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Journal of Physics Condensed Matter

Vortex lines in4He clusters

John Lekner

Published May 10, 2000• 3 citations
Publisher

Abstract

We present translationally invariant many-body wavefunctions which represent vortices in boson clusters. These functions are orbital angular momentum eigenstates, and a particular one within the class is an exact energy eigenstate for a cluster interacting with harmonic pairwise potentials. The results are valid for N atoms, with N ⩾2. Implications for superfluidity in 4 He nanodroplets are discussed.

DOI
10.1088/0953-8984/12/19/303

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We present translationally invariant many-body wavefunctions which represent vortices in boson clusters. These functions are orbital angular momentum eigenstates, and a particular one within the class is an exact energy eigenstate for a cluster interacting with harmonic pairwise potentials. The results are valid for N atoms, with N ⩾2. Implications for superfluidity in 4 He nanodroplets are discussed.
Though vortices are an unavoidable presence in bulk superfluid helium, they have not been observed to date in helium nanodroplets. The energy and angular momentum of both straight and curved vortex states of a helium nanodroplet are examined as a function of droplet size. For droplets in the size range of many experiments, it is found that during the pickup of heavy solutes, a significant fraction of events deposit sufficient energy and angular momentum to form a straight vortex line. Curved vortex lines exist down to nearly zero angular momentum and energy, and thus could in principle form in almost any collision. Further, the coalescence of smaller droplets during the cooling by expansion could also deposit sufficient angular momentum to form vortex lines. Despite their high energy, most vortices are predicted to be stable at the final temperature (0.38 K) of helium nanodroplets due to lack of decay channels that conserve both energy and angular momentum.
We discuss the rotational excitations of highly quantum many-body systems (for example, polyatomic molecules or microdroplets of helium). For a general system, the states Fί, where and ί is a rotationally invariant ground or vibrational state, are shown to be eigenfunctions of L 2 and Lz , with eigenvalues L(L+1)ħ 2 and Lħ (for even L). These wavefunctions preserve the translational invariance and the permutation and inversion symmetries of the N-particle state ί. For harmonic pair interactions, the f = 1 wavefunctions are shown to be exact eigenstates of the N-body hamiltonian. For large N, the states Fί(f=1) represent surface oscillations of the type first proposed by Bohr. An inequality for the rotational excitation energy is obtained variationally; it depends on two, three, and four-particle correlations. Other translationally invariant angular momentum eigenfunctions are also discussed.
The properties of a special class of correlated many-body wave functions, named rotating vortex clusters (RVCs), that preserve the total angular momentum of a small cloud of trapped rotating bosons are investigated. They have lower energy and provide a superior description for the formation of vortices compared to the mean-field Gross-Pitaevskii (GP) states that break the rotational symmetry. The GP vortex states are shown to be wave packets composed of such RVC states. Our results suggest that, for a small number of bosons, the physics is different from that of ideal Bose-Einstein condensates which characterize larger assemblies.
We study low-lying excited states of 4He{}^{4}\mathrm{He}4He clusters up to a cluster size of 40 atoms in a variational framework. The ansatz wave function combines two- and three-body correlations, coming from a translationally invariant configuration interaction description, and Jastrow-type short-range correlation. We have previously used this scheme to determine the ground-state energies of 4He{}^{4}\mathrm{He}4He and 3He{}^{3}\mathrm{He}3He clusters. Here we present an extension of this ansatz wave function having a good quantum angular momentum L. The variational procedure is applied independently to the cases with L=0,2,4,L=0,2,4,L=0,2,4, and upper bounds for the corresponding energies are thus obtained. Moreover, centroid energies for L excitations are calculated through the use of sum rules. A comparison with previous calculations is also made.
Helium nanodroplets are considered ideal model systems to explore quantum hydrodynamics in self-contained, isolated superfluids. However, exploring the dynamic properties of individual droplets is experimentally challenging. In this work, we used single-shot femtosecond x-ray coherent diffractive imaging to investigate the rotation of single, isolated superfluid helium-4 droplets containing ~10(8) to 10(11) atoms. The formation of quantum vortex lattices inside the droplets is confirmed by observing characteristic Bragg patterns from xenon clusters trapped in the vortex cores. The vortex densities are up to five orders of magnitude larger than those observed in bulk liquid helium. The droplets exhibit large centrifugal deformations but retain axially symmetric shapes at angular velocities well beyond the stability range of viscous classical droplets.