Geometric Scaling of Efimov States in a ${}^6\mathrm{Li}\text{−}{}^{133}\mathrm{Cs}$ Mixture

In few-body physics, Efimov states are an infinite series of three-body bound states that obey universal discrete scaling symmetry when pairwise interactions are resonantly enhanced. Despite abundant reports of Efimov states in recent cold atom experiments, direct observation of the discrete scaling symmetry remains an elusive goal. Here we report the observation of three consecutive Efimov resonances in a heteronuclear Li-Cs mixture near a broad interspecies Feshbach resonance. The positions of the resonances closely follow a geometric series 1, $\lambda$, ${\lambda }^2$. The observed scaling constant ${\lambda }_{\text{exp}}=4.9(4)$ is in good agreement with the predicted value of 4.88.

Figure 1

Discrete scaling symmetry of Efimov states. A series of Efimov states (dashed curves) exists near a Feshbach resonance located at $B_0$. Away from the resonance, on the side with scattering length $a<0$, they merge into the three-body scattering continuum (crosses). Physical observables in the Efimov scenario—including the molecular size $R_n$, the binding energies $E_n$, and the Efimov resonance positions at the scattering length $a_n$ (associated with the magnetic field $B_n$)—show discrete scaling symmetry. The discrete scaling law is graphically represented by the location of the resonances on the $B$ axis and the size of the spheres, respectively.

Figure 2

Atom number decay of single-species and Li-Cs mixture samples. (a) At $B=848.0\mathrm{G}$, $\sim 5\mathrm{G}$ away from the Li-Cs Feshbach resonance ($a_{\mathrm{Li}\mathrm{Cs}}=-354a_0$, $a_{\mathrm{Cs}\mathrm{Cs}}=-1240a_0$) Li loss increases significantly when Cs is introduced (left panel). Cs loss is dominated by Cs-Cs-Cs recombination (right panel). Sample temperature is $T=390\mathrm{nK}$. (b) Near the Li-Cs Feshbach resonance $B=842.7\mathrm{G}$ ($a_{\mathrm{Cs}\mathrm{Cs}}=-1570a_0$), enhanced atom loss is evident in both the Li and Cs atom number evolution when both species are present. Sample temperature is $T=340\mathrm{nK}$. Data in (a) and (b) are scaled to the initial atom numbers, $N_{\mathrm{Li}}=2\sim 3\times 10^4$ and $N_{\mathrm{Cs}}=4\sim 5\times 10^4$, obtained from double exponential fits (continuous and dotted lines), which also serve as guides to the eye.

Figure 3

Observation of three Li-Cs-Cs Efimov resonances. (a) Scaled Li number versus magnetic field showing the first Li-Cs-Cs Efimov resonance, from the average of 13 individual scans. Here, $N_{\mathrm{Li}}=1.3\times 10^4$ and $N_{\mathrm{Cs}}=2.7\times 10^4$ with typical temperature $T=800\mathrm{nK}$ and hold time 225 ms. (b) Scaled Li and Cs [inset, (d)] numbers near the second and third Li-Cs-Cs Efimov resonances, from the average of 68 scans with typical temperature $T=360\mathrm{nK}$ and hold time 115 ms. The mean atom numbers are $N_{\mathrm{Li}}=1.4\times 10^4$ and $N_{\mathrm{Cs}}=2.1\times 10^4$. (c) Scaled Li and Cs [inset, (e)] numbers close to the third Li-Cs-Cs Efimov resonance and the Li-Cs Feshbach resonance, from the average of 327 scans with typical temperature $T=270\mathrm{nK}$ and hold time 115 ms. The mean atom numbers are $N_{\mathrm{Li}}=9.1\times 10^3$ and $N_{\mathrm{Cs}}=1.4\times 10^4$. The scaled atom numbers come from the average of the individual scans divided by their respective mean values. The vertical dashed lines indicate the Feshbach resonance and arrows indicate the Efimov resonances. The dashed curves correspond to an interpolation of the data and serve as a guide to the eye.

Figure 4

Feshbach and Efimov resonance structure in the recombination coefficient. We extract and normalize the recombination coefficient $K_3$ from the atom number measurement based on a rate equation model [28]. Using the data in Fig. 3 ($T=800\mathrm{nK}$, open circles) and Fig. 3 ($T=360\mathrm{nK}$, solid circles), we show that the extracted $K_3$ displays four peaks. The three peaks at magnetic fields $B_1$, $B_2$, and $B_3$ are associated with Efimov resonances, and the global maximum at the lower field $B_0$ represents the Feshbach resonance. $K_0=10^{-25}{\mathrm{cm}}^6/\mathrm{s}$ is the loss coefficient we obtain at $\sim 852\mathrm{G}$. The inset shows the zoom-in view of the resonance structure from the lower temperature measurement, which offers higher resolution to the higher order Efimov resonances. The solid line represents a four-Lorentzian fit to the data, and serves as a guide to the eye.