#  Forward-backward asymmetry induced $CP$ asymmetry of
$B^{\pm}\rightarrow\pi^{\pm}\pi^{+}\pi^{-}$

Ya-Rui Wei  Zhen-Hua Zhang  School of Nuclear Science and Technology,
University of South China, Hengyang, Hunan, 421001, China.

###### Abstract

$CP$ violation of the decay $B^{\pm}\rightarrow\pi^{\pm}\pi^{+}\pi^{-}$ in the
$f_{0}(500)-\rho(770)^{0}$ interfering region is analyzed. The forward-
backward asymmetries (FBAs) and the corresponding $CP$ asymmetries FB-$CP$As
are particularly investigated. To isolate the $CP$V caused by the interference
of different partial wave more cleanly, we also introduce the
direct-$CP$V-subtracted FB-$CP$A. Based on the LHCb data, we extract the FBAs,
FB-$CP$As, direct-$CP$V-subtracted FB-$CP$A, as well as the regional $CP$As
with invariant mass of the $\pi^{+}\pi^{-}$ pair in the range 0.2
GeV/$c^{2}<\sqrt{s_{\text{low}}}<1.8$ GeV/$c^{2}$. It is found that the
(direct-$CP$V-subtracted) FB-$CP$As are quite large in the
$f_{0}(500)-\rho(770)^{0}$ interfering region, which confirms that the
interference of the intermediate resonances $f_{0}(500)$ and $\rho(770)^{0}$
plays an important role for the $CP$ violation of the three-body decay channel
$B^{\pm}\rightarrow\pi^{\pm}\pi^{+}\pi^{-}$.

##  I Introduction

Charge-parity ($CP$) violation was discovered by J.W.Cronion and V.L.Fitch in
the neutral kaon system in 1964 Christenson  _et al._ (1964). It is closely
related to the matter-antimatter symmetry in our universe Sakharov (1967), and
is one of the most basic and important properties of the weak interaction. In
the Standard Model (SM), $CP$ violation ($CP$V) results from the weak complex
phase in the Cabibbo-Kobayashi-Maskawa (CKM) matrix that reflects the
transitions of different generations of quarks Cabibbo (1963); Kobayashi and
Maskawa (1973). To date, $CP$V has been observed in the $K$, $B$ and $D$ meson
systems Christenson  _et al._ (1964); Alavi-Harati  _et al._ (1999); Aubert
_et al._ (2001); Abe  _et al._ (2001); Poluektov  _et al._ (2010); del Amo
Sanchez  _et al._ (2010); Aaij  _et al._ (2013a, 2019), all of which are
consistent with the KM mechanism of SM blueKobayashi and Maskawa (1973).

The study on $CP$V in multi-body decays of the bottom and the charmed hadrons
plays an increasingly important role in both testing the KM mechanism of SM
and looking for new sources of $CP$V. Interestingly, large $CP$As regionalized
in part of the phase space in some three-body decay channels of $B$ mesons
have been reported by LHCb Aaij  _et al._ (2013b, 2014, 2020, 2022).
Meanwhile, the integrated $CP$As are relatively small due to the cancellation
among different parts of the phase space. Among these three-body decay
channels, $B^{\pm}\rightarrow\pi^{\pm}\pi^{+}\pi^{-}$ is one of the most
extensively studied. Amplitude analysis of this decay channel shows that
$\rho(770)^{0}$ is the dominant resonance Jessop  _et al._ (2000); Gordon  _et
al._ (2002); Aubert  _et al._ (2005, 2009). The large regional $CP$A observed
by LHCb located right in part of the $\rho(770)^{0}$ region where the angle
(in the rest frame of $\rho(770)^{0}$) of the two pions with the same charge
with $B$—which will be denoted as $\theta$ in this paper—is smaller that
$90^{0}$ Aaij  _et al._ (2014, 2020). Theoretical analysis indicates that the
aforementioned large regional $CP$A can be explained by the interference of
$\rho(770)^{0}$ with the nearby resonance $f_{0}(500)$, where the
corresponding amplitudes are respectively $P$\- and $S$-waves Zhang  _et al._
(2013); Dedonder  _et al._ (2011); Alvarenga Nogueira  _et al._ (2015);
Bediaga and Magalhães (2015); Zhang  _et al._ (2015); Cheng  _et al._ (2021).
The interference behavior around $\rho(770)^{0}$ can be well explained based
on a QCD factorization approach for the weak amplitude of
$B^{-}\rightarrow\rho(770)^{0}\pi^{-}$ and $B^{-}\rightarrow
f_{0}(500)\pi^{-}$ Cheng:2020ipp.

Although it can be well explained by the interference of $f_{0}(500)$ and
$\rho(770)^{0}$, the large regional $CP$A in
$B^{\pm}\rightarrow\pi^{\pm}\pi^{+}\pi^{-}$ entangles all kinds of
contributions other than the aforementioned $f_{0}(500)-\rho(770)^{0}$
interference, such as contributions from interference between the tree and
penguin part corresponding to $\rho(770)^{0}$. Recently, one of the authors
(Z.H.Z.) has proposed to study $CP$V induced by the interference of the nearby
resonance through the forward-backward asymmetry (FBA) of the final particle
and the FBA induced $CP$ Asymmetry (FB-$CP$A) Zhang (2021). One advantage of
this method is that it can isolated $CP$V caused by the interfering effect of
the nearby resonances (respectively corresponding to even- and odd-waves) Hu
and Zhang (2022).

The large data sample allows LHCb to study the $CP$V caused by the
interference of the $S$\- and $P$-waves in more detail. To do this, the event
yields are allocated into bins according to $\cos\theta>0$ or $\cos\theta<0$.
In this way, the corresponding regional $CP$As can be systematics studied Aaij
_et al._ (2014). However, the analysis of the FBAs and the FB-$CP$As is absent
in the LHCb’s previous works. This motivates us to performed the analysis of
the FBAs and the FB-$CP$As of $B^{\pm}\rightarrow\pi^{\pm}\pi^{+}\pi^{-}$ in
this paper, based on the LHCb’s data in Refs. Aaij  _et al._ (2014) and Aaij
_et al._ (2020).

This paper is organized as follows. In Sec. II, we first illustrate the
definitions of FBA and FB-$CP$A in detail. In Sec. III, based on the data
sample corresponding to an integrated luminosity of 3.0 fb-1 collected by the
LHCb detector Aaij  _et al._ (2014), we extract the regional $CP$As, the FBAs,
the FB-$CP$As, as well as the newly introduced $CP$ observables
direct-$CP$V-subtracted FB-$CP$As. A fit of $\cos\theta$-dependence of $CP$A
with only the inclusion of the amplitudes corresponding to $f_{0}(500)$ and
$\rho(770)^{0}$ is presented at the end of this section. In Sec. IV, we
briefly give the conclusion.

##  II  THE FORWARD-BACKWARD ASYMMETRY INDUCED $CP$ ASYMMETRY

We first illustrate the definition of several $CP$A observables for the decay
process $B^{-}\rightarrow\pi^{-}\pi^{+}\pi^{-}$. In the c. m. frame of one of
the $\pi^{+}\pi^{-}$ pair with low invariant mass, the aforementioned angle
$\theta$ between the two $\pi^{-}$’s are illustrated in Fig. 1. In practice,
the invariant mass of the $\pi^{+}\pi^{-}$ pair is separated into small
intervals. For each interval, it can be further separated into two small
regions according to $\cos\theta>0$ or $\cos\theta<0$, which will be denoted
as $\Omega_{i}^{+}$ or $\Omega_{i}^{-}$, respectively, where the subscript $i$
is the label of the small interval. The regional $CP$As, which have been
repeatedly dealt with in the literature both theoretically and experimentally,
are defined as

$\displaystyle
A_{CP}^{\Omega_{i}^{\pm}}=\frac{N_{\Omega_{i}^{\pm}}-\overline{N}_{\Omega_{i}^{\pm}}}{N_{\Omega_{i}^{\pm}}+\overline{N}_{\Omega_{i}^{\pm}}},$
(1)

for the region $\Omega_{i}^{\pm}$, where $N_{\Omega_{i}^{\pm}}$ and
$\overline{N}_{\Omega_{i}^{\pm}}$ are the event yields of
$B^{-}\rightarrow\pi^{-}\pi^{-}\pi^{+}$ and
$B^{+}\rightarrow\pi^{+}\pi^{+}\pi^{-}$ in the region $\Omega_{i}^{\pm}$,
respectively. Of course, the regional $CP$A for
$\Omega_{i}\equiv\Omega_{i}^{+}+\Omega_{i}^{-}$ is defined as

$\displaystyle
A_{CP}^{\Omega_{i}}=\frac{N_{\Omega_{i}}-\overline{N}_{\Omega_{i}}}{N_{\Omega_{i}}+\overline{N}_{\Omega_{i}}},$
(2)

where $N_{\Omega_{i}}$ and $\overline{N}_{\Omega_{i}}$ are respectively the
event yields of $B^{-}\rightarrow\pi^{-}\pi^{-}\pi^{+}$ and
$B^{+}\rightarrow\pi^{+}\pi^{+}\pi^{-}$ in $\Omega_{i}$.

Figure 1: The definition of $\theta$.

The FBA of the interval $i$ of $B^{-}\rightarrow\pi^{-}\pi^{+}\pi^{-}$ is
defined as the relative event yields difference between $\Omega_{i}^{+}$ and
$\Omega_{i}^{-}$:

$\displaystyle
A_{i}^{FB}\equiv\frac{N_{\Omega_{i}^{+}}-N_{\Omega_{i}^{-}}}{N_{\Omega_{i}^{+}}+N_{\Omega_{i}^{-}}}.$
(3)

The FB-$CP$A of the interval $i$ is defined as

$\displaystyle A^{FB}_{CP,i}$
$\displaystyle=\frac{1}{2}\left(A_{i}^{FB}-\overline{A_{i}^{FB}}\right),$ (4)

where $\overline{A^{FB}}$ represents the FBA of the interval $i$ for
$B^{+}\rightarrow\pi^{+}\pi^{+}\pi^{-}$. Comparing with the regional $CP$As,
the FB-$CP$A is free from the assumption of equal production of $B^{-}$ and
$B^{+}$ Aaij  _et al._ (2017), which reduces the corresponding systematic
uncertainties.

Alternatively, one can define the $CP$A corresponding to FBA as
$A_{CP,i}^{FB,\text{alt.}}=\frac{A_{i}^{FB}-\overline{A_{i}^{FB}}}{A_{i}^{FB}+\overline{A_{i}^{FB}}}$,
which is similar with the $CP$As defined by the decay width. However, there
are good reasons for us not to use this definition here. Mathematically, since
neither $A^{FB}$ nor $\overline{A^{FB}}$ are positive-definite, the $CP$A
defined in the above equation is not bounded in $(-1,1)$. In other words,
$A_{CP,i}^{FB}$ is normalized, while $A_{CP,i}^{FB,\text{alt.}}$ is not. To be
more specific, $A_{CP,i}^{FB,\text{alt.}}$ is in fact the relative FB-$CP$A
with respect to the $CP$-averaged FBA,
$A_{i}^{FB,CP-\text{av.}}\equiv\frac{1}{2}(A_{i}^{FB}+\overline{A_{i}^{FB}})$,
which can be clearly seen from a transformed expression:
$A_{CP,i}^{FB,\text{alt.}}={A_{CP,i}^{FB}}/{A_{i}^{FB,CP-\text{av.}}}$. Hence
the aforementioned alternative definition is questionable when one wants to
make a comparison with other $CP$As, such as the regional ones. From this
perspective, the definition of Eq. (4) is more reasonable for the usage in
this paper 111Similar story happens in other cases, such as the hyperon
decays, where the $CP$A corresponding to the decay parameter $\alpha$ are
defined in the literature as
$A_{CP}^{\alpha}=\frac{\alpha+\bar{\alpha}}{\alpha-\bar{\alpha}}$, while an
alternative definition which is similar with Eq. (4) was presented in Ref.
Zhang (2022). According to the logic here, the former should be view as the
relative $\alpha$-induced $CP$A with respect to the $CP$-averaged decay
parameter $\alpha^{CP-\text{av.}}\equiv\frac{1}{2}(\alpha-\bar{\alpha})$,
while the latter is the $\alpha$-induced $CP$A:
$A_{CP}^{\alpha-\text{ind.}}\equiv\frac{1}{2}(\alpha+\bar{\alpha})$. Of
course, we are not saying that the latter definition of $CP$A is better. On
the contrary, despite non-normalized, the former does have some distinct
advantages. For example, the former defined relative $CP$A can be measured
through $\Lambda_{c}^{+}\rightarrow\Lambda(\rightarrow p\pi^{-})h^{+}$ Wang
and Yu (2022); Bel (2022) based only on the $CP$ symmetry assumption for the
decay $\Lambda_{c}^{+}\rightarrow\Lambda h^{+}$ ($h^{+}=K^{+}$ or $\pi^{+}$),
while the latter cannot. .

The nonzero of FBA is caused by the interference of the odd- and even-waves Hu
and Zhang (2022). To see this, one express the decay amplitude as

$\displaystyle A$ $\displaystyle=\sum\limits_{l}a_{l}P_{l}(\cos\theta),$ (5)

the FBA is then

$\displaystyle A^{FB}$
$\displaystyle=\frac{1}{\sum\limits_{j}\left[\langle|a_{j}|^{2}\rangle/(2j+1)\right]}\mathop{\sum\limits_{\text{even}\;l}}\limits_{\mathop{\text{odd}\;k}}f_{lk}\mathfrak{R}\left(\langle
a_{l}a_{k}^{*}\rangle\right),$ (6)

where
$f_{lk}=\frac{(-)^{(l+k+1)/2}l!k!}{2^{l+k-1}(l-k)(l+k+1)[(l/2)!]^{2}\\{[(k-1)/2]!\\}^{2}}$
Byerly (2003). Consequently, the FB-$CP$A provide an effective procedure to
isolate $CP$V corresponding to the interference of odd- and even-waves.
Strictly speaking, however, the FB-$CP$A contains also $CP$Vs corresponding to
the difference between the decay width of $CP$-conjugate processes. This can
been seen from the denominator of the above equation, which is proportional to
the decay width. Hence, the difference between the decay width of the
$CP$-conjugate processes, which is in fact the origin of the direct $CP$V, can
also contribute to FB-$CP$A. To eliminate this, one can introduce an
observable, which will be called as the direct-$CP$V-subtracted FB-$CP$A,
taking the form

$\displaystyle\tilde{A}_{CP}^{FB}$
$\displaystyle\equiv\frac{\mathop{\sum\limits_{\text{even}\;l}}\limits_{\mathop{\text{odd}\;k}}f_{lk}\mathfrak{R}\left(\langle
a_{l}a_{k}^{*}\rangle-\langle\bar{a}_{l}\bar{a}_{k}^{*}\rangle\right)}{\sum\limits_{j}\left[\langle|a_{j}|^{2}\rangle/(2j+1)\right]+\sum\limits_{j}\left[\langle|\bar{a}_{j}|^{2}\rangle/(2j+1)\right]}.$
(7)

For the current situation, the direct-$CP$V-subtracted FB-$CP$A of the
interval $i$ can be expressed as

$\displaystyle\tilde{A}_{CP,i}^{FB}$
$\displaystyle=\frac{\left(N_{\Omega_{i}^{+}}-N_{\Omega_{i}^{-}}\right)-\left(\overline{N}_{\Omega_{i}^{+}}-\overline{N}_{\Omega_{i}^{-}}\right)}{N_{\Omega_{i}}+\overline{N}_{\Omega_{i}}},$
(8)

based on the assumption that the $B^{-}$ and $B^{+}$ are equally produced.

##  III  FBA AND FB-$CP$A ANALYSIS BASED ON LHCb DATA IN
$B^{\pm}\rightarrow\pi^{\pm}\pi^{+}\pi^{-}$

Detailed analysis of the event distributions of the decay
$B^{\pm}\rightarrow\pi^{\pm}\pi^{+}\pi^{-}$ has been performed by LHCb based
on a data sample corresponding to an integrated luminosity of 3.0 fb-1 Aaij
_et al._ (2014, 2020). Based on the data in Ref. Aaij  _et al._ (2014), we can
extract the regional $CP$As, the FBAs, the FB-$CP$As, as well as the
direct-$CP$V-subtracted FB-$CP$As in a wide region of the $\pi^{+}\pi^{-}$
pair with lower invariant mass: 0.2 GeV/$c^{2}<\sqrt{s_{\text{l}ow}}<$1.8
GeV/$c^{2}$, which are presented in Figs. 2 and 3.

Figure 2: The regional $CP$As, $A_{CP}^{\Omega_{i}^{+}}$, and
$A_{CP}^{\Omega_{i}^{-}}$, and $A_{CP}^{\Omega_{i}}$ of the decay channel
$B^{\pm}\rightarrow\pi^{\pm}\pi^{+}\pi^{-}$ extracted from the LHCb data in
Ref. Aaij  _et al._ (2014) for $\sqrt{s_{\text{low}}}$ from 0.2 GeV/$c^{2}$ to
1.8 GeV/$c^{2}$. The dotted, dashed, and solid lines are
$A_{CP}^{\Omega_{i}^{+}}$, $A_{CP}^{\Omega_{i}^{-}}$, and
$A_{CP}^{\Omega_{i}}$, respectively.

FIG. 2 shows the corresponding regional $CP$As of each bins with width of 0.05
GeV/$c^{2}$ lying in the range 0.2 GeV/$c^{2}<\sqrt{s_{\text{low}}}<1.8$
GeV/$c^{2}$, while FIG. 3 shows the FBAs, the FB-$CP$As, and the
direct-$CP$V-subtracted FB-$CP$As. The errors are estimated with only the
inclusion of the statistical uncertainties of the event yields estimated
according to $\sqrt{N}$. Both of the two figures show interesting behavior
around the $f_{0}(500)-\rho(770)^{0}$ interference region, i.e., 0.45
GeV/$c^{2}<\sqrt{s_{\text{low}}}<0.75$ GeV/$c^{2}$.

Figure 3: The FBAs ($A_{i}^{FB}$), the FB-$CP$As ($A_{CP,i}^{FB}$), and the
direct-$CP$V-subtracted FB-$CP$As ($\tilde{A}_{CP,i}^{FB}$) of the decay
channel $B^{\pm}\rightarrow\pi^{\pm}\pi^{+}\pi^{-}$ for
$\sqrt{s_{\text{low}}}$ from 0.2 GeV/$c^{2}$ to 1.8 GeV/$c^{2}$. The dotted
and dashed lines are $A_{i}^{FB}$ and $\overline{A_{i}^{FB}}$, respectively.
The solid and dash-dotted lines are $A_{CP,i}^{FB}$ and
$\tilde{A}_{CP,i}^{FB}$, respectively.
Figure 4: The $CP$As, $A_{CP,i}^{FB}$, $\tilde{A}_{CP,i}^{FB}$,
$A_{CP}^{\Omega_{i}}$, $A_{CP}^{\Omega_{i}^{+}}$, and
$A_{CP}^{\Omega_{i}^{-}}$ of the decay channel
$B^{\pm}\rightarrow\pi^{\pm}\pi^{+}\pi^{-}$ for $\sqrt{s_{\text{low}}}$ from
0.45 GeV/$c^{2}$ to 0.75 GeV/$c^{2}$. The upper and lower solid lines are
$A_{CP,i}^{FB}$ and $\tilde{A}_{CP,i}^{FB}$, respectively. The dotted, dashed,
and dash-dotted lines are $A_{CP}^{\Omega_{i}}$, $A_{CP}^{\Omega_{i}^{+}}$,
and $A_{CP}^{\Omega_{i}^{-}}$, respectively.

One can see from Fig. 2 that the regional $CP$As $A_{CP}^{\Omega_{i}^{+}}$ are
quite large in the $f_{0}(500)-\rho(770)^{0}$ interference region. For the
FBAs, the FB-$CP$As, and the direct-$CP$V-subtracted FB-$CP$As in Fig. 3, one
can see that there are big differences between $A_{i}^{FB}$ and
$\overline{A_{i}^{FB}}$, resulting in a large $A_{CP,i}^{FB}$. For a more
detailed and transparent comparison, we present all the five $CP$As,
$A_{CP}^{\Omega_{i}}$, $A_{CP}^{\Omega_{i}^{+}}$, $A_{CP}^{\Omega_{i}^{-}}$,
$A_{CP,i}^{FB}$, and $\tilde{A}_{CP,i}^{FB}$ in Fig. 4.

The first thing one can see from Fig. 4 is that $A_{CP}^{\Omega_{i}^{+}}$,
$A_{CP,i}^{FB}$, and $\tilde{A}_{CP,i}^{FB}$ are quite large throughout the
whole $f_{0}(500)-\rho(770)^{0}$ interference region. Besides, the differences
between $A_{CP}^{\Omega_{i}^{+}}$ and $A_{CP}^{\Omega_{i}^{-}}$ are very large
too. However, when summed up the event yields of the region $\Omega_{i}$, the
resulting $CP$As $A_{CP}^{\Omega_{i}}$ are much smaller. This means that the
$CP$ violation in this region is dominated by the interference of $S$\- and
$P$-waves amplitudes, i.e., the interference between the amplitudes
corresponding to $f_{0}(500)$ and $\rho(770)^{0}$ respectively. This
conclusion can also be verified from the small difference between
$A_{CP,i}^{FB}$ and $\tilde{A}_{CP,i}^{FB}$, since their difference reflects
the $CP$V within the $S$\- or $P$-waves.

Table 1:  The averaged $CP$As, $A_{CP}^{FB}$, $\tilde{A}_{CP}^{FB}$, $A_{CP}^{\Omega}$, $A_{CP}^{\Omega^{+}}$ and $A_{CP}^{\Omega^{-}}$ of the whole $f_{0}(500)-\rho(770)^{0}$ interference region, for 0.45 GeV/$c^{2}<\sqrt{s_{\text{low}}}<0.75$ GeV/$c^{2}$, where the uncertainties are statistical only. $A_{CP}^{FB}$ | $\tilde{A}_{CP}^{FB}$ | $A_{CP}^{\Omega}$ | $A_{CP}^{\Omega^{+}}$ | $A_{CP}^{\Omega^{-}}$
---|---|---|---|---
$0.224\pm 0.012$ | $0.194\pm 0.013$ | $0.099\pm 0.013$ | $0.405\pm 0.020$ | $-0.074\pm 0.017$

Amplitude analysis of this decay channel
$B^{\pm}\rightarrow\pi^{-}\pi^{+}\pi^{\pm}$ showed that $\rho(770)^{0}$ plays
a dominant role and that the interference of $\rho(770)^{0}$ and $f_{0}(500)$
also contributes significantly Aaij  _et al._ (2020). The $CP$Vs,
correspondingly, get contribution from the interference of the tree and
penguin parts of the $S$\- or $P$-wave amplitudes, which are in fact the
origin of direct $CP$As of the decay channel
$B^{\pm}\rightarrow\pi^{\pm}f_{0}(500)$ or
$B^{\pm}\rightarrow\pi^{\pm}\rho(770)^{0}$, and the interference between $S$\-
and $P$-waves amplitudes. All of the aforementioned contributions of $CP$V
present in the regional $CP$As, $A_{CP}^{\Omega_{i}^{+}}$ and
$A_{CP}^{\Omega_{i}^{-}}$. While for the region $\Omega_{i}$, the contribution
from the $S$\- and $P$-waves interfering term cancel out totally. Meanwhile,
theoretical analysis in Sec. III shows that the (direct-$CP$V-subtracted)
FB-$CP$As contain only the contributions from the interference between the
$S$\- and $P$-waves amplitudes. The large FB-$CP$As and the
direct-$CP$V-subtracted FB-$CP$As in Fig. 4 indicates with no doubt that the
$CP$Vs corresponding to the interference between $S$\- and $P$-waves
amplitudes dominant in the $f_{0}(500)-\rho(770)^{0}$ interference region.
This can also be seen from the numerical values of $A_{CP}^{FB}$,
$\tilde{A}_{CP}^{FB}$, $A_{CP}^{\Omega}$, $A_{CP}^{\Omega^{+}}$ and
$A_{CP}^{\Omega^{-}}$, of the whole region presented in Table. 1, from which
one can see that the $A_{CP}^{FB}$ and $\tilde{A}_{CP}^{FB}$ are much larger
than $A_{CP}^{\Omega}$.

The interference of the $S$\- and $P$-waves can also explain the fine
structures of the $\cos\theta$-dependencies of the regional $CP$As. Figure 5
presents the $\cos\theta$-dependencies of the regional $CP$As measured by LHCb
for 0.62 GeV/$c^{2}<\sqrt{s_{\text{low}}}<0.78$ GeV/$c^{2}$, where
$\cos\theta$ is equally divided into $30$ bins Aaij  _et al._ (2020). From
Fig. 5 one can clearly see that the regional $CP$As depend on $\cos\theta$
strongly. It has been pointed out that the tendency of the regional $CP$As of
taking opposite signs for $\cos\theta>0$ and $\cos\theta<0$ is due to the
interference of $f_{0}(500)$ and $\rho(770)^{0}$ Aaij  _et al._ (2014).
Indeed, one can fit the data in Fig. 5 quite well by the inclusion of only
$f_{0}(500)$ and $\rho(770)^{0}$. The fitted curve is also shown in Fig. 5.

To fit the data, the decay amplitude is parametrized as

$\displaystyle\mathcal{A}_{B^{-}\rightarrow\pi^{-}\pi^{+}\pi^{-}}$
$\displaystyle=\cos\theta\mathcal{R}_{1}\frac{c_{\rho}e^{i\delta_{\rho}}}{s_{\text{low}}-m_{\rho}^{2}+im_{\rho}\Gamma_{\rho}}+\mathcal{R}_{2}\frac{c_{f}e^{i\delta_{f}}}{s_{\text{low}}-m_{f}^{2}+im_{f}\Gamma_{f}},$
(9)
$\displaystyle\mathcal{\overline{A}_{B^{+}\rightarrow\pi^{-}\pi^{+}\pi^{+}}}$
$\displaystyle=\cos\theta\mathcal{R}_{1}\frac{\bar{c}_{\rho}e^{i\bar{\delta}_{\rho}}}{s_{\text{low}}-m_{\rho}^{2}+im_{\rho}\Gamma_{\rho}}+\mathcal{R}_{2}\frac{\bar{c}_{f}e^{i\bar{\delta}_{f}}}{s_{\text{low}}-m_{f}^{2}+im_{f}\Gamma_{f}},$
(10)

where $c_{i}$ and $\delta_{i}$ ($i=\rho,f$) represent the corresponding
amplitudes and the relative phases contribution of component $\rho(770)^{0}$
and $f_{0}(500)$, respectively,
$\mathcal{R}_{1}=\sqrt{s_{\text{low}}-m_{\pi}^{2}}\sqrt{\frac{(m_{B}^{2}-s_{\text{low}}+m_{\pi}^{2})^{2}}{s_{\text{low}}}-m_{\pi}^{2}}$,
$\mathcal{R}_{2}=\sqrt{s^{*}-m_{\pi}^{2}}\sqrt{\frac{(m_{B}^{2}-s^{*}+m_{\pi}^{2})^{2}}{s^{*}}-m_{\pi}^{2}}$,
where $s^{*}=\frac{m_{\rho}^{2}+m_{f}^{2}}{2}$ and $m_{B}$, $m_{\pi}$,
$m_{\rho}$, and $m_{f}$ are the masses of $B$, $\pi$, $\rho(770)^{0}$, and
$f_{0}(500)$, respectively. The parameters used in Eqs. (9) and (10) are
listed in Table 2, and are borrowed from Ref. Workman (2022). Our fit is
performed in a least square method. To fit the data, the three parameters
$c_{\rho}$, $\delta_{\rho}$, and $\bar{\delta}_{\rho}$ are set fixed at
$c_{\rho}=1$, $\delta_{\rho}=\bar{\delta}_{\rho}=0$. The fitting results for
the rest five of the parameters are presented in Table 3. The goodness for
this fit is $\chi^{2}/d.o.f.=1.10$, where $\chi^{2}$ is the Pearson’s chi-
square and is defined as
$\chi^{2}\equiv\sum_{j=1}\frac{(O_{j}-E_{j})^{2}}{E_{j}}$ with $O_{j}$ and
$E_{j}$ being the observed and expected event yields of bin $j$, and $d.o.f.$
is the degrees of freedom.

Table 2:  Input parameters used in Eqs. (9) and (10). Parameters | Value (in GeV)
---|---
$m_{B^{\pm}}$ | $5.279$
$m_{\pi}$ | $0.139$
$m_{\rho(770)^{0}}$ | $0.775$
$\Gamma_{\rho(770)^{0}}$ | $0.149$
$m_{f_{0}(500)}$ | $0.563$
$\Gamma_{f_{0}(500)}$ | $0.350$

Figure 5: Dependence of regional $CP$As on $\cos\theta$ for 0.62 GeV/$c^{2}<\sqrt{s_{\text{low}}}<$0.78 GeV/$c^{2}$, where $\cos\theta$ is divided into 30 bins. The fitted curve is also shown in this figure. Table 3:  Parameters values obtained by the fitting. Parameter | Value | Parameter | Value
---|---|---|---
$c_{\rho}$ | 1 | $\bar{c}_{\rho}$ | $1.05\pm 0.02$
$\delta_{\rho}$ | $0$ | $\bar{\delta}_{\rho}$ | $0$
$c_{f}$ | $0.49\pm 0.07$ | $\bar{c}_{f}$ | $0.50\pm 0.06$
$\delta_{f}$ | $-0.64\pm 0.06$ | $\bar{\delta}_{f}$ | $-1.36\pm 0.31$

##  IV SUMMARY AND CONCLUSION

In this paper, the FBA, the FB-$CP$A, the direct-$CP$V-subtracted FB-$CP$As
and the regional $CP$As for the three-body decay of $B$ meson
$B^{\pm}\rightarrow\pi^{\pm}\pi^{+}\pi^{-}$ are analyzed based on the data of
LHCb Aaij  _et al._ (2014, 2020). We focus on the $CP$As in the
$f_{0}(500)-\rho(770)^{0}$ interfering region. It is found that the FB-$CP$As
are quite large in this region. According to the theoretical analysis, the
large FB-$CP$A and the direct-$CP$V-subtracted FB-$CP$As can be explained by
the interference of the amplitudes corresponding to $f_{0}(500)$ and
$\rho(770)^{0}$. Moreover, the analysis indicates that the interference
between the amplitudes of $f_{0}(500)$ and $\rho(770)^{0}$ is the main
contribution to $CP$Vs in this region. The aforementioned interference can
even explain more detailed structures of $CP$Vs in this region, according to
the fitting result of Fig. 5.

In conclusion, the interference of intermediate resonances can play an
important role in $CP$V of three-body decay of bottom meson. The FB-$CP$A and
the direct-$CP$V-subtracted FB-$CP$A, according the theoretical analysis and
the data-based analysis of this paper, are ideal observables to study $CP$Vs
coursed by the interference of intermediate resonances.

###### Acknowledgements.

This work was supported by National Natural Science Foundation of China under
Grants No. 12192261, and Natural Science Foundation of Hunan Province under
Grants No. 2022JJ30483.

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#  Accelerating many-body entanglement generation by dipolar interactions in
the Bose-Hubbard model

Marlena Dziurawiec  Institute of Physics PAS, Aleja Lotnikow 32/46, 02-668
Warszawa, Poland  Tanausú Hernández Yanes  Institute of Physics PAS, Aleja
Lotnikow 32/46, 02-668 Warszawa, Poland  Marcin Płodzień  ICFO - Institut de
Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860
Castelldefels, Barcelona, Spain  Mariusz Gajda  Institute of Physics PAS,
Aleja Lotnikow 32/46, 02-668 Warszawa, Poland  Maciej Lewenstein  ICFO -
Institut de Ciencies Fotoniques, The Barcelona Institute of Science and
Technology, 08860 Castelldefels, Barcelona, Spain  ICREA, Pg. Lluís Companys
23, 08010 Barcelona, Spain  Emilia Witkowska  Institute of Physics PAS, Aleja
Lotnikow 32/46, 02-668 Warszawa, Poland

###### Abstract

The spin squeezing protocols allow the dynamical generation of massively
correlated quantum many-body states, which can be utilized in entanglement-
enhanced metrology and technologies. We study a quantum simulator generating
twisting dynamics realized in a two-component Bose-Hubbard model with dipolar
interactions. We show that the interplay of contact and long-range dipolar
interactions between atoms in the superfluid phase activates the anisotropic
two-axis counter-twisting mechanism, accelerating the spin squeezing dynamics
and allowing the Heisenberg-limited accuracy in spectroscopic measurements.

##  I Introduction

The second Quantum Revolution’s main objective lies in multipartite entangled
states: their production, storage, certification, and application. Such
states, i.e., many-body entangled and many-body Bell correlated states, are
essential resources for quantum-based technologies and quantum-enhancement
metrology Acín  _et al._ (2018); Eisert  _et al._ (2020); Kinos  _et al._
(2021); Laucht  _et al._ (2021); Becher  _et al._ (2022); Fraxanet  _et al._
(2022). As such, a general protocol allowing the controlled generation of such
states is an extensive research direction in modern quantum science. Spin
squeezing represents such a protocol paving the way for high-precision
measurements, allowing overcoming the shot-noise limit Kitagawa and Ueda
(1993); Wineland  _et al._ (1994), generate many-body entangled Fadel  _et
al._ (2018); Evrard  _et al._ (2019); Hosten  _et al._ (2016); Pedrozo-
Peñafiel  _et al._ (2020); Bao  _et al._ (2020), and many-body Bell correlated
states Tura  _et al._ (2014); Schmied  _et al._ (2016); Aloy  _et al._ (2019);
Baccari  _et al._ (2019); Tura  _et al._ (2019); Müller-Rigat  _et al._
(2021); Płodzień  _et al._ (2022). The spin squeezing applies to a system of
$N$ quantum in two internal states corresponding to a spin-1/2 degree of
freedom, and further described by the collective spin of the quantum number
$S=N/2$. The uncertainty of spectroscopic measurements is $\xi/\sqrt{N}$,
where

$\xi^{2}=\frac{N\Delta^{2}\hat{S}_{\perp\min}}{\langle S\rangle^{2}},$ (1)

is the spin squeezing parameter, and $\Delta^{2}\hat{S}_{\perp\min}$ is the
minimal variance in the plane perpendicular to total spin vector. The spin
squeezing is a witness of entanglement-depth, i.e. quantum state is not
$k$-producible, when $\xi<1/k$ Sørensen  _et al._ (2001); Pezzé and Smerzi
(2009); Hyllus  _et al._ (2012); Tóth (2012).

The paradigmatic theoretical models realizing spin squeezing through unitary
evolution are given by the so-called One-Axis Twisting (OAT), and Two-Axis
Counter-Twisting (TACT) Hamiltonians Kitagawa and Ueda (1993). The OAT
Hamiltonian has the form of the non-linear operator, often cast as

$\hat{H}_{\rm OAT}=\hbar\chi\hat{S}_{z}^{2},$ (2)

where $z$ is the twisting axis, and $\chi^{-1}$ is the time-scale on which
spin squeezing parameter has the lowest value $\xi^{2}_{\rm best}$. The lowest
value of the squeezing parameter scales with particle numbers, and for OAT it
is $\xi_{\rm best}\propto N^{-1/3}$ at $\chi t_{\rm best}\simeq N^{-2/3}$
Kitagawa and Ueda (1993). The TACT Hamiltonian reads

$\hat{H}_{\rm TACT}=\hbar\chi(\hat{S}^{2}_{z}-\hat{S}^{2}_{x}),$ (3)

where the clockwise and counter-clockwise twisting take places around two
orthogonal axes $z$ and $x$. The advantage of the TACT is that it gives the
Heisenberg limited level of the best squeezing, namely $\xi_{\rm best}\propto
N^{-1/2}$. In addition, the time scale of the best squeezing is accelerated
with respect to OAT, and it is given by $\chi t_{\rm best}\sim N^{-1}\log(2N)$
Kajtoch and Witkowska (2015) .

Realizing quantum simulators of OAT or TACT dynamics is essential for quantum
enhancement metrology. Ultra-cold atoms form a perfect platform for quantum
simulators mimicking such twisting dynamics. OAT has been realized with Bose-
Einstein condensates utilizing atom-atom collisions Sørensen and Mølmer
(1999); Sørensen  _et al._ (2001); Riedel  _et al._ (2010); Gross  _et al._
(2010); Hamley  _et al._ (2012); Qu  _et al._ (2020), and atom-light
interactions Leroux  _et al._ (2010); Maussang  _et al._ (2010). Another
research directions are ultra-cold platforms simulating the Hubbard and
Heisenberg models, which generate twisting dynamics. In the case of bosons,
twisting dynamics is induced by atom-atom collisions Kajtoch  _et al._ (2018);
He  _et al._ (2019a); Płodzień  _et al._ (2020); Mamaev  _et al._ (2021a),
while for spinful fermions a synthetic spin-orbit coupling is necessary Wall
_et al._ (2016); Kolkowitz  _et al._ (2017a, b); Bromley  _et al._ (2018); He
_et al._ (2019b); Mamaev  _et al._ (2021b); Yanes  _et al._ (2022). Finally,
twisting dynamics can be activated with long-range interacting bosons, what
provides a platform for spin squeezing simulators by casting original
Hamiltonian onto long-range interacting spin-chain Civitarese  _et al._
(2010); Perlin  _et al._ (2020); Bilitewski  _et al._ (2021); Roscilde  _et
al._ (2021); Wu  _et al._ (2022); Comparin  _et al._ (2022a, b).

In this work, we propose a quantum simulator for the TACT model realized in a
one-dimensional two-component Bose-Hubbard model in the superfluid phase,
considering both contact and dipolar interactions. With the help of full many-
body dynamics and an effective two-mode model (TMM) description, we show that
the realized squeezing dynamics capture properties of the anisotropic TACT
model where the clockwise and counter-clockwise twisting take place with
different rates. Next, we show that scaling with the system size of the best
squeezing parameter and best squeezing time is equivalent to the scaling
obtained for the TACT model. Our results show the significant acceleration of
the spin squeezing dynamics by dipolar interactions, which is an essential
effect from the experimental point of view

The paper is organized as follows. In Sec.II we introduce the considered
model. Starting with general many-body description of the system we provide an
effective two mode model accounting for both contact and long-range dipolar
interactions. Next, in Sec.III we perform analysis of the mean-field phase
space of the anisotropic TACT model. In Sec.IV with the help of Bogoliubov-
Born-Green-Kirkwood-Yvon (BBGKY) hierarchy of equations Anglin and Vardi
(2001); André and Lukin (2002), we analyze the scaling of the best squeezing
and the best squeezing time with the system size. We conclude in Sec.V.

##  II Exact and effective models

We consider $N$ bosonic atoms in the two internal states $\ket{\uparrow}$,
$\ket{\downarrow}$ which corresponds to the ensemble of $N$ spin-1/2 particles
(qubits). The atoms are described by the following Hamiltonian:

$\displaystyle\hat{H}=\hat{H}_{0}+\hat{H}_{\rm d},$ (4)
$\displaystyle\hat{H}_{0}=\int
d^{3}\bm{r}\hat{\Psi}^{\dagger}(\bm{r})\left(-\frac{\hbar^{2}\nabla^{2}}{2m}+V_{\rm
latt}\right)\hat{\Psi}(\bm{r}),$ (5) $\displaystyle\hat{H}_{\rm d}=\int
d^{3}\bm{r}_{1}\int
d^{3}\bm{r}_{2}\hat{\Psi}^{\dagger}(\bm{r}_{1})\hat{\Psi}^{\dagger}(\bm{r}_{2})V_{12}\hat{\Psi}(\bm{r}_{2})\hat{\Psi}(\bm{r}_{1}),$
(6)

where the vector of bosonic field operators is
$\hat{\Psi}^{T}(\bm{r})=(\hat{\Psi}_{\uparrow}({\bf
r}),\hat{\Psi}_{\downarrow}(\bm{r}))$ with $\hat{\Psi}_{\sigma}(\bm{r})$
describing an atom at the position $\bm{r}$ in the state
$\sigma=\uparrow,\downarrow$. The interaction potential $V_{12}$ is a sum of
two terms, $V_{12}=V_{\rm c}+V_{\rm d}$, the short range contact interaction

$\displaystyle V_{\rm c}$
$\displaystyle=\frac{4\pi\hbar^{2}a_{s}}{m}\delta\left(\bm{r}_{1}-\bm{r}_{2}\right),$
(7)

and the long range dipolar interaction

$\displaystyle V_{\rm d}$
$\displaystyle=\frac{\bm{\mu}_{1}\cdot\bm{\mu}_{2}}{|\bm{r}_{1}-\bm{r}_{2}|^{3}}-\frac{3[\bm{\mu}_{1}\cdot(\bm{r}_{1}-\bm{r}_{2})][\bm{\mu}_{2}\cdot(\bm{r}_{1}-\bm{r}_{2})]}{|\bm{r}_{1}-\bm{r}_{2}|^{5}},$
(8)

where $\bm{\mu}_{1,2}$ is the dipole moment, $m$ is the atomic mass and
$a_{s}$ is the s-wave scattering length.

The atoms are loaded into the one-dimensional optical lattice potential
$V_{\rm latt}=V_{0}\sin^{2}{kx}$, where $k=2\pi/\lambda_{\rm latt}$ is a wave-
vector associated with the lattice wave-length $\lambda_{\rm latt}$. We
consider the unit filling, so the number of lattice sites $M$ equals the total
number of atoms $N$ ($M=N$). We assume the atoms are in the superfluid phase
and occupy the lowest Bloch band. In the tight-binding approximation, the
field operators is conveniently expanded in the basis of the Wannier
functions, and the system Hamiltonian (4) reduces to the two-component Bose-
Hubbard model (BHM) extended by the dipolar term, namely

$\hat{H}=\hat{H}_{\rm BH}+\hat{H}_{\rm d}\equiv\hat{H}_{\rm dBH}.$ (9)

The two-component Bose-Hubbard Hamiltonian $\hat{H}_{BH}$ reads

$\displaystyle\hat{H}_{\rm
BH}=-J\sum_{\sigma=\uparrow,\downarrow}\sum_{j}\left(\hat{a}^{\dagger}_{\sigma,j}\hat{a}_{\sigma,j+1}+\hat{a}^{\dagger}_{\sigma,j}\hat{a}_{\sigma,j-1}\right)$
$\displaystyle+\sum_{j}\left(\frac{U_{\uparrow\uparrow}}{2}\hat{n}_{j\uparrow}(\hat{n}_{j\uparrow}-1)+\frac{U_{\downarrow\downarrow}}{2}\hat{n}_{j\downarrow}(\hat{n}_{j\downarrow}-1)+U_{\uparrow\downarrow}\hat{n}_{j\uparrow}\hat{n}_{j\downarrow}\right),$
(10)

where $\hat{a}_{\sigma,j}$ and $\hat{n}_{\sigma,j}$ are the on-site
annihilation and number operators of atoms in the state $|\sigma\rangle$ at
the site $j$. The hopping amplitude $J$ does not depend on the spin state
$\sigma$, while the interaction coefficients $U_{\sigma\sigma^{\prime}}$
contain the contributions of both the on-site contact and the on-site dipolar
interaction Dutta  _et al._ (2015). The dipolar interaction term $\hat{H}_{d}$
reads

$\displaystyle\hat{H}_{\rm d}$ $\displaystyle=\sum_{j,k\neq
j}\frac{\gamma^{2}}{4|j-k|^{3}}(\hat{S}_{z,j}\hat{S}_{z,k}-2\hat{S}_{x,j}\hat{S}_{x,k}+\hat{S}_{y,j}\hat{S}_{y,k}),$
(11)

where the dipole moment was associated with the spin operators,
$\bm{\mu}=-\gamma\bm{S}$, and where the on-site spin operators are
$\hat{S}^{+}_{j}=\hat{a}^{\dagger}_{\uparrow,j}\hat{a}_{\downarrow,j}$,
$\hat{S}^{-}_{j}=\hat{a}^{\dagger}_{\downarrow,j}\hat{a}_{\uparrow,j}$,
$\hat{S}^{\pm}_{j}=\hat{S}_{x,j}\pm i\hat{S}_{y,j}$ and
$\hat{S}_{z,j}=(\hat{n}_{j,\uparrow}-\hat{n}_{j,\downarrow})/2$, while the
collective spin operators read $\hat{S}_{x}=\frac{1}{2}\sum_{j}\hat{S}_{x,j}$,
$\hat{S}_{y}=\frac{1}{2i}\sum_{j}\hat{S}_{y,j}$, and
$\hat{S}_{z}=\frac{1}{2}\sum_{j}\hat{S}_{z,j}$. The range of dipole potential
extends over several lattice sites under typical experimental conditions.
Therefore, it is approximately constant on scales comparable to the spatial
localization of Wannier functions. Under this condition the dipolar part of
the Hamiltonian can be simplified in the form (11), see Appendix A for
derivation.

We consider the dynamical generation of spin squeezed states from an initial
spin coherent state when the system is in the superfluid phase,
$U_{\sigma\sigma}\ll J$, and contact interactions compete with the long-range
one. Such the regime corresponds to the situation when the wave-functions of
atoms are delocalized over the entire lattice and the condensate fraction,
$f_{c}\equiv\frac{1}{N^{2}}\sum_{i,j}\sum_{\sigma=\uparrow,\downarrow}\langle\hat{a}^{\dagger}_{\sigma,j}\hat{a}_{\sigma,j}\rangle$,
approximately equals one.

The two-component Bose-Hubbard model can simulate the OAT dynamics via contact
interactions among bosons in the superfluid phase Płodzień  _et al._ (2020).
Here we show that the system can simulate the anisotropic TACT dynamics when
dipolar interactions between the bosonic atoms are taken into account. To
understand why the twisting mechanism is simulated by the system Hamiltonian
(9) we perform the following analysis. First, we consider $\hat{H}_{\rm dBH}$
in the quasi-momentum representation by using the Fourier transforms,
$\hat{a}_{\sigma,j}=\frac{1}{\sqrt{N}}\sum_{n}e^{i\frac{2\pi}{M}jn}\hat{a}_{\sigma,q_{n}}$
and
$\hat{S}_{\sigma,j}=\frac{1}{\sqrt{N}}\sum_{n}e^{i\frac{2\pi}{M}jn}\hat{S}_{\sigma,q_{n}}$,
where the quasi-momentum reads $q_{n}=\frac{2\pi}{M}n$ for
$n=0,1,2,\cdots,N-1$. Next, by keeping the zero-momentum mode contributions
only $q_{n}=0$ one can show that $\hat{H}_{\rm dBH}$ reduces to the effective
model that is a sum of two terms

$\hat{H}_{{\rm dBH},q_{n}=0}=\hat{H}_{{\rm BH},q_{n}=0}+\hat{H}_{{\rm
d},q_{n}=0}.$ (12)

The first term, $\hat{H}_{\rm BH,n=0}$, comes from the zero quasi-momentum
mode of the Bose-Hubbard Hamiltonian,

$\displaystyle\hat{H}_{{\rm BH},q_{n}=0}$
$\displaystyle=-2J\hat{N}_{q_{n}=0}+\Omega_{NN}\hat{N}^{2}_{q_{n}=0}$
$\displaystyle+\Omega_{SN}\hat{S}_{z,q_{n}=0}\hat{N}_{0}+\Omega_{SS}\hat{S}_{z,q_{n}=0}^{2},$
(13)

where

$\displaystyle\Omega_{NN}$
$\displaystyle=\frac{U_{\uparrow\uparrow}+U_{\downarrow\downarrow}+4U_{\uparrow\downarrow}}{8N},$
(14) $\displaystyle\Omega_{SN}$
$\displaystyle=\frac{U_{\uparrow\uparrow}-U_{\downarrow\downarrow}}{2N},$ (15)
$\displaystyle\Omega_{SS}$
$\displaystyle=\frac{U_{\uparrow\uparrow}+U_{\downarrow\downarrow}-2U_{\uparrow\downarrow}}{2N},$
(16)

and realizes OAT dynamics Płodzień  _et al._ (2020). The second term,
$\hat{H}_{d,q_{n}=0}$, consists of zero momentum component of the dipolar
interaction:

$\displaystyle\hat{H}_{{\rm d},q_{n}=0}$
$\displaystyle=2\frac{\gamma^{2}h^{(3)}_{\left\lfloor{N/2}\right\rfloor}}{N}\left(\hat{S}^{2}_{z,q_{n}=0}-3\hat{S}^{2}_{x,q_{n}=0}\right),$
(17)

with
$h^{(3)}_{\left\lfloor{N/2}\right\rfloor}=\sum_{d=1}^{{\left\lfloor{N/2}\right\rfloor}}1/d^{3}$.
Finally, by collecting the particular terms in (12) we obtain

$\hat{H}_{{\rm
dHB},q_{n}=0}=\frac{U-U_{\uparrow\downarrow}}{N}\hat{S}^{2}_{z,q_{n}=0}-6\frac{\gamma^{2}h^{(3)}_{\left\lfloor{N/2}\right\rfloor}}{N}\hat{S}^{2}_{x,q_{n}=0},$
(18)

where we neglected the constant energy terms assuming $\hat{N}_{0}=N$ and
$U_{\uparrow\uparrow}=U_{\downarrow\downarrow}=U$. Note here, the zero quasi-
momentum component of the spin operators correspond to the collective spin
operators in the position representation, namely
$\hat{S}_{\beta,q_{n}=0}=\frac{1}{\sqrt{N}}\sum_{j}\hat{S}_{\beta,j}$ with
$\beta=x,y,z$. As such, we can replace $\hat{S}_{\beta,0}$ by the collective
spin operators $\hat{S}_{\beta}$ in (18). Taking this into account, we
identify the effective two-mode model (TMM):

$\hat{H}_{\rm TMM}=\hbar\chi\left(\hat{S}_{z}^{2}-\eta\hat{S}_{x}^{2}\right)$
(19)

which is the anisotropic TACT with

$\displaystyle\hbar\chi$ $\displaystyle=\frac{U-U_{\uparrow\downarrow}}{N}$
(20) $\displaystyle\hbar\chi\eta$
$\displaystyle=6\frac{\gamma^{2}h^{(3)}_{\left\lfloor{N/2}\right\rfloor}}{N}$
(21)

where $\eta$ is the anisotropy parameter and $\chi$ sets the energy scale. In
the two limit cases $\eta=0$ and $\eta=1$ the effective model (19) reduces to
the OAT and TACT model, respectively.

(a)(b) Figure 1: (a) Time evolution of the spin squeezing parameter $\xi^{2}$
defined in (1) for different values of anisotropy parameter
$\eta=\\{0.0,0.5,1.0\\}$ (lines from right to left). The two limiting cases,
i.e. $\eta=0$ and $\eta=1$ correspond to OAT and TACT dynamics, respectively.
Points correspond to the results from the exact many-body numerical simulation
of $\hat{H}_{\rm dBH}$ given by (9), while solid lines to the numerical
results from the effective two-mode model (19) when $N=M=10$, $U=0.01$,
$J=1.0$ and $U_{\uparrow\downarrow}=0.95U$. (b) Color encoded values of the
spin squeezing parameter $\xi^{2}$ versus $\chi t$ and $\eta$ obtained from
the numerical simulations of the two mode model (19) for $N=10^{3}$ atoms. The
solid red line indicates the best squeezing time. (a)(b)(c)(d) Figure 2:
Mean-field phase portraits indicating geometrical representations of
trajectories and directions of evolution for the effective two-mode model (19)
for the anisotropy parameter $\eta=\\{10^{-5},0.04,0.4,1.0\\}$ (from left to
right). The angles $\theta$ between in-going and out-going flow along constant
energy lines are marked by red (see main text). The corresponding values of
angles $\theta$ are approximately $\pi$, $0.87\pi$, $0.64\pi$, $0.5\pi$ (from
left to right).

In Fig.1 (a) we show spin squeezing parameter (1) obtained from the exact
many-body numerical simulation of the system dynamics under the dipolar Bose-
Hubbard Hamiltonian (9) and the effective two-mode model (19), see Appendix B
for more details concerning numerical simulations. For the chosen set of
parameters, the condensate fraction is approximately one, $f_{c}\approx 1$, at
the time scale corresponding to the best squeezing, which justifies our two-
mode approximation. The overall agreement between the two models can be
noticed. The acceleration of the squeezing dynamics is visible by increasing
the value of the anisotropy parameter. In Fig.1(b) we present a variation of
the spin squeezing parameter $\xi^{2}$ in time and anisotropy parameter $\eta$
from the two-mode model (19) for $N=10^{3}$. One can observe the two limiting
cases corresponding to OAT and TACT dynamics for $\eta=0$ and $\eta=1$,
respectively. We observe a smooth transition between OAT and TACT dynamics in
the intermediate region.

In the next paragraph, we provide an intuitive explanation for the OAT-TACT
crossover with the help of phase portrait analysis of the two-mode model (19).
In Section IV we derive the scaling of the best squeezing and the best
squeezing time with $N$ showing the acceleration of squeezing dynamics by the
dipolar interactions.

##  III Mean-field phase portraits

The activation of the TACT dynamics by dipolar interactions can be intuitively
explained by analyzing the structure of the mean-field phase space of the two-
mode model (19). It is a good navigator for the dynamical spin squeezing
Kajtoch and Witkowska (2015) as the eigenstates of quantum Hamiltonian
localize on classical phase space energy contours Trimborn  _et al._ (2009)
and quantum evolution distinguishes between stable and unstable classical
fixed-points Shchesnovich and Konotop (2008).

The analysis of the mean-field phase space is performed by replacing the
annihilation and creation operators by complex numbers Smerzi  _et al._
(1997), $\hat{a}\rightarrow\sqrt{N\rho_{a}}e^{i\phi_{a}}$,
$\hat{b}\rightarrow\sqrt{N\rho_{b}}e^{i\phi_{b}}$ what transforms the spin
operators to $\hat{S}_{x}\rightarrow N\frac{\sqrt{1-z^{2}}}{2}\cos\phi$,
$\hat{S}_{y}\rightarrow N\frac{\sqrt{1-z^{2}}}{2}\sin\phi$,
$\hat{S}_{z}\rightarrow\frac{N}{2}z$. This allows introducing the new
canonical variables $z=\rho_{a}-\rho_{b}$ and $\phi=\phi_{a}-\phi_{b}$. The
Hamiltonian (19) takes the form of the energy functional $\epsilon(\phi,z)$:

$\epsilon(\phi,z)=\frac{N}{4}z^{2}-\frac{\eta N}{4}(1-z^{2})\cos^{2}\phi.$
(22)

Equations of motion for the canonical position $\phi$ and the conjugate
momentum $z$ are set by the Hamilton equations:

$\displaystyle\dot{\phi}$
$\displaystyle=\frac{\partial\epsilon(\phi,z)}{\partial
z}=\frac{N}{2}z+\frac{\eta N}{2}z\cos^{2}\phi$ $\displaystyle\dot{z}$
$\displaystyle=-\frac{\partial\epsilon(\phi,z)}{\partial\phi}=-\frac{\eta
N}{2}(1-z^{2})\cos\phi\sin\phi.$ (23)

In the following we will analyze the topology of phase portraits which are a
geometrical representation of trajectories of a dynamical system in the phase
space. In our case, trajectories are tangent to the velocity field
$(\dot{\phi},\dot{z})$. The phase portrait consists of fixed points or closed
orbits corresponding to a steady state, and satisfies
$(\dot{\phi},\dot{z})=(0,0)$. Spin squeezing takes place in the vicinity of
unstable fixed points. We are interested in the fixed point located at $z=0$
and $\phi=\frac{\pi}{2}$ according to the location of our initial spin
coherent state. The classification of fixed points can be found by analysis of
the eigenproblem of the stability matrix ${\cal M}$ which in our case is

${\cal M}=\begin{bmatrix}\frac{\partial^{2}\epsilon}{\partial
z\partial\phi}&\frac{\partial^{2}\epsilon}{\partial^{2}z}\\\
-\frac{\partial^{2}\epsilon}{\partial^{2}\phi}&-\frac{\partial^{2}\epsilon}{\partial
z\partial\phi}\end{bmatrix}=\frac{N}{2}\begin{bmatrix}0&1\\\
\eta&0\end{bmatrix}.$ (24)

When $\eta\neq 0$ then the matrix ${\cal M}$ has two non-degenerate real
eigenvalues of the oposite sign
$(\lambda_{1},\lambda_{2})=\frac{1}{2}\sqrt{\eta}N(-1,1)$ and two real
eigenvectors

$v_{1}=\begin{bmatrix}-\frac{1}{\sqrt{\eta}}\\\
1\end{bmatrix},v_{2}=\begin{bmatrix}\frac{1}{\sqrt{\eta}}\\\ 1\end{bmatrix}.$
(25)

Scalar product of the two eigenvectors $\\{v_{1},v_{2}\\}$ defines the angle
$\theta$ between in-going and out-going trajectories crossing at the centre of
the unstable saddle fixed points

$\theta=\arccos{\frac{\langle v_{1}|v_{2}\rangle}{|v_{1}||v_{2}|}}.$ (26)

In Fig. 2 we show examples of the mean-field phase portraits, i.e. the
constant energy lines for $\eta=\\{10^{-5},0.04,0.4,1\\}$. The arrows indicate
the direction of the evolution, and visualize the dynamics in the vicinity of
the fixed point. For $\eta\approx 0$ the angle between in-going and out-going
trajectories is $\theta\approx\pi$ and corresponds to the pure OAT dynamics
with non-isolated unstable fixed point. For $\eta>0$ the nature of fixed point
changes to the unstable saddle fixed point, see panels (b)-(d), which in the
limiting case $\eta=1$ corresponds to the TACT dynamics, panel (d). Note, the
angle $\theta$ is approximately $\pi/2$ when the value of anisotropy parameter
$\eta$ is one.

##  IV Scaling with the system size

In this paragraph we study the scaling of the best squeezing for the
anisotropic TACT model (19). We apply the Gaussian approach within the
Bogoliubov-Born- Green-Kirkwood-Yvon (BBGKY) hierarchy Anglin and Vardi
(2001); André and Lukin (2002) which was used in Kajtoch and Witkowska (2015)
to explain the scaling for the TACT model. Here, we generalize the theory
taking into account the values of parameter $\eta$ different than one.

We start with equations of motion for expectation values of spin operators
$\langle\dot{\hat{S}}_{j}\rangle$ which involve terms that depend on the
first-order moments $\langle\hat{S}_{j}\rangle$ and second-order moments
$\langle\hat{S}_{i}\hat{S}_{j}\rangle$. Subsequently, the time evolution of
the second-order moments depends on themselves and on third-order moments, and
so on. It leads to the BBGKY hierarchy of equations of motion for expectation
values of operator products. The hierarchy is then truncated by keeping the
first- and the second-order moments,

$\displaystyle\langle\hat{S}_{i}\hat{S}_{j}\hat{S}_{k}\rangle$
$\displaystyle\simeq\langle\hat{S}_{i}\hat{S}_{j}\rangle\langle\hat{S}_{k}\rangle+\langle\hat{S}_{j}\hat{S}_{k}\rangle\langle\hat{S}_{i}\rangle+\langle\hat{S}_{j}\hat{S}_{i}\rangle\langle\hat{S}_{j}\rangle$
$\displaystyle-\langle\hat{S}_{i}\rangle\langle\hat{S}_{j}\rangle\langle\hat{S}_{k}\rangle.$
(27)

To perform the scaling analysis, we first introduce a small parameter
$\epsilon=1/N$, and transform the spin components into
$\hat{J}_{j}=\sqrt{\epsilon}\hat{S}_{j}$ which obey cyclic commutation
relations $[\hat{J}_{x},\hat{J}_{y}]=i\sqrt{\epsilon}\hat{J}_{z}$. The
Hamiltonian (19) then reads
$\hat{H}=\frac{\hbar\chi}{\epsilon}\left(\hat{J}_{z}^{2}-\eta\hat{J}_{x}^{2}\right)$.
Equations of motion for expectation values of the spin operators
$\langle\hat{J}_{j}\rangle\equiv h_{j}$, second order moments
$\langle\hat{J}_{i}\hat{J}_{j}\rangle\equiv\Delta_{ij}$ and
$\langle\hat{J}_{j}^{2}\rangle-\langle\hat{J}_{j}\rangle^{2}\equiv\delta_{j}$
read

$\displaystyle\dot{h}_{y}$ $\displaystyle=2(1+\eta)\Delta_{xz},$ (28)
$\displaystyle\dot{\Delta}_{xz}$
$\displaystyle=-2(\delta_{z}+\eta\delta_{x})h_{y},$ (29)
$\displaystyle\dot{\delta}_{z}$ $\displaystyle=-4\eta\Delta_{xz}h_{y},$ (30)
$\displaystyle\dot{\delta}_{x}$ $\displaystyle=-4\Delta_{xz}h_{y},$ (31)

where time is measured in dimensioneless unit $\tau=\chi t/\sqrt{\epsilon}$.
The initial coherent state at the unstable saddle fixed point,
$|\Psi(0)\rangle=|\theta=\pi/2,\varphi=\pi/2\rangle$, gives the following
initial conditions: $h_{y}(0)=(2\sqrt{\epsilon})^{-1}$,
$\delta_{z}(0)=\delta_{x}(0)=1/4$ and $\Delta_{xz}(0)=0$. In order to find the
approximate solution we introduce the two quadratures:
$X=\delta_{z}+\sqrt{\eta}\Delta_{xz}$ and
$Y=\delta_{z}-\sqrt{\eta}\Delta_{xz}$ obeing the dynamical equations
$\dot{X}=-4\sqrt{\eta}Xh_{y}$ and $\dot{Y}=-4\sqrt{\eta}Yh_{y}$ which have the
following solutions:

$\displaystyle X(t)=X(0)e^{-4\sqrt{\eta}f(\tau)},$ (32) $\displaystyle
Y(t)=Y(0)e^{4\sqrt{\eta}f(\tau)},$ (33)

where $f(\tau)=\int_{0}^{\tau}h_{y}(\tau^{\prime})d\tau^{\prime}$ for
$\eta\neq 0$. This gives

$\displaystyle\delta_{z}(\tau)$
$\displaystyle=\delta_{z}(0)\cosh{\left[4\sqrt{\eta}f(\tau)\right]},$ (34)
$\displaystyle\Delta_{xz}(\tau)$
$\displaystyle=-\frac{\delta_{z}(0)}{\sqrt{\eta}}\sinh{\left[4\sqrt{\eta}f(\tau)\right]},$
(35) $\displaystyle h_{y}(\tau)-h_{y}(0)$
$\displaystyle=-\frac{\delta_{z}(0)}{\sqrt{\eta}}\int_{0}^{\tau}\sinh{\left[4\sqrt{\eta}f(\tau^{\prime})\right]}d\tau^{\prime}.$
(36)

In principle, the solution for $h_{y}$ can be find in self-consistent way,
here however, we approximate it by taking the first iteration, namely
$f(\tau)\simeq f(0)+f^{\prime}(0)\tau$, which results in

$h_{y}(\tau)=\frac{1}{2\sqrt{\epsilon}}\left[1+\frac{(1+\eta)\epsilon}{2\eta}\left(1-\cosh(2\tau\sqrt{\eta/\epsilon})\right)\right].$
(37)

(a)(b) Figure 3: (a) Examples of the scaling of the best squeezing
$\xi^{2}_{\rm best}$ and the best squeezing time $\chi t_{best}$ with the
number of atoms $N$ for various values of the parameter of anisotropy
$\eta=\\{1,\,0.5,\,10^{-3}\\}$. (b) Scaling exponents $\alpha$ obtained by
fitting the function $\propto N^{-\alpha}$ to the best squeezing $\xi^{2}_{\rm
best}$ is shown by the red points (red solid line is added to guide the eye).
The angle $\theta$ between the in-going and out-going mean-field trajectories
(see main text) is shown by the dashed blue line.

Next, one evaluates the evolution of (34) and (35) by taking (37) in
$f(\tau)$. Finally, noting that the spin squeezing parameter (1) is determined
by the quadrature $X(t)$, namely $\xi^{2}\approx X(t)$, when approximating
$\langle S\rangle\approx h_{y}(0)/\sqrt{\epsilon}$, we obtain the scaling of
the best squeezing and the best squeezing time with $N$ by keeping the leading
order terms in $\epsilon$, what

$\xi^{2}_{\rm best}\sim 1/N,\,\,\,\,\,\chi t_{\rm best}\sim\frac{\ln(\eta
N)}{\sqrt{\eta}N},$ (38)

when $\eta$ is of the order of one. We compared the above analytical
predictions with the numerically solved set of differential equations
(28)-(31) and confirmed the scaling (38) when $\eta\in(0.3,1)$.

The quantitative illustration of the above results can be provided by
analyzing the scaling of the best squeezing with $N$ obtained from the
numerical time evolution of the TMM Hamiltonian (19). Fig. 3 (a) presents the
best squeezing $\xi^{2}_{\rm best}$ and the best squeezing time $\chi t_{\rm
best}$ as a function of particle number $N$. Power-law behaviour can be
noticed for various $\eta$. Therefore, for each value of the anisotropy
parameter $\eta$ we extracted the scaling exponent $\alpha$ by fitting
$\xi^{2}_{\rm best}\sim N^{-\alpha}$. Panel (b) of Fig. 3 shows the change of
the fitted exponent $\alpha$ as the function of anisotropy parameter $\eta$
and is compared to the variation of the angle $\theta$. A characteristic
feature is a change in the value of $\alpha$ when $\eta\in(10^{-3},10^{-1})$.
In the same range $\theta$ diminishes from $\pi$ to approximately $\pi/2$. We
conclude, the variation of $\alpha$ is driven by the change in the structure
of the unstable fixed point. It is worth to mention here, that $\alpha\approx
1$ when $\eta\approx 1$. Our results show that the Heisenberg limited level of
squeezing is possible in the anisotropic TACT model.

##  V Conclusions

In this work, we show how the OAT mechanism, generating many-body
entanglement, can be accelerated by the long-range interactions via activation
of the anisotropic TACT mechanisms. We explain the activation of the TACT
mechanism during competition of contact and dipolar interactions between
bosons in a superfluid phase. We propose the feasible experimentally quantum
simulator for the amisotropic TACT dynamics based on dipolar two-component
Bose-Hubbard in a one-dimensional optical lattice. With the help of the
scaling analysis, we show that it is possible to obtain a Heisenberg limited
level of squeezing for a weak anisotropy. The anisotropic TACT model
accelerates the spin squeezing dynamics compared to OAT with the improvement
of the level of squeezing. Our protocol allows for fast generation of many-
body entangled states with entanglement depth larger than in a standard OAT
scenario.

Our work provides an essential step toward generating many-body entangled
states during two-axis counter-twisting protocol in state-of-the-art
experimental setups, paving the way for obtaining the Heisenberg limit of
spectroscopic measurements in ultracold atoms systems.

## ACKNOWLEDGMENTS

We gratefully acknowledge discussions with B. B. Laburthe-Tolra. This work was
supported by the Polish National Science Centre projects
DEC-2019/35/O/ST2/01873 (T.H.Y.), DEC-2020/38/L/ST2/00375 (M.D.), and Grant
No. 2019/32/Z/ST2/00016 through the project MAQS under QuantERA, which has
received funding from the European Union’s Horizon 2020 research and
innovation program under grant agreement no 731473 (E.W. and M.G.). M.P.
acknowledges the support of the Polish National Agency for Academic Exchange,
the Bekker programme no: PPN/BEK/2020/1/00317. ICFO group acknowledges support
from: ERC AdG NOQIA; Ministerio de Ciencia y Innovation Agencia Estatal de
Investigaciones (PGC2018-097027-B-I00/10.13039/501100011033,
CEX2019-000910-S/10.13039/501100011033, Plan National FIDEUA
PID2019-106901GB-I00, FPI, QUANTERA MAQS PCI2019-111828-2, QUANTERA DYNAMITE
PCI2022-132919, Proyectos de I+D+I “Retos Colaboración” QUSPIN
RTC2019-007196-7); European Union NextGenerationEU (PRTR); Fundació Cellex;
Fundació Mir-Puig; Generalitat de Catalunya (European Social Fund FEDER and
CERCA program (AGAUR Grant No. 2017 SGR 134, QuantumCAT & U16-011424, co-
funded by ERDF Operational Program of Catalonia 2014-2020); Barcelona
Supercomputing Center MareNostrum (FI-2022-1-0042); EU Horizon 2020 FET-OPEN
OPTOlogic (Grant No 899794); National Science Centre, Poland (Symfonia Grant
No. 2016/20/W/ST4/00314); European Union’s Horizon 2020 research and
innovation programme under the Marie-Skłodowska-Curie grant agreement No
101029393 (STREDCH) and No 847648 (“La Caixa” Junior Leaders fellowships
ID100010434: LCF/BQ/PI19/11690013, LCF/BQ/PI20/11760031, LCF/BQ/PR20/11770012,
LCF/BQ/PR21/11840013). A part of computations were carried out at the Centre
of Informatics Tricity Academic Supercomputer & Network.

##  Appendix A Dipolar interaction

To obtain the lattice version of dipolar interaction one starts with the
Hamiltonian (6) and (8), associates the dipole moment with the Pauli matrices
as $\bm{\mu}_{1}=-\gamma\bm{\sigma}_{1}$ with
$\bm{\sigma}_{1}=(\sigma_{x},\sigma_{y},\sigma_{z})$, and the same at ${\bf
r}_{2}$. Then one obtains

$\displaystyle\hat{H}_{\rm d}=\int d^{3}\bm{r}_{1}\int
d^{3}\bm{r}_{2}\frac{\gamma^{2}}{|\bm{r}_{1}-\bm{r}_{2}|^{3}}$
$\displaystyle\times\left[(1-3\cos^{2}\theta_{12})\left(\hat{J}^{z}_{1}\hat{J}^{z}_{2}-\frac{\hat{J}^{+}_{1}\hat{J}^{-}_{2}+\hat{J}^{-}_{1}\hat{J}^{+}_{2}}{4}\right)\right.$
$\displaystyle-\frac{3}{4}\sin^{2}\theta_{12}\left(e^{2i\phi_{12}}\hat{J}^{-}_{1}\hat{J}^{-}_{2}+{\rm
h.c.}\right)$ $\displaystyle\left.-\frac{3}{4}\sin
2\theta_{12}\left(e^{i\phi_{12}}(\hat{J}^{z}_{1}\hat{J}^{-}_{2}+\hat{J}^{-}_{1}\hat{J}^{z}_{2})+{\rm
h.c.}\right)\right],$ (39)

with ${\bf r}_{1}\neq{\bf r}_{2}$, and where
$\hat{J}^{+}_{1}=\hat{\Psi}_{\uparrow}^{\dagger}({\bf
r}_{1})\hat{\Psi}_{\downarrow}({\bf r}_{1})$,
$\hat{J}^{-}_{1}=\hat{\Psi}_{\downarrow}^{\dagger}({\bf
r}_{1})\hat{\Psi}_{\uparrow}({\bf r}_{1})$,
$\hat{J}^{z}_{1}=(\hat{\Psi}^{\dagger}_{\uparrow}({\bf
r}_{1})\hat{\Psi}_{\uparrow}({\bf
r}_{1})-\hat{\Psi}_{\downarrow}^{\dagger}({\bf
r}_{1})\hat{\Psi}_{\downarrow}({\bf r}_{1}))/2$, and similarly at ${\bf
r}_{2}$. The two angles $\phi,\,\theta$ parameterize the normal vector along
${\bf r}_{1}-{\bf r}_{2}$ direction, namely $\vec{n}_{12}=\frac{{\bf
r}_{1}-{\bf r}_{2}}{|{\bf r}_{1}-{\bf
r}_{2}|}=(\cos\phi_{12}\sin\theta_{12},\sin\phi_{12}\sin\theta_{12},\cos\theta_{12})$.

We assume the system is loaded into one-dimensional optical lattice potential
$V_{\rm latt}=V_{0}\sin^{2}({2\pi x/\lambda_{\rm latt}})$, $\lambda_{\rm
latt}$ is the lattice wave-length, while remains in its ground state in
transverse directions. We assume also the atomic gas is polarized initially
along the $z$-axis and the polarization axis sets the quantization axis, as
illustrated in Fig. 4. Therefore, we consider the following form of the field
operator

$\displaystyle\hat{\Psi}_{\uparrow}({\bf r})$
$\displaystyle=\hat{\Phi}_{\uparrow}(x)\phi(y)\phi(z),$ (40)

and we expand $\hat{\Phi}_{\uparrow}(x)$ in the basis of Wannier functions
$w(x-x_{j})$ localized around lattice sites, where $x_{j}$ denotes position of
the j-th site in the lowest energy band,

$\hat{\Phi}_{\uparrow}(x)=\sum_{j}\hat{a}_{\uparrow,j}w(x-x_{j}),$ (41)

where $\hat{a}_{j,\uparrow}$ annihilates an atom in the single-particle Wan-
nier state $w(x-x_{j})$ of the lowest energy band localized on the j-th site,
in the internal state $\uparrow$. In (40) we assume $\phi(y)$ and $\phi(z)$
are the ground state wave-functions of the system in the $y$ and $z$
directions. The same applies for the $\downarrow$ operator.

Figure 4: Schematic of the geometry of the system: grey arrows indicates
initial configuration of the elementary dipols and the green curve the optical
lattice potential.

The geometry of the system, we have chosen, determines the normal vector,
$\vec{n}=(1,0,0)$, and sets the value of $\theta_{12}=\pi/2$ and
$\phi_{12}=0$. Taking this into account, in the tight-binding limit when the
lattice height is larger than the recoil energy
$E_{R}=(2\pi)^{2}/(2m\lambda_{\rm latt}^{2})$ and the Wannier functions are
well localized around each lattice site, the dipolar Hamiltonian reduces to

$\hat{H}_{\mathrm{d}}=\sum_{j,k\neq
j}\frac{\gamma^{2}}{d^{3}\left|j-k\right|^{3}}\left(\hat{S}_{z,j}\hat{S}_{z,k}-2\hat{S}_{x,j}\hat{S}_{x,k}+\hat{S}_{y,j}\hat{S}_{y,k}\right),$
(42)

due to normalization of the wave functions, and were $d=\lambda_{\rm latt}/2$
will be absorbed in the parameter $\gamma^{2}$ in the main part of the paper.

It is worth commenting here about the importance of the geometry chosen. There
is a symmetry between the $x$ and $y$ axis, i.e., if the lattice would be
along the $y$ axis the resulting Hamiltonian (42) would have the factor minus
two in the front of the $\hat{S}^{y}_{j}\hat{S}^{y}_{k}$ term. On the other
hand, if the lattice would be along $z$ axis, the factor $-2$ appears in the
front of $\hat{S}^{z}_{j}\hat{S}^{z}_{k}$. This has an important consequence
in the resulting effective model (19) which would be the OAT one.

##  Appendix B Numerical evaluation of spin squeezing parameter

###  B.1 Dipolar Bose-Hubbard model

We performed the full many-body numerical simulations of $\hat{H}_{\rm
dBH}=\hat{H}_{\rm BH}+\hat{H}_{\rm d}$ with (II) and (11). To this end we
constructed the Fock states basis, as described in Płodzień  _et al._ (2020).
We implemented numerically the matrix representations of the Hamiltonian
$\hat{H}_{\rm dBH}$, and the initial spin coherent state is

$|\Psi(0)\rangle=|\theta,\psi\rangle=e^{-i\hat{S}_{z}\psi}e^{-i\hat{S}_{y}\theta}|\Psi_{a}\rangle,$
(43)

where $|\Psi_{a}\rangle$ is the ground state of the system when all atoms are
in the $|\uparrow\rangle$ state. The system evolves according to the unitary
operator, namely

$|\Psi(t)\rangle=e^{-i\hat{H}_{\rm dBH}t/\hbar}|\Psi(0)\rangle,$ (44)

and the spin squeezing parameter (1) is calculated.

###  B.2 Two-mode model

In order to find the scaling exponents we perform numerical time evolution of
the two-mode model. We express Hamitlonian (19) in the Fock state basis
consisting the vectors of the form $|n,N-n\rangle$, where $N$ is the total
number of atoms, $n$ is the number of the particles in the $|\uparrow\rangle$
state and $N-n$ is the number of the particles in the $|\downarrow\rangle$
state.

Our initial state is the spin coherent state, which we obtain as a double
rotation of the state $|N,0\rangle$ according to:

$|\Psi(0)\rangle=|\theta,\psi\rangle=e^{-i\hat{S}_{z}\psi}e^{-i\hat{S}_{y}\theta}|N,0\rangle.$
(45)

Next, we apply the unitary evolution

$|\Psi(t)\rangle=e^{-i\hat{H}_{\rm TMM}t/\hbar}|\Psi(0)\rangle,$ (46)

and calculate the spin squeezing parameter (1) and find its first minimum
$\xi^{2}_{\rm best}$, as well as the time at which it occurs $\chi t_{\rm
best}$.

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