Time-of-flight experiments in superfluid helium-4 have revealed, alongside the well-understood normal electron bubble (a roughly 19 angstrom cavity enclosing one electron), at least 18 discrete 'exotic' negative ions with higher mobilities - hence smaller radii - together with a continuous background of faster carriers. First seen in 1971, they remain the only charge carriers in the most thoroughly studied quantum liquid whose structure is unidentified. The leading proposal, the fission model, holds that when an electron's wave function is divided - for example by partial transmission at the liquid-vapor interface - each part can be trapped in its own cavity, so that a bubble contains only a fraction F of a single electron's probability; the exotic ions would then be the first observed mesoscopic objects each carrying a fraction of one electron's wave function, and their stability would show that the helium boundary does not act as a measuring apparatus that collapses the superposition. This problem, posed by Wei, Xie, Cooper and Maris (2016) and still open in the 2022 literature, is to determine what the exotic ions are: to confirm or refute the fractional-bubble identification against the alternatives (impurity anions, helium anions, electrons self-trapped on vortex rings, quantum-dot-like bubbles). Two connected questions bear on this identification: whether the coupled electron-helium equations admit multi-cavity states and what mechanism selects the discrete fractions; and, as a direct corollary, whether partial transmission produces stable fractional bubbles rather than collapsing to complete ones.

Background

Superfluid helium-4 (the common isotope of helium, which becomes a frictionless quantum liquid below the lambda transition at 2.17 K) is the most thoroughly studied quantum liquid, and it supports mobile charged particles that can be manipulated and timed with high precision. An excess electron injected into the liquid is repelled by the electrons of the helium atoms (the Pauli exclusion principle acting between the injected electron and the closed atomic 1s shells), an effect represented macroscopically by a potential barrier of about 1 eV that raises the electron's energy inside the liquid above its vacuum value. The electron therefore localizes in a spherical cavity free of helium atoms - the normal electron bubble (NEB) - whose radius R of about 19 angstroms at zero pressure follows from minimizing the bubble energy

E(R)=h28mR2+4πR2α+4π3R3P,E(R) = \frac{h^2}{8 m R^2} + 4 \pi R^2 \alpha + \frac{4 \pi}{3} R^3 P,

where the first term is the electron's zero-point confinement energy (m is the electron mass and h is Planck's constant), alpha of about 0.375 dyn/cm is the liquid-vapor surface tension, and P is the external pressure.

The standard probe of such carriers is the mobility (drift velocity per unit electric field), measured by time-of-flight: a pulse of ions crosses a known distance in a known electric field and the arrival-time spectrum is recorded. Near 1 K the liquid contains only dilute thermal excitations, chiefly rotons (elementary excitations near the minimum of helium's phonon-roton energy-momentum curve), whose scattering off an ion limits its drift; the mobility of a compact spherical ion decreases with its size, so an arrival-time spectrum maps directly onto a size spectrum. Ever since the experiments of Ihas and Sanders (1971), such spectra have shown, besides the NEB peak, a single 'fast ion' with roughly six times the NEB mobility (radius around 8 angstroms) and a family of at least 18 additional discrete peaks - the exotic ions - spread between the fast ion and the NEB, on top of a continuous background of carriers; the same structure appears with different source-tip materials (Wei, Xie, Seidel and Maris 2013; Wei et al. 2015; Xing and Maris 2020). After five decades these are the only charge carriers in superfluid helium whose structure remains unidentified.

The leading proposal is the fission model of Maris (2000). In its original form, an electron bubble optically excited from its spherically symmetric 1S ground state to the 1P excited state deforms towards a dumbbell shape, and the inertia of the moving liquid wall can pinch the cavity in two, after which the electron wave function divides between the daughter cavities; Maris called the resulting fractionally occupied bubbles 'electrinos'. In the form most relevant to the plasma experiments, an electron arriving at the liquid-vapor interface with energy only slightly above the entry threshold is partially transmitted (quantum reflection gives a transmission coefficient below unity), so the part of the wave function that enters the liquid can nucleate a cavity containing only a fraction F of the total probability. Minimizing the energy above with the confinement term scaled by F gives a radius R proportional to the fourth root of F at zero pressure, so fractional bubbles are smaller and hence faster than the NEB, in the observed range. Wei et al. (2015) also proposed a mechanism for the discreteness of the observed peaks - matching of the electronic eigenvalues between daughter cavities, giving preferred fractions such as f=1/(1+n2)f = 1/(1+n^2) and of order tens of distinct sizes - and, with Xie et al. (2014), computed the F-dependent optical absorption energies (1S to 1P and 1S to 2P transitions) and critical explosion pressures (the external tension at which a bubble becomes mechanically unstable and expands without bound) that would discriminate the model directly.

The proposal immediately met foundational objections. Rae and Vinen (2001) argued from a product-ansatz wave function that a two-bubble state would rapidly collapse to a single bubble ('electrino collapse'); Elser (2001) argued that the adiabatic (Born-Oppenheimer) approximation fails along the fission path; Jackiw, Rebbi and Schrieffer (2001) analyzed whether 'fractional electrons' are well defined in this setting. The source paper of the present problem, Wei, Xie, Cooper and Maris (2016), rebuts the Rae-Vinen argument by showing that their ansatz, if required to satisfy the full Schrodinger equation with the actual electron-helium interaction, forces that interaction to separate into electron-only and helium-only parts - eliminating the very repulsion that stabilizes bubbles - and it formulates the coupled system as two Gross-Pitaevskii equations (the Gross-Pitaevskii equation is the nonlinear mean-field Schrodinger equation governing a condensate order parameter, written here for both the helium condensate and the electron wave function, mutually coupled through a contact repulsion calibrated to the 1 eV barrier). Within this framework the paper verifies the single-bubble states and estimates that two bubbles separated by more than about 30 angstroms of liquid are stable against tunneling on experimental timescales, but it reports that eigenstates in which the electron is divided among multiple cavities had never been systematically investigated, and it observes that the proposed two-bubble states are not energy eigenstates (fragments confined in cavities of different sizes oscillate at different frequencies).

Competing identifications remain in play. Khrapak and Schmidt (2011) proposed that the exotic ions are stable negative impurity ions (such as the molecular and atomic anions O2-, O- and H-), which are always present in small amounts in discharge plasmas and are surrounded by smaller bubbles or clusters, and that the fast ion is the negative ion of a helium excimer molecule (an excimer being a short-lived excited dimer); Dyugaev, Grigor'ev and Lebedeva (2012) similarly proposed rare-gas anions. Elser (2015) and Khrapak and Bronin (2017, 2019) proposed that an electron can self-trap on a quantized vortex ring (a circular superfluid vortex line whose circulation is quantized in units of h divided by the mass of a helium atom) of radius around 7 angstroms; Lawandy (2019) proposed core-shell quantum-dot-like electron bubbles. Wei et al. (2015) argue that the impurity-anion route would require some eighteen distinct impurity species in comparable amounts, with no impurity emission lines seen in the discharge spectrum, and that the measured lifetimes of helium anions, of order tens to hundreds of microseconds, are far shorter than the roughly 0.1 s flight time in these experiments; but no alternative has been quantitatively fit to all of the peaks and the background, and none has been excluded at the data level.

The question also has a quantum-measurement dimension. A state in which one electron's wave function is divided between two well-separated cavities in a macroscopic liquid is a laboratory realization of the 'Einstein's boxes' thought experiment (a single particle's wave function split between two containers): under purely unitary Schrodinger evolution both parts persist, whereas if the liquid surface acts as a measuring apparatus, the superposition collapses and only one complete bubble survives. Xing, Sirisky, Wei, Cooper and Maris (2022) reported the first mobility-level test: electrons from a plasma in the vapor are partially transmitted at the liquid surface, and the measured mobility distribution of the resulting carriers is consistent with simulations of fractional bubbles, supporting the conclusion that entry into the liquid does not itself act as an immediate measurement that collapses the electron's wave function. A companion experiment (Xing and Maris 2022) found that alpha particles, which inject electrons directly into the liquid, produce no high-mobility ions - consistent with the partial-transmission production mechanism. The review of Xing and Maris (2020), however, still records the structure of the exotic ions as unknown, and the 2022 papers themselves present their results as consistent with, rather than decisive for, the fractional-bubble identification.

Problem Statement

Determine the physical nature of the exotic negative ions in superfluid helium-4 - in particular, establish or refute the identification, proposed by the fission model, of each exotic ion as a bubble containing a fraction F < 1 of the wave function of a single electron. This question was raised by Wei, Xie, Cooper and Maris (2016) and remains open in the most recent literature (2022). Three closely connected aspects of the search bear on this one objective:

(1) Experimental identification. Measurements that discriminate between the fission model and the competing identifications (impurity anions, helium anions, electrons self-trapped on vortex rings, core-shell quantum-dot bubbles).

(2) Theoretical foundation. Whether the coupled electron-helium equations - the coupled Gross-Pitaevskii system of the 2016 paper or more realistic finite-range density functionals - admit stationary or long-lived multi-cavity states in which the electron's wave function is divided among separated cavities; the stability of such states against tunneling and relaxation; and the mechanism that selects discrete fractions F, including whether it reproduces the observed number and spacing of the exotic-ion peaks.

(3) Measurement corollary. Whether the electron-helium interaction at a bubble boundary constitutes a quantum measurement - that is, whether an electron whose wave function is partially transmitted into the liquid forms a stable fractional bubble (no collapse) or always ends up in a single complete bubble (collapse).

The question is resolved by establishing what the exotic ions are - or by excluding all proposed identifications, itself a decisive result that would redirect the search - with quantitative evidence assessed against the candidate models.

References

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