I will give an introduction to turbulent cascades statistical theories in classical homogeneous and isotropic, incompressible flows. I will discuss separately the three-dimensional and the two dimensional phenomenologies, and then discuss how in the presence of anisotropy or spatial confinement, we may observe features of both.

Discovered in 1937, the superfluid phase of liquid ⁴He (He II) remains a paradigm for exploring the rich and still only partially understood dynamics of quantum fluids. Among its most striking features are topological defects known as quantum vortices. In He II, applying heat flux in a closed end of a flow channel open at the other end generates counterflow turbulence, a unique form of turbulence characterized by a tangle of vortices where energy is transferred through vortex reconnections—with no classical equivalent.

Breaking the symmetry of this system by imposing rotation polarizes the vortex tangle, revealing a fundamentally new type of vortex dynamics. The initial state, a triangular vortex lattice governed by Feynman’s rule [1], evolves into distinct hydrodynamical regimes when counterflow is imposed to the rotating superfluid sample. These were first investigated by Swanson et al. [2] using second-sound attenuation measurements, which identified two regimes separated by critical heat-flux thresholds, marked by an increase of vortex-line density.

In this course, we will revisit these seminal experiments and complement them with direct visualization of quantum vortices using the CryoLEM (Cryogenic Lagrangian Exploration Module). This state-of-the-art, temperature-controlled rotating cryostat enables the direct observation of vortices in He II by injecting micron-sized solid H₂ or D₂ tracers that attach to the vortex cores [1,3]. A heater inside the experimental volume allows us to quantitatively study counterflow regimes under rotation.

Through direct imaging and advanced image analysis, we demonstrate that the first regime identified by Swanson et al. [2] corresponds to a wave-like deformation of the vortex lattice (see Fig. 1). The presence of inertial waves is confirmed by frequency analysis and comparison with the theoretical dispersion relation for vortex waves in rotating He II [4]. Preliminary observations also suggest the presence of Kelvin waves, warranting further investigation.

Above the second critical threshold, we show that the system transitions to a turbulent state with strongly interacting vortices. We also explore how temperature and rotation influence the critical thresholds for this transition. For temperatures above 2 K, our measurements align qualitatively with the extrapolated results from [2], highlighting the complementarity of distinct approaches. Together, these experiments provide new pathways toward the exploration of rotating quantum turbulence.

[1] C. Peretti et al., Sci. Adv. 9, eadh2899 (2023).
[2] C.E. Swanson et al., Phys. Rev. Lett. 50(3), 190 (1983).
[3] J. Vessaire et al., Rev. Sci. Instrum. 97, 025206 (2026).
[4] K. L. Henderson, C. F. Barenghi, Europhys. Lett. 67(1), 56 (2004).

Mechanical oscillators have probed superfluid helium since the field’s origins—classic examples include Andronikashvili’s torsional oscillator and Vinen’s vibrating wire. Some modern experiments instead require a macroscopic object to move through the superfluid at a controlled, constant speed, which is technically challenging when energy dissipation must be strictly limited at (sub)millikelvin temperatures. I will present examples of measurements made possible by constant-speed motion [1] in both superfluid ⁴He [2] and superfluid ³He [3], and discuss practical solutions used to realize low-dissipation motion. I will also outline several experiments utilizing motion at a constant speed that could become feasible in the future.

[1] Zmeev, D. E., J. Low Temp. Phys. 175, 480 (2014).
[2] Zmeev, D. E. et al., Phys. Rev. Lett. 115, 155303 (2015).
[3] Bradley, D. I. et al., Nature Phys. 12, 1017 (2016); Autti, S. et al., Nature Comm. 11, 4742 (2020); Autti, S. et al., Nature Comm. 14, 6819 (2023).

I will begin with a general outline of the specifics of quantum turbulence in ⁴He in the T = 0 limit as different from the T > 1 K regime. Existing paradigms and experimental evidence for them will be presented. Open questions and possible ways forward will be discussed. Experimental techniques, already available and speculative will be described.

Manchester experiments on the generation and characterization of the dynamics of various types of turbulence in ⁴He at T < 1 K will be presented:

• generation of vortex tangles by ion jet, towed grid, rotational agitation, and impulsive spin-down
• measurements of the vortex line density by scattering charged vortex rings off them
• experiments with vortex arrays and tangles in rotation
• experiments with the motion of electrons along straight vortex lines and through a vortex tangle
• detection of a propagating turbulent front using vortex-trapped electrons and He₂^* excimers
• experiments with unidirectional vortex rings and their mutual scatterings
• visualization of the motion of micron-sized polymer particles through vortex tangles.

The contribution of visualisation techniques to the phenomenological description of flows of superfluid helium-4 is summarised. The focus is on flows characterised by relatively high fluid velocity, occurring in the temperature range where viscous effects are not negligible. In particular, similarities and differences with related flows of Newtonian fluids are reported, including investigations on thermally generated vortex rings and on starting vortex flows at high Reynolds numbers. Special emphasis is given to the dependence of the observed features on the probed flow scales and on the presence of thermal gradients.

During this lecture, I will give an introduction to helium-3, discuss its basic properties and connections to various fields of physics, cover the experimental techniques required to cool a sample of helium-3 down to sub-mK temperatures and for studying it, and go through a few examples from recent research development.

We discuss experimental studies of quantum turbulence in superfluid ³He-B, the coldest currently accessible fermionic liquid. In addition to normal scattering, fermionic excitations can undergo Andreev reflection. At low temperatures, where the thermal excitations in the superfluid comprise ballistic quasiparticles, this process underpins the non-invasive imaging of topological structures such as quantum vortices or textures.

These structures can be produced via analogues of cosmological processes—for example, the Kibble mechanism—or by exceeding the Landau critical velocity to break the condensate. We have developed a 5x5 pixel quasiparticle camera operating at 150 microkelvin and demonstrate two-dimensional images of a quasiparticle beam and a tangle of quantized vortices (quantum turbulence) generated by a mechanical oscillator.

To fully understand this turbulence, the ability to study single vortices is essential. Nano-electromechanical systems (NEMS) are strong candidates for probing single-vortex dynamics, as they exhibit low power consumption and minute energy dissipation. Here, we utilize a Si₃N₄ beam with aluminium metallization to investigate phenomena occurring in superfluid ⁴He at temperatures as low as 10 mK. Our results demonstrate that these nanoscale beams are ideally suited for the real-time detection of individual quantum vortices. We will present the interaction between our beam and a captured single vortex while offering insights into the dissipative processes that occur during this interaction.

In this lecture I will review the main experimental technique to produce and probe a superfluid of ultracold atoms, highlighting the main characteristic of the bosonic and fermionic cases. I will then focus on the study of vortex dynamics on these platforms, reviewing the main experimental tools to inject and detect vortices together with some recent results achieved in the atomic superfluid community.