Four fermions in a one-dimensional harmonic trap: Accuracy of a variational-ansatz approach

A detailed analysis of a system of four interacting ultracold fermions confined in a one-dimensional harmonic trap is performed. The analysis is done in the framework of a simple variational ansatz for the many-body ground state, and its predictions are compared with the results of numerically exact diagonalization of the many-body Hamiltonian. A short discussion of the role of the quantum statistics, i.e., Bose-Bose and Bose-Fermi mixtures, is also presented. It is concluded that the variational ansatz, although appearing to be oversimplified, gives surprisingly good predictions of many different quantities for mixtures of equal as well as different mass systems. The result may have some experimental importance since it gives a quite simple and validated method for describing experimental outputs.

Figure 1

Ground-state energy as a function of interactions predicted by the interpolatory ansatz equation (2) (solid thin and solid thick lines for $\mu =1$ and $\mu =10$, respectively) and numerically exact diagonalization of the Hamiltonian (crosses and squares for $\mu =1$ and $\mu =10$, respectively). Predictions of the numerical ansatz are clearly overestimated, and convergence to exact results is rather poor. Dashed lines correspond to the modified ansatz, which gives much better predictions. See the text for details. The energies and the interaction strength are measured in units of $\hslash \omega$ and ${\hslash }^{3/2}{\omega }^{1/2}{m}_b^{-1/2}$, respectively.

Figure 2

The fidelity equation (6) between ground-state wave functions obtained variationally and with the exact diagonalization of the Hamiltonian (thin black and thick black lines for systems with $\mu =1$ and $\mu =10$, respectively). By construction, in the limiting cases of vanishing or very strong interactions the fidelity is equal to 1. For intermediate interactions, where the predictions of the ansatz are not exact, fidelity drops down. These results suggest that for systems of different masses inaccuracy is larger than for systems of the same mass. The variational ansatz results displayed here correspond to the original ansatz as discussed in the text. The interaction strength $g$ is measured in units of ${\hslash }^{3/2}{\omega }^{1/2}{m}_b^{-1/2}$.

Figure 3

Single-particle density profile for intermediate interactions, $g=2$, calculated from the exact ground state (solid lines) and from the variational ground state (dashed lines). Density profiles are reproduced by the ansatz method quite well for an equal-mass system $\mu =1$ (left panel) as well as for different-mass systems $\mu =10$ (right panel). In the latter case the predictions are much better for heavier components (thick lines) than for lighter components (thin lines). A comparison of profiles for the lighter components suggests that the single-particle density undergoes a separation which is much sharper than that predicted by the variational method at the same interaction strength. As in Fig. 2, the ansatz results are based on the original ansatz. The positions and the densities are measured in units of $\sqrt{\hslash /\left(m_b\omega \right)}$ and $\sqrt{m_b\omega /\hslash }$, respectively.

Figure 4

Two-body density profile of opposite fermions calculated in the ground state of the system calculated with both methods for intermediate interaction $g=2$. Predictions of the variational method (right panels) are consistent with predictions of the exact diagonalization (left panels) for equal-mass systems ($\mu =1$; top panels) as well as different-mass systems ($\mu =10$; bottom panels). However, in the latter case, i.e., for the mass imbalance $\mu =10$, the probability of finding both fermions in the middle of the trap is overestimated. This fact has consequences in the single-particle profiles (right panel of Fig. 3), where incomplete separation of densities is predicted by the variational method. Again, the ansatz results are based on the original ansatz as in Figs. 2 and 3. The positions $q_1,q_2$ are measured in natural units of harmonic oscillator $\sqrt{\hslash /\left(m_b\omega \right)}$, and the two-body density is measured in units of $m_b\omega /\hslash$.

Figure 5

The probabilities (9) of finding a single fermion in a given single-particle orbital of the harmonic confinement as functions of interactions. In the limit of vanishing interactions the fermions can be found only in the two lowest orbitals. When the interactions are present other orbitals contribute to the ground state of the system. The predictions of the variational method (solid grey line for $k=0$, dashed black line for $k=1$, solid black line for $k=2$, and dashed grey line for $k=3$) are roughly consistent with the exact-diagonalization results (crosses for $k=0$, squares for $k=1$, circles for $k=2$, and triangles for $k=3$). However, for the mass-imbalanced systems and stronger interactions the variational method predicts too rapid a drop in the ground-orbital contribution below the contribution of the second excited orbital. Here the results are also obtained with the original ansatz, as done in Figs. 2, 3, and 4. The interaction strength $g$ is measured in units of ${\hslash }^{3/2}{\omega }^{1/2}{m}_b^{-1/2}$.