Abstract
The field of fewbody physics has originally been motivated by understanding nuclear matter, but in the past few years ultracold gases with tunable interactions have emerged as model systems to experimentally explore fewbody quantum systems^{1,2,3}. Even though the energy scales involved are vastly different for ultracold and nuclear matter (picoelectronvolt as compared with megaelectronvolt), fewbody phenomena acquire universal properties for nearresonant twobody interactions^{2}. Socalled Efimov states represent a paradigm for universal quantum states in the threebody sector^{4}. After decades of theoretical work, a first experimental signature of such a weakly bound trimer state was recently found under conditions where a weakly bound dimer state is absent^{5,6,7}. Here, we report on a trimer state in the opposite regime, where such a dimer state exists. The trimer state manifests itself in a resonant enhancement of inelastic collisions in a mixture of atoms and dimers. Our observation is closely related to an atom–dimer resonance as predicted by Efimov^{8,9,10}, but occurs in the theoretically challenging regime where the trimer spectrum reveals effects beyond the universal limit.
Main
Trimer states arise as a natural consequence of twobody binding forces, but the general understanding of a quantum system of three interacting particles is a remarkably difficult task. For resonant twobody interactions, however, the energy spectrum follows simple, yet surprising rules, as manifested in Efimov’s scenario describing a series of trimer states^{4}. Systems in nuclear physics^{5} and molecular physics^{11,12} were considered as candidates for Efimov states, but only recently ultracold atomic gases have opened up the possibility to realize and explore the required interaction conditions in a controlled way^{1,3}. In view of these new developments, a particularly important question is how idealized fewbody scenarios are connected to nearuniversal systems existing in the real world.
A universal threebody system of identical bosons can be fully characterized by two parameters, the twobody scattering length a and an extra threebody parameter; the latter results from shortrange physics^{2}. In principle, knowledge on one Efimov trimer state, for example its binding energy for a given value of a, is sufficient to determine the threebody parameter and thus to predict the complete spectrum. A test of universality in a real threebody system is possible, when at least two different pieces of information on the trimer spectrum become experimentally available. For the caesium system, information was obtained by measuring threebody recombination ^{1}. The observation of a triatomic resonance marked the particular value of the negative scattering length where a trimer state of Borromean character^{5} reaches the threshold for dissociation into three free atoms. Observations at positive scattering lengths revealed a decay minimum, but an interpretation in terms of universal arguments is questionable because of ambiguities concerning the origin of this feature^{13}.
Here, we follow a new experimental approach and show that an atom–dimer mixture provides experimental access to the situation where a nonBorromean trimer state couples to the threshold for dissociation into a free atom and a dimer. The phenomenon that we observe is a resonance in atom–dimer scattering, which manifests itself in resonantly enhanced inelastic decay. The resonance location provides an unambiguous piece of information that complements the previous results on caesium and facilitates a comparison with universal predictions concerning the spectrum of trimer states.
For caesium atoms in the lowest internal state (hyperfine and projection quantum numbers F=3 and m_{F}=3), the swave scattering length a shows a pronounced dependence on the magnetic field in the lowfield region below 50 G (see Fig. 1, inset). Over a wide range, a is very large and exceeds the range of the attractive van der Waals potential, which can be characterized^{3} by a length r_{vdW}≃100 a_{0}, where a_{0} is Bohr’s radius, and a corresponding energy E_{vdW}≃h×2.7 MHz. Universality in general requires scattering lengths much larger than r_{vdW} and energies much smaller than E_{vdW}. For caesium, a nearuniversal halo dimer state ^{5,6,7} exists for large positive a with a binding energy of E_{b}=ℏ/(m a^{2})≪E_{vdW}, where m is the atomic mass.
A schematic diagram of the relevant threebody energy spectrum is shown in Fig. 1, illustrating the energies of trimer states (red dashed curves) and atom–dimer thresholds (blue solid curves). The energy dependencies of the thresholds are well known, because of the precise knowledge of the caesium twobody spectrum^{14}. The dimer state that corresponds to the atom–dimer threshold at positive magnetic fields has nearuniversal halo character in a wide magneticfield range above 20 G (ref. 7). The trimer states are located in the regime where a exceeds r_{vdW}, with binding energies well below E_{vdW}. We therefore refer to them as Efimov states ^{4}, although sometimes more strict definitions are used^{15}. An Efimov trimer intersects the threeatom threshold, at which three free atoms couple resonantly to a trimer. Similarly, an Efimov trimer couples to a halo dimer and a free atom at the atom–dimer threshold.
The energy spectrum of trimer states is not precisely known, but their appearance at the thresholds can give clear signatures of their locations. The observation of a giant threebody recombination loss resonance in an ultracold atomic caesium sample at 7.5 G, corresponding to a=−850 a_{0}, has pinpointed the location at which one of the Efimov states (labelled n′=0) hits the threeatom threshold^{1} (open arrow in Fig. 1). The next Efimov resonance in threebody recombination loss (as caused by the state with n′=1) is predicted at negative magnetic fields, in principle accessible with atoms in the F=3, m_{F}=−3 state. Unfortunately, in practice, its observation will be obscured by fast twobody losses^{3}. Several studies have suggested the intersection of a trimer state with the atom–dimer threshold for positive magnetic fields below 50 G (T. Köhler, private communication, 2006; B. D. Esry, private communication, 2007; ref. 16; see filled arrow in Fig. 1). Note that in our case, the regions with a<0 and a>0 are connected through a zero crossing and not through a pole. Therefore, the states with n=1 and n′=1 are not adiabatically connected as they would in a complete realization of Efimov’s scenario.
The appearance of an Efimov trimer at the atom–dimer threshold is predicted to manifest itself in a resonant enhancement of atom–dimer relaxation^{9,10}. Relaxation is energetically possible because of the presence of deeply bound dimer states and leads to loss of both the atom and the dimer from the trap, as the corresponding release of energy generally exceeds the trap depth. The resonant coupling of an atom and a dimer to a trimer opens up strong loss channels as the trimer state decays rapidly into a deeply bound dimer state plus a free atom. The particle loss is described by the rate equation , where n_{D}(n_{A}) is the molecular (atomic) density and β denotes the loss rate coefficient for atom–dimer relaxation. In the nonuniversal regime, relaxation loss in ultracold atom–dimer samples has been studied in various systems^{17,18,19,20} and was found to be essentially independent of the magnetic field. In the universal regime, suppression of loss has been observed in systems involving fermions^{21}.
The experimental realization of an ultracold sample of simultaneously trapped atoms and dimers is a challenging task and requires special trap conditions (see the Methods section). We prepare an ultracold atomic sample in a crossedbeam optical dipole trap, after which a part of the atomic ensemble is converted into dimers by means of Feshbach association^{6,22} using a 200mGwide Feshbach resonance at 48 G (refs 7, 14). For our lowest temperatures of 30 nK, we obtain a mixture of about 3×10^{4} atoms and 4×10^{3} dimers. After preparation of the mixture, we ramp to a certain magnetic field and wait for a variable storage time. Then we switch off the trap and let the sample expand before ramping back the magnetic field over the 48 G resonance to dissociate the molecules, after which standard absorption imaging is carried out. During the expansion, a magnetic field gradient is applied to spatially separate the atomic and molecular cloud (Stern–Gerlach separation)^{22}. In this way, we simultaneously monitor the number of remaining atoms and dimers (see Fig. 2a). A typical loss measurement is shown in Fig. 2b. We observe loss of dimers on a timescale of a few tens of milliseconds. To obtain β, we have set up a dimer loss model based on the abovementioned rate equation (see the Methods section). Because the number of atoms greatly exceeds the number of dimers, a simple analytic expression can be derived, which is fitted to the data. The loss of dimers due to dimer–dimer relaxation is small and is taken into account; the corresponding loss rate for this process was measured independently using a pure dimer sample^{7}.
The relaxation rate coefficient β is shown in Fig. 3 as a function of the twobody scattering length a; the inset shows the same data as a function of the magnetic field. For a<0 (B<17 G), we observe an essentially constant β of about 1.5×10^{−11} cm^{3} s^{−1}. In this region, the atom–dimer system is nonuniversal and its properties are not directly related to the scattering length. With increasing a, β exhibits a strikingly different behaviour. We first observe a sharp rise in β, which reaches its maximum value at about a=400 a_{0} (B=25 G), and a subsequent smooth decrease towards values similar to those in the nonuniversal region. We interpret the observed resonant enhancement as being caused by the appearance of a threebody bound state at the atom–dimer threshold. Owing to the presence of the nearthreshold trimer state, the atom–dimer scattering length is expected to diverge^{2}. In analogy with a usual twobody Feshbach resonance, such a threebody resonance could offer the unique possibility to tune the atom–dimer interaction from attractive to repulsive while the atomic twobody scattering length a stays always positive.
An intriguing question is whether the observed resonance is related to a trimer state that crosses the atom–dimer threshold or emerges from it. The behaviour of the relaxation rate with temperature can provide further insight on this subject. For crossing states, the trimers also exist in quasibound states in the continuum above the atom–dimer threshold, and the location of the loss maximum will show a related shift with temperature. For a trimer state merging with the threshold, such a shift will not occur. Efimov’s scenario predicts that in the Borromean region an Efimov trimer crosses the threeatom threshold, and measurements on the triatomic resonance position have indeed revealed a shift with temperature^{23}. In contrast, for the nonBorromean region an Efimov trimer is expected to merge with the atom–dimer threshold, as illustrated in Fig. 1. Figure 3 shows two data sets at different temperatures, namely at 40(10) nK (blue open triangles) and 170(20) nK (red closed squares). We observe β to be independent of the temperature of the mixture. We do not observe any shift of the resonance position, supporting the expectation from Efimov’s scenario that the trimer state exists only below the atom–dimer threshold. In addition, also the magnitude of the loss rate is not affected by the temperature change, indicating that the measurements are in the threshold regime and not unitarity limited.
At large scattering length, trimer states are expected to have Efimovian character. In the nonBorromean region, Efimov physics manifests itself as a series of asymmetric resonances in the atom–dimer relaxation rate, in the universal limit separated by the factor e^{π/s}_{0}≈22.7, where s_{0}=1.00624. In the zerotemperature universal limit, an analytic expression of β has been found within an effective field theory^{10}. The loss rate coefficient has the form β=C_{AD}(a)ℏa/m with C_{AD}(a)=D[sinh(2η_{*})/(sin^{2}[s_{0}ln(a/a_{*})]+sinh^{2}η_{*})]. The parameters a_{*} and η_{*}, corresponding to the resonance position and the decay parameter respectively, are both free in the theory because the shortrange physics of realistic threebody system is usually largely unknown. We compare our findings with the universal predictions by fitting the analytic expression of β to our 170 nK data in the region where a>r_{vdW}=100a_{0}, with a_{*}, η_{*} and D as free parameters. As shown in Fig. 3, we observe a good qualitative agreement, in particular regarding the characteristic shape of the observed loss feature. From the fit, we obtain a_{*}=367(13) a_{0}, η_{*}=0.30(4) and D=2.0(2). In ref. 10, D is a fixed value, which is predicted to be 20.3 in the zerotemperature universal limit.
In the ideal Efimov scenario, the locations of the resonance features in atom–dimer relaxation at a_{*}^{(n)}>0 and those in threebody recombination at a_{−}^{(n′)}<0 are connected through the relation a_{*}^{(n)}/a_{−}^{(n′)}≈1.06×22.7^{(n−n′−1)} (refs 2, 24). With the observation of a_{−}^{(0)}=−850(20) a_{0} (ref. 1) and our present finding of a_{*}^{(1)}=367(13) a_{0}, we obtain a_{*}^{(1)}/a_{−}^{(0)}=0.43(2), which is significantly smaller than the value 1.06 in the ideal scenario. Theoretical models that take the finite range of the twobody potential into account^{13,25,26,27} have shown that finiterange corrections, which are particularly important for lowlying Efimov states, lead to shifts of the resonance positions. A downshift of a_{*}^{(n)} is expected along with an upshift of a_{−}^{(n′)}. This would indeed result in a smaller a_{*}^{(1)}/a_{−}^{(0)}. An alternative explanation would be a change in the threebody parameter between the a<0 and a>0 regions, which may occur in our case, where these regions are connected through a zero crossing instead of a pole in a.
We have observed a strong loss resonance in an ultracold atom–dimer mixture, induced by a weakly bound trimer state. Our work demonstrates that atom–dimer relaxation measurements can provide information on the threebody spectrum in a complementary way to threebody atomic recombination. To resolve the open issues regarding the relation between different resonance features, more efforts, both experimentally and theoretically, are necessary. On the experimental side, the realization of the complete Efimov scenario requires a Feshbach resonance in which both sides of the resonance are accessible. For caesium, a broad Feshbach resonance at 800 G is an excellent candidate for this purpose^{15}. A full understanding of the threebody sector in realworld systems near universality is required as a prerequisite to explore more complex fewbody phenomena^{2,28,29,30}, such as the fourbody scenario on which present experiments are beginning to shed new light^{7,30}.
Methods
Preparation.
Our ultracold atom–dimer mixture is trapped in a crossedbeam optical dipole trap generated by two 1,064 nm laser beams with waists of about 250 and 36 μm (ref. 7). As atoms and dimers in general have different magnetic moments, the application of a levitation field is not appropriate and a sufficiently high optical gradient in the vertical direction to hold the atoms and dimers against gravity is required. However, to obtain very low temperatures and not too high densities, a tight trap is not advantageous. Here, we use an adjustable elliptic trap potential with weak horizontal confinement and tight confinement in the vertical direction. The ellipticity is introduced by a rapid spatial oscillation of the 36 μm waist beam in the horizontal plane with the use of an acoustooptic modulator, creating a timeaveraged optical potential. The final temperature of the atomic and molecular sample can be set by varying the ellipticity and the laser power of the laser beam in the final trap configuration, and is in the range of 30–250 nK. For the lowest temperature samples, the final timeaveraged elliptic potential is characterized by trap frequencies of 10 and 20 Hz in the horizontal plane, and 80 Hz in the vertical direction.
Dimer loss model.
We measure the atom–dimer relaxation loss rate β by recording the time evolution of the dimer number N_{D} and atom number N_{A}. In a harmonic trap, the atomic and molecular samples can be described by Gaussian density distributions, where the width depends on the trap frequencies, the temperature and the mass. Because the polarizability of the halo dimers is twice that of the atoms, the trap frequencies of the atoms and the dimers are the same. We find that the atomic and molecular samples have the same temperature^{7}. The time evolution of N_{D} can then be described by the following rate equation:
with and being the mean atomic and molecular density, respectively, m the atomic mass, the geometric mean of the trap frequencies and T the temperature. Here, loss of dimers due to dimer–dimer relaxation is also taken into account through the dimer–dimer relaxation loss rate coefficient α. Because of the unequal mass, the density distributions of the atomic and molecular samples are not the same. As a result, an effective atomic density experienced by the molecular cloud has to be considered, which is taken into account by the factor in front of the atom–dimer loss term^{18}.
Our experiments are carried out in the regime in which N_{A}≫N_{D} and loss of atoms as a result of atom–dimer relaxation is negligible. Threebody recombination leads to atom loss on a much longer timescale compared with the molecular lifetime. Therefore, N_{A} can be taken as a constant and equation (1) has the following solution:
where N_{D,0}≡N_{D}(t=0), and . If β N_{A}≫α N_{D}, that is, dimer–dimer relaxation loss is negligible compared with atom–dimer relaxation loss, equation (2) simplifies to
and N_{D} shows an exponential decay with a 1/e lifetime of (b N_{A})^{−1}. In our experiments, dimer–dimer relaxation loss can be neglected for B>20 G and equation (3) is fitted to the data. For B<20 G, β is much smaller than α (ref. 7) and the application of equation (2) is required, taking α from independent loss measurements of a pure dimer sample (ref. 7). For each measurement of β, the trap frequencies and the temperature are determined by sloshing mode and timeofflight measurements, respectively.
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Acknowledgements
We thank T. Köhler, B. D. Esry and P. Massignan for many fruitful discussions. We acknowledge support by the Austrian Science Fund (FWF) within SFB 15 (project part 16). S.K. is supported within the Marie Curie IntraEuropean Program of the European Commission. F.F. is supported within the Lise Meitner program of the FWF.
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Knoop, S., Ferlaino, F., Mark, M. et al. Observation of an Efimovlike trimer resonance in ultracold atom–dimer scattering. Nature Phys 5, 227–230 (2009). https://doi.org/10.1038/nphys1203
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DOI: https://doi.org/10.1038/nphys1203
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