# Controlling entanglement in a triple-well system of dipolar atoms

K. Wittmann W<EMAIL_ADDRESS>Instituto de Física da UFRGS, Porto Alegre,
RS, Brazil  L. H. Ymai  Universidade Federal do Pampa, Bagé, RS, Brazil.  B.
H. C. Barros  Instituto de Física da UFRGS, Porto Alegre, RS, Brazil  J. Links
School of Mathematics and Physics - The University of Queensland, Brisbane,
Australia.  A. Foerster<EMAIL_ADDRESS>Instituto de Física da UFRGS,
Porto Alegre, RS, Brazil

###### Abstract

We study the dynamics of entanglement and atomic populations of ultracold
dipolar bosons in an aligned three-well potential described by an extended
Bose-Hubbard model. We focus on a sufficiently strong interacting regime where
the couplings are tuned to obtain an integrable system, in which the time
evolution exhibits a resonant behavior that can be exactly predicted. Within
this framework, we propose a protocol that includes an integrability breaking
step by tilting the edge wells for a short time through an external field,
allowing the production of quantum states with a controllable degree of
entanglement. We analyze this protocol for different initial states and show
the formation of highly entangled states as well as NOON-like states. These
results offer valuable insights into how entanglement can be controlled in
ultracold atom systems that may be useful for the proposals of new quantum
devices.

##  I Introduction

Quantum entanglement is a phenomenon discovered in the foundations of quantum
physics that paved the way for a new era of technological advances. It
represents non-local correlations between separate parts of a quantum system.
As a resource, entanglement has been proven to be very useful for performing
numerous tasks that face barriers in a classical setting, finding broad
applications in quantum information processing [1, 2, 3, 4, 5, 6], quantum
teleportation [7, 8, 9, 10, 11], and quantum metrology and sensing [12, 13,
14].

Entangled states are key ingredients in the proposals of protocols for the
development of new quantum devices [15, 16, 17, 18, 19, 20, 21], and hence
understanding the mechanisms for producing and controlling entangled states
with a high degree of precision is of fundamental importance. In this context,
the search for highly entangled states is the aim of many technological
quantum applications [22], which can be exploited through different platforms.
Among these, ultracold atoms are especially interesting because they enable
the manipulation of atoms arranged in optical potentials, with astonishing
precision and versatility of the operating control [23, 24, 25].

In recent experiments on ultracold quantum gases, dipolar bosons are loaded
into optical lattices to generate long-range dipole-dipole interaction (DDI),
allowing access to fascinating novel quantum properties and phases [26]. The
dynamics of such dipolar boson systems have been intensively studied and
described, with good results, by an extended Bose-Hubbard model (EBHM) [27,
28]. One interesting feature of the EBHM with few bosonic modes is that the
couplings of interactions can be tuned to achieve an integrable regime, which
is particularly suited for the design of quantum devices. For instance, in
[29], the conserved charge provided by integrability plays a crucial role when
examining the quantum dynamics of a dipolar Bose-Einstein condensate (BEC) on
a three-well aligned system, making it a potential candidate for constructing
an atomic transistor [30]. Other integrable quantum systems are being recently
utilized to support the development of quantum technologies. These include
quantum circuits created from transfer matrices [31] and those created through
the star-triangle relation [32], central spin models for quantum sensors [33],
and the preparation of Bethe states on a quantum computer [34, 35, 36, 37].

Here we consider an integrable triple-well model of dipolar bosons and propose
a protocol to create quantum states with controllable entanglement level. The
control is realized by breaking the integrability for a short time and the
resulting entanglement is characterized by the von Neumann entropy and
correlation functions. We test the protocol for a range of different initial
states, demonstrating how to produce highly entangled states as well as other
important quantum states such as NOON-like states 111NOON state, belonging to
the class of Schrödinger cat states, is an “all and nothing” superposition of
two different modes. See for example [51, 55, 56, 21, 57].

The paper is organized as follows. In section II, we describe the system and
discuss the conditions for obtaining an effective description of the
integrable system in the resonant regime. In section III, we analyse the
dynamics of the system and the entanglement behaviour in the resonant regime.
In Section IV, we propose a protocol for controlling entanglement by briefly
tilting the edge sites of the system. In sections V-VII, we analyse the action
of the protocol on different initial states. A discussion on interferometric
applications of the protocol and details on the ground state structure are
given in the appendices. The conclusions are given in section VIII.

##  II System description

We consider a system of dipolar atoms in an aligned triple-well potential
described by the following extended Bose-Hubbard model:

$\displaystyle H$ $\displaystyle=$
$\displaystyle\frac{U_{0}}{2}\sum_{i=1}^{3}N_{i}(N_{i}-1)+\sum_{i=1}^{3}\sum_{j=1;j\neq
i}^{3}\frac{U_{ij}}{2}N_{i}N_{j}$ (1)
$\displaystyle-\frac{J}{\sqrt{2}}(a_{1}^{\dagger}a_{2}+a_{2}^{\dagger}a_{1}+a_{2}^{\dagger}a_{3}+a_{3}^{\dagger}a_{2}),$

where $a_{i}$, $a_{i}^{\dagger}$ and $N_{i}=a_{i}^{\dagger}a_{i}$ are the
bosonic annihilation, creation, and number operators of the well (or site)
$i=1,2,3$, respectively. The coupling $J$ denotes the hopping rate of atoms
between neighboring wells and $U_{0}$ and $U_{ij}=U_{ji}$ set the on-site and
long-range interactions, respectively. The on-site interaction
$U_{0}=U_{sr}+U_{dd}$ results from short range interaction $U_{sr}$ and on-
site dipole-dipole interaction (DDI) $U_{dd}$. The short-range interaction
$U_{sr}\propto 4\pi\hbar^{2}a/m$ is determined by the $s$-wave scattering
length $a$ which is controlled through a magnetic field via Feshbach resonance
and $m$ is the mass of atom. The on-site DDI $U_{dd}\propto\mu^{2}$ and long
range interactions $U_{ij}\propto\mu^{2}$ obey an inverse cubic law whose
strength is determined by the permanent magnetic dipole moment $\mu$ of
dipolar atom considered and highly depends on the geometry of potential trap
and the polarization direction of dipoles [27, 39]. A schematic representation
of this system is presented in Figure 1.

Figure 1: Schematic representation of the triple well system. The arrows
represent the dipoles of atoms oriented along the direction of polarization.
The coupling $J$ represents the hopping rate, $U_{0}$ characterizes on-site
interactions, while $U_{ij}$ characterizes the DDI between particles on
different sites.

For the particular case when $U_{12}=U_{23}$ and $U_{13}=U_{0}$, the
Hamiltonian given by Eq. (1) is integrable [40] and can be reduced, up to a
global constant, to

$\displaystyle H$ $\displaystyle=$ $\displaystyle U(N_{1}-N_{2}+N_{3})^{2}$
(2)
$\displaystyle-\frac{J}{\sqrt{2}}(a_{1}^{\dagger}a_{2}+a_{2}^{\dagger}a_{1}+a_{2}^{\dagger}a_{3}+a_{3}^{\dagger}a_{2}),$

where $N=N_{1}+N_{2}+N_{3}$ is the total number of particles and
$U=(U_{0}-U_{12})/4$ represents the effective interaction energy, which can be
experimentally tuned by controlling the polarization direction and depth of
the well potential trap. A discussion on the feasibility of a physical
realization of this system by means of Bose-Einstein condensates of dipolar
atoms can be found in [29]. In this integrable case, the model can be
formulated and solved using the quantum inverse scattering method and Bethe
ansatz methods [40]. It acquires the additional conserved operator

$\displaystyle
Q=\frac{1}{2}(N_{1}+N_{3}-a_{1}^{\dagger}a_{3}-a_{3}^{\dagger}a_{1}),$ (3)

besides the Hamiltonian $H$ and the total number of particles $N$, resulting
in three independent conserved operators in an equal number of system modes.
The conserved charge $Q$ plays an important role in the resonant regime,
characterized by the tunneling of atoms between the wells at the edges
(labeled by $i=1$ and $3$), while the number of particles in the middle well
($i=2$) remains approximately constant. This resonant behavior is a
consequence of a second-order process that occurs in a relatively strong
interaction regime [27, 41]. More specifically, when $|U(N-2l)/J|\gg 1$ and
for the initial state

$|\Psi_{0}\rangle=|N-k-l,l,k\rangle,$ (4)

where $l$ ($l=0,...,N$) and $k$ ($k=0,...,N-l$) represent the number of atoms
initially at wells $2$ and $3$, respectively, the quantum dynamics of the
Hamiltonian (2) can be well described by the effective Hamiltonian [29],

$\displaystyle H_{\text{eff}}=\omega_{l}Q,$ (5)

where constant $\omega_{l}$ is given by

$\omega_{l}=\lambda_{l}J^{2}$ (6)

with
$\lambda_{l}=\frac{1}{4|U|}\left(\frac{l+1}{N-2l-1}-\frac{l}{N-2l+1}\right)$
depending on the initial number $l$ of bosons in the middle well. The constant
$\omega_{l}$ will play the role of the resonant tunneling frequency, with
period $T_{l}=2\pi/\omega_{l}$. For the case where $l=0$, let us simply denote
it by $T\equiv T_{0}$.

In the following sections, we first discuss the dynamical quantities that
characterize the behavior of the system and provide information about its
quantum entanglement. After that, we provide a protocol that briefly tilts
wells 1 and 3 to control the entanglement of the quantum state. Then, we
analyze the effects of the protocol on different initial states.

##  III Dynamics of populations and entanglement

We start by considering the dynamics of the system described above in the
integrable and resonant regime, and for convenience, we set $\hbar=1$. We
focus on the time evolution of the average number of particles per well

$\langle N_{i}\rangle=\langle\Psi(t)|N_{i}|\Psi(t)\rangle$ (7)

and the von-Neumann entanglement entropy

$S_{i}(t)=-\operatorname{Tr}\left[\rho_{i}(t)\log\rho_{i}(t)\right]$ (8)

where the density matrix is defined as
$\rho(t)=|\Psi(t)\rangle\langle\Psi(t)|$ and $\rho_{i}(t)$ is the reduced
density matrix of site $i$ where the remaining subsystem is traced out. The
von-Neumann entropy quantifies the bipartite entanglement between the site $i$
and the subsystem of other two sites. In the integrable regime, an initial
state described by $|\Psi_{0}\rangle$ will evolve in time according to

$|\Psi(t)\rangle=\mathcal{U}(t)|\Psi_{0}\rangle$ (9)

where $\mathcal{U}(t)\equiv e^{-iHt}$ is the time-evolution operator. In what
follows, we will use $|\Psi\rangle$ to refer to states obtained using
Hamiltonian (2), and we will use the notation $|\widetilde{\Psi}\rangle$ with
a tilde to denote analytic states obtained using the effective Hamiltonian
(5), from which analytic results can be derived. A comparison between the
quantum states $|\Psi\rangle$ and $|\widetilde{\Psi}\rangle$ obtained for the
same set of parameters will be quantified through the fidelity defined as
$F=|\langle\Psi|\widetilde{\Psi}\rangle|^{2}$. We will consider that the state
$|\Psi\rangle$ theoretically approaches the analytic state
$|\widetilde{\Psi}\rangle$ when $F>0.95$.

For the case of initial state (4), the state $|\widetilde{\Psi}(t)\rangle$
predicts that $\langle N_{2}\rangle=l$ remains constant, while the atoms
oscillate harmonically between sites $i=1$ and $i=3$, according to the
expectation values given by [29]

$\langle
N_{i}\rangle=\frac{1}{2}\left[N-l+(2-i)(N-l-2k)\cos(\omega_{l}t)\right].$ (10)

The expression above shows a maximum amplitude with oscillations of period
$T=2\pi/\omega_{0}$ when all atoms are initially located in one of the edge
wells (i.e., when $|\Psi_{0}\rangle=|N,0,0\rangle$ or
$|\Psi_{0}\rangle=|0,0,N\rangle$ ) and an equilibrium with $\langle
N_{1}\rangle=\langle N_{3}\rangle=N/2$ remaining constant when the edge wells
initially have the same number of atoms (i.e., when
$|\Psi_{0}\rangle=|l,0,l\rangle$, with $N=2l$). These two extreme cases will
be the subject of our study later on. In Figure 2 are shown some numerical
results for the case $|\Psi_{0}\rangle=|N,0,0\rangle$ using the Hamiltonian
(2). Figure 2-a shows the perfect agreement between the results of the
numerical simulation and the expectation values given in (10). In Fig. 2-b the
time evolution of the entanglement entropy $S_{1}$ presents a period of $T/2$
and its first maximum occurring at $t=T/4$, exactly when the populations reach
an equilibrium with $\langle N_{1}\rangle=\langle N_{3}\rangle$. Nevertheless,
despite the von-Neumann entanglement entropy is the most frequently used
measure to quantify entanglement, it does not depend on any particular
observable, making it difficult to perform a direct experimental measurement
of its magnitude. In order to generate signatures to indicate the formation of
highly entangled states, in addition to enabling experimental measurements, we
also evaluate the two-site correlation function defined as

$C_{ij}\equiv|\langle N_{i}\rangle\langle N_{j}\rangle-\langle
N_{i}N_{j}\rangle|.$ (11)

Using the state $|\widetilde{\Psi}(t)\rangle$, we can derive the correlation
function in the closed form $C_{13}=(N/4)\sin^{2}(t\,\omega_{0})$, from which
a maximum value is directly obtained at $t=T/4$. The agreement between
numerical simulation of $C_{13}$ and its analytic formula can be seen in Fig.
2-c. From Figures 2-b and c it is clear that the maximum values of the two-
site correlation functions and the entanglement entropy occur simultaneously.
This result shows that the two-site correlation function is also able to
reveal information about the quantum entanglement of the subsystem of wells.

Figure 2: (a) Expectation value of $N_{1}/N$ (green line) and $N_{3}/N$
(yellow line) for the initial state $|\Psi_{0}\rangle=|20,0,0\rangle$, $U=-2$
and $J=1$. The marks represent the analytic expression (10). (b) Time
evolution of the entanglement entropy $S_{1}$ in units of
$S_{1}^{max}=\ln(N+1)$. (c) Time evolution of correlation functions $C_{13}$
in units of $C_{13}^{max}=N/4$. The numerical simulation (11) is represented
by the solid line, while the marks results from the closed form expression.
The vertical dotted lines mark the instant $t=T/4$, where $\langle
N_{1}\rangle=\langle N_{3}\rangle$ and the values of $S_{1}$ and $C_{13}$ are
maximum.

It is worth noting that the entanglement entropy $S_{2}$ vanishes, since the
state of site 2 remains constant in resonant regime, while $S_{1}=S_{3}\neq 0$
showing that bipartite entanglement is present only in the subsystem of sites
1 and 3.

##  IV Protocol for quantum entanglement control

We now focus on establishing a protocol for generating and controlling
maximally entangled states. The control of quantum entanglement can be
achieved by tilting wells 1 and 3 through the action of an additional coherent
light beam superimposed on the triple well system designed on an optical trap.
During the presence of tilt on wells 1 and 3 the dynamics is governed by the
Hamiltonian [30]

$\mathcal{H}(\epsilon)=H+\epsilon(N_{3}-N_{1}),$ (12)

where $H$ is the integrable Hamiltonian (2) and the parameter $\epsilon$
characterizes the energy offset between edge potential wells (see Figure 3).
Other properties of this Hamiltonian (12) can be found in [29, 42]. Here we
will examine the case where the tilt is introduced into the protocol as a
short-duration square pulse just after the initial state evolves to the state
with maximum correlation at $t=T/4$, as identified in the previous section. It
will be seen that the amount of quantum entanglement is completely determined
by the duration of the square pulse.

The full description of the protocol can be represented as follows

$\displaystyle|\Psi(t)\rangle=|\Psi_{k}(t)\rangle,\quad t_{k-1}\leq t\leq
t_{k},$

where the states for steps $k=1,2,$ and $3$ of the protocol are given
sequentially by

$\displaystyle|\Psi_{1}(t)\rangle$ $\displaystyle=$
$\displaystyle\mathcal{U}(t-t_{0},0)|\Psi_{0}\rangle,$
$\displaystyle|\Psi_{2}(t)\rangle$ $\displaystyle=$
$\displaystyle\mathcal{U}(t-t_{1},\epsilon)|\Psi_{1}(t_{1})\rangle,$
$\displaystyle|\Psi_{3}(t)\rangle$ $\displaystyle=$
$\displaystyle\mathcal{U}(t-t_{2},0)|\Psi_{2}(t_{2})\rangle.$

Here, $\mathcal{U}(t,\epsilon)\equiv e^{-i\mathcal{H}(\epsilon)t}$ is the time
evolution operator. This sequence is depicted in Figure 3 below, illustrating
the dependence on the parameter $\epsilon$.

Figure 3: Schematic representation of the control protocol sequence. The tilt
$\Delta V$ between the potentials of sites 1 and 3 is induced through an
external field. The duration of the field allows the entanglement in the
subsystem consisting of sites 1 and 3 to be controlled.

In what follows, we continue to adopt the notation $|\Psi\rangle$ (without
tilde) for states obtained using the Hamiltonian (12), and
$|\widetilde{\Psi}\rangle$ for analytic states obtained using the effective
Hamiltonian
$\mathcal{H}_{\text{eff}}(\epsilon)=H_{\text{eff}}+\epsilon(N_{3}-N_{1})$
where $H_{\text{eff}}$ is given by (5).

At the end of the whole process, the protocol generates the state

$\displaystyle|\Psi_{\text{out}}\rangle$ $\displaystyle\equiv$
$\displaystyle|\Psi_{3}(t_{3})\rangle$ (13) $\displaystyle=$
$\displaystyle\mathcal{U}(\Delta t_{3},0)\mathcal{U}(\Delta
t_{2},\epsilon)\mathcal{U}(\Delta t_{1},0)|\Psi_{0}\rangle,$

where $\Delta t_{k}=t_{k}-t_{k-1}$, is the duration of $k$-th ($k=1,2,3$) step
of the protocol. As mentioned earlier, we are assuming that $t_{0}=0$, $\Delta
t_{1}=T/4$ and $\Delta t_{2}\ll\Delta t_{1,3}$ such that the breaking of
integrability is the dominant effect in the second step of protocol. In the
following sections, the action of the protocol on different initial input
states will be investigated in detail.

##  V Input Fock state

We start by first considering the case of a completely localized initial state
given by $|\Psi_{0}\rangle=|N,0,0\rangle$. In Figure 4, it is shown the effect
of the protocol on the dynamics for different values of duration of a square
pulse, $\Delta t_{2}$, counted in units of period
$T_{\epsilon}=2\pi/\Omega_{\epsilon}$, where $\Omega_{\epsilon}=2\epsilon$.

In the first line of the Fig. 4, after the action of the square pulse ($t\geq
t_{2}$), it is shown the expectation value of the fractional population of
sites $i=1,3$, which is given by

$\langle N_{i}\rangle/N=1/2-(1-i/2)\sin[\omega_{0}(t-t_{2})]\cos\phi,$ (14)

and $\phi$ is a dimensionless parameter defined as

$\displaystyle\phi=2\epsilon\Delta t_{2}.$

We observe that the amplitude of the expectation values of $N_{i}/N$ decrease
gradually with increasing the pulse duration $\Delta t_{2}$ until the dynamics
becomes stationary balanced for a long time at $\Delta t_{2}=T_{\epsilon}/4$
(or $\phi=\pi/2$) and completely reversed at $\Delta t_{2}=T_{\epsilon}/2$ (or
$\phi=\pi$). In the second line of Fig. 4, the range of values of entanglement
entropy gradually decreases with increasing duration of the pulse, becoming
stationary at its maximum value at $\Delta t_{2}=T_{\epsilon}/4$. The dynamics
of entanglement of the state $|\Psi(t)\rangle$ along the control process is
also signaled in the third line of Fig. 4 through the correlation function of
sites 1 and 3.

Figure 4: Time evolution of the expectation values (row a) $N_{1}/N$ (green
line), $N_{2}/N$ (dotted line), $N_{3}/N$ (yellow line); entanglement entropy
in units of $S_{1}^{max}=\log(N+1)$ (row b) and, two-site correlation function
in units of $C_{13}^{max}=N/4$ (row c). Each column represents a different
value of $\,\Delta t_{2}$. The first column represents the integrable case,
where $\,\Delta t_{2}=0$. The other columns show the cases where $\,\Delta
t_{2}=T_{\epsilon}/8$, $\Delta t_{2}=T_{\epsilon}/4$ and $\,\Delta
t_{2}=T_{\epsilon}/2$, in sequence. In all cases, initial state
$|\Psi_{0}\rangle=|20,0,0\rangle$, $U=2$, $J=1$ and $\epsilon=1$. The vertical
dashed lines represent the instant $t=t_{2}$.

In Figure 5 we present the entanglement entropy of state
$|\Psi_{\text{out}}\rangle$ as a function of $\phi$ for three time intervals
$\Delta t_{3}=T/16,\,T/8,\,T/4$.

Figure 5: Entanglement entropy of state $|\Psi_{\text{out}}\rangle$ as
function of $\phi\in[0,2\pi]$ for $\Delta t_{3}=T/16$ (green) , $\Delta
t_{3}=T/8$ (blue) and $\Delta t_{3}=T/4$ (magenta), using the initial state
$|\Psi_{0}\rangle=|20,0,0\rangle$, $U=2$, $J=1$ and $\epsilon=1$. The dashed
vertical line represents $\Delta t_{2}=T_{\epsilon}/2$.

We observe that entanglement entropy can be controlled over a larger range of
values at $\Delta t_{3}=T/4$. Therefore, for fixed duration $\Delta
t_{3}=T/4$, the protocol predicts the following quantum state

$\displaystyle|\widetilde{\Psi}_{\text{out}}(\phi)\rangle=\frac{\left[\sin(\phi/2)\,a_{1}^{\dagger}+\cos(\phi/2)\,a_{3}^{\dagger}\right]^{N}}{\sqrt{N!}}|0,0,0\rangle,$
(15)

where $|0,0,0\rangle$ is the vacuum state. From the above expression, the
correlation function of sites 1 and 3 can be determined analytically as a
function of parameter $\phi$ and it is given by $C_{13}=(N/4)\sin^{2}\phi$.
When performing the control within the interval $\Delta
t_{2}\in[0,T_{\epsilon}]$, the expression above shows that the maximized
correlation $C_{13}^{max}=N/4$ occurs at $\phi=\pi/2\,(3\pi/2)$, when the
state $|\Psi_{\text{out}}\rangle$ has the maximum entanglement entropy with
all atoms into the (anti)symmetric coherent state with fidelity
$F=0.99827\,(0.999509)$:

$\displaystyle|\widetilde{\Psi}_{\text{out}}\rangle=\frac{\left(a_{1}^{\dagger}\pm
a_{3}^{\dagger}\right)^{N}}{\sqrt{2^{N}N!}}|0,0,0\rangle.$

If all atoms are initially loaded into site 3 (i.e.,
$|\Psi_{0}\rangle=|0,0,N\rangle$), the states with maximum correlation are
generated with symmetry reversed compared to the case where
$|\Psi_{0}\rangle=|N,0,0\rangle$. In the next section, we consider the case
where initially both sites 1 and 3 have the same number of atoms.

##  VI Twin-Fock input state

In this section we investigate the quantum entanglement control for the case
of the initial twin-Fock state in the sites 1 and 3, given by
$|\Psi_{0}\rangle=|l,0,l\rangle$, for which $N=2l$ and the expectation values
$\langle N_{1}\rangle=\langle N_{3}\rangle=l$ remain constant under integrable
time evolution in the resonant regime. Figure 6 presents the dynamics of the
entanglement entropy of state $|\Psi(t)\rangle$ for three different duration
of a square pulse.

Figure 6: Entanglement entropy (a) for $\,\Delta t_{2}=0$ (blue line),
$\,\Delta t_{2}=T_{\epsilon}/8$ (green line) and $\,\Delta
t_{2}=T_{\epsilon}/4$ (dashed line); and correlation (b) for $\,\Delta
t_{2}=0$ (brown line), $\,\Delta t_{2}=T_{\epsilon}/8$ (pink line) and
$\,\Delta t_{2}=T_{\epsilon}/4$ (dashed line). For all the cases, it was used
the initial state $|\Psi_{0}\rangle=|10,0,10\rangle$, $U=2$, $J=1$, and
$\epsilon=1$. The vertical dashed lines represent the instant $t=t_{2}$.

Again, for $\Delta t_{2}=T_{\epsilon}/4$, the entanglement entropy is
stationary.

The Figure 7 shows the entanglement entropy $S_{1}$ as function of
dimensionless parameter $\phi$ and three time intervals $\Delta
t_{3}=T/16,\,T/8$, and $T/4$.

Figure 7: Entanglement entropy of state $|\Psi_{\text{out}}\rangle$ as
function of $\phi\in[0,2\pi]$ for $\Delta t_{3}=T/4$ (magenta), $\Delta
t_{3}=T/8$ (blue) and $\Delta t_{3}=T/16$ (green), using the initial state
$|\Psi_{0}\rangle=|10,0,10\rangle$, $U=2$, $J=1$, $\epsilon=1$ and $\Delta
t_{1}=T/4$. The dashed vertical line represents $\Delta t_{2}=T_{\epsilon}/2$.

In this case, the output state $|\Psi_{\text{out}}\rangle$ presents high
entanglement entropy with a small dip at $\Delta t_{2}=T_{\epsilon}/4$ in its
signature and for $\Delta t_{3}=T/4$ the state predicted by our protocol is
given by

$\displaystyle|\widetilde{\Psi}_{\text{out}}(\phi)\rangle=\frac{\left(2\cos\phi\,a_{1}^{\dagger}a_{3}^{\dagger}+\sin\phi[(a_{1}^{\dagger})^{2}-(a_{3}^{\dagger})^{2}]\right)^{l}}{2^{l}l!}|0,0,0\rangle.$

The above state allows determining analytically the correlation function of
sites 1 and 3 as a function of parameter $\phi$, given by

$\displaystyle C_{13}=\frac{l(l+1)}{2}\sin^{2}\phi.$

In particular, for $\phi=\pi/2$ and $\phi=3\pi/2$, the correlation achieves
its maximum value $C_{13}=l(l+1)/2$ and the state
$|\widetilde{\Psi}_{\text{out}}\rangle$ is highly entangled with the
respective fidelities $F=0.998315$ and $F=0.998771$, given by (up to global
phase)

$\displaystyle|\widetilde{\Psi}_{\text{out}}\rangle=\frac{\left[(a_{1}^{\dagger})^{2}-(a_{3}^{\dagger})^{2}\right]^{l}}{2^{l}l!}|0,0,0\rangle.$

The above state shows that the protocol acts on the initial twin-Fock state by
performing a discrete Fourier transform on the modes 1 and 3 defined as
$a_{1(3)}^{\dagger}\to(a_{1}^{\dagger}\pm a_{3}^{\dagger})/\sqrt{2}$ [43],
which leads to a quantum state with only an even number of particles at sites
1 and 3. This result can be interpreted as a destructive interference process
on the odd number of particles, similar to the well-known Hong-Ou-Mandel (HOM)
effect [44, 45].

It is worth noting that, in the resonant regime, the time evolution operators
$\mathcal{U}(T/4,0)$ and $\mathcal{U}(\Delta t_{2},\epsilon)$ used to generate
the output state $|\Psi_{\text{out}}\rangle$ play an analogous role of the
$50:50$ beam-splitter and phase shifter operations in a Mach-Zehnder (MZ)
interferometer [46, 47] This shows the protocol is capable of performing
interferometric operations in which the phase estimation sensitivity depends
on the choice of the initial state and the observable to be detected. (see
Appendix A).

##  VII Entangled input state

In the previous sections, we considered a class of non-entangled initial
states in which the state of well 2 remains constant over time, and therefore
remains disentangled from the rest of the system. Now we will consider an
entangled initial state in which quantum entanglement between well 2 and the
subsystem composed of the other two wells is also manifest. To this end, let
us analyze the effect of the protocol on the initial state defined as

$\displaystyle|\Psi_{0}\rangle=\frac{1}{\sqrt{2}}|0,N,0\rangle+\frac{1}{\sqrt{2N!}}\left(\frac{a_{1}^{\dagger}+a_{3}^{\dagger}}{\sqrt{2}}\right)^{N}|0,0,0\rangle.$
(17)

The above state has a NOON-like state (NLS) structure, in the sense it is a
superposition between the state with all particles in well 2 and the state
with all atoms in the subsystem of wells 1 and 3. However, the state of the
subsystem of well 1 and 3 has all particles into a coherent state
$|\text{CS}\rangle=\frac{1}{\sqrt{N!}}\left(\frac{a_{1}^{\dagger}+a_{3}^{\dagger}}{\sqrt{2}}\right)^{N}|0,0,0\rangle$.
The motivation for its study is directly related to the ground state of the
integrable Hamiltonian (see Appendix B).

Now, considering the case of initial state (17), the effective Hamiltonian is
still given by (5) with $l=0$, since $\omega_{N}=\omega_{0}$. Then, the
protocol predicts the following quantum state

$\displaystyle|\widetilde{\Psi}_{\text{out}}(\phi)\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}|0,N,0\rangle$ (18)
$\displaystyle+\frac{1}{\sqrt{2N!}}\left(c_{\phi}a_{1}^{\dagger}-s_{\phi}a_{3}^{\dagger}\right)^{N}|0,0,0\rangle,$

where we define

$\displaystyle c_{\phi}$ $\displaystyle=$
$\displaystyle\cos\left(\frac{\phi}{2}-\frac{\pi}{4}\right),\quad
s_{\phi}=\sin\left(\frac{\phi}{2}-\frac{\pi}{4}\right).$

Figure 8 presents the change of entanglement entropies $S_{1}$, $S_{2}$, and
$S_{3}$ with respect to the parameter $\phi$. The figure clearly shows that
the entropy $S_{2}$ remains constant at $S_{2}=\log 2$ while the other
entropies exhibit a dip at $\phi=\pi/2(3\pi/2)$ with the typical value
$S_{1(3)}=\log 2$ of a NOON state.

Figure 8: Entanglement entropies (in units os $S_{\text{NOON}}=\log 2$)
$S_{1}$ (solid line), $S_{2}$ (dot line) and $S_{3}$ (dot-dashed line) of
state $|\widetilde{\Psi}_{\text{out}}\rangle$ as function of
$\phi\in[0,2\pi]$, for $N=10$ and $U=-1.3$.

In addition, the two-site correlation functions obtained from the quantum
state (18) are given by (see Figure 9)

$\displaystyle C_{13}$ $\displaystyle=$
$\displaystyle\frac{N(N-2)}{16}\cos^{2}\phi,$ $\displaystyle C_{12}$
$\displaystyle=$
$\displaystyle\frac{N^{2}}{4}\cos^{2}\left(\frac{\phi}{2}-\frac{\pi}{4}\right),$
$\displaystyle C_{23}$ $\displaystyle=$
$\displaystyle\frac{N^{2}}{4}\sin^{2}\left(\frac{\phi}{2}-\frac{\pi}{4}\right).$

Figure 9: Correlation functions (in units of $C^{max}=N^{2}/4$) $C_{12}$
(solid line), $C_{23}$ (dot-dashed line) and $C_{13}$ (dot line) for $N=10$
and $U=-1.3$.

From the Figure 9, it is clear the occurrence of maximum of $C_{12}$ coincides
with the cancellation of $C_{23}$ and vice-versa when $C_{13}=0$ at
$\phi=\pi/2$ and $\phi=3\pi/2$, producing the corresponding states

$\displaystyle|\widetilde{\Psi}_{\text{out}}(\pi/2)\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}|0,N,0\rangle+\frac{1}{\sqrt{2}}|N,0,0\rangle,$
$\displaystyle|\widetilde{\Psi}_{\text{out}}(3\pi/2)\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}|0,N,0\rangle+\frac{(-1)^{N}}{\sqrt{2}}|0,0,N\rangle,$

with the fidelities $F=0.996123$ and $F=0.973085$, respectively. The above
NOON states can be seen as the result of an entanglement deconcentration
process [48, 49, 50] through unitary transformation on the NLS state, since
they are produced in the subsystems 12 and 23 with less entanglement entropy
$S_{1,3}$ than the initial state. This result shows that the protocol controls
the transition between the bipartite and tripartite entanglement of the
quantum state. It also suggests that the triple-well system can be thought of
as a potential shared router operating at the interface between two individual
quantum devices to perform a transfer of a NOON state.

##  VIII Conclusion

We have proposed a protocol to generate states with controlled levels of
entanglement, where the control is realized by breaking the integrability for
a short period of time. Our study provides closed formulas for correlation
functions to characterize the entanglement in terms of the integrability
breaking time, which allowed us to predict the time required to generate
highly entangled states. In the action of protocol on one of the initial
states, the maximum correlation predicts the formation of NOON states,
whereas, for other unentangled initial states, the maximum correlations are
closely related to interference processes.

Our results have the potential to open new avenues for the manipulation and
short-range transfer of entangled states within multimode sytems. These may
find applications in quantum routing processes of new devices based on
ultracold quantum technology.

##  IX Acknowledgments

The authors acknowledge support from CNPq (Conselho Nacional de
Desenvolvimento Científico e Tecnológico) - Edital Universal 406563/2021-7. AF
and JL are supported by the Australian Research Council through Discovery
Project DP200101339. We thank Rafael Barfknecht for helpful discussions.

##  Appendix A Interferometry

In this section, we discuss some interferometric aspects of the protocol
proposed in section IV.

First, we consider the state produced in equation 15 to calculate the
imbalance population between sites 1 and 3. This provides an interference
pattern as a function of parameter $\phi$ according to equation

$\displaystyle\langle
N_{1}-N_{3}\rangle\equiv\langle\widetilde{\Psi}_{\text{out}}(\phi)|N_{1}-N_{3}|\widetilde{\Psi}_{\text{out}}(\phi)\rangle=-N\cos\phi.$

Note the unconventional negative sign can be changed by extending the duration
of the last operation to $\Delta t_{3}=3T/4$. The phase uncertainty can be
obtained using the error propagation theory [51] and is given by

$\displaystyle\Delta\phi=\frac{\Delta(N_{1}-N_{3})}{|\partial_{\phi}\langle
N_{1}-N_{3}\rangle|}=\frac{1}{\sqrt{N}}.$

where the notation $\Delta X=\sqrt{\langle X^{2}\rangle-\langle X\rangle^{2}}$
is the standard deviation of operator $X$. The above result shows the
uncertainty of parameter $\phi$ is the shot noise limited.

The sensitivity of parameter $\phi$ can be improved for the case of initial
twin-Fock state $|l,0,l\rangle$ at sites 1 and 3. This can be achieved by
detecting the parity operator $\Pi_{1}=e^{-i\pi N_{1}}$ [52], whose
expectation value for the output state generated in equation VI is given by

$\displaystyle\langle\Pi_{1}\rangle\equiv\langle\widetilde{\Psi}_{\text{out}}(\phi)|\Pi_{1}|\widetilde{\Psi}_{\text{out}}(\phi)\rangle=P_{l}(\cos(2\phi-\pi)),$

where

$\displaystyle P_{l}(x)=\sum_{k=0}^{\lfloor
l/2\rfloor}\frac{(-1)^{k}}{2^{l}}\binom{l}{k}\binom{2l-2k}{l}x^{l-2k},$

are the Legendre polynomials. The sensitivity of parameter $\phi$ can be
estimated by

$\displaystyle\Delta\phi=\frac{\Delta\Pi_{1}}{|\partial_{\phi}\langle\Pi_{1}\rangle|},$
(19)

which shows the uncertainty of parameter $\phi$ approaches to the Heisenberg
limit $\Delta\phi\approx 1/(2l)$ when $\phi\approx\pi/2$ (see [52] for
details).

##  Appendix B Ground state

In this section, we discuss the structure of the ground state of integrable
Hamiltonian (2) in the resonant regime with $U<0$. To this end, we first note
that the Hamiltonian (2) can be reduced to a Bose-Hubbard Hamiltonian of a
two-site structure (see, for instance [53])

$\displaystyle
H=U(N_{13}-N_{2})^{2}-J(a_{2}^{\dagger}a_{13}+a_{13}^{\dagger}a_{2}),$

by identifying the single mode operator $\displaystyle
a_{13}=\frac{a_{1}+a_{3}}{\sqrt{2}}$ and the total number of particles
$N_{13}=N_{1}+N_{3}$ in the subsystem of sites 1 and 3. On the other hand, for
a small number of atoms, it is known that the ground state of two-site Bose-
Hubbard Hamiltonian admits the generation of NOON state
$\displaystyle|\text{NOON}\rangle=\frac{1}{\sqrt{2}}(|N,0\rangle+|0,N\rangle)$
in strong repulsive interaction regime [54], which has the entanglement
entropy $S_{\text{NOON}}=-Tr(\rho_{1}\log\rho_{1})=\log 2$ due to it having
only one pair of equally likely base Fock state. Likewise, in the resonant
regime (with $U<0$) for small $N\sim 10$, the ground state $|\text{GS}\rangle$
of three modes integrable Hamiltonian (2) presents high fidelity (above 0.99)
to the NOON-like state (NLS)

$\displaystyle|\text{NLS}\rangle=\frac{1}{\sqrt{2}}|0,N,0\rangle+\frac{1}{\sqrt{2}}\frac{(a_{13}^{\dagger})^{N}}{\sqrt{N!}}|0,0,0\rangle.$

The Figure 10 presents the fidelity
$F=|\langle\text{NLS}|\text{GS}\rangle|^{2}$ as a function of $|U/J|$ and $N$
for $U<0$.

Figure 10:  (a) Fidelity vs $|U/J|$ for $U<0$ and $N=5$ (turquoise), $N=10$
(magenta) and $N=12$ (pink). (b) Fidelity vs $N$ for $U=-1.0$ (dashed line),
$U=-1.3$ (solid line) and $U=-2.0$ (dot dashed line). The vertical lines mark
$|U/J|=1.3$ and $N=10$, values used in section VII.

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