Evidence for Universal FourBody States Tied to an Efimov Trimer
Fewbody physics produces bizarre and counterintuitive phenomena, with the Efimov effect representing the major paradigm of the field [1]. Early in the 1970s, Efimov found a solution to the quantum threebody problem, predicting the existence of an infinite series of universal weakly bound threebody states. Surprisingly, these Efimov trimers can even exist under conditions where a weakly bound dimer state is absent [2–4]. An essential prerequisite for the Efimov effect is a large twobody scattering length $a$, far exceeding the characteristic range of the interaction potential. Ultracold atomic systems with tunable interactions [5] have opened up unprecedented possibilities to explore such fewbody quantum systems under well controllable experimental conditions. In particular, $a$ can be made much larger than the van der Waals length ${r}_{\mathrm{vdW}}$ [6], the range of the interatomic interaction.
In the past few years, signatures of Efimov states have been observed in ultracold atomic and molecular gases of cesium atoms [7,8], and recently in threecomponent Fermi gases of ${}^{6}\mathrm{Li}$ [9,10], in a Bose gas of ${}^{39}\mathrm{K}$ atoms [11], and in mixtures of ${}^{41}\mathrm{K}$ and ${}^{87}\mathrm{Rb}$ atoms [12]. In all these experiments, Efimov states manifest themselves as resonantly enhanced losses, either in atomic threebody recombination or in atomdimer relaxation processes.
As a next step in complexity, a system of four identical bosons with resonant twobody interaction challenges our understanding of fewbody physics. The extension of universality to fourbody systems has been attracting increasing interest both in theory [13–18] and experiment [19]. A particular question under debate is the possible relation between universal three and fourbody states [13–16,18]. In this context, Hammer and Platter predicted the fourbody system to support universal tetramer states in close connection with Efimov trimers [16].
Recently, von Stecher, D’Incao, and Greene presented key predictions for universal fourbody states [18]. For each Efimov trimer, they demonstrate the existence of a pair of universal tetramer states according to the conjecture of Ref. [16]. Such tetramer states are tied to the corresponding trimer through simple universal relations that do not invoke any fourbody parameter [13,15,18]. The authors of Ref. [18] suggest resonantly enhanced fourbody recombination in an atomic gas as a probe for such universal tetramer states. They also find hints on the existence of one of the predicted fourbody resonances by reinterpreting our earlier recombination measurements on ${}^{133}\mathrm{Cs}$ atoms at large negative scattering lengths [7]. In this Letter, we present new measurements on the Cs system dedicated to fourbody recombination in the particular region of interest near a triatomic Efimov resonance. Our results clearly verify the central predictions of Ref. [18]. We observe two loss resonances as a signature of the predicted tetramer pair and we find strong evidence for the fourbody nature of the underlying recombination process.
The fourbody extended Efimov scenario [16,18] is schematically illustrated in Fig. 1, where the tetramer states (Tetra1 and Tetra2) and the relevant thresholds are depicted as a function of the inverse scattering length $1/a$. Within the fourbody scenario, the Efimov trimers ( $T$) are associated with trimeratom thresholds ( $T+A$, dashed lines). The pair of universal tetramer states (solid lines) lies below the corresponding $T+A$ threshold. For completeness, we also show the $a>0$ region. Here, the picture is even richer because of the presence of the weakly bound dimer state, which leads to the dimeratomatom threshold ( $D+A+A$) and the dimerdimer threshold ( $D+D$). In the fourbody scenario, the tetramer states emerge at the atomic threshold for $a<0$ and connect to the $D+D$ threshold for $a>0$.
The Efimov trimer intersects the atomic threshold at $a={a}_{T}^{*}$, which leads to the observed triatomic resonance [7]. The corresponding tetramer states are predicted [18] to intersect the atomic threshold at scattering length values
These universal relations, linking three and fourbody resonances, express the fact that no additional parameter, namely, the socalled fourbody parameter, is needed to describe the system behavior. In contrast to the connection between two and threebody systems, where a threebody parameter is required to locate the trimer states, the universal properties of the fourbody system are thus directly related to the threebody subsystem.
In analogy to the wellestablished fact that Efimov trimers lead to loss resonances in an atomic gas [7,20], universal fourbody states can also be expected to manifest themselves in a resonant increase of atomic losses [18]. Resonant coupling between four colliding atoms and a tetramer state ( $a\simeq {a}_{\mathrm{Tetra}}^{*}<0$) drastically enhances fourbody recombination to lower lying channels. Possible decay channels are trimeratom, dimerdimer, and dimeratomatom channels. In each of these recombination processes, we expect all the particles to rapidly escape from the trap, as the kinetic energy gained usually exceeds the trap depth.
We prepare an ultracold optically trapped atomic sample in the lowest hyperfine sublevel ( $F=3$, ${m}_{F}=3$), as described in Ref. [19]. By varying the magnetic field between 6 and 17 G, the scattering length $a$ can be tuned from $1100$ to $0{a}_{0}$ [21], where ${a}_{0}$ is Bohr’s radius. For presenting our experimental data in the following, we convert the applied magnetic field into $a$ using the fit formula of Ref. [7]. After several cooling and trapping stages [21], the atoms are loaded into an optical trap, formed by crossing two infrared laser beams [19]. The trap frequencies in the three spatial directions are about $({\omega}_{x},{\omega}_{y},{\omega}_{z})=2\pi \times (10,46,65)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{Hz}$. Similar to [21], we support the optical trap by employing a magnetic levitation field acting against gravity. Evaporative cooling in the levitated trap is stopped just before the onset of BoseEinstein condensation in order to avoid implosion of the gas. For our typical temperature of 50 nK, we obtain about $8\times {10}^{4}$ noncondensed atoms with a peak density of about $7\times {10}^{12}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\mathrm{cm}}^{3}$.
In a first set of experiments, we record the atom number after a fixed storage time in the optical trap for variable scattering length in the $a<0$ region. Figure 2 shows the observed losses, containing both three and fourbody contributions. The threebody part consists of a background that follows a general ${a}^{4}$scaling behavior [20,22,23] and resonant losses caused by the triatomic Efimov resonance, which for a 50nK sample was observed to occur at ${a}_{T}^{*}=870(10){a}_{0}$ [7,24]; this is consistent with the large losses shown in Fig. 2(a). Beside this expected behavior of the threebody subsystem, we clearly observe two additional loss features, one located at about $410{a}_{0}$ [Fig. 2(a)] and one at about $730{a}_{0}$ [Fig. 2(b)]. The observation of the resonance at $730{a}_{0}$ is particularly demanding and requires a careful choice of parameters as the signal needs to be discriminated against the very strong background that is caused by threebody losses. Here we use a much shorter hold time of 8 ms, which is the shortest possible time required to ensure precise magnetic field control in our apparatus.
We interpret the two observed resonant loss features as the predicted pair of fourbody resonances [18]. For the resonance positions we find ${a}_{\mathrm{Tetra}1}^{*}/{a}_{T}^{*}\simeq 0.47$ and ${a}_{\mathrm{Tetra}2}^{*}/{a}_{T}^{*}\simeq 0.84$, which are remarkably close to the predictions of Eq. (1).
In a second set of experiments, we study the time dependence of the atomic decay in the optical trap. Here we focus on the region around the resonance at ${a}_{\mathrm{Tetra}1}^{*}\simeq 410{a}_{0}$, where the threebody losses are comparatively weak and thus allow for a detailed analysis of the loss curves. Representative loss measurements for three different values of $a$ are shown in Fig. 3.
The observed decay can be fully attributed to threebody and fourbody recombination collisions. This is due to the fact that inelastic twobody collisions of atoms in the lowest Zeeman sublevel are energetically suppressed, and onebody losses, such as background collisions or lightinduced losses, can be completely neglected under our experimental conditions. The corresponding differential equation for the decaying atom number reads as
where ${L}_{3}$ and ${L}_{4}$ denote the three and the fourbody recombination rate coefficient, respectively. The average density is calculated by integrating the density over the volume $\u27e8{n}^{2}\u27e9=(1/N)\int {n}^{3}{d}^{3}\mathbf{r}$ and $\u27e8{n}^{3}\u27e9=(1/N)\int {n}^{4}{d}^{3}\mathbf{r}$. By considering a thermal density distribution of Gaussian shape in the threedimensional harmonic trap, we obtain $\u27e8{n}^{2}\u27e9={n}_{p}^{2}/\sqrt{27}$ and $\u27e8{n}^{3}\u27e9={n}_{p}^{3}/8$, with ${n}_{p}=N[m{\overline{\omega}}^{2}/2\pi {k}_{B}T{]}^{3/2}$ the peak density. Here, $m$ is the atomic mass, $T$ the temperature, and $\overline{\omega}=({\omega}_{x}{\omega}_{y}{\omega}_{z}{)}^{1/3}$ the mean trap frequency. We determine the trap frequencies and the temperature by sloshing mode and timeofflight measurements, respectively.
In general Eq. (2) is not analytically solvable. An analytic solution can be found in the limit of either pure threebody losses or pure fourbody losses. Therefore we fit our decay curves with a numerical solution of Eq. (2), keeping both ${L}_{3}$ and ${L}_{4}$ as free parameters. Note that we have not included antievaporation heating [23] in our model because we do not observe the corresponding temperature increase in our experiments. We believe that, for the fast decay observed here, the sample may not have enough time to thermalize.
Our experimental data clearly reveal a qualitative change of the decay curves when $a$ is tuned between ${a}_{T}^{*}$ and ${a}_{\mathrm{Tetra}1}^{*}$. Figure 3(a) shows that for $a\approx {a}_{T}^{*}$ the loss is dominated by threebody recombination; here the full numerical fitting curve follows the pure threebody solution. A different situation is found at $410{a}_{0}$; see Fig. 3(c). Here a pure threebody analysis cannot properly describe the observed behavior and the full numerical solution reveals a predominant fourbody character. In intermediate situations, for which an example is shown in Fig. 3(b), both three and fourbody processes significantly contribute to the observed decay.
From the decay curves taken at different values of $a$ we determine ${L}_{3}$ and ${L}_{4}$; the results are shown in Figs. 4(a) and 4(b), respectively. The threebody contribution ${L}_{3}$ follows previously observed behavior [7], as dictated by the ${a}^{4}$ scaling in combination with the Efimov effect.
Our major result is shown in Fig. 4(b), where we plot the rate coefficient ${L}_{4}$. Our data provide the first available quantitative information on ${L}_{4}$, establishing the role of fourbody collisions in ultracold gases. For $a<{a}_{\mathrm{Tetra}1}^{*}$, where no universal tetramer states exist, the fourbody losses are typically very weak. Here, we measure ${L}_{4}\simeq 0.2\times {10}^{37}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\mathrm{cm}}^{9}/\mathrm{s}$. With increasing $a$, the system undergoes a significant change in its behavior, with fourbody collisions dominating the atomic decay; see Fig. 3(c). We observe a sharp increase of ${L}_{4}$, which reaches its maximum value at $a=412(2){a}_{0}$. This observation is another strong piece of evidence for the predicted universal fourbody state at ${a}_{\mathrm{Tetra}1}^{*}$ [18]. To directly estimate the relative contributions of three and fourbody recombination, one can compare ${L}_{3}$ with ${n}_{0}{L}_{4}$, where ${n}_{0}\simeq 1.0\times {10}^{13}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\mathrm{cm}}^{3}$ is the initial peak density at 40 nK. At resonance, ${n}_{0}{L}_{4}$ exceeds ${L}_{3}$ by more than 1 order of magnitude, with ${L}_{3}=0.7\times {10}^{25}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\mathrm{cm}}^{6}/\mathrm{s}$ and ${n}_{0}{L}_{4}=3\times {10}^{24}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\mathrm{cm}}^{6}/\mathrm{s}$. With further increasing $a$, ${L}_{4}$ decreases and ${L}_{3}$ increases such that ${L}_{3}>{n}_{0}{L}_{4}$. For $a>700{a}_{0}$, the very fast threebody decay renders the analysis of the loss curves in terms of ${L}_{4}$ unreliable. In addition to the large statistical fit errors seen in our data in this region, other systematic error sources like a nonthermal evolution of the atomic density distribution can have a strong influence.
Figure 4 also includes the theoretical predictions for ${L}_{3}$ and ${L}_{4}$ at 40 nK and demonstrates a remarkable qualitative agreement with our experimental results. The theoretical approach utilizes a solution of the fourbody problem in the hyperspherical adiabatic representation [18]; the derivation and associated calculations of ${L}_{4}$, adapted from [25], provide the first quantitative description of the fourbody recombination rate. The calculations only require to fix the position of the triatomic Efimov resonance as determined in the previous experiment at 10 nK of Ref. [7]. The difference in the width and the amplitude of the fourbody resonance between experimental and theoretical data may be explained by different coupling to possible decay channels.
Our work leads to important conclusions related to the concept of universality for quantum systems with increasing complexity. The observation of the two fourbody resonances close to the predicted positions [18] points to the universal character of the underlying states. This also supports the view of Refs. [13,15,18] that a fourbody parameter is not required to describe the system. Universal fourbody states then emerge as a genuine consequence of the Efimov spectrum. This also provides a novel way to test Efimov physics. The Efimovian character of a threebody resonance can be probed by observing the universal tetramer resonances tied to it, without the necessity to explore the full geometric scaling of Efimov physics by changing the scattering length by orders of magnitude.
While our present work has focussed on fourbody phenomena at $a<0$, a further exciting step will be the exploration of the entire fourbody spectrum. For $a>0$, the spectrum becomes richer and new phenomena can be expected such as resonant interactions between fourbody states and twodimer states. In this way, experiments on fewbody phenomena in ultracold atoms will keep on challenging our understanding of the universal physics of a few resonantly interacting particles.
We are aware of related results in ${}^{39}\mathrm{K}$, in which enhanced losses near a Feshbach resonance may be interpreted as a fourbody resonance [26].
References
 V. Efimov, Phys. Lett. B 33, 563 (1970).
 A. S. Jensen, K. Riisager, D. V. Fedorov, and E. Garrido, Rev. Mod. Phys. 76, 215 (2004).
 T. Köhler, K. Góral, and P. S. Julienne, Rev. Mod. Phys. 78, 1311 (2006).
 E. Braaten and H.W. Hammer, Phys. Rep. 428, 259 (2006).
 C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, arXiv:0812.1496.

The van der Waals length is defined as ${r}_{\mathrm{vdW}}=\frac{1}{2}(m{C}_{6}/{\hslash}^{2}{)}^{1/4}$, where ${C}_{6}$ is the van der Waals dispersion coefficient [3]. For Cs, ${r}_{\mathrm{vdW}}=100{a}_{0}$.
 T. Kraemer et al., Nature (London) 440, 315 (2006).
 S. Knoop et al., Nature Phys. 5, 227 (2009).
 T. B. Ottenstein et al., Phys. Rev. Lett. 101, 203202 (2008).
 J. H. Huckans et al., arXiv:0810.3288.
 M. Zaccanti, G. Modugno, C. D‘Errico, M. Fattori, G. Roati, and M. Inguscio, in Proceedings of DAMOP2008, State College, PA, 2008 (unpublished).
 G. Barontini et al., arXiv:0901.4584.
 L. Platter, H.W. Hammer, and UlfG. Meißner, Phys. Rev. A 70, 052101 (2004).
 M. Yamashita, L. Tomio, A. Delfino, and T. Frederico, Europhys. Lett. 75, 555 (2006).
 G. J. Hanna and D. Blume, Phys. Rev. A 74, 063604 (2006).
 H. Hammer and L. Platter, Eur. Phys. J. A 32, 113 (2007).
 Y. Wang and B. D. Esry, arXiv:0809.3779 [Phys. Rev. Lett. (to be published)].
 J. von Stecher, J. P. D’Incao, and C. H. Greene, arXiv:0810.3876.
 F. Ferlaino et al., Phys. Rev. Lett. 101, 023201 (2008).
 B. D. Esry, C. H. Greene, and J. P. Burke, Phys. Rev. Lett. 83, 1751 (1999).
 T. Weber et al., Science 299, 232 (2003).
 P. O. Fedichev, M. W. Reynolds, and G. V. Shlyapnikov, Phys. Rev. Lett. 77, 2921 (1996).
 T. Weber et al., Phys. Rev. Lett. 91, 123201 (2003).
 H.C. Nägerl et al., AIP Conf. Proc. 869, 269 (2006).
 N. P. Mehta et al., arXiv:0903.4145.
 M. Zaccanti (private communication).
 C. Chin et al., Phys. Rev. A 70, 032701 (2004).
 ResamplingBased Multiple Testing, P. H. Westfall and S. S. Young (John Wiley and Sons, New York, 1993).