Do attractive bosons condense?

Motivated by experiments on bose atoms in traps which have attractive interactions (e.g. 7Li), we consider two models which may be solved exactly. We construct the ground states subject to the constraint that the system is rotating with angular momentum proportional to the number of atoms. In a conventional system this would lead to quantised vortices; here, for attractive interactions, we find that the angular momentum is absorbed by the centre of mass motion. Moreover, the state is uncondensed and is an example of a ‘fragmented’ condensate discussed by Nozières and Saint James. The same models with repulsive interactions are fully condensed in the thermodynamic limit. Pacs Numbers:03.75.Fi,67.90.+z Typeset using REVTEX 1 One of the most novel aspects of the creation of Bose condensates with neutral atoms in traps is the possibility of observing a bose gas with attractive interactions (negative scattering lengths). The case of Li has been studied both experimentally [1,2] and theoretically. Condensation has been predicted to be stable for a sufficiently small number of particles or sufficiently weak interactions [3,4]. The instability to collapse when these conditions are not obeyed has also been discussed by several authors [5–9]. In this Letter we show, using two exactly soluble models, that there may be other possibilities for non-condensed states with attractive interactions. The states are the ‘fragmented’ condensates discussed by Nozières and Saint James [10] in the context of excitonic bose condensates. The possibility of such states emerges from the realisation [11] that it is the exchange interaction which causes bosons with repulsive interactions to condense into a single one-particle state, if there are several one-particle ground states. Conversely for attractive interactions, the exchange term is negative and may prefer ‘fragmented’ [10] condensation into more than one state if there is a degeneracy (or perhaps if the interactions are sufficiently strong). Kagan et al. [4] argue that trapped gases with sufficiently large negative scattering lengths are unstable to the formation of clusters using a somewhat different argument, but with the same physical origin. The two models we examine are: particles in a harmonic trap with L quanta of angular momenta and attractive contact interactions treated as a degenerate perturbation [12]; rotating particles in a harmonic trap interacting with harmonic interactions [13–16]. (Both of these cases have been of interest for fermions [12,14], where rotation is replaced by a magnetic field and the phenomena are related to the fractional quantum Hall effect.) Rotation is considered in both cases, partly because the non-rotating ground state, in the thermodynamic limit, is trivial in both cases (for different reasons) and partly because the response to rotation is characteristic of superfluidity in the system [17,18]. Consider the two-dimensional Hamiltonian H = H0 +H1