Independent Control of Scattering Lengths in Multicomponent Quantum Gases

We develop a method of simultaneous and independent control of different scattering lengths in ultracold atomic gases, such as ${}^{40}\mathrm{K}$ or a ${}^{40}\mathrm{K}\mathrm{\text{−}}{}^6\mathrm{Li}$ mixture. Our method can be used to engineer multicomponent quantum phases and Efimov trimer states.

Figure 1

(a) Schematic hyperfine levels used for the control of a single scattering length. The states $|f_{1,2}⟩$ are coupled to form two dressed states $|f_{h,l}⟩$. (b) Bare channels with hyperfine states $|1⟩\equiv |f_1⟩|g⟩$ and $|2⟩\equiv |f_2⟩|g⟩$ are coupled with parameters ($\Omega$, $\Delta$), and $|2⟩$ is coupled to the bound state $|{\varphi }_{\mathrm{res}})$ in the closed channel with hyperfine state $|c⟩$ via interaction $W$. The state $|1⟩$ can decay via a hyperfine relaxation process with two-body loss rate $K_1$. (c) Dressed channels. $a_{lg}$ is resonantly enhanced when the threshold energy of the channel with $|f_l⟩|g⟩$ crosses the bound state $|{\Phi }_{\mathrm{res}})$ in $|c⟩$. (d),(e) Hyperfine states used for the control of two scattering lengths with our first (d) and second (e) methods. All the levels are plotted in the rotating frame of reference. In (b) and (c), $r$ indicates the interatomic distance.

Figure 2

(a) Independent control of two scattering lengths in the three-component gases of a ${}^{40}\mathrm{K}\mathrm{\text{−}}{}^6\mathrm{Li}$ mixture. The scattering length $a_{lg}$ is plotted in units of the Bohr radius $a_0$ as a function of $\Delta$ with $\Omega =40\mathrm{MHz}$ and $B=157.3$ (blue dashed curve), 157.6 (black solid curve), and 157.9 G (red dash-dotted curve). (b) Independent control of two scattering lengths in the three-component gases of ${}^{40}\mathrm{K}$ with $B=200\mathrm{G}$ and $\Omega =2\mathrm{MHz}$, ${\Omega }^{\prime }=9\mathrm{MHz}$. $\mathrm{Re}[a_{lg}]$ (blue solid curve) and $K_l$ (green dash-dotted curve) are plotted as functions of $\Delta$; $\mathrm{Re}[a_{l^{\prime }g}]$ (red dashed curve) and $K_{l^{\prime }}$ (black dotted curve) are plotted as functions of ${\Delta }^{\prime }$. Since the scattering lengths between $|f_1(f_1^{\prime })⟩$ and $|g⟩$ are not available, we take the values to be the same as those between $|f_2(f_2^{\prime })⟩$ and $|g⟩$, which are given in Refs. 12, 16, 17.