Analytical solutions for the dynamics of two trapped interacting ultracold atoms

We discuss exact solutions of the Schrödinger equation for the system of two ultracold atoms confined in an axially symmetric harmonic potential. We investigate different geometries of the trapping potential, in particular we study the properties of eigenenergies and eigenfunctions for quasi-one-dimensional and quasi-two-dimensional traps. We show that the quasi-one-dimensional and the quasi-two-dimensional regimes for two atoms can be already realized in the traps with moderately large (or small) ratios of the trapping frequencies in the axial and the transverse directions. Finally, we apply our theory to Feshbach resonances for trapped atoms. Introducing in our description an energy-dependent scattering length we calculate analytically the eigenenergies for two trapped atoms in the presence of a Feshbach resonance.

Figure 1

Eigenenergies of the relative motion for two atoms interacting via $s$-wave pseudopotential and confined in a harmonic trap with $\eta ={\omega }_{\perp }∕{\omega }_z=5$. The scattering length $a$ is scaled in units of the harmonic oscillator length $d=\sqrt{\hslash ∕\left(\mu {\omega }_z\right)}$.

Figure 2

Eigenenergies of the relative motion for two atoms interacting via $s$-wave pseudopotential and confined in a harmonic trap with $\eta ={\omega }_{\perp }∕{\omega }_z=1.1$ (upper plot) and $\eta =1$ (lower plot). The scattering length $a$ is scaled in units of the harmonic oscillator length $d=\sqrt{\hslash ∕\left(\mu {\omega }_z\right)}$.

Figure 3

Eigenenergies of the relative motion for two atoms interacting via $s$-wave pseudopotential and confined in a harmonic trap with $\eta ={\omega }_{\perp }∕{\omega }_z=10$. The exact energy levels (solid lines) are compared with predictions for the one-dimensional model with energy-dependent (dotted line, indistinguishable from the solid one) and with the standard, energy-independent renormalized scattering length (dashed lines). The three-dimensional scattering length $a$ is scaled by the harmonic oscillator length $d=\sqrt{\hslash ∕\left(\mu {\omega }_z\right)}$.

Figure 4

Eigenenergies of the relative motion for two atoms interacting via $s$-wave pseudopotential and confined in a harmonic trap with $\eta ={\omega }_{\perp }∕{\omega }_z=1∕5$. The scattering length $a$ is scaled in units of the harmonic oscillator length $d=\sqrt{\hslash ∕\left(\mu {\omega }_z\right)}$.

Figure 5

Eigenenergies of the relative motion for two atoms interacting via $s$-wave pseudopotential and confined in a harmonic trap with $\eta ={\omega }_{\perp }∕{\omega }_z=0.1$. The exact energy levels (solid lines) are compared with predictions of the one-dimensional model with the energy-dependent (dotted line, indistinguishable from the solid one) and with the standard, energy-independent renormalized scattering length (dashed lines). The three-dimensional scattering length $a$ is scaled by the harmonic oscillator length $d=\sqrt{\hslash ∕\left(\mu {\omega }_z\right)}$.

Figure 6

Exact wave function $r\Psi \left(\mathbf{r}\right)$ for two atoms interacting via $s$-wave pseudopotential and trapped in a harmonic potential with $\eta ={\omega }_{\perp }∕{\omega }_z=100$. The figure shows the ground state for the scattering length $a=\pm \infty$. All lengths are scaled in units of $a_z=\sqrt{\hslash ∕\left(\mu {\omega }_z\right)}$.

Figure 7

Exact wave function $r\Psi \left(\mathbf{r}\right)$ for two atoms interacting via $s$-wave pseudopotential and trapped in a harmonic potential with $\eta ={\omega }_{\perp }∕{\omega }_z=100$. The figure presents the first excited state for the scattering length $a=\pm \infty$. All lengths are scaled in units of $a_z=\sqrt{\hslash ∕\mu {\omega }_z}$.

Figure 8

The axial profiles of the wave function presented in Fig. 7, evaluated for $\rho =0$, $\rho =0.08$, and $\rho =0.16$. The exact profiles (solid lines) are compared with the predictions of the quasi-one-dimensional approximation given by Eq. (65). All lengths are scaled in units of $a_z=\sqrt{\hslash ∕\left(\mu {\omega }_z\right)}$.

Figure 9

The radial profiles of the wave function presented in Fig. 7, evaluated for $z=0$, $z=0.5$, and $z=1.0$. The exact profiles (solid lines) are compared with the predictions of the quasi-one-dimensional approximation given by Eq. (65). The inset shows the details of $z=0$ profile for small values of $\rho$. All lengths are scaled in units of $a_z=\sqrt{\hslash ∕\left(\mu {\omega }_z\right)}$.

Figure 10

Exact wave function $r\Psi \left(\mathbf{r}\right)$ for two atoms interacting via $s$-wave pseudopotential and trapped in a harmonic potential with $\eta ={\omega }_{\perp }∕{\omega }_z=0.01$. The figure shows the ground state for the scattering length $a=\pm \infty$. All lengths are scaled in units of $a_z=\sqrt{\hslash ∕\left(\mu {\omega }_z\right)}$.

Figure 11

Exact wave function $r\Psi \left(\mathbf{r}\right)$ for two atoms interacting via $s$-wave pseudopotential and trapped in a harmonic potential with $\eta ={\omega }_{\perp }∕{\omega }_z=0.01$. The figure presents the first excited state for the scattering length $a=\pm \infty$. All lengths are scaled in units of $a_z=\sqrt{\hslash ∕\left(\mu {\omega }_z\right)}$.

Figure 12

The radial profiles of the wave function presented in Fig. 11, evaluated for $z=0$, $z=1$, and $z=2$. The exact profiles (solid lines) are compared with predictions of the quasi-two-dimensional approximation (68). All lengths are scaled in units of $a_z=\sqrt{\hslash ∕\left(\mu {\omega }_z\right)}$.

Figure 13

The axial profiles of the wave function presented in Fig. 11, evaluated for $\rho =0.1$, $\rho =5$, and $\rho =10$. The exact profiles (solid lines) are compared with predictions of the quasi-two-dimensional approximation (68). All lengths are scaled in units of $a_z=\sqrt{\hslash ∕\left(\mu {\omega }_z\right)}$.

Figure 14

Energy spectrum for two ${}^{87}\mathrm{Rb}$ atoms versus magnetic field $B$ near the Feshbach resonance at $100\mathrm{mT}$. The atoms are confined in an axially symmetric trap with ${\omega }_z=5\mathrm{kHz}$ and ${\omega }_{\perp }=500\mathrm{kHz}$. The dotted line shows the value of the magnetic field at which the energy-dependent scattering length $a_{\mathrm{eff}}\left(E\right)$ diverges.

Figure 15

Energy spectrum for two ${}^{87}\mathrm{Rb}$ atoms versus magnetic field $B$ near a Feshbach resonance at $100\mathrm{mT}$. The atoms are confined in an axially symmetric trap with ${\omega }_z=500\mathrm{kHz}$ and ${\omega }_{\perp }=5\mathrm{kHz}$. The dotted line shows the value of the magnetic field at which the energy-dependent scattering length $a_{\mathrm{eff}}\left(E\right)$ diverges.