Effect of Background Scattering on Efimov Scenario for Overlapping Narrow Feshbach Resonances

Abstract

Efimov physics in the vicinity of two overlapping narrow Feshbach resonances can be explored within a framework of a three-channel model where a non-interacting open channel is coupled to two closed molecular channels. Here, we determine how it compares to the extended two-channel model, which includes an open channel with finite background scattering and a single molecular channel. We identify the parameter range in which the three-channel model surpasses the extended two-channel model. Furthermore, the three-channel model is extended to include background scattering, and then both models are applied to the experimentally relevant system of bosonic lithium atoms polarized on two different energy levels, with an isolated and two overlapping narrow Feshbach resonances, respectively. We confirm, in agreement with previous studies, that being small, the background scattering length in lithium has a negligible effect on the Efimov features in the case of isolated resonance. However, in the case of overlapping Feshbach resonances, the inclusion of background scattering improves the performance of the theory with respect to the experimentally measured position of the Efimov resonance.

1 Introduction

Magnetic Feshbach resonances are one of the most salient features underlying ultracold collisions [1]. Intuitively, they can be understood within a simple two-channel model. Consider a pair of atoms that interact in an open channel at such a low kinetic energy that their scattering is fully described by the s-wave scattering length a. If a closed channel with a nearly degenerate bound state is coupled to the open channel, the scattering length is resonantly enhanced. The relative energy between the closed and open channels is tuned by the external magnetic field, and the functional form of a(B) assumes the following form [1]:

$$\begin{aligned} a=a_{bg}\left( 1-\frac{\Delta B}{B-B^{(res)}}\right) , \end{aligned}$$

(1)

where $a_{bg}$ represents the off-resonant background scattering length, and $\Delta B$ indicates the magnetic field interval between the position of the resonance $B^{(res)}$ and $a=0$ (the resonance width).

Ultracold atoms exhibiting Feshbach resonances have opened a new avenue for studying the universal spectrum of three-body Efimov states. This effect was first proposed by Efimov in the context of nuclear physics [2], and predicts that near-resonant two-body interactions induce an infinite ladder of three-body bound states separated by a universal scaling factor. Consequently, a single three-body parameter determines the entire three-body spectrum. The three-body parameter is often determined by the scattering length value $a_{-}^{(0)}$ at which the lowest Efimov trimer state meets the three-body continuum and leads to an Efimov resonance. Extensive experimental efforts across many ultracold atomic species have focused on measuring $a_{-}^{(0)}$ and mapping the Efimov spectrum [3,4,5].

The basic phenomenology of an isolated s-wave Feshbach resonance and Efimov-related physics can be described within the framework of a two-channel model. In its simplest form, an open channel with no background scattering ($a_{bg}=0$) is coupled to a closed channel that can be fine-tuned by a magnetic-field-dependent binding energy level [6]. Although $a_{bg}=0$ leads to the absence of the magnetic field at which $a=0$, Eq. (1) remains valid because when $a_{bg}\rightarrow 0$, the resonance width $\Delta B\rightarrow \infty$ such that $\Delta =a_{bg}\Delta B$ remains finite. The two-channel model has been intensively studied in the past, and various few- and many-body scenarios in bosonic and fermionic gases have been investigated within its framework [7,8,9,10,11,12]. The model can be extended to include background scattering in the open channel [13,14,15], thus reintroducing the $a=0$ condition and improving its performance in the low scattering length limit.

The two-channel model remains a simplified framework for a generally multi-channel two-body scattering problem. For example, a frequently occurring situation of overlapping Feshbach resonances is already beyond the scope of the two-channel model. Detailed incorporation of all possible channels in different theoretical approaches usually leads to heavy numerical calculations [16,17,18,19,20,21,22,23]. In contrast, the simplest way to extend the phenomenology of the two-channel model to three channels has been suggested in [24] and successfully applied to the Efimov scenario of two overlapping Feshbach resonances in $^6$Li–$^{133}$Cs mixture [25]. Although an open channel with no interaction is used in this model, an additional close channel regenerates the $a=0$ limit between the overlapping resonances and improves the performance of the model [24].

A direct question which arises from these studies is to what extent the results of the three-channel model can be reproduced by those of the two-channel model extended to include the background scattering. In what limits does the three-channel model go beyond the latter? Here, we investigate these questions by directly comparing the results of both approaches. We identify the region where the three-channel model surpasses the performance of the extended two-channel model. Moreover, we generalize the three-channel model to include the background scattering and apply it to the relevant scenario in lithium atoms.

The paper is organized as follows. We begin in Sect. 2 by comparing the performance of the three-channel model versus the two-channel model with background scattering applied to model systems of two overlapping resonances. In Sect. 3 we extend the three-channel model to include the background scattering, and in Sect. 4 we apply the two- and three-channel models with background scattering to the Efimov scenarios in lithium atoms. In Sect. 5 we summarize our results.

2 Two-Channel Model with Background Versus Three-Channel Model

In this section we recollect, in general lines, the three-channel model (3CM) developed in [24]. We then define the two model systems of two overlapping Feshbach resonances, which we examine with the two-channel model extended to include nonzero background scattering (2CMbg). The latter is inspired by the approach described in [15]. Note that in [24] the same model systems are approximated by the two-channel model without the background scattering (2CM), and significant deviations in both the two- and three-body sectors are reported. For completeness, the 3CM is reviewed in Appendix A, while the 2CMbg is summarized in Appendix B.

2.1 Overview of Models and Model Systems

2.1.1 Three-Channel Model

The 3CM is described by the Hamiltonian ${\hat{H}}={\hat{H}}_{0}+{\hat{H}}_{int}$, where ${\hat{H}}_{int}={\hat{H}}_{1}+{\hat{H}}_{2}+{\hat{H}}_{12}$. The first term ${\hat{H}}_{0}$ is the bare Hamiltonian of three channels:

$$\begin{aligned} {\hat{H}}_{0}=\int \frac{d^{3}k}{(2\pi )^{3}}\left[ \frac{\hbar ^{2}k^{2}}{2m}{\hat{a}}_{\vec {k}}^{\dagger }{\hat{a}}_{\vec {k}}+\left( E_{b,1}+\frac{\hbar ^{2}k^{2}}{4m}\right) {\hat{b}}_{\vec {k}}^{\dagger }{\hat{b}}_{\vec {k}} +\left( E_{b,2}+\frac{\hbar ^{2}k^{2}}{4m}\right) {\hat{c}}_{\vec {k}}^{\dagger }{\hat{c}}_{\vec {k}}\right] , \end{aligned}$$

(2)

where ${\hat{a}}_{\vec {k}}$ annihilates free particles of mass m, ${\hat{b}}_{\vec {k}}({\hat{c}}_{\vec {k}})$ annihilates molecules in the first (second) channel, and $E_{b,1}(E_{b,2})$ is the bare energy of the first (second) molecular channel, which we assume to be a linear function of the magnetic field: $E_{b,i}=\mu _{i}\left( B_{i}-B\right)$. Without loss of generality, we assume $B_{1}<B_{2}$. Both molecular channels are coupled to the open channel by means of:

$$\begin{aligned} {\hat{H}}_{1}=\Lambda _{1}\int \frac{d^{3}k}{\left( 2\pi \right) ^{3}}\int \frac{d^{3}q}{\left( 2\pi \right) ^{3}}\left[ {\hat{b}}_{\vec {k}}^{\dagger }{\hat{a}}_{\vec {q}+\frac{\vec {k}}{2}}{\hat{a}}_{-\vec {q}+\frac{\vec {k}}{2}} +{\hat{a}}_{-\vec {q}+\frac{\vec {k}}{2}}^{\dagger }{\hat{a}}_{\vec {q}+\frac{\vec {k}}{2}}^{\dagger }{\hat{b}}_{\vec {k}}\right] , \end{aligned}$$

(3)

$$\begin{aligned} {\hat{H}}_{2}=\Lambda _{2}\int \frac{d^{3}k}{\left( 2\pi \right) ^{3}}\int \frac{d^{3}q}{\left( 2\pi \right) ^{3}}\left[ {\hat{c}}_{\vec {k}}^{\dagger }{\hat{a}}_{\vec {q}+\frac{\vec {k}}{2}}{\hat{a}}_{-\vec {q}+\frac{\vec {k}}{2}} +{\hat{a}}_{-\vec {q}+\frac{\vec {k}}{2}}^{\dagger }{\hat{a}}_{\vec {q}+\frac{\vec {k}}{2}}^{\dagger }{\hat{c}}_{\vec {k}}\right] . \end{aligned}$$

(4)

In addition, the two molecular channels can be directly coupled to each other via:

$$\begin{aligned} {\hat{H}}_{12}=\Lambda _{12}\int \frac{d^{3}k}{\left( 2\pi \right) ^{3}}\left[ {\hat{c}}_{\vec {k}}^{\dagger }{\hat{b}}_{\vec {k}}+{\hat{b}}_{\vec {k}}^{\dagger }{\hat{c}}_{\vec {k}}\right] . \end{aligned}$$

(5)

Solutions of the Schrödinger equation $({\hat{H}}-E)|\psi \rangle =0$ in two- and three-body sectors (with respective wave functions $|\psi \rangle$) can be found in [24] and are summarized in Appendix A.

Fig. 1

Two-body sectors of the NB (a, b) and BN (c, d) models. The scattering length (a, c) and dimer binding energy (b, d) of the three-channel model (solid lines) are compared with the two-channel model including $a_{bg}$ (dashed lines). The gray vertical lines indicate the resonance positions $B_{1}^{(res)}$ and $B_{2}^{(res)}$. The shaded region shows the range of the Efimov ground state trimer $B_{*}^{(0)}<B<B_{-}^{(0)}$ associated with $B_{2}^{(res)}$

2.1.2 Definition of Model Systems

Table 1 Parameters of the two model systems: NB and BN [24]

We use the 3CM to define two model systems of overlapping s-wave Feshbach resonances. To facilitate comparison between the performance of 2CM and 2CMbg in model systems, the latter are chosen to be the same as in [24] where they are compared to only 2CM. The model alkali-like atom is considered to have purely singlet molecular bound states and to be subject to a high enough magnetic field to ensure linear Zeeman shift of atomic energy levels to a good approximation. The value of the magnetic moments of both closed channels with respect to the open channel is thus $\mu _1=\mu _2=-2\mu _B$, where $\mu _B=1.4$ MHz/G is the Bohr magneton. The first (second) system is characterized by $\Delta _{1}\ll \Delta _{2}$ ($\Delta _{2}\ll \Delta _{1}$) where $\Delta _{1}$ and $\Delta _{2}$ are the resonance widths (see Eq. (A10)). For both systems, denoted as narrow-broad (NB) and broad-narrow (BN), respectively, the two Feshbach resonances are separated by a distance of $B_{2}^{(res)}-B_{1}^{(res)}=20$G. As indicated in Table 1, for the NB (BN) system, the resonance at $B_{1}^{(res)}$ is narrow (broad) with $\Delta _{1}/a_{0}=150$G ($\Delta _{1}/a_{0}=1000$G) while the resonance at $B_{2}^{(res)}$ is broad (narrow) with $\Delta _{2}/a_{0}=1000$G ($\Delta _{2}/a_{0}=150$G). Efimov energy levels are explored only across the resonance at higher magnetic field for both scenarios. This is because they are true bound states, while the states across the other resonance are embedded into the atom-dimer continuum [24]. By comparing the Efimov physics in the NB and BN scenarios, the influence of overlapping resonances can be revealed for both configurations.

2.1.3 Two-Channel Model with Background Scattering

The Hamiltonian of the 2CMbg can be written as the sum of two terms: ${\hat{H}}^{\text {2bg}}={\hat{H}}^{\text {2bg}}_{0}+{\hat{H}}_{int}^{\text {2bg}}$. The first term is the bare Hamiltonian of free interacting atoms and closed channel molecules:

$$\begin{aligned} {\hat{H}}_{0}^{\text {2bg}}= & {} \int \frac{d^{3}k}{(2\pi )^{3}}\left[ \frac{\hbar ^{2}k^{2}}{2m}{\hat{a}}_{\vec {k}}^{\dagger }{\hat{a}}_{\vec {k}}+\left( E_{\text {b}}+\frac{\hbar ^{2}k^{2}}{4m}\right) {\hat{b}}_{\vec {k}}^{\dagger }{\hat{b}}_{\vec {k}}\right] \nonumber \\{} & {} +\Lambda _{bg}\int \frac{d^{3}k}{\left( 2\pi \right) ^{3}}\int \frac{d^{3}q}{\left( 2\pi \right) ^{3}}\int \frac{d^{3}p}{\left( 2\pi \right) ^{3}}\chi _{\vec {q}}^{*}\; \chi _{\vec {p}}\; {\hat{a}}_{\vec {p}+\frac{\vec {k}}{2}}^{\dagger }{\hat{a}}_{-\vec {p}+\frac{\vec {k}}{2}}^{\dagger }{\hat{a}}_{\vec {q}+\frac{\vec {k}}{2}}{\hat{a}}_{-\vec {q}+\frac{\vec {k}}{2}}. \end{aligned}$$

(6)

The background scattering between the atoms in the open channel is modeled by a separable potential: It is characterized by the coupling constant $\Lambda _{bg}$ and a Gaussian function $\chi _{\vec {q}}$ with a cutoff length b [15]:

$$\begin{aligned} \chi _{\vec {q}}=\exp \left( -\frac{q^{2}b^{2}}{2}\right) . \end{aligned}$$

(7)

The cutoff length b represents the potential range, and it is on the order of the van der Waals length ($r_{\text {vdW}}$). $E_{\text {b}}$ is the bare molecular energy, which, as in the 3CM, we assume to be a linearly function in the external magnetic field: $E_{b}=\mu \left( B_{0}-B\right)$. The second term, ${\hat{H}}_{int}^{\text {2bg}}$, couples the two channels:

$$\begin{aligned} {\hat{H}}_{int}^{\text {2bg}}=\Lambda \int \frac{d^{3}k}{\left( 2\pi \right) ^{3}}\int \frac{d^{3}q}{\left( 2\pi \right) ^{3}}\left( \chi _{\vec {q}}^{*} \; {\hat{b}}_{\vec {k}}^{\dagger }{\hat{a}}_{\vec {q}+\frac{\vec {k}}{2}}{\hat{a}}_{-\vec {q}+\frac{\vec {k}}{2}} +\chi _{\vec {q}}\;{\hat{a}}_{-\vec {q}+\frac{\vec {k}}{2}}^{\dagger }{\hat{a}}_{\vec {q}+\frac{\vec {k}}{2}}^{\dagger }{\hat{b}}_{\vec {k}}\right) . \end{aligned}$$

(8)

Solutions of the Schrödinger equation $({\hat{H}}^{\text {bg}}-E)|\psi \rangle =0$ in two- and three-body sectors (with respective wave functions $|\psi \rangle$) are discussed in [15] and summarized in Appendix B. Here we provide the main results of the two-body sector only, namely the scattering length:

$$\begin{aligned} a=a_{bg}\left( 1-\frac{\frac{2\pi \hbar ^{2}\Lambda ^{2}a_{bg}}{m\Lambda _{bg}^{2}}}{E_{b}-\frac{\Lambda ^{2}}{\Lambda _{bg}}+\frac{2\pi \hbar ^{2}\Lambda ^{2}a_{bg}}{m\Lambda _{bg}^{2}}}\right) , \end{aligned}$$

(9)

where $a_{bg}$ is

$$\begin{aligned} a_{bg}=\frac{b\sqrt{\pi }\Lambda _{bg}}{\Lambda _{bg}+\frac{2\pi ^{3/2}\hbar ^{2}}{m}b}, \end{aligned}$$

(10)

and the binding energy (denoted $E_{D}=-\frac{\hbar ^{2}q_{D}^{2}}{m}<0$) which satisfies the following equation:

$$\begin{aligned} \frac{a_{bg}\exp (b^{2}q_{D}^{2})q_{D}\text {erfc}(bq_{D})}{a_{bg}\exp (b^{2}q_{{D}}^{2})q_{D}\text {erfc}(bq_{D})-1}+\frac{\hbar ^{2}q_{D}^{2}}{m\mu \Delta B}=\frac{B-B^{(res)}}{\Delta B}. \end{aligned}$$

(11)

$E_D$ is obtained from the numerical solution of this equation.

2.2 Approximation of the Model Systems by the Two-Channel Model with Background Scattering

Here we investigate how well the model systems can be approximated by the 2CMbg. As the latter describes only an isolated Feshbach resonance, each resonance in the 3CM is considered independently, and the resonance positions $B_{1}^{(res)}$ and $B_{2}^{(res)}$ are fixed to those defined by the 3CM.

In 2CMbg there are four bare parameters: $b,\Lambda _{bg},\Lambda$ and $B_{0}$. The scattering length (see Eq. (1)) is described by three observables: $a_{bg}, \Delta B, B^{(res)}_i$ and another observable is provided by the binding energy. Analytical expressions between the bare parameters and the scattering length observables can be achieved (see Eqs. (B8) and (B9)) by direct comparison between Eq. (9) and Eq. (1). There are different available strategies that can be followed to fix the bare parameters of the model. Here we choose to fit $E_D$ of the 3CM with Eq. (11) and to extract the values of $a_{bg},\Delta B,\mu$ and b and the remaining bare parameters are deduced from Eqs. (10, B8, B9). The resulting parameters are summarized in Table 2. Note that in this approach, we define all unknowns, including the magnetic moment $\mu$ and the potential range b, as fitting parameters. By allowing a maximum number of parameters to be adjusted, we might obtain an underdetermined system. However, we find that the fitting procedure converges and, as a self-consistency check, we calculate the parameter $R^*$ that is related to the resonance strength (see Eq. (B10)). For narrow resonances, we expect that $R^*\gg b$ and $R^*\gg |a_{bg}|$, and as shown in the last line of Table 2 both conditions are satisfied for all resonances, although at a different level of accuracy. The fitting procedure thus supports the fact that the underlining resonances are narrow.

Once all parameters are determined, we calculate the binding energies in the three-body sector and the results obtained from the 2CMbg are compared with those of the model systems.

Finally, we comment on other strategies to fix the bare parameters of the 2CMbg. One possibility is to fit the scattering length with Eq. (9) and to extract the remaining bare parameter (b) through the corresponding analytic expression. Another approach can follow the idea that the two- and three-body sectors are renormalizable with respect to the cutoff parameter (b) [24] leading to the elimination of the latter. However, to obtain consistent results in this case, $B^{(res)}_i$ must be released as a fitting parameter. Yet another strategy is to make a global fit of both the scattering length and the binding energy together and to extract all parameters of the model. Although different approaches lead to some variations of the results in the three-body sector, the main conclusions of this study remain invariant to the specifics of this choice. We emphasize that the main motivation here is to identify universal properties of the results which will be highlighted when presented.

Table 2 2CMbg parameters obtained from the fit to the 3CM binding energies for two model systems. The $R^*$ parameter in the last line is defined according to Eq. (B10)

2.2.1 Two-Body Sector

Figure 1a, b compares the scattering length and the binding energies of the 2CMbg with the NB model system. As the three-body sector is calculated across the resonance at a higher magnetic field only ($B^{(res)}_2$) we discuss it first. The fitting of the binding energy is shown in Fig. 1b as a green dashed line and indicates a relatively poor reproduction of the model system. Note that the obtained absolute value of $|\mu |=4.514$ MHz/G (see Table 2, “NB-Res2”) is significantly higher than that of the model system ($|\mu |=2.8$ MHz/G). Note also that the opposite situation occurs for the second resonance (see Table 2, “NB-Res1”). This behavior reflects the presence of strong mutual repulsion between the binding energy levels in the NB model system induced by coupling through a continuum [24]. Although deviations of the 2CMbg binding energy from the model system vary as a function of the magnetic field fitting range, the main failure can be observed when the scattering length dependencies on the magnetic field are compared (see Fig. 1a). It is clear that the 2CMbg has no chance to reproduce a sharp turn toward the scattering length zero crossing failing to fulfill the main motivation behind this model (see green dashed line versus gray solid line in Fig. 1a). We emphasize that this conclusion is independent of the applied strategy to fix bare parameters and that we expect the three-body sector of the 2CMbg to miss the NB model system behavior.

In contrast to the previous case, the binding energy of the lower magnetic field resonance ($B=B^{(res)}_1$) is reproduced quite well by the 2CMbg (see the magenta dotted line in Fig. 1b). However, the scattering length behavior still fails to reproduce the position of the zero crossing and shows significant deviations from the model system (see the magenta dotted line versus the gray solid line in Fig. 1a). The reason of this distortion can be attributed to the above-mentioned strong energy level repulsion. Note, though, that this result is not universal. If a different strategy is chosen and the scattering length is fitted instead of the binding energy, the zero crossing will be captured by the 2CMbg. This is, however, irrelevant for the results in the three-body sector which are calculated for the higher magnetic field resonance only.

Figure 1c, d compares the scattering length and the binding energies of 2CMbg to the BN model system. Here, the success of the 2CMbg is obvious in both the binding energies (Fig. 1d) and the scattering length (Fig. 1c) behavior. The binding energy for both resonances follows the model system almost perfectly. More importantly, the scattering length for the higher field resonance reproduces the zero crossing quite precisely. We thus expect that the 2CMbg will be as successful in the three-body sector as it is in the two-body sector. Note that the obtained absolute value of $|\mu |$ (see Table 2, “BN-Res1” and “BN-Res2”) is very close to that of the model system. In the case of the BN system, the repulsion of energy levels is considerably weaker as the bare states are farther apart from each other [24].

Fig. 2

Three-body binding energies for the a–c NB and d–f BN models. Solid (dashed) lines indicate the results of 3CM (2CMbg) for the dimer (gray), Efimov ground (magenta) and the first excited (violet) states. a and d delineate the entire spectrum as a function of the inverse scattering length. In particular in (a), the ground-state trimer extends into the regime of zero scattering length and the lower resonance at $B_{1}^{res}$ (indicated by an arrow). b, e show the $a>0$ regions as a function of the dimer energy. c and f represent the trimer-dimer energy difference in the same region. All energies are normalized to a characteristic energy scale $E^{*}=\hbar ^{2}/(mR^{*})^{2}$ [26]

2.2.2 Three-Body Sector

Figure 2a–c compares the ground and first excited Efimov states calculated within the framework of the 2CMbg (dashed lines) and the 3CM of the NB configuration (solid lines). Efimov spectrum is calculated across the Feshbach resonance centered at $B_2^{(res)}$ only. Figure 2a (Fig. 2b) shows two binding energies as a function of the inverse scattering length (dimer binding energy $E_D$). Figure 2c represents the difference between trimer and dimer binding energies as a function of $E_D$. The failure of the 2CMbg to reproduce the 3CM binding energy of the ground state trimer can be identified either in the positive scattering length region (Fig. 2a) or deep dimer binding energies (Fig. 2b, c). The 2CMbg trimer simply merges with the dimer state (see dashed line), while the 3CM trimer extends across the $B_1^{(res)}$, which is indicated by solid lines in Fig. 2a that exit the figure towards $+\infty$ and reenter it back from $-\infty$. The black arrow in the figure indicates where the ground state Efimov trimer crosses $B_1^{(res)}$ ($1/(a-a_{bg})=0$). It then continues to the lower scattering length region and only then merges with the dimer state (see the shaded area in Fig. 1a, b which indicates the region of extension of the ground state Efimov trimer). The large discrepancy between the merging points can be clearly identified in Fig. 2a and c. Note that the same behavior is observed in comparison of 3CM with 2CM ($a_{bg}=0$) in [24]. Here we see that the inclusion of the background scattering length into the two-channel model does not solve this problem, indicating a failure of the 2CMbg to reproduce the results of the 3CM. Finally, note that the first excited state is well captured by the 2CMbg.

Figure 2d–f compares the ground and first excited Efimov states calculated within the framework of the 2CMbg (dashed lines) and the 3CM for the BN configuration (solid lines). It is clear that the results of both models are nearly identical in this case, and the results of 3CM can be reproduced well by the 2CMbg.

3 Three-Channel Model with Background Scattering

A real-world example of the NB model system is $^7$Li bb-channel [27, 28]. However, the above analysis shows that the NB system is poorly approximated by the 2CMbg, and the 3CM is a better approach. However, comparing the 3CM to the bb-channel reveals significant disagreement with the coupled-channel calculations [24] and the experimental results. Thus, the extension of the 3CM framework to non-zero background interactions has a potential to improve the performance of the model while keeping it minimalistic and avoiding heavy numerical calculations associated with the full treatment of van der Waals interactions. Our goal is to apply this model to the $^7$Li bb-channel, which will be the subject of Sect. 4. Here, we elaborate on the extension of the three-channel model to include background scattering (3CMbg).

3.1 Hamiltonian

The Hamiltonian for the three-channel model with background scattering is ${\hat{H}}^{\text {3bg}}={\hat{H}}_{0}^{\text {3bg}}+{\hat{H}}_{int}^{\text {3bg}}$, where ${\hat{H}}_{int}^{\text {3bg}}={\hat{H}}_{1}^{\text {3bg}}+{\hat{H}}_{2}^{\text {3bg}}$, and the three terms are:

$$\begin{aligned} {\hat{H}}_{0}^{\text {3bg}}= & {} \int \frac{d^{3}k}{\left( 2\pi \right) ^{3}}\left[ \frac{\hbar ^{2}k^{2}}{2\,m}{\hat{a}}_{\vec {k}}^{\dagger }{\hat{a}}_{\vec {k}}+\left( E_{b,1}+\frac{\hbar ^{2}k^{2}}{4\,m}\right) {\hat{b}}_{\vec {k}}^{\dagger }{\hat{b}}_{\vec {k}}+\left( E_{b,2}+\frac{\hbar ^{2}k^{2}}{4\,m}\right) {\hat{c}}_{\vec {k}}^{\dagger }{\hat{c}}_{\vec {k}}\right] \nonumber \\{} & {} +\Lambda _{\text {bg}}\int \frac{d^{3}p}{\left( 2\pi \right) ^{3}}\int \frac{d^{3}k}{\left( 2\pi \right) ^{3}}\int \frac{d^{3}q}{\left( 2\pi \right) ^{3}}\chi _{\vec {q}}^{*}\,\chi _{\vec {p}}\,{\hat{a}}_{\vec {p}+\frac{\vec {k}}{2}}^{\dagger }{\hat{a}}_{-\vec {p}+\frac{\vec {k}}{2}}^{\dagger }{\hat{a}}_{\vec {q}+\frac{\vec {k}}{2}}{\hat{a}}_{-\vec {q}+\frac{\vec {k}}{2}},\end{aligned}$$

(12)

$$\begin{aligned} {\hat{H}}_{1}^{\text {3bg}}= & {} \Lambda _{1}\int \frac{d^{3}k}{\left( 2\pi \right) ^{3}}\int \frac{d^{3}q}{\left( 2\pi \right) ^{3}}\left[ \chi _{\vec {q}}^{*}\,{\hat{b}}_{\vec {k}}^{\dagger }{\hat{a}}_{\vec {q}+\frac{\vec {k}}{2}}{\hat{a}}_{-\vec {q}+\frac{\vec {k}}{2}}+\chi _{\vec {q}}\,{\hat{a}}_{-\vec {q}+\frac{\vec {k}}{2}}^{\dagger }{\hat{a}}_{\vec {q}+\frac{\vec {k}}{2}}^{\dagger }{\hat{b}}_{\vec {k}}\right] , \end{aligned}$$

(13)

$$\begin{aligned} {\hat{H}}_{2}^{\text {3bg}}= & {} \Lambda _{2}\int \frac{d^{3}k}{\left( 2\pi \right) ^{3}}\int \frac{d^{3}q}{\left( 2\pi \right) ^{3}}\left[ \chi _{\vec {q}}^{*}\,{\hat{c}}_{\vec {k}}^{\dagger }{\hat{a}}_{\vec {q}+\frac{\vec {k}}{2}}{\hat{a}}_{-\vec {q}+\frac{\vec {k}}{2}}+\chi _{\vec {q}}\,{\hat{a}}_{-\vec {q}+\frac{\vec {k}}{2}}^{\dagger }{\hat{a}}_{\vec {q}+\frac{\vec {k}}{2}}^{\dagger }{\hat{c}}_{\vec {k}}\right] , \end{aligned}$$

(14)

where $\chi _{\vec {q}}$ retains the same definition as in the 2CMbg (see Eq. (7)). The second term in Eq. (12) models the background scattering as a contact interaction between atoms in the open channel. Furthermore, we set the intermolecular coupling at zero $(\Lambda _{12}=0)$, since it is a redundant parameter and can essentially be set at will (see [24]).

3.2 Two-Body Sector

Solving the Schrödinger equation $({\hat{H}}_{2}^{\text {3bg}}-E)|\psi _{2B}\rangle =0$ with the two-body ansatz $|\psi _{2B}\rangle$ defined in Eq. (A5), leads to the following three coupled equations:

$$\begin{aligned} \left( \frac{\hbar ^{2}k^{2}}{m}-E\right) \alpha _{\vec {k}} +2\Lambda _{bg}\chi _{\vec {k}}\int \frac{d^{3}q}{\left( 2\pi \right) ^{3}}\chi _{\vec {q}}^{*}\,\alpha _{\vec {q}}+\Lambda _{1}\chi _{\vec {k}}\,\beta +\Lambda _{2}\chi _{\vec {k}}\,\gamma =0, \end{aligned}$$

(15)

$$\begin{aligned} \left( E_{b,1}-E\right) \beta +2\Lambda _{1}\int \frac{d^{3}q}{\left( 2\pi \right) ^{3}}\chi _{\vec {q}}^{*}\,\alpha _{\vec {q}}=0, \end{aligned}$$

(16)

$$\begin{aligned} \left( E_{b,2}-E\right) \gamma +2\Lambda _{2}\int \frac{d^{3}q}{\left( 2\pi \right) ^{3}}\chi _{\vec {q}}^{*}\,\alpha _{\vec {q}}=0. \end{aligned}$$

(17)

To determine the scattering amplitude, we follow the same procedure described in 3CM and 2CMbg models (see Appendices A and B). For $E=\frac{\hbar ^{2}k_{0}^{2}}{m}>0$, in analogy to Eq. (B6), the scattering amplitude f(E) satisfies the following equation:

$$\begin{aligned} \frac{m\left| \chi _{\vec {k}_{0}}\right| ^{2}}{4\pi \hbar ^{2}f(E)}=\int \frac{d^{3}k}{\left( 2\pi \right) ^{3}}\frac{\left| \chi _{\vec {k}}\right| ^{2}}{(E-\frac{\hbar ^{2}k^{2}}{m}+i\eta )}-\frac{1}{2\Lambda _{bg}}\frac{\left( E_{b,1}-E\right) }{\left( E_{b,1}-E\right) -\frac{\Lambda _{1}^{2}}{\Lambda _{bg}}-\frac{\Lambda _{2}^{2}}{\Lambda _{bg}}\frac{\left( E_{b,1}-E\right) }{\left( E_{b,2}-E\right) }}. \end{aligned}$$

(18)

It is easy to verify that setting $\Lambda _{2}=0$ in Eq. (18) recovers Eq. (B6), which expresses the scattering amplitude in the 2CMbg. In the absence of open-to-closed channel couplings, i.e., for $\Lambda _{i}=0$ (where $i=\{1,2\}$) and at $E=0$ ($k_{0}=0)$, the scattering amplitude reduces to the background scattering length: $f(E=0)=-a_{bg}$. As a result, $a_{bg}$ for 3CMbg is the same as for 2CMbg (see Eq. (10)). The scattering length is then calculated along the same lines as in 2CMbg (see Appendix B) and yields the following:

$$\begin{aligned} a=a_{bg}\left( 1+\frac{\left( \frac{a_{bg}}{\sqrt{\pi }b\Lambda _{bg}}-\frac{1}{\Lambda _{bg}}\right) \frac{\Lambda _{1}^{2}}{\mu _{1}}\left( B_{\text {2}}-B\right) +\left( \frac{a_{bg}}{\sqrt{\pi }b\Lambda _{bg}}-\frac{1}{\Lambda _{bg}}\right) \frac{\Lambda _{2}^{2}}{\mu _{2}}\left( B_{1}-B\right) }{\left( B_{1}-B\right) \left( B_{\text {2}}-B\right) -\frac{a_{bg}}{\sqrt{\pi }b\Lambda _{bg}}\frac{\Lambda _{2}^{2}}{\mu _{2}}\left( B_{1}-B\right) -\frac{a_{bg}}{\sqrt{\pi }b\Lambda _{bg}}\frac{\Lambda _{1}^{2}}{\mu _{1}}\left( B_{\text {2}}-B\right) }\right) . \end{aligned}$$

(19)

The scattering length can be parametrized as following [27]:

$$\begin{aligned} a=a_{bg}\left( 1-\frac{\Delta B_{1}}{B-B_{1}^{(res)}}-\frac{\Delta B_{2}}{B-B_{2}^{(res)}}\right) , \end{aligned}$$

(20)

where $a_{bg}$, $\Delta B_i$ and $B_i^{(res)}$ are five observable parameters which can be directly related to bare parameters of the model by a direct comparison with Eq. (19).

To complete the two-body sector, Eqs. (15)–(17) are solved for the binding energy ($E=-\hbar ^{2}\lambda _{D}^{2}/m<0$), which is found to satisfy the following equation:

$$\begin{aligned}{} & {} \left( 1-\frac{\Lambda _{1}^{2}}{\Lambda _{bg}\left( E_{b,1}+\hbar ^{2}\lambda _{D}^{2}/m\right) }-\frac{\Lambda _{2}^{2}}{\Lambda _{bg}\left( E_{b,2}+\hbar ^{2}\lambda _{D}^{2}/m\right) }\right) \nonumber \\{} & {} \qquad \left( -\pi \lambda \exp (b^{2}\lambda _{D}^{2})\text {erfc}(\lambda _{D}b)+\frac{\sqrt{\pi }}{b}\right) =-\frac{2\pi ^{2}\hbar ^{2}}{m\Lambda _{bg}}. \end{aligned}$$

(21)

3.3 Three-Body Sector: Efimov Trimers

The trimer binding energy $E_{T}=-\hbar ^{2}\lambda _{T}^{2}/m$ is found by solving the Schrödinger equation $({\hat{H}}- E_{T})|\psi _{3B}\rangle =0$ with the three-body ansatz defined in Eq. (A11), leads to the following coupled equations:

$$\begin{aligned}{} & {} \left( \frac{\hbar ^{2}q^{2}}{m}+\frac{3}{4}\frac{\hbar ^{2}k^{2}}{m}-E\right) \alpha _{\vec {k},\vec {q}}+\Lambda _{1}\chi _{\vec {q}}\,\beta _{\vec {k}}+\Lambda _{2}\chi _{\vec {q}}\,\gamma _{\vec {k}}\nonumber \\{} & {} \qquad +2\Lambda _{bg}\chi _{\vec {q}}\int \frac{d^{3}q}{\left( 2\pi \right) ^{3}}\chi _{\vec {q}}^{*}\left[ \alpha _{\vec {k},\vec {q}}+2\alpha _{\vec {q}-\frac{\vec {k}}{2},-\frac{\vec {q}}{2}-\frac{3\vec {k}}{4}}\right] =0, \end{aligned}$$

(22)

$$\begin{aligned}{} & {} \left( \frac{3}{4}\frac{\hbar ^{2}k^{2}}{m}+E_{b,1}-E\right) \beta _{\vec {k}}+2\Lambda _{1}\int \frac{d^{3}q}{\left( 2\pi \right) ^{3}}\chi _{\vec {q}}^{*}\left[ \alpha _{\vec {k},\vec {q}}+2\alpha _{\vec {q}-\frac{\vec {k}}{2},-\frac{\vec {q}}{2}-\frac{3\vec {k}}{4}}\right] =0, \end{aligned}$$

(23)

$$\begin{aligned}{} & {} \left( \frac{3}{4}\frac{\hbar ^{2}k^{2}}{m}+E_{b,2}-E\right) \gamma _{\vec {k}}+2\Lambda _{2}\int \frac{d^{3}q}{\left( 2\pi \right) ^{3}}\chi _{\vec {q}}^{*}\left[ \alpha _{\vec {k},\vec {q}}+2\alpha _{\vec {q}-\frac{\vec {k}}{2},-\frac{\vec {q}}{2}-\frac{3\vec {k}}{4}}\right] =0. \end{aligned}$$

(24)

Solving this set of equations to find the trimer binding energy, we get:

$$\begin{aligned}{} & {} \left( \frac{1}{\frac{\Lambda _{1}^{2}}{\frac{3}{4}\frac{\hbar ^{2}k^{2}}{m}+E_{b,1}+\hbar ^{2}\lambda ^{2}/m}+\frac{\Lambda _{2}^{2}}{\frac{3}{4}\frac{\hbar ^{2}k^{2}}{m}+E_{b,2}+\hbar ^{2}\lambda ^{2}/m}-\Lambda _{bg}}\right) \beta _{k}^{\text {eff}}\nonumber \\{} & {} \qquad =\frac{m}{\hbar ^{2}}\frac{\beta _{k}^{\text {eff}}}{2\pi ^{2}b}\left( -\pi b\left( \lambda ^{2}+\frac{3}{4}k^{2}\right) ^{\frac{1}{2}}\exp \left( b^{2}\left( \lambda ^{2}+\frac{3}{4}k^{2}\right) \right) \text {erfc}\left( b\left( \lambda ^{2}+\frac{3}{4}k^{2}\right) {}^{\frac{1}{2}}\right) +\sqrt{\pi }\right) \nonumber \\{} & {} \qquad \qquad +\frac{m}{\pi ^{2}\hbar ^{2}}\int dq\beta _{q}^{\text {eff}}q^{2}\exp \left( -\frac{5}{8}q^{^{2}}b^{2}\right) \exp \left( -\frac{5}{8}k^{^{2}}b^{2}\right) \frac{\exp \left( b^{2}\left( \lambda ^{2}+k^{2}+q^{2}\right) \right) }{kq}\nonumber \\{} & {} \qquad \qquad [E_{1}\left( v(-1)\right) -E_{1}\left( v(1)\right) ],\nonumber \\ \end{aligned}$$

(25)

where $E_{1}(z)=\int _{1}^{\infty }ds\frac{e^{-sz}}{s}$ is the exponential integral and

$$\begin{aligned} v(j)=b^{2}\left( \lambda ^{2}+k^{2}+q^{2}+jkq\right) . \end{aligned}$$

(26)

The steps are similar for both the two-channel and the three-channel models. First, we define $\psi (k)=k\beta _{k}^{\text {eff}}$, switch variables using Eq. (B18) and rescale $\psi (\xi )\rightarrow \frac{\psi (\xi )}{\cosh (\xi )}.$ Limiting ourselves to odd solutions in $\psi (\xi )$, we can extend the integration range to $-\infty$ and get the integral equation in the form: $\int _{-\infty }^{\infty }d\xi ^{\prime }{\mathcal {M}}_{\lambda _{T}}(\xi ,\xi ^{\prime })\psi (\xi ^{\prime })=0$ where the matrix elements are given by:

$$\begin{aligned} M\left( \xi ,\xi ^{\prime }\right) =\delta \left( \xi -\xi ^{\prime }\right) F\left( \xi ^{\prime }\right) -\delta \left( \xi -\xi ^{\prime }\right) f\left( \xi ^{\prime }\right) -L\left( \xi ,\xi ^{\prime }\right) , \end{aligned}$$

(27)

where:

$$\begin{aligned}{} & {} F\left( \xi \right) =\frac{\hbar ^{2}}{m\lambda _{T}\cosh \xi } \left[ \left( \frac{\Lambda _{1}^{2}}{\frac{\hbar ^{2}}{m}\lambda _{T}^{2}\cosh ^{2}\xi + \mu _{1}\left( B_{1}-B\right) } \right) + \left( \frac{\Lambda _{2}^{2}}{\frac{\hbar ^{2}}{m}\lambda _{T}^{2}\cosh ^{2}\xi + \mu _{2}\left( B_{2}-B\right) } \right) - \Lambda _{bg} \right] ^{-1},\nonumber \\ \end{aligned}$$

(28)

$$\begin{aligned}{} & {} f\left( \xi \right) =\frac{1}{2\pi ^{2}b}\left[ -\pi b\exp \left( \left( b\lambda _{T}\cosh \xi \right) ^{2}\right) \text {erfc}\left( b\lambda _{T}\cosh (\xi )\right) +\frac{\sqrt{\pi }}{\lambda _{T}\cosh (\xi )}\right] ,\end{aligned}$$

(29)

$$\begin{aligned}{} & {} L\left( \xi ,\xi ^{\prime }\right) =\frac{1}{\sqrt{3}\pi ^{2}}\exp \left( \left( b\lambda _{T}\right) ^{2}\left( 1+\frac{1}{2}\sinh ^{2}(\xi )+\frac{1}{2}\sinh ^{2}(\xi ^{\prime })\right) \right) {{\cdot }}\nonumber \\{} & {} \qquad \left( E_{1}\left( \left( b\lambda _{T}\right) ^{2}\left( t(-1)\right) \right) -E_{1}\left( \left( b\lambda _{T}\right) ^{2}\left( t(1)\right) \right) \right) , \end{aligned}$$

(30)

where t(j) is defined in Eq. (B20). The requirement of a vanishing determinant, $\text {det}M_{\lambda _{T}}(\xi ,\xi ^{\prime })=0$, provides an implicit equation for $\lambda _{T}$ which can be solved numerically as discussed in Appendix A. Among many possible solutions, $\lambda _{T}$ should satisfy the condition $\lambda _{T}>\text {max}\{0,\lambda _{D}\}$.

4 Efimov Scenario in Bosonic Lithium

In this section, we apply the 3CMbg on $^7$Li bb-channel which provides an example of two overlapping narrow Feshbach resonances extensively studied in recent experiments [29,30,31,32,33,34]. An example of an isolated narrow Feshbach resonance is provided by the $^7$Li aa-channel, which has also been experimentally studied in [35,36,37]. We apply the 2CMbg to the latter and compare it with the performance of the 3CMbg on the bb-channel. For both models, we fix the magnetic moment to $\mu =-2.66$ MHz/G for the aa-channel and $\mu _{1}=\mu _{2}=\mu$ for the bb-channel to match the coupled-channel calculations. The characteristic range of the interactions of $^7$Li atoms (the van der Waals length $r_{\text {vdW}}=(mC_6/\hbar ^2)^{1/4}/2=32a_0$, where $C_6$ is the dispersion coefficient and $a_0$ is the Bohr radius) and the background scattering length $a_{bg}$ are small.

Fig. 3

Two-body sector of $^{7}$Li bb-channel: a the scattering length, and b the dimer binding energy, as obtained from the 3CMBg model (dashed magenta) are compared with the exact coupled-channel calculations (solid gray)

4.1 $^{7}$Li bb-channel

The bb-channel ($|F=1,m_f=0\rangle$) features two Feshbach resonances characterized by the dimensionless strength parameters $s_{\text {res},1}=0.0411$ ($R_{1}^{*}=722 a_{0}$) and $s_{\text {res},2}=0.493$ ($R_{2}^{*}=60 a_{0}$) [27]. $s_{\text {res},1}\ll s_{\text {res},2}<1$ ($R_{1}^{*}\gg R_{2}^{*}>r_{\text {vdW}}$) indicates that both resonances are narrow and can be classified as NB systems.

4.1.1 Two-Body Sector

To determine the bare parameters of our model, we fit the coupled-channel calculations scattering length with Eq. (20) as shown in Fig. 3a, and summarize the fitting parameters in Table 3. Then, the analytic expressions shown in Appendix C are used to extract the bare parameters of the model. However, there are only four equations for five bare parameters, and we are free to choose one of them to complete the settings by fitting the binding energy. The most reasonable choice is to fit b as it signifies the range of the model interaction potential. It is expected though that the Gaussian approximation of the real van der Waals potential will be applicable to a limited range only. To verify the range of applicability, we chose to fit the position of the Feshbach resonance $B_{2}^{\text {(res)}}$ as an additional parameter. Figure 3b shows the result of the binding energy fitting with the fitting range limited to $-5$ MHz which is the maximal range we used here. Very good agreement with the coupled-channel calculations is confirmed for binding energies down to $\sim -10$ MHz. We obtain the position of the “broad” resonance at $B_{2}^{\text {(res)}} = 893.83$ G which deviates from the position defined by the scattering length by a tiny amount of $\sim 5\times 10^{-5}$ G (see Table 3). The corresponding potential range is $b = 23.76\,a_{0}$ (see Table 4). Then, using the analytical expressions outlined in Appendix C, the bare parameters are determined and shown in Table 4.

Table 3 Fitting parameters of the $^{7}$Li coupled-channel calculations

Table 4 Bare parameters of the 3CMBg for $^{7}$Li bb- and aa-channels

4.1.2 Three-Body Sector

Once the parameters are fixed in the two-body sector, the Efimov trimer energies can be calculated according to Sect. 3.3. Our primary interest lies in the determination of the scattering length $a_{-}$, which corresponds to the intersection of the ground state Efimov trimer with the three-body continuum, and $a_{*}^{(1)}$, marking the merging of the first excited Efimov trimer state with the atom-dimer continuum. These Efimov resonances can be directly compared with the experimental results.

As discussed, the value of the parameter b depends on the fitting range, which affects other parameters and ultimately the position of the Efimov resonances. Figure 4a and b show the positions of $a_{-}$ and $a_{*}^{(1)}$ as a function of the fitting ranges down to $-5$ MHz. The general trend for both resonances to move to smaller absolute values is in agreement with [26] which indicates that the final background scattering length leads to the same qualitative behavior. However, the experimental value $a_{-}=-280\, a_{0}$ [30] is impossible to achieve with this theory. Comparison of the experimental value of $a_{*}^{(1)}= 170\, a_{0}$ [31, 34] leads to the same conclusion (see Fig. 4b). The quality of the approximation is visible in Fig. 4c where $\Delta B^{(res)}_2=(B^{(res)}_2|_{E_{D}=0}-B^{(res)}_2|_{a=\infty })$ is plotted. The fitting of the binding energy is limited to $-5$ MHz because $\Delta B^{(res)}_2$ exceeds the fitting error of the resonance position extracted from the scattering length (see Fig. 3c), which is equal to 40 mG [28]. To correctly predict the position of Efimov resonances, the use of the van der Waals potential is unavoidable, leading to heavy multi-channel numerical calculations [23]. In conclusion, the Feshbach resonance in the bb-channel is not narrow enough to allow for simplified treatment within the 3CMBg framework to provide us with quantitative predictions.

Fig. 4

$^{7}$Li bb-channel: variations of a $a_{-}$ and b $a_{*}^{(1)}$ as a function of the binding energy fitting range derived from the 3CMbg model; c $\Delta B^{(res)}_2=(B^{(res)}_2|_{E_{D}=0}-B^{(res)}_2|_{a=\infty })$ and d change in the interaction range b as a function of the fitting range

4.2 $^{7}$Li aa-channel

$^7$Li aa-channel ($|F=1, m_f=1\rangle$), having an isolated Feshbach resonance, can be treated within the 2CMbg framework. The resonance is characterized by the $s_{\text {res}}= 0.56$ ($R^*=54.6\, a_{0}$) which indicates that it is rather similar to the “broad” resonance in the bb-channel.

The bare parameters of the model are obtained in the same way as before. The fitting of the coupled-channel calculations is shown in Fig. 5a and b and the parameters are summarized in Table 3 while the derived bare parameters of the model are found in Table 4. The agreement of the 2CMbg with the coupled-channel calculations is excellent as can be seen especially well in the fitting of the binding energy which is done down to $-5$ MHz but shown in Fig. 5b all the way down to $-200$ MHz ($\sim 40\%$ of the van der Waals energy).

Fig. 5

Two-body sector of $^{7}$Li aa-channel: a the scattering length, and b the dimer binding energy, as obtained from the 2CMbg model (dashed magenta) are compared with the exact coupled-channel calculations (solid gray)

Despite this excellent agreement in the two-body sector, the three-body energy levels miss the experimental results even more significantly than for the case of the bb-channel. Figure 6 illustrates the position of the Efimov resonance ($a_{-}$) obtained as a function of the fitting range. As before, we find that the values are far from the experimental results reported in [31], where $a_{-}$ was estimated to be $a_{-} \approx -200\,a_{0}$. Interestingly, the prediction for the bb-channel agrees somewhat better with the experiment compared to the aa-channel. We thus speculate that this might be attributed to the influence of the second, truly narrow resonance for which the 3CMbg is expected to work better.

Fig. 6

$^{7}$Li aa-channel: variation of $a_{-}$ as a function of the binding energy fitting range derived from the 2CMbg model

5 Conclusions

In conclusion, we examine how the three-channel model (3CM) developed to describe two overlapping Feshbach resonances compares with the two-channel model extended to include background scattering (2CMbg) for different model systems. Specifically, we find that while 2CMbg is a viable substitute for 3CM for the broad-narrow (BN) model system, it fails to correctly reproduce the 3CM for the NB one.

To further investigate the contribution of the background scattering length, we extend the 3CM framework to include the background interactions (3CMbg) and apply it to the $^7\text {Li}$ bb-channel. Furthermore, we apply the 2CMbg model to the $^7\text {Li}$ aa-channel and compare it to the performance of the 3CMbg for the bb-channel. Despite the simplicity of our models, we find good agreement with the coupled-channel calculations of the binding energies in the two-body sector. Although inclusion of the background scattering length improves the performance of models in the three-body sector, significant discrepancies remain between the predicted and experimentally found positions of the Efimov resonances. These discrepancies can be attributed to the intermediate character of the relevant $^7$Li Feshbach resonances ($s_{\text {res}}\sim 1$). Finally, we note a curious observation that the 3CMbg for the bb-channel performs better than the 2CMbg for the aa-channel which might be attributed to the influence of the second resonance in the overlapping pair of resonances in the bb-channel.

Appendices

Appendix A: Three-Channel Model

Here we briefly summarize the main results of the three-channel model [24]. The Hamiltonian of the system consists of an open channel, the two closed channels, and the interaction term: ${\hat{H}}={\hat{H}}_{0}+{\hat{H}}_{int}$, where ${\hat{H}}_{int}={\hat{H}}_{1}+{\hat{H}}_{2}+{\hat{H}}_{12}$. The first term ${\hat{H}}_{0}$ is the bare Hamiltonian of the three channels:

$$\begin{aligned} {\hat{H}}_{0}=\int \frac{d^{3}k}{(2\pi )^{3}}\left[ \frac{\hbar ^{2}k^{2}}{2m}{\hat{a}}_{\vec {k}}^{\dagger }{\hat{a}}_{\vec {k}}+\left( E_{b,1}+\frac{\hbar ^{2}k^{2}}{4m}\right) {\hat{b}}_{\vec {k}}^{\dagger }{\hat{b}}_{\vec {k}}\right. +\left. \left( E_{b,2}+\frac{\hbar ^{2}k^{2}}{4m}\right) {\hat{c}}_{\vec {k}}^{\dagger }{\hat{c}}_{\vec {k}}\right] , \end{aligned}$$

(A1)

where ${\hat{a}}_{\vec {k}}$ annihilates free particles of mass m, ${\hat{b}}_{\vec {k}}({\hat{c}}_{\vec {k}})$ annihilates molecules in the first (second) channel, and $E_{b,1}(E_{b,2})$ is the bare energy of the first (second) molecular channel: $E_{b,i}=\mu _{i}\left( B_{i}-B\right)$. Without loss of generality, we assume $B_{1}<B_{2}$. Both molecular channels are coupled to the open channel by means of:

$$\begin{aligned} {\hat{H}}_{1}=\Lambda _{1}\int \frac{d^{3}k}{\left( 2\pi \right) ^{3}}\int \frac{d^{3}q}{\left( 2\pi \right) ^{3}}\left[ {\hat{b}}_{\vec {k}}^{\dagger }{\hat{a}}_{\vec {q}+\frac{\vec {k}}{2}}{\hat{a}}_{-\vec {q}+\frac{\vec {k}}{2}}+{\hat{a}}_{-\vec {q}+\frac{\vec {k}}{2}}^{\dagger }{\hat{a}}_{\vec {q}+\frac{\vec {k}}{2}}^{\dagger }{\hat{b}}_{\vec {k}}\right] , \end{aligned}$$

(A2)

$$\begin{aligned} {\hat{H}}_{2}=\Lambda _{2}\int \frac{d^{3}k}{\left( 2\pi \right) ^{3}}\int \frac{d^{3}q}{\left( 2\pi \right) ^{3}}\left[ {\hat{c}}_{\vec {k}}^{\dagger }{\hat{a}}_{\vec {q}+\frac{\vec {k}}{2}}{\hat{a}}_{-\vec {q}+\frac{\vec {k}}{2}}+{\hat{a}}_{-\vec {q}+\frac{\vec {k}}{2}}^{\dagger }{\hat{a}}_{\vec {q}+\frac{\vec {k}}{2}}^{\dagger }{\hat{c}}_{\vec {k}}\right] . \end{aligned}$$

(A3)

In addition, the two molecular channels are coupled to each other via:

$$\begin{aligned} {\hat{H}}_{12}=\Lambda _{12}\int \frac{d^{3}k}{\left( 2\pi \right) ^{3}}\left[ {\hat{c}}_{\vec {k}}^{\dagger }{\hat{b}}_{\vec {k}}+{\hat{b}}_{\vec {k}}^{\dagger }{\hat{c}}_{\vec {k}}\right] . \end{aligned}$$

(A4)

1.1 1. Two-Body Observables

To describe the two-body sector we solve the Schrödinger equation $({\hat{H}}-E)|\psi _{2B}\rangle =0$ with the most general two-body ansatz in the center-of-mass frame:

$$\begin{aligned} |\psi _{2B}\rangle =\gamma {\hat{c}}_{\vec {k}=0}^{\dagger }|0\rangle +\beta {\hat{b}}_{\vec {k}=0}^{\dagger }|0\rangle +\int \frac{d^{3}k}{(2\pi )^{3}}\alpha _{\vec {k}}{\hat{a}}_{\vec {k}}^{\dagger }{\hat{a}}_{-\vec {k}}^{\dagger }|0\rangle . \end{aligned}$$

(A5)

Once plugged into the Schrödinger equation the following system of coupled equations is obtained:

$$\begin{aligned} \left( \frac{\hbar ^{2}k^{2}}{m}-E\right) \alpha _{\vec {k}}+\Lambda _{1}\beta +\Lambda _{2}\gamma =0, \end{aligned}$$

(A6)

$$\begin{aligned} \left( E_{b,1}-E\right) \beta +2\Lambda _{1}\int \frac{d^{3}q}{\left( 2\pi \right) ^{3}}\alpha _{\vec {q}}+\Lambda _{12}\gamma =0, \end{aligned}$$

(A7)

$$\begin{aligned} \left( E_{b,2}-E\right) \gamma +2\Lambda _{2}\int \frac{d^{3}q}{\left( 2\pi \right) ^{3}}\alpha _{\vec {q}}+\Lambda _{12}\beta =0. \end{aligned}$$

(A8)

1.1.1 a. Scattering Length

The open channel coefficient $\alpha _{\vec {k}}$ can be extracted from Eq. (A6) when positive energy is considered $E=\hbar ^{2}k_{0}^{2}/m>0$. By comparing the expression for $\alpha _{\vec {k}}$ with the scattering Green’s function, the scattering amplitude $f_{k_{0}}$ can be obtained in terms of the molecular amplitudes $\beta$ and $\gamma$:

$$\begin{aligned} f_{k_{0}}=-\frac{m}{4\pi \hbar ^{2}}\left( \Lambda _{1}\beta +\Lambda _{2}\gamma \right) . \end{aligned}$$

(A9)

Furthermore, the coupled Eqs. (A6, A7, A8) can be numerically solved to obtain $\beta$ and $\gamma$ as functions of the bare parameters such as coupling strengths $\Lambda _{1}$, $\Lambda _{2}$ and bare molecular energies $E_{b,1}$, $E_{b,2}$. Thus, by combining these numerical solutions for $\beta$ and $\gamma$ with Eq. (A9) the full scattering amplitude $f_{k_{0}}$ can be determined solely in terms of the microscopic bare parameters of the model (see Eq. (6) in [24]).

1.1.2 b. Binding Energy

For the binding energy solution, the coupled Eqs. (A6)–(A8) are solved assuming $E=-\hbar ^{2}\lambda _{D}^{2}/m<0$ $(\lambda _{D}>0)$ (See Eq. (8) [24]).

1.1.3 c. Relating the Bare Parameters to the Observable Parameters

Observable parameters such as resonance positions $B_{i}^{(res)}$ and widths $\Delta _{i}$ can be explicitly related to the bare parameters of the model, including coupling strengths $\Lambda _{i}$ and bare molecular energies $E_{b,i}$. The analytical expression for the scattering length a(B) can be derived in terms of the bare parameters at the zero energy limit ($f_{k_{0}=0}=-a$). Then it can be compared to a familiar parametrization form of the coupled-channels calculations of multiple Feshbach resonances [27]:

$$\begin{aligned} a(B)=\frac{\Delta _{1}}{B_{1}^{(res)}-B}+\frac{\Delta _{2}}{B_{2}^{(res)}-B}. \end{aligned}$$

(A10)

establishing the relation between the observable parameters of Eq. (A10) and the underlying bare quantities of the model (see appendix in [24]). Since there are five bare parameters $(\Lambda _{1},\Lambda _{2},\Lambda _{12},B_{1},B_{2})$ but only four observables $(\Delta _{1},\Delta _{2},B_{1}^{(res)},B_{2}^{(res)})$, one bare parameter, such as the intermolecular coupling $\Lambda _{12}$ remains unconstrained. As shown in [24], it can be determined at will within a certain limit (for example set to zero) or constrained by another known quantity (such as bare resonance spacing). This choice, of course, influences the values of the other parameters.

1.2 2. Three-Body Sector: Efimov Trimers

Having determined the bare parameters in the two-body sector, we now calculate the Efimov energy levels in the three-body sector without additional adjustments.

The trimer binding energy $E_{T}=-\hbar ^{2}\lambda _{T}^{2}/m$ is determined by solving the Schrödinger equation $({\hat{H}}-E_{T})|\psi _{3B}\rangle =0$, provided the condition $\lambda _{T}>\text {max}\{0,\lambda _{D}\}$ is satisfied. The latter signifies that only trimers associated with the higher magnetic field resonance are properly determined by the solution. The trimers of the lower resonances are embedded into the atom-dimer continuum of the higher resonance.

Substitution of $|\psi _{3B}\rangle$:

$$\begin{aligned} |\psi _{3B}\rangle =\int \frac{d^{3}k}{\left( 2\pi \right) ^{3}}\gamma _{\vec {k}}{\hat{c}}_{\vec {k}}^{\dagger }{\hat{a}}_{-\vec {k}}^{\dagger }|0\rangle +\int \frac{d^{3}k}{\left( 2\pi \right) ^{3}}\beta _{\vec {k}}{\hat{b}}_{\vec {k}}^{\dagger }{\hat{a}}_{-\vec {k}}^{\dagger }|0\rangle +\int \frac{d^{3}k}{\left( 2\pi \right) ^{3}}\int \frac{d^{3}q}{\left( 2\pi \right) ^{3}}\alpha _{\vec {k},\vec {q}}{\hat{a}}_{\vec {q}+\frac{\vec {k}}{2}}^{\dagger }{\hat{a}}_{-\vec {q}+\frac{\vec {k}}{2}}^{\dagger }{\hat{a}}_{-\vec {k}}^{\dagger }|0\rangle \nonumber \\ \end{aligned}$$

(A11)

into the Schrödinger equation leads to three coupled equations:

$$\begin{aligned} \left( \frac{\hbar ^{2}q^{2}}{m}+\frac{3}{4}\frac{\hbar ^{2}k^{2}}{m}-E_{T}\right) \alpha _{\vec {k},\vec {q}}+\Lambda _{1}\beta _{\vec {k}}+\Lambda _{2}\gamma _{\vec {k}}=0, \end{aligned}$$

(A12)

$$\begin{aligned} \left( \frac{3}{4}\frac{\hbar ^{2}k^{2}}{m}+E_{b,1}-E_{T}\right) \beta _{\vec {k}}+\Lambda _{12}\gamma _{\vec {k}} +2\Lambda _{1}\int \frac{d^{3}q}{\left( 2\pi \right) ^{3}}\left[ \alpha _{\vec {k},\vec {q}}+2\alpha _{\vec {q}-\frac{\vec {k}}{2},-\frac{\vec {q}}{2}-\frac{3\vec {k}}{4}}\right] =0, \end{aligned}$$

(A13)

$$\begin{aligned} \left( \frac{3}{4}\frac{\hbar ^{2}k^{2}}{m}+E_{b,2}-E_{T}\right) \gamma _{\vec {k}}+\Lambda _{12}\beta _{\vec {k}} +2\Lambda _{2}\int \frac{d^{3}q}{\left( 2\pi \right) ^{3}}\left[ \alpha _{\vec {k},\vec {q}}+2\alpha _{\vec {q}-\frac{\vec {k}}{2},-\frac{\vec {q}}{2}-\frac{3\vec {k}}{4}}\right] =0, \end{aligned}$$

(A14)

which are reduced to two by eliminating the free particle amplitude $\alpha _{\vec {k},\vec {q}}$.

To solve the coupled equations, a high momentum cutoff $k_{c}$ is introduced to regularize potential divergences. The quantities are renormalized to be dimensionless, denoted by a tilde, except for the magnetic moment $\mu$, which is renormalized to units of 1/G.

Defining $\psi (k)=(\beta _{k},\gamma _{k})^{T}$, the procedure is similar to that of the two-channel model [7]. The amplitudes are rescaled as ${\tilde{\beta }}_{k}\rightarrow \frac{{\tilde{\beta }}_{k}}{{\tilde{k}}},{\tilde{\gamma }}_{k}\rightarrow \frac{{\tilde{\gamma }}_{k}}{{\tilde{k}}}.$ The variables are then changed to $(\xi ,\xi ^{\prime })$ using the variables switch in Eq. (B18) and rescaled again using ${\tilde{\beta }}_{\xi }\rightarrow \frac{{\tilde{\beta }}_{\xi }}{\cosh \xi },{\tilde{\gamma }}_{\xi }\rightarrow \frac{{\tilde{\gamma }}_{\xi }}{\cosh \xi }$. After these steps, the Schrödinger equation takes the form:

$$\begin{aligned} \int _{-\infty }^{\infty }d\xi ^{\prime }\begin{pmatrix}{\mathcal {M}}_{11}\left( \xi ,\xi ^{\prime }\right) &{} {\mathcal {M}}_{12}\left( \xi ,\xi ^{\prime }\right) \\ {\mathcal {M}}_{21}\left( \xi ,\xi ^{\prime }\right) &{} {\mathcal {M}}_{22}\left( \xi ,\xi ^{\prime }\right) \end{pmatrix}\begin{pmatrix}{\tilde{\beta }}_{\xi }\\ {\tilde{\gamma }}_{\xi } \end{pmatrix}=0. \end{aligned}$$

(A15)

The matrix elements $({\mathcal {M}}_{\lambda _{T}})_{ij}$ are given by:

$$\begin{aligned} ({\mathcal {M}}_{\lambda _{T}})_{ij}=[\delta _{ij}(f_{i}(\xi ')-h(\xi '))+h(\xi ')-{\tilde{\Lambda }}_{i}{\tilde{\Lambda }}_{j}g(\xi ')]\delta (\xi -\xi ') -\tilde{\Lambda _{i}}{\tilde{\Lambda }}_{j}L(\xi ,\xi '), \end{aligned}$$

(A16)

where we have defined:

$$\begin{aligned} f_{i}\left( \xi \right) ={\tilde{\lambda }}_{T}\cosh \xi +\frac{\tilde{\mu _{i}}}{{\tilde{\lambda }}_{T}\cosh \xi }\left( B_{i}-B\right) , \end{aligned}$$

(A17)

$$\begin{aligned} g\left( \xi \right) =\frac{1}{\pi ^{2}}\left( \frac{1}{{\tilde{\lambda }}_{T}\cosh \xi }-\frac{\pi }{2}\right) , \end{aligned}$$

(A18)

$$\begin{aligned} h\left( \xi \right) =\frac{{\tilde{\Lambda }}_{12}}{{\tilde{\lambda }}_{T}\cosh \xi }, \end{aligned}$$

(A19)

$$\begin{aligned} L\left( \xi ,\xi ^{\prime }\right) =\frac{2}{\sqrt{3}\pi ^{2}}\ln \left( \frac{e^{2\left( \xi -\xi ^{\prime }\right) }+e^{\xi -\xi ^{\prime }}+1}{e^{2\left( \xi -\xi ^{\prime }\right) }-e^{\xi -\xi ^{\prime }}+1}\right) . \end{aligned}$$

(A20)

Requiring the determinant to vanish, $\det {\mathcal {M}}_{\lambda _{T}}(\xi ,\xi ^{\prime })=0,$ provides an implicit equation for the binding energies of the Efimov trimer. The mathematical solutions to this equation must be validated by analyzing the associated eigenfunctions. Only odd eigenfunctions with the correct nodal structure represent the physical solution. This procedure identifies the spectrum of the bound state [24].

Appendix B: Two-Channel Model with Background

The problem is formulated in Sect. 2.1.3. Here we show a solution of the Schrödinger equation in two- and three-body sectors following [15].

1.1 1. Two-Body Sector

In the two-body sector we substitute the following ansatz in the Schrödinger equation $({\hat{H}}^{\text {2bg}}-E)|\psi _{2B}\rangle =0$, where ${\hat{H}}^{\text {2bg}}$ is defined in Sect. 2.1.3:

$$\begin{aligned} |\psi _{2B}\rangle =\int \frac{d^{3}k}{(2\pi )^{3}}\alpha _{\vec {k}}\hat{a}_{\vec {k}}^{\dagger }\hat{a}_{\vec {-k}}^{\dagger }|0\rangle +\beta {\hat{b}}_{\vec {k}=0}^{\dagger }|0\rangle . \end{aligned}$$

(B1)

Solving the equation leads to a system of two coupled integral equations for amplitudes $\alpha _{\vec {k}}$ and $\beta$:

$$\begin{aligned} \left( \frac{\hbar ^{2}k^{2}}{m}-E\right) \alpha _{\vec {k}}+\Lambda \chi _{\vec {k}}\,\beta +2\Lambda _{bg}\chi _{\vec {k}}\int \frac{d^{3}q}{(2\pi )^{3}}\chi _{\vec {q}}^{*}\,\alpha _{\vec {q}}=0, \end{aligned}$$

(B2)

$$\begin{aligned} \left( E_{b}-E\right) \beta +2\Lambda \int \frac{d^{3}q}{(2\pi )^{3}}\chi _{\vec {q}}^{*}\,\alpha _{\vec {q}}=0, \end{aligned}$$

(B3)

where $E_b$ is the bare molecular level.

1.1.1 a. Scattering Amplitude

To compute the scattering amplitude, we express $\alpha _{\vec {k}}$ from Eqs. (B2) and (B3) and compare it with the Green scattering function for $E=\frac{\hbar ^{2}k_{0}^{2}}{m}>0$:

$$\begin{aligned} \alpha _{\vec {k}}=\left( 2\pi \right) ^{3}\delta \left( \vec {k}-\vec {k}_{0}\right) +\frac{\Lambda _{bg}\chi _{k}}{\Lambda }\frac{E-E_{b}+\frac{\Lambda ^{2}}{\ \Lambda _{bg}}}{E-\frac{\hbar ^{2}k^{2}}{m}+i\eta }\beta . \end{aligned}$$

(B4)

Comparing Eq. (B4) to the scattering theory allows determination of the scattering amplitude f(E):

$$\begin{aligned} f(E) =-\frac{m}{4\pi \hbar ^2}\frac{\Lambda _{bg}\chi _{k}}{\Lambda }(E-E_{b}+\frac{\Lambda ^{2}}{\Lambda _{bg}})\beta . \end{aligned}$$

(B5)

If amplitude $\beta$ is expressed explicitly, f(E) satisfies the following relation:

$$\begin{aligned} \frac{m\left| \chi _{k_0}\right| ^{2}}{4\pi \hbar ^{2}f(E)}=\int \frac{d^{3}k}{\left( 2\pi \right) ^{3}}\frac{\left| \chi _{k}\right| ^{2}}{(E-\frac{\hbar ^{2}k^{2}}{m}+i\eta )} +\frac{\Lambda ^{2}}{2\Lambda _{bg}^{2}(E-E_{b} +\frac{\Lambda ^{2}}{\Lambda _{bg}})}-\frac{1}{2\Lambda _{bg}}. \end{aligned}$$

(B6)

1.1.2 b. Relating the Bare Parameters to Observable Parameters

Here we link the observable parameters $({{a_{bg}}, \Delta B, B^{(res)}, E_{D}})$ to the model parameters $({\Lambda _{bg}, \Lambda , b, B_{0}, \mu )}$. In the limit of the vanishing coupling strength between channels ($\Lambda =0$), the scattering amplitude reduces to the background scattering length: $f\left( E\rightarrow 0\right) =-a_{\text {bg}}$. Then, from Eq. (B6) one derives Eq. (10) of the main text.

The scattering amplitude which includes the resonance contribution ($\Lambda \ne 0$) is calculated for negative energies $E=\frac{\hbar ^{2}k_{0}^{2}}{m}<0$ and extended to positive energies by the standard analytical continuation ($k_{0}=iq$ with $q>0$). The scattering amplitude in this limit converges to the scattering length $f\left( E\right) =-a$ and Eq. (B6) gives:

$$\begin{aligned} \frac{1}{f(E)}=q\,\text {erfc}(bq)-\frac{\exp \left( -b^{2}q^{2}\right) }{a_{bg}}\left( 1-\frac{\frac{2\pi \hbar ^{2}\Lambda ^{2}a_{bg}}{m\Lambda _{bg}^{2}}}{E-E_{b}+\frac{\Lambda ^{2}}{\Lambda _{bg}}}\right) . \end{aligned}$$

(B7)

The scattering length expression (Eq. (9) of the main text) is derived directly from this expression and when compared to the well-known parametrization equation (Eq. (1)), the desired relations between the observables and model parameters can be obtained:

$$\begin{aligned} \Delta B=-\frac{2\pi \hbar ^{2}\Lambda ^{2}a_{bg}}{\mu m\Lambda _{bg}^{2}}, \end{aligned}$$

(B8)

$$\begin{aligned} B^{(res)}=B_{0}-\frac{\Delta B}{1-b\frac{\sqrt{\pi }}{a_{bg}}}. \end{aligned}$$

(B9)

1.1.3 c. Binding Energy

The binding energy (denoted $E_{D}=\frac{\hbar ^{2}q_{D}^{2}}{m}>0$) is given by the poles of the scattering amplitude of Eq. (B7) with negative energy $(E=-E_{dim}<0)$ leading to the expression for $q_D$ (Eq. (11) in the main text). Denoting the $R^{\star }$ parameter as a new length scale related to the resonance strength [1]:

$$\begin{aligned} R^{\star }=\frac{\hbar ^{2}}{m|a_{\text {bg}}\mu \Delta B|}, \end{aligned}$$

(B10)

Equation (11) can be renormalized as following:

$$\begin{aligned} \frac{q_{D}^{*}\exp \left( \left( q_{D}^{*}b^*\right) ^{2}\right) \text {erfc}\left( q_{D}^{*}b^*\right) }{q_{D}^{*}a_{bg}^*\exp \left( \left( q_{D}^{*}b^*\right) ^{2}\right) \text {erfc}\left( q_{D}^{*}b^*\right) -1}+(-1)^j q_{D}^{*2}=\frac{1}{a_{bg}^*-a^*} \end{aligned}$$

(B11)

where $q_{D}^*=q_{D} R^*$, $b^*=b/R^*$, $a_{bg}^*=a_{bg}/R^*$, $a^*=a/R^*$ and $j=0$ if $\frac{\hbar ^{2}}{ma_{bg}\mu \Delta B} < 0$, or $j=1$ if $\frac{\hbar ^{2}}{ma_{bg}\mu \Delta B} > 0$. Note that these dimensionless quantities signify different types of Feshbach resonances. Specifically, $|a_{bg}^*|\ge 1$ corresponds to a broad resonance while $|a_{bg}^*|\ll 1$ signifies a narrow resonance. $|a_{bg}|\gg b$ denotes a shape resonance [15].

1.2 2. Three-Body Sector: Efimov Trimers

Having fixed the bare parameters in the two-body sector, we proceed to the three-body sector with no more adjustable parameters. To find the binding energy of the Efimov trimers, we look for a negative energy solution $E=-\frac{\hbar ^{2}\lambda _{T}^{2}}{m}$ where $\lambda _{T}>\text {max}\{0,\lambda _{D}\}$, of $({\hat{H}}^{2\text {bg}}-E)|\psi _{3B}\rangle =0$ with:

$$\begin{aligned} {}|\psi _{3B}\rangle= & {} \int \frac{d^{3}k}{(2\pi )^{3}}\beta _{\vec {k}}{\hat{b}}{}_{\vec {k}}^{\dagger }{\hat{a}}_{-\vec {k}}^{\dagger }|0\rangle \nonumber \\{} & {} +\int \frac{d^{3}k}{(2\pi )^{3}}\int \frac{d^{3}q}{(2\pi )^{3}}\alpha _{\vec {k},\vec {q}}{\hat{a}}_{\vec {q}+\frac{\vec {k}}{2}}^{\dagger }{\hat{a}}_{-\vec {q}+\frac{\vec {k}}{2}}^{\dagger }{\hat{a}}_{-\vec {k}}^{\dagger }|0\rangle . \end{aligned}$$

(B12)

We work in the center of mass frame and have chosen $\vec {k}$ and $\vec {q}$ to be a set of Jacobi momenta. The Schrödinger equation leads to the coupled equations:

$$\begin{aligned} (E_{b}-E_{rel})\beta _{\vec {k}} +2\Lambda \frac{1}{(2\pi )^{3}}\int d^{3}q\chi _{\vec {q}}^{*}\left( \alpha _{\vec {k},\vec {q}}+2\alpha _{\vec {q}-\frac{\vec {k}}{2},-\frac{\vec {q}}{2}-\frac{3}{4}\vec {k}}\right) =0, \end{aligned}$$

(B13)

$$\begin{aligned} \varLambda \chi _{\vec {q}}\beta _{\vec {k}}+\alpha _{\vec {k},\vec {q}}\left( \frac{\hbar ^{2}q^{2}}{m}-E_{rel}\right) +2\Lambda _{bg}\frac{1}{(2\pi )^{3}}\chi _{\vec {q}}\int d^{3}q\chi _{\vec {q}}^{*}\left( \alpha _{\vec {k},\vec {q}}+2\alpha _{\vec {q}-\frac{\vec {k}}{2},-\frac{\vec {q}}{2}-\frac{3}{4}\vec {k}}\right) =0, \end{aligned}$$

(B14)

where $E_{rel}=E-\frac{3}{4}\frac{\hbar ^{2}k^{2}}{m}$ is the collisional (or relative) energy of an atomic pair of total momentum $\vec {k}$. To simplify Eqs. (B13, B14), we introduce a dressed atom-molecule wave function $\beta _{\vec {k}}^{\text {eff}}$ [15]:

$$\begin{aligned} \beta _{\vec {k}}^{\text {eff}}=\frac{\Lambda _{bg}\beta _{\vec {k}}}{\Lambda ^{2}}[E_{rel}-E_{b}+\frac{\Lambda ^{2}}{\Lambda _{bg}}]. \end{aligned}$$

(B15)

Substituting Eq. (B15) into Eqs. (B13, B14) leads to:

$$\begin{aligned} \left( \frac{\hbar ^{2}q^{2}}{m}-E_{rel}\right) \alpha _{\vec {k},\vec {q}}+\Lambda \chi _{\vec {q}}\beta _{\vec {k}}^{\text {eff}}=0. \end{aligned}$$

(B16)

One eliminates $\alpha _{\vec {k},\vec {q}}$ from Eq. (B16) and insert it in Eq. (B13) to obtain a closed integral equation verified by the effective atom-molecule wave function:

$$\begin{aligned} \frac{\exp \left( -mE_{rel}b^{2}/\hbar ^{2}\right) }{4\pi f\left( E_{rel}\right) }\beta _{\vec {k}}^{\text {eff}} = \frac{2}{(2\pi )^{3}}\int d^{3}q\frac{\beta _{\vec {q}}^{\text {eff}}\exp \left( -\left( \vec {q}+\frac{\vec {k}}{2}\right) ^{2} \frac{b^{2}}{2}\right) \exp \left( -\left( \vec {k}+\frac{\vec {q}}{2}\right) ^{2}\frac{b^{2}}{2}\right) }{\lambda _{T}^{2}+q^{2}+\vec {q} \cdot \vec {k}+k^{2}}. \end{aligned}$$

(B17)

To solve Eq. (B17) we define $\psi (k)=k\beta _{k}^{\text {eff}}$ and change variables using:

$$\begin{aligned} {k=\frac{2}{\sqrt{3}}\lambda _{T}\textrm{sinh}(\xi ), q=\frac{2}{\sqrt{3}}\lambda _{T}\mathrm{\sinh }(\xi ^{\prime }).} \end{aligned}$$

(B18)

Then we rescale $\psi (\xi )\rightarrow \frac{\psi (\xi )}{\cosh (\xi )}$ and look only to odd solutions $\psi (\xi )$. The lower integration limit may be extended to $-\infty$ provided we divide the entire integral by two:

$$\begin{aligned}{} & {} -\frac{2}{\pi \sqrt{3}}\int _{-\infty }^{\infty }d\xi ^{\prime }\psi (\xi ^{\prime })\exp \left( b^{2}\lambda _{T}^{2}\left( 1+\frac{1}{2}\sinh ^{2}(\xi )+\frac{1}{2}\sinh ^{2}(\xi ^{\prime })\right) \right) \nonumber \\{} & {} \qquad \left( E_{1}\left( b^{2}\lambda _{T}^{2}\left( t(-1)\right) \right) -E_{1}\left( b^{2}\lambda _{T}^{2}\left( t(1)\right) \right) \right) +\frac{\exp (b^{2}\lambda _{T}^{2}\cosh ^{2}\xi )}{\lambda _{T}\cosh (\xi )}\psi (\xi )\nonumber \\{} & {} \qquad \left( \lambda _{T}\cosh (\xi )\text {erfc}(b\lambda _{T}\cosh (\xi ))-\frac{\exp (-b^{2}\lambda _{T}^{2}\cosh ^{2}\xi )}{a_{bg}}\left( 1+\frac{\frac{2\pi \hbar ^{2}\Lambda ^{2}a_{bg}}{m\varLambda _{bg}^{2}}}{\lambda _{T}^{2}\cosh ^{2}\xi +E_{b}-\frac{\varLambda ^{2}}{\varLambda _{bg}}}\right) \right) =0,\qquad \quad \end{aligned}$$

(B19)

where $E_{1}(z)=\int _{1}^{\infty }ds\frac{e^{-sz}}{s}$ is the exponential integral and

$$\begin{aligned} t(j)=1+\frac{4}{3}\sinh ^{2}(\xi )+\frac{4}{3}\sinh ^{2}(\xi ^{\prime })+j\frac{4}{3}\sinh (\xi )\sinh (\xi ^{\prime }). \end{aligned}$$

(B20)

In the limit of $b\rightarrow 0,$ the kernel in the integral coincides with the integral of the STM equation [38]. By inserting $\intop d\xi ^{\prime }\delta \left( \xi -\xi ^{\prime }\right)$ into the first term, this expression can be incorporated into the integral and one obtains:

$$\begin{aligned} \int _{-\infty }^{\infty }d\xi ^{\prime }{\mathcal {M}}_{\lambda _{T}}(\xi ,\xi ^{\prime })\psi (\xi ^{\prime })=0, \end{aligned}$$

(B21)

with $\psi (\xi )$ being an odd function. The matrix ${\mathcal {M}}_{\lambda _{T}}(\xi ,\xi ^{\prime })$ can be expressed as:

$$\begin{aligned} M\left( \xi ,\xi ^{\prime }\right) =\delta \left( \xi -\xi ^{\prime }\right) K\left( \xi ^{\prime }\right) -\delta \left( \xi -\xi ^{\prime }\right) H\left( \xi ^{\prime }\right) -2\pi L\left( \xi ,\xi ^{\prime }\right) , \end{aligned}$$

(B22)

where

$$\begin{aligned}{} & {} H(\xi ) = \frac{1}{\lambda _{T}^{*} a_{bg}^{*} \cosh (\xi )} \left( 1 - \frac{1}{\left( \lambda _{T}^{*} \cosh (\xi )\right) ^{2} (-1)^{j} a_{bg}^{*} + 1 + \frac{1}{\frac{a^{*}}{a_{bg}^{*}} - 1}}\right) , \end{aligned}$$

(B23)

$$\begin{aligned}{} & {} K\left( \xi ^{\prime }\right) =\exp \left( \left( b^{*}\lambda _{T}^{*}\cosh (\xi )\right) ^{2}\right) \text {erfc}\left( b^{*}\lambda _{T}^{*}\cosh (\xi )\right) , \end{aligned}$$

(B24)

$$\begin{aligned}{} & {} L\left( \xi ,\xi ^{\prime }\right) =\frac{1}{\sqrt{3}\pi ^{2}}\exp \left( \left( b^{*}\lambda _{T}^{*}\right) ^{2}\left( 1+\frac{1}{2}\sinh ^{2}(\xi )+\frac{1}{2}\sinh ^{2}(\xi ^{\prime })\right) \right) \nonumber \\{} & {} \qquad \qquad \left( E_{1}\left( \left( b^{*}\lambda _{T}^{*}\right) ^{2}t(-1)\right) -E_{1}\left( \left( b^{*}\lambda _{T}^{*}\right) ^{2}t(1)\right) \right) , \end{aligned}$$

(B25)

where $\lambda _{T}^{*}=\lambda _{T}R^{*}$, $b^*=b/R^*$, $a_{bg}^*=a_{bg}/R^*$, $a^*=a/R^*$ and $j=0$ if $\frac{\hbar ^{2}}{ma_{bg}\mu \Delta B} < 0$, or $j=1$ if $\frac{\hbar ^{2}}{ma_{bg}\mu \Delta B} > 0$, and t(i) is defined in Eq. (B20). We then renormalized Eq. (B19) with respect to $R^{*}$.

The requirement of a vanishing determinant, $\det M_{\lambda _{T}}(\xi ,\xi ^{\prime })=0$, provides an implicit equation for $\lambda _{T}$. This transcendental equation can be solved numerically by discretizing $\xi$ and $\xi ^{\prime }$ on a grid. One searches for solutions $\lambda _{T}=\lambda _{T}^{(sol)}$ where the determinant changes sign as $\lambda _{T}$ passes through $\lambda _{T}^{(sol)}$. Note that for given $\frac{R^{*}}{a_{bg}}$ and $\frac{b}{R^{*}},$ there are typically many solutions $\lambda _{T}^{(sol)}$. To verify a particular solution is physical, the obtained $\lambda _{T}^{(sol)}$ is substituted back into $M_{\lambda _{T}}(\xi ,\xi ^{\prime })$, and its eigenvalues and eigenfunctions $\psi (\xi )$ are computed. The solution corresponds to a bound state only if one eigenvalue is zero (within numerical precision) and the associated eigenfunction $\psi (\xi )$ is an odd function of $\xi$ (has an odd number of nodes).

Appendix C: Relating the Bare Parameters to the Observable Parameters in 3CMbg

We aim to link the five bare parameters {$\Lambda _{1},\Lambda _{2},b,B_{2},B_{1}$} to the four observable parameters $\{B_{1}^{res},B_{2}^{res},\Delta B_{1},\Delta B_{2}\}$.

This is accomplished by comparing Eq. (19) to Eq. (20), which yields:

$$\begin{aligned} B_{1,2}^{res}= & {} \frac{-C_{4} C_{1}}{2 C_{3}} + \frac{1}{2}(B_1 + B_2) \pm \frac{R}{2 \sqrt{\pi }} \end{aligned}$$

(C1)

$$\begin{aligned} \Delta B_1= & {} -\frac{\sqrt{\pi }(C_{4}-1)}{2 C_{3}^2 R}\left( C_{4} C_{1}^2 - C_{3} C_{2}( B_2 - B_1) -\frac{C_{3} C_{1} R}{\sqrt{\pi }}\right) \end{aligned}$$

(C2)

$$\begin{aligned} \Delta B_2= & {} \frac{\sqrt{\pi }(C_{4}-1)}{2 C_{3}^2 R}\left( C_{4} C_{1}^2 - C_{3} C_{2}( B_2 - B_1) +\frac{C_{3} C_{1} R}{\sqrt{\pi }}\right) \end{aligned}$$

(C3)

where we have defined:

$$\begin{aligned} R= \sqrt{\frac{C_{1}^2 C_{4}^2 \pi }{C_{3}^2} + \frac{2 \pi C_{4} (B_1 - B_2) C_{2} }{ C_{3}} + (B_1 - B_2)^2 \pi }) \end{aligned}$$

(C4)

and $C_{1} = \Lambda _2^2 \mu _1 + \Lambda _1^2 \mu _2$, $C_{2} = \Lambda _2^2 \mu _1 - \Lambda _1^2 \mu _2$, $C_{3}= \Lambda _{bg} \mu _1 \mu _2$, $C_{4}= \frac{a_{bg}}{b \sqrt{\pi }}$. From Eq. (10), it is clear that $\Lambda _{bg}$ is not an independent parameter. It is rather defined by $a_{bg}$ and $b$ through the relation:

$$\begin{aligned} \Lambda _{bg}=\frac{2\hbar ^{2}\pi ^{\frac{3}{2}}}{m}\left( \frac{ba_{bg}}{b\sqrt{\pi }-a_{bg}}\right) \end{aligned}$$

(C5)

The model has five bare but only four observable parameters. Thus, there is one free parameter that we chose to be b (as explained in the main text). By fixing the value of b, we can establish a mapping between the bare and observable parameters using the relations outlined above. This provides a straightforward way to connect our theoretical model with the experimental observables.