Three-body crossover from a Cooper triple to a bound trimer state in three-component Fermi gases near a triatomic resonance

We theoretically investigate ground-state properties of a three-component Fermi gas with pairwise contact interactions between different components near a triatomic resonance where bound trimers are about to appear. Using variational equations for in-medium two- and three-body cluster states in three dimensions, we elucidate the competition of pair and triple formations due to the Fermi surface effects. We present the ground-state phase diagram that exhibits transition from a Cooper pair to Cooper triple state and crossover from a Cooper triple to tightly bound trimer state at negative scattering lengths. This three-body crossover is analogous to the Bardeen-Cooper-Schrieffer to Bose-Einstein condensation crossover observed in a two-component Fermi gas. We predict that the threshold scattering length $a_{-}$ for three-body states can be shifted towards the weak-coupling side due to the emergence of Cooper triples.

Figure 1

In-medium binding energies $-E_3$ (solid curve) and $-E_2$ (dashed curve) of a trimer and a pair, respectively, calculated at $k_{\mathrm{F}}{r}_{\mathrm{\Lambda }}=0.127$, 0.170, and 0.255 (corresponding to $\mathrm{\Lambda }/k_{\mathrm{F}}=10$, 7.5, and 5). For reference, we plot a cutoff energy scale $0.12{\mathrm{\Lambda }}^2/m$ (horizontal dotted lines) corresponding to the in-vacuum trimer binding energy with each cutoff at unitarity. $a_{\mathrm{m}-}$ shows the threshold scattering length for the in-medium trimer formation at given $k_{\mathrm{F}}{r}_{\mathrm{\Lambda }}$.

Figure 2

Three-body pole $E_{\mathrm{pole}}=3E_{\mathrm{F}}+E_3$ (solid curve) for $k_{\mathrm{F}}{r}_{\mathrm{\Lambda }}=0.127$. For comparison, we plot the three-body energy $E_{3,\mathrm{v}}$ (dashed-dotted curve) in vacuum. We find four regimes consisting of Cooper pairs [CP, $1/\left(k_{\mathrm{F}}a\right)<1/\left(k_{\mathrm{F}}{a}_{\mathrm{m}-}\right)$], Cooper triples [CT, $1/\left(k_{\mathrm{F}}{a}_{\mathrm{m}-}\right)\le 1/\left(k_{\mathrm{F}}a\right)<1/\left(k_{\mathrm{F}}{a}_{\mathrm{m}-}\right)$], squeezed Cooper triples [SCT, $1/\left(k_{\mathrm{F}}{a}_{-}\right)\le 1/\left(k_{\mathrm{F}}a\right)<1/\left(k_{\mathrm{F}}{a}_0\right)$], and bound trimers [BT, $1/\left(k_{\mathrm{F}}a\right)\ge 1/\left(k_{\mathrm{F}}{a}_0\right)$].

Figure 3

Ground-state phase diagram in the plane of the range parameter $k_{\mathrm{F}}{r}_{\mathrm{\Lambda }}$ and the inverse scattering length $1/\left(k_{\mathrm{F}}a\right)$. The characteristic couplings $1/\left(k_{\mathrm{F}}{a}_{\mathrm{m}-}\right), 1/\left(k_{\mathrm{F}}{a}_{-}\right)$, and $1/\left(k_{\mathrm{F}}{a}_0\right)$ that distinguish between CP, CT, SCT, and TB are plotted.

Figure 4

Three-body loss rate $\mathrm{\Gamma }$ in the crossover from the Cooper triple to the bound trimer state (solid curve) at $k_{\mathrm{F}}{r}_{\mathrm{\Lambda }}=0.127$. The dashed-dotted curve shows the result in the absence of the Fermi surface. The calculations are stopped where the three-body bound-state solution disappears. We normalize $\mathrm{\Gamma }$ with ${\mathrm{\Gamma }}_0$, which is the three-body loss rate at $a=a_{-}$ in the vacuum case.