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# Emergence of N-body tunable interactions in
universal few-atom systems

M. T. Yamashita1 T. Frederico2 Lauro Tomio1 1Instituto de Física Teórica,
UNESP, 01140-700, São Paulo, Brazil.
2Instituto Tecnológico de Aeronáutica, 12228-900, São José dos Campos, Brazil.

###### Abstract

A three-atom molecule AAB, formed by two identical bosons A and a distinct one
B, is studied by considering coupled channels close to a Feshbach resonance.
It is assumed that the subsystems AB and AA have, respectively, one and two
channels, where, in this case, AA has open and closed channels separated by an
energy gap. The induced three-body interaction appearing in the single channel
description is derived using the Feshbach projection operators for the open
and closed channels. An effective three-body interaction is revealed in the
limit where the trap setup is tuned to vanishing scattering lengths . The
corresponding homogeneous coupled Faddeev integral equations are derived in
the unitarity limit. The s-wave transition matrix for the AA subsystem is
obtained with a zero-range potential by a subtractive renormalization scheme
with the introduction of two finite parameters, besides the energy gap. The
effect of the coupling between the channels in the coupled equations is
identified with the energy gap, which essentially provides an ultraviolet
scale that competes with the van der Waals radius - this sets the short-range
physics of the system in the open channel. The competition occurring at short
distances exemplifies the violation of the “van der Waals universality” for
narrow Feshbach resonances in cold atomic setups. In this sense, the active
role of the energy gap drives the short-range three-body physics.

Few-atom systems and Universality and Feshbach resonances

##  I Introduction - an overview of recent problems involving universality in
few-body systems

The existence of several nuclear potentials with many free parameters, usually
set to reproduce scattering observables, raises the question about the
possibility to make something relevant with a zero-range potential. From the
absolute point-of-view it is not possible to describe the energy spectrum or
the nuclei structure with a such quasi-no-parameter potential. However, the
description of physical observables for specific nuclear systems, after
identifying the relevant scales, is completely suitable for a Dirac-delta
potential: this is in the core of the universality concept 2000aDelfino ;
2006braaten ; 2011Frederico .

When applied to few-body systems, universality means, grosso modo,
independence on details of the short-range part of the interactions which are
describing such systems. These systems appear in several few-body areas, being
characterized for having their typical sizes, represented by the absolute
value of the two-body scattering length $a_{0}$ and potential range $r_{0}$,
such that the first is much greater that the second, $|a|\gg r_{0}$ \- this
property defines a weakly-bound system. There are many molecules in the atomic
context, which satisfy the relation $|a|/r_{0}\gg 1$ 2010Chin . In the nuclear
context, these weakly-bound structures are well represented by halo nuclei, in
which we have one or more halo nucleons weakly-bound to a core nucleus.
fredericoppnp2012 .

Universality in few-body physics provided explanation to many interesting
phenomena, which have been extensively studied in recent years. The remarkable
one is the Efimov effect 1970efimov , which was stablished by studying
bosonic-like three-body system with at least two subsystems having infinite
scattering length. This effect became a paradigm when mentioning universal
aspects of few-body systems - nowadays, the study of the relevant physical
scales close to the unitary limit is known as “Efimov physics”.

A natural three-body system with two-body subsystems exactly at zero energy
does not exist. One of the systems found in nature that approaches the ideal
situation is the helium trimer. It was suggested that the Efimov effect could
be verified in this molecule 1984CG . The lack of any other natural three-body
system close to the Efimov limit was the main reason why the original article
from Efimov was considered for thirty years uniquely as a theoretical
allegory. However, with the recent modern techniques involving ultracold
traps, it was possible to measure the ground and first excited states of the
helium trimer 2015-kunitski , confirming the original suggestion and many
other theoretical studies from different groups 1985Huber ; 1996Esry ;
1997Kolganova ; 2001Motovilov ; 2004Kolganova ; 2000Delfino ; 2000Gianturco ;
2002Gianturco ; 2012Roudnev ; 2014Wu ; 2017Kolganova ; 2018Kolganova .

The confirmation of Efimov states is also possible by producing them
artificially. Some experiments with cold-atom systems were able to observe
signals of Efimov resonant levels by manipulating the two-body interactions.
The possibility to freely tune the two-body interaction in ultracold atomic
traps brought unprecedented possibilities to artificially create the ideal
condition for Efimov states 2006Kraemer . The use of Feshbach resonances
technique 1958Feshbach  to alter the two-body energies 1998Inouye ;
1998Courteille ; 1999Timmermans  in atomic traps produced a rich playground
with plenty of possibilities to investigate the correlations between the
physical scales. Experimentalists are now able to continuously vary $a_{0}$
from large positive to large negative values, across the threshold at which a
bound state turns into a resonance or a virtual state (see e.g. Ref.
fredericoppnp2012 ).

The continuous change of spatial dimension is also another interesting aspect
2019Garrido  that can now be explored in actual experimental realizations
2017Klauss . The study of one, two or three-dimensional solitons has already
started long ago (See 2000Gammal ; 2005Abdullaev  and references therein).
However, it still missing systematic experimental studies about the effect of
the dimensional change in few-body observables, e.g., the Efimov effect is
drastically affected by the spatial dimensional reduction. The disappearance
of Efimov effect in fractional dimensions between three and two dimensions,
including another few-body aspects involving a continuous change of spatial
dimensions, has recently being investigated by some groups 2018Sandoval ;
2018aRosa ; 2018bRosa ; 2018Christensen ; 2018Abhishek ; 2019Yama ; 2019Beane
; 2019Garrido .

All the experimental possibilities regarding the change of dimensionality and
atomic interactions are not possible in the nuclear context where the nucleon-
nucleon interaction is rigid. A conjecture was raised in Ref. 2008Higa  that a
resonant Hoyle state in 12C could emerge from an Efimov bound state. In order
to investigate this conjecture, a recent work studied the screening of the
Coulomb potential in three alphas 2020Phyu .

The strong repulsion at short distances coming from the Coulomb potential is
what prevents the appearance of Efimov effect in three alphas. However, in hot
dense plasmas the electrons can, in principle, shield the repulsion coming
from the protons favoring the formation of the Hoyle resonance. A continuous
increase of the electronic density can also produce a similar effect,
occurring in atomic traps, that is the change of the effective two-body
interaction of the system. However, recent calculations made in Ref. 2020Phyu
exclude the possibility of a Hoyle state emerging from an Efimov bound state.

Another interesting aspect in the context of few-atom systems, which is the
central point of this article, is that, at least theoretically, it is possible
that induced few-atom (beyond two) forces appear YamaEPL2006  when the cold
atomic trap set-up is tuned near a narrow Feshbach resonance, where the closed
channel is strongly coupled to the open channel WangNPHYS2014 . If the control
of only two-body interactions 1999Timmermans  revealed a wide horizon to study
correlations among their observables, due to the change of few-body scales
2020Paula , the possibility to control three and more atomic interaction would
give an unprecedented freedom to few-body community.

Formally, the origin of the induced few-particle effective interactions in a
single channel representation of few-atom system goes back to the Feshbach
decomposition of the Hilbert space in open channels ($P$-space) and closed
channels ($Q$-space) , given that $P+Q=1$(details on the coupled-channel
formalism applied to potential scattering can be found in the textbook by
Canto and Hussein in Ref. Canto2013 ). The $Q$-space represents the state
where the two atoms interacts in a region of the potential where the
scattering states are closed. The $P$-space corresponds to potential well
lower in energy where the scattering states are open. Actually, $P$ and $Q$
are associated with different spin states of the low partial waves of the
atom-atom system (see e.g. WangNPHYS2014 ). In such framework, attractive
effective few-body forces arise from connected diagrams with the intermediate
virtual propagation of the system in the $Q$-space as illustrated in
YamaEPL2006 . The strength is enhanced for narrow resonances, as the coupling
between open and closed channels is larger in this case WangNPHYS2014 .

The experimental evidences suggesting the possibility of few-atom forces come
from the so called “van der Waals universality” BerPRL2011 ; ChinArXiv2011 ;
WangPRL2012 ; SchmidtEPJB2012 ; NaidonPRL2014 ; NaidonPRA2014  of Efimov
states across broad and narrow Feshbach resonances 2014chin ; 2017chin  with
Lithium-Caesium (6Li-133Cs) mixtures. The “van der Waals universality”
associates the position $(a_{0}<0)$ of the resonant three-atom recombination
peak, originated when an Efimov state dives into the continuum, with van der
Waals radius $(\ell_{vdW})$, such that the ratio $a_{0}/\ell_{vdW}\sim-9$ is
verified for broad resonances BerPRL2011 ; 2017chin . However, the
experimental results from Ref. 2017chin  indicated a dependence of the
position of the Efimov resonance on the Feshbach resonance strength, deviating
from the prediction of the single channel “van der Waals” universality. Such
observation suggests that close to narrow Feshbach resonances a single channel
description may be poor, and beyond the expected large variations of the atom-
atom scattering length, the three-body scale can also change, as also
supported by the observation of resonant recombinations in the 6Li-133Cs2
experiments 2017chin .

Note that the length scale associated with the position of the triatomic
recombination resonance is observed to be larger than the van der Waals length
for narrow resonances 2017chin . This was interpreted in Ref. 2020Paula  as a
manifestation of the attraction due to the induced three-atom interaction,
which dislocates the effective repulsive barrier jonsell , where the triatomic
continuum resonance 2004Bringas  is formed, to distances larger than
$\ell_{vdW}$. Therefore, the position of the narrow recombination resonance
appears dislocated towards larger absolute values of the scattering length
with respect to broad Feshbach resonances, as experimentally observed 2017chin
.

One should expect that, in few-atom systems driven by the Feshbach resonance
mechanism, which can induce from three up to N-body interactions, systems
beyond the atom-atom scattering length can be manipulated by tuning also the
short-range scales associated with three, four and more particles YamaEPL2006
; 2011hadizadeh ; 2019bazak ; 2020Paula . Such exciting possibilities
motivated us to explore the ideas we have sketched in Ref. YamaEPL2006 . We
write here the formalism of the three-body Faddeev equations for the bound
state in the presence of open and closed channels. Following that, a practical
framework can be formulated by identifying the relevant parameters of the
Feshbach resonance, which control the induced three-body interaction and the
associated short-range scale.

In this contribution, the AAB atomic bound state problem with coupled open-
closed channels close to a Feshbach resonance is formulated through the
Faddeev equations for the wave function. The effective three-body force
appearing in the single channel description is detailed by using the Feshbach
projection operators in the open and closed channels. We discuss the
interesting case where the scattering lengths vanish, such that in the open
channel the direct interaction vanishes, while we show formally that the
Faddeev components of the wave function in the open channel survives, allowing
the system to effectively interact in a single channel description through the
coupling of the open channel with the closed one. Furthermore, the Faddeev
formalism is derived explicitly for a zero-range model in one particular
example of three-particle AAB system, where the AA subsystem has open and
closed channels, by formulating in practice the ideas proposed in Ref.
YamaEPL2006 . The dependence of the short-range physics on the parameters of
the Feshbach resonance in the AA subsystem is obtained with the corresponding
meaning being explored qualitatively.

This work is organized as follows. In the next Sect. II, the Faddeev equation
for the bound state wave function for a general three-body model with open-
closed channels is formulated. It is also discussed the single channel
reduction and induced three-body interaction in the open channel. In addition,
the effective three-body interaction appearing when the system has vanishing
scattering length is derived within the closed channel description of the
three-body bound state. In Sect. III, the AAB Faddeev equations for the bound
state within a zero-range model with open and closed channels is derived in
the case that the AA bosonic subsystem has one open and one closed channels,
and by considering the AB interaction acting only in a single channel. Once
obtained the two-body T-matrix for the AA subsystem within the zero-range
interaction, the integral equations for the AAB coupled channel model in the
unitarity limit $(a\to\pm\infty)$ are derived, by considering the dependence
on the Feshbach resonance parameters. In Sect. IV, we present our final
considerations.

##  II Open-closed channels three-body model

The open-closed channels model for three-atom interactions has altogether
eight channels, as each pair can interact in an open or closed channel. The
notation has to reflect such different physical situations, with an index
$\alpha$ or $\beta$ for the corresponding channel wave function, in which the
atom pairs can be in open or closed channels, being indicated by
$\alpha_{ij}$, which runs over the two-body channels. Then, the possibilities
are

$\alpha=(\alpha_{ij},\alpha_{jk},\alpha_{ki}),\,$ (1)

and in a situation where only two of them are possible, namely the open and
closed one, it allows altogether eight three-body channels. Assuming, two-atom
interactions, the potential is an operator that allows transitions between the
open and closed channels: $V^{a,b}_{ij}$, which for the moment we are not
specifying.

After setting the structure of the Hilbert space where the wave function is
defined, we write the three-body Hamiltonian as a matrix, with operators as
matrix elements in the open-closed channel two-atom states, as

$H_{\alpha\beta}=(H_{0}+\Delta_{\alpha})\delta_{\alpha\beta}+\sum_{i>j}V^{\alpha_{ij},\beta_{ij}}_{ij}\delta_{\alpha_{jk},\beta_{jk}}\,\delta_{\alpha_{ki},\beta_{ki}}\,,$
(2)

where, to simplify the presentation, we introduce the following notation

$V^{\alpha\beta}_{ij}:=V^{\alpha_{ij},\beta_{ij}}_{ij}\,\delta_{\alpha_{jk},\beta_{jk}}\,\delta_{\alpha_{ki},\beta_{ki}}\,,$
(3)

for the potential. The kinetic energy operator is $\hat{H}_{0}$ and
$\Delta_{\alpha}$ is the energy level of channel $\alpha$. The energy of the
channels are given by

$\Delta_{\alpha}=\sum_{i>j}e(\alpha_{ij})\,,$ (4)

where $e(a_{ij})$ is the $a_{ij}$ channel energy. For the open channel we set
it as $e=0$, therefore the three-body open channel continuum has $\Delta=0$.
Furthermore, the standard translational invariance applies to
$H_{\alpha\beta}$ as long as the two-atom potential depends only on relative
coordinates.

The eigenvalue equation for the energy states of the system reads:

$\sum_{\beta}H_{\alpha\beta}|\Psi^{\beta}\rangle=\sum_{\beta}\bigg{[}(H_{0}+\Delta_{\alpha})\delta_{\alpha\beta}+\sum_{i>j}V^{\alpha\beta}_{ij}\bigg{]}|\Psi^{\beta}\rangle=E|\Psi^{\alpha}\rangle\,.$
(5)

We write the Faddeev components of the particular case of the bound-state wave
function as Fad  ,

$|\Psi^{\alpha}\rangle=\frac{1}{E-H_{0}-\Delta_{\alpha}+i\varepsilon}\sum_{\beta}\sum_{i>j}V^{\alpha\beta}_{ij}|\Psi^{\beta}\rangle\,,$
(6)

where

$G^{\alpha}_{0}(E)\equiv\frac{1}{E-H_{0}-\Delta_{\alpha}+i\varepsilon}\,$ (7)

is the resolvent. The corresponding Faddeev equations can be written as

$|\Psi^{\alpha}_{i}\rangle=G^{\alpha}_{0}(E)t^{\alpha\beta}_{jk}(E)\big{(}|\Psi^{\beta}_{j}\rangle+|\Psi^{\beta}_{k}\rangle\big{)}\,,$
(8)

with the two-body T-matrix within the three-body system being a solution of

$t^{\alpha\beta}_{jk}(E)=V^{\alpha\beta}_{jk}+V^{\alpha\gamma}_{jk}G^{\gamma}_{0}(E)t^{\gamma\beta}_{jk}(E)\,.$
(9)

For the sake of clarity, when written solely for the two-body subsystem, it
reads:

$t^{ab}_{jk}(E)=V^{ab}_{jk}+V^{ac}_{jk}\frac{1}{E-H_{0}^{(2)}-e(c)+i\varepsilon}t^{cb}_{jk}(E)\,,$
(10)

where $H_{0}^{(2)}$ is the two-body kinetic energy operator.

Let us consider the most simple AAB case, when only the AA subsystem has open
and closed channels, while AB does not interact in a closed channel. In such
simplified situation the three-body system has only two channels. Then, close
to a s-wave Feshbach resonance, the two-body scattering amplitude (evidently
associated with the open channel $o$ and $a=b=o$) is customarily approximated
by the corresponding effective range expansion:

$\langle\vec{k}^{\prime}|t^{oo}_{jk}(E)|\vec{k}\rangle=-\frac{1}{(2\pi)^{2}\mu_{ij}}\left[-a^{-1}+\frac{1}{2}r_{0}k^{2}+\cdots-
ik\right]^{-1},$ (11)

where $k=\sqrt{2\mu_{ij}E}$, $a_{0}$ is the scattering length and $r_{0}$ the
effective range.

###  II.1 Single channel reduction and induced three-atom interaction

It is possible to reduce the Hamiltonian eigenvalue equation to the open
channel, at the expenses of introducing an effective Hamiltonian, which can be
derived by using the Feshbach projection operators to the open channels
($P$-space) and closed channels ($Q$-space), respectively, such that $P+Q=1$.
In our notation,

$P_{\alpha\beta}=\delta_{\alpha o}\delta_{o\beta}\quad\text{and}\quad
Q_{\alpha\beta}=\delta_{\alpha\beta}-P_{\alpha\beta}\,,$ (12)

where $o$ indicates that the three pair of particles are in two-body open
channels. However, if we do the common procedure, we loose the re-sum of two-
body intermediate state propagation in the closed channels which builds the
two-body T-matrix in the open channel. This motivates us to use the Feshbach
projection $P$ and $Q$ operators directly in the Faddeev equations. We can
exemplify the procedure by considering the bound-state in (8), such that

$\displaystyle|\Psi^{\alpha}_{i}\rangle$ $\displaystyle=$ $\displaystyle
G^{\alpha}_{0}(E)t^{\alpha\beta}_{jk}(E)(P_{\beta\gamma}+Q_{\beta\gamma})\big{(}|\Psi^{\gamma}_{j}\rangle+|\Psi^{\gamma}_{k}\rangle\big{)}$
(13) $\displaystyle=$ $\displaystyle G^{\alpha}_{0}(E)t^{\alpha
o}_{jk}(E)\big{(}|\Psi^{o}_{j}\rangle+|\Psi^{o}_{k}\rangle\big{)}+G^{\alpha}_{0}(E)t^{\alpha\beta}_{jk}(E)\big{(}Q_{\beta\gamma}|\Psi^{\gamma}_{j}\rangle+Q_{\beta\gamma}|\Psi^{\gamma}_{k}\rangle\big{)}.$

For the open channel we have that

$\displaystyle|\Psi^{o}_{i}\rangle$ $\displaystyle=$ $\displaystyle
G^{o}_{0}(E)t^{oo}_{jk}(E)\big{(}|\Psi^{o}_{j}\rangle+|\Psi^{o}_{k}\rangle\big{)}+G^{o}_{0}(E)t^{o\beta}_{jk}(E)\big{(}Q_{\beta\gamma}|\Psi^{\gamma}_{j}\rangle+Q_{\beta\gamma}|\Psi^{\gamma}_{k}\rangle\big{)}.$
(14)

For the closed channels, the Faddeev equations can be written as an
inhomogeneous integral equations, given by

$\displaystyle Q_{\alpha\beta}|\Psi^{\beta}_{i}\rangle$ $\displaystyle=$
$\displaystyle Q_{\alpha\beta}G^{\beta}_{0}(E)t^{\beta
o}_{jk}(E)\big{(}|\Psi^{o}_{j}\rangle+|\Psi^{o}_{k}\rangle\big{)}+Q_{\alpha\delta}G^{\delta}_{0}(E)t^{\delta\beta}_{jk}(E)\big{(}Q_{\beta\gamma}|\Psi^{\gamma}_{j}\rangle+Q_{\beta\gamma}|\Psi^{\gamma}_{k}\rangle\big{)}.$
(15)

Due to the elimination of the closed channels to describe the dynamics of the
system only using the open channel, the interactions in the Faddeev equations
for the open channel gain new terms coming from the virtual propagation of the
three atoms in a closed channel.

Such interactions are given automatically by three-particle connected
operators, that means an induced three-atom interaction. To illustrate this,
we introduce the iterative solution corresponding to Eq. (15), given by

$\displaystyle
Q_{\alpha\beta}|\Psi^{\beta}_{i}\rangle=Q_{\alpha\beta}G^{\beta}_{0}(E)t^{\beta
o}_{jk}(E)\big{(}|\Psi^{o}_{j}\rangle+|\Psi^{o}_{k}\rangle\big{)}+\cdots,$
(16)

in Eq. (14), which results in the single channel Faddeev equations:

$\displaystyle|\Psi^{o}_{i}\rangle$ $\displaystyle=$ $\displaystyle
G^{o}_{0}(E)t^{oo}_{jk}(E)\big{(}|\Psi^{o}_{j}\rangle+|\Psi^{o}_{k}\rangle\big{)}+G^{o}_{0}(E)t^{o\beta}_{jk}(E)Q_{\beta\gamma}G^{\gamma}_{0}(E)t^{\gamma
o}_{ki}(E)\big{(}|\Psi^{o}_{i}\rangle+|\Psi^{o}_{k}\rangle\big{)}$ (17)
$\displaystyle+$ $\displaystyle
G^{o}_{0}(E)t^{o\beta}_{jk}(E)Q_{\beta\gamma}G^{\gamma}_{0}(E)t^{\gamma
o}_{ij}(E)\big{(}|\Psi^{o}_{i}\rangle+|\Psi^{o}_{j}\rangle\big{)}\big{)}+\cdots.$

The new connected operators, contributing to the kernel of the first terms of
Eq. (17) are

$\displaystyle t^{o\beta}_{jk}(E)Q_{\beta\gamma}G^{\gamma}_{0}(E)t^{\gamma
o}_{ki}(E)\;\;{\rm
and}\;\;t^{o\beta}_{jk}(E)Q_{\beta\gamma}G^{\gamma}_{0}(E)t^{\gamma
o}_{ij}(E)\,,$ (18)

these operators clearly indicate the propagation in closed channels, which
characterizes the nature of a three-body force as sketched in Ref. YamaEPL2006
.

It is not a complex exercise to extrapolate the above result to the full
series of connected kernel, considering any number of intermediate propagation
in closed channels. Therefore, it is clear that the Feshbach resonance also
drives a triatomic interaction, besides the two-atom scattering length, and
potentially could be controlled, at least from the theoretical point-of-view.
The framework developed so far can also be generalized to four atoms with the
corresponding Faddeev-Yakubovski equations Yak .

###  II.2 Effective three-body interaction and vanishing scattering length

In atomic traps, the scattering length can be tuned by Feshbach resonance
techniques, as shown in Ref. 1998Inouye , such that one can reach a particular
situation is which the two-body scattering length is exactly zero. In this
limit, the dynamics of the Bose-Einstein condensate, as described by the
Gross-Pitaesvki equation, says that the interaction between identical bosonic
atoms ceases to exist. However, as discussed in Sect. II.1, in the single
channel Faddeev equation we have the contribution of an effective three-body
force when the closed channels are eliminated in favor of the open channel.

By assuming that the s-wave two-atom T-matrix in the open channel is obtained
for a short-range interaction, close to a zero-range form, its matrix element
will not depend on the relative momentum, except for the dependence on the
energy of the system. In this case, the matrix elements will resemble the
amplitude written in Eq. (11), such that in the $a_{0}\to 0$ limit, the two-
body T-matrix in the open channel vanishes, i.e. $t^{oo}_{ij}(E)=0$, and for
the identical bosons system, the coupled open-closed channels Faddeev equation
(14) will simplify and given by

$|\Psi^{o}_{i}\rangle=G^{o}_{0}(E)t^{o\beta}_{jk}(E)\big{(}Q_{\beta\gamma}|\Psi^{\gamma}_{j}\rangle+Q_{\beta\gamma}|\Psi^{\gamma}_{k}\rangle\big{)}\,,$
(19)

where the term carrying the open-channel two-body T-matrix disappear, allowing
to determine the open channel Faddeev components of the wave function from the
ones in the closed channel.

By eliminating the open channel components in the closed-channel Faddeev
equations, (13), by using Eq. (19) written above, one gets the following set
of coupled equations in the closed channels:

$\displaystyle Q_{\alpha\beta}|\Psi^{\beta}_{i}\rangle$ $\displaystyle=$
$\displaystyle Q_{\alpha\beta}G^{\beta}_{0}(E)t^{\beta
o}_{jk}(E)G^{o}_{0}(E)t^{o\gamma}_{ki}(E)\big{(}Q_{\gamma\delta}|\Psi^{\delta}_{k}\rangle+Q_{\gamma\delta}|\Psi^{\delta}_{i}\rangle\big{)}$
(20) $\displaystyle+$ $\displaystyle Q_{\alpha\beta}G^{\beta}_{0}(E)t^{\beta
o}_{jk}(E)G^{o}_{0}(E)t^{o\gamma}_{ij}(E)\big{(}Q_{\gamma\delta}|\Psi^{\delta}_{i}\rangle+Q_{\gamma\delta}|\Psi^{\delta}_{j}\rangle\big{)}$
$\displaystyle+$ $\displaystyle
Q_{\alpha\delta}G^{\delta}_{0}(E)t^{\delta\beta}_{jk}(E)\big{(}Q_{\beta\gamma}|\Psi^{\gamma}_{j}\rangle+Q_{\beta\gamma}|\Psi^{\gamma}_{k}\rangle\big{)}.$

The connected operators in the first two terms of the equation, which can be
interpreted as an effective three-body interaction, come from the coupling of
the closed and open channels with a virtual propagation of the system to the
open channel coming back to the closed one. This is can be seen, for example,
in the first term of the kernel

$Q_{\alpha\beta}G^{\beta}_{0}(E)t^{\beta
o}_{jk}(E)G^{o}_{0}(E)t^{o\gamma}_{ki}(E)Q_{\gamma\delta}.$ (21)

The indices $\beta$ and $\gamma$ in $t^{\beta o}_{jk}(E)$
$G^{o}_{0}(E)t^{o\gamma}_{ki}(E)$ are referring to the closed channels and
$G^{o}_{0}(E)$ corresponds to the virtual propagation of the three-body system
in the open channel.

This extreme situation can be studied in schematic models in the limit of
zero-range interactions, for example. In what follows we will exemplify
another case modeled with a two-channel s-wave zero-range interaction model,
close to the Feshbach resonance. This setup allows the derivation of an
analytically two-body T-matrix and the built of the bound-state Faddeev
equations.

##  III  AAB system with open and closed channels: zero-range model

We substantiate the model proposed in Ref. YamaEPL2006  by assuming a Feshbach
resonance in only one pair. The two identical bosons A interact with a third
particle B, and all s-wave interactions have zero-range. First, we derive the
two-body T-matrix for the AA system, and then the bound-state Faddeev
equations for the open-closed channel AAB model are derived in the unitarity
limit, $a_{0}\to\pm\infty$. The parametric dependencies on the Feshbach
resonance are pointed out.

###  III.1 Two-channel zero range interaction model

The T-matrix of a two channel zero-range interaction in s-wave can be
analytically derived, having a simple form when the renormalized interaction
strength in the open channel vanishes. Such assumption, taken for simplicity,
allows us to study the three-body dynamics close to the Feshbach resonance,
where the coupling with the closed channel puts the system in the unitarity
limit.

This model can be renormalized by subtracting the resolvent at some given
scale (see e.g. fredericoppnp2012 ), which for convenience we choose for zero
energy. The two-body T-matrix elements of the Lippmann-Schwinger equation can
be written, in this model, in an operatorial-matrix notation as

$\displaystyle T(E)$ $\displaystyle=$ $\displaystyle\begin{bmatrix}0&\eta\\\
\eta&\lambda\end{bmatrix}+\int d^{3}p\begin{bmatrix}0&\eta
g_{0}(p^{2},E-e;0)\\\ \eta g_{0}(p^{2},E;0)&\lambda
g_{0}(p^{2},E-e;0)\end{bmatrix}T(E)\,,$ (22)

where the renormalized strengths $\lambda$ and $\eta$ refer to the interaction
in the closed channel and the coupling between the open and close channels,
respectively. The resolvent, with outgoing boundary condition, is given by

$g_{0}(p^{2},E;\mu^{2})=\frac{E+\mu^{2}}{(\mu^{2}+\frac{p^{2}}{2m_{r}}-i\varepsilon)(E-\frac{p^{2}}{2m_{r}}+i\varepsilon)}\,,$
(23)

where $m_{r}$ is the reduced mass of the two atoms. Note that, the resolvent
in the closed channel carries the energy $e$, which allows it to open only
when $E>e$ and when the scattered particles can transitioned from one channel
to the other one. In what follows, we will study the situation where $E<e$.
The operator $T(E)$, in momentum space, has matrix elements that do not depend
on any momenta, only depending on $E$ \- this is a consequence of the zero-
range interaction, used in this example.

The T-matrix equation (22) allows analytical solution in the following form,
with the channel terms given by

$\displaystyle t^{oo}(E)$ $\displaystyle=$
$\displaystyle\left[{\frac{1-\lambda
B(E-e)}{\eta^{2}B(E-e)}-B(E)}\right]^{-1},$ (24) $\displaystyle t^{oc}(E)$
$\displaystyle=$ $\displaystyle
t^{co}(E)={\eta}\left[{1-B(E-e)(\eta^{2}B(E)+\lambda)}\right]^{-1},$ (25)
$\displaystyle t^{cc}(E)$ $\displaystyle=$
$\displaystyle\left[{\frac{1}{\eta^{2}B(E)+\lambda}-B(E-e)}\right]^{-1},$ (26)

where

$B(E)\equiv\int
d^{3}p\frac{E}{(\frac{p^{2}}{2m_{r}}-i\varepsilon)(E-\frac{p^{2}}{2m_{r}}+i\varepsilon)}=-i(2\pi)^{2}\sqrt{2m_{r}^{3}E}\,,$
(27)

where we work with the hypothesis of a closed channel, $E<e$, such that
$B(E-e)=(2\pi)^{2}\sqrt{2m_{r}^{3}(e-E)}$. Therefore,

$\displaystyle t^{oo}(E)$ $\displaystyle=$
$\displaystyle\left\\{4\pi^{2}m_{r}\left[{\frac{1}{a_{0}(E)}+i\sqrt{2m_{r}E}}\right]\right\\}^{-1}\,,$
(28) $\displaystyle t^{oc}(E)$ $\displaystyle=$ $\displaystyle
t^{co}(E)={\eta}\left[{1-\zeta(E)\sqrt{2m_{r}(e-E)}}\right]^{-1}\,,$ (29)
$\displaystyle t^{cc}(E)$ $\displaystyle=$
$\displaystyle\left\\{4\pi^{2}m_{r}\left[{\frac{1}{\zeta(E)}-\sqrt{2m_{r}(e-E)}}\right]\right\\}^{-1},$
(30)

where

$\displaystyle a_{0}(E)$ $\displaystyle\equiv$
$\displaystyle\frac{(2\pi)^{4}m^{2}_{r}\eta^{2}\,\sqrt{2m_{r}(e-E)}}{1-(2\pi)^{2}m_{r}\lambda\,\sqrt{2m_{r}(e-E)}},$
(31) $\displaystyle\zeta(E)$ $\displaystyle\equiv$ $\displaystyle
4\pi^{2}m_{r}\left[\lambda-i\eta^{2}(2\pi)^{2}m_{r}\sqrt{2m_{r}E}\right].$
(32)

This example shows that by varying $e$ one can tune $a_{0}(0)$, namely the
scattering length. The model could be enriched by also considering the
interaction strength in the open channel. This is left for another work.

One interesting situation that emerges in this model corresponds to the
unitarity limit, $a_{0}(0)\to\pm\infty$. In this situation
$\lambda^{-1}=(2\pi)^{2}m_{r}\sqrt{2m_{r}e}$, giving

$\displaystyle t^{oo}(E)$ $\displaystyle=$
$\displaystyle\left\\{4\pi^{2}m_{r}\left[{\frac{1}{a_{U}(E)}+i\sqrt{2m_{r}E}}\right]\right\\}^{-1},$
(33) $\displaystyle t^{oc}(E)$ $\displaystyle=$ $\displaystyle
t^{co}(E)=-{i}\left[{2(2\pi)^{4}m_{r}^{3}\eta\sqrt{E(e-E)}}\right]^{-1}\,,$
(34) $\displaystyle t^{cc}(E)$ $\displaystyle=$
$\displaystyle\frac{1}{(2\pi)^{2}m_{r}}\left[\frac{1-i2(2\pi)^{4}\eta^{2}m_{r}^{3}\sqrt{E(e-E)}}{i\eta^{2}2(2\pi)^{4}m_{r}^{3}(e-E)\sqrt{2m_{r}E}}\right],$
(35)

where

$\displaystyle
a_{U}(E)\equiv\frac{(2\pi)^{4}m^{2}_{r}\eta^{2}\,\sqrt{2m_{r}e(e-E)}}{\sqrt{e}-\sqrt{e-E}}\,.$
(36)

For $E<0$, which corresponds to the three-body bound state,

$t^{cc}(E)=-\frac{1}{(2\pi)^{2}m_{r}}\left[\frac{1+\eta^{2}2(2\pi)^{4}m_{r}^{3}\sqrt{|E|(e-E)}}{\eta^{2}2(2\pi)^{4}m_{r}^{3}(e-E)\sqrt{2m_{r}|E|}}\right],$
(37)

leading to a strong attraction in the closed channel within the kernel of the
Faddeev equations. The other term $t^{oc}(E)$, from the coupling between the
open and closed channels, has an attractive effect in three atom system. This
will be discussed in the next section.

###  III.2 AAB coupled channel model in the unitarity limit

Our starting point is Eq. (8) and the two-body T-matrix (33) for
$a_{0}=\pm\infty$. We are assuming that the subsystem AA has two channels and
$AB$ has only one open channel. The first observation is that the matrix
elements of the T-matrix for the AA system, derived in Sect. III.1 in momentum
space, depends only on the energy, as a consequence of the of the zero-range
interaction s-wave. This is also the case for the single channel AB T-matrix.
For this particular AAB system, in units of $m_{A}=m_{B}=1$ and $\hbar=1$, it
follows that

$\langle\vec{q}_{A(B)}\vec{p}_{A(B)}|\Psi^{\alpha}_{A(B)}\rangle=\frac{f^{\alpha}_{A(B)}(\vec{q}_{A(B)})}{E-\frac{3}{4}q_{A(B)}^{2}-p_{A(B)}^{2}-\Delta_{\alpha}}\,,$
(38)

where $\vec{q}_{A(B)}$ is the relative momentum of particle $A(B)$ with
respect to the center of mass of the pair $A^{\prime}B(AA)$, with
$\vec{p}_{A(B)}$ the relative momentum of the particles $A^{\prime}$ and $B$
($A$ and $A^{\prime}$).

Considering that the bosonic wave function is symmetric by the exchange of the
identical bosons A and A’, it follows that the spectator functions
$f^{\alpha}_{i}$ should be:

$f^{\alpha}_{A}(\vec{q}_{A})=f^{\alpha}_{A^{\prime}}(\vec{q}_{A^{\prime}})\equiv
f^{\alpha}_{A}(\vec{q})\,,$ (39)

for $\vec{q}_{A}=\vec{q}_{A^{\prime}}\equiv\vec{q}$. For the AAB system only
two channels are present: AA standing in the open or closed channels. The
correspondent spectator functions are $f^{\alpha}$, with $\alpha=0,1$,
respectively, for the open, (o), and closed, (c), channels, indicating the
pair AA propagating in the open or closed channels, and correspondingly
$\Delta_{0}=\alpha\,e$. Altogether for this model, the spectator functions are
four: $f^{0}_{A}(\vec{q})\,,~{}f^{1}_{A}(\vec{q})\,,~{}f^{0}_{B}(\vec{q})\,$
and $f^{1}_{B}(\vec{q})$.

In the unitary limit, i.e., by assuming $a_{AB}=a_{AA}\to\pm\infty$ for both
interactions, the set of four coupled Faddeev equations are derived from Eq.
(8) by using Eq. (38):

$\displaystyle
f^{0}_{A}(\vec{q})=\frac{1}{2\pi^{2}\sqrt{|E|+\frac{3}{4}q^{2}}}\int
d^{3}k\,{\cal
K}_{E}^{0}(\vec{q},\vec{k})\bigg{(}f^{0}_{A}(\vec{k})+f^{0}_{B}(\vec{k})\bigg{)},$
(40) $\displaystyle
f^{0}_{B}(\vec{q})=\frac{1/(2\pi^{4}\eta)}{\sqrt{\left(|E|+e+\frac{3}{4}q^{2}\right)\left(|E|+\frac{3}{4}q^{2}\right)}}\int
d^{3}k\,{\cal K}^{1}_{E}(\vec{q},\vec{k})f^{1}_{A}(\vec{k})$
$\displaystyle+\frac{1/\pi^{2}}{\left[\frac{-1}{a_{U}(-|E|-\frac{3}{4}q^{2})}+\sqrt{|E|+\frac{3}{4}q^{2}}\,\right]}\int
d^{3}k\,{\cal K}_{E}^{0}(\vec{q},\vec{k})f^{0}_{A}(\vec{k})\,,$ $\displaystyle
f^{1}_{A}(\vec{q})=\frac{1/(2\pi^{2})}{\sqrt{|E|+e+\frac{3}{4}q^{2}}}\int
d^{3}k\,{\cal
K}_{E}^{1}(\vec{q},\vec{k})\bigg{(}f^{1}_{A}(\vec{k})+f^{1}_{B}(\vec{k})\bigg{)},$
$\displaystyle
f^{1}_{B}(\vec{q})=\frac{1/(2\pi^{4}\eta)}{\sqrt{\left(|E|+e+\frac{3}{4}q^{2}\right)\left(|E|+\frac{3}{4}q^{2}\right)}}\int
d^{3}k\,{\cal K}^{0}_{E}(\vec{q},\vec{k})f^{0}_{A}(\vec{k})$
$\displaystyle+\frac{{1}/{(2\pi^{2}\eta)^{2}}+\sqrt{\left(|E|+e+\frac{3}{4}q^{2}\right)\left(|E|+\frac{3}{4}q^{2}\right)}}{\pi^{2}(|E|+e+\frac{3}{4}q^{2})\,\sqrt{|E|+\frac{3}{4}q^{2}}}\int
d^{3}k\,{\cal K}^{1}_{E}(\vec{q},\vec{k})f^{1}_{A}(\vec{k})\,,$

where the three-body resolvent in terms of the spectator momenta is given by:

${\cal
K}^{\alpha}_{E}(\vec{q},\vec{k})=\frac{1}{|E|+\alpha\,e+q^{2}+k^{2}+\vec{q}.\vec{k}}\,,$
(44)

which takes into account the energy gap between the two channels.

The coupled set of equations (40) and (III.2) for the spectator functions
$f^{0}(\vec{q})$ and $f^{0}_{B}(\vec{q})$ reduces to the Skorniakov and Ter-
Martirosian equations STM  for an AAB system, which for three atoms in s-wave
present both to the Efimov effect and to the Thomas collapse. The Efimov
effect and Thomas collapse are related to the breaking of continuous scale
invariance to a discrete one. This demands the necessity of an ultraviolet
(UV) scale associated to the three-body one, which carries all correlations of
physical observables of the s-wave three-particle system. These correlations
can easily be represented by scaling functions (see fredericoppnp2012 ).

We have discussed in Sect. II.1 the appearance of an effective three-body
interaction, when the Faddeev equations in the open and closed channels are
reduced to a single one. The set of coupled integral equations for the
spectator functions of the AAB system, (40)-(III.2), illustrates such
dynamical mechanism. The coupling with $f^{1}_{A}$ in the open channel
equation (III.2) could be translated to an effective three-body interaction
acting in the open channel equations for $f^{0}_{A}(\vec{q})$ and
$f^{0}_{B}(\vec{q})$.

The coupled set of equations (40)-(III.2) needs to be regularized at the UV
region to avoid the collapse of the AAB system for vanishing total angular
momentum. The regularization can be done by resorting to a subtraction in the
kernel. Then instead of (44) we can use the subtracted resolvent AdhPRL1995 :

$\displaystyle{\cal K}^{\alpha}_{E}(\vec{q},\vec{k})$ $\displaystyle=$
$\displaystyle\frac{1}{|E|+\alpha\,e+q^{2}+k^{2}+\vec{q}.\vec{k}}-\frac{1}{\mu^{2}+\alpha\,e+q^{2}+k^{2}+\vec{q}.\vec{k}}\,$
$\displaystyle=$
$\displaystyle\frac{\mu^{2}-E}{\left(|E|+\alpha\,e+q^{2}+k^{2}+\vec{q}.\vec{k}\right)\left(\mu^{2}+\alpha\,e+q^{2}+k^{2}+\vec{q}.\vec{k}\right)}\,,$

such that $\mu^{2}>>e$. It will imply that, even with $\mu^{2}$ fixed, the
effect of the coupling between the open and closed channels, even at the
unitarity limit, allows to drive the Efimov states by changing $\eta$ and/or
$e$. In this model, the relevant dimensionless quantities for driving the
Efimov states at the unitarity are $e/\mu$ and $\eta\,\mu$. The energy of the
AAB system is given in units of $\mu$.

The short-range scale is related to the van der Waals interaction as $\mu\sim
1/\ell_{vdW}$. At unitarity, the single channel description of the AAB system
makes that all s-wave three-body observables be scaled with powers of $\mu$.
This is a direct consequence of the possibility to find the particles
simultaneously at the same position, i.e., depending on the configuration of
the constituents of the system, this collapse can exist for bosonic or
fermionic systems. For example, in the single channel framework given by Eqs.
(40) and (III.2), with the subtracted kernel (III.2), once the coupling with
the closed channel is disregarded, one can easily verify that at the unitarity
the three-body binding energy is proportional to $\mu^{2}$, as no other scales
are present in the integral equations in limit of a zero-range interaction.
The separation and coupling between the open and closed channel should
represent other length scales, namely $\eta$ and $1/\sqrt{e}$, which controls
the scattering length for $E=p^{2}<e$

$\displaystyle p\cot\delta^{U}_{0}$ $\displaystyle=$
$\displaystyle-\frac{\sqrt{e}-\sqrt{e-p^{2}}}{4\pi^{4}\eta^{2}\,\sqrt{e}\,\sqrt{e-p^{2}}}\Bigg{|}_{p^{2}\to
0}=-\frac{1}{2}R^{*}p^{2}\,\,\text{with}\,\,\,R^{*}=\frac{1}{4\pi^{4}\eta^{2}e\sqrt{e}}\,,$
(46)

where the parameter $R^{*}$ is controlled by the length scale
$\gamma=\eta^{-2}e^{-\frac{3}{2}}$, with $R^{*}$ related to the width of the
Feshbach resonance 2004Petrov .

The separation energy $e$ between the two channels gives the momentum scale
$\sqrt{e}$, which competes in the UV region with the subtraction scale, as it
is seen in the coupling term between the open and closed channels in Eq.
(III.2). Note that for momentum $k\sim\sqrt{e}$, the virtual propagation of
the system brings to the bound state a new scale in the UV region competing
with $\mu^{2}$. The van der Waals radius is associate to an equivalent energy
scale of $\hbar^{2}/(m\ell_{vdW}^{2})$. For example, in the 133Cs2 system,
where $\ell_{vdW}=101\,a_{0}$ 2010Chin  ($a_{0}$ the Bohr radius) the energy
$\mu^{2}\,\sim\,127~{}\mu$K sets the UV scale for the 133Cs3 system. The
energy gap between the open and closed channels will compete, in the UV
region, with the subtraction scale that can drive the triatomic system, beyond
the scattering and effective range, by moving the Feshbach resonance
parameters.

##  IV Final considerations

In summary, the three-atom bound state problem with coupled open-closed
channels close to a Feshbach resonance is formulated through the Faddeev
equations for the wave function. The effective three-body force appearing in
the single channel description was derived using the Feshbach projection
operators in the open and closed channels. Following this formal presentation,
we investigated the interesting case for which the trap setup is tuned to
vanishing scattering lengths, such that the direct open channel interaction
disappears. In this case, we demonstrated that an effective interaction in the
open channel appears due to an effective three-body interaction built from the
coupling between the open and closed channels.

We have provided one explicit example of a three-body system composed by two
identical bosons and a third different particle (AAB system) with open and
closed channels, which realizes in practice the schematic discussion we have
performed in Ref. YamaEPL2006 . In this illustrative case, the subsystem AA
has an open and a closed channel separated by an energy gap, and interacting
through a zero range s-wave potential. The transition matrix for the AA
subsystem is derived resorting to a subtractive renormalization scheme, at the
expense of introducing two finite parameters, besides the energy gap. The
subsystem AA and AB are tuned to the unitary limit, and the Faddeev equations
for the bound AAB system was derived explicitly, to expose the dependences of
the kernel with the two body parameters and energy gap. The set of coupled
equations allowed to identify the effect of the coupling between the Faddeev
components in the open channel with the closed channel ones. The energy gap
between the channels essentially provides an UV scale probed by the virtual
propagation of the three-body system in the closed channel. This short-range
scale competes with the van der Waals radius that sets the short-range physics
of the system in the open channel. Therefore, we have clearly illustrated how
the three-atom system can be tuned by the Feshbach resonance beyond the
scattering length and van der Waals radius.

As a final comment, considering the different possibilities to study few-body
physics in atomic traps, we should mention that few-atom system can also be
driven by the change in the effective dimensions, by squeezing the trap, while
tuning both the two and few-body parameters controlling the Feshbach
resonance. Controlled cold chemical reactions 2008Krems ; 2009Carr ;
2010Ospelkaus  with forces driven by external fields is also a field of
intense activity (see e.g. 2017Rui ; 2018Hoffmann ; 2018Gao ; 2019Yang ;
2019Li ; 2019Wang ; 2020Reynolds ; 2020Makrides ).

## Acknowledgements

This article is dedicated to the memory of Prof. Mahir Saleh Hussein, for his
relevant contributions in Physics. We thank Vanderlei Bagnato for calling our
attention to the possibility of zero scattering length in atomic traps. The
following Brazilian agencies are being acknowledge for their partial support:
Conselho Nacional de Desenvolvimento Científico e Tecnológico [Procs.
303579/2019-6 (MTY), 308486/2015-3 (TF), 304469-2019-0(LT) and INCT-FNA
Proc.464898/2014-5] and Fundação de Amparo à Pesquisa do Estado de São Paulo
[Projs. 2019/00153-8 (MTY), 2017/05660-0 (LT and TF) and 2019/07767-1 (TF)].

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