Hyperspherical lowest-order constrained-variational approximation to resonant Bose-Einstein condensates

We study the ground-state properties of a system of $N$ harmonically trapped bosons of mass $m$ interacting with two-body contact interactions, from small to large scattering lengths. This is accomplished in a hyperspherical coordinate system that is flexible enough to describe both the overall scale of the gas and two-body correlations. By adapting the lowest-order constrained-variational method, we are able to semiquantitatively attain Bose-Einstein condensate ground-state energies even for gases with infinite scattering length. In the large-particle-number limit, our method provides analytical estimates for the energy per particle $E_0/N\approx 2.5N^{1/3}\hslash \omega$ and two-body contact $C_2/N\approx 16N^{1/6}\sqrt{m\omega /\hslash }$ for a Bose gas on resonance, where $\omega$ is the trap frequency.

Figure 1

Angular wave function ${\varphi }_{\nu }$ as a function of $r_{12}$ for $N=10$ (a) at small scattering length and (b) on resonance $a=\infty$. All length scales are in units of $a_{\text{HO}}=\sqrt{\hslash /m\omega }$. The insets show the zero crossings of the curves. The crossing occurs at approximately $a$ for small scattering length and at some finite value in the unitary limit. Note that ${\varphi }_{\nu }$ is actually a function of the hyperangle $\alpha$. The relation $r_{12}=\sqrt{2}\rho sin\alpha$ is used to convert the angle $\alpha$ to the pair distance $r_{12}$. Here $\rho$ has its value at the minimum of the corresponding hyperradial potential.

Figure 2

Zero crossing $a_c$ of the angular wave function ${\varphi }_{\nu }$ for $N=10$ as a function of the scattering length $a$. The inset shows the value ${\rho }_{\text{min}}$ that is used to compute $a_c$; ${\rho }_{\text{min}}$ is taken to be the value of $\rho$ where the effective potential $V^{\mathrm{eff}}$, discussed in the next section, is at its minimum.

Figure 3

Effective potential $V^{\mathrm{eff}}$ as a function of the hyperradius $\rho$ for $a=0$ (noninteracting case), $a>0$ (repulsive interaction), and $a<0$ (attractive interaction). For any $a>0$ and $N$, a local minimum always exists. However, for a given $a<0$, there exists a maximum $N$ when the local minimum of $V^{\mathrm{eff}}$ starts to disappear. The highest $V^{\mathrm{eff}}$ curve corresponds to the $a\to +\infty$ case.

Figure 4

Ground-state energy $E_0$ as a function of the scattering length $a$ for $N=10$. The left and right panels consider negative and positive scattering lengths.

Figure 5

Ground-state energy per particle $E_0/N$ as a function of the scattering length $a$ for $N=4$.

Figure 6

Ground-state energy per particle $E_0/N$ as a function of the scattering length $a$ for $N=10$. The inset shows a blowup of the weakly interacting regime.

Figure 7

(a) Ground-state energy per particle, in units of $\hslash \omega$. The inset shows the various perturbative versions (small-$a$ region). (b) Two-body contact per particle, in units of $a_{\text{HO}}^{-1}$, as a function of the scattering length $a$ for $N=1000$.