The diffusion Monte Carlo method is applied to describe a trapped atomic Bose-Einstein condensate at zero temperature, fully quantum mechanically and nonperturbatively. For low densities, ${n(0)a}^{3}<~2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$
[n(0), peak density; a, s-wave scattering length], our calculations confirm that the exact ground-state energy for a sum of two-body interactions depends to a good approximation on only the atomic-physics parameter a, and no other details of the two-body model potential. Corrections to the mean-field Gross-Pitaevskii energy range from being essentially negligible to about 20% for
N=2−−50 particles in the trap with positive s-wave scattering length
a=100−−10000a.u. Our numerical calculations confirm that inclusion of an additional effective potential term in the mean-field equation, which accounts for quantum fluctuations [see, e.g., E. Braaten and A. Nieto, Phys. Rev. B 56, 14 745 (1997)], leads to a greatly improved description of trapped Bose gases.
We present a Monte Carlo method for the direct simulation of evaporative cooling and the formation of the Bose-Einstein condensate of magnetically trapped atoms. The motion of the atoms in the trap is treated classically, while quantum-mechanical probability is used to describe binary elastic collisions between bosons. Two implementations of Bose statistics in the frame of a classical Monte Carlo simulation have been developed. The one involves the evaluation of the number of quantum states in single-atom phase space and the other makes use of the quantum degeneracy of energy bands. Both techniques give results compatible with the analytic results for an ideal trapped Bose gas and with the theory of a weakly interacting Bose gas, as concerns the equilibrium condensate fraction. Our simulation method also makes it possible to study the dynamic formation of the condensate, giving characteristic condensation times consistent with the experiments.
We investigate the properties of hard-core bosons in harmonic traps over a wide range of densities. Bose-Einstein condensation is formulated using the one-body density matrix (OBDM) which is equally valid at low and high densities. The OBDM is calculated using diffusion Monte Carlo methods and it is diagonalized to obtain the ``natural'' single-particle orbitals and their occupation, including the condensate fraction. At low boson density, ${\mathrm{na}}^{3}<{10}^{\ensuremath{-}5},$ where
n=N/V and a is the hard-core diameter, the condensate is localized at the center of the trap. As
na3 increases, the condensate moves to the edges of the trap. At high density, it is localized at the edges of the trap. At ${\mathrm{na}}^{3}<~{10}^{\ensuremath{-}4},$ the Gross-Pitaevskii theory of the condensate describes the whole system within
1 At ${\mathrm{na}}^{3}\ensuremath{\approx}{10}^{\ensuremath{-}3},$ corrections are
3 to the Grass-Pitaevskii energy but
30 to the Bogoliubov prediction of the condensate depletion. At ${\mathrm{na}}^{3}\ensuremath{\gtrsim}{10}^{\ensuremath{-}2},$ mean-field theory fails. At ${\mathrm{na}}^{3}\ensuremath{\gtrsim}0.1,$ the bosons behave more like a liquid
4He droplet than a trapped boson gas.
Dipolar condensates have recently been coaxed to form the long-sought supersolid phase. While one-dimensional supersolids may be prepared by triggering a roton instability, we find that such a procedure in two dimensions (2D) leads to a loss of both global phase coherence and crystalline order. Unlike in 1D, the 2D roton modes have little in common with the supersolid configuration. We develop a finite-temperature stochastic Gross-Pitaevskii theory that includes beyond-mean-field effects to explore the formation process in 2D and find that evaporative cooling directly into the supersolid phase-hence bypassing the first-order roton instability-can produce a robust supersolid in a circular trap. Importantly, the resulting supersolid is stable at the final nonzero temperature. We then experimentally produce a 2D supersolid in a near-circular trap through such an evaporative procedure. Our work provides insight into the process of supersolid formation in 2D and defines a realistic path to the formation of large two-dimensional supersolid arrays.