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Bose atoms in a trap a variational Monte Carlo formulation for the universal behavior at th.pdf

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Bose atoms in a trap a variational Monte Carlo formulation for the universal behavior at th

a r X i v : c o n d - m a t / 0 6 0 3 0 5 6 v 1 [ c o n d - m a t .s t a t - m e c h ] 2 M a r 2 0 0 6 Bose atoms in a trap: a variational Monte Carlo formulation for the universal behavior at the Van der Waals length scale Imran Khan and Bo Gao? Department of Physics and Astronomy, University of Toledo, Toledo, Ohio 43606 (Dated: February 4, 2008) We present a variational Monte Carlo (VMC) formulation for the universal equations of state at the Van der Waals length scale [B. Gao, J. Phys. B 37, L227 (2004)] for N Bose atoms in a trap. The theory illustrates both how such equations of state can be computed exactly, and the existence and the importance of long-range atom-atom correlation under strong confinement. Explicit numerical results are presented for N = 3 and 5, and used to provide a quantitative understanding of the shape-dependent confinement correction that is important for few atoms under strong confinement. PACS numbers: 05.30.Jp,21.45.+v,03.75.Nt I. INTRODUCTION Few atoms in a trap, which can in principle be real- ized, e.g., through a Mott transition from a degenerate quantum gas in a optical lattice [1, 2], is a fundamental system for studying atomic interactions and correlations. It has many features of a many-atom system, yet still suf- ficiently simple to be amenable to a number of different theoretical approaches that can check against and learn from each other. These methods include, for example, the Monte Carlo methods (see, e.g., [3, 4, 5, 6, 7]), and the hyperspherical methods [8, 9, 10, 11, 12, 13, 14]. Unlike the problem of two identical atoms in a sym- metric harmonic trap, which can be reduced to a 1-D problem and be solved exactly [15, 16, 17, 18, 19], the problem of N atoms in a trap (N 2) is much more complex. Partly due to this complexity, and partly due to our previously limited understanding of atomic inter- action based on the effect-range theory [20, 21], existing studies of N atoms in a trap, with virtually no exception,