Universality in Three- and Four-Body Bound States of Ultracold Atoms

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Science  18 Dec 2009:
Vol. 326, Issue 5960, pp. 1683-1685
DOI: 10.1126/science.1182840


Under certain circumstances, three or more interacting particles may form bound states. Although the general few-body problem is not analytically solvable, the so-called Efimov trimers appear for a system of three particles with resonant two-body interactions. The binding energies of these trimers are predicted to be universally connected to each other, independent of the microscopic details of the interaction. By exploiting a Feshbach resonance to widely tune the interactions between trapped ultracold lithium atoms, we find evidence for two universally connected Efimov trimers and their associated four-body bound states. A total of 11 precisely determined three- and four-body features are found in the inelastic-loss spectrum. Their relative locations on either side of the resonance agree well with universal theory, whereas a systematic deviation from universality is found when comparing features across the resonance.

One of the most notable few-body phenomena is the universally connected series of three-body bounds states first predicted by Efimov (1) in 1970. Efimov showed that three particles can bind in the presence of resonant two-body interactions, even in circumstances where any two of the particles are unable to bind. When the two-body scattering length a is much larger than the range of the interaction potential r0, the three-body physics becomes independent of the details of the short-range interaction. Surprisingly, if one three-body bound state exists, then another can be found by increasing a by a universal scaling factor, and so on, resulting in an infinite number of trimer states (2). Universality is expected to persist with the addition of a fourth particle (37), with two four-body states associated with each trimer (5, 7). Intimately tied to the three-body state, these tetramers do not require any additional parameters to describe their properties.

Ultracold atoms are ideal systems for exploring these weakly bound few-body states because of their inherent sensitivity to low-energy phenomena, as well as the ability afforded by Feshbach resonances to continuously tune the interatomic interactions. Pioneering experiments with trapped, ultracold atoms have obtained signatures of individual Efimov states (812)—as well as two successive Efimov states (13, 14)—via their effect on inelastic collisions that lead to trap loss. Evidence of tetramer states associated with the trimers has also been found (13, 15). Although the locations of successive features are consistent with the predicted universal scaling, systematic deviations as large as 60% were observed and attributed to nonuniversal short-range physics (13). In the work presented here, we use a Feshbach resonance in 7Li for which a/r0 can be tuned over a range spanning three decades (16). This enables the observation of multiple features that are compared to universal theory.

We confine 7Li in the |F = 1, mF = 1〉 (where F is the total spin quantum number and mF is its projection) hyperfine state in an elongated, cylindrically symmetric, hybrid magnetic—plus—optical dipole trap, as described previously (16). A set of Helmholtz coils provides an axially oriented magnetic bias field that we used to tune the two-body scattering length a via a Feshbach resonance located near 737 G (17). For a > 0, efficient evaporative cooling is achieved by setting the bias field to 717 G, where a ~ 200a0 (a0 is the Bohr radius), and reducing the optical-trap intensity. Depending on the final trap depth, we create either an ultracold thermal gas just above the condensation temperature TC or a Bose-Einstein condensate (BEC) with >90% condensate fraction. For investigations with a < 0, we first set the field to 762 G where a ~ −200a0 and proceed with optical-trap evaporation, which is stopped at a temperature T slightly above TC. In both cases the field is then adiabatically ramped to a final value and held for a variable hold time. The fraction of atoms remaining at each time is measured via in situ polarization phase-contrast imaging (18) for clouds where the density is high, or absorption imaging in the case of lower densities.

Analyzing the time evolution of the number of atoms in the trap determines the three-body loss coefficient L3 (8, 13, 19), as well as the four-body loss coefficient L4 (15). Recombination into a dimer is a three-body process because a third atom is needed to conserve both momentum and energy. For a > 0, the dimer can be weakly bound with binding energy ε = ħ2/(ma2) (where m is the atomic mass and ħ is Planck’s constant h divided by 2π), whereas for a < 0 there are only deeply bound molecular dimers. The recombination energy released in the collision is sufficient to eject all three atoms from the trap for a < 0, as well as for a > 0 when ε>˜U (where U is the trap depth). In the case of the BEC data, this latter condition holds for a<˜5000a0. Nonetheless, we assume that all three atoms are lost for any recombination event because, even for a larger than 5000a0, we observe rapid three-body loss. We ascribe this observation to a high probability for dimers to undergo vibrational relaxation collisions that result in kinetic energies much greater than U. Four-body processes proceed in a similar fashion (6, 15).

The equation describing the dynamics of three- and four-body loss is1NdNdt=g(3)3!L3n2g(4)4!L4n3(1)where N is the total number of atoms in the trap at time t, and the brackets denote averages over the density distribution n (17). For a thermal gas, the spatial correlation coefficients g(3) and g(4) are, respectively, 3! and 4!, whereas for a BEC, both are set to 1 (20, 21). We have verified that heating from recombination is small for our short observation times and therefore omit this effect in our analysis (15, 19). By fitting the time evolution of the number of atoms to the solution of Eq. 1, we extract L3 and L4 as a function of a. Figure S1 shows the loss of atoms as a function of time in regimes where either L3 or L4 dominates (17). Four-body loss is readily distinguished from three-body loss by the shape of the loss curve.

Figure 1 shows the extracted values of L3 across the Feshbach resonance, exhibiting the expected a4 scaling (22, 23), but with several dips and peaks punctuating this trend. Two prominent peaks, labeled a1 and a2 in Fig. 1A, dominate the landscape for a < 0. We attribute these peaks to the crossings of the energies of the first two trimer states with the free-atom threshold, thus providing additional pathways into deeply bound molecular states (23). For a > 0, the dominant features are dips, indicated in Fig. 1A as a1+ and a2+, corresponding to recombination minima. These minima are associated with the merging of the same two trimer states into the atom-dimer continuum and have been attributed to destructive interference between two different decay pathways into weakly bound dimers (22, 23). We fit the data to L3(a) = 3C(a)ħa4/m, where C(a) is a logarithmically periodic function characterizing effects from the Efimov states (17). The analytic expression for C(a) contains the location of one universal trimer resonance a < 0 or recombination minimum a+ > 0 and an inelasticity parameter η related to the lifetime of the Efimov state (2). The observed features are fit individually to extract these parameters (Table 1). The universal theory describing Efimov physics (2) predicts a logarithmic spacing in the two-body scattering length between trimer states of eπ/s022.7, where s0 = 1.00624 is a universal parameter (1). Table 2 shows that the ratios a2+/a1+ and a2/a1 agree well with the universal theory.

Fig. 1

(A) L3 as a function of a. Data shown with purple diamonds correspond to a thermal gas with N ~ 106, T ~ 1 to 3 μK (31), and U ~ 6 μK and were taken with radial and axial trapping frequencies ωr = (2π) 820 Hz and ωz = (2π) 7.3 Hz, respectively. The remaining data correspond to a BEC with N ~ 4 × 105, T < 0.5 TC, U ~ 0.5 μK, and ωr = (2π) 236 Hz. We adjust ωz (17) to enhance or reduce three-body loss, where ωz = (2π) 1.6 Hz (red triangles), ωz = (2π) 4.6 Hz (blue circles), and ωz = (2π) 16 Hz (green squares). The black dashed lines show an a4 scaling, and the thick black solid lines are fits to an analytic theory (2, 17). The thin green lines show the square of the energies (in arbitrary units) of the first and second Efimov states, as predicted from the universal theory (2), where we have fixed the location of the first Efimov state to overlap with Embedded Image, and the atom-dimer continuum is coincident with the dashed line for a > 0. Several representative error bars indicating the SE from the fit are shown (17). (B to D) Detail around the loss features associated with the atom-dimer and two possible dimer-dimer resonances. The black dotted lines are fits to eq. S4, whereas the black solid lines include additional superimposed Gaussian fits to account for the features not described by eq. S4.

Table 1

Locations (in a0) of three- and four-body loss features and inelasticity parameters (dimensionless) (17). The features Embedded Image and Embedded Image are tentatively assigned. The first number in parentheses characterizes the range over which χ2 of the fit to theory increases by one while simultaneously adjusting the other parameters in the fit. The second number characterizes the systematic uncertainties in the determination of a (17).

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Table 2

Relative locations of loss features, those predicted by theory, and the percent difference Δ = (data/theory – 1). The uncertainties are those propagated from Table 1.

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A local maximum in L3, indicated as a2 and shown in detail in Fig. 1B, can be discerned between the two recombination minima a1+ and a2+. We associate this feature with an atom-dimer resonance, given its location with respect to the nearby minima. A simple model (13) has been proposed to explain the enhanced losses present at the atom-dimer resonance. This model describes an avalanche process whereby a single dimer traveling through a collisionally thick gas shares its kinetic energy with multiple atoms, thereby increasing from three the effective number of atoms lost for each dimer formed (24).

For a < 0, L3 achieves its maximum value of ~10−19 cm6/s at a2. This value is reasonably consistent with the expected unitarity limit (19, 25). At even larger values of |a|, L3 saturates to a value below the unitarity limit, a behavior previously seen in experiments (8) and numerical calculations (25, 26).

The four-body loss coefficient (L4) for a < 0 was also extracted from the data, and the results are presented in Fig. 2. Three resonant peaks in L4 are observed, which we associate with the crossings of tetramer states with the free-atom continuum (37, 13, 15, 27). Two universal tetramers are predicted to accompany each Efimov trimer (5, 7). The black solid line in Fig. 2 is calculated using only the observed three-body locations and widths, in addition to an overall scaling, without any other free parameters (17). The agreement between this curve and the data lead us to assign the peaks to the second tetramer of the first Efimov trimer, a1,2T, and both tetramers of the second Efimov trimer, a2,1T and a2,2T (15). Although we do not have the resolution to detect an enhancement in L4 at the expected location of the first tetramer a1,1T, an enhancement of L3 is observed at the expected location (Fig. 1A), which we tentatively identify with a1,1T (7, 13). The existence of two tetramer states tied to a single trimer state has also been verified in 133Cs (15) and 39K (13).

Fig. 2

L4 extracted from a thermal gas. The black solid curve is motivated by theory (17, 27), and the blue dashed curve is the solid curve divided by a7 (6). The uncertainty in L4 from the fit is a factor of 2, whereas the systematic uncertainty is a factor of 3 due to uncertainties in ωr, ωz, N, and T. For |a| > 2 × 104a0, differentiation between three- and four-body losses becomes unreliable because of the very fast decay rates. Data with L4 < 10−36 cm9/s are consistent with no four-body loss.

Two additional peaks in L3 are observed on the a > 0 side of the resonance (Fig. 1, C and D). Features at these relative positions have not been previously observed or predicted, although they occur very close to where the two tetramer states associated with the second trimer are expected to merge with the dimer-dimer continuum (28). We have no explanation of how a dimer-dimer resonance would affect the inelastic-loss rate, as we expect the dimer fraction to be small and, consequently, the probability of dimer-dimer collisions to be negligible. One possibility is that they arise because of an interference effect, similar to that occurring in the three-body process at a1+ and a2+. Presently, we tentatively associate these features with dimer-dimer resonances located at a2,1 and a2,2.

In Table 2, we present the relative spacings of observed loss features along with those predicted by the universal theory. Universal scaling is expected when |a| >> r0, where r0 is the van der Waals radius (33a0 for Li) (29). Another requirement for universality is that |a| >> |Re|, where Re is the effective range (14). Figure S4 shows that Re is relatively small over the relevant field range and is ~ −10a0 on resonance (17). For comparison, in the |1, 0〉 state of 7Li, Re ~ −30a0 at the resonance near 894 G (14). Both conditions for universality are well-satisfied for the second Efimov state, but the requirement that |a| >> r0 is only marginally satisfied for the first. Nonetheless, we find good agreement with the universal scaling relations between features on each side of the Feshbach resonance separately.

The features across a Feshbach resonance are also thought to be universally connected (2, 26). However, when we compare features across the Feshbach resonance, we find a systematic discrepancy with theory of a factor of 2 (Table 2). This discrepancy can be expressed as a difference in the three-body short-range phase between the two sides of the Feshbach resonance ΔΦ = s0ln(|a|/a+) (22, 26). The locations of the features reported here result in phase differences of 0.92(10)(0) and 0.86(4)(17) (the uncertainties are defined in Table 1) for the first and second trimer, respectively, whereas the universal prediction is 1.61(3) (2). Finite temperature causes the trimer resonances to broaden and shift toward smaller |a| (8, 25, 26). This would decrease the values of ΔΦ, because we extract L3 from a thermal cloud at a and a much colder BEC at a+. Measurements of 39K also show a discrepancy with theory across the resonance, but with ΔΦ = 1.91(7) (13). On the other hand, measurements of the first trimer resonance and second trimer recombination minimum in the |1, 0〉 state of 7Li result in ΔΦ = 1.7(2), in good agreement with universal theory, assuming the universal scaling of 22.7 between trimer states (14). These variations in ΔΦ may indicate the need for additional physics to be included in the universal model (26, 30).

Supporting Online Material

Materials and Methods

Figs. S1 to S4


References and Notes

  1. Materials and methods are available as supporting material on Science Online.
  2. In ramping from −200a0 to a < −3000a0, we observe an increase in the axial size of the thermal cloud that is consistent with a temperature increase of the cloud to ~3 μK. During the trap-loss measurements, we observe negligible change in the Gaussian width of the thermal cloud (17).
  3. We thank E. Olson for his contributions to this project and acknowledge useful discussions with E. Braaten, J. P. D’Incao, C. H. Greene, N. P. Mehta, and H. T. C. Stoof. Support for this work was provided by the NSF, Office of Naval Research, the Keck Foundation, and the Welch Foundation (grant C-1133).
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