Session H16: Quantum Fluids: Mainly Helium, Vortices, and Solitons

Weak to strong limits in confined superfluid helium

\newcommand{\boxes}[1]{$\left( #1~\mu \rm{m} \right)^3$} Experiments with $^4$He confined in \boxes{2} boxes connected via a thin film 33~nm thick have shown that the boxes act as isolated entities when spaced 4~$\mu$m edge-to-edge [1], whereas when spaced 2~$\mu$m edge-to-edge, they are strongly coupled to each other [2]. To investigate the spatial dependence of this coupling we are currently measuring \boxes{2} boxes spaced 1~$\mu$m edge-to-edge. We report measurements of the specific heat and superfluid density of helium confined in this geometry. These new data will help us to map the transition between fully isolated to fully coupled boxes which, in this limit, should behave like a 2 $\mu$m thick film. Questions involving our understanding of the correlation length $\xi$ arise, since it is observed that coupling is manifest over much larger distances than $\xi$.\\[4pt] [1] Perron J~K, Kimball M~O, Mooney K~P and Gasparini F~M 2010 {\em Nat. Phys.\/} {\bf 6} 499--502\\[0pt] [2] Perron J~K, and Gasparini F~M 2011 {\em submitted to PNAS\/}   

Critical point coupling in liquid helium and the significance of the correlation length

\newcommand{\boxes}[1]{$\left( #1~\mu \rm{m} \right)^3$} Recent measurements of liquid helium confined to \boxes{2} boxes connected through a 33~nm film have shown coupling effects between boxes spaced distances much larger than the correlation length $\xi(t,L)$[1,2] and proximity effects on the connecting film[3]. An analysis of data suggests that $\xi(t,L)$ is the relevant parameter in these effects. This dependence on $\xi(t,L)$ is used to argue that the enhancement in the specific heat due to coupling is a reflection of the finite-size correlation length in the boxes and hence its scaling function. All this raises some profound questions about our physical understanding of $\xi(t,L)$. \\[4pt] [1] Perron J~K, Kimball M~O, Mooney K~P and Gasparini F~M 2010 {\em Nat. Phys.\/} {\bf 6} 499--502\\[0pt] [2] Perron J~K, and Gasparini F~M 2011 {\em submitted to PNAS\/}\\[0pt] [3] Perron J~K, and Gasparini F~M 2011 {to be published in \em JPCS\/}   

Creating Only Isotropic Homogeneous Turbulence in Liquid Helium near Absolute Zero

Flow through a grid is a standard method to produce isotropic, homogeneous turbulence for laboratory study. This technique has been used to generate quantum turbulence (QT) above 1 K in superfluid helium\footnote{S. R. Stalp, L. Skrbek, and R. J. Donnelly, \textit{Phys. Rev. Lett}. \textbf{\textit{82}}, 4831 (1999).} where QT seems to mimic classical turbulence. Efforts have been made recently\footnote{G.~G.~Ihas, G.~Labbe, S-c.~Liu, and K.~J.~Thompson\textit{, J. Low Temp. Phys}. \textbf{150}, 384 (2008).} to make similar measurements near absolute zero, where there is an almost total absence of normal fluid and hence classical viscosity. This presents the difficulty that most motive force devices produce heat which overwhelms the phenomena being investigated. The process of designing and implimenting a ``dissipation-free'' motor for pulling a grid through superfluid helium at millikelvin temperatures has resulted in the development of new techniques which have broad application in low temperature research. Some of these, such as Meissner-affect magnetic drives, capacitive and inductive position sensors, and magnetic centering devices will be described. Heating results for devices which can move in a controlled fashion from very low speed up to 10 cm/s will be presented. Acknowledgement: We thank W.F. Vinen for many useful discussions.   

Superfluid Onset and 2D phase transitions of Helium-4 on Lithium and Sodium

We have fabricated lithium and sodium films on quartz crystal microbalances (QCM) using in situ low temperature pulsed laser deposition. The frequency shift and dissipation of the QCM was measured as a function of helium pressure and chemical potential and used to construct the phase diagram of helium films on these substrates. Pressure measurement techniques based on an RGA mass spectrometer, which provides accurate measurement below 10$^{-8}$ Torr will be described. Lithium and sodium are predicted to be intermediate strength substrates which are strong enough to be wetted by He-4 but weak enough that solid-like layers do not form, so they are candidates for observing sub-monolayer superfluidity in direct contact with a metallic surface. Helium adsorption isotherms and quenches between 0.5K and 1.6K on both lithium and sodium indicated continuous, sub-monolayer helium film growth and superfluid onsets in sub-monolayer films. Features below 1K indicate a collision between a classical 2D liquid/vapor phase transition and the Kosterlitz-Thouless superfluid phase transition. We see no evidence for the pre-wetting step instability predicted for helium on sodium.   

Discrepancies Between Theory and Experiment for Field-Dependence and $f$-wave Interactions in Superfluid $^3$He-$B$

We have performed transverse acoustics experiments in superfluid $^3$He-B, exploring the magnetic field splitting of the imaginary squashing mode (ISQ), a collective mode of the order parameter labelled by its total angular momentum $J=2$. We have compared theoretical calculations\footnote{J.A. Sauls and J.W. Serene, Phys. Rev. Lett. {\bf 49}, 1183 (1982).} of the Zeeman splitting, $g_{2^-}$, and its dependence on the strength of $f$-wave pairing interactions, $x_3^{-1}$, with our recent experimental data, showing unexpected discrepancies. We suggest that the origin of these discrepancies can be traced to limits on the applicability of the theoretical calculations at high magnetic field and at frequencies some distance from the order parameter collective mode. We discuss the analysis done by Davis \textit{et al.} in light of those limitations.\footnote{J.P. Davis {\it et al.}, Phys. Rev. Lett. {\bf 100}, 015301 (2008).} This work was supported by the National Science Foundation DMR-1103625.   

Role of the order parameter manifold on surface Majorana fermions and spin susceptibility of superfluid $^{3}$He-B

Here, we theoretically investigate surface Andreev bound states (SABS) in superfluid $^{3}$He-B confined to a slab geometry. It is known that the Majorana property gives rise to the Ising anisotropy of spin susceptibility on the surface, which reflects the assumption that the order parameter manifold of the B-phase is restricted to the subspace. In this talk, we first demonstrate that the SO(3) manifold, which describes the relative rotation of spin and orbital, plays a critical role on the various properties associated with the SABS, such as the Majorana nature, gapless dispersion, topological invariant, and spin susceptibility. Then, based on the quasiclassical Eilenberger theory which takes account of the dipole interaction, we quantitatively discuss thermodynamics and the Majorana property of the SABS in superfluid $^{3}$He-B under a magnetic field, where the Majorana property and spin susceptibility is determined by the interplay of the magnetic field and dipole interaction.   

Macroscopic quantum tunneling of a single vortex in a rotating Bose-Einstein condensate

A vortex can be created in a metastable state near the center of a harmonically trapped Bose condensate when it is rotated with suitable velocities. This state has a finite lifetime before which the vortex tunnels outwards to the edge of the trap. We estimate this tunneling rate semiclassically with the vortex treated as a point particle. This is followed by a more exact treatment incorporating the dynamics and mass of the vortex. The calculation is based on a Thomas-Fermi approximation, relevant to the typically large atomic densities used in experiments, but we also analyze the low density limit in a ``weakly nonlinear'' perturbative framework. We discuss the feasibility of detecting this effect with currently achievable experiments.   

Expansion Dynamics of a Ring Bose--Einstein Condensate

We studied the dynamics of BECs when released from a ring trap under conditions similar to those that obtained in a recent experiment done at NIST. In that experiment a ring--shaped BEC was formed in an all--optical trap created by intersecting a horizontal light sheet and a vertical Laguerre-Gaussian beam. Condensates were created in these traps and then ``stirred'' by applying Raman pulses having orbital angular momentum (OAM). We modeled the dynamics of condensates formed under these conditions by first solving the 2D time--dependent Gross--Pitaevskii equation (GPE) in imaginary time to obtain the initial condensate shape. We accounted for the OAM by applying a phase imprint to this wave function and then propagated it using the GPE in real time with the trap off. We found that, after release, the condensate expands both inward and outward. When no OAM was applied, this inward expansion causes the hole in the ring to close up entirely in turn causing a buildup of atom density there. Inflow and outflow of atoms from the center caused expanding interference rings to form. With non-zero applied initial OAM similar behavior was observed except that the central hole never closes with hole size increasing with increasing initial OAM. We compare our results with the NIST experiment.   

Quantum Dynamics of a Bose Superfluid Vortex

Quantum vortex dynamics remain poorly understood despite decades of theoretical investigation. The vortex is a topological soliton, arising from the same medium as the quasiparticles with which it interacts. Hence the coupling between the vortex ``zero mode'' and the quasiparticles has no term linear in the quasiparticle variables -- the lowest order coupling is quadratic. We present a fully quantum-mechanical derivation of the vortex equation of motion valid at low temperatures where the normal fluid density is small. The resulting equation of motion is naturally expressed as a function of the dimensionless frequency $\tilde \Omega = \hbar \Omega/k_BT$. The usual Hall-Vinen/Iordanskii equations are valid when $\tilde \Omega \ll 1$ (the ``classical regime''), but elsewhere, the equations are strongly memory dependent. We will discuss the experimental implications of this frequency dependence in Bose superfluids and cold atomic gases.   

Fragmented Many-Body states of definite angular momentum and stability of attractive 3D Condensates

We consider a 3D Bose-Einstein Condensate (BEC), with attractive interparticle interactions, embedded in a harmonic, spherically symmetric trap. This system is metastable only if the total number of bosons $N$ and the interaction strength $lambda_0$ do not exceed some critical values. Otherwise the system collapses. Gross-Pitaevskii (GP) theory predicts the maximum (critical) number of bosons $N_{cr}^{GP}$ that, for a given $\lambda_0$, can be loaded to the system, without its collapse. But, what happens to the excited states? To investigate the structure and stability of these states we must go beyond GP theory; these states have definite values of angular momentum (AM) L, are highly fragmented and can support number of bosons much greater than $N_{cr}^{GP}$. Secondly, we investigate the impact of external rotation of the trap to the AM and stability of the gas. We find that, for all allowed values of rotation frequency no significant stabilization occurs. The symmetry of the ground state does not change and no AM is transferred to the gas. This behaviour is attributed to the attractive nature of the interparticle interaction.   

Origins of bright soliton transparency to Bogoliubov quasi-particles

Bogoliubov quasi-particles can pass through a one-dimensional bright soliton without reflection at all energies.\footnote{D. J. Kaup, Phys. Rev. A {\bf 42}, 5689 (1990).} Reflectionless properties of this kind usually originate from a supersymmetric structure of the corresponding Hamiltonian.\footnote{E. Witten, Nucl. Phys. B {\bf 188}, 513 (1981).}$^,$\footnote{C. V. Sukumar, J. Phys. A {\bf 18}, 2917 (1985).} However, we give a strong indication that in this case\footnotemark[1], the mathematical mechanism enabling full spectrum transparency of a scattering object does not fall into any of the conventional paradigms.   

Supersymmetric Structure of two Families of Solitons

Solitons have generated considerable interest in the cold atoms and condensed matter communities. We demonstrate that two families of $n$-soliton solutions (with $n$ an integer) -- one for the attractive nonlinear Schr\"{o}dinger (NLS) equation, and one for the sine-Gordon (sG) equation -- originate from a quantum-mechanical supersymmetric (QM-SUSY) chain connecting a set of reflectionless operators $\hat{H}_n$. The families consist of breather-type solitons for NLS\footnote{D. Schrader, IEEE J. Quantum Electron. {\bf 31}, 2221 (1995).} and multi-(anti)kink solitons with specific velocities for sG. The operators $\hat{H}_n$, which we refer to as Akulin`s Hamiltonians\footnote{V. M. Akulin, \underline{Coherent Dynamics of Complex Quantum Systems} (Springer, Heidelberg, 2006).}, form reflectionless direct-scattering initial conditions for the inverse scattering method. Such a QM-SUSY chain is analogous to the known connection between QM-SUSY chains of P\"{o}schl-Teller potentials and solitons of the Korteweg-de Vries (KdV) equation\footnote{Sukumar, J. Phys. A {\bf 19}, 2297 (1986)}. The existence of QM-SUSY chains connecting soliton solutions, now for three different integrable nonlinear equations, sheds light on the underlying mechanisms responsible for soliton generation.   

From Dark Solitons to Vortex Clusters in Bose-Einstein Condensates

In this talk, we 'll start by considering the experimental, theoretical and numerical dynamical properties of dark solitons in quasi-one-dimensional trapped Bose-Einstein condensates. We will identify the oscillations and interactions of such coherent structures and we will then aim towards generalizing the corresponding notions to quasi-two-dimensional vortex states. In the latter setting, we will use two approaches: one from the linear limit of the underlying system that will permit us to identify the bifurcations of multi-vortex cluster states, while the second in the large density limit will enable our consideration of the vortices as precessing and interacting particles within the condensate. We will corroborate our analytical predictions within these limits and numerical results bridging the limits with experimental observations for the dynamics of few-vortex clusters.   

Multifractals in Soliton sea on Fibonacci Lattice

Systems exhibiting Bose-Einstein condensation are suitable for fabricating artificially designed structures, e.g., the bichromatic potentials have attracted much attention. More exotic structures also appear to be experimentally feasible in the optical imaging system with high resolution. We consider one of exotic structures, Fibonacci potential, which is quasiperiodic. On the Fibonacci potential without nonlinear interaction, it is known that the spectrum is singular continuous and all the eigenvectors are called a critical state, in which multifractal states are included. On the other hand, the physical properties of the Bose-Einstein condensates confined in a optical lattice can be described in terms of the Schr\"odinger equation with nonlinear term via particle-particle interactions. We numerically and mathematically investigated nonlinear Schr\"odinger equation on Fibonacci potential focusing on the competition between nonlinear fluctuation and criticality~[M.~Takahashi {\it et al.}, arXiv:1110.6328]. The conclusion is that the critical states with the spectrum in the Cantor set retains their profile irrespective of the strength of the nonlinearity. The spectrum for the critical states is in a sea of ``stationary solitons" which appear as a result of nonlinear effects.