Abstract
In this article, we revisit the heteronuclear Efimov effect in a BoseFermi mixture with large mass difference in the BornOppenheimer picture. As a specific example, we consider the combination of bosonic ^{133}Cs and fermionic ^{6}Li. In a system consisting of two heavy bosons and one light fermion, the fermionmediated potential between the two heavy bosons gives rise to an infinite series of threebody bound states. The intraspecies scattering length determines the threebody parameter and the scaling factor between consecutive Efimov states. In a second scenario, we replace the single fermion by an entire Fermi Sea at zero temperature. The emerging interaction potential for the two bosons exhibits longrange oscillations leading to a weakening of the binding and a breakup of the infinite series of Efimov states. In this scenario, the binding energies follow a modified Efimov scaling law incorporating the Fermi momentum. The scaling factor between deeply bound states is governed by the intraspecies interaction, analogous to the Efimov states in vacuum.
Introduction
The late Mahir S. Hussein, to whom this special issue is dedicated, is best known for his groundbreaking contributions in nuclear physics, e.g., to the understanding of the fusion and breakup of loosely bound nuclei [1]. From a very early stage on, Mahir also recognized the importance of ultracold atomic gases, in particular BoseEinstein condensates, for reaching a deeper understanding of strongly correlated manybody quantum systems. In particular, based on his solid background in nuclear physics, he was among the first to realize the immense potential of magnetically tuned Feshbach resonances for tuning effective interactions over a wide range [2]. Therefore, it appeared appropriate to us to discuss one specific, rather spectacular example of the application of tunable interparticle interactions on the quantum level, namely the socalled Efimov effect. This effect, originally proposed in the context of nuclear binding [3], nicely demonstrates the fruitful crossfertilization of concepts between nuclear physics and ultracold atomic quantum gases, which was so dear to Mahir S. Hussein.
The unexpected emergence of an infinite series of bound threeparticle states via resonant pairwise interaction was first predicted theoretically by Vitaly Efimov in 1970 [3]. Since then, the Efimov effect has become a prime example for studying universality in quantum fewbody systems. In the universal regime, i.e., when the scattering length a between two particles exceeds the characteristic range of interparticle interaction r_{0}, there exists a series of three body bound states which follow a simple discrete scaling law. Remarkably, these threebody bound states can form, even though the twobody interactions are too weak to support a twobody bound state. Initially, Efimov’s prediction raised serious doubts about the applicability of his concepts, but theorists trying to prove him wrong had to finally concede that he might actually be right. While Efimov proposed the effect to be observed in nuclear systems, such as in ^{3}He^{+} or in the Hoyle state of ^{12}C, the existence of a series of threebound states requires the twobody interactions to be close to resonance. It was only in 2006, 36 years after Efimov’s theoretical prediction, that first experimental evidence was eventually found in a gas of ultracold ^{133}Cs atoms [4]. The precise tuning of twobody interactions from infinite repulsion to attraction via Feshbach resonances opened up unique opportunities to study the range of universality in the Efimov scenario. Following the investigations in homonuclear systems, the Efimov effect was later also observed in heteronuclear systems [5,6,7,8,9,10,11] where, e.g., a large mass ratio [8, 9] leads to a denser Efimov spectrum which allowed for the observation of up to three consecutive Efimov resonances.
Besides its relevance in the field of fewbody physics, the Efimov effect plays an important role in the understanding of manybody systems in the presence of threebody bound states. Similarly, one can consider the Efimov effect from a fewbody perspective and study the influence of a surrounding manybody background to the trimer. The latter has been studied in different scenarios with either a Fermi Sea [12,13,14,15] or a BEC [16, 17] serving as a background. The former has been studied in [18,19,20], following recent experimental advances in creating Bose [21,22,23,24,25] and Fermi polarons [26,27,28,29,30,31,32]. More details on different aspects of the Efimov effect are covered extensively in several reviews, which have appeared in the years after its first realization [33,34,35,36,37,38,39].
In this work, we will focus on the specific case of two heavy bosons and a light fermion, which allows to apply the BornOppenheimer (BO) approximation. We investigate two limiting cases of the Efimov scenario, first, in vacuum, and second, in the presence of a Fermi Sea. While the first case reproduces the wellknown features of the Efimov effect, the second case provides novel insights serving as a precursor to understand effective interactions of Fermi polarons [40], i.e., strongly correlated impurities in a Fermi sea. As a specific example, we consider the heteronuclear mixture of bosonic ^{133}Cs and fermionic ^{6}Li, for which experiments have been performed by us and others. Providing a simple and intuitive access to understand the Efimov effect in a massimbalanced system, the BO approximation not only does capture the existence of an infinite series of threebody bound states [33, 41,42,43] but also can account for intraspecies interactions via shortrange van der Waals (vdW) potentials [44]. Our paper is structured as follows: after giving a basic introduction to the Efimov scenario in Section 2, we describe how we solve the Schrödinger equation by employing the BO approximation in Section 3. In Section 4, we present our results on the Efimov energy spectrum for a system consisting of two Cs atoms and one Li atom, taking into account finite CsCs swave interaction. We extend our results to a system of two Cs atoms immersed in a Fermi Sea (Section 5), before concluding in Section 6.
The Efimov Scenario
The Efimov scenario for the heteronuclear case can be visualized in an energy diagram (Fig. 1) which is plotted against the inverse interspecies scattering length 1/a between the identical bosons B and the distinguishable particle X. For simplicity, we first consider the scenario in which the two bosons resonantly interact (\(a_{\text {BB}} \rightarrow \infty \)). Then the energy diagram can be divided into three different regions representing the threebody scattering states, Efimov trimers, and atomdimer states. For positive energies (E > 0), the three atoms are unbound and possess a finite kinetic energy. For negative energies (E < 0), one needs to distinguish between the sign of the scattering length. On the positive scattering length side (a > 0), the system supports a weakly bound dimer state BX with an energy of 1/2μa^{2}, where μ is the reduced mass. Above the atomdimer threshold, the dimer state BX coexists with a free atom B. The Efimov trimers exist in the region below the threebody dissociation threshold at a < 0 and the atomdimer threshold at a > 0. In this region, an infinite number of Efimov trimers with energies E_{n} exist which cross the threebody dissociation threshold at values of \(a_{}^{(n)}\) and \(a_{*}^{(n)}\) on the negative and positive scattering length side, respectively. These crossings can be described by the discrete scaling laws
where the scaling factor \(\lambda = e^{\pi /s_{0}}\) with the dimensionless parameter s_{0} is dependent on the mass ratio, the number of resonant interactions and the quantum statistics of the particles [39].
In real systems, interactions have a finite range, given by nonzero interparticle distances in the underlying interaction potentials. Therefore, in the infinite progression of Efimov trimers a ground state has to be considered. This can be done by introducing a threebody parameter (3BP) which defines the position of the energy E_{0} or the scattering length \(a_{}^{(0)}\) of the lowest Efimov trimer. The 3BP is determined by shortrange twobody interactions and can be expressed in terms of vdW units (see Eq. 4). As indicated in Fig. 1, the universal Efimov scaling therefore only holds if \(a \gg max(r_{vdW}^{BX}, r_{vdW}^{BB})\) due to finite range effects.
BornOppenheimer Approximation
In the BornOppenheimer approximation, we can solve the threebody Schrödinger equation for two heavy bosons with mass M and a light atom with mass m in a twostep approach. We assume that the mass ratio is M/m ≫ 1, such that the light atom immediately follows the motion of the heavy ones. In the first step, the Schrödinger equation is solved for the light atom with a potential created by the two heavy scatterers at fixed distances. The resulting energy serves as an interaction potential V_{E}(R), induced by the presence of the light particle, in the Schrödinger equation for the heavy bosons
where R denotes the distance between the two heavy bosons. The bosonboson interaction is modeled by a van der Waals potential with a hard core of the form [45, 46]
The C_{6} coefficient allows us to naturally introduce the van der Waals radius r_{vdW} and energy E_{vdW} via [47]
and
The cutoff radius R_{0} in Eq. 3 determines the 3BP and can be directly related to the bosonboson scattering length a_{BB} via [45]
where J_{ν}(x) and N_{ν}(x) are Bessel functions of the first and second kind, respectively. As Eq. 6 has more than one solution, the value of R_{0} does determine not only a_{BB} but also the number of bound states supported by the vdW potential V_{BB}(R). However, we note that the exact number of dimer states is irrelevant for our purposes as it has no significant effect on the longrange part of the Efimov wavefunctions [44].
The induced interaction potential V_{E}(R) possesses a symmetric and an antisymmetric solution [43]. In the limit of \(a \rightarrow \infty \), only the former is relevant for the discussion of bound states. It reads
where c ≈ 0.567 is the solution of c = e^{−c} and is connected to the scaling factor s_{0} via
With the interaction potentials V_{E}(R) and V_{BB}(R) we can solve Eq. 2 in order to understand the Efimov scenario for a CsCsLi system and its scaling behavior taking into account bosonboson interactions.
Two Bosons Meet One Fermion
In Fig. 2 we show the solution of Eq. 2 using the LiCs mass ratio of M_{Cs}/m_{Li} = 22.1 and a shortrange cutoff R_{0} for which the energy spectrum supports up to two Cs_{2} dimer states. It is instructive to consider the two cases

(a)
only bosonboson interaction V_{BB}(R)

(b)
total potential V_{E}(R) + V_{BB}(R).
For case a, i.e., pure twobody vdW interaction, we identify two weakly bound dimer states (orange dashed lines). The energy of the least bound state approaches the binding energy of the Cs_{2} dimer \(E_{b} = 1/M a_{\text {CsCs}}^{2}\) for positive and increasing a_{CsCs}. The energy of the most deeply bound state crosses the CsCs resonance and shows a steplike behavior around a_{CsCs} = r_{vdW}. This behavior marks the crossover between a vdWdominated (a_{CsCs} < r_{vdW}) dimer and a halo state (a_{CsCs} > r_{vdW}). For case b) (blue lines), we find that the energy of the most deeply bound state closely follows case a, such that we can assign this state to the Cs_{2} dimer state. The next bound state E_{n= 0} does not follow the respective Cs_{2} dimer state anymore, but persists also across the CsCs resonance. This clearly shows the effect of the mediated interaction necessary to form Efimov trimer states. The following bound states with energies E_{n≥ 1} correspond to the infinite progression of Efimov states. They show a gradual step around a_{CsCs} = r_{vdW} which we assign to an overlap of the wavefunctions of the Efimov trimers with the Cs_{2} halo dimer. This crossover between states originating from a shortrange molecular vdW potential and a longrange ∝− 1/R^{2} potential is one of the main results of the finite range BO approximation, providing an intuitive access to the threebody problem. From the calculated energy spectrum, we can additionally extract the scaling factor between adjacent energy levels. For \({\lambda _{n}^{2}} := E_{n}/E_{n1}\) the scaling factor between the two most deeply bound states amounts to \(\lambda _{n=1}^{2} = 42.9\) at resonance. When crossing the resonance towards negative a_{CsCs}, the scaling factor approaches the universal value of \(\lambda _{n \rightarrow \infty }^{2} = (5.63)^{2} = 31.7\) for a pure ∝− 1/R^{2} potential. The deviation close to resonance can again be explained by the existence of the weakly bound Cs_{2} dimer state. We find that already the second deepest scaling factor \(\lambda _{n=2}^{2} \approx 31.9\) is close to the universal value.
Two Bosons Meet the Fermi Sea
Let us now consider two heavy bosons immersed in a Fermi Sea. In the BO approximation the effective Limediated potential between the two bosons can be calculated by [48]:
Here ΔE(R) denotes the energy reduction of the interacting system compared to the free system
with the bound state wavevectors and scattering phase shifts described by κ_{±} and δ_{±}, respectively [48]. In the asymptotic limit, the energy reduction \({\varDelta } E(R \rightarrow \infty )\) is equivalent to the chemical potential of two free, heavy atoms in a Fermi Sea [49]. The length scale of the Fermi Sea is governed by the Fermi wavevector k_{F} which is related to the atomic density n via k_{F} = (6π^{2}n)^{1/3}.
In Fig. 3 (left panel) we plot the total potential V (R) = V_{eff}(R) + V_{BB}(R) for which we solve the Schrödinger equation for unitarity \(a_{\text {LiCs}} = a_{\text {CsCs}} = \infty \). The gray shaded area marks the hard wall for R < R_{0}. If we set k_{F}r_{vdW} = 0 (blue line), we recover the potential from the Efimov scenario consisting of a shortrange ∝− 1/R^{6} vdW potential (gray dashed line) and a longrange Efimov potential ∝− 1/R^{2} (gray dotted line). For increasing k_{F}, the effective potential starts to grow a repulsive barrier around \(R \approx k_{F}^{1}\) showing damped oscillations. In the inset of Fig. 3, this behavior can be seen more clearly for even larger values of k_{F}. The form of the potential is reminiscent of the form of Friedel oscillations [50] which arise due to the sharp edge of the Fermi distribution. In fact, in the limit of \(a_{\text {LiCs}} \ll R, k_{F}^{1}\), V_{eff}(R) takes the same form of the Friedel oscillations or of the RKKY interaction [51] in the context of magnetic interactions.
We now use the effective potential V_{eff}(R) to replace the Efimov potential V_{E}(R) in Eq. 2 and solve for the eigenenergies of the system (right panel of Fig. 3). We note that the chosen values of k_{F}r_{vdW} = 0.01 and k_{F}r_{vdW} = 0.05 correspond to densities between 10^{11} cm^{− 3} and 10^{13} cm^{− 3}. For k_{F}r_{vdW} = 0.01, the Cs_{2} dimer state and the deepest two Efimov states remain unchanged. For the following state, we see that the effective potential starts to deviate from the Efimov potential around R ≈ 70r_{vdW} leading to a smaller binding energy. As the potential gets more repulsive for increasing R, the formation of bound states is completely suppressed. Similarly, for k_{F}r_{vdW} = 0.05, the weakening of the binding can be understood from a deviation of the Efimov potential around R ≈ 10r_{vdW} and the suppression of the infinite series of Efimov states begins one state earlier.
Analogous to the case of two Cs atoms and one Li atom, we want to investigate the role of the intraspecies interaction, as shown in Fig. 4. The energies in the case k_{F}r_{vdW} = 0 again represent the previous Efimov bound state energies. Introducing a Fermi Sea to the system (k_{F}r_{vdW} = 0.05) we see that, on the positive scattering length side, the n = 1 state shows first deviations from the Efimov scaling when a_{CsCs} takes values larger than r_{vdW} at the step going from the vdWdominated to the longrange regime. Crossing the resonance towards negative scattering lengths, this deviation gets larger again at the step − a_{CsCs} ≈ r_{vdW}. Following the energy line further to the positive scattering length side, the state rapidly dissociates before it reaches the longrange regime. In the same way, with a smaller density of the Fermi Sea of k_{F}r_{vdW} = 0.01, the system supports one more bound state before the binding energy vanishes.
We calculate the density dependence of the binding energies (Fig. 5) and find that they remain nearly constant before they rapidly go to the continuum. The number of bound states is given by the Fermi wavevector k_{F} and the shape of the lines suggest a similar scaling as we have seen in Fig. 1 following Eq. 1. Indeed, in the presence of the Fermi Sea, the two heavy bosons follow a new discrete scaling law including the additional length scale k_{F} [12, 14]:
In the case of finite intraspecies interactions we find that this scaling is fulfilled for highlying bound states n > 1 (inset of Fig. 5). Analogous to the scenario of two Cs atoms and one Li atom in Section 4, the first excited bound state (n = 1, orange line) shows only a small deviation from the scaling, whereas the scaling is broken for the ground state due to finite range effects.
Conclusions
In summary, we have calculated the binding energies of a CsCsLi system and of a system of two Cs atoms in a Li Fermi Sea and studied the influence of the intraspecies scattering length using the BO approximation. In the CsCsLi system, the intraspecies interaction leads to a steplike behavior in the energy spectrum and the existence of weakly bound Cs_{2} dimers influence the scaling factor. Immersing the two Cs atoms in a Li Fermi Sea suppresses the formation of bound states for sufficiently high k_{F} and breaks the discrete Efimov scaling law. Instead, a new scaling law can be formulated which takes into account the wavevector k_{F}. This additional length scale of the Fermi Sea may also be used to define a new window of universality which is not only determined by shortrange interactions, but also by the Fermi wavevector. In an experiment, the shifted position of the Efimov states in the Fermi Sea, which can also be interpreted as bipolaronic states, may be observed by means of threebody loss measurements, similar to the previous Efimov experiments [52] while now lower temperatures and higher densities are required. The influence of the intraspecies scattering length can be studied in the LiCs system which features two interspecies Feshbach resonances around 843 G and 889 G with negative and positive sign of the intraspecies scattering length, respectively. However, we note that our simple BO approximation does only provide qualitative results. For the CsCsLi system, more quantitative results beyond the BO approximation can be obtained by means of a spinless vdW theory [53] where the threebody problem is solved in the hyperspherical formalism with twobody interactions modeled by a single channel Lennard Jones potential. Also finite temperature effects as well as scattering of trimers by the Fermi Sea which may lead to the excitation of particlehole pairs and a change of the effective interaction potential [13] have to be considered for a more realistic description of the system.
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Acknowledgments
E.L. acknowledges support by the IMPRSQD.
Funding
Open Access funding enabled and organized by Projekt DEAL. This work is supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)  ProjectID 273811115  SFB 1225 ISOQUANT and by DFG under Germany’s Excellence Strategy EXC2181/1  390900948 (Heidelberg STRUCTURES Excellence Cluster).
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Tran, B., Rautenberg, M., Gerken, M. et al. Fermions Meet Two Bosons—the Heteronuclear Efimov Effect Revisited. Braz J Phys 51, 316–322 (2021). https://doi.org/10.1007/s13538020008115
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DOI: https://doi.org/10.1007/s13538020008115
Keywords
 Efimov physics
 Lithiumcesium
 Fermi Sea
 BornOppenheimer approximation