next up previous contents
Next: N-body system Up: Energetics of quasi-one-dimensional Bose Previous: Energetics of quasi-one-dimensional Bose   Contents


Two-body system

Section 4.2 discusses the calculation of the energy spectrum related to the internal motion of two bosons under highly-elongated confinement, Eq. 4.7, using a B-spline basis, and the eigenspectrum related to the internal motion of two bosons under 1D confinement, Eq. 4.9, using Eqs. 4.11 through (4.13). We now use these essentially exact eigenenergies to benchmark our FN-MC calculations. Toward this end, we solve the 3D Schrödinger equation, Eq. 4.18, and the 1D Schrödinger equation, Eq. 4.19, for various interaction strengths for $N=2$ and $\lambda =0.01$ using FN-MC techniques. The resulting MC energies $E_{3D}$ and $E_{1D}$ include the center of mass energy of $(1 + \lambda/2)\hbar \omega_\perp$. To compare with the internal eigenenergies discussed in Sec. 4.2, we subtract the center of mass energy from the FN-MC energies.

For $N=2$, the lowest-lying gas-like state of the 3D Hamiltonian for the short-range potential $V^{SR}$ is the first excited eigenstate. Consequently, we solve the 3D Schrödinger equation by the FN-MC technique using the trial wave function $\psi _T$ given by Eqs. 2.37 and 2.91. Figure 4.4 shows the elliptical nodal surface of the trial wave function $\psi _T$ (solid lines) together with the essentially exact nodal surface calculated using a B-spline basis set (symbols; see also Sec. 4.2) for $\lambda =0.01$ and three different scattering lengths, $a_{3D}/a_\perp=1,6$, and $-4$. Notably, the semi-axes $a$ along the $\rho $ coordinate is larger than that along the $z$ coordinate ($a/b >1$), ``opposing'' the shape of the confining potential, for which the characteristic length along the $\rho $ coordinate is smaller than that along the $z$ coordinate ($a_\perp/a_z<1$). Figure 4.4 indicates good agreement between the essentially exact nodal surface and the parameterization of the nodal surface by an ellipse for $a_{3D}/a_\perp =1$ and $6$; some discrepancies become apparent for negative $a_{3D}$. Since the FN-MC method results in the exact energy if the nodal surface of $\psi _T$ coincides with the nodal surface of the exact eigenfunction, comparing the FN-MC energies for two particles with those obtained from a B-spline basis set calculation provides a direct measure of the quality of the nodal surface of $\psi _T$. Figure 4.3(c) compares the internal 3D energy of the lowest-lying gas-like state calculated using a B-spline basis (diamonds, see Sec. 4.2) with that calculated using the FN-MC technique (asterisks). The agreement between these two sets of energies is -- within the statistical uncertainty -- excellent for all scattering lengths $a_{3D}$ considered. We conclude that our parameterization of the two-body nodal surface, Eq. 2.91, is accurate over the whole range of interaction strengths $a_{3D}$ considered.

Figure 4.4: Nodal surface of the trial wave function $\psi _T$ (solid lines, Eq. 2.91) together with the essentially exact nodal surface calculated using a B-spline basis set (symbols) for $\lambda =0.01$, $N=2$, and three different scattering lengths, $a_{3D}/a_\perp =1$ (pluses), $a_{3D}/a_\perp =6$ (asterisks), and $a_{3D}/a_\perp =-4$ (diamonds). The nodal surface is shown as a function of the internal coordinates $z$ and $\rho $. Good agreement between the elliptical nodal surface (solid lines) and the essentially exact nodal surface (symbols) is visible for $a_{3D}/a_\perp =1$ and $6$. Small deviations are visible for $a_{3D}/a_\perp =-4$.
\includegraphics[width=0.57\columnwidth]{jpb4.eps}

We expect that our parameterization of the two-body nodal surface is to a good approximation independent of the confining potential in $z$ (for small enough $\lambda $). In fact, we expect our nodal surface to closely resemble that of the scattering wave function at low scattering energy of the 3D wave guide Hamiltonian given by Eq. 4.1. To quantify this statement, Fig. 4.5 shows the semi-axes $a$ and $b$ (pluses and asterisks, respectively) obtained by fitting an ellipse, see Eq. 2.91, to the nodal surface obtained by solving the Schrödinger equation for the two-body Hamiltonian, Eq. 4.7, using a B-spline basis for various aspect ratios $\lambda=0.001,...,1$, and fixed scattering length, $a_{3D}=2a_\perp$ (similar results are found for other scattering lengths). Indeed, the nodal surface for a given $a_{3D}/a_\perp $ is nearly independent of the aspect ratio $\lambda $ for $\lambda \le 0.01$. These findings for two particles imply that the parameterization of the nodal surface of $\psi _T$ used in the FN-MC many-body calculations should be good as long as the density along $z$ is small.

Figure 4.5: Semi-axes $a$ (pluses) and $b$ (asterisks) obtained by fitting an ellipse (see Eq. 2.91) to the essentially exact nodal surface for two bosons under cylindrical confinement, calculated using a B-spline basis set as a function of the anisotropy parameter $\lambda $, for $a_{3D}/a_\perp =2$. Dotted lines are shown to guide the eye. For $\lambda \le 0.01$, the nodal surface is nearly independent of the anisotropy parameter $\lambda $.
\includegraphics[width=0.55\columnwidth]{jpb5.eps}

Next, consider the 1D Hamiltonian, Eq. 4.19, for $N=2$. For positive $g_{1D}$, the lowest-lying gas-like state is the ground state of the two-body system and we hence use the DMC technique with $\psi _T$ given by Eqs. 2.64 and 2.65; for $g_{1D}<0$, the lowest-lying gas-like state is the first excited state, and we instead use the FN-MC technique with $\psi _T$ given by Eqs. 2.64 and 2.66. Figure 4.3 shows the 1D energies of the lowest-lying gas-like state calculated using Eqs. 4.11 through 4.13 (solid line), together with those calculated by the FN-MC technique (squares). We find excellent agreement between these two sets of 1D energies.

The comparison for two bosons between the FN-MC energies and the energies calculated by alternative means serves as a stringent test of our MC codes, since these codes are implemented such that the number of particles enters simply as a parameter.


next up previous contents
Next: N-body system Up: Energetics of quasi-one-dimensional Bose Previous: Energetics of quasi-one-dimensional Bose   Contents
G.E. Astrakharchik 15-th of December 2004