Dipolar Bose-Einstein condensates with dipole-dependent scattering length

We consider a Bose-Einstein condensate of polar molecules in a harmonic trap, where the effective dipole may be tuned by an external field. We demonstrate that taking into account the dependence of the scattering length on the dipole moment is essential to reproducing the correct energies and for predicting the stability of the condensate. We do this by comparing Gross-Pitaevskii calculations with diffusion Monte Carlo calculations. We find very good agreement between the results obtained by these two approaches once the dipole dependence of the scattering length is taken into account. We also examine the behavior of the condensate in nonisotropic traps.

Figure 1

(Color online) Absolute values of selected reduced $T$-matrix elements $t_{l{l}^{\prime }}\equiv t_{l0}^{l^{\prime }0}$ (symbols) for the OH-OH model potential as a function of the relative scattering energy, compared with the first-order Born approximation (dashed lines). Note, the first-order Born approximation to $t_{00}$ diverges and is not shown.

Figure 2

(Color online) Scattering length $a\left(d\right)$ versus dipole length $D_{*}$ for the dipolar potential with hard wall cutoff $b$ given in Eq. (2).

Figure 3

(Color online) Energy per particle $E∕N$ as a function of the dipole length $D_{*}$ for $N=10$. The symbols with error bars show DMC results. The solid line is the solution of the GPE. The dotted line shows a GP calculation without taking the dipole dependence of the scattering length into account, setting it equal to the hardwall cutoff. The dashed line shows a GP calculation with the dipole dependent scattering length, but omitting the long range dipolar term. The lines terminate at the collapse point of the condensate.

Figure 4

(Color online) Energy per particle $E∕N$ as a function of the dipole length $D_{*}$ for $N=10$, 20, and 50 (from top to bottom). Solid lines show mean field GP energies. Dots $(N=10)$, circles $(N=20)$, and triangles $(N=50)$ show DMC energies. Error bars indicate the statistical uncertainties of the DMC energies.

Figure 5

Variational many-body energies $E_V∕N$ for $N=20$, $b=0.0137a_{\mathrm{HO}}$ and $D_{*}=0.079a_{\mathrm{HO}}$ (crosses), $D_{*}=0.084a_{\mathrm{HO}}$ (squares), and $D_{*}=0.093a_{\mathrm{HO}}$ (triangles) as a function of the Gaussian width $b_r$, where $b_r=b_{\rho }=b_z$.

Figure 6

(Color online) Condensate sizes $X=\sqrt{⟨x^2⟩}$ and $Z=\sqrt{⟨z^2⟩}$, for 10 particles with hard wall cutoff $b=0.0137a_{\mathrm{HO}}$ in a spherical trap. Solid and dashed lines show $X$ and $Z$ obtained by solving the GP equation using $V_{\mathrm{eff}}$ with the dipole-normalized scattering length. Symbols with error bars show the corresponding DMC results. The dotted lines show results of a GP calculation with constant scattering length $a=b$. The dash-dotted line shows results of a GP calculation with dipole dependent scattering length that omits the long range dipolar interaction. Inset: The aspect ratio $Z∕X$ for the solution to the GP equation using $V_{\mathrm{eff}}$ (solid line), for the solution to the GP equation with constant scattering length $a=b$ (dotted line), and for the DMC solution to the $N$-body Schrödinger equation (symbols with error bars).

Figure 7

(Color online) Energy per particle $E∕N$ obtained by solving the GP equation using $V_{\mathrm{eff}}$ as a function of the dipole length $D_{*}$ for $N=10$ in a pancake-shaped trap with ${\omega }_z∕{\omega }_{\rho }=10$ (dashed line) and in a cigar-shaped trap with ${\omega }_z∕{\omega }_{\rho }=0.1$ (solid line). Symbols with error bars show the corresponding DMC results. $a_{\mathrm{HO}}$ is defined by the shortest dimension in each case (see text). The hard wall cutoff is $b=0.0137a_{\mathrm{HO}}$.

Figure 8

(Color online) $E∕N$ for two particles with hard wall cutoff $b=0.0137a_{\mathrm{HO}}$ in a spherical trap as a function of the dipole length $D_{*}$ obtained by solving the GP equation (solid lines) and the Schrödinger equation (dashed lines). The GP line terminates at points of collapse. For the Schrödinger equation, the three lowest eigenvalues are displayed (negative energies are not shown). The dotted line shows the results from a GP calculation with a constant scattering length $a=b$. The vertical dash-dotted lines indicate the resonance positions of $V$, Eq. (3) (see also Fig. 2).

Figure 9

(Color online) Energy per particle $E∕N$ versus dipole length $D_{*}$ with hard-core radius $b=0.0433a_{\mathrm{HO}}$. Solid lines show the GP energies for $N=10$, 20, and 50 particles (from bottom to top). Symbols with error bars show the DMC energies for $N=10$ (dots), $N=20$ (circles), and $N=50$ (triangles).