Quantum hydrodynamics in superfluid helium and atomic Bose-Einstein condensates (BECs) has been recently one of the most important topics in low temperature physics. In these systems, a macroscopic wave function (order parameter) appears because of Bose-Einstein condensation, which creates quantized vortices. Turbulence consisting of quantized vortices is called quantum turbulence (QT). The study of quantized vortices and QT has increased in intensity for two reasons. The first is that recent studies of QT are considerably advanced over older studies, which were chiefly limited to thermal counterflow in 4 He, which has no analogue with classical traditional turbulence, whereas new studies on QT are focused on a comparison between QT and classical turbulence. The second reason is the realization of atomic BECs in 1995, for which modern optical techniques enable the direct control and visualization of the condensate and can even change the interaction; such direct control is impossible in other quantum condensates like superfluid helium and superconductors. Our group has made many important theoretical and numerical contributions to the field of quantum hydrodynamics of both superfluid helium and atomic BECs. In this article, we review some of the important topics in detail. The topics of quantum hydrodynamics are diverse, so we have not attempted to cover all these topics in this article. We also ensure that the scope of this article does not overlap with our recent review article (arXiv:1004.5458), "Quantized vortices in superfluid helium and atomic Bose-Einstein condensates", and other review articles.

We theoretically study the Kelvin-Helmholtz instability in phase-separated two-component Bose-Einstein condensates using the Gross-Pitaevskii and Bogoliubov-de Gennes models. A flat interface between the two condensates is shown to deform into sawtooth or Stokes-type waves, leading to the formation of singly quantized vortices on the peaks and troughs of the waves. This scenario of interface instability in quantum fluids is quite different from that in classical fluids.

We study the asymptotic interaction between two half-quantized vortices in two-component Bose-Einstein condensates. When two vortices in different components are placed at distance 2R, the leading order of the force between them is found to be (ln R/ξ − 1/2)/R 3 , in contrast to 1/R between vortices placed in the same component. We derive it analytically using the Abrikosov ansatz and the profile functions of the vortices, confirmed numerically with the Gross-Pitaevskii model. We also find that the short-range cutoff of the intervortex potential linearly depends on the healing length.

We theoretically study the development of quantum turbulence from two counter-propagating superfluids of miscible Bose-Einstein condensates by numerically solving the coupled Gross-Pitaevskii equations. When the relative velocity exceeds a critical value, the countersuperflow becomes unstable and quantized vortices are nucleated, which leads to isotropic quantum turbulence consisting of two superflows. It is shown that the binary turbulence can be realized experimentally in a trapped system.PACS numbers: 03.75. Mn, 67.25.dk, 47.27.Cn Hydrodynamic instability develops into turbulence or turbulent flow due to the complicated dynamics of eddies of various length scales [1]. This phenomenon appears in diverse fields of physics, such as magneto-hydrodynamic fluids in plasma physics [2], relativistic fluids in astrophysics [3], and quantum fluids in low temperature physics [4,5]. Turbulence has been studied thoroughly in classical fluid dynamics as the most important unsolved problem of classical physics. Recently, quantum turbulence (QT), turbulence in superfluids, has attracted interest in low temperature physics [6]. QT consists of topological defects, vortices with definite quantized circulation, and can provide a simple prototype for understanding classical turbulence. QT has been historically studied in superfluid 4 He and 3 He and has been numerically studied in atomic Bose-Einstein condensates (BECs) [7,8] and has recently been realized in experiments [9].Early studies on QT were devoted primarily to thermal counterflow in superfluid 4 He, where the hydrodynamics is usually described using a two-fluid model in which the system consists of an inviscid superfluid component and a viscous normal fluid component [10]. Thermal counterflow is driven by an injected heat current, where the two components flow in opposite directions. For a sufficiently large relative velocity between the two components, the laminar counterflow state becomes unstable, developing into a superfluid turbulent state. In the transition, remnant vortices, quantized vortices attached to the container wall, are stretched by the mutual friction between the superfluid and normal fluid components, growing into a tangle through reconnections with other vortices [11,12].We consider a similar case in two-component BECs, QT arising from the instability of the counterflow of two superfluids, namely, countersuperflow instability. Counter-superflow instability has been theoretically studied in several multicomponent miscible superfluid systems, such as helium superfluids [13][14][15][16], mixture BECs of cold atoms [17], and nucleon superflows in rotating neutron stars [18]. Atomic BEC systems have great advantages over other superfluid systems, because modern experimental techniques enable us to easily control superflows and directly visualize the dynamics of topological defects therein. Recently, Hamner et al. experimentally realized countersuperflow instability for the first time, and observed that shocks and dark-bright solitons nucleate via the in...

Dynamical instabilities at the interface between two Bose-Einstein condensates that are moving relative to each other are investigated using mean-field and Bogoliubov analyses. Kelvin-Helmholtz instability is dominant when the interface thickness is much smaller than the wavelength of the unstable interface mode, whereas the counter-superflow instability becomes dominant in the opposite case. These instabilities emerge not only in an immiscible system but also in a miscible system where an interface is produced by external potential. Dynamics caused by these instabilities are numerically demonstrated in rotating trapped condensates.