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arXiv:2203.10943 · Journal of Experimental and Theoretical Physics Letters

Quantum Turbulence and Planckian Dissipation

Григорий Ефимович Воловик

Published Apr 1, 2022• 6 citations
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Abstract

The notion of the Planckian dissipation is extended to the system of the Caroli–de Gennes–Matricon discrete energy levels in the vortex core of superconductors and fermionic superfluids. In this extension, the Planck dissipation takes place when the relaxation time τ is comparable with the quantum Heisenberg time $${{t}_{H}} = \hbar {\text{/}}\Delta E$$ , where $$\Delta E$$ is the interlevel distance in the vortex core (the minigap). This type of Planck dissipation has two important physical consequences. First, it determines the regime, when the effect of the axial anomaly becomes important. The anomalous spectral flow of the energy levels along the chiral branch of the Caroli–de Gennes–Matricon states becomes important in the super-Planckian region, i.e., when $$\tau < \hbar {\text{/}}\Delta E$$ . Second, the Planck dissipation separates the laminar flow of the superfluid liquid at $$\tau < \hbar {\text{/}}\Delta E$$ and the vortex turbulence regime at $$\tau > \hbar {\text{/}}\Delta E$$ .

DOI
10.1134/s0021364022100344

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The notion of the Planckian dissipation is extended to the system of the Caroli–de Gennes–Matricon discrete energy levels in the vortex core of superconductors and fermionic superfluids. In this extension, the Planck dissipation takes place when the relaxation time τ is comparable with the quantum Heisenberg time tH=ℏ/ΔE{{t}_{H}} = \hbar {\text{/}}\Delta EtH​=ℏ/ΔE , where ΔE\Delta EΔE is the interlevel distance in the vortex core (the minigap). This type of Planck dissipation has two important physical consequences. First, it determines the regime, when the effect of the axial anomaly becomes important. The anomalous spectral flow of the energy levels along the chiral branch of the Caroli–de Gennes–Matricon states becomes important in the super-Planckian region, i.e., when τ<ℏ/ΔE\tau < \hbar {\text{/}}\Delta Eτ<ℏ/ΔE . Second, the Planck dissipation separates the laminar flow of the superfluid liquid at τ<ℏ/ΔE\tau < \hbar {\text{/}}\Delta Eτ<ℏ/ΔE and the vortex turbulence regime at τ>ℏ/ΔE\tau > \hbar {\text{/}}\Delta Eτ>ℏ/ΔE .
We argue that turbulence in superfluids is governed by two dimensionless parameters. One of them is the intrinsic parameter q which characterizes the friction forces acting on a vortex moving with respect to the heat bath, with q −1 playing the same role as the Reynolds number Re= UR/ν in classical hydrodynamics. It marks the transition between the “laminar” and turbulent regimes of vortex dynamics. The developed turbulence described by Kolmogorov cascade occurs when Re≫1 in classical hydrodynamics, and q ≪1 in superfluid hydrodynamics. Another parameter of superfluid turbulence is the superfluid Reynolds number Re s = UR/κ , which contains the circulation quantum κ characterizing quantized vorticity in superfluids. This parameter may regulate the crossover or transition between two classes of superfluid turbulence: (i) the classical regime of Kolmogorov cascade where vortices are locally polarized and the quantization of vorticity is not important; (ii) the quantum Vinen turbulence whose properties are determined by the quantization of vorticity. A phase diagram of the dynamical vortex states is suggested.
Identified as a unique regime of fluid flow some 500 years ago, turbulence has fascinated and confounded for centuries. Indeed, a fundamental understanding of the turbulence phenomenon remains one of the most important unsolved problems in classical physics. Interestingly, superfluid atomic Bose-Einstein condensates (BECs) can also be agitated into highly excited states of chaotic vortex motion, suggestive of turbulence. However, in contrast to classical fluids, the vortices in these quantum systems exhibit topological stability, the restriction of quantized circulation, and a well defined length scale characterized by the so-called healing length of the superfluid. Such simplifying aspects motivate the study of turbulence in these systems, which may shed new light on the longstanding classical problem. \n\nWhilst a considerable number of studies have been conducted on turbulence in three-dimensional BEC systems, and such studies have uncovered similarities between the classical and quantum counterparts, including the presence of the Kolmogorov k−5/3k^{-5/3}k−5/3 law in the kinetic energy spectrum and the associated direct energy cascade, comparatively few studies have been conducted on two-dimensional quantum turbulence. An important question for such systems is whether the inverse energy cascade, a defining feature of classical two-dimensional turbulent fluid flows, may occur in these systems. In addition to a Kolmogorov kinetic energy spectrum, this process is signified by the clustering of same-circulation vortices, which results in the emergence of macroscopic vortex structures from the turbulent flow. \n\nAlthough intermittent few-vortex clustering has been observed in the break-down of superfluid flow around an obstacle, current attempts to observe a sustained inverse cascade have been unsuccessful, largely due to complications which arise due to the compressibility of the dilute Bose gas - a feature not considered in the classical theory.\n\nWe present a study of two-dimensional quantum turbulence in atomic BECs within the framework of damped Gross-Pitaevskii theory. With the motivation of determining the existence of the inverse cascade phenomenon in such systems, we focusing on experimentally realistic forcing procedures, and study the details of kinetic energy spectra, the statistics of vortex clustering, and the transport of kinetic energy through scale space in the resulting turbulent flows.\nWe investigate a range of vortex injection regimes produced through the circular stirring of a harmonically confined system with a repulsive Gaussian obstacle potential, identifying distinct regimes of turbulence. In the incompressible regime relevant to the inverse cascade, we identify numerous problems with the system and forcing procedure. Motivated by these findings, we investigate turbulence in a minimal model of the spindown of a toroidal persistent current. In this system, we identify a regime of forcing and damping in which the inverse cascade is corroborated by three independent measures: I) merging of vortex clusters; II) a Kolmogorov scaling in the incompressible kinetic energy spectrum; and III) Spectral condensation, associated with the emergence of large scale rotating structures from the turbulent flow. We provide clear evidence that the inverse cascade phenomenon, previously observed in a range of classical fluid systems, can also occur in quantum turbulence.
The microscopic mechanism of thermal dissipation in quantum turbulence is numerically studied by solving the coupled system involving the Gross-Pitaevskii equation and the Bogoliubov-de Gennes equation. At low temperatures, the obtained dissipation does not work at scales greater than the vortex core size. However, as the temperature increases, dissipation works at large scales and it affects the vortex dynamics. We successfully obtain the mutual friction coefficients of the vortex in dilute Bose-Einstein condensates dynamics as functions of temperature.
In quantum fluids, the quantization of circulation forbids the diffusion of a vortex swirling flow seen in classical viscous fluids. Yet, accelerating quantum vortices may lose their energy into acoustic radiations(1,2), similar to the way electric charges decelerate on emitting photons. The dissipation of vortex energy underlies central problems in quantum hydrodynamics(3), such as the decay of quantum turbulence, highly relevant to systems as varied as neutron stars, superfluid helium and atomic condensates(4,5). A deep understanding of the elementary mechanisms behind irreversible vortex dynamics has been a goal for decades(3,6), but it is complicated by the shortage of conclusive experimental signatures(7). Here we address this challenge by realizing a programmable vortex collider in a planar, homogeneous atomic Fermi superfluid with tunable inter-particle interactions. We create on-demand vortex configurations and monitor their evolution, taking advantage of the accessible time and length scales of ultracold Fermi gases(8,9). Engineering collisions within and between vortex-antivortex pairs allows us to decouple relaxation of the vortex energy due to sound emission and that due to interactions with normal fluid (that is, mutual friction). We directly visualize how the annihilation of vortex dipoles radiates a sound pulse. Further, our few-vortex experiments extending across different superfluid regimes reveal non-universal dissipative dynamics, suggesting that fermionic quasiparticles localized inside the vortex core contribute significantly to dissipation, thereby opening the route to exploring new pathways for quantum turbulence decay, vortex by vortex.By controlling the generation and collision of individual vortices in atomic Fermi superfluids, a study provides a comprehensive view of vortex decay due to mutual friction and vortex-sound interaction.
An out-of-equilibrium two-dimensional superfluid relaxes towards equilibrium via a process of coarsening, driven by the annihilation of vortices with antivortices. Here we present a comparison of two different numerical models of this process, a dissipative point vortex model and Fokker-Planck evolution, across a wide range of initial configurations and levels of dissipation. We find that our dissipative point vortex model is well approximated by Fokker-Planck evolution only for very low initial energies per vortex, E 0 / N v ≲ − 4 , when almost all vortices and antivortices are closely bound into dipoles. We observe that the dynamical critical exponent z in the dissipative point vortex model undergoes a crossover, from a roughly constant value close to 2 for E 0 / N v ≳ − 4 , to a value which depends explicitly on the initial conditions for E 0 / N v ≲ − 5 .
In Fermi superfluids, such as superfluid (3)He, the viscous normal component can be considered to be stationary with respect to the container. The normal component interacts with the superfluid component via mutual friction, which damps the motion of quantized vortex lines and eventually couples the superfluid component to the container. With decreasing temperature and mutual friction, the internal dynamics of the superfluid component becomes more important compared with the damping and coupling effects from the normal component. As a result profound changes in superfluid dynamics are observed: the temperature-dependent transition from laminar to turbulent vortex motion and the decoupling from the reference frame of the container at even lower temperatures.