Universal properties of Bose systems with van der Waals interaction

We show that universal properties of a many-atom quantum system exist at a much higher density, or a much shorter length scale, than those implied by traditional theories for a dilute Bose gas. In particular, a universal equation of state at length scale β6 = (mC6/h̄) (corresponding to the van der Waals interaction) is defined and investigated by combining the concept of effective potential, the constrained variational method and the analytic solution for the van der Waals potential. In one application, we show that a many-atom Bose system with negative scattering length may form a metastable liquid Bose– Einstein condensate (BEC) state with an equilibrium density that is controlled by a scaled scattering length. The realization of gaseous Bose–Einstein condensate (BEC) [1–3] has brought an explosion of activities in the study of many-atom quantum systems [4, 5]. Considerable advances have been made in using the Gross–Pitevaskii (GP) equation to simulate a vast variety of interesting phenomena that characterize a macroscopic quantum system [4, 5]. In terms of microscopic understanding, however, the rate of advance has been more limited. Much of what we know can still be summarized by the famous equation of state for a quantum gas of hard spheres [6, 7]: