have shown, when the wave amplitudes exceed critical values, waves on the surface of He‑II can excite not only classical vortices in the normal helium but also quantum vortices under the surface of the liquid layer.

This work is devoted to studying the penetration depth of quantum vortices, which are formed during the nonlinear interaction of two mutually orthogonal capillary waves on the surface of the He‑II layer, into the liquid bulk at temperatures  $T $ = 1.5 and 1.7 K. To detect the excitation of quantum vortices, negative charges (“electron bubbles” with a radius of 17 Å) were injected into liquid helium were used as tracer particles, which can be captured by quantum vortices at liquid temperatures below 1.7 K. It has been established that the quantum vortices generated by capillary waves with a frequency of $ f = 49.9 $ Hz are concentrated under the liquid surface in a narrow layer less than 50 nm deep, which is much less than the depth of the viscous sublayer of normal helium,  $\delta \sim 5 μm$.

[1] P. Moroshkin, P. Leiderer, K. Kono, S. Inui and M. Tsubota, "Dynamics of the Vortex-Particle Complexes Bound to the Free Surface of Superfluid Helium", Phys. Rev. Lett. 122, 174502 (2019);

[2] P. M. Walmsley, A. I. Golov, H. E. Hall, A. A. Levchenko and W. F. Vinen, "Dissipation of Quantum Turbulence in the Zero Temperature Limit", Phys. Rev. Lett. 99, 265302 (2007);

[3] S. S. Vergeles, A. A. Levchenko and V. M. Parfenyev, "Nonlinear generation of currents by waves on liquid surface", Phys. Usp. 68, 1131 (2025);

[4] P. G. Selin, I. A. Remizov, M. R. Sultanova, E. V. Lebedeva, A. A. Levchenko and L. P. Mezhov-Deglin, "Threshold of Generation of Quantum Vortices by Waves on the Surface of Superfluid He-II", JETP Lett. 122, 178 (2025).

I.A. Remizov, P.G. Selin, E.V. Lebedeva et al.
JETP Letters 124, issue 8 (2026)