Universality of an impurity in a Bose-Einstein condensate

Universality is a powerful concept in physics, allowing one to construct physical descriptions of systems that are independent of the precise microscopic details or energy scales. A prime example is the Fermi gas with unitarity limited interactions, whose universal properties are relevant to systems ranging from atomic gases at microkelvin temperatures to the inner crust of neutron stars. Here we address the question of whether unitary Bose systems can possess a similar universality. We consider the simplest strongly interacting Bose system, where we have an impurity particle ("polaron") resonantly interacting with a Bose-Einstein condensate (BEC). Focusing on the ground state of the equal-mass system, we use a variational wave function for the polaron that includes up to three Bogoliubov excitations of the BEC, thus allowing us to capture both Efimov trimers and associated tetramers. Unlike the Fermi case, we find that the length scale associated with Efimov trimers (i.e., the three-body parameter) can strongly affect the polaron's behaviour, even at boson densities where there are no well-defined Efimov states. However, by comparing our results with recent quantum Monte Carlo calculations, we argue that the polaron energy is a \emph{universal} function of the Efimov three-body parameter for sufficiently low boson densities. We further support this conclusion by showing that the energies of the deepest bound Efimov trimers and tetramers at unitarity are universally related to one another, regardless of the microscopic model. On the other hand, we find that the quasiparticle residue and effective mass sensitively depend on the coherence length $\xi$ of the BEC, with the residue tending to zero as $\xi$ diverges, in a manner akin to the orthogonality catastrophe.