Multichannel effects in the Efimov regime from broad to narrow Feshbach resonances

As originally analyzed by Efimov, three particles that interact with resonant pairwise interactions show universal behavior independent of the details of the interaction potentials Efimov (1970). Whereas Efimov first analyzed this effect in the context of nuclear physics, over the last decades ultracold alkali atoms have proven to be an ideal platform to study Efimov physics experimentally for Bose gases Knoop et al. (2009); Zaccanti et al. (2009); Gross et al. (2009); Pollack et al. (2009); Gross et al. (2010), Fermi gases Ottenstein et al. (2008); Lompe et al. (2010); Huckans et al. (2009); Williams et al. (2009); Nakajima et al. (2011) and mixtures Barontini et al. (2009); Pires et al. (2014a, b); Tung et al. (2014); Ulmanis et al. (2015); Johansen et al. (2017); Ulmanis et al. (2016a, b); Wacker et al. (2016). In these atomic systems, the pairwise interaction can be tuned into the resonant regime close to Feshbach resonances by applying external magnetic fields Chin et al. (2010). This tunability is a consequence of Zeeman shifts in the atomic hyperfine states of the individual atoms. Combinations of these internal hyperfine states on the two and three atom level form the different scattering channels of the system which are coupled by a multichannel interaction potential when the atoms approach each other.

The strength of the interaction in the underlying two-body system can be parametrized by the $s$-wave scattering length $a$ in the ultracold regime. The behavior of $a$ near a Feshbach resonance can then be characterized by a background scattering length $a_{\mathrm{bg}}$ and a resonance width parameter $r^{*}$ Chin et al. (2010). The parameter $r^{*}$ is related to the width of the resonance in magnetic field $\Delta B=\hbar^{2}/m\,d\mu\,a_{\mathrm{bg}}\,r^{*}$, with $m$ the mass of the atoms and $d\mu$ the magnetic moment of the bound state associated with the resonance. Large values of $r^{*}$ therefore correspond to narrow Feshbach resonances, whereas small values correspond to broad ones. The term $\Delta B$ can be determined from the magnetic field dependence of $a$

where $B$ and $B_{0}$ denote the magnetic field strength and the resonance position in magnetic field, respectively.

The universal Efimov regime is characterized by large absolute values of the scattering length, which diverges on resonance ($|a|\gg r_{\mathrm{vdW}}$), where $r_{\mathrm{vdW}}=(mC_{6}/\hbar^{2})^{1/4}/2$ is the range associated with the $-C_{6}/r^{6}$ van der Waals tail of the atomic interaction. In the universal Efimov regime a single three-body parameter determines the location of three-body features, e.g. the binding energies $E_{n}$ of an infinite sequence of weakly bound three-body states, referred to as Efimov trimers, that emerge successively when the interaction is tuned to resonance. The binding energies follow the universal scaling relation, $E_{n+1}/E_{n}=e^{-2\pi/s_{0}}$ with $s_{0}\approx 1.00624$ for identical bosons Efimov (1970); Braaten and Hammer (2006). 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.

The three-body parameter $a_{-}^{(0)}$ has been measured for many species and Feshbach resonances Kraemer et al. (2006); Pollack et al. (2009); Gross et al. (2009, 2010); Zaccanti et al. (2009); Wild et al. (2012); Ferlaino et al. (2011); Berninger et al. (2011a). Surprisingly, most early experiments for identical bosons found the three-body parameter to be close to $a_{-}^{(0)}/r_{\mathrm{vdW}}\approx-9$ over different species and Feshbach resonances Kraemer et al. (2006); Gross et al. (2009, 2010); Wild et al. (2012); Ferlaino et al. (2011); Berninger et al. (2011a). Following the experiments, this universal value of $a_{-}^{(0)}$ could be theoretically explained Wang et al. (2012); Naidon et al. (2014a, b) relying on single-channel models for the interaction potential. However, such single-channel models cannot correctly describe the two-body physics close to narrow Feshbach resonances.

For narrow resonances $a_{-}^{(0)}$ can deviate from the universal value as has been predicted theoretically Petrov (2004); Gogolin et al. (2008); Schmidt et al. (2012); Langmack et al. (2018); Sørensen et al. (2012); Chapurin et al. (2019); Secker et al. (2020a) and observed experimentally Roy et al. (2013); Chapurin et al. (2019); Gross et al. (2011); Dyke et al. (2013). However, agreement between theory and experiment has been achieved only in a few instances Chapurin et al. (2019); Secker et al. (2020a). The deviation in $a_{-}^{(0)}$ from the universal value thus indicates the importance of multichannel effects for narrow resonances. To represent narrow resonances on the two-body level simple two-channel models can be used to correctly account for the resonance width parameter $r^{*}$. Generalizing those models to the three-body case resulted in studies of $a_{-}^{(0)}$ as a function of $r^{*}$ Petrov (2004); Gogolin et al. (2008); Schmidt et al. (2012); Sørensen et al. (2012), as well as of both $r^{*}$ and $a_{\mathrm{bg}}$ Langmack et al. (2018). All those models find a dependence of $a_{-}^{(0)}$ on $r^{*}$, however opposite trends for the behavior of $|a_{-}^{(0)}|$ have been obtained. On the one hand $|a_{-}^{(0)}|$ can increase with increasing $r^{*}$ reaching a narrow resonance limit ($r^{*}\rightarrow\infty$) where $r^{*}$ sets the three-body universal regime as well as the three-body parameter Petrov (2004); Gogolin et al. (2008); Schmidt et al. (2012) or on the other hand $|a_{-}^{(0)}|$ can decrease with increasing $r^{*}$ Sørensen et al. (2012). Interestingly both increasing Roy et al. (2013); Chapurin et al. (2019) and decreasing Gross et al. (2011); Dyke et al. (2013) trends have also been observed experimentally. One of the major differences in the models that predict opposite behavior is the way the two-body multichannel structure is embedded into the three-body multichannel space. This indicates the importance of an accurate representation of the multichannel structure of the three-atom system.

On the two-body level, a symmetric spin channel is in general either of the form $|\mathpzc{c}\mathpzc{c}\rangle$ or $|\mathpzc{c}\mathpzc{c}^{\prime}\rangle_{S}=(|\mathpzc{c}\mathpzc{c}^{\prime}\rangle+|\mathpzc{c}^{\prime}\mathpzc{c}\rangle)/\sqrt{2}$, with $\mathpzc{c}$ and $\mathpzc{c}^{\prime}$ labeling the different internal spin states of the individual atoms. For identical bosons such symmetric spin channels need to be considered when combined with even partial wave components of the interaction. In this paper we analyze both realizations $|\mathpzc{c}\mathpzc{c}\rangle$ and $|\mathpzc{c}\mathpzc{c}^{\prime}\rangle_{S}$ for the closed channel and are thereby extending earlier studies Langmack et al. (2018); Schmidt et al. (2012); Gogolin et al. (2008); Petrov (2004) to the $|\mathpzc{c}\mathpzc{c}^{\prime}\rangle_{S}$ scenario. We consider pairwise separable interaction potentials that have been used to study the dynamics of the many-body system Colussi et al. (2018); Musolino et al. (2019) and generalize them to a multi-channel interaction. We find that the realization of the two-channel model in spin space affects the Efimov spectrum in the broad and especially intermediate resonance width regime, leading to both increasing and decreasing trends of $|a_{-}|$ depending on the realization, while the narrow resonance limit can be recovered for $r^{*}\rightarrow\infty$ in all considered cases. Additionally, we find the narrow resonance limit to be independent of the form of the interaction we consider.

The paper is outlined as follows. In Sec. II, we start with the analysis of the three-body bound state equations in momentum representation for a general multichannel system. To keep the system clean whilst retaining the multichannel characteristics, we proceed to analyze a simple two-channel model in Sec. III, where we restrict ourselves to the case of separable $s$-wave interactions and two internal spin states $|a\rangle$ and $|b\rangle$ per particle. Here, we distinguish between the two different closed-channel realizations $\ket{bb}$ and $\ket{ab}_{S}$ we mentioned earlier. The two-body details are discussed in section IV. The results of our analysis and a comparison of the different closed-channel realizations are presented in Sec. V.1. For the $\ket{bb}$ realization they resemble earlier multichannel studies and allow for an effective field theory description containing a dimer field Gogolin et al. (2008); Schmidt et al. (2012); Langmack et al. (2018) as we discuss in appendix C. In Sec. V.2 we analyze the narrow resonance limit of the two realizations. We then conclude in Sec. VI with a summary of the most important findings and suggestions for future research.

For the three-body multichannel model system with separable interactions that we want to focus on in Secs. III to V, we choose to work in a momentum space representation. This enables us to study Efimov physics by solely considering a one-dimensional integral equation. Therefore we outline the general multichannel version of the three-body bound state equations for three identical bosons in momentum space here. Since we do not yet restrict ourselves to any special kind of model system, the equations presented should apply to any short range interaction including those coupled-channels models, which currently provide the most accurate theoretical description of the interatomic interaction.

We consider identical bosons with pairwise potentials $V_{ij}$ describing the interaction between particles $i$ and $j$. The three-body bound state equation can then be formulated in the following form Merkuriev and Faddeev (1993)

where $\mathcal{T}_{ij}(E)$ is a generalized two-body transition operator, $G_{0}(E)$ is the free three-body Green’s operator, that in the multichannel context also accounts for the asymptotic energies of the channels, and where $P_{+}$ and $P_{-}$ denote the two cyclic permutation operators. The index $(ij)$ specifies a choice of a two particle subsystem formed by particle $i$ and $j$. We introduce the corresponding system of Jacobi momenta

where $\mathbf{k}_{i}$, $\mathbf{k}_{j}$ and $\mathbf{k}_{k}$ denote the momentum of particle $i$, $j$ and $k$ respectively, such that the relative momentum between particles $i$ and $j$ is now related to $\mathbf{p}$. The particular choice of the pair $(ij)$ is abitrary since we are considering identical bosons. Eq. (2) has a solution only when $E$ is an eigenenergy of the three-body Hamiltonian. The corresponding bound state wave function is then given by $\Psi=(1+P_{+}+P_{-})G_{0}(E)\Phi_{ij}$.

In the following we discuss the operators introduced in Eq. (2) in more detail. We start with the analysis of the generalized two-body transition operator $\mathcal{T}_{ij}(E)$. Given the interaction $V_{ij}$, we can use the Lippmann-Schwinger equation in order to define $\mathcal{T}_{ij}$ Merkuriev and Faddeev (1993)

$\mathcal{T}_{ij}(E)$ is related to the two-body $t$ operator $t(z)$ by Merkuriev and Faddeev (1993)

where $\mathpzc{C}$ and $\mathpzc{C}^{\prime}$ represent any symmetric or antisymmetric combination of the product of internal states of particle $i$ and $j$ and where $\mathpzc{c}$ and $\mathpzc{c}^{\prime}$ represent the spin of particle $k$.

Since we consider a three-body system of identical bosons, where each particle $i$ can occupy several internal spin states labeled by $\mathpzc{c}_{i}$, we can define the permutation operators $P_{\pm}$ as products of permutation operators $P_{\pm}^{c}$ acting only on coordinates and permutation operators $P_{\pm}^{s}$ acting only on internal states respectively, such that

where the coordinate permutation operators can be written in integral form as

with momentum states normalized according to $\langle\mathbf{p}^{\prime}|\mathbf{p}\rangle=\delta(\mathbf{p}^{\prime}-\mathbf{p})$. The internal state permutation operators can be expressed using a summation over all available internal spin states, such that

The sum of permutation operators can then be written as Glöckle (1983)

where

are the symmetrization and antisymmetrization operators in particles $i$ and $j$ and where $P_{ij}$ is the operator exchanging the particles in the pair $(ij)$. Eq. (12) then follows from the identity

For identical bosons we need $\Phi_{ij}=P_{ij}\Phi_{ij}$ such that $\Psi$ is totally symmetric and therefore Eq. (2) simplifies to

Using this we can give Eq. (2) in $|\mathpzc{C},\mathpzc{c},\mathbf{p},\mathbf{q}\rangle$ representation, such that

Note how the Green’s function has evaluated to

where $E(\mathpzc{C},\mathpzc{c})$ represents the asymptotic energy of the channel $|\mathpzc{C},\mathpzc{c}\rangle$. In appendix A we give more details on how to work out the elements $\langle\mathpzc{C}^{\prime},\mathpzc{c}|P_{+}^{s}|\mathpzc{C}^{\prime\prime},\mathpzc{c}^{\prime\prime}\rangle$.

To explore possible multichannel effects we proceed with the analysis of a simple model potential, which has a single separable $s$-wave component

with form factor $\zeta$ and only symmetric spin combinations $\mathpzc{C}$ and $\mathpzc{C}^{\prime}$. In addition we assume just two internal spin states per particle, which we label with $a$ and $b$ ($\mathpzc{c}_{i},\mathpzc{c}_{j},\mathpzc{c}_{k}\in\{a,b\}$) and assume a difference in channel energy of $\epsilon_{ab}$. Since the interaction is separable, so is the two-body $t$-operator

that we work out explicitly for a step-function shaped form factor in the following section. Searching for solutions with zero total angular momentum, we can then make the ansatz $\left\langle\mathpzc{C},\mathpzc{c},\mathbf{p},\mathbf{q}\left|\Phi_{ij}\right\rangle\right.=\zeta(p)\langle\mathpzc{C},\mathpzc{c},q|\phi_{ij}\rangle$. Evaluating this ansatz in Eq. (II) and dividing both sides by $\zeta(p)$, we find

where we implicitly defined the operator $K$ in the last line. We implement Eq. (III) numerically by replacing the $q^{\prime}$ integration by a summation over a finite $q^{\prime}$-grid.

Applying the relations as presented in appendix A in order to analyze the elements $\langle\mathpzc{C}^{\prime}\mathpzc{c}|P_{+}^{s}|\mathpzc{C}^{\prime\prime}\mathpzc{c}^{\prime\prime}\rangle$ and dropping the index in $\phi_{ij}$, Eq. (22) can be rewritten into the following matrix form

where we write out the symmetric spin states in the pair ($ij$) explicitly as $\mathpzc{C}\in\{\underline{aa},\underline{ab},\underline{bb}\}$. This equation indicates, that even a model with a single $s$-wave component and two internal states, yields a system with six coupled channels for the Faddeev component $\phi$.

By setting $\langle\mathpzc{C}^{\prime}|v|\mathpzc{C}\rangle=0$ when $\mathpzc{C}$ or $\mathpzc{C}^{\prime}$ equals $\underline{ab}$, or alternatively setting $\langle\mathpzc{C}^{\prime}|v|\mathpzc{C}\rangle=0$ when $\mathpzc{C}$ or $\mathpzc{C}^{\prime}$ equals $\underline{bb}$ we can study scenarios where the closed symmetric spin channel $Q$ is either of the form $|\mathpzc{c}\mathpzc{c}\rangle$ or $|\mathpzc{c}\mathpzc{c}^{\prime}\rangle_{S}=(|\mathpzc{c}\mathpzc{c}^{\prime}\rangle+|\mathpzc{c}^{\prime}\mathpzc{c}\rangle)/\sqrt{2}$, respectively. Thereby we reduce to a two-channel model on the two-body level with open channel $P=\underline{aa}$ in both of these cases. As we will see in the following section $\langle\mathpzc{C}^{\prime}|\tau(z)|\mathpzc{C}\rangle=0$ whenever $\langle\mathpzc{C}^{\prime}|v|\mathpzc{C}\rangle=0$. Therefore some of the elements $\langle\mathpzc{C}^{\prime}\mathpzc{c}^{\prime}|K|\mathpzc{C}\mathpzc{c}\rangle$ in Eq. (III) evaluate to zero and the three-body equations can be simplified.

Considering Eq. (29) for the situation with closed channel $Q=\underline{bb}$, we find that the three-body equations involving the open channel $\underline{aa}a$ can be reduced to

The solution is thus solely determined by the open channel part

Since just the open-channel component of the $t$-operator is needed in this equation, we can later on use the Feshbach formalism Feshbach (1992), which we generalize in appendix B to an off shell version, to construct an approximate model system, in which the closed-channel is modelled by a separable energy-dependent interaction term added to the open channel. We analyze this model and its narrow resonance limit in more detail in Sec. V. We note that this model system is similar to the ones discussed in effective field theory Gogolin et al. (2008); Schmidt et al. (2012); Langmack et al. (2018) and the ones considered for the narrow resonance limit Petrov (2004); Gogolin et al. (2008). In appendix C we show that the $Q=\underline{bb}$ model can be approximated by the effective field theory models used in Gogolin et al. (2008); Schmidt et al. (2012); Langmack et al. (2018).

If we alternatively fix the closed channel $Q$ to correspond to $\underline{ab}$, we find that Eq. (29) can be expressed as

Contrary to the realization where $Q=\underline{bb}$, this equation indicates that the realization $Q=\underline{ab}$ results in coupling terms to the closed channels. Consequently it is no longer possible to describe the model in terms of the Feshbach formalism. As such, the system does no longer resemble the ones discussed in effective field theory Gogolin et al. (2008); Schmidt et al. (2012); Langmack et al. (2018) and requires a careful analysis in terms of all four coupled-channels as presented in Eq. (63). The results of this analysis will be presented in Sec. V.1.

In the following we derive the two-body $t$-operator for the separable system explicitly. We consider the cases of open channel $P=\underline{aa}$ and closed channel $Q=\underline{bb}$ as well as open channel $P=\underline{aa}$ and closed channel $Q=\underline{ab}$. Since we can reduce to a two-body two-channel system in both cases, the $t$-operators are identical, when the difference $\epsilon$ in closed and open channel energy and the coupling strengths $\langle P|v|P\rangle,\,\langle Q|v|Q\rangle,\,\langle Q|v|P\rangle=\langle Q|v|P\rangle^{*}$ are the same. We get $\epsilon=2\epsilon_{ab}$ for $Q=\underline{bb}$ and $\epsilon=\epsilon_{ab}$ for $Q=\underline{ab}$. According to Eq. (19) we have a two-body interaction of the form

with potential interaction and coupling strength parameters $\bar{v}_{PP},\,\bar{v}_{QQ},\,\bar{v}_{PQ}=\bar{v}_{QP}\in\mathbb{R}$. We define the form factor $\zeta$ as

where $\Lambda$ is a momentum cut-off scale. For such an interaction, the two-body transition operator can be obtained analytically in a straightforward fashion. Resembling Eq. (5) the two-body operator is defined as

with $g_{0}$ the free Green’s function of the two-body system. Since we are considering the interaction between just two channels, we can fix the zero energy to equal the asymptotic energy of the open channel $P$. By doing so, the asymptotic energy of the closed-channel $Q$ reduces to the energy difference $\epsilon$ between the open and closed channel. Consequently, the free Green’s operator of the two-body system $g_{0}(z)$ can be expressed as

Next, the separable interactions allow us to express the $t$-operator in the following separable form

where the energy dependent terms $t_{\mathpzc{C}\mathpzc{C}^{\prime}}(z)$ can be computed explicitly from

For step function shaped form factors considered in this section, the previous equation can be solved analytically using the identity

to solve for $\langle\zeta|g_{0}(z)|\zeta\rangle$.

Here we would like to point out that it is possible to generalize this method to any finite number of channels and form factors. Furthermore, the model can be adjusted to match the low energy scattering properties of a given system. The scattering length is then given by

and depends on the parameters $\bar{v}_{PP}$, $\bar{v}_{QQ}$, $\bar{v}_{PQ}=\bar{v}_{QP}$ and $\epsilon$, so that we have $a=a(\bar{v}_{PP},\bar{v}_{QQ},\bar{v}_{PQ},\epsilon)$. We then define the background scattering length by

the resonance energy $\epsilon_{0}$ by

and the resonance width parameter

With those definitions we can map any given set of $(a_{\mathrm{bg}},r^{*},\epsilon_{0})$ to a set of system parameters $(\bar{v}_{PP},\bar{v}_{QQ},\bar{v}_{PQ})$.

To analyze the narrow resonance limit of $\tau_{PP}(z)$ we first use the Feshbach formalism outlined in appendix B to approximate the system. We recognize that the transition matrix is the only operator in the three-body equation that depends on the form of the two-body interactions. Following the procedure as outlined in appendix B, we find that the transition matrix element $\tau_{PP}$ as introduced in Eq. (69) reduces to the following simple form in the narrow resonance limit

where we have used system parameters in units of $r^{*}$, such that $\tilde{t}_{PP}=t_{PP}/(r^{*}/m\hbar)$, and where we have introduced the dimensionless scattering length $\tilde{a}=a/r^{*}$ and energy $\tilde{z}=z\,m\,r^{*2}/\hbar^{2}$. The above expression is valid for arbitrary form factors $\zeta$ and leads to a narrow resonance limit of the Efimov spectrum which we discuss in detail in section V.2. From that we conclude that the above limit also holds in the general setting with non separable interaction potentials. The $t$-operator for those general potentials can be expanded in separable terms, with a single separable term representing the resonant component Mestrom et al. (2019); Secker et al. (2020b, a). Only the open-open component of the resonant term will approach the limit in Eq. (76). All other terms in the open-open component are finite even on resonance in units related to the range of the interaction and therefore vanish according to Eq. (75) in units of the resonance width parameter $r^{*}$ when taking it to infinity.

We study the dissociation scattering lengths $a_{-}^{(n)}$ as well as the binding energies of the three deepest trimer states for varying values of the resonance width parameter $r^{*}$. To completely determine the system we fix the threshold difference on resonance $\epsilon_{0}$ and the background scattering length $a_{bg}$, such that we can map the scattering length $a$ to the threshold difference $\epsilon$ between the open and closed channel. In the following we denote quantities made dimensionless in units of the cut-off scale $\Lambda$ with a bar. This means that all lengths are given in multiples of $\hbar/\Lambda$ and all energies are given in multiples of $\Lambda^{2}/m$. We choose $\bar{a}_{bg}=-0.2$ to have no additional bound background dimer states in our model and set the resonance position to a value of $\bar{\epsilon}_{0}=1.5$ to just have a single closed channel trimer for $Q=\underline{ab}$ as will be discussed below in more detail.