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arXiv:cond-mat/0603056 · Physical Review A

Bose atoms in a trap: A variational Monte Carlo formulation for the universal behavior at the van der Waals length scale

Imran Khan, Bo Gao

Published Jun 23, 2006• 11 citations
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Abstract

We present a variational Monte Carlo 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.

DOI
10.1103/physreva.73.063619

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We present a variational Monte Carlo formulation for the universal equations of state at the van der Waals length scale [B. Gao, J. Phys. B 37, L227 (2004)] for NNN 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=3N=3N=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.
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 β 6 = ( mC 6 /ℏ 2 ) 1/4 (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.
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 = (mC6/2)1/4 (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 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),[n(0),[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−−50N=2--50N=2−−50 particles in the trap with positive s-wave scattering length a=100−−10000a.u.a=100--10000\mathrm{a}.\mathrm{u}.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 variational Monte Carlo (VMC) method that works equally well for the ground and the excited states of a quantum system. The method is based on the minimization of the variance of energy, as opposed to the energy itself in standard methods. As a test, it is applied to the investigation of the universal spectrum at the van der Waals length scale for two identical Bose atoms in a symmetric harmonic trap, with results compared to the basically exact results obtained from a multiscale quantum-defect theory.
We investigate the long-range behavior of the induced Casimir interaction between two spinless heavy impurities, or polarons, in superfluid cold atomic gases. With the help of effective field theory (EFT) of a Galilean invariant superfluid, we show that the induced impurity-impurity potential at long distance universally shows a relativistic van der Waals-like attraction (∼1/r^{7}) resulting from the exchange of two superfluid phonons. We also clarify finite temperature effects from the same two-phonon exchange process. The temperature T introduces the additional length scale c_{s}/T with the speed of sound c_{s}. Leading corrections at finite temperature scale as T^{6}/r for distances r≪c_{s}/T smaller than the thermal length. For larger distances the potential shows a nonrelativistic van der Waals behavior (∼T/r^{6}) instead of the relativistic one. Our EFT formulation applies not only to weakly coupled Bose or Fermi superfluids but also to those composed of strongly correlated unitary fermions with a weakly coupled impurity. The sound velocity controls the magnitude of the van der Waals potential, which we evaluate for the fermionic superfluid in the BCS-BEC crossover.
The equation of state of a weakly interacting two-dimensional Bose gas is studied at zero temperature by means of quantum Monte Carlo methods. Going down to as low densities as $n{a}^{2}\ensuremath{\propto}{10}^{\ensuremath{-}100}$ permits us to obtain agreement on beyond mean-field level between predictions of perturbative methods and direct many-body numerical simulation, thus providing an answer to the fundamental question of the equation of state of a two-dimensional dilute Bose gas in the universal regime (i.e., entirely described by the gas parameter na2n{a}^{2}na2). We also show that the measure of the frequency of a breathing collective oscillation in a trap at very low densities can be used to test the universal equation of state of a two-dimensional Bose gas.