Abstract
The problem of three interacting quantal bodies, in its various guises, seems deceptively simple, but it has also provided striking surprises, such as the Efimov effect^{1,2}, which was confirmed experimentally^{3} only more than 35 years after its initial prediction. The importance of understanding the threebody problem was magnified by the explosion of ultracold science following the formation of Bose–Einstein condensates in 1995 (ref. 4). For ultracold gases, threebody recombination (where B+B+B collide to form B_{2}+B) was quickly recognized as the main loss process and connected^{5,6,7,8} with the Efimov effect in the ‘universal’ realm of very large atom–atom scattering lengths a. The problem of four interacting bodies challenges theory far more than the threebody quantal problem. Some key insights have been achieved in recent years^{9,10,11,12,13,14,15,16}. Here, we present a major extension of our understanding of the fourbody problem in the universal largea regime. Our results support a previous conjecture^{10} that two resonantly bound fourbody states are attached to every universal threebody Efimov resonance and they improve the calculated accuracy of their universal properties. A hitherto unanalysed feature found in ultracoldgas experiments^{3} supports this universal prediction, and it provides the first evidence of fourbody recombination (where B+B+B+B form B_{3}+B, B_{2}+B+B or B_{2}+B_{2}).
Main
The experiment^{3} that observed strong evidence for the longpredicted Efimov effect^{1,2} has spawned a new level of confidence in our theoretical understanding of the threebody problem with shortrange forces. However, even though in some respects the threebody problem is beginning to seem ‘almost solved’, the next step in complexity—to four interacting particles—remains at a primitive stage, comparatively speaking. Although a few studies have been pursued^{9,10,11,12}, the nonperturbative fourboson problem is still largely uncharted territory, especially for processes that begin or end with three or four free particles. Our present study relates most to the pioneering work of Hammer, Platter and Meißner^{9,10}, and of Yamashita et al. ^{11}. It concerns a key question in strongly interacting fewbody systems: are universal principles of Efimov physics relevant for the fourboson problem? In the early 1970s, the nuclear physicist Vitaly Efimov^{1,2} predicted on very general grounds that three neutral bosons, whose mutual interaction is characterized by a large value for the twobody swave scattering length a, can form a large number of weakly bound states whenever a≫r_{0}, where r_{0} is the characteristic range of the interaction. Surprisingly, this could happen even when none of the pairs can bind (a<0). Here we provide an analysis that convincingly demonstrates the existence of a class of universal fourboson states that are intimately related to the Efimov effect. Our results connect with and extend previous analyses^{9,10,11} and provide a more complete landscape of the universal fourboson phenomena. In addition, we demonstrate how fourboson universal states can be seen (and in retrospect, apparently have already been seen) in ultracoldgas experiments.
Our theoretical model hinges on the tunable interaction strength achievable in ultracoldgas experiments. For alkali atoms, when an external B field is placed near a Fano–Feshbach resonance^{17} a small change of B can cause a to vary from to , allowing for the exploration of a vast range of interatomic interaction strengths. We mimic such variations in a by explicitly modifying the interatomic interactions^{5}. In our framework, the solution of the fourbody problem culminates with the solution of the ‘hyperradial’ Schrödinger equation:
where the hyperradius R describes the overall size of the system. Here, m is the atomic mass, E the total energy and F_{ν}(R) the hyperradial wavefunction, with ν representing the set of quantum numbers needed to label each channel. In the hyperspherical adiabatic representation,^{5,18} most of the complexity of the problem arises in solving the hyperangular equations to determine the effective potentials W_{ν ν}(R) and couplings W_{ν ν′}(R), in this case using a correlated Gaussian hyperspherical approach^{19}. The reduction of the problem to the hyperradial Schrödinger equation (1) then leads to a simple, intuitive picture: the effective potentials W_{ν ν}(R) support all bound and quasibound states of the system, and the offdiagonal nonadiabatic couplings W_{ν ν′}(R) drive inelastic transitions among different channels.
We explore the universality of the fourboson system and its relation to Efimov physics by solving the Schrödinger equation (1) for different model potentials using two complementary numerical techniques: the adiabatic hyperspherical approximation^{18,20}, and the correlated Gaussian basis set expansion^{14,21}. We base our conclusions on the analysis of energies, potential curves and wavefunctions that describe ground and excited states (see Supplementary Information for details) in the universal regime a≫r_{0}, where r_{0} is the characteristic length scale of the twobody interaction, usually associated with the van der Waals length. Figure 1a shows the ‘generalized Efimov plot’^{10}, with all important features that relate two, three and fourbody physics. Figure 1a shows our numerical results for the fourboson energies (solid black lines) along with the dimer–dimer and dimer–atom–atom breakup thresholds for a>0 (solid red lines) and the energies of the Efimov states (dashed green lines) representing the atom–trimer breakup threshold. Figure 1c,d conveys the geometrical nature of the fourbody states taking into account the extremely ‘floppy’ nature of the Efimov trimer states, in which all possible triangular shapes, and even linear configurations, are comparably probable^{22,23}.
Figure 1a implies that the fourboson spectrum, throughout the range r_{0}/a≪1, is characterized by precisely two tetramer states that are associated with each threebody Efimov state, confirming the ref. 10 prediction. In fact, our extensive numerical tests show that these fourboson energies obey a universal relationship to the corresponding Efimov state energy, which at unitarity () can be expressed as
where E_{3b}^{(n)} is the energy of the nth Efimov state, n=0,1,2,…, and E_{4b}^{(n,m)}, m=1 and 2, are the two tetramer energies associated with it (see Supplementary Information for details). Here, we find that the universal relation between three and fourbody energies is characterized by these two universal numbers, equal to: c_{1}≈4.58 and c_{2}∼1.01. For the lowest two fourboson states ref. 10 obtained c_{1}≈5 and c_{2}∼1.01. Similar values, less deep in the universal regime, can be extracted from the small B_{2} limit of equations (39) and (41) of ref. 9. Our own calculations differ from the universal values if we consider the lowest fourbody states (see Supplementary Information). In fact, we believe that the ability of the current method to calculate many more weaklybound energy levels than previous techniques has been decisive, and it permits us to verify the universal numbers up to 2% accuracy and resolve a previouslyexisting disagreement in the literature, between the results of Hammer, Platter and Meißner^{9,10} and those of Yamashita et al.^{11}.
The resolution of this controversy is related to the fundamental question of whether or not an additional ‘fourbody parameter’, encapsulating nonuniversal aspects from the details of the interactions, is required to specify the nature of the fourboson spectrum and scattering observables, akin to the usual threebody parameter^{1,2,5,6,7,24,25}. Specifically, we support the conclusions of refs 9, 10, 12 that no additional fourbody parameter enters at leading order in the description of universal fourbody spectra and scattering properties. Our result also explains the observations of Yamashita et al. ^{11} that the energies of more compact fourboson states can vary depending on the details of the interatomic interactions. Our understanding emerges from Fig. 2, showing the fourbody effective potentials calculated at unitarity, . We have verified that the effectivepotential curves scale with the size of the trimer state. Therefore, if the lowest Efimov state has a size that exceeds r_{0} only marginally, the minimum of the fourbody potential is close enough to r_{0} and its fourbody states are affected by the shape of the twobody interaction, that is, nonuniversal physics. On the other hand, if the lowest Efimov state is large compared with r_{0}, then the minimum of the fourbody potential lies at R≫r_{0} and the fourboson states probe almost no nonuniversal effects. Furthermore, the scaling behaviour of the potential curves implies that the threebody Efimov effect controls the fourboson spectrum. As a consequence, the fourboson states follow the same geometric scaling as the threeboson states, with successive energies related by the factor e^{−2π/s}_{0}≈1/515 and successive radii expanding by the factor e^{π/s}_{0}≈22.7 (where s_{0}≈1.00624). These fourboson states are not true bound states, of course, as was pointed out in ref. 10, so the preceding discussion relates to the real part of their energies.
Parenthetically, we also confirm the existence of a class of fourbody states that represent the Efimov effect for three bodies. As pointed out in ref. 8, these fouratom states occur whenever an Efimov state is created for a>0 (see Fig. 1b). In this case the atom–dimer scattering length, a_{ad}, has a pole and whenever a_{ad}≫a a series of fouratom, threebody Efimov states, namely dimer–atom–atom states. Numerically, we observe the emergence of an attractive dipole potential (∝−1/R^{2}) in the dimer–atom–atom channel, for a≪R≪a_{ad}, confirming the existence of such fourboson states. In fact, the recent observation of an Efimov resonance in ref. 26 in an ultracold atom–molecule mixture could enable the probing of such fourbody states experimentally. We also confirm the nonexistence of a ‘true Efimov effect’ for four bosons in the spirit of the ref. 27 prediction (see Supplementary Information).
The universal fourbody physics discussed above can readily be observed in ultracold quantum gases. In general, weakly bound states deeply affect the collisional properties of ultracold gases, enhancing the atomic and/or molecular losses. The relative importance of fourbody processes, however, remains largely unexplored, and we could argue that such processes should be far less likely than twobody or threebody collisions in a typical lowdensity gas. On the other hand, near the threshold for formation of any fourboson states, the fourbody scattering observables should show a resonant enhancement that dramatically affects the collisional behaviour of the gas, even at low densities. The results of the Innsbruck experiment^{3}, realized at atomic densities of n(0)≈3×10^{12} cm^{−3}, were interpreted under the premise that the atom loss stems just from threebody recombination, B+B+B→B_{2}+B+E_{rel.}, which releases enough kinetic energy to eject the collision products. Sure enough, the experimental data show a resonant peak in the threebody recombination rate K_{3}, more specifically at a threebody recombination length of , at a=−850 a.u., in agreement with theoretical expectations for the manifestation of Efimov physics through threebody recombination^{5,6,7}.
Although the experimental Innsbruck data^{3} are reasonably well understood^{24,28,29,30}, distinguishing three and fourbody losses is difficult, and fourbody processes could still be embedded in the observed decay rates. Accordingly, we have reanalysed the Innsbruck data^{3}, looking for possible signatures of new fourboson states. The key observation from our results is that for a<0, when an Efimov state is created, say at a=a_{3b}^{*}, it is accompanied by the creation of two fourboson states at slightly less negative values of a, and those states should enhance fourbody processes in this experimentally explored region of a. Moreover, our calculations indicate that, once the scattering length a_{3b}^{*} is known, we know the scattering lengths at which such fourbody states appear. This universal relation is determined by the energy spectrum (see Fig. 1a), namely
From our numerical calculations we have found that these relations are approximately fulfilled even when a_{3b}^{*} is not deep in the universal regime, suggesting an insensitivity to nonuniversal effects.
The main process where such states should appear is fourbody recombination, where the four initially free atoms collide to recombine into the dimer–dimer channel, the dimer–atom–atom channel and/or the atom–trimer channel. Figure 3a,b, respectively, depicts this process through the effective potentials at scattering lengths very close to the threshold a_{3b}^{*} for formation of an Efimov state (green dashed line) and just past the point (a<a_{3b}^{*}) of its creation. When a fourboson state resides energetically close to the collision threshold, we expect a resonant enhancement to the fourbody recombination rate, K_{4}. A straightforward Wigner thresholdlaw analysis demonstrates that K_{4} approaches a constant as the collision energy is tuned to zero^{16,31}, and thus fourbody recombination can indeed potentially compete with threebody recombination in causing atomic losses.
To assess the importance of K_{4} and quantify our predictions, we have calculated K_{4} by numerically solving equation (1) using a formula for K_{4} derived elsewhere^{31}. The main difficulty in comparing our results with data is that existing experiments are probably unable to distinguish three and fourbody losses. We therefore introduce an effective threebody recombination rate, in which both three and fourbody physics are included:
where n(t) is the peak atomic density in the trap at time t, calculated by solving the time evolution rate equations. Figure 3c shows our recombination length^{5} for t=20 ms. For K_{3} we use the thermally averaged results of ref. 24 calculated for temperatures of 10, 200 and 250 nK, and adjust it to fit the Efimov resonance at a=a_{3b}^{*}=−850 a.u. and the experimental data for a>a_{3b}^{*}. Our results show that for this range of a threebody recombination is indeed the dominant loss process. For a<a_{3b}^{*}, however, we find much better agreement by assuming that fourbody recombination is the dominant loss process—the dashed curve in Fig. 3c is the 10 nK contribution from K_{3}. For this range of a, as indicated in Fig. 3c by the vertical dashed lines at a=a_{4b1}^{*} and a=a_{4b2}^{*}, K_{4} is resonantly enhanced when the two fourboson states are created (see the circled region in Fig. 1a). This agreement strongly suggests that the 2006 Innsbruck experiment^{3} also offers the first experimental evidence for the universal fourboson states we discussed here, although the agreement with the second resonance predicted for a=a_{4b2}^{*} and 10 nK requires some imagination—and for temperatures of 200 and 250 nK this resonance feature is washed out owing to thermal effects. Nevertheless, the verification of the universal constraint between three and fourbody physics (equation (3)) strengthens the conclusion that the main resonant feature^{3} at −850 a.u. is indeed an Efimov resonance.
Note also that for a>a_{3b}^{*} fourbody recombination to Efimov states, B+B+B+B→B_{3}+B (see Fig. 3b), is likely to be the dominant fourbody decay pathway. Although threebody recombination tends to dominate the atom loss, the formation of Efimov states through fourbody recombination is nonnegligible. In Fig. 3d we show the atomic density n_{B} and the density of trimers n_{B}_{3} at 10 nK for a up to −10,000 a.u. Near the threshold for Efimov state formation little energy is released through fourbody recombination (approximately the trimer binding energy) and the Efimov state can remain trapped. In this case, our results indicate that about 10% of the atoms will form trimers. For larger a, however, the trimer formation is strongly enhanced by a resonance associated with the lowest fourboson state attached to the second Efimov trimer (see Fig. 1a). In this case, we find that about 50% of the atoms will form trimers. Here, however, the energy released through fourbody recombination ejects both the atom and the Efimov trimer. Nevertheless, in an experiment where only atoms are visible, the magnetic field could be set to a value such that a>a_{3b}^{*}, the number of atoms measured and the field ramped back to a value such that a<a_{3b}^{*}, where no trimers exist. The reappearance of atoms after this ramp would be a convincing signature of the first experimental realization of an ultracold gas of Efimov trimers.
Finally, the Innsbruck group has observed two resonant loss features in fourbody recombination^{32} that satisfy our relations in equation (3), offering stronger experimental evidence of the universal fourbody physics demonstrated here.
Methods
Effective threebody recombination.
The idea behind the definition of K_{3}^{eff}(a,t) is to obtain a quantity that encapsulates both three and fourbody contributions. Accordingly to our findings, the atomic losses can indeed be seen as an effective threebody recombination within a characteristic timescale, as explained below. That can presumably facilitate a comparison with the available experimental data.
The key ingredient in the derivation of K_{3}^{eff}(a,t) is the relation between the loss coefficient and the loss rates as given by
where N is the number of atoms involved in the recombination event. Note that K_{3} (or K_{4}) represents a fundamental fewbody entity, namely, the recombination probability per unit time for a single triad (or tetrad) in a unit volume squared (cubed)^{17}. The first factor on the righthand side of this equation represents the number of atoms lost in the Natom recombination process, whereas the 1/N! factor accounts for the indistinguishability of the collision partners. The atoms are assumed not to be in a Bose–Einstein condensate.
The rate equation that governs the time evolution of the atomic density can be written as
or alternatively,
from which we define the effective L_{3}^{eff}(a,t). From this definition, and using the relation between L_{N} and K_{N} above, we can easily arrive at our definition of K_{N}^{eff} following the steps below:
We have in fact verified that for times t<∼t_{0}=[n(0)^{2}(K_{3}+n(0)K_{4}/3)]^{−1} (≈50 ms for our case) the time dependence of n(t) can be described as a result of the effective threebody rate in equation (4), by setting n(t)=n(0). For longer times, n(t) is affected primarily by three or fourbody processes depending on whether or not K_{3}≫n(0)K_{4}.
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Acknowledgements
This work was supported in part by the National Science Foundation. We are indebted to N. Mehta and S. Rittenhouse for extensive discussions and for access to their unpublished derivations before publication. We also thank F. Ferlaino, S. Knoop, H.C. Nägerl and R. Grimm from the Innsbruck group for discussions about their experimental data.
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von Stecher, J., D’Incao, J. & Greene, C. Signatures of universal fourbody phenomena and their relation to the Efimov effect. Nature Phys 5, 417–421 (2009). https://doi.org/10.1038/nphys1253
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