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Introduction to Quantum Turbulence SPECIAL FEATURE: INTRODUCTION INTRODUCTION SPECIAL FEATURE: Introduction to quantum turbulence Carlo F. Barenghia,1, Ladislav Skrbekb, and Katepalli R. Sreenivasanc aJoint Quantum Centre Durham-Newcastle and School of Mathematics and Statistics, Newcastle University, Newcastle upon Tyne NE1 7RU, United Kingdom; bFaculty of Mathematics and Physics, Charles University, 12116 Prague, Czech Republic; and cDepartment of Physics and Courant Institute of Mathematical Sciences, New York University, New York, NY 10012 The term quantum turbulence denotes the turbulent motion of quantum fluids, systems such as superfluid helium and atomic Bose–Einstein condensates, which are characterized by quantized vorticity, superfluidity, and, at finite temperatures, two-fluid behavior. This article intro- duces their basic properties, describes types and regimes of turbulence that have been observed, and highlights similarities and differences between quantum turbulence and classical turbulence in ordinary fluids. Our aim is also to link together the articles of this special issue and to provide a perspective of the future development of a subject that contains aspects of fluid mechanics, atomic physics, condensed matter, and low-temperature physics. quantized vortices | absolute zero | Kolmogorov spectrum Turbulence is a spatially and temporally com- Quantum Fluids resulting either in superconductivity in plex state of fluid motion. Five centuries ago, In this series of articles, we shall be concerned charged systems of electrons in crystal lattices Leonardo da Vinci noticed that water falling almost exclusively with superfluid 4He, the or in superfluidity in systems consisting of into a pond creates eddies of motion. Today, B-phase of superfluid 3He,and,toalesser neutral atoms. The outcome is surprisingly turbulence still provides physicists, applied extent, with ultracold atomic gases. These similar to what happens for bosons: Super- mathematicians, and engineers with a con- systems exist as fluids at temperatures on the fluidity arises from the formation of a co- tinuing challenge. Leonardo realized that the order of a Kelvin, milliKelvin, and micro- herent particle field that can be described ‡ motion of water shapes the landscape. Today’s Kelvin, respectively. Their constituents are using the formalism of the order parameter researchers appreciate that many physical either bosons (such as 4He atoms with zero or a condensate macroscopic wave function. processes, from the generation of the Galactic spin) or fermions (such as 3He atoms with Superfluidity of 4He was experimentally magnetic field to the efficiency of jet engines, spin 1/2). This difference is fundamental: discovered by Kapitza and by Allen and depend on turbulence. the former obey Bose–Einstein statistics and Misener in 1938 although it is now believed The articles in this collection are devoted the latter Fermi–Dirac quantum statistics. that Kamerlingh Onnes must have had su- to a special form of turbulence known as Letusconsideranideal(noninteracting) perfluid helium in his apparatus when he first quantum turbulence (1–3), which appears gas of bosons first. Under normal conditions liquefiedheliuminLeidenin1908.Heand in quantum fluids. Quantum fluids differ at room temperature, the de Broglie wave- other pioneers of low-temperature physics from ordinary fluids in three respects: (i) length λ of each atom is much smaller than soon discovered that, below a critical tem- they exhibit two-fluid behavior at nonzero the average separation d between the atoms; perature Tλ ≅ 2:17 K, liquid helium displays temperature or in the presence of impu- if the temperature T is lowered, λ increases unusual behavior. They therefore called it rities, (ii) they can flow freely, without the until, if T is sufficiently small, λ becomes helium I and helium II, respectively, above dissipative effect of viscous forces, and larger than d, and the quantum–mechanical and below this temperature. Still, in 1938, (iii) their local rotation is constrained to wave aspects become dominant. The result- London linked the properties of helium II to discrete vortex lines of known strength ing phase transition, called the Bose–Einstein Bose–Einstein condensation. Further prog- (unlike the eddies in ordinary fluids, which condensation (5), is characterized by a mac- ress in understanding superfluidity of 4He are continuous and can have arbitrary size, roscopic number of bosons occupying the was driven by the work of Landau based on shape, and strength). Superfluidity and quan- state of zero energy. Although the possibility different considerations. tized vorticity are extraordinary manifesta- of Bose–Einstein condensation was raised in The physical properties of normal liquid tions of quantum mechanics at macroscopic- 1924–1925, its direct experimental demon- 3He at milliKelvin temperatures are well de- length scales. stration in dilute ultracold atomic gases oc- scribed in the frame of the Fermi liquid theory Recent experiments have highlighted quan- curred only in 1995. To achieve superfluidity titative connections, as well as fundamental (flow without friction), another ingredient is Author contributions: C.F.B., L.S., and K.R.S. performed research differences, between turbulence in quantum necessary: The particles must interact with and wrote the paper. fluids and turbulence in ordinary fluids each other. The authors declare no conflict of interest. (classical turbulence). The relation between In fermionic systems, at temperatures 1To whom correspondence should be addressed. E-mail: c.f. thetwoformsofturbulenceisindeeda much lower than a characteristic Fermi tem- [email protected]. † common theme in the articles collected here. perature, particles occupy the interior of The term quantum turbulence was introduced into the literature Because different scientific communities (low- the Fermi sphere in the momentum space, in 1986 by R. J. Donnelly in a symposium dedicated to – the memory of G. I. Taylor (4). temperature physics, condensed-matter phys- with only relatively few particle hole pairs, ‡ What determines the need for a quantum mechanical description ics, fluid dynamics, and atomic physics) have called excitations. Attractive interaction be- is not the absolute value of temperature but whether it is contributed to progress in quantum turbu- tween fermions leads to an instability and lower than a certain characteristic temperature of the system (for † example, the Fermi temperature); instances of quantum fluids lence, the aim of this article is to introduce the formation of Cooper pairs, which are at high temperature are exciton-polariton condensates ( ≈ 300 K) the main ideas in a coherent way. bosons undergoing Bose condensation, and neutron stars (106 to 109 K). www.pnas.org/cgi/doi/10.1073/pnas.1400033111 PNAS | March 25, 2014 | vol. 111 | suppl. 1 | 4647–4652 Downloaded by guest on September 29, 2021 of Landau. Note the striking difference in emission of quasiparticles called rotons). In mechanism when helium is cooled through kinematic viscosities of 4He (the lowest of a more general sense, the Landau criterion the superfluid second-order phase transition: all known fluids, two orders of magnitude less applies to any superfluid: On exceeding cer- The phase of ψ, unable to adjust everywhere than water’s) and of 3He near the superfluid tain critical velocity (which in fermionic at the same time, leaves vortex lines as defects. transition (comparable with that of air or superfluids is called the Landau pair-braking The possibility of quantum turbulence was olive oil). Superfluidity of 3He was theoreti- velocity), it becomes energetically favorable first raised by Feynman (6); soon afterward, cally predicted by Pitaevskii and experimen- to generate quasiparticles, which means the Vinen showed that a turbulent vortex tangle tally discovered by Osheroff, Richardson, and onset of dissipation. can be generated in the laboratory (7) by Lee in 1973. Cooper pairs consisting of two What makes superfluid hydrodynamics applying a heat flux to helium II. Two 3 He atoms (which themselves are fermions), particularly interesting is that the circulation properties of vortex lines are important for # v · dr rotating about their center of mass, are bosons integral C s is equal either to the quantum turbulence. The first is the mutual of total spin and orbital numbers equal to quantum of circulation κ = h=m or to zero, friction force (7, 8), which couples the su- one. This property allows three different depending on whether or not the integration perfluid and the normal fluid. It arises from projections on quantization axes in or- path C encloses a quantized vortex line; here, the scattering of thermal quasiparticles (con- h ’ m bital and spin spaces, and, as a conse- is Planck sconstantand the mass of the stituents of the normal fluid) by the velocity 4 quence, several different superfluid phases relevant boson (one atom in He, a Cooper field of the vortex lines. The second property of 3He exist. The classification of them and pair in 3He-B). This quantization condition, 3 comprises Kelvin waves, which are helical of the types of quantized vorticity in He is suggested by Onsager and experimentally displacements of the vortex core that rotate beyond the scope of this article (in partic- confirmed by Vinen, arises from the exis- with angular velocity ω ∼ κk2, where k is the ular, because quantum turbulence has been tence and the single-valuedness of a complex, one-dimensional wavenumber (shorter waves ψ studied only in
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