Topology in superfluid3He: Recent results

We discuss: the topological phase transition in nonsingular vortices in³He-A; vortices in³He-B and solitons terminating on strings; topological defects of the A-B interface: the interaction of continuous A-phase vortices with singular B-phase vortices across the interface; extended degeneracy and topology of Larmor precession; and internal topology in thin³He-A film, responsible for chiral edge states of fermions and QHE.     

Textures and Exotic Vortices in Neutral Fermion Superfluids

There has been intense interest in various Fermion superfluids in neutral atom liquids and gases, including chiral p-wave pairing in ³He-A phase and Feshbach-resonanced ⁶Li atom gases and d-wave pairing in atom gases. It is particularly interesting to find exotic vortices and associated low-lying Fermionic excitations under rotation. Here we report on our efforts of those topics: (1) Majorana Fermion in chiral superfluids near a p-wave Feshbach resonance. (2) Possible half-quantum vortices in p-wave superfluids of trapped Fermion atom gases. (3) Stability of a half-quantum vortex in rotating superfluid ³He-A between parallel plates. (4) Majorana bound state in rotating superfluid ³He-A between parallel plates. (5) Non-Abelian Fractional vortex in d-wave Feshbach resonance superfluids. We will summarize some of those works in a coherent manner in order to bridge the understanding between cold atom community and superfluid ³He community by stressing the importance of cross fertilization between them.     

An introduction to quantum turbulence

This paper provides a brief introduction to quantum turbulence in simple superfluids, in which the required rotational motion in the superfluid component is due entirely to the topological defects that are identified as quantized vortices. Particular emphasis is placed on the basic dynamical behaviour of the quantized vortices and on turbulent decay mechanisms at a very low temperature. There are possible analogies with the behaviour of cosmic strings.     

Stabilization and Pumping of Giant Vortices in Dilute Bose–Einstein Condensates

Recently, it was shown that giant vortices with arbitrarily large quantum numbers can possibly created in dilute Bose–Einstein condensates by cyclically pumping vorticity into the condensate. However, multiply quantized vortices are typically dynamically unstable in harmonically trapped nonrotated condensates, which poses a serious challenge to the vortex pump procedure. In this theoretical study, we investigate how the giant vortices can be stabilized by the application of a Gaussian potential peak along the vortex core. We find that achieving dynamical stability is feasible up to high quantum numbers. To demonstrate the efficiency of the stabilization method, we simulate the adiabatic creation of an unsplit 20-quantum vortex with the vortex pump.     

Switching and self-trapping dynamics of bose-Einstein solitons

Abstract

We investigate the dynamics of two weakly coupled Bose condensates in long cigar-shaped traps. The Bose condensates are characterized by attractive mean-field interaction and consequently can be studied in terms of bright solitons. We exploit the analogy with directional fibre couplers in nonlinear fibre optics to uncover interesting dynamical regimes like switching of condensates from one trap to another and self-trapping of condensates. We also discuss the analogy between two weakly coupled Bose condensates and the Josephson junction in superfluids and superconductors.     

Vortex sheet in superfluid3He-A

A new state of rotating superfluid³He-A has been found recently. Usually superfluids respond to rotation by creating an array of vortex lines, which are parallel to the rotation axis, and the circulation around them is quantized. In the new state the vorticity is located on a 2 dimensional sheet instead of 1 D lines. The sheet is parallel to the rotation axis z but in the x — y plane it folds to equidistant layers. The distance between the layers is larger but on the same order of magnitude as the distance between vortex lines. In contrast to other superfluids, the sheet is stable in the A phase because of its internal structure. The sheet has as a backbone a topologically stable domain wall called soliton, to which non-singular vorticity is bound. Thus it can exist in spite of its presumably higher energy. The vortex sheet is distinguished by its NMR response, in particular because of its higher absorption at a characteristic frequency. Experiment and theory on the vortex sheet are in good agreement.     

Numerical Studies of Quantum Turbulence

We review numerical studies of quantum turbulence. Quantum turbulence is currently one of the most important problems in low temperature physics and is actively studied for superfluid helium and atomic Bose–Einstein condensates. A key aspect of quantum turbulence is the dynamics of condensates and quantized vortices. The dynamics of quantized vortices in superfluid helium are described by the vortex filament model, while the dynamics of condensates are described by the Gross–Pitaevskii model. Both of these models are nonlinear, and the quantum turbulent states of interest are far from equilibrium. Hence, numerical studies have been indispensable for studying quantum turbulence. In fact, numerical studies have contributed to revealing the various problems of quantum turbulence. This article reviews the recent developments in numerical studies of quantum turbulence. We start with the motivation and the basics of quantum turbulence and invite readers to the frontier of this research. Though there are many important topics in the quantum turbulence of superfluid helium, this article focuses on inhomogeneous quantum turbulence in a channel, which has been motivated by recent visualization experiments. Atomic Bose–Einstein condensates are a modern issue in quantum turbulence, and this article reviews a variety of topics in the quantum turbulence of condensates, e.g., two-dimensional quantum turbulence, weak wave turbulence, turbulence in a spinor condensate, some of which have not been addressed in superfluid helium and paves the novel way for quantum turbulence researches. Finally, we discuss open problems.     

Transition to Superfluid Turbulence

Turbulence in superfluids depends crucially on the dissipative damping in vortex motion. This is observed in the B phase of superfluid ³He where the dynamics of quantized vortices changes radically in character as a function of temperature. An abrupt transition to turbulence is the most peculiar consequence. As distinct from viscous hydrodynamics, this transition to turbulence is not governed by the velocity-dependent Reynolds number, but by a velocity-independent dimensionless parameter 1/q which depends only on the temperature-dependent mutual friction—the dissipation which sets in when vortices move with respect to the normal excitations of the liquid. At large friction and small values of the dynamics is vortex number conserving, while at low friction and large vortices are easily destabilized and proliferate in number. A new measuring technique was employed to identify this hydrodynamic transition: the injection of a tight bundle of many small vortex loops in applied vortex-free flow at relatively high velocities. These vortices are ejected from a vortex sheet covering the AB interface when a two-phase sample of ³He-A and ³He-B is set in rotation and the interface becomes unstable at a critical rotation velocity, triggered by the superfluid Kelvin–Helmholtz instability.      

Vortex Molecules in Thin Films of Layered Superconductors

Both the equilibrium and transport properties of the vortex matter are essentially affected by the behavior of the intervortex interaction potential. In isotropic bulk superconductors this potential is well known to be repulsive and is screened at intervortex distances R greater than the London penetration depth λ. As a result, in perfect crystals quantized Abrikosov vortices form a triangular lattice. In thin films of anisotropic superconductors this standard interaction potential behavior appears to be strongly modified because of the interplay between the long-ranged repulsion predicted in the pioneering work by J. Pearl and the attraction caused by the tilt of the vortex lines with respect to the anisotropy axes. This interplay results in a new type of vortex arrangement formed by finite-size vortex chains, i.e., vortex molecules. Tilted vortices with such unusual interaction potential form clusters with the size depending on the field tilting angle and film thickness or/and can arrange into multiquanta flux lattice. The magnetic flux through the unit cells of the corresponding flux line lattices equals to an integer number N of flux quanta. Thus, the increase in the field tilting (or varying temperature) should be accompanied by the series of the phase transitions between the vortex lattices with different N. A similar scenario should be realized in strongly anisotropic BSCCO high-T _c superconductors where in tilted field a crossing lattice of Abrikosov vortices (the stacks of pancakes in this case) and Josephson vortices appears. This crossing leads to the zigzag deformation of the pancakes stacks which is responsible for the attraction interaction competing with the long-ranged Pearl’s repulsion.     

Baryon Asymmetry of Universe: View from Superfluid 3He

The origin of the excess of matter over antimatter in our Universe remains one of the fundamental problems. Dynamical baryogenesis in the process of the broken symmetry electroweak transition in the expanding Universe is the widely discussed model where the baryonic asymmetry is induced by the quantum chiral anomaly. We discuss the modelling of this phenomenon in superfluid ³ He and superconductors where the chiral anomaly is realized in the presence of quantized vortex, which introduces nodes into the energy spectrum of the fermionic quasiparticles. The spectral flow of fermions through the nodes during the vortex motion leads to the creation of fermionic charge from the superfluid vacuum and to transfer of the superfluid linear momentum into the heat bath, thus producing an extra force on the vortex, which in some cases compensates the Magnus force. This spectral-flow force was calculated 20 years ago by Kopnin and Kravtsov for s-wave superconductors, but only recently was it measured in a broad temperature range in Manchester experiments on rotating superfluid ³ He. The momentogenesis observed in ³ He is analogous to the dynamical production of baryons by cosmic strings. Some other possible scenaria of baryogenesis related to superfluid ³ He are discussed.     

Counterflow Quantum Turbulence and the Instability in Two-component Bose-Einstein Condensates

We theoretically study the nonlinear dynamics of the instability of counter-superflow in two miscible Bose-Einstein condensates. The condensates become unstable when the relative velocity exceeds a critical value, which is called counter-superflow instability. We reveal that the counter-superflow instability can lead to quantum turbulence by numerically solving the coupled Gross-Pitaevskii equations. The modes amplified by the instability grow into solitons and decay into quantized vortices. Eventually, the vortices become tangled and quantum turbulence of two superfluids. We show that this process may occur in experiments by investigating the dynamics in a 2D trapped system.     

Rapidly Rotating Bose-Einstein Condensates

No Heading How does a rapidly rotating condensed Bose gas carry extreme amounts of angular momentum? The energetically favored state of a not-too-rapidly rotating Bose condensed gas is, as observed, a triangular lattice of singly quantized vortices. This paper describes the fates of the vortex lattice in both harmonic and anharmonic traps when condensates are rotated extremely rapidly.PACS numbers: 03.75.Lm, 67.40.Db, 67.40.Vs, 05.30.Jp     

Iordanskii and Lifshitz-Pitaevskii Forces in the Two-Fluid Model

It has been known since the pioneering work of Onsager and Feynman that the statistical mechanics and dynamics of vortices play an essential role in the behavior of superfluids and superconductors. However, the theory of vortices in quantum fluids remains in a most unsatisfactory state, with many conflicting results in the literature. In this paper we review the theory of Thouless, Ao and Niu, which gives an expression for the total transverse force acting on a quantized vortex that is in apparent disagreement with the word of lordanskii and of Lifshitz and Pitaevskii. In particular, no transverse force proportional to the asymptotic normal fluid velocity was found. We use two-fluid hydrodynamics to study this discrepancy.     

Triplet Superconductivity in a Nutshell

The triplet superconductors have been around us since 1980, when Jerome et al. discovered the Bechgaard salts (TMTSF)₂PF₆ and (TMTSF)₂ClO₄. Now there are more than 20 or so triplet superconductors discovered. Recently we found that most of them with the known gap symmetries can be mapped to the superfluid phase of ³He-A and ³He-A₁. Further, in most of them, (the chiral vector, i.e. the quantization axis of the pair angular momentum) is fixed parallel to one of the crystal axes and all the topological defects are considered in terms of -textures, where is the spin vector. Then Sr₂RuO₄, UPt₃, PrOs₄Sb₁₂ and (TMTSF)₂ClO₄ belong to type-A, which is an analog of superfluid ³He-A. In these superconductors, a vortex splits into a pair of half quantum vortices (HQVs) at low temperatures. On the other hand, CePt₃Si, CeIrSi₃, CeRhSi₃, UIr and Li₂Pt₃B (those in non-centrosymmetric crystals) belong to type-A₁, which is an analog of superfluid ³He-A₁. In all of these triplet superconductors, vortices harbor the zero mode or the Majorana fermions, the implications of which deserves further exploration.     

Phase Slips and Josephson Weak Links in Superfluid Helium

When superfluid ⁴He flows through a submicron aperture, the velocity is limited by a critical value which marks the onset of quantized vortex creation. The evolution of the vortices causes the quantum phase across the aperture to change by 2π, leading to a detectable drop in flow energy. Recent studies of these phase slip events have provided new insights into the nucleation mechanisms for quantum vortices. By contrast, superfluid ³He passing through a submicron aperture exhibits nonlinear hydrodynamics, characterized by a Josephson-like current phase relation. Recent experiments have revealed a multitude of effects analogous to phenomena observed in superconductors. The experiments also reveal unexpected effects such as bistability, π-states, and novel dissipation mechanisms. PACS numbers: 67.57.-z, 74.50.+r, 67.40.Hf, 67.40. Vs.     

Vortex textures in3He-A near the transition to the A1 phase

The decay of the nonsingular doubly quantized vortex in³He-A into a pair of Mermin-Ho vortex textures in the immediate vicinity of the transition to theA ₁phase is confirmed using the variational approach.     

Relation Between Vortex Pinning Energy and Anderson's Theorem

We discuss the elementary vortex pinning in type-II superconductors in connection with the Anderson's theorem for nonmagnetic impurities. We address the following two issues. One is an enhancement of the vortex pinning energy in the unconventional superconductors. This enhancement comes from the pair-breaking effect of a nonmagnetic defect as the pinning center far away from the vortex core (i.e., the pair-breaking effect due to the non-applicability of the Anderson's theorem in the unconventional superconductors). The other is an effect of the chirality on the vortex pinning energy in a chiral p-wave superconductor. The vortex pinning energy depends on the chirality. This is related to the cancellation of the angular momentum between the vorticity and chirality in a chiral p-wave vortex core, resulting in local applicability of the Anderson's theorem (or local recovery of the Anderson's theorem) inside the vortex core.     

Instability of Overlapped Vortices Rotating in Opposite Directions in Binary Bose–Einstein Condensates

We theoretically study the instability of vortices in pancake-shaped trapped binary Bose–Einstein condensates. We consider that a quantized vortex is at the center of each condensate and two condensates rotate in the opposite directions. The total circulation is zero in BECs having the overlapped vortices because there is relative rotation, which is the rotation of one component in relation to the rotation of the other, but no total rotation, which is the sum of the rotation in both components. We think that the zero-quantum vortices are unstable because this system is locally countersuperflow, two counterpropagating miscible superflows. In a uniform system, the countersuperflow is unstable when the relative velocity between the two condensates exceeds a critical value. We investigate the dynamics of the zero-quantum vortices by numerically solving the Gross–Pitaevskii equations. To understand the results of the numerical calculations, we apply the countersuperflow instability to our present system.     

On Developed Superfluid Turbulence

Superfluid turbulence is governed by two dimensionless parameters. One of them is the intrinsic parameter q which characterizes the relative value of the friction force acting on a vortex with respect to the non-dissipative forces. The inverse parameter q^?1 plays the same role as the Reynolds number Re = U R/ν in classical hydrodynamics. It marks the transition between the “laminar” and turbulent regimes of vortex dynamics. The developed turbulence, described by a Kolmogorov cascade, occurs when Re ? 1 in classical hydrodynamics. In superfluids, the developed turbulence occurs at q ? 1. Another parameter of superfluid turbulence is the superfluid Reynolds number Re_s = U R/κ, which contains the circulation quantum κ characterizing quantized vorticity in superfluids. The two parameters q and Re_s control the crossover or transition between two classes of superfluid turbulence: (i) the classical regime, where the Kolmogorov cascade, probably modifed by the non-canonical dissipation due to mutual friction, is effective, vortices are locally polarized, and the quantization of vorticity is not important; and (ii) the Vinen quantum turbulence where the properties are determined by the quantization of vorticity. The phase diagram of these dynamical vortex states is suggested. PACS numbers: 43.37.+q, 47.32.Cc, 67.40.Vs, 67.57.Fg.     

Trapping of Vorticity by Îl-textures in 3He-A

No Heading Using a rotating cryostat, we have observed nucleation, annihilation and strong intrinsic pinning of continuous vortices in a slab of superfluid ³He-A containing stable textural defects. A model of a critical state set by either the critical velocity for vortex nucleation or pinning strength is developed. It predicts a hysteretic dependence of trapped vorticity on the angular velocity of rotation, in agreement with the observations. We argue that the static defects responsible for nucleation and trapping of vorticity are networks of domain walls between regions of opposite orientation of the Îl-vector.PACS numbers: 76.40.11f, 67.40.Bz, 67.40.Pm.