Excited Thomas-Efimov levels in ultracold gases

Since the early days of quantum physics, the complex behavior of three interacting particles has been the subject of numerous experimental and theoretical studies. In a recent Letter to Nature, Kraemer et al. [Nature (London) 440, 315 (2006)] report on experimental “evidence for Efimov quantum states” in an ultracold gas of cesium atoms. Such quantum states refer to an infinite series of energy levels of three identical Bose particles, accumulating at the threshold for dissociation as the scattering length of each pair is tuned to infinity. Whereas the existence of a single Efimov state has been predicted for three helium atoms, earlier experimental studies concluded that this elusive state had not been found. In this paper we show by an intuitive argument and full numerical calculations that the helium and cesium experiments actually provide evidence of the same, ground state of this trimer spectrum, which the helium experimentalists and pioneering theoretical studies had not associated with Efimov’s effect. Unlike the helium trimer, the observed ${}^{133}\mathrm{Cs}_3$ resonance refers to a Borromean molecular state. We discuss how as yet unobserved, excited Efimov quantum states might be detected in ultracold gases of ${}^{85}\mathrm{Rb}$ and of ${}^{133}\mathrm{Cs}$ at magnetic field strengths in the vicinity of $0.08\mathrm{T}$ $\left(800\mathrm{G}\right)$.

Figure 1

(Color online) Efimov plot [1] illustrating the wave number $K=-{(m\mid E_n\mid ∕{\hslash }^2)}^{1∕2}$ $(n=1,2,3,\dots )$ indicated by solid curves vs the inverse scattering length $1∕a$. Here $m$ is the single-particle mass and $E_n$ is the energy of the $n\text{th}$ three-particle level. The wave number labeled dimer (dotted line) is associated with the two-body bound state emerging in the limit $1∕a\to 0$ (i.e., $\mid a\mid \to \infty$) at the vertical dashed line. In accordance with Ref. 1, its energy is universal, i.e., $E_b=-{\hslash }^2∕\left(m{a}^2\right)$. The limits $1∕a\to \pm \infty$ of the horizontal axis refer to zeros of the scattering length. Ideally, each Efimov level emerges in a three-body zero-energy resonance at negative $a$ and eventually vanishes into the continuum leading to an atom-dimer resonance at positive $a$. The ground state of the Thomas-Efimov spectrum $(n=1)$ is not displayed.

Figure 2

(Color online) Comparison between full coupled-channels (circles) and two-channel (curves) calculations of the ${}^{133}\mathrm{Cs}$ $6s$-dimer energies (upper panel) [17, 25, 27], as well as their associated closed-channel admixtures [19], $Z\left(B\right)$ (lower panel), vs the magnetic-field strength $B$. For each magnetic-field strength zero energy is chosen to coincide with the scattering threshold. The resonance position $B_0$ is indicated by the dotted line. Curves labeled $E_{-1}^d$ and $E_{-2}^d$ refer to the two highest excited vibrational bound states for $B>B_0$ and the levels they adiabatically correlate with at $B<B_0$, whereas the Feshbach molecular energy $E_b$ existing for $B<B_0$ is not resolved in the upper panel. Its closed-channel admixture in the lower panel is negligible.

Figure 3

(Color online) Calculated three-body recombination loss-rate constants $K_3$ (upper panel) and energies of metastable trimer levels of ${}^{133}\mathrm{Cs}$ atoms labeled $E_1$ and $E_2$ (lower panel) vs the magnetic-field strength $B$. For each magnetic-field strength zero energy coincides with the scattering threshold for three asymptotically free atoms. Only ranges of $B>0$ are accessible to experiments with atoms in the Zeeman ground state, whereas $B<0$ would be accessible to experiments with the $(F=3,m_F=-3)$ level. Measurements of $K_3$ [8, 28] performed at $10\mathrm{nK}$ and about $200\mathrm{nK}$ are indicated by circles and squares, respectively. The dashed and dot-dashed curves labeled $E_{-1}^d$ and $E_{-2}^d$ in the lower panel refer to the $6s$-dimer levels [17, 25, 27] of Fig. 2, constituting the target states included in our calculations. Dashed and dot-dashed curves in the upper panel indicate their associated contributions to the total three-body recombination loss-rate constant (solid curve). The inset of the upper panel illustrates the variation of the scattering length with magnetic field.

Figure 4

(Color online) Comparison between Efimov plots associated with ${}^{133}\mathrm{Cs}$ in the vicinity of zero field (upper panel) and ${}^4\mathrm{He}$ (lower panel). The filled circles refer to the degree of excitation $(n=1)$ of the trimer states detected in the experiments of Refs. 8, 14 at the inverse scattering lengths indicated on the horizontal axis. Dotted curves indicate energies of dimer levels causing diatomic zero-energy resonances at $1∕a=0$. The range of scattering lengths in the upper panel extends from the observed [8] three-body zero-energy resonance position to $a=a_{\mathrm{bg}}$ at field strengths $B<0$. In accordance with Refs. 12, 13, the curves labeled $E_1$ illustrate the increasing separation of the trimer ground level from the dimer energy as $1∕a$ is increased. By contrast, the $E_2$ level in the lower panel approaches the atom-dimer threshold. It eventually gets overrun by the dimer energy causing an atom-dimer resonance as the potential is scaled to enhance the pairwise attraction [10, 11]. For this reason, only the excited $E_2$ level has been referred to in Refs. 5, 10, 11, 15 as a genuine Efimov state.

Figure 5

(Color online) Scattering length (upper panel) and diatomic bound-state energies (lower panel) of ${}^{133}\mathrm{Cs}$ atoms vs magnetic-field strength in the vicinity of $800\mathrm{G}$. The solid curve in the upper panel refers to exact coupled-channels calculations [27] including basis states of $s$- and $d$-wave symmetries, while the dashed curve is a fit of Eq. (2) to the calculated data. According to this fit, the position of the broadest singularity of the scattering length is located at $B_0=800.6\mathrm{G}$ (vertical dotted line). Comparatively narrow zero-energy resonances near 735 and $825\mathrm{G}$ are due to dimer levels of predominantly $d$-wave symmetry shown in the lower panel. Their energies with respect to the zero-energy dissociation threshold are indicated by long-dashed, dashed, and dot-dashed curves. These states are more weakly coupled to the continuum of free atoms than $s$-wave dimer levels. For convenience, their mutual coupling is neglected in these calculations. Solid curves in the lower panel refer to those $s$-wave dimer levels that constitute the target states included in our three-body recombination calculations. The $6s$-dimer level [27] causing the broadest singularity of the scattering length in the upper panel is unresolved on the energy scale chosen in the figure.

Figure 6

(Color online) Calculated three-body recombination loss-rate constants $K_3$ (upper panel) and energies of metastable trimer levels of ${}^{133}\mathrm{Cs}$ atoms labeled $E_1$ and $E_2$ (lower panel) vs the magnetic-field strength $B$ on the high-field side of the $800.6\mathrm{G}$ diatomic zero-energy resonance. For each magnetic-field strength zero energy in the lower panel coincides with the scattering threshold for three asymptotically free atoms. Similarly to Fig. 3, the dashed and dot-dashed curves labeled $E_{-1}^d$ and $E_{-2}^d$ in the lower panel refer to the $s$-wave dimer levels shown in the lower panel of Fig. 5, which constitute the target states included in our calculations of $K_3$. Dashed and dot-dashed curves in the upper panel indicate their associated contributions to the total three-body recombination loss-rate constant (solid curve). Three-body zero-energy resonance peaks are labeled by the degree of excitation $(n=1,2)$ of their associated metastable Thomas-Efimov trimer levels of the lower panel.

Figure 7

(Color online) Comparison between the calculated total three-body recombination loss-rate constants $K_3$ associated with the low-field (solid curve) and high-field (dashed curve) diatomic zero-energy resonances of ${}^{133}\mathrm{Cs}$ vs the scattering length $a$. Filled circles and squares refer to experimental data of Refs. 8, 28 observed at low magnetic-field strengths (see Fig. 3) in dilute gases of 10 and $200\mathrm{nK}$, respectively. The range of scattering lengths displayed covers the ground-state $(n=1)$ and excited-state $(n=2)$ three-body zero-energy resonances of the Thomas-Efimov spectrum. The inset shows an enlargement of the presently experimentally accessed [8, 28] range of $a\left(B\right)$ as well as the associated calculated and measured $K_3$ loss-rate constants.

Figure 8

(Color online) Efimov plot associated with ${}^{85}\mathrm{Rb}$ trimers [20] in the vicinity of the $155\mathrm{G}$ diatomic zero-energy resonance. The solid and dashed curves refer to the $E_2$ and $E_1$ levels, respectively, while the dotted line indicates the energy of the Feshbach molecular dimer state. The inset shows the magnetic-field dependence of the scattering length $a\left(B\right)$ for this atomic species with a negative background-scattering length of $a_{\mathrm{bg}}=-443a_0$ [47].

Figure 9

Jacobi coordinates of the relative motion of three atoms. The set of coordinates $\mathbf{\rho }={\mathbf{\rho }}_1$ and $\mathbf{r}={\mathbf{r}}_1$ is selected in such a way that it describes the hypothetical situation of an interacting pair of atoms (2,3), whereas atom 1 is considered as a spectator. Two other sets of these coordinates describing different arrangements of the atoms can be obtained through cyclic permutations of the atomic indices.

Figure 10

(Color online) Comparison between coupled-channels (filled circles) and two-channel single-resonance (long-dashed and dot-dashed curves) approaches to the highest excited vibrational $s$-wave states of ${}^{133}\mathrm{Cs}_2$ vs the magnetic-field strength $B$ in the vicinity of $800\mathrm{G}$. The dashed line labeled $E_{\mathrm{res}}\left(B\right)$ indicates the bare Feshbach resonance-state energy, while the horizontal dashed line refers to the second highest excited bare vibrational level of the background-scattering potential $E_{-2}^{\mathrm{bg}}$. The energies of the associated highest excited bare level $E_{-1}^{\mathrm{bg}}$ and of the Feshbach molecular dimer state $E_b\left(B\right)$ are not resolved. The point of degeneracy of $E_b$ with the scattering threshold (zero energy) is indicated by the vertical dotted line labeled $B_0$. It coincides with the position of the broad singularity of the scattering length in Fig. 5. This measurable zero-energy resonance position is distinct from the zero-energy crossing point of the bare Feshbach-resonance level $B_{\mathrm{res}}$. The deviation at lower $B$ fields between the coupled-channels and two-channel $E_{-2}^d$ levels is mainly due to a strong avoided crossing of dressed, coupled-channels $s$ states in the vicinity of $600\mathrm{G}$, which is not included in the two-channel approach.