Energy spectrum of harmonically trapped two-component Fermi gases: Three- and four-particle problem

Trapped two-component Fermi gases allow the investigation of the so-called BCS-BEC crossover by tuning the interspecies atom-atom $s$-wave scattering length $a^{(\mathit{aa})}$ from attractive to repulsive, including vanishing and infinitely large values. Here, we numerically determine the energy spectrum of the equal-mass spin-balanced four-fermion system—the smallest few-particle system that exhibits BCS-BEC crossoverlike behavior—as a function of $a^{(\mathit{aa})}$ using the stochastic variational approach. For comparative purposes, we also treat the two- and three-particle systems. States with vanishing and finite total angular momenta as well as with natural and unnatural parities are considered. In addition, the energy spectrum of weakly attractive and weakly repulsive gases is characterized by employing a perturbative framework that utilizes hyperspherical coordinates. The hyperspherical coordinate approach allows the straightforward assignment of quantum numbers and furthermore provides great insight into the strongly interacting unitary regime.

Figure 1

Relative $s$-wave energies $E_{\mathrm{rel}}(1,1)$ of the trapped atom-atom system, obtained by solving Eq. (15) for $k=\mathit{aa}$, as a function of $a_{\mathrm{ho}}/a^{(\mathit{aa})}$. The lowest $s$-wave eigenenergy (solid line) is referred to as $E_{\mathrm{dimer}}$ in the text. Here, $E_{\mathrm{ho}}$ denotes the harmonic oscillator energy, $E_{\mathrm{ho}}=\hslash \omega$.

Figure 2

Three-fermion energies for (a) $(L_{\mathrm{rel}},{\Pi }_{\mathrm{rel}})=(0,+1)$ and (b) $(L_{\mathrm{rel}},{\Pi }_{\mathrm{rel}})=(1,-1)$ as a function of the inverse $s$-wave scattering length $a_{\mathrm{ho}}/a^{(\mathit{aa})}$. Symbols show the essentially exact zero-range energies obtained by solving Eq. (33). Solid lines show the energies obtained by treating the noninteracting atomic Fermi gas perturbatively for negative and positive $a^{(\mathit{aa})}$, while dashed lines show the energies obtained by treating the effective atom plus dimer system perturbatively.

Figure 3

Three-fermion energies as a function of the absolute value of the $s$-wave scattering length $\left|a^{(\mathit{aa})}\right|$ for small $\left|a^{(\mathit{aa})}\right|$, $a^{(\mathit{aa})}⩽0$, and (a) $E_{\mathrm{rel}}(2,1)\approx 4\hslash \omega$, (b) $E_{\mathrm{rel}}(2,1)\approx 5\hslash \omega$, (c) $E_{\mathrm{rel}}(2,1)\approx 6\hslash \omega$, and (d) $E_{\mathrm{rel}}(2,1)\approx 7\hslash \omega$. Squares, pluses, diamonds, crosses, and circles show the essentially exact zero-range energies obtained by solving Eq. (33) for $L_{\mathrm{rel}}=0–4$. For comparison, solid lines show the perturbative results for natural parity states. The solid lines correspond to (a) $(L_{\mathrm{rel}},{\Pi }_{\mathrm{rel}})=(1,-1)$; and from bottom to top to (b) $(L_{\mathrm{rel}},{\Pi }_{\mathrm{rel}})=(0,+1)$ and $(2,+1)$; (c) $(L_{\mathrm{rel}},{\Pi }_{\mathrm{rel}})=(1,-1)$, $(3,-1)$, and $(1,-1)$; and (d) $(L_{\mathrm{rel}},{\Pi }_{\mathrm{rel}})=(0,+1)$, $(2,+1)$, $(0,+1)$, $(4,+1)$, and $(2,+1)$.

Figure 4

Frequency of the coefficients $c^{(1)}$, which characterize the energy shifts $E^{(1)}$ of the weakly interacting atomic Fermi gas, for the three- and four-fermion systems. Panels (a) and (b) show the distribution of the $c^{(1)}$ coefficients for $E_{\mathrm{rel}}(2,1)\approx 7\hslash \omega$ (fourth energy manifold of the three-fermion system) and for $E_{\mathrm{rel}}(2,1)\approx 8\hslash \omega$ (fifth energy manifold of the three-fermion system). Panels (c) and (d) show the distribution of the $c^{(1)}$ coefficients for $E_{\mathrm{rel}}(2,2)\approx 19\hslash \omega /2$ (fourth energy manifold of the four-fermion system) and for $E_{\mathrm{rel}}(2,2)\approx 21\hslash \omega /2$ (fifth energy manifold of the four-fermion system). Note the log scale of the vertical axis.

Figure 5

Symbols show the scaled energies $E_{\mathrm{rel}}(2,1)-E_{\mathrm{dimer}}$ as a function of the inverse scattering length $a_{\mathrm{ho}}/a^{(\mathit{aa})}$ for (a) $(L_{\mathrm{rel}},{\Pi }_{\mathrm{rel}})=(0,+1)$, (b) $(L_{\mathrm{rel}},{\Pi }_{\mathrm{rel}})=(1,-1)$, (c) $(L_{\mathrm{rel}},{\Pi }_{\mathrm{rel}})=(2,+1)$, and (d) $(L_{\mathrm{rel}},{\Pi }_{\mathrm{rel}})=(3,-1)$. For comparison, dashed lines show the energies predicted by the effective atom-dimer model, which provides a semiquantitative description of the states belonging to the “atom plus dimer” family (see text for details).

Figure 6

Symbols show the coefficients $K_{\mathrm{unit}}$ that correspond to natural parity states for (a) the three-fermion system and (b) the four-fermion system as a function of the angular momentum quantum number $L_{\mathrm{rel}}$. Dotted lines show the coefficients $K_{\mathrm{unit},\mathrm{model}}$ (see text for details).

Figure 7

Four-fermion energies $E_{\mathrm{rel}}(2,2)$ as a function of the inverse $s$-wave scattering length $a_{\mathrm{ho}}/a^{(\mathit{aa})}$ with $1/a^{(\mathit{aa})}⩽0$. For $a^{(\mathit{aa})}\to 0^{-}$, all energy levels corresponding to natural parity states around $E_{\mathrm{ni},\mathrm{rel}}=13\hslash \omega /2$, $15\hslash \omega /2$, and $17\hslash \omega /2$ are shown. Solid, dotted, dashed, dash-dotted, and dash-dash-dotted lines show the zero-range energies corresponding to natural parity states with $L_{\mathrm{rel}}=0$ to $L_{\mathrm{rel}}=4$. Pluses show the energy of the energetically lowest-lying unnatural parity state with $L_{\mathrm{rel}}=1$.

Figure 8

Four-fermion energies $E_{\mathrm{rel}}(2,2)$ as a function of the absolute value of the $s$-wave scattering length $\left|a^{(\mathit{aa})}\right|$ for small $\left|a^{(\mathit{aa})}\right|$, $a^{(\mathit{aa})}⩽0$, and $E_{\mathrm{rel}}(2,2)\approx 13\hslash \omega /2$. Squares, diamonds, and pluses show the numerically determined four-fermion energies for $(L_{\mathrm{rel}},{\Pi }_{\mathrm{rel}})=(0,+1)$, $(L_{\mathrm{rel}},{\Pi }_{\mathrm{rel}})=(2,+1)$, and $(L_{\mathrm{rel}},{\Pi }_{\mathrm{rel}})=(1,+1)$, respectively. Solid and dashed lines show the perturbative results for natural and unnatural parity states, respectively.

Figure 9

Four-fermion energies $E_{\mathrm{rel}}(2,2)$ as a function of the absolute value of the $s$-wave scattering length $\left|a^{(\mathit{aa})}\right|$ for small $\left|a^{(\mathit{aa})}\right|$ ($a^{(\mathit{aa})}⩽0$), $E_{\mathrm{rel}}(2,2)\approx 17\hslash \omega /2$ and $(L_{\mathrm{rel}},{\Pi }_{\mathrm{rel}})=(2,+1)$. Diamonds show the numerically determined four-fermion energies, while solid lines show the perturbative results. Note that the fourth and fifth states (counted from the bottom) are nearly degenerate for the scattering length range depicted.

Figure 10

Scaled four-fermion energies $E_{\mathrm{rel}}(2,2)-2E_{\mathrm{dimer}}$ as a function of $a_{\mathrm{ho}}/a^{(\mathit{aa})}$ for (a) $(L_{\mathrm{rel}},{\Pi }_{\mathrm{rel}})=(0,+1)$, (b) $(L_{\mathrm{rel}},{\Pi }_{\mathrm{rel}})=(2,+1)$, and (c) $(L_{\mathrm{rel}},{\Pi }_{\mathrm{rel}})=(4,+1)$. Symbols show the numerically determined energies, while dashed lines show the energies predicted by the effective dimer-dimer model (see text for details).

Figure 11

Symbols show the difference between the lowest coefficient $K_{\mathrm{unit}}$ obtained from our numerically determined four-fermion energies and the lowest coefficient $K_{\mathrm{unit},\mathrm{model}}$ predicted by the simple dimer-dimer model discussed in the text as a function of $L_{\mathrm{rel}}$ ($L_{\mathrm{rel}}$ even). To guide the eye symbols are connected by dotted lines.