## Abstract

The quantum mechanical three-body problem is one of the fundamental challenges of few-body physics. When the two-body interactions become resonant, an infinite series of universal three-body bound states is predicted to occur, whose properties are determined by the strength of the two-body interactions. We used radio-frequency fields to associate Efimov trimers consisting of three distinguishable fermions. The measurements of their binding energy are consistent with theoretical predictions that include nonuniversal corrections.

Under certain conditions, the long-range behavior of a physical system can be described without detailed knowledge of its short-range properties; few-body systems with resonant interactions are a prime example of this concept of universality (*1*). Ultracold gases, where resonant scattering may be achieved by tuning the interactions with the use of Feshbach resonances (*2*), have been used extensively to test the predictions of universal theory.

If the parameter describing the interactions, the s-wave scattering length *a*, is much larger than the characteristic length scale *r*_{0} of the interaction potential, the few-body physics in such ultracold gases is predicted to become universal. For two particles with a large positive scattering length, there is a weakly bound universal dimer state whose binding energy scales as 1/*a*^{2}. For three particles with large interparticle interactions, there is a series of weakly bound trimer states, which are called Efimov trimers (*3*). For negative scattering lengths, these trimer states become bound at critical values of the interaction strength, which are spaced by a universal scaling factor. For diverging interaction strength, this results in an infinite series of Efimov states. Their absolute position depends on short-range three-body physics, which can be described by a single three-body parameter. For positive scattering lengths, the trimer states disappear when they cross the atom-dimer threshold; that is, their binding energy becomes degenerate with the binding energy of a dimer state (fig. S1). These crossings of trimer states with the continuum threshold can be observed experimentally as resonant enhancements of the rate constants for inelastic three-atom and atom-dimer collisions, respectively. This made it possible to obtain the first convincing evidence for the existence of Efimov trimers by tuning the interparticle interaction in an ultracold atomic gas using a Feshbach resonance, which enabled the observation of signatures of Efimov trimers in the rate of inelastic three-body collisions (*4*). Since then, this technique has been used with great success, but it is limited to observations of the crossings of Efimov states with the continuum (*5*–*17*).

The predictions of universal theory can be tested by observing multiple crossings of trimer states with the continuum in a single system. Some experiments are consistent with these predictions (*11*, *13*), whereas others see larger deviations (*9*) or even a systematic shift across a Feshbach resonance (*12*).

We studied Efimov physics in a conceptually different manner by directly measuring the binding energy of an Efimov state as a function of interaction strength. Our experiments use an ultracold Fermi gas consisting of fermionic ^{6}Li atoms in the three energetically lowest Zeeman substates. We label these states |1〉, |2〉, and |3〉 (*6*). Because of Pauli blocking, only atoms in different states interact with each other at ultracold temperatures. These interactions are described by three different interparticle scattering lengths *a*_{12}, *a*_{23}, and *a*_{13} for the respective combinations of atoms in different states (*18*) (Fig. 1A). For each of these combinations, there is a broad Feshbach resonance leading to three different weakly bound dimer states, which we label |12〉, |23〉, and |13〉 (Fig. 1B). If the corresponding scattering length is much larger than the characteristic radius *r*_{0} ≈ 60 *a*_{0} of the ^{6}Li interaction potential (where *a*_{0} is Bohr’s radius), the dimer binding energy *E _{ij}* is given by the universal relation

*E*=

_{ij}*ħ*

^{2}/

*ma*

_{ij}^{2}, where

*m*is the mass of a

^{6}Li atom,

*ħ*is Planck’s constant divided by 2π, and the indices

*i*,

*j*= 1, 2, 3 indicate the state of the atoms. Because the Feshbach resonances for the three combinations overlap, all scattering lengths can be tuned to large values simultaneously, leading to the appearance of Efimov states. However, as the different scattering lengths do not diverge at the same magnetic field, the series of trimer states is finite. There are two Efimov trimer states in this system (

*19*) whose crossings with the continuum have already been located by observing enhanced inelastic collision rates (

*6*,

*7*,

*10*,

*14*,

*15*) (Fig. 1B). From these measurements, the binding energies of these trimer states according to universal theory have been calculated (

*15*,

*19*), where (

*15*) also included finite range corrections to the two-body interactions.

To directly measure the binding energy of one of these Efimov states, we used radio-frequency (rf) spectroscopy, a technique that has been extensively used to study weakly bound dimer states (*18*, *20*). This technique is based on the use of rf fields to drive transitions between different internal states of the atoms, where in both the initial and final state the atoms can be either free or bound in a molecule. The difference in binding energy between the initial and final state can be measured as a shift of the transition frequency from the bare atomic transition.

The most straightforward way to perform rf spectroscopy of a bound state is to dissociate the molecule into free atoms. In this case, the rf transition is simply shifted by the binding energy of the molecule. However, because Efimov trimers are highly unstable—with lifetimes expected to range from a few nanoseconds to tens of microseconds (*19*)—it is impossible to prepare macroscopic samples of trimers in current experiments, and dissociation spectroscopy is not technically feasible. This can be overcome by using rf fields to associate trimers from free atoms or from atoms and dimers, but obtaining observable association rates is challenging for several reasons.

Let us first consider the initial states available for the association. Starting from one atom in state |1〉 and two atoms in state |2〉, one can use an rf transition to drive the atoms in state |2〉 to state |3〉. However, this is a coherent process that affects both atoms in state |2〉 in the same way, so they remain identical particles. Thus, this process cannot lead to the formation of trimers. Instead we can bind two of the atoms into a weakly bound |12〉 dimer, which breaks the symmetry between the two atoms in state |2〉, because for one of them the rf transition is now shifted by the binding energy of the dimer (*21*).

Next, the wave function overlap of the initial and final state must be considered. Close to the crossing of the trimer state with the atom-dimer threshold, the size of the dimer and the trimer are on the order of the scattering lengths. Hence, the spatial wave function of a dimer and an atom has a finite overlap with the spatial wave function of the trimer when the atom approaches the dimer to a distance on the order of the scattering length. Therefore, the association rate depends critically on the phase-space density of the initial atom-dimer mixture. If we were to use a degenerate gas, it would spatially separate into a molecular Bose-Einstein condensate of dimers and an outer shell of unbound fermionic atoms (*22*), which would reduce the spatial overlap between atoms and dimers in the trap. Therefore, we must use a thermal mixture (*T* ≈ 1 μK; *T*/*T*_{F} ≈ 0.7, where *T*_{F} is the Fermi temperature of the unpaired atoms) of atoms in state |2〉 and |12〉 dimers, which limits the efficiency of the association. A similar limitation has already been observed in the rf association of weakly bound dimers (*23*). Finally, the short lifetime of the Efimov trimer leads to a suppression of trimer formation analogous to the quantum Zeno effect. This can in principle be avoided by driving the rf transition with a Rabi frequency that is large relative to the decay rate of the trimer. However, this is not feasible with our current experimental setup, which further reduces the association rate.

From these considerations, it follows that the trimer state with the most favorable conditions for rf association in our system is the second trimer state, as it is larger in size and has a longer lifetime (*19*). This state crosses into the |1〉-|23〉 atom-dimer continuum at *B* = 685 G (Fig. 1B). To measure its binding energy, we prepared mixtures of atoms in state |2〉 and |12〉 dimers at magnetic fields between 670 and 740 G and applied rf fields at frequencies around the |2〉-|3〉 transition.

At the frequency ν_{0} that corresponds to the bare atomic transition, the free atoms are driven from state |2〉 to state |3〉 while the dimer remains unaffected (Fig. 2B). If the RF is blue-detuned by the binding energy *E*_{12} of the dimers, the dimers are dissociated (Fig. 2A). The trimer is associated at the frequency ν = ν_{0} − (*E*_{123}/*h*) + (*E*_{12}/*h*), where *E*_{123} is the binding energy of the Efimov trimer with respect to the |1〉-|2〉-|3〉 continuum. Therefore, the association is red-shifted from the bare transition by the difference between the dimer and trimer binding energies (Fig. 2C). In each of these cases, we observe a loss of atoms from the trap, either through decay in inelastic |3〉-|12〉 or |1〉-|2〉-|3〉 collisions or through the decay of the associated trimers. If the rf is not resonant for either of these processes, the atom number is not affected. Sample spectra for different magnetic fields are shown in Fig. 3.

Because of the limited wave function overlap of the initial and final state and the quantum Zeno suppression of the association, we apply strong rf pulses with a duration of 35 to 50 ms and a Rabi frequency of Ω ≈ 2π × 7 kHz to associate enough trimers to obtain an observable decrease in the atom number. Collisions lead to decoherence and thus a strong broadening of the bare transition. However, as the association is offset from the free-free transition by (*E*_{123} − *E*_{12})/*h*, the association features can still be observed (Fig. 3). For magnetic fields below the crossing of the trimer and the |1〉-|23〉 atom-dimer threshold at 685 G, there is no trimer state and consequently we observe no association peak. At *B* = 695 G, the association signal is clearly visible. For higher magnetic fields, *E*_{123} − *E*_{12} becomes smaller, and the association peak moves closer to the free-free transition. Because we do not have a better model for the line shapes of the broadened transitions, we fit the spectra with two overlapping Gaussians to determine the position of the association dips. From these we calculate the binding energy of the trimer using the values of the binding energy of the |12〉 dimer known from previous experiments (*18*). The resulting binding energies of the trimer are plotted in Fig. 4. To confirm that the results were independent of the initial system and the transition used, we performed the same measurement starting from a |2〉-|23〉 mixture at magnetic fields between 705 and 725 G and driving the |1〉-|2〉 transition (fig. S2). The two sets of measurements agree within experimental uncertainties (Fig. 4).

The measured binding energy of the trimer increases with decreasing scattering lengths while the difference to the binding energy of the |23〉 dimer decreases, as expected from the universal scenario. At *B* = 690 G, the dimer and trimer binding energies are the same within our experimental resolution, and for *B* ≤ 685 G, the trimer association feature vanishes. This coincides with the previously observed enhancement of inelastic atom-dimer collisions at 685 ± 2 G (*14*), which confirms the interpretation of such loss resonances as signatures of a crossing of a trimer state with the continuum.

We also compared our data to the predictions from a theoretical model by Nakajima *et al*. that uses the observed crossings of the trimer states as inputs to calculate the trimer binding energies (*15*). This model includes finite range corrections to the two-body interactions known from previous experiments (*18*). To exclude a dependence of their results on the specific model, Nakajima *et al*. used two different parameterizations of the non-universal corrections (A and B) that yielded essentially identical results. Within the systematic uncertainties due to temperature effects and the broadened line shapes, our data are in good agreement with their predictions. Additionally, their model includes an energy dependence of the three-body parameter. However, in the magnetic field region of interest, the difference to a model with a constant three-body parameter derived from the atom-dimer resonance at 685 G is too small to be resolved with our current resolution (*15*, *24*).

By increasing the resolution of the measurements and reducing the systematic uncertainties, it should be possible to use the techniques developed in this work to do precision tests of few-body physics and to directly determine the lifetime of Efimov trimers from the width of the association peaks. Driving the rf transition with a Rabi frequency larger than the decay rate of the trimers should allow coherent population of the trimer state, as was done for universal dimers (*23*). This would be a major step toward the preparation of macroscopic samples of Efimov trimers.

## Supporting Online Material

www.sciencemag.org/cgi/content/full/330/6006/940/DC1

Materials and Methods

Figs. S1 and S2

## References and Notes

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- ↵ The lifetime of this atom-dimer mixture is on the order of several seconds; as for two-component Fermi gases, inelastic processes are suppressed by Pauli blocking. In bosonic systems, such mixtures decay much more quickly and this scheme is probably inapplicable.
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- We thank E. Braaten, J. P. D’Incao, and H. W. Hammer for inspiring discussions; P. Naidon for providing his data on the trimer binding energies; and J. Ullrich and his group for their support. Supported by the IMPRS-QD (G.Z. and A.N.W.), the Helmholtz Alliance HA216/EMMI, and the Heidelberg Center for Quantum Dynamics.