The energy spectrum of superfluid turbulence is studied numerically by solving the GrossPitaevskii equation. We introduce the dissipation term which works only in the scale smaller than the healing length, to remove short wavelength excitations which may hinder the cascade process of quantized vortices in the inertial range. The obtained energy spectrum is consistent with the Kolmogorov law.PACS numbers: 67.40. Vs, 47.37.+q, 67.40.Hf The physics of quantized vortices in liquid 4 He is one of the most important topics in low temperature physics [1]. Liquid 4 He enters the superfluid state at 2.17 K. Below this temperature, the hydrodynamics is usually described using the two-fluid model in which the system consists of inviscid superfluid and viscous normal fluid. Early experimental works on the subject focused on thermal counterflow in which the normal fluid flowed in the opposite direction to the superfluid flow. This flow is driven by the injected heat current, and it was found that the superflow becomes dissipative when the relative velocity between two fluids exceeds a critical value [6] in superfluid 4 He at temperatures above 1 K, and their results were consistent with the Kolmogorov law. The Kolmogorov law is one of the most important statistical laws [7] of fully developed CT, so these experiments show a similarity between ST and CT. This can be understood using the idea that the superfluid and the normal fluid are likely to be coupled together by the mutual friction between them and thus to behave like a conventional fluid [8]. Since the normal fluid is negligible at very low temperatures, an important question arises: even without the normal fluid, is ST still similar to CT or not?To address this question, we consider the statistical law of CT [7]. The steady state for fully developed turbulence of an incompressible classical fluid follows the Kolmogorov law for the energy spectrum. The energy is injected into the fluid at some large scales in the energycontaining range. This energy is transferred in the inertial range from large to small scales without been dissipated. The inertial range is believed to be sustained by the self-similar Richardson cascade in which large eddies are broken up to smaller ones, having the Kolmogorov lawHere the energy spectrum E(k) is defined as E = dk E(k), where E is the kinetic energy per unit mass and k is the wave number from the Fourier transformation of the velocity field. The energy transferred to smaller scales in the energy-dissipative range is dissipated by the viscosity with the dissipation rate, which is identical with the energy flux ǫ of Eq. (1) in the inertial range. The Kolmogorov constant C is a dimensionless parameter of order unity.In CT, the Richardson cascade is not completely understood, because it is impossible to definitely identify each eddy. In contrast, quantized vortices in superfluid are definite and stable topological defects. A BoseEinstein condensed system yields a macroscopic wave function Φ(x, t) = ρ(x, t)e iφ(x,t) , whose dynamics is g...

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 study quantum turbulence in trapped Bose-Einstein condensates by numerically solving the Gross-Pitaevskii equation. Combining rotations around two axes, we successfully induce quantum turbulent state in which quantized vortices are not crystallized but tangled. The obtained spectrum of the incompressible kinetic energy is consistent with the Kolmogorov law, the most important statistical law in turbulence. The study of turbulence has a very long history, going back at least to Leonardo da Vinci, and understanding and controlling turbulence are great dreams of science and technology. Classical turbulence (CT) exhibits highly complicated configurations of eddies. Many studies have been devoted to the dynamical and statistical properties of CT after Kolmogorov's pioneering work [1,2] on flow at very high Reynolds number, namely, fully developed CT. The characteristic behavior of CT has been believed to be sustained by the Richardson cascade of eddies from large to small scales. However, in CT, there is no universal way to identify each eddy, because they continue to nucleate, diffuse, and disappear. As a result, many aspects of CT are still not perfectly understood.Turbulence is also possible in superfluids, such as the superfluid phases of 4 He and 3 He. Such quantum turbulence (QT) consists of definite topological defects known as quantized vortices and has recently attracted interest as a way to better understand turbulence [3].Superfluid 4 He has been extensively studied, in particular with relation to quantized vortices [4]. Below the lambda temperature T λ = 2.17 K, liquid 4 He enters the superfluid state through Bose-Einstein condensation. The hydrodynamics of superfluid 4 He is strongly influenced by quantum effects; any rotational motion is sustained by quantized vortices with quantized circulation κ = /m, where m is the particle mass. There are two typical cooperative phenomena of quantized vortices. One is a vortex lattice under rotation in which straight quantized vortices form a triangular lattice along the rotation axis [5]. The other is a vortex tangle in QT in which vortices become tangled in a flow [6,7].QT has been studied as a problem in low temperature physics since its discovery some 50 years ago. Its study has recently entered a new stage beyond low temperature physics. One of the main motivations of recent studies is to investigate the relationship between QT and CT. Some similarities between the two types of turbulence have been experimentally observed in superfluid 4 He [8, 9] and 3 He [10,11], and have been theoretically confirmed by numerical simulations of the quantized vortex-filament model [12] and a model using the Gross-Pitaevskii (GP) equation [13,14,15,16]. In particular, we have successfully obtained the Kolmogorov law for QT, which is one of the most important statistical laws in CT [17] by a numerical simulation of the GP equation [14,15].The similarity between QT and CT means that QT is an ideal prototype to study the statistics and vortex dynamics of turbulence, because QT ...

We have studied a Bose-Einstein condensate of 87 Rb atoms under an oscillatory excitation. For a fixed frequency of excitation, we have explored how the values of amplitude and time of excitation must be combined in order to produce quantum turbulence in the condensate. Depending on the combination of these parameters different behaviors are observed in the sample. For the lowest values of time and amplitude of excitation, we observe a bending of the main axis of the cloud. Increasing the amplitude of excitation we observe an increasing number of vortices. The vortex state can evolve into the turbulent regime if the parameters of excitation are driven up to a certain set of combinations. If the value of the parameters of these combinations is exceeded, all vorticity disappears and the condensate enters into a different regime which we have identified as the granular phase. Our results are summarized in a diagram of amplitude versus time of excitation in which the different structures can be identified. We also present numerical simulations of the Gross-Pitaevskii equation which support our observations.

There is a growing interest in the relation between Bose-Einstein condensation (BEC) and superfluidity. A Bose system confined in random media such as porous glass is suitable for studying this relation because BEC and superfluidity can be suppressed and controlled in such a disordered environment. However, it is not clear how this relation is affected by disorder and there are few theoretical studies that can be quantitatively tested by experiment. In this work, we develop the dilute Bose gas model with a random potential that takes into account the pore size dependence of porous glass. Then we compare our model with the measured low-temperature specific heat, condensate density, and the superfluid density of 4 He in Vycor glass. This comparison uses no free parameters. We predict phenomena at low temperatures that have not yet been observed. First, the random potential causes a T -linear specific heat instead of the T 3 dependence that is usually caused by phonons. Second, the BEC can remain even when the superfluidity disappears at low densities. And third, the system makes a reentrant transition at low densities; that is, the superfluid phase changes to the normal phase again as the temperature is reduced. This reentrant transition is more likely to be observed when the strength of the random potential is increased.PACS numbers:

We investigate the collision dynamics of two non-Abelian vortices and find that, unlike Abelian vortices, they neither reconnect themselves nor pass through each other, but create a rung between them in a topologically stable manner. Our predictions are verified using the model of the cyclic phase of a spin-2 spinor Bose-Einstein condensate.PACS numbers: 03.75.Mn, 05.30.Jp, 11.27.+d Quantized vortices are topological defects in the superfluid order-parameter, and their character depends on the topology of the order-parameter manifold of the system. In the case of single component Bose-Einstein Condensates (BECs), for example, the order-parameter manifold is U (1) and the quantized vortices are characterized by an additive group of integers. However, the situation is dramatically different for spinor BECs, where some phases accommodate non-Abelian vortices where the collision dynamics shows markedly different behavior from that of Abelian vortices. In this Letter, we investigate the collisional properties of non-Abelian vortices and their unique topological properties. In particular, we find that a rung vortex that bridges the colliding vortices is always formed upon collision, regardless of the ranges of the kinematic parameters. We verify these ideas with numerical simulations for a spin-2 BEC.The topological charge of a vortex can be identified by studying how the order-parameter changes along a closed path encircling the vortex. For U (1) vortices, the U (1) phase changes around the vortex by an integer multiple of 2π, and the topological charge is expressed by integers. In spinor BECs, on the other hand, the change of the orderparameter around the vortex involves not only the U (1) phase but also an SO(3) rotation of the spin. Therefore, there are phases in which topological charges of vortices do not commute with each other and form a non-Abelian group. We define such vortices with noncommutative topological charges as non-Abelian vortices [1].A salient feature of non-Abelian vortices manifests itself in the collision dynamics. In Abelian vortices, the following three types of collisions are possible: reconnection, passing through, and a formation of a rung vortex that bridges the colliding vortices. The reconnection of Abelian U (1) vortices has been studied theoretically [2], and observed recently in superfluid 4 He [3]. Moreover, there are some theoretical works which predict the rung structure that connects two attracting U (1) vortices [4] or U (1) × U (1) vortices [5].When the vortices are non-Abelian, the situation changes dramatically. It was predicted that the collision of two non-Abelian vortices produces a tangled and con- nected rung vortex [6]. In fact, for the case of two vortices with noncommutative topological charges, reconnection and passing through are topologically forbidden and only the formation of a rung vortex is allowed. This can be understood by considering algebraic and geometric structures of the vortices as shown in Fig 1. Let us consider two colliding vortices with topologi...

There is a growing interest in the relation between classical turbulence and quantum turbulence. Classical turbulence arises from complicated dynamics of eddies in a classical fluid. In contrast, quantum turbulence consists of a tangle of stable topological defects called quantized vortices, and thus quantum turbulence provides a simpler prototype of turbulence than classical turbulence. In this paper, we investigate the dynamics and statistics of quantized vortices in quantum turbulence by numerically solving a modified Gross-Pitaevskii equation. First, to make decaying turbulence, we introduce a dissipation term that works only at scales below the healing length. Second, to obtain steady turbulence through the balance between injection and decay, we add energy injection at large scales. The energy spectrum is quantitatively consistent with the Kolmogorov law in both decaying and steady turbulence. Consequently, this is the first study that confirms the inertial range of quantum turbulence.Comment: 14pages, 24 figures and 1 table. Appeared in Journal of the Physical Society of Japan, Vol.74, No.12, p.3248-325

We develop Bogoliubov theory of spin-1 and spin-2 Bose-Einstein condensates (BECs) in the presence of a quadratic Zeeman effect, and derive the Lee-Huang-Yang (LHY) corrections to the ground-state energy, pressure, sound velocity, and quantum depletion. We investigate all the phases of spin-1 and spin-2 BECs that can be realized experimentally. We also examine the stability of each phase against quantum fluctuations and the quadratic Zeeman effect. Furthermore, we discuss a relationship between the number of symmetry generators that are spontaneously broken and that of Nambu-Goldstone (NG) modes. It is found that in the spin-2 nematic phase there are special Bogoliubov modes that have gapless linear dispersion relations but do not belong to the NG modes.