# Borromean states in a one-dimensional three-body system

Tobias Schnurrenberger  German Aerospace Center (DLR), Institute of Quantum
Technologies, 89081 Ulm, Germany  Lucas Happ  Few-body Systems in Physics
Laboratory, RIKEN Nishina Center, Wakō 351-0198, Japan  Maxim A. Efremov
German Aerospace Center (DLR), Institute of Quantum Technologies, 89081 Ulm,
Germany

(August 28, 2024)

###### Abstract

We show the existence of Borromean bound states in a one-dimensional quantum
three-body system composed of two identical bosons and a distinguishable
particle. It is assumed that there is no interaction between the two bosons,
while the mass-imbalanced two-body subsystems can be tuned to be either bound
or unbound. Within the framework of the Faddeev equations, the three-body
spectrum and the corresponding wave functions are computed numerically. In
addition, we identify the parameter-space region for the two-body interaction,
where the Borromean states occur, evaluate their geometric properties, and
investigate their dependence on the mass ratio.

##  I Introduction

Few-body physics plays a central role in the fields of ultra-cold quantum
gases [1], nuclear physics [2] and hadron physics [3]. Special attention is
given to three-body systems that have resonantly interacting two-body
subsystems because they often show universal properties that are independent
of the details of their short range potentials [4]. Three-body bound states
can even exist in situations when all of the two-body subsystems are unbound
and those states are then named Borromean states [5, 6].

In three spatial dimensions, a two-body system with an overall attractive
potential may be unbound, when the interaction is weak enough. A system of
three identical bosons with the same pairwise interaction can however be
bound. Therefore there is a window for the coupling constant where Borromean
states can occur [7]. The Efimov effect is a well known example to show this
behaviour [8, 9] and the theoretical predictions have been verified
experimentally [6]. Moreover, such Borromean binding plays an essential role
in subatomic physics, e.g. in halo nuclei [5, 10].

When the particles are restricted to a two dimensional plane, the situation is
different. Here a two-body interaction with an overall attractive contribution
always supports a bound state [11]. Therefore the bound states of a three-body
system with such two-body interactions are not Borromean. Nevertheless it has
been shown that Borromean three-body states can exist in two dimensions by
adding a repulsive contribution to the two-body interaction potential [12,
13].

Similar to two dimensions, a one-dimensional two-body system with an overall
attractive potential always has a bound state [11]. This makes it difficult to
find situations where Borromean states can occur. To the best of our
knowledge, Borromean states have not yet been predicted or observed in a one-
dimensional setup. In this paper we consider a two-body interaction that has
tunable attractive and repulsive contributions, which are separated in space.
In this way, the parameters for this potential can be chosen such that the
two-body system is either bound or unbound. Further, our three-body system
consist of two types of particles with different masses. We solve this three-
body system numerically with the Faddeev equations in momentum space [14] and
find that this three-body system indeed has Borromean states. Moreover, we
identify their region of existence in the parameter-space of the two-body
interaction and analyze their geometric properties. In addition, we find that
the number of Borromean states increases for larger mass ratios.

Considering the experimental feasibility, quasi one-dimensional systems in the
form of cigar shaped traps for ultra cold quantum gases have been realized
[15]. In addition, the interaction between different types of atoms can be
tuned with Feshbach resonances [16, 17] or confinement-induced resonances [18,
19]. We have performed our calculations for the mass ratio corresponding to a
Cs-Li mixture [20, 21]. By identifying the rate of the one-dimensional three-
body recombination processes [22], we think it will be possible to examine the
predicted Borromean states in future experimental studies.

The article is organized as follows. In section II we introduce the underlying
two-body subsystems as well as the corresponding three-body system. We also
discuss the Faddeev equations that we use in our numerical calculations. In
section III we show from our numerical results the existence of Borromean
states and their dependence on the system parameters. In section IV we
summarize our findings. The Appendices A, B and C provide further details on
our calculations.

##  II Two- and three-particle systems

###  II.1 Two-particle subsystem

Figure 1:  The parameter-space $(\alpha,v_{0})$ of the two-body subsystems
divided into four separate regions with respect to the number of bound and
virtual states. From top-left to bottom-right: one virtual state (region I),
one bound state (region II), one virtual and one bound state (region III), two
bound states (region IV). The dashed line indicates $\alpha_{c}(v_{0})$ given
by Eq. (3).

We start from a one-dimensional two-particle system consisting of
distinguishable particles with masses $M$ and $m$. The stationary Schrödinger
equation for the two-particle wave function $\psi^{(2)}(x)$ is then given by

$\displaystyle\left[-\frac{1}{2}\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}+v(x)\right]\psi^{(2)}(x)=\mathcal{E}^{(2)}\psi^{(2)}(x),$
(1)

with an interaction potential $v(x)$. A two-body potential in one-dimension
with $\int\mathrm{d}x\leavevmode\nobreak\ v(x)<0$ always has a bound state
[11]. In contrast, a single repulsive barrier does not support a bound state.
In order to be able to tune the system between a bound and unbound regime, we
need both, an attractive and repulsive term. For simplicity, and in order to
treat the two-body system analytically, we consider a potential consisting of
two delta distributions

$\displaystyle
v(x)=-v_{0}\left[\delta\left(x-\frac{1}{2}\right)-\alpha\delta\left(x+\frac{1}{2}\right)\right]$
(2)

with the parameters $v_{0}>0$ and $\alpha$, describing the magnitudes of the
overall potential and its relative repulsive contribution respectively.

Both equations (1) and (2) are presented in dimensionless variables. The
position coordinate $x$ is measured in units of the distance $a$ between the
two delta functions. The potential strength $v_{0}$ and the two-particle
energy $\mathcal{E}^{(2)}$ are both given in units of a characteristic energy
$\hbar^{2}/\mu a^{2}$, where $\mu=Mm/(M+m)$ is the reduced mass.

In Appendix A we solve Eq. (1) analytically, derive a transcendental equation
for the two-particle energy $\mathcal{E}^{(2)}$ and discuss the number of
solutions depending on $\alpha$. For $v_{0}>0$ we can distinguish four regions
in the parameter-space $(\alpha,v_{0})$, Fig. 1, corresponding to different
numbers of bound and virtual states [23] in our two-body subsystem. For
$\alpha>0$ the line

$\displaystyle\alpha_{c}(v_{0})=\frac{1}{1-2v_{0}}$ (3)

corresponds to $\mathcal{E}^{(2)}=0$ and separates the region I with a single
virtual state and region II with a single bound state. To explore the
Borromean states in the three-particle system, we focus on the transition
between these two regions. For $\alpha<0$, the two-particle subsystem supports
either one virtual and one bound state (region III), or two bound states
(region IV).

###  II.2 Three-particle system

Figure 2:  The Jacobi coordinates $x_{1}$ and $y_{23}$ for the one-dimensional
three-particle (BBX) system. The dot shows the center-of-mass of two identical
bosons, particles $2$ and $3$.

Next, we consider a three-body system in one dimension with two identical
bosons (B) of mass $M$ and one distinguishable particle (X) of mass $m$. We
assume no interaction between the two bosons, while the BX subsystems interact
via the potential $v(x)$, Eq. (2).

By eliminating the center-of-mass motion of the three-body BBX system,
depicted in Fig. 2, we arrive at the stationary Schrödinger equation

$\displaystyle\left[H_{0}+V_{21}+V_{31}\right]\psi^{(3)}=\mathcal{E}^{(3)}\psi^{(3)}$
(4)

for the three-particle wave function $\psi^{(3)}(x_{1},y_{23})$, where the
Hamiltonian of the free motion is given by

$\displaystyle
H_{0}=-\frac{\alpha_{x}}{2}\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}_{1}}-\frac{\alpha_{y}}{2}\frac{\mathrm{d}^{2}}{\mathrm{d}y_{23}^{2}}$
(5)

and both of its coefficients

$\displaystyle\alpha_{x}=\frac{m+2M}{2(m+M)}\quad\text{and}\quad$
$\displaystyle\alpha_{y}=\frac{2m}{m+M}$ (6)

depend only on the mass ratio $M/m$. The two-body potential terms

$\displaystyle
V_{21}=v\left(x_{1}+\frac{y_{23}}{2}\right)\quad\text{and}\quad$
$\displaystyle V_{31}=v\left(x_{1}-\frac{y_{23}}{2}\right)$ (7)

for the BX subsystems are given by the function $v(x)$, Eq. (2), with the
arguments being the relative distances between the respective particles B and
X.

In this article, we are interested in finding the Borromean states and their
main features in our system. A three-body state is called Borromean, if it is
bound while all of its two-body subsystems are unbound by themselves [5, 6].
For our interaction potential $v(x)$, Eq. (2), the Borromean states should
therefore lie in the region I of the parameter-space $(\alpha,v_{0})$,
displayed in Fig. 1, where there is no two-body bound state.

###  II.3 Faddeev equations

We employ the Faddeev equations [24, 25, 26] in order to calculate the bound-
state spectrum and the wave functions of the BBX system. For a derivation of
the explicit form of the Faddeev equations for our type of system, we refer to
Appendix A of the paper [14]. In practice, all information is encapsulated in
the set of one-dimensional integral equations

$\displaystyle\varphi_{\lambda}(p,\mathcal{E})=\leavevmode\nobreak\ $
$\displaystyle\sum_{\nu}\int_{\mathbb{R}}\frac{\mathrm{d}q}{2\pi}K_{\lambda\nu}(p,q,\mathcal{E})\varphi_{\nu}(q,\mathcal{E})$
(8)

for the functions $\varphi_{\nu}(p,\mathcal{E})$ with the kernel

$\displaystyle K_{\lambda\nu}(p,q,\mathcal{E})=\leavevmode\nobreak\ $
$\displaystyle\frac{g_{\lambda}(q+\beta
p,\mathcal{E}_{p})\,g^{\ast}_{\nu}(p+\beta
q,\mathcal{E}_{q})}{\mathcal{E}-\frac{1}{2}q^{2}-\frac{1}{2}p^{2}-\beta
pq}\tau_{\nu}(\mathcal{E}_{q})$ (9)

and the shorthand notation

$\displaystyle\beta=\frac{M}{M+m}\quad\mathrm{and}\quad\mathcal{E}_{p}=\mathcal{E}-\frac{1}{2}\alpha_{x}\alpha_{y}p^{2}.$
(10)

The functions $g_{\nu}(k,\mathcal{E})$ and $\tau_{\nu}(\mathcal{E})$ originate
from the separable expansion for the off-shell $t$-matrix of the BX subsystem,
whereas the indices $\lambda$ and $\nu$ denote the number of terms in this
expansion. As shown in Appendix B, for the specific form of our interaction
potential $v(x)$, Eq. (2), the separable expansion has exactly two terms,
$\nu\in\\{-,+\\}$, and $g_{\nu}(k,\mathcal{E})$ and $\tau_{\nu}(\mathcal{E})$
can be derived analytically.

We solve Eq. (8) numerically, Appendix C, and obtain the energy $\mathcal{E}$
of three-body bound states together with their functions
$\varphi_{\nu}(p,\mathcal{E})$. The latter can be used to construct the
Faddeev component

$\Phi(k,p)=\sum_{\nu}g_{\nu}(k,\mathcal{E}_{p})\tau_{\nu}(\mathcal{E}_{p})\varphi_{\nu}(p,\mathcal{E})$
(11)

and finally the wave function

$\displaystyle\psi(p_{1},k_{23})$ $\displaystyle=\leavevmode\nobreak\
G_{0}(p_{1},k_{23},\mathcal{E})$
$\displaystyle\times\biggl{[}\Phi\left(-\alpha_{x}p_{1}-\frac{\alpha_{y}}{2}k_{23},-\frac{1}{2}p_{1}+k_{23}\right)$
$\displaystyle+\hskip
5.26376pt\Phi\left(-\alpha_{x}p_{1}+\frac{\alpha_{y}}{2}k_{23},-\frac{1}{2}p_{1}-k_{23}\right)\biggl{]}$
(12)

of the three-body system in momentum space. Here,

$G_{0}(p_{1},k_{23},\mathcal{E})=\frac{(2\pi)^{2}}{\mathcal{E}-\frac{1}{2}\alpha_{x}p_{1}^{2}-\frac{1}{2}\alpha_{y}k_{23}^{2}}$
(13)

is the Green function of the free three-body system, corresponding to the
Hamiltonian $H_{0}$, Eq. (5).

In this way, the wave function in position space can be obtained from the
Fourier transform

$\psi^{(3)}(x_{1},y_{23})=\int_{\mathbb{R}}\frac{\mathrm{d}p_{1}}{2\pi}\int_{\mathbb{R}}\frac{\mathrm{d}k_{23}}{2\pi}\mathrm{e}^{\mathrm{i}(p_{1}x_{1}+k_{23}y_{23})}\psi^{(3)}(p_{1},k_{23})$
(14)

of the wave function in momentum space.

##  III Borromean states

In this section we study the three-body spectrum of the BBX system and the
conditions for the parameters $\alpha$ and $v_{0}$ under which the Borromean
states exist, as well as the ground state wave function and its geometric
properties. Further, we examine the dependence of Borromean states on the mass
ratio $M/m$.

The goal is to observe the behavior of the three-body BBX system, while its BX
subsystems undergo a transition from supporting exactly one bound state to
supporting only a single virtual state. To do this, we are tuning the
parameters $\alpha$ and $v_{0}$ for both two-body interactions $V_{21}$ and
$V_{31}$ simultaneously, so that both BX subsystems are identical at all
times.

###  III.1 Existence and Borromean window

Figure 3:  Three-body spectrum as a function of the repulsion parameter
$\alpha$ for the coupling constant $v_{0}=0.32$ and the mass ratio $M/m=22.2$.
The dashed line shows the energy $\mathcal{E}^{(2)}_{0}$ of the two-body bound
state (left) or virtual state (right), whereas the dotted lines display the
energies $\mathcal{E}^{(3)}_{n}$ of the three-body bound states. As $\alpha$
approaches $\alpha_{c}$ from below (left), the energies
$\mathcal{E}^{(3)}_{1}$ and $\mathcal{E}^{(3)}_{2}$ of the excited three-body
states follow the two-body energy described by the power law, Eq. (15), and
vanish at $\alpha=\alpha_{c}$. On the contrary, the three-body ground state
has a finite, negative energy $\mathcal{E}^{(3)}_{0}$ at $\alpha=\alpha_{c}$
and remains bound even for $\alpha>\alpha_{c}$ (right), until
$\alpha=\alpha_{w}$. As for $\alpha>\alpha_{c}$ there is no two-body bound
state, this three-body state is Borromean, and the interval
$\alpha_{c}<\alpha<\alpha_{w}$ defines the Borromean window.  Figure 4:
Contour plots of the Borromean ground state wave function
$|\psi^{(3)}(x_{1},y_{23})|$ in position space spanned by the Jacobi
coordinates $x_{1}$ and $y_{23}$, Fig. 2, for three different values of the
repulsion parameter: $\alpha=0$ (a), $\alpha=2.11$ (b), $\alpha=3.84$ (c), as
well as $v_{0}=0.32$ and $M/m=22.2$. With increasing $\alpha$, the wave
function starts to become increasingly skewed towards the positive $x_{1}$
direction, whereas it remains symmetric in $y_{23}$ direction. Similarly to
the energy spectrum shown in Fig. 3, this trend is continuous and no sudden
change occurs when the state becomes Borromean. For all values of $\alpha$,
the derivative of the wave function jumps at the positions of the attractive
(repulsive) contact interactions, highlighted by purple (red) dashed lines.
Figure 5:  Upper-left quadrant of Fig 1, displaying the parameter space
$(\alpha,v_{0})$. We highlight two curves: (i) the two-body threshold
$\alpha_{c}$, given by Eq. (3), separates regions (I) and (II) with one
virtual and one bound state in the BX subsystems, respectively. (ii) The
three-body threshold $\alpha_{w}$ marks the line, above which there is no
three-body bound state. It generally depends on the mass ratio, here
$M/m=22.2$, and is determined numerically. Together, both lines define an area
(“Borromean window”) between the dotted and dashed curves in which our system
has Borromean states.  Figure 6:  The expectation values $\braket{x_{1}}$,
$\braket{y_{23}}$ (a) and the standard deviations $\sigma_{x_{1}}$,
$\sigma_{y_{23}}$ (b), Eq. (17), as functions of $\alpha$. Top: With
increasing $\alpha$, also $\braket{x_{1}}$ increases, meaning that the
distinguishable particle is found on average further away from the center-of-
mass of the two bosons. Close to the dissociation point $\alpha_{w}$,
$\braket{x_{1}}$ becomes very large. As the other two particles are identical,
$\braket{y_{23}}$ remains zero for all values of $\alpha$. Bottom: both
standard deviations $\sigma_{x_{1}}$, $\sigma_{y_{23}}$ grow with increasing
$\alpha$. Overall, the effect is stronger for the $x_{1}$ coordinate and for
both directions strongly enhanced close to the dissociation point $\alpha_{w}$
where the three-body state becomes very dilute.

We start in parameter-region II, Fig. 1, at the point $(v_{0}=0.32,\alpha=0)$,
where the potential $v(x)$, Eq. (2), has only a single delta well and
therefore the BX subsystem supports exactly one bound state. We confirm the
correctness of our calculations by setting the mass ratio to $M/m=22.2$,
corresponding to a Cs-Li mixture, and reproducing the energy spectrum

$\displaystyle\mathcal{E}^{(3)}_{0}=$ $\displaystyle\enspace
2.7515\;\mathcal{E}^{(2)}$ $\displaystyle\mathcal{E}^{(3)}_{1}=$
$\displaystyle\enspace 1.3604\;\mathcal{E}^{(2)}$
$\displaystyle\mathcal{E}^{(3)}_{2}=$ $\displaystyle\enspace
1.0525\;\mathcal{E}^{(2)}$

of three-body bound states reported previously in Ref. [27] and obtained with
the Skorniakov-Ter-Martirosian method [28]. Here, $\mathcal{E}^{(3)}_{0}$,
$\mathcal{E}^{(3)}_{1}$, and $\mathcal{E}^{(3)}_{2}$ denote the energies of
the ground, first excited, and second excited three-body state, respectively,
in units of the two-body energy $\mathcal{E}^{(2)}$.

Positive values for $\alpha$ introduce a repulsive barrier in the potential
$v(x)$, Eq. (2). Therefore, increasing $\alpha$ pushes the two-body energy
$\mathcal{E}^{(2)}$ of the single bound state in the BX subsystems closer to
the threshold, $\mathcal{E}^{(2)}=0$, which is reached at $\alpha=\alpha_{c}$.
For the corresponding BBX system, increasing the barrier strength $\alpha$
results in larger energies $\mathcal{E}^{(3)}_{n}$, i.e. weaker binding, for
all three bound states, as displayed in Fig. 3. When $\alpha$ approaches
$\alpha_{c}$, only the energies $\mathcal{E}^{(3)}_{1}$ and
$\mathcal{E}^{(3)}_{2}$ of the excited states vanish together with the energy
$\mathcal{E}^{(2)}$, depicted by the dashed line. The linear behavior in the
log-log scale of Fig. 3 suggests a power law dependence, and indeed in
Appendix A we find

$\mathcal{E}^{(2)}\approx-\frac{1}{2}\left[\frac{v_{0}(1-2v_{0})^{2}}{(1-v_{0})^{2}+v_{0}^{2}}\right]^{2}(\alpha-\alpha_{c})^{2}$
(15)

as $\alpha\rightarrow\alpha_{c}$. In sharp contrast, the three-body ground
state does not dissociate. Instead its energy $\mathcal{E}^{(3)}_{0}$ takes a
finite, negative value at $\alpha=\alpha_{c}$, Fig. 3.

By increasing $\alpha$ beyond $\alpha_{c}$, we enter region I of Fig. 1, where
the two-body systems become unbound and each only has one virtual state. Here,
the BBX system is however able to retain the bound ground state. We emphasize
that both BX subsystems are unbound, hence the three-body state is Borromean.
Moreover, we note that the energy $\mathcal{E}^{(3)}_{0}$ of the Borromean
state as a function of $\alpha$ changes continuously at the point
$\alpha=\alpha_{c}$. In addition, we observe from Fig. 3, that for even larger
values of $\alpha$, the Borromean state eventually disappears at
$\alpha=\alpha_{w}$, above which there is no three-body bound state anymore.
For $\alpha\rightarrow\alpha_{w}$, we numerically find that
$\mathcal{E}^{(3)}_{0}$ follows the power law

$\mathcal{E}^{(3)}_{0}\approx-0.0014471\;(\alpha_{w}-\alpha)^{1.0417}$ (16)

with $\alpha_{w}=3.8951$. Here the energy $\mathcal{E}^{(3)}_{0}$ is no longer
smaller than the energy $\mathcal{E}^{(2)}$ of the corresponding virtual state
in the two-body subsystem.

In summary, we find that for a given potential strength $v_{0}$ and mass ratio
$M/m$, a Borromean state exists in the BBX system in the window
$\alpha_{c}<\alpha<\alpha_{w}$. By determining $\alpha_{c}$ and $\alpha_{w}$
for multiple values of $v_{0}$, we identify an area in the parameter space
$(\alpha,v_{0})$, Fig. 1, where the Borromean state occurs, as depicted in
Fig. 5. Here it becomes evident that a larger magnitude $v_{0}$ leads to a
wider “Borromean window” $\left(\alpha_{c},\alpha_{w}\right)$ for $\alpha$.

###  III.2 Geometric properties

Studying the geometric properties of the three-body ground state gives us
insight into its particle configuration. For that we calculate the wave
function $\psi^{(3)}(x_{1},y_{23})$ of the BBX system and present a contour
plot of $|\psi^{(3)}(x_{1},y_{23})|$ in Fig. 4 for three different values of
the parameter $\alpha$: $\alpha=0$, (a), $\alpha=2.11$, (b), and
$\alpha=3.84$, (c). To visualize the interaction potentials in both BX
subsystems, we plot additionally the dashed lines $x_{1}\pm y_{23}/2=1/2$
(purple) and $x_{1}\pm y_{23}/2=-1/2$ (red) describing the attractive and
repulsive delta wells in $v(x)$, Eq. (2), respectively. In addition, we
calculate the expectation values $\braket{x_{1}}$ and $\braket{y_{23}}$, as
well as the standard deviations $\sigma_{x_{1}}$ and $\sigma_{y_{23}}$ as
functions of $\alpha$, with

$\sigma_{z}=\sqrt{\braket{z^{2}}-\braket{z}^{2}},$ (17)

and display them in Figs. 6 (a) and (b), accordingly.

For zero repulsion in the interaction ($\alpha=0$), Fig. 4 (a) shows that the
wave function $|\psi^{(3)}(x_{1},y_{23})|$ is point symmetric with respect to
its expectation values $\braket{x_{1}}=1/2$ and $\braket{y_{23}}=0$. Since
$y_{23}$ is the distance between the identical bosonic particles, Fig. 2, the
wave function $\psi^{(3)}(x_{1},y_{23})$ is symmetric in this coordinate, i.e.
$\psi^{(3)}(x_{1},-y_{23})=\psi^{(3)}(x_{1},y_{23})$, giving rise to
$\braket{y_{23}}=0$ for all parameters. Apart from the shift by 1/2 in $x_{1}$
direction, this result coincides with the one found previously [27] for the
same three-body system but with the single contact interaction being centered.

Next, we increase the repulsion, by setting $\alpha=2.11$, a bit below
$\alpha_{c}=2.78$, and present the resulting wave function in Fig. 4 (b). Here
we have a shift of the expected position $\braket{x_{1}}=2.31$ of the
distinguishable particle, whereas $\braket{y_{23}}=0$, as shown in Fig. 6 (a).
Moreover, the wave function becomes broader compared to the case with
$\alpha=0$, Fig. 4 (a), which is also indicated by the scale of
$|\psi(x_{1},y_{23})|$ as well as the standard deviations $\sigma_{x_{1}}$ and
$\sigma_{y_{23}}$, presented in Fig. 6 (b). The result also agrees with the
fact that the ground state energy $\mathcal{E}^{(3)}_{0}$ at $\alpha=2.11$ is
smaller by about one order in comparison with $\mathcal{E}^{(3)}_{0}$ at
$\alpha=0$, as displayed by Fig. 3.

Now increasing $\alpha$ to the value slightly below $\alpha_{w}$, which
identifies the Borromean three-body threshold, Fig. 4 (c), the wave function
$\psi(x_{1},y_{23})$ shows that the distinguishable particle moves on average
even further away, $\braket{x_{1}}=19.5$, from the center-of-mass of the
identical particles, whereas $\braket{y_{23}}=0$, Fig. 6 (a). In addition, the
wave function becomes again broader and extremely delocalized with very weak
binding energy $\mathcal{E}^{(3)}_{0}$ as $\alpha\rightarrow\alpha_{w}$, Fig.
3, before the Borromean ground state dissociates completely. This observation
is in line with both standard deviations $\sigma_{x_{1}}$ and
$\sigma_{y_{23}}$ becoming very large when $\alpha$ approaches $\alpha_{w}$,
Fig. 6 (b).

It is important to note, that the wave function continuously changes as the
function of $\alpha$ throughout Figs. 4 (a), (b) and (c), as do the
expectation values presented in Fig. 6.

###  III.3 Dependence on the mass ratio

Figure 7:  Three-body spectrum as a function of the mass ratio $M/m$ and with
the coupling constant $v_{0}=0.32$. We choose the repulsion parameter $\alpha$
just above $\alpha_{c}$, i.e. inside the Borromean window, Fig. 5, and
therefore the displayed states are Borromean. With increasing mass ratio, the
system goes from supporting zero, to one and to two Borromean bound states.
Figure 8:  Three-body spectrum as a function of $\alpha$ with constant
$v_{0}=0.32$ for the mass ratios $M/m=0.2$ (a) and $M/m=720$ (b). Top: The BBX
systems supports only a single three-body bound state for $\alpha<\alpha_{c}$
and it dissociates at $\alpha=\alpha_{c}$, leading to no Borromean state for
the small mass ratio. Bottom: For the much larger mass ratio, the three-body
system supports many bound states for $\alpha<\alpha_{c}$. Here, only the
first three bound states with the lowest energies are presented. The two
deepest bound states remain bound also for $\alpha>\alpha_{c}$ and are hence
Borromean states.

Finally, we analyze the dependence of the Borromean three-body energy on the
mass ratio $M/m$. This is shown in Fig. 7. Here we choose $\alpha$ to be just
slightly above $\alpha_{c}$, i.e. we are just inside the Borromean window,
where the Borromean states are relatively deeply bound. Nevertheless, we see
that for smaller mass ratios, the binding energy of the Borromean ground state
(blue line) quickly approaches zero. On the other hand, as the mass ratio
increases, this Borromean state becomes more strongly bound. Moreover, when
the mass ratio is sufficiently large ($M/m\gtrsim 600$), a second (excited)
Borromean state appears, as depicted by a green line. We note that the energy
$\mathcal{E}^{(3)}$ is expressed in units of $\hbar^{2}/(\mu a^{2})$ with the
reduced mass $\mu$. Overall, for larger mass ratios the three-body states
become more bound, and the Borromean states appear one by one.

To trace back the origin of the second Borromean state, we analyze the three-
body spectrum for different values of $\alpha$ and $M/m$, as demonstrated in
Fig. 8. In subfigure (a) we choose $M/m=0.2$ and we cannot find a Borromean
state. There is just a single non-Borromean bound state for
$\alpha<\alpha_{c}$. That’s similar to the case, when the subsystems have a
zero-range interaction, then the three-body system does not support more than
one bound state for $M/m\leq 1$ [29]. At an intermediate value of the mass
ratio, e.g. $M/m=22.2$ in Fig. 3, we clearly see the three-body ground state
in the Borromean region, $\alpha_{c}<\alpha<\alpha_{w}$. Finally, for
$M/m=720$, Fig. 8 (b), the ground state is quite deeply bound and we see the
second state (green line) in the Borromean window. Again, the energy of all
three-body bound states changes smoothly as a function of $\alpha$ when
crossing the value $\alpha=\alpha_{c}$.

##  IV Conclusion

In this article we have investigated Borromean states in a three-body BBX
system confined in one dimension, provided no interaction in the BB subsystem
and each BX subsystem supports only a single virtual state. By solving the
Faddeev equations numerically, we have calculated the spectrum and the
corresponding wave functions of the BBX system. We have demonstrated for the
first time the existence of Borromean states in 1D and, for a given mass
ratio, identified an area in the parameter-space, where these states occur.
Further, we have found that the number of Borromean states and their binding
energies strongly depend on the mass ratio of the two particles types. In
addition, we have shown that in all cases these novel states originate from
ordinary bound ones with the lowest three-body binding energies.

As an outlook, for experimental verification of our theoretical predictions
and their applications in physics of quantum gases, it is useful to calculate
the rate of the one-dimensional three-body recombination processes for the BBX
system considered here in detail. Moreover, further theoretical studies of
Borromean states in one dimension might involve the use of more realistic
atom-atom potentials, like the Lennard-Jones potential, that has an attractive
and repulsive contribution.

## Acknowledgment

We thank A. Volosniev and P. Belov for fruitful discussions and advice. The
work of L. H. is gratefully supported by the RIKEN special postdoctoral
researcher program. The authors gratefully acknowledge the scientific support
and HPC resources provided by the German Aerospace Center (DLR). The HPC
system CARA is partially funded by “Saxon State Ministry for Economic Affairs,
Labour and Transport“ and „Federal Ministry for Economic Affairs and Climate
Action”.

##  Appendix A BX subsystem

Figure 9:  The left- and right-hand sides of the transcendental equation (25)
as functions of $\kappa_{I}$ with $v_{0}=0.32$. We look for solutions
$\kappa_{I}\neq 0$. In subfigure (a) $\alpha=0.4$ and the intersection of
these lines is marked by a dot and occurs for $\kappa_{I}>0$, corresponding to
the bound state. In subfigure (b) $\alpha=10$, the intersection occurs for
$\kappa_{I}<0$ and corresponds to the virtual state.  Figure 10:  Comparison
of the two-body energy $\mathcal{E}^{(2,a)}$, given by the asymptotic
expansion, Eq. (31), to $\mathcal{E}^{(2,n)}$ obtained as the numerical
solution of the transcendental equation (25). Subfigure (a) shows the solution
for the bound state with $\alpha<\alpha_{c}$, whereas subfigure (b) displays
the solution for the virtual state with $\alpha_{c}<\alpha$.

In this Appendix we derive the transcendental equation for the spectrum of the
BX subsystem. Moreover, we discuss the number of solutions as a function of
$\alpha$ and derive the asymptotic behavior of the two-body energy near the
threshold $\alpha\to\alpha_{c}$.

We start from the stationary Schrödinger equation

$\displaystyle\left[-\frac{1}{2}\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}+v(x)\right]\psi^{(2)}(x)=\mathcal{E}^{(2)}\psi^{(2)}(x),$
(18)

where the two-body interaction potential $v(x)$ is given by the potential

$\displaystyle
v(x)=-v_{0}\left[\delta\left(x-\frac{1}{2}\right)-\alpha\delta\left(x+\frac{1}{2}\right)\right]$
(19)

with positive $v_{0}$.

With $\kappa$ defined as $\mathcal{E}^{(2)}=\kappa^{2}/2$, the solution of Eq.
(18) can be written in the form

$\displaystyle\psi^{(2)}(x)=\begin{cases}A\mathrm{e}^{\mathrm{i}\kappa
x}+A^{\prime}\mathrm{e}^{-\mathrm{i}\kappa x},&x<-\frac{1}{2}\\\
B\mathrm{e}^{\mathrm{i}\kappa x}+B^{\prime}\mathrm{e}^{-\mathrm{i}\kappa
x},&-\frac{1}{2}<x<\frac{1}{2}\\\ C\mathrm{e}^{\mathrm{i}\kappa
x}+C^{\prime}\mathrm{e}^{-\mathrm{i}\kappa x},&\frac{1}{2}<x,\end{cases}$ (20)

where $A$, $A^{\prime}$, $B$, $B^{\prime}$, $C$, and $C^{\prime}$ are
constants.

To determine them, we first incorporate the outgoing-wave boundary condition
[30], that is $A=C^{\prime}=0$. Next, we require the wave function to be
continuous at $x=-\frac{1}{2}$ and $x=\frac{1}{2}$ and arrive at two
conditions

$\displaystyle A^{\prime}\mathrm{e}^{\mathrm{i}\kappa/2}=$
$\displaystyle\;B\mathrm{e}^{-\mathrm{i}\kappa/2}+B^{\prime}\mathrm{e}^{\mathrm{i}\kappa/2}$
(21) $\displaystyle
B\mathrm{e}^{\mathrm{i}\kappa/2}+B^{\prime}\mathrm{e}^{-\mathrm{i}\kappa/2}=$
$\displaystyle\;C\mathrm{e}^{\mathrm{i}\kappa/2}.$ (22)

Integrating the Schrödinger equation (18) on the intervals
$x\in\left[-\frac{1}{2}-\epsilon,-\frac{1}{2}+\epsilon\right]$ and
$x\in\left[\frac{1}{2}-\epsilon,\frac{1}{2}+\epsilon\right]$, and then taking
the limit $\epsilon\to 0$, gives us expressions for the jump of the derivative
of the wave function at $x=-\frac{1}{2}$ and $x=\frac{1}{2}$, respectively.
These expressions yield the two additional conditions

$\displaystyle
A^{\prime}\left(1+2\mathrm{i}\frac{v_{0}\alpha}{\kappa}\right)\mathrm{e}^{\mathrm{i}\kappa/2}+B\mathrm{e}^{-\mathrm{i}\kappa/2}-B^{\prime}\mathrm{e}^{\mathrm{i}\kappa/2}=$
$\displaystyle\;0$ (23)
$\displaystyle-B\mathrm{e}^{\mathrm{i}\kappa/2}+B^{\prime}\mathrm{e}^{-\mathrm{i}\kappa/2}+C\left(1-2\mathrm{i}\frac{v_{0}}{\kappa}\right)\mathrm{e}^{\mathrm{i}\kappa/2}=$
$\displaystyle\;0.$ (24)

The system of four linear equations (21)–(24) for the coefficients
$A^{\prime}$, $B$, $B^{\prime}$, and $C$ has non-trivial solutions only if the
determinant of this system equals zero, giving rise to a transcendental
equation for $\kappa=\kappa_{R}+\mathrm{i}\kappa_{I}$. By setting
$\kappa_{R}=0$ we restrict the possible solutions to virtual ($\kappa_{I}<0$)
and bound ($\kappa_{I}>0$) states and obtain the transcendental equation

$\displaystyle(\kappa_{I}-v_{0})(\kappa_{I}+v_{0}\alpha)=-v_{0}^{2}\alpha\mathrm{e}^{-2\kappa_{I}}$
(25)

for $\kappa_{I}$.

We are not interested in the trivial solution $\kappa_{I}=0$ of Eq. (25). In
Fig. 9, both the left (gray line) and right (black line) sides of Eq. (25) are
depicted as a function of $\kappa_{I}$. We see that for $\alpha>0$ there is
one non-trivial solution (black dot). Whether this root is located on the
positive or negative $\kappa_{I}$-axis depends on the difference of the
derivatives of the left- and right-hand sides of Eq. (25) at $\kappa_{I}=0$,
that is $v_{0}(\alpha-1)-2v_{0}^{2}\alpha=v_{0}(\alpha-1-2v_{0}\alpha)$. This
difference vanishes for

$\displaystyle\alpha=\alpha_{c}=\frac{1}{1-2v_{0}}.$ (26)

In this way, for $0<\alpha<\alpha_{c}$ there is only one bound state (with
$\kappa_{I}>0$) in the BX subsystem, whereas for $0<\alpha_{c}<\alpha$, the
subsystem supports only one virtual state (with $\kappa_{I}<0$). In the case
$\alpha<0$, there are one bound and one virtual state for
$\alpha_{c}<\alpha<0$ and two bound states for $\alpha<\alpha_{c}<0$. All
these cases are summarized in the parameter space of the BX subsystem, Fig. 1.

In addition, we now derive the analytical formula for the non-trivial solution
of Eq. (25) as $\alpha\rightarrow\alpha_{c}$. In this limit the non-trivial
root $\kappa_{I}\rightarrow 0$. Therefore, we can use the asymptotic expansion

$\displaystyle
e^{-2\kappa_{I}}=1-2\kappa_{I}+2\kappa_{I}^{2}+{\mathcal{O}}(\kappa_{I}^{3})$
(27)

in Eq. (25) and solve the resulting equation for $\kappa_{I}\neq 0$, to arrive
at

$\displaystyle\kappa_{I}(\alpha)\approx
v_{0}\frac{1+\alpha(2v_{0}-1)}{1+2v_{0}^{2}\alpha}.$ (28)

Next, we expand $\alpha$ around $\alpha_{c}$ to the first order and obtain

$\displaystyle\kappa_{I}(\alpha)\approx c_{1}(\alpha-\alpha_{c})$ (29)

with the coefficient

$\displaystyle c_{1}=-\frac{v_{0}(1-2v_{0})^{2}}{(1-v_{0})^{2}+v_{0}^{2}}.$
(30)

As a result, we obtain the approximate behavior

$\displaystyle\mathcal{E}^{(2)}(\alpha)=-\frac{1}{2}\kappa_{I}^{2}\approx-\frac{c_{1}^{2}}{2}(\alpha-\alpha_{c})^{2}$
(31)

of the two-body energy as $\alpha\rightarrow\alpha_{c}$.

##  Appendix B Separable expansion

In this Appendix we derive the analytical form of the separable expansion [25]
for the two-body $t$-matrix, corresponding to the two-body potential $v(x)$,
Eq. (2).

We expand the $t$-matrix in separable terms

$\displaystyle
t(k,k^{\prime},\mathcal{E})=\sum_{\nu}\tau_{\nu}(\mathcal{E})g^{\ast}_{\nu}(k,\mathcal{E})g_{\nu}(k^{\prime},\mathcal{E})$
(32)

with

$\displaystyle\tau_{\nu}(\mathcal{E})=\frac{\eta_{\nu}(\mathcal{E})}{\eta_{\nu}(\mathcal{E})-1},$
(33)

where the functions $\eta_{\nu}(k,\mathcal{E})$ and $g_{\nu}(\mathcal{E})$ are
determined by the integral equation

$\displaystyle\int_{\mathbb{R}}\frac{\mathrm{d}k^{\prime}}{2\pi}\frac{v(k,k^{\prime})}{\mathcal{E}-\frac{1}{2}k^{\prime
2}}g_{\nu}(k^{\prime},\mathcal{E})=\eta_{\nu}(\mathcal{E})g_{\nu}(k,\mathcal{E}).$
(34)

Here,

$\displaystyle
v(k,k^{\prime})=\int_{\mathbb{R}}\mathrm{d}x\,v(x)\,\mathrm{e}^{-\mathrm{i}(k-k^{\prime})x}$
(35)

is the momentum representation of the two-body interaction potential $v(x)$
and the functions $g_{\nu}(k,\mathcal{E})$ are orthogonal and normalized
according to the condition

$\displaystyle\int_{\mathbb{R}}\frac{\mathrm{d}k}{2\pi}\frac{g_{\nu}(k,\mathcal{E})\,g_{\nu^{\prime}}^{\ast}(k,\mathcal{E})}{\mathcal{E}-\frac{1}{2}k^{2}}=-\delta_{\nu,\nu^{\prime}}.$
(36)

By inserting the potential defined in Eq. (2) into Eq. (35), we find

$\displaystyle
v(k,k^{\prime})=-v_{0}\left[e^{-\mathrm{i}(k-k^{\prime})/2}-\alpha\,\mathrm{e}^{\mathrm{i}(k-k^{\prime})/2}\right],$
(37)

resulting in a separable kernel of the integral equation (34). This allows us
to rewrite Eq. (34) in the form

$\displaystyle-
v_{0}\left[\mathrm{e}^{-\mathrm{i}k/2}G_{\nu}^{(+)}(\mathcal{E})-\alpha\,\mathrm{e}^{\mathrm{i}k/2}G_{\nu}^{(-)}(\mathcal{E})\right]=\eta_{\nu}(\mathcal{E})g_{\nu}(k,\mathcal{E}),$
(38)

where

$\displaystyle
G_{\nu}^{(\pm)}(\mathcal{E})=\int_{\mathbb{R}}\frac{\mathrm{d}k^{\prime}}{2\pi}\frac{\mathrm{e}^{\pm\mathrm{i}k^{\prime}/2}}{\mathcal{E}-\frac{1}{2}k^{\prime
2}}g_{\nu}(k^{\prime},\mathcal{E})$ (39)

are single-argument functions of the energy $\mathcal{E}$.

In order to determine $G_{\nu}^{(\pm)}(\mathcal{E})$, we insert the function
$g_{\nu}(k,\mathcal{E})$ given by Eq. (38) into Eq. (39) and obtain

$\displaystyle\left(1+\frac{v_{0}}{\eta_{\nu}}{\mathcal{A}}\right)G_{\nu}^{(+)}-\frac{v_{0}\alpha}{\eta_{\nu}}{\mathcal{B}}G_{\nu}^{(-)}=0$
(40)
$\displaystyle\frac{v_{0}}{\eta_{\nu}}{\mathcal{B}}^{*}G_{\nu}^{(+)}+\left(1-\frac{v_{0}\alpha}{\eta_{\nu}}{\mathcal{A}}\right)G_{\nu}^{(-)}=0$
(41)

with

$\displaystyle{\mathcal{A}}=$
$\displaystyle\int_{\mathbb{R}}\frac{\mathrm{d}k^{\prime}}{2\pi}\frac{1}{\mathcal{E}-\frac{1}{2}k^{\prime
2}}=-\frac{1}{\sqrt{-2\mathcal{E}}}$ (42) and $\displaystyle{\mathcal{B}}=$
$\displaystyle\int_{\mathbb{R}}\frac{\mathrm{d}k^{\prime}}{2\pi}\frac{\mathrm{e}^{\mathrm{i}k^{\prime}}}{\mathcal{E}-\frac{1}{2}k^{\prime
2}}=-\frac{1}{\sqrt{-2\mathcal{E}}}\mathrm{e}^{-\sqrt{-2\mathcal{E}}},$ (43)

valid for $\mathcal{E}<0$.

The system of algebraic equations (40)–(41) for $G_{\nu}^{(-)}$ and
$G_{\nu}^{(+)}$ has non-trivial solutions only when the determinant of this
system is zero, that is

$\left(\eta_{\nu}+v_{0}{\mathcal{A}}\right)\left(\eta_{\nu}-v_{0}\alpha{\mathcal{A}}\right)+v_{0}^{2}\alpha|{\mathcal{B}}|^{2}=0,$
(44)

resulting in two eigenvalues

$\displaystyle\eta_{\pm}(\mathcal{E})=$
$\displaystyle\frac{v_{0}}{2\sqrt{-2\mathcal{E}}}\left[1-\alpha\pm\mathcal{S}(\mathcal{E})\right]$
(45)

with

$\mathcal{S}(\mathcal{E})=\sqrt{(1-\alpha)^{2}+4\alpha\left(1-e^{-2\sqrt{-2\mathcal{E}}}\right)}$
(46)

Finally, using Eq. (38) together with the normalization condition Eq. (36), we
obtain the expression

$\displaystyle g_{\nu}(k,\mathcal{E})=$
$\displaystyle\frac{\frac{v_{0}}{\eta_{\nu}}\left|\frac{\eta_{\nu}}{v_{0}}\right|(-2\mathcal{E})^{1/4}}{\sqrt{\mathrm{e}^{2\sqrt{-2\mathcal{E}}}\mathcal{P}^{2}_{\nu}(\mathcal{E})-2\mathcal{P}_{\nu}(\mathcal{E})+1}}$
$\displaystyle\times\left[\mathrm{e}^{\sqrt{-2\mathcal{E}}}\mathcal{P}_{\nu}(\mathcal{E})\mathrm{e}^{\mathrm{i}k/2}-\mathrm{e}^{-\mathrm{i}k/2}\right]$
(47)

with

$\mathcal{P}_{\nu}(\mathcal{E})=\left(1-\frac{\eta_{\nu}}{v_{0}}\sqrt{-2\mathcal{E}}\right)=\frac{\alpha\mathrm{e}^{-2\sqrt{-2\mathcal{E}}}}{\left(\frac{\eta_{\nu}}{v_{0}}\sqrt{-2\mathcal{E}}+\alpha\right)}$
(48)

for the corresponding eigenfunctions. Where we used Eq. (44) in the second
step.

##  Appendix C Numerics

In this Appendix we discuss our approach for the numerical solution of the
integral Eq. (8). We use discrete momenta $p_{i}$ and $q_{j}$ with the indices
$i,j\in\\{0,\dots,N-1\\}$, with $N$ being the number of discretization points,
to define the vector elements

$\displaystyle\left[\varphi_{\lambda}(\mathcal{E})\right]_{i}=\leavevmode\nobreak\
$ $\displaystyle\varphi_{\lambda}(p_{i},\mathcal{E})$ (49) and the matrix
elements
$\displaystyle\left[\tilde{K}_{\lambda\nu}(\mathcal{E})\right]_{ij}=\leavevmode\nobreak\
$ $\displaystyle\frac{w_{j}}{2\pi}K_{\lambda\nu}(p_{i},q_{j},\mathcal{E}),$
(50)

respectively. The integral weights $w_{j}$ depend on the exact form of the
discretization. We use the Gauss-Legendre quadrature rule [31, 32]. The
indices $\lambda$ and $\nu$ label the terms from the separable expansion given
in Eq. (32). In practice, the sum over $\nu$ in Eq. (8) needs to be finite or
truncated. In the case of the two-body potential, Eq. (2), we have exactly two
terms. The integral equation now takes the form of an eigenvalue problem

$\underbrace{\begin{bmatrix}\varphi_{0}\\\
\varphi_{1}\end{bmatrix}}_{=\mathbf{v}}=\pm\underbrace{\begin{bmatrix}\tilde{K}_{00}&\tilde{K}_{01}\\\
\tilde{K}_{10}&\tilde{K}_{11}\end{bmatrix}}_{=\mathbf{W}}\begin{bmatrix}\varphi_{0}\\\
\varphi_{1}\end{bmatrix}$ (51)

with fixed eigenvalue $+1$ $(-1)$ for bosons (fermions). Equation (51) has a
solution when

$\mathrm{det}\left[\pm\mathbf{W}(\mathcal{E})-\mathbb{I}\right]=0$ (52)

is fulfilled. The determinant in Eq. (52) is calculated for a range of energy
values $\mathcal{E}$ and each zero point of the determinant corresponds to an
eigen-energy $\mathcal{E}^{(3)}$ of the three-body spectrum of Eq. (4).

To find the wave function, we calculate the eigenvector
$\mathbf{v}(\mathcal{E}^{(3)})$ of the matrix $\mathbf{W}(\mathcal{E}^{(3)})$
for the corresponding eigen-energy $\mathcal{E}^{(3)}$. From the subvectors
$\varphi_{\nu}(\mathcal{E}^{(3)})$ of the eigenvector
$\mathbf{v}(\mathcal{E}^{(3)})$ we can get the Faddeev component from its
separable terms via Eq. (11). Finally, we use Eq. (II.3) to obtain the wave
function in momentum space.

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