# Entanglement Swapping Using Hyperentangled Pairs of Two-Level Neutral Atoms

Sajal Hasan  Department of Physics, COMSATS University Islamabad, Islamabad,
Pakistan  Syed M. Arslan  0000-0002-5820-6576<EMAIL_ADDRESS>College of
Science and Engineering, Qatar Center of Quantum Computing (QC2), Hamad bin
Khalifa University, Doha, Qatar  Muhammad Imran  National Institute of Lasers
and Optronics College Pakistan Institute of Engineering and Applied Sciences,
Nilore, Islamabad, Pakistan  Rameez-ul Islam  National Institute of Lasers and
Optronics College Pakistan Institute of Engineering and Applied Sciences,
Nilore, Islamabad, Pakistan  Saif Al-Kuwari  0000-0002-4402-7710 College of
Science and Engineering, Qatar Center of Quantum Computing (QC2), Hamad bin
Khalifa University, Doha, Qatar  Tasawar Abbas  0000-0002-6048-7346 Department
of Physics, COMSATS University Islamabad, Islamabad, Pakistan

# Entanglement Swapping Using Hyperentangled Pairs of Two-Level Neutral Atoms

Sajal Hasan  Department of Physics, COMSATS University Islamabad, Islamabad,
Pakistan  Syed M. Arslan  0000-0002-5820-6576<EMAIL_ADDRESS>College of
Science and Engineering, Qatar Center of Quantum Computing (QC2), Hamad bin
Khalifa University, Doha, Qatar  Muhammad Imran  National Institute of Lasers
and Optronics College Pakistan Institute of Engineering and Applied Sciences,
Nilore, Islamabad, Pakistan  Rameez-ul Islam  National Institute of Lasers and
Optronics College Pakistan Institute of Engineering and Applied Sciences,
Nilore, Islamabad, Pakistan  Saif Al-Kuwari  0000-0002-4402-7710 College of
Science and Engineering, Qatar Center of Quantum Computing (QC2), Hamad bin
Khalifa University, Doha, Qatar  Tasawar Abbas  0000-0002-6048-7346 Department
of Physics, COMSATS University Islamabad, Islamabad, Pakistan

Entanglement Swapping Using Hyperentangled Pairs of Two-Level Neutral Atoms

## ABSTRACT

Hyperentangled swapping is a quantum communication technique that involves the
exchange of hyperentangled states, which are quantum states entangled in
multiple degrees of freedom, to enable secure and efficient quantum
information transfer. In this paper, we demonstrate schematics for the
hyperentanglement swapping between separate pairs of neutral atoms through the
mathematical framework of atomic Bragg diffraction, which is efficient and
resistant to decoherence, yielding deterministic results with superior overall
fidelity. The utilized cavities are in superposition state and interact with
the incoming atoms off-resonantly. Quantum information carried by the cavities
is swapped through resonant interactions with two-level auxiliary atoms. We
also discuss entanglement swapping under a delayed-choice scenario and provide
a schematic generalization covering multiple-qubit scenarios. Finally, we
introduce specific experimental parameters to demonstrate the experimental
feasibility of the scheme.

Keywords: Quantum Communication, Quantum Information, Quantum Optics

##  I Introduction

Entanglement is an essential component in many quantum information
technologies, including quantum computing [1], quantum sensing[2, 3], and,
most notably, quantum communication [4, 5, 6, 7].

Many quantum communication protocols based on entanglement have been developed
and experimentally demonstrated [8, 9, 10, 11, 12, 13, 14]. One of these
protocols is Entanglement Swapping, which is a quantum phenomenon that allows
the entanglement of two particles, even if they have never directly
interacted. In fact, it has many applications in quantum information[8, 9,
10]. Recently, many quantum communication protocols began to use a variant of
the normal quantum entanglement, known as Hyperentanglement, where the
entanglement takes place between multiple degrees of freedom. Hyperentangled
states improve channel capacity by minimizing the physical resources, and
hence, in certain situations, lower the decoherence linked to the number of
quantum entities involved [15, 16, 17]. The swapping of such hyperentangled
states presents a viable solution for efficient quantum information transfer
that requires minimal physical resources, especially in the context of
information distribution over complex quantum networks [18, 19].

In this paper, we engineer hyperentanglement with its internal degree of
freedom being molded by the atom’s ability to be in the ground or in the
excited state, and use the transverse quantized momentum state of the atom as
its external degree of freedom. We use Atomic Bragg diffraction to generate
hyperentangled atomic states i.e. states entangled simultaneously in atomic
internal and external degrees of freedom and their further manipulation for
intended entanglement swapping. Atomic Bragg diffraction is one of the most
exciting applications of cavity quantum electrodynamics, which is often used
for quantum state engineering and atomic interferometry and can be used to
generate hyperentanglement with external and internal degrees of freedom of
atoms [20, 18]. Atomic Bragg diffraction is a technique through which one can
develop different types of optical gadgets such as atomic beam splitters and
atomic mirrors [21, 22, 23, 24].

Our scheme involves three parties: Alice, Bob, and Carol. First, Alice and Bob
mutually generate a hyperentangled pair of two-level atoms. Engineering of
such states involves off-resonant atomic Bragg diffractions with the cavity
field, which is initially in a superposition state of zero and one followed by
exposing a momentum arm to a classical laser. Atoms are initially in the
ground state $\ket{g}$ with the momentum state $\ket{P_{0}}$. The interactions
are first-order Bragg diffractions. Consequently, after controlled exposure to
the laser field, the hyperentangled atomic state is produced, where the
internal and external (transverse-momentum) degrees of freedom are correlated.
Once Alice and Bob generate such state, each one sends 1 qubit (atom) to a
third-party, Carol, who possesses two further cavity fields in superposition.
Carol adjusts the cavities so that the deflected arm of each atom passes
through one cavity, and the undeflected arm of each atom passes through
another cavity. Finally, Alice and Bob detect the internal and external states
of the two atoms through four different state-selective detectors after
passing them through a Ramsey zone. Therefore, this results in the remaining
two atoms, which never interacted with each other, sharing an entangled state.
This completes the entanglement swapping process, in which the entanglement is
swapped between the entangled pairs that were initially prepared.

The rest of this paper is organized as follows: In section II, review the
mathematical framework of atomic Bragg diffraction along with the generation
of hyperentangled Bell state among couple of two-level atoms. In Section III,
we describe our entanglement swapping scheme. Section IV describes the
delayed-choice hyperentangled swapping along with its due generalization.
Section V briefly elaborates on the experimental feasibility envisioned for
the proposal under the prevailing experimental scenario. Finally, Section VI
concludes the paper.

##  II Atomic Bragg Diffraction

Atomic Bragg diffraction is a Raman scattering phenomenon that follows the
general conditions of Bragg diffraction. Thus, the conservation of momentum
and energy of the system remains intact. Momentum exchange occurs between the
quantized field trapped in a cavity and the atom, where the magnitude remains
constant, and only the direction of the external momenta changes. The atom’s
momentum consists of two components: transverse and longitudinal. The atom
travels at a slight angle relative to the axis of the cavity, which results in
the quantization of the transverse component of the momentum, and the
longitudinal component, still large, may be treated classically. When such an
atom interacts with the quantized cavity field, the atom experiences a
momentum kick and, consequently, discrete multiples of quantized atomic
momentum are transferred along the k-vector of the field. This is provided by
the relation $P_{out}=P_{in}+l\hbar k$, where $P_{in}=P_{0}=l_{0}\hbar
k\big{/}2$ is the initial momentum of the atom and $l_{0}$ is an even integer
that refers to the order of Bragg diffraction. Each interaction transfers a
momentum kick to the atom of magnitude 0 or $2\hbar k$ and thus $P_{out}$ is
the resultant momentum after $l$ such interactions [25]. Furthermore, the
relation of energy conservation is
$|p_{in}|^{2}\big{/}2M=|p_{out}|^{2}\big{/}2M$, where $M$ is the mass of the
atom. Using the relations for conservation of momentum and energy, we obtain
the fundamental principle of atomic Bragg diffraction:
$l(l+l_{0})\hbar^{2}k^{2}/2M=0$, which consists of two solutions, the first is
$l=0$ representing the undeflected momentum state and the second is $l=-l_{0}$
representing the deflected momentum state. Similarly, first, second and third-
order Bragg diffractions will be represented by higher-order of even integers
2, 4 and 6, respectively [26, 27].

To demonstrate the phenomena of first-order Bragg diffraction, consider a two-
level atom in its ground state $\ket{g}$ and quantized external momentum state
$\ket{P_{0}=\hbar k}$ on the x-axis, i.e., cavity axis, with the atom moving
in the z-axis direction with longitudinal classical momentum, as shown in Fig.
(1). Meanwhile, the cavity is initially taken to be in superposition state
$(\ket{0}+\ket{1})/\sqrt{2}$. The interaction picture Hamiltonian under dipole
and rotating wave approximation is [26, 27, 28, 29, 30, 31, 32, 33, 34, 35,
15, 36, 37, 38, 32],

$H_{I}=\frac{P_{x}^{2}}{2M}+\frac{\hbar\Delta}{2}\sigma_{z}+\hbar\mu\cos(kx)\left[b\sigma_{+}+\sigma_{-}b^{\dagger}\right].$
(1)

Here, the term $P_{x}^{2}/2M$ represents the kinetic energy of the atom
associated with the quantized momentum component, and $\sigma_{-}$,
$\sigma_{+}$ and $\sigma_{z}$ are the atomic lowering, raising, and inversion
operators, respectively. $\Delta$ represents atom-field de-tuning and $\mu$ is
the coupling constant between the atom and field with $b$ and $b^{\dagger}$
being the annihilation and creation field ladder operators.

Figure 1: Schematics of the hyperentangled Bell state generation using off-
resonant Bragg diffraction of two atoms initially in states
$\ket{g_{1},P_{0}^{1}}$ and $\ket{g_{2},P_{0}^{2}}$ injected sequentially in a
cavity which is in superposition state $(\ket{0}+\ket{1})/\sqrt{2}$.

In order to build a correlation between the internal and external degrees of
freedom of a pair of atoms, we begin with consideration of an atom in initial
ground state $\ket{g}$ with quantized momentum state $\ket{P_{0}}$ which
interacts with a cavity field in a superposition state
$(\ket{0}+\ket{1})/\sqrt{2}$ as represented in Fig. (1). The initial state
vector is $\ket{\psi(t=0)}=\ket{g,P_{0}}\otimes(\ket{0}+\ket{1})/\sqrt{2}$.
This specific initial state can be used to engineer the hyper-superposition
state of a single atom [8, 18]. For this the proposed state vector can be
written as,

$\begin{split}\left|\psi(t)\right>=&e^{-\dot{\dot{\iota}}\left(\frac{P_{0}^{2}}{2M\hbar}-\frac{\Delta}{2}\right)t}\sum_{l=-\infty}^{\infty}\left[C_{g,0}^{P_{l}}(t)\left|g,0,P_{l}\right>\right.\\\
&\left.+C_{g,1}^{P_{l}}(t)\left|g,1,P_{l}\right>+C_{e,0}^{P_{l}}(t)\left|e,0,P_{l}\right>\right],\end{split}$
(2)

where $C_{g,0}^{P_{l}}(t),C_{g,1}^{P_{l}}(t)$ and $C_{g,1}^{P_{l}}(t)$ are the
unknown time-evolved probability amplitudes of the respective states. It
should be noted that $C_{g,0}^{P_{l}}(t)$ represents the possibility that the
atom remains in the ground state $\ket{g}$ leaving the cavity in fock state
$\ket{0}$, while $C_{g,1}^{P_{l}}(t)$ represents the possibility that the atom
leaves the cavity in fock state $\ket{1}$ and still remains in ground state
$\ket{g}$ and finally $C_{g,1}^{P_{l}}(t)$ represents the case where the
internal state of the atom gets excited $\ket{e}$ leaving the cavity in fock
state $\ket{0}$, after $l$ interactions. Solving the Schrodinger equation
using the proposed state vector given in Eq. (2) with the Hamiltonian
expressed in Eq. (1), we obtain the following differential equations for the
rate of change of the probability amplitudes. We note that expression (3) is a
simple first-order differential equation, while Eq. (4) and (5) represent an
infinite set of coupled differential equations,

$\dot{\dot{\iota}}\frac{\partial}{\partial
t}C_{g,0}^{P_{l}}(t)=\left(\frac{l(l_{0}+l)\hbar
k^{2}}{2M}\right)C_{g,0}^{P_{l}}(t),$ (3)
$\begin{split}\dot{\dot{\iota}}\frac{\partial}{\partial
t}C_{g,1}^{P_{l}}(t)=&\left(\frac{l(l_{0}+l)\hbar
k^{2}}{2M}\right)C_{g,1}^{P_{l}}(t)\\\
&+\frac{\mu}{2}\left(C_{e,0}^{P_{l+1}}(t)+C_{e,0}^{P_{l-1}}(t)\right),\end{split}$
(4) $\begin{split}\dot{\dot{\iota}}\frac{\partial}{\partial
t}C_{e,0}^{P_{l}}(t)=&\Delta C_{e,0}^{P_{l}}(t)\\\
&+\frac{\mu}{2}\left[C_{g,1}^{P_{l+1}}(t)+C_{g,1}^{P_{l-1}}(t)\right].\end{split}$
(5)

In this case, the atom-cavity interaction is considered to be off-resonant and
obeys the Bragg diffraction condition, i.e $\Delta>\omega_{r}$ where
$\omega_{r}=\hbar k^{2}/2M$ is the recoil frequency imparted to the atom. The
adiabatic assumption is applied where the contribution of the slowly varying
probability amplitudes is considered negligible [38]. Initially, the atom is
in the ground state $\ket{g}$, therefore, Eq. (4) shows an even number of
interactions and Eq. (5) shows an odd number of interactions. Setting
$l_{0}=2$ for first order Bragg diffraction, we get,

For l=1,

$\begin{split}\dot{\dot{\iota}}\frac{\partial}{\partial
t}C_{e,0}^{P_{1}}(t)=\Delta
C_{e,0}^{P_{1}}(t)+\frac{\mu}{2}\left(C_{g,1}^{P_{2}}(t)+C_{g,1}^{P_{0}}(t)\right),\end{split}$
(6)

For l=0,

$\begin{split}\dot{\dot{\iota}}\frac{\partial}{\partial
t}C_{g,1}^{P_{0}}(t)=\frac{\mu}{2}\left(C_{e,0}^{P_{1}}(t)+C_{e,0}^{P_{-1}}(t)\right),\end{split}$
(7)

For l=-1,

$\begin{split}\dot{\dot{\iota}}\frac{\partial}{\partial
t}C_{e,0}^{P_{-1}}(t)=\Delta
C_{e,0}^{P_{-1}}(t)+\frac{\mu}{2}\left(C_{g,1}^{P_{0}}(t)+C_{g,1}^{P_{-2}}(t)\right),\end{split}$
(8)

For l=-2,

$\begin{split}\dot{\dot{\iota}}\frac{\partial}{\partial
t}C_{g,1}^{P_{-}2}(t)=\frac{\mu}{2}\left(C_{e,0}^{P_{-1}}(t)+C_{e,0}^{P_{-3}}(t)\right),\end{split}$
(9)

For l=-3,

$\begin{split}\dot{\dot{\iota}}\frac{\partial}{\partial
t}C_{e,0}^{P_{-3}}(t)=\Delta
C_{e,0}^{P_{-3}}(t)+\frac{\mu}{2}\left(C_{g,1}^{P_{-2}}(t)+C_{g,1}^{P_{-4}}(t)\right).\end{split}$
(10)

Furthermore, under the adiabatic approximation, the time evolution of the
probability amplitudes $\partial C_{g,1}^{P_{1}}(t)/\partial t,\partial
C_{g,1}^{P_{-1}}(t)/\partial t$ and $\partial C_{g,1}^{P_{-3}}(t)\partial t$
can be ignored on practical grounds and the higher order amplitudes
$C_{g,1}^{P_{2}}(t)$ and $C_{g,1}^{P_{-4}}(t)$ may also be skipped due to
meager contributions [35, 15, 39, 37, 38, 32]. Under these approximations, the
above infinite set of coupled equations reduces to the following two coupled
differential equations:

$\frac{\partial}{\partial t}C_{g,1}^{P_{0}}(t)=\dot{\dot{\iota}}\beta
t\left[2C_{g,1}^{P_{0}}(t)+C_{g,1}^{P_{-}2}(t)\right],$ (11)
$\frac{\partial}{\partial t}C_{g,1}^{P_{-2}}(t)=\dot{\dot{\iota}}\beta
t\left[C_{g,1}^{P_{-}2}(t)+C_{g,1}^{P_{0}}(t)\right],$ (12)

with $\beta=\mu^{2}/4\Delta$. By solving these equations, we retrieve the
time-evolved probability amplitudes associated with the state vector given in
Eq. (2):

$\begin{split}C_{g,1}^{P_{0}}(t)=&e^{2\dot{\dot{\iota}}\beta
t}\left[C_{g,1}^{P_{0}}(0)\cos\left(\beta
t\right)+\dot{\dot{\iota}}C_{g,1}^{P_{-}2}(0)\sin\left(\beta
t\right)\right],\end{split}$ (13)
$\begin{split}C_{g,1}^{P_{-2}}(t)=&e^{2\dot{\dot{\iota}}\beta
t}\left[C_{g,1}^{P_{-}2}(0)\cos\left(\beta
t\right)+\dot{\dot{\iota}}C_{g_{1},1}^{P^{1}_{0}}(0)\sin\left(\beta
t\right)\right].\end{split}$ (14)

By selecting the appropriate initial probability amplitudes, one can engineer
the hyper-superposition state. In this specific case, we have selected
$C_{g,1}^{P_{0}}(0)=C_{g,0}^{P_{0}}(0)=1/\sqrt{2}$ and $C_{g,1}^{P_{-2}}(0)=0$
together with the appropriate interaction time as $t=2\pi\Delta/\mu^{2}$ we
get the state [20, 18]:

$\ket{\psi(t=2\pi\Delta/\mu^{2})}=\frac{1}{\sqrt{2}}\left[\left|0,g_{1},P_{0}\right>-\dot{\dot{\iota}}\left|1,g_{1},P_{-2}\right>\right].$
(15)

However, if initially the atom would have been considered to be in excited
state, the same mathematical procedure would be followed, which is presented
in Appendix A.

To generate the hyper-superposition state between the internal and external
degrees of freedom of a single atom, the lower deflected momenta arm i.e.
$\ket{P_{-2}}$ is exposed to the classical laser beam along the y-axis, as
depicted in Fig. (1). This semi-classical resonant interaction [40] is
governed by the Hamiltonian,
$(\hbar\Omega/2)[e^{-\dot{\dot{\iota}}\phi}\sigma_{+}+e^{\dot{\dot{\iota}}\phi}\sigma_{-}]$,
where $\Omega$ is the Rabi frequency of the atom and choosing the interaction
time as $t=\pi/\Omega$, the state becomes:

$\ket{\psi(t=\pi/\Omega)}=\frac{1}{\sqrt{2}}\left[\left|0,g_{1},P^{1}_{0}\right>-\dot{\dot{\iota}}\left|1,e_{1},P^{1}_{-2}\right>\right].$
(16)

Here, it is apparent that a correlation has been built between the atom’s
external and internal states, and the cavity field. The subscripts of the
internal states and the superscripts of the external states represent the
first atom to pass through the cavity field. Passing a second atom, initially
in state $\ket{g_{2},P_{0}^{2}}$ from the same atom-cavity system, a
correlation will be generated between the external states of both atoms with
the cavity field. This second atom will follow similar conditions [20, 18] to
off-resonant Bragg diffraction as described earlier, and following the same
mathematical procedure, which will obtain the probability amplitudes as
described in the equation. (13) and Eq. (14). This second atom will again pass
through the semi-classical laser field obeying resonant interaction and the
state of the system will become:

$\left|\psi(t)\right>=\frac{1}{\sqrt{2}}\left(\left|0,g_{1},g_{2},P_{0}^{1},P_{0}^{2}\right>-\left|1,e_{1},e_{2},P_{-2}^{1},P_{-2}^{2}\right>\right).$
(17)

When the cavity of the atomic system is traced out, a hyperentangled bipartite
Bell state [20] will be generated. For this, an auxiliary two-level atom
initially in the ground state interacts resonantly with the cavity field. This
resonant interaction is governed by the Hamiltonian,
$H_{R}=\hbar\mu\left[\sigma_{+}b+\sigma_{-}b^{\dagger}\right]$, with
$b(b^{\dagger})$ being the ladder operators for the field and
$\sigma_{+}(\sigma_{-})$ are transition operators for the atom. This resonant
auxiliary atom leaves the cavity field into the vacuum state and is detected
in either of its internal states after passing through the Ramsey zone, which
allows us to engineer two mutually orthogonal equi-probable states. For
brevity, we take the following engineered state:

$\left|\psi(t)\right>=\frac{1}{\sqrt{2}}\left(\left|g_{1},g_{2},P_{0}^{1},P_{0}^{2}\right>-\dot{\dot{\iota}}\left|e_{1},e_{2},P_{-2}^{1},P_{-2}^{2}\right>\right).$
(18)

The above equation represents the hyperentangled Bell state, which shows that
the internal and external degrees of freedom of both atoms are entangled [28].
This basic hyperentangled state acts as a building block for our proposal of
hyper-entanglement swapping.

##  III Entanglement Swapping

Fig. (2) visually describes our proposed scheme, which consists of two stages:
1) the generation of a bipartite hyperentangled state, 2) swapping to another
bipartite hyperentangled state. At stage 1 in Fig. (2), two independent
parties, Alice and Bob, generate hyperentangled pairs of atoms 1 and 2, and
atoms 3 and 4, respectively, as described by Eq. 18. Once these pairs are
generated, the initial cavities (Cavity 1 and 2) are traced out. This produces
two pairs of hyperentangled atoms, which can be jointly written as:

$\displaystyle\ket{\psi_{0}}=\frac{1}{2}\big{[}\big{(}\ket{g_{1},g_{2},P_{0}^{1},P_{0}^{2}}-\dot{\dot{\iota}}\ket{e_{1},e_{2},P_{-2}^{1},P_{-2}^{2}}\big{)}$
$\displaystyle\otimes\big{(}\ket{g_{3},g_{4},P_{0}^{3},P_{0}^{4}}-\dot{\dot{\iota}}\ket{e_{3},e_{4},P_{-2}^{3},P_{-2}^{4}}\big{)}\big{]}.$
(19)

Eq. III show that there is no correlation between atoms 1 and 4 and similarly
between atoms 2 and 3. In stage 2, we swap this hyper-entanglement on the pair
of atom 1 and atom 4 by performing Bell basis measurement of the internal and
external states of atoms 2 and 3. We consider two cavities (A and B) that
exist in the superposition state $(\ket{0}+\ket{1})/\sqrt{2}$. Then, the joint
state of the two pairs of hyperentangled atoms, together with the cavities,
can be written as:

$\begin{split}\left|\psi\right>=&\frac{1}{2}\left[\left|g_{2},P_{0}^{2}\right>\left|g_{3},P_{0}^{3}\right>\left|g_{1},g_{4},P_{0}^{1},P_{0}^{4}\right>\right.\\\
&\left.-\dot{\dot{\iota}}\left|g_{2},P_{0}^{2}\right>\left|e_{3},P_{-2}^{3}\right>\left|g_{1},e_{4},P_{0}^{1},P_{-2}^{4}\right>\right.\\\
&\left.-\dot{\dot{\iota}}\left|e_{2},P_{-2}^{2}\right>\left|g_{3},P_{0}^{3}\right>\left|e_{1},g_{4},P_{-2}^{1},P_{0}^{4}\right>\right.\\\
&\left.-\left|e_{2},P_{-2}^{2}\right>\left|e_{3},P_{-2}^{3}\right>\left|e_{1},e_{4},P_{-2}^{1},P_{-2}^{4}\right>\right]\\\
&\otimes\frac{1}{\sqrt{2}}\left(\left|0_{A}\right>+\left|1_{A}\right>\right)\otimes\frac{1}{\sqrt{2}}\left(\left|0_{B}\right>+\left|1_{B}\right>\right).\end{split}$
(20)

Figure 2: Illustration for implementing Quantum Swapping.

The undeflected atomic wavepackets of atom 2 and atom 3, namely
$\ket{g_{1},P_{0}^{2}}$ and $\ket{g_{3},P_{0}^{3}}$, respectively, interact
with cavity A. Similarly, the deflected atomic wavepacket, namely
$\ket{e_{1},P_{-2}^{2}}$ and $\ket{e_{3},P_{-2}^{3}}$, respectively, interact
with cavity B. This is illustrated in Fig. (2). Hence, Eq. 20 can be re-
ordered as:

$\begin{split}\ket{\psi}=&\frac{1}{2}\Big{[}\ket{g_{2},P_{0}^{2}}\ket{g_{3},P_{0}^{3}}\ket{A}-\dot{\dot{\iota}}\ket{g_{2},P_{0}^{2}}\ket{e_{3},P_{-2}^{3}}\ket{B}\\\
&-\dot{\dot{\iota}}\ket{e_{2},P_{-2}^{2}}\ket{g_{3},P_{0}^{3}}\ket{C}-\ket{e_{2},P_{-2}^{2}}\ket{e_{3},P_{-2}^{3}}\ket{D}\Big{]}\\\
&\otimes\prod_{i=A,B}\frac{1}{\sqrt{2}}\big{(}\ket{0_{i}}+\ket{1_{i}}\big{)}.\end{split}$
(21)

Here, atoms 1 and 4 will not interact with cavities A and B; therefore, for
simplicity, we have denoted the non-interactive atomic states by

$\displaystyle\left|A\right>$
$\displaystyle=\left|g_{1},g_{4},P_{0}^{1},P_{0}^{4}\right>$
$\displaystyle\left|B\right>$
$\displaystyle=\left|g_{1},e_{4},P_{0}^{1},P_{-2}^{4}\right>$
$\displaystyle\left|C\right>$
$\displaystyle=\left|e_{1},g_{4},P_{-2}^{1},P_{0}^{4}\right>$
$\displaystyle\left|D\right>$
$\displaystyle=\left|e_{1},e_{4},P_{-2}^{1},P_{-2}^{4}\right>$

According to our scheme, the undeflected arms of atoms 2 and 3,
$\left|g_{2},P_{0}^{2}\right>$ and $\left|g_{3},P_{0}^{3}\right>$, will now
interact with cavity A, while the deflected arms
$\left|e_{2},P_{-2}^{2}\right>$ and $\left|e_{3},P_{-2}^{3}\right>$ interact
with cavity B; this is illustrated in Fig. (2). These interactions are off-
resonant Bragg diffractions, governed by the Hamiltonian given in Eq. (1).
States of atoms in ground state interacting with cavity A obey similar initial
conditions as discussed in section II. For this case, the governing equations
are presented in Eq. (3), Eq. (4) and Eq. (5), and the resulting probability
amplitudes are given in Eq. (13) and Eq. (14). However, the governing
equations corresponding to Eq. (3), Eq. (4) and Eq. (5) for the split
wavepackets of atoms 2 and 3 ($\left|e_{2},P_{-2}^{2}\right>$ and
$\left|e_{3},P_{-2}^{3}\right>$), in which the internal states of the atoms
are initially in the excited states, will follow from Eq. (27), Eq. (28) and
Eq. (49) in Appendix A. In this case, the time-evolved probability amplitudes
are also given in Appendix A. This completes the interaction of the respective
states of atoms 2 and 3 with cavity A and B, respectively, and the time-
evolved state at this stage becomes:

$\begin{split}\ket{\psi}=&\frac{1}{4}\bigg{[}\ket{0_{A},0_{B}}\Big{(}\ket{g_{2},g_{3},P_{0}^{2},P_{0}^{3}}\ket{A}\\\
&-\ket{g_{2},e_{3},P_{0}^{2},P_{0}^{3}}\ket{B}-\ket{e_{2},g_{3},P_{0}^{2},P_{0}^{3}}\ket{C}\\\
&+\ket{e_{2},e_{3},P_{0}^{2},P_{0}^{3}}\ket{D}\Big{)}\\\
&+\ket{0_{A},1_{B}}\Big{(}\ket{g_{2},g_{3},P_{0}^{2},P_{0}^{3}}\ket{A}\\\
&-\dot{\iota}\ket{g_{2},e_{3},P_{0}^{2},P_{-2}^{3}}\ket{B}+\dot{\iota}\ket{e_{2},g_{3},P_{-2}^{2},P_{0}^{3}}\ket{C}\\\
&-\ket{e_{2},e_{3},P_{-2}^{2},P_{-2}^{3}}\ket{D}\Big{)}\\\
&+\ket{1_{A},0_{B}}\Big{(}-\ket{g_{2},g_{3},P_{-2}^{2},P_{-2}^{3}}\ket{A}\\\
&+\dot{\iota}\ket{g_{2},e_{3},P_{-2}^{2},P_{0}^{3}}\ket{B}-\dot{\iota}\ket{e_{2},g_{3},P_{0}^{2},P_{-2}^{3}}\ket{C}\\\
&+\ket{e_{2},e_{3},P_{0}^{2},P_{0}^{3}}\ket{D}\Big{)}\\\
&+\ket{1_{A},1_{B}}\Big{(}-\ket{g_{2},g_{3},P_{-2}^{2},P_{-2}^{3}}\ket{A}\\\
&-\ket{g_{2},e_{3},P_{-2}^{2},P_{-2}^{3}}\ket{B}-\ket{e_{2},g_{3},P_{-2}^{2},P_{-2}^{3}}\ket{C}\\\
&-\ket{e_{2},e_{3},P_{-2}^{2},P_{-2}^{3}}\ket{D}\Big{)}\bigg{]}.\end{split}$
(22)

This equation shows that cavities A and B and both atoms 2 and 3 are mutually
sharing information. To swap the information of the cavities only to atoms 2
and 3, an auxiliary atom initially in the ground state ($\ket{g_{s}}$ and
$\ket{g_{t}}$) passes through each cavity in turn while obeying the resonant
interaction with the Hamiltonian
$H_{R}=\hbar\mu\left(\sigma_{+}b+\sigma_{-}b^{\dagger}\right)$ [27]. These
auxiliary atoms are detected in one of the possible combinations after passing
through the Ramsey zone and traced out from the whole system. Here, we have
detected them in their ground states, but the other combinations also yield
similar physical results. Now, we are only left with a state in which atoms
are correlated with each other, whereas all the cavities are traced out from
the state vector. The state of the system becomes:

$\begin{split}\ket{\psi}=&\frac{1-\dot{\dot{\iota}}}{8}\bigg{[}\Big{(}\ket{g_{2},g_{3},P_{0}^{2},P_{0}^{3}}+\dot{\dot{\iota}}\ket{g_{2},g_{3},P_{-2}^{2},P_{-2}^{3}}\Big{)}\bigg{]}\\\
&\otimes\ket{A}\\\
&-\frac{1}{8}\bigg{[}\Big{(}\ket{g_{2},e_{3},P_{0}^{2},P_{0}^{3}}+\ket{g_{2},e_{3},P_{0}^{2},P_{-2}^{3}}\\\
&-\ket{g_{2},e_{3},P_{-2}^{2},P_{0}^{3}}-\ket{g_{2},e_{3},P_{-2}^{2},P_{-2}^{3}}\Big{)}\bigg{]}\\\
&\otimes\ket{B}\\\
&-\frac{1}{8}\bigg{[}\Big{(}\ket{e_{2},g_{3},P_{0}^{2},P_{0}^{3}}-\ket{e_{2},g_{3},P_{0}^{2},P_{-2}^{3}}\\\
&+\ket{e_{2},g_{3},P_{-2}^{2},P_{0}^{3}}-\ket{e_{2},g_{3},P_{-2}^{2},P_{-2}^{3}}\Big{)}\bigg{]}\\\
&\otimes\ket{C}\\\
&+\frac{1-\dot{\dot{\iota}}}{8}\bigg{[}\Big{(}\ket{e_{2},e_{3},P_{0}^{2},P_{0}^{3}}+\dot{\iota}\ket{e_{2},e_{3},P_{-2}^{2},P_{-2}^{3}}\Big{)}\bigg{]}\\\
&\otimes\ket{D}.\end{split}$ (23)

The final step of our scheme is to detect the internal and external states of
atoms 2 and 3 after they are passed through a Ramsey zone. Depending on the
result of our detection, atoms 1 and 4 may share a different entangled state.
The following state represents all the possible entangled states corresponding
to the detections on atoms 2 and 3.

$\begin{split}&\ket{\psi}=\frac{1}{16}\sum_{b=1,3}\sum_{a=1,3}\\\
&\bigg{[}\Big{(}(1-\dot{\dot{\iota}})\ket{A}-\iota^{a}\ket{B}-\iota^{b}\ket{C}+(1-\dot{\dot{\iota}})\ket{D}\Big{)}\\\
&\ket{b_{2},a_{3},P_{0}^{2},P_{0}^{3}}\\\
&+\Big{(}(1+\dot{\dot{\iota}})\ket{A}+\iota^{a}\ket{B}+\iota^{b}\ket{C}+(\dot{1+\dot{\iota}})\ket{D}\Big{)}\\\
&\ket{b_{2},a_{3},P_{-2}^{2},P_{-2}^{3}}\\\
&+\Big{(}(-\iota^{a}\ket{B}+\iota^{b}\ket{C}\Big{)}\ket{b_{2},a_{3},P_{0}^{2},P_{-2}^{3}}\\\
&+\Big{(}(\iota^{a}\ket{B}-\iota^{b}\ket{C}\Big{)}\ket{b_{2},a_{3},P_{-2}^{2},P_{0}^{3}}\bigg{]}.\end{split}$
(24)

In Eq. 24, the internal states $\ket{1}$ and $\ket{3}$ refer to the states
$\ket{g}$ and $\ket{e}$ of both atoms, respectively. Similarly, we have
different possibilities of entangled states of atoms 1 and 4 according to the
detections of atoms 2 and 3. For example, assume that detections on atoms 2
and 4 yields $\ket{g_{2},g_{3},P_{0}^{2},P_{-2}^{3}}$, then we are left with
the following state:

$\begin{split}\ket{\psi}=-\ket{B}+\ket{C},\end{split}$ (25)

which, after expanding the states $\ket{B}$ and $\ket{C}$, becomes:

$\begin{split}\ket{\psi}=(\ket{e_{1},g_{4},P_{-2}^{1},P_{0}^{4}}-\ket{g_{1},e_{4},P_{0}^{1},P_{-2}^{4}})/\sqrt{2}.\end{split}$
(26)

Eq. 26 represents hyperentanglement among the internal and external degrees of
freedom of atoms 1 and 4. We note that atoms 1 and 4 do not share a direct
interaction with each other in our scheme, and we have successfully swapped
hyper-entanglement from atoms 1 and 2 and atoms 3 and 4 to atoms 1 and 4,
eventually.

##  IV Delayed-Choice Entanglement Swapping

The idea of delayed choice entanglement swapping revolves around the
manipulation of entangled particles in a way that their observed properties
are seemingly influenced by decisions made at a later time and even at a
distant location. Here, we will explore two key variants: bipartite
entanglement swapping, which involves the interaction of two entangled pairs,
and n-partite entanglement swapping, where the phenomenon extends to multiple
entangled particles. These variations shed light on the interconnected nature
of quantum systems and the profound influence of observation on the behavior
of entangled particles.

###  IV.1 Bipartite Entanglement

Entanglement swapping as described in Section III can also be performed in a
delayed-choice manner [41], which is more interesting in the context of
quantum information, as it highlights the importance of information and
information eraser in various quantum protocols. It also explicitly
demonstrates the irrelevance of temporal ordering as quantum systems are being
processed under unitary dynamics [9]. In our context, we adopt the same
procedure as discussed in Section III to obtain two states of entangled
bipartite atomic external momenta, provided for further manipulation as
required for delay choice swapping. This pair of entangled states may be
expressed in a single mathematical form as follows:

$\begin{split}\ket{\psi^{1,2}(t)}=&\frac{1}{\sqrt{2}}\big{[}\ket{0_{1},g_{1},g_{2},P_{0}^{1},P_{0}^{2}}\\\
&+\ket{1_{1},g_{1},g_{2},P_{-2}^{1},P_{-2}^{2}}\big{]}\end{split}$ (27)

and

$\begin{split}\ket{\psi^{3,4}(t)}=&\frac{1}{\sqrt{2}}\big{[}\ket{0_{2},g_{3},g_{4},P_{0}^{3},P_{0}^{4}}\\\
&+\ket{1_{2},g_{3},g_{4},P_{-2}^{3},P_{-2}^{4}}\big{]}.\end{split}$ (28)

Here cavity 1 and cavity 2 are still entangled with their respective atoms via
their quantized external momenta states:
$\ket{P_{0}^{(j)}},\ket{P_{-2}^{(j)}}$ with $j=1,2,3,4$. Also, all atoms in
free flight are still in their individual ground states $g_{ij}$, again with
$j=1,2,3,4$. These control variables are kept unchanged so they can be used in
future delayed-choice manipulation.

In this section, we only demonstrate the basic process of delayed-choice
swapping, and therefore we will not discuss the phases involved in various
stages. For a symmetric Ramsey Zone (RZ), we take the transformations:

$\begin{split}\ket{g_{A}}\rightarrow\frac{1}{\sqrt{2}}\big{(}\ket{g_{A_{j}}}+\ket{e_{A_{j}}}\big{)}\end{split}$
(29)

and

$\begin{split}\ket{e_{A}}=\frac{1}{\sqrt{2}}\big{(}\ket{g_{A_{j}}}-\ket{e_{A_{j}}}\big{)}.\end{split}$
(30)

The subscript $A_{j}$ designates the two-level atoms that will be used later
to erase the quantum information carried by cavity 1 and cavity 2 (in this
case, $j=1,2$). Similarly, the symmetric Hadamard Transformations (HT) for
atomic quantized external momenta states (atomic momenta beam splitter
transformations) may also be expressed as:

$\begin{split}\ket{P_{0}^{(j)}}\rightarrow\frac{1}{\sqrt{2}}\big{(}\ket{P_{0}^{(j)}+\ket{P_{-2}^{(j)}}}\end{split}$
(31)

and

$\begin{split}\ket{P_{-2}^{(j)}}\rightarrow\frac{1}{\sqrt{2}}\big{(}\ket{P_{0}^{(j)}-\ket{P_{-2}^{(j)}}},\end{split}$
(32)

evidently with $j=1,2,3,4$.
For the initial state, we tensor product expressions (Eq. 27) and (28) to get:

$\begin{split}&\ket{\psi^{1,2,3,4}}=\ket{\psi^{1,2}}\otimes\ket{\psi^{3,4}}=\\\
&\frac{1}{\sqrt{2}}.\frac{1}{\sqrt{2}}\Bigg{[}\ket{0_{1},0_{2},g_{1},g_{2},g_{3},g_{4},P_{0}^{1},P_{0}^{2},P_{0}^{3},P_{0}^{4}}\\\
&+\ket{0_{1},0_{2},g_{1},g_{2},g_{3},g_{4},P_{0}^{1},P_{0}^{2},P_{-2}^{3},P_{-2}^{4}}\\\
&+\ket{0_{1},0_{2},g_{1},g_{2},g_{3},g_{4},P_{-2}^{1},P_{-2}^{2},P_{0}^{3},P_{0}^{4}}\\\
&+\ket{0_{1},0_{2},g_{1},g_{2},g_{3},g_{4},P_{-2}^{1},P_{-2}^{2},P_{-2}^{3},P_{-2}^{4}}\Bigg{]}.\end{split}$
(33)

For the intended entanglement swapping, we project both cavity fields
simultaneously over a symmetric beam splitter as shown in figure 3.

Figure 3: Schematics for implementing Entanglement Swapping: A delayed-choice
scenario.

As illustrated in Fig. 3, this swapping procedure is inherently probabilistic
because no photon and two-photon detection events at either of the detectors
mark a failure case, which happens half of the time (i.e. with 50 percent
probability). Thus, the initial state leading to successful entanglement
swapping will be left with

$\begin{split}&\ket{\psi^{1,2,3,4}}=\\\
&\frac{1}{\sqrt{2}}\Bigg{[}\ket{0_{1},1_{2},g_{1},g_{2},g_{3},g_{4},P_{0}^{1},P_{0}^{2},P_{-2}^{3},P_{-2}^{4}}\\\
&+\ket{1_{1},0_{2},g_{1},g_{2},g_{3},g_{4},P_{-2}^{1},P_{-2}^{2},P_{0}^{3},P_{0}^{4}}\Bigg{]}.\end{split}$
(34)

Projection over the beam splitter is, in fact, the Bell-basis measurement and
it effectively erases which-path or which-cavity information carried a priori
by the photons. Furthermore, the photons have identical parameters and are
simultaneously subjected to the beam splitter and detected at $D_{1}$ or
$D_{2}$. Thus, after emerging from the beam splitter, the photon subscripts 1
or 2, designating the individual cavities, are no longer relevant to further
calculations as they are traced out of the system.

The symmetric beams splitter is one of the most simple quantum devices for the
coupling of the inputs to the outputs (optical in and out posts) and is mainly
characterised by the following transformations: [42],

$\begin{split}\hat{a}_{2}=\frac{1}{\sqrt{2}}(\hat{a}_{0}+\dot{\iota}\hat{a}_{1}),\quad\hat{a}_{3}=\frac{1}{\sqrt{2}}(\dot{\iota}\hat{a}_{0}+\hat{a}_{1}).\end{split}$
(35)

Simply stated, an incident single atom, whenever deflected through a beam
splitter, symmetric or otherwise, emerges with the phase $\pi/2$, while on the
other hand, the transmitted component receives no phase as it passes through
without deflection.

Thus the state expressed in Eq. (34) after the applications of the above
mentioned beam splitter transformation comes to be,

$\begin{split}&\ket{\psi^{1,2,3,4}}_{BS}=\\\
&\frac{1}{\sqrt{2}}\Bigg{[}\frac{1}{\sqrt{2}}\big{(}\ket{g_{1},g_{2},g_{3},g_{4},P_{0}^{1},P_{0}^{2},P_{-2}^{3},P_{-2}^{4}}\\\
&+\dot{\iota}\ket{g_{1},g_{2},g_{3},g_{4},P_{-2}^{1},P_{-2}^{2},P_{0}^{3},P_{0}^{4}}\big{)}\otimes\ket{1_{D_{1}}}\\\
&+\frac{1}{\sqrt{2}}\big{(}\dot{\iota}\ket{g_{1},g_{2},g_{3},g_{4},P_{0}^{1},P_{0}^{2},P_{-2}^{3},P_{-2}^{4}}\\\
&+\ket{g_{1},g_{2},g_{3},g_{4},P_{-2}^{1},P_{-2}^{2},P_{0}^{3},P_{0}^{4}}\big{)}\otimes\ket{1_{D_{2}}}\Bigg{]}.\end{split}$
(36)

In Eq. (36), we traced out the common vacuum state. This completes the simple
entanglement swapping protocol in the external quantized momenta states.

At the receiver’s side, the receiver can locally transform this four-partite
entangled momenta state into a hyperentangled state (i.e., state entangled in
two degrees of freedom: internal and external states) by exposing a pair of
momenta components of any of the two atom pairs (atom 1, atom 2) or (atom 3,
atom 4) to a classical Ramsey field in a delayed-choice manner. This type of
interaction is governed by the resonant semi-classical interaction Hamiltonian
$\nu=-h\Omega_{R}(\sigma_{+}+\sigma_{-})/2$ with $\Omega_{R}$ being the
resonant Rabi frequency and $\sigma+(\sigma_{-})$ is the atomic
raising/lowering operator [40].

From expression (36), the entangled state engineered through the swapping when
the photon is detected at the detector $D_{1}$, is given by:

$\begin{split}\ket{\psi^{1,2,3,4}}=&\frac{1}{\sqrt{2}}\Bigg{[}\ket{g_{1},g_{2},g_{3},g_{4},P_{0}^{1},P_{0}^{2},P_{-2}^{3},P_{-2}^{4}}\\\
&+\dot{\iota}\ket{g_{1},g_{2},g_{3},g_{4},P_{-2}^{1},P_{-2}^{2},P_{0}^{3},P_{0}^{4}}\Bigg{]},\end{split}$
(37)

Exposing the atomic momenta components $\ket{P_{0}^{1},P_{0}^{2}}$ of atom 1
and atom 2 as well as $\ket{P_{-2}^{3},P_{-2}^{4}}$ of atom 3 and atom 4 to a
classical laser field with interaction time corresponding to $\pi$ Rabi pulse
causes the transformations of the internal state of the atom,

$\begin{split}\ket{g}\rightarrow\dot{\iota}\ket{e},\quad\ket{e}\rightarrow\dot{\iota}\ket{g}.\end{split}$
(38)

Hence, expression (37), under laser-atom interaction for the specified time,
transforms to the following four-partite hyperentangled state, i.e. the state
entangled in the internal as well as the external momenta states in accordance
with the systematic depicted in figure (3). Finally, we obtain the delayed-
choice engineered hyperentangled state with the receiver opting for atom-laser
field interactions with appropriate later time. The desired state, will,
therefore be:

$\begin{split}\ket{\psi^{1,2,3,4}}^{hypent}=&\frac{1}{\sqrt{2}}\Bigg{[}\ket{e_{1},e_{2},e_{3},e_{4},P_{0}^{1},P_{0}^{2},P_{-2}^{3},P_{-2}^{4}}\\\
&+\dot{\iota}\ket{g_{1},g_{2},g_{3},g_{4},P_{-2}^{1},P_{-2}^{2},P_{0}^{3},P_{0}^{4}}\Bigg{]}.\end{split}$
(39)

This is because four atoms have successively gone through atom-field
interactions leading to the change of the respective ground states into the
corresponding excited state, each producing a local phase of $\dot{\iota}$ and
collectively $\dot{\iota}^{4}=1$.

###  IV.2 n-partite Entanglement

We can generalize bipartite entanglement swapping and its delayed-choice based
conversion into n-partite hyperentanglement swapping. Experimentally, it has
been well demonstrated that a stream of thousands of atoms can interact with a
cavity field consecutively before the appreciable inset of decoherence [43,
39, 37]. This is mainly due to the availability of high-Q superconducting
cavities with useful life as high as a fraction of a second [44, 45]. Thus,
instead of Bragg diffracting only a single pair of atoms from each cavity, we
may diffract $n/2$ atoms, where $n\geq 500$, from each cavity [46]. Following
the procedure described in section IV.1, the pair of the generated entangled
states of the $n/2$-partite may be expressed as follows.

$\begin{split}\ket{\psi}^{n/2}_{C_{1}}&=\frac{1}{\sqrt{2}}\Bigg{[}\ket{0_{1}}\otimes\prod_{i=1}^{n/2}\bigg{(}\ket{g_{i},P_{0}^{(i)}}\bigg{)}\\\
&\quad+\ket{1_{1}}\otimes\prod_{i=1}^{n/2}\bigg{(}\ket{g_{i},P_{-2}^{(i)}}\bigg{)}\Bigg{]}\end{split}$
(40)

and,

$\begin{split}\ket{\psi}^{n/2}_{C_{2}}=&\frac{1}{\sqrt{2}}\Bigg{[}\ket{0_{2}}\otimes\Pi^{n}_{j=\frac{n}{2}+1}\bigg{(}\ket{g_{j},P_{0}^{(j)}}\bigg{)}\\\
&+\ket{1_{2}}\otimes\Pi^{n}_{j=\frac{n}{2}+1}\bigg{(}\ket{g_{j},P_{-2}^{(j)}}\bigg{)}\Bigg{]}.\end{split}$
(41)

Again, by employing the same procedure cited earlier, i.e. by projecting
cavity fields simultaneously over a symmetric beam splitter, we can engineer
an n-partite entangled state through swapping. This procedure yields the
state,

$\begin{split}&\ket{\psi}^{n}=\\\
&\frac{1}{\sqrt{2}}\Bigg{[}\bigg{[}\Pi^{n/2}_{j=1}\bigg{(}\ket{g_{j},P_{0}^{(j)}}\bigg{)}\otimes\Pi^{n}_{k=\frac{n}{2}+1}\bigg{(}\ket{g_{k},P_{-2}^{(k)}}\bigg{)}\bigg{]}\\\
&+\dot{\iota}\bigg{[}\Pi^{n/2}_{j=1}\bigg{(}\ket{g_{j},P_{-2}^{(j)}}\bigg{)}\otimes\Pi^{n}_{k=\frac{n}{2}+1}\bigg{(}\ket{g_{k},P_{0}^{(k)}}\bigg{)}\bigg{]}\Bigg{]}\end{split}$
(42)

This state, once in the access of the receiver, can be transformed into the
hyperentangled state in a delayed-choice manner by passing the specifically
selected momentum components through classical Ramsey fields, where the split
wave packet of each atom is excited from its respective ground state following
the procedure described at length in the previous section. The procedure will
duly yield the following hyperentangled state,

$\begin{split}&\ket{\psi}^{n-hypent}=\\\
&\frac{1}{\sqrt{2}}\Bigg{[}\bigg{[}(\dot{\iota}^{n}\Pi^{n/2}_{j=1}\bigg{(}\ket{e_{j},P_{0}^{(j)}}\bigg{)}\otimes\Pi^{n}_{k=\frac{n}{2}+1}\bigg{(}\ket{e_{k},P_{-2}^{(k)}}\bigg{)}\bigg{]}\\\
&+\dot{\iota}\bigg{[}\Pi^{n/2}_{j=1}\bigg{(}\ket{g_{j},P_{-2}^{(j)}}\bigg{)}\otimes\Pi^{n}_{k=\frac{n}{2}+1}\bigg{(}\ket{g_{k},P_{0}^{(k)}}\bigg{)}\bigg{]}\Bigg{]}\end{split}$
(43)

or,

$\begin{split}&\ket{\psi}^{n-hypent}=\\\
&\frac{1}{\sqrt{2}}\Bigg{[}\bigg{[}\dot{\iota}^{n-1}\Pi^{n/2}_{j=1}\bigg{(}\ket{e_{j},P_{0}^{(j)}}\bigg{)}\otimes\Pi^{n}_{k=\frac{n}{2}+1}\bigg{(}\ket{e_{k},P_{-2}^{(k)}}\bigg{)}\bigg{]}\\\
&+\bigg{[}\Pi^{n/2}_{j=1}\bigg{(}\ket{g_{j},P_{-2}^{(j)}}\bigg{)}\otimes\Pi^{n}_{k=\frac{n}{2}+1}\bigg{(}\ket{g_{k},P_{0}^{(k)}}\bigg{)}\bigg{]}\Bigg{]}.\end{split}$
(44)

The procedure adopted for the generation of entanglement, swapping and the
subsequent delayed-choice transformation into the n-partite hyperentanglement
engineering, is universal and hence can be utilized for diverse entanglement
morphology including cluster and graph states [47, 48]. Thus, the protocol
presented here may contribute to the realization of complex quantum entangled
networks and other applications requiring information distribution over
complex quantum structures, such as many-body systems, and simulation of
biological systems. [49].

##  V Experimental Feasibility

Quantum entanglement, along with its characteristic traits, is the most
prominent resource in the field of quantum information [50, 51, 52, 53, 54,
55]. Similarly, entanglement swapping is a central process that has been used
to clarify a foundational issue of entanglement engineering, i.e. the
demonstration of entanglement between two parties without any direct temporal
or spatial interaction of the parties involved. This was carried out through
Bell-basis measurements on two independent but entangled pairs, which
consequently placed information (and information eraser) as the core theme in
quantum information theory [41]. The advent of hyperentanglement, and its
utilization in quantum information, further fortified the quantum information
domain, as it significantly enhances the channel capacity without invoking any
further physical resources, which minimizes resource-related quantitative
decoherence. Cavity QED-based atom-field systems have been historically the
pioneering gadgets for efficient and meaningful demonstrations and
experimental implementations of many quantum information proposals. [43, 21].
However, the main problem with such multi-partite quantum states is the
decoherence. This is because many quantum information experiments generally
require precise interactions under a scenario that is not ideally closed,
eventually leading to decoherence.

Our proposed hyperentanglement swapping fundamentally utilizes external
momenta states of the neutral atoms, and such states are well known to be
resistant to decoherence under what is generally termed quantum Darwinism
[56]. For state engineering, we rely on the off-resonant Bragg diffraction of
the neutral atoms from the cavity fields. This subsequently generates the
desired states in momentum space without exchanging any real photons between
the atom and the field. Therefore, in our work, the pathway for the
decoherence emanating from spontaneous emission is also effectively closed.
Currently, we can engineer cavities with lifetimes close to a fraction of a
second [57, 58], which successfully tackles decoherence. This means that
several atoms can interact with a quantized cavity field without raising
substantial threats to the coherence of the system [44]. It is also reasonable
to treat the system as closed because the average interaction time between an
atom and a cavity field is in the range of microseconds. Over the years,
numerous experimental validations for atomic Bragg diffraction have been
performed, with even $8^{th}$ order atomic Bragg diffraction generating
experimentally fruitful results [59, 60, 61]. Mach-Zehnder-Bragg
interferometry has also been applied experimentally with results involving
splitting of momenta of 102 photons [62]. Furthermore, the transmission of the
resulting momentum of around 112 $\hbar k$ has also been experimentally
observed with high visibility of the fringes [63, 64]. Therefore, in light of
these experimental demonstrations, it may be safely assumed that our proposed
scheme can be realized experimentally.

Moreover, the proposed scheme holds practical grounds due to the fact that
high-fidelity systems have previously been proposed and executed efficiently
in the microwave Bragg regime. In the optical regime, experiments have also
shown generally good and reliable Bragg diffraction of atoms. In terms of
experimental efficiency, variational errors in interaction time can be avoided
by utilizing ultra-cold atoms trapped by magneto-optics with a negligible
spread in velocity [59, 61, 65, 66]. To be more specific, the atomic Bragg
diffraction of 85Rb atoms by $780$ nm optical laser beams [57, 58, 61]
provides a glaring experimental realization where atoms with mass $M=85$ amu
are exploited with recoil frequency $\omega_{r}=2.4$ x $10^{4}$ rad/s and Rabi
frequency $\Omega=2\pi$ x $16.4$ MHz. The parameters related to the cavity
include detuning of $1$ GHz and finesse $4.4$ x $10^{5}$. As an alternative,
4He atoms with $M=4$ amu, $\omega_{r}=1.06$ MHz, $\lambda=543.5$ nm,
$\Delta=6.28$ GHz and the effective Rabi frequency
$\frac{\mu^{2}}{4\Delta}=120$ KHz can also be used for an interaction time of
$13\mu$s the finesse of the cavities, i.e $F=7.85$ x $10^{6}$, leading to the
lifetimes up to milliseconds [37, 67]. Consequently, these atomic and cavity
parameters maintain the conditions for off-resonant atomic Bragg diffraction,
i.e $\Delta>>\omega_{r}$ and $\omega_{r}+\Delta=\mu\sqrt{n}/2$, where the
interaction time is kept far lower than the lifetime of the cavity (around
$0.5\mu$s) [59, 68, 69, 70].

##  VI Conclusion

In this paper, we proposed a framework for entanglement swapping of
hyperentanglement between two hyperentangled pairs. One atom from each pair is
manipulated while we do not interfere with the two entangled atoms. Passing
the undeflected arms from Cavity A and deflected arms from Cavity B, we
produce a hyper-correlation between the two atoms. Finally, detection of these
atom’s internal and external states after passing them through Ramsey zone,
swaps the entanglement on to the atoms which passed without interactions
initially. Such a protocol, if implemented experimentally, will ensure the
safe decoherence-free and high-capacity information transfer over two quantum
nodes or quantum information processing units that were independent and
uncorrelated initially.

## References

  * [1] Mayank Gupta and Manisha J Nene.  Quantum computing: An entanglement measurement.  In 2020 IEEE International Conference on Advent Trends in Multidisciplinary Research and Innovation (ICATMRI), pages 1–6. IEEE, 2020.
  * [2] Stefano Pirandola, B Roy Bardhan, Tobias Gehring, Christian Weedbrook, and Seth Lloyd.  Advances in photonic quantum sensing.  Nature Photonics, 12(12):724–733, 2018.
  * [3] Quntao Zhuang, Zheshen Zhang, and Jeffrey H Shapiro.  Distributed quantum sensing using continuous-variable multipartite entanglement.  Physical Review A, 97(3):032329, 2018.
  * [4] Steven B Giddings and Massimiliano Rota.  Quantum information or entanglement transfer between subsystems.  Physical Review A, 98(6):062329, 2018.
  * [5] Mohammed Zidan, A Abdel-Aty, A Younes, EA Zanaty, I El-Khayat, and Mahmoud Abdel-Aty.  A novel algorithm based on entanglement measurement for improving speed of quantum algorithms.  Appl. Math. Inf. Sci, 12(1):265–269, 2018.
  * [6] Alaa Sagheer, Mohammed Zidan, and Mohammed M Abdelsamea.  A novel autonomous perceptron model for pattern classification applications.  Entropy, 21(8):763, 2019.
  * [7] Peter W Shor.  Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer.  SIAM review, 41(2):303–332, 1999.
  * [8] Sami Ul Haq and Aeysha Khalique.  Long distance cavity entanglement by entanglement swapping using atomic momenta.  Optics Communications, 334:290–293, 2015.
  * [9] Jian-Wei Pan, Dik Bouwmeester, Harald Weinfurter, and Anton Zeilinger.  Experimental entanglement swapping: entangling photons that never interacted.  Physical review letters, 80(18):3891, 1998.
  * [10] Sougato Bose, Vlatko Vedral, and Peter L Knight.  Multiparticle generalization of entanglement swapping.  Physical Review A, 57(2):822, 1998.
  * [11] Xiao-song Ma, Stefan Zotter, Johannes Kofler, Rupert Ursin, Thomas Jennewein, Časlav Brukner, and Anton Zeilinger.  Experimental delayed-choice entanglement swapping.  Nature Physics, 8(6):479–484, 2012.
  * [12] Alexander M Goebel, Claudia Wagenknecht, Qiang Zhang, Yu-Ao Chen, Kai Chen, Jörg Schmiedmayer, and Jian-Wei Pan.  Multistage entanglement swapping.  Physical Review Letters, 101(8):080403, 2008.
  * [13] Liaqat Ali, Manzoor Ikram, Tasawar Abbas, Iftikhar Ahmad, et al.  Hyperentanglement teleportation through external momenta states.  Journal of Physics B: Atomic, Molecular and Optical Physics, 54(23):235501, 2022.
  * [14] Liaqat Ali, Manzoor Ikram, Tasawar Abbas, and Iftikhar Ahmad.  Teleportation of atomic external states on the internal degrees of freedom.  Quantum Information Processing, 21(2):55, 2022.
  * [15] Rameez-ul Islam.  Engineering entanglement in cavity quantum electrodynamical systems.  PhD thesis, Quaid-i-Azam University Islamabad, Pakistan, 2008.
  * [16] Yuan-Hao Sun, Yu-Qing Guo, and Cong Cao.  Complete and fidelity-robust hyperentangled-state analysis of photon systems with single-sided quantum-dot-cavity systems under the balance condition.  Quantum Information Processing, 22(9):344, 2023.
  * [17] Zhi Zeng and Ka-Di Zhu.  Complete hyperentangled bell state analysis assisted by hyperentanglement.  Laser Physics Letters, 17(7):075203, 2020.
  * [18] SM Arslan Anis, Tasawar Abbas, Muhammad Imran, et al.  Engineering quantum networks through bragg diffracted hyperentangled atoms.  Physica Scripta, 96(12):125102, 2021.
  * [19] Pei-Ying Xiong, Xu-Tao Yu, Hai-Tao Zhan, and Zai-Chen Zhang.  Multiple teleportation via partially entangled ghz state.  Frontiers of Physics, 11:1–8, 2016.
  * [20] Sami ul Haq and Farhan Saif.  Extended entanglement to quantum networks.  Optik, 124(23):5914–5917, 2013.
  * [21] Serge Haroche.  Nobel lecture: Controlling photons in a box and exploring the quantum to classical boundary.  Reviews of Modern Physics, 85(3):1083, 2013.
  * [22] Manzoor Ikram, Muhammad Imran, Guo-Qin Ge, et al.  Entanglement and the paradox of untying the defining feature from a quantum entity.  Physical Review A, 100(5):052122, 2019.
  * [23] Muhammad Imran, Rameez-ul Islam, Muhammad Haroon Saeed, and Manzoor Ikram.  Quantum three-box paradox: a proposal for atom optics implementation.  Quantum Information Processing, 20:1–18, 2021.
  * [24] Izma Qureshi, Tasawar Abbas, Muhammad Imran, and Rameez-ul Islam.  Exploring wave–particle behaviors of entangled bragg diffracted neutral atoms.  Journal of Mathematical Physics, 64(1), 2023.
  * [25] DS Krähmer, AM Herkommer, E Mayr, VM Akulin, I Sh Averbukh, T van Leeuwen, VP Yakovlev, and W Schleich.  Atom optics in quantized light fields.  In Quantum Optics VI: Proceedings of the Sixth International Symposium on Quantum Optics, Rotorua, New Zealand, January 24–28, 1994, pages 87–102. Springer, 1994.
  * [26] Arbab A Khan and M Suhail Zubairy.  Quantum logic gate operating on atomic scattering by standing wave field in bragg regime.  Fortschritte der Physik: Progress of Physics, 46(4-5):417–422, 1998.
  * [27] Arbab A Khan and M Suhail Zubairy.  Quantum non-demolition measurement of fock states via atomic scattering in bragg regime.  Physics Letters A, 254(6):301–306, 1999.
  * [28] Mehwish Nawaz, Tasawar Abbas, Manzoor Ikram, et al.  Engineering quantum hyperentangled states in atomic systems.  Journal of Physics B: Atomic, Molecular and Optical Physics, 50(21):215502, 2017.
  * [29] Mehwish Nawaz, Manzoor Ikram, et al.  Remote state preparation through hyperentangled atomic states.  Journal of Physics B: Atomic, Molecular and Optical Physics, 51(7):075501, 2018.
  * [30] Mehwish Nawaz, Tasawar Abbas, Manzoor Ikram, et al.  Atomic cheshire cat: untying energy levels from the de broglie motion.  Journal of Physics B: Atomic, Molecular and Optical Physics, 52(10):105501, 2019.
  * [31] Tasawar Abbas, Manzoor Ikram, et al.  Biasing a coin after the toss: asymmetric delayed choice quantum eraser via bragg regime cavity qed.  Laser Physics Letters, 12(1):015203, 2014.
  * [32] Manzoor Ikram, Muhammad Imran, Tasawar Abbas, et al.  Wheeler’s delayed-choice experiment: A proposal for the bragg-regime cavity-qed implementation.  Physical Review A, 91(4):043636, 2015.
  * [33] Rameez-ul Islam, Ashfaq H Khosa, Farhan Saif, and János A Bergou.  Generation of atomic momentum cluster and graph states via cavity qed.  Quantum information processing, 12:129–148, 2013.
  * [34] Farhan Saif, Ashfaq H Khosa, et al.  An engineering two-mode field noon state in cavity qed.  Journal of Physics B: Atomic, Molecular and Optical Physics, 43(1):015501, 2009.
  * [35] Manzoor Ikram, Farhan Saif, et al.  Engineering maximally entangled n-photon noon field states using an atom interferometer based on bragg regime cavity qed.  Journal of Physics B: Atomic, Molecular and Optical Physics, 40(7):1359, 2007.
  * [36] R Ul-Islam, M Ikram, R Ahmed, AH Khosa, and F Saif.  Atomic state teleportation: from internal to external degrees of freedom.  Journal of Modern Optics, 56(7):875–880, 2009.
  * [37] Ashfaq H Khosa, Manzoor Ikram, and M Suhail Zubairy.  Measurement of entangled states via atomic beam deflection in bragg’s regime.  Physical Review A, 70(5):052312, 2004.
  * [38] Aeysha Khalique and Farhan Saif.  Engineering entanglement between external degrees of freedom of atoms via bragg scattering.  Physics Letters A, 314(1-2):37–43, 2003.
  * [39] Rameez-ul Islam, Ashfaq H Khosa, and Farhan Saif.  Generation of bell, noon and w states via atom interferometry.  Journal of Physics B: Atomic, Molecular and Optical Physics, 41(3):035505, 2008.
  * [40] Muhammad Suhail Zubairy.  Quantum optics.  Cambridge University Press, 1997.
  * [41] Asher Peres.  Delayed choice for entanglement swapping.  Journal of Modern Optics, 47(2-3):139–143, 2000.
  * [42] CC Gerry and P Knight.  Introductory quantum optics cambridge univ, 2005.
  * [43] Serge Haroche and J-M Raimond.  Exploring the quantum: atoms, cavities, and photons.  Oxford university press, 2006.
  * [44] Samuel Deleglise, Igor Dotsenko, Clement Sayrin, Julien Bernu, Michel Brune, Jean-Michel Raimond, and Serge Haroche.  Reconstruction of non-classical cavity field states with snapshots of their decoherence.  Nature, 455(7212):510–514, 2008.
  * [45] Stefan Kuhr, Sébastien Gleyzes, Christine Guerlin, Julien Bernu, U Busk Hoff, Samuel Deléglise, Stefano Osnaghi, Michel Brune, J-M Raimond, Serge Haroche, et al.  Ultrahigh finesse fabry-pérot superconducting resonator.  Applied Physics Letters, 90(16), 2007.
  * [46] Liaqat Ali, Muhammad Imran, Manzoor Ikram, Iftikhar Ahmad, et al.  Generation of maximally entangled n-photon field w-states via cavity qed.  The European Physical Journal Plus, 137(11):1–12, 2022.
  * [47] Damian Markham and Barry C Sanders.  Graph states for quantum secret sharing.  Physical Review A, 78(4):042309, 2008.
  * [48] Chao-Yang Lu, Xiao-Qi Zhou, Otfried Gühne, Wei-Bo Gao, Jin Zhang, Zhen-Sheng Yuan, Alexander Goebel, Tao Yang, and Jian-Wei Pan.  Experimental entanglement of six photons in graph states.  Nature physics, 3(2):91–95, 2007.
  * [49] Diego Santiago-Alarcon, Horacio Tapia-McClung, Sergio Lerma-Hernández, and Salvador E Venegas-Andraca.  Quantum aspects of evolution: a contribution towards evolutionary explorations of genotype networks via quantum walks.  Journal of the Royal Society Interface, 17(172):20200567, 2020.
  * [50] Albert Einstein, Boris Podolsky, and Nathan Rosen.  Can quantum-mechanical description of physical reality be considered complete?  Physical review, 47(10):777, 1935.
  * [51] John S Bell.  On the einstein podolsky rosen paradox.  Physics Physique Fizika, 1(3):195, 1964.
  * [52] Alain Aspect, Philippe Grangier, and Gérard Roger.  Experimental realization of einstein-podolsky-rosen-bohm gedankenexperiment: A new violation of bell’s inequalities.  Physical review letters, 49(2):91, 1982.
  * [53] Michael A Nielsen and Isaac L Chuang.  Quantum computation and quantum information.  Cambridge university press, 2010.
  * [54] Eli Megidish, A Halevy, T Shacham, T Dvir, L Dovrat, and HS Eisenberg.  Entanglement swapping between photons that have never coexisted.  Physical review letters, 110(21):210403, 2013.
  * [55] Jean-Michel Raimond, Michel Brune, and Serge Haroche.  Manipulating quantum entanglement with atoms and photons in a cavity.  Reviews of Modern Physics, 73(3):565, 2001.
  * [56] Philip Ball.  Quantum all the way: how does our classical world emerge from the counterintuitive principles of quantum theory? can we even be sure that the world doesn’t’go quantum’when no one is watching? philip ball talks to the theorists and experimentalists trying to find out.  Nature, 453(7191):22–26, 2008.
  * [57] Gerhard Rempe, F Schmidt-Kaler, and Herbert Walther.  Observation of sub-poissonian photon statistics in a micromaser.  Physical review letters, 64(23):2783, 1990.
  * [58] Christopher C Gerry.  Proposal for a mesoscopic cavity qed realization of the greenberger-horne-zeilinger state.  Physical Review A, 54(4):R2529, 1996.
  * [59] Stephan Dürr, Stefan Kunze, and Gerhard Rempe.  Pendellösung oscillations in second-order bragg scattering of atoms from a standing light wave.  Quantum and Semiclassical Optics: Journal of the European Optical Society Part B, 8(3):531, 1996.
  * [60] DW Vernooy, Vladimir S Ilchenko, H Mabuchi, EW Streed, and HJ Kimble.  High-q measurements of fused-silica microspheres in the near infrared.  Optics letters, 23(4):247–249, 1998.
  * [61] S Dürr, T Nonn, and G Rempe.  Origin of quantum-mechanical complementarity probed by a ‘which-way’experiment in an atom interferometer.  Nature, 395(6697):33–37, 1998.
  * [62] Sheng-wey Chiow, Tim Kovachy, Hui-Chun Chien, and Mark A Kasevich.  102 $\hbar$ k large area atom interferometers.  Physical review letters, 107(13):130403, 2011.
  * [63] Benjamin Plotkin-Swing, Daniel Gochnauer, Katherine E McAlpine, Eric S Cooper, Alan O Jamison, and Subhadeep Gupta.  Three-path atom interferometry with large momentum separation.  Physical Review Letters, 121(13):133201, 2018.
  * [64] Andrew G Manning, Roman I Khakimov, Robert G Dall, and Andrew G Truscott.  Wheeler’s delayed-choice gedanken experiment with a single atom.  Nature Physics, 11(7):539–542, 2015.
  * [65] S Kunze, S Dürr, and G Rempe.  Bragg scattering of slow atoms from a standing light wave.  Europhysics Letters, 34(5):343, 1996.
  * [66] David M Giltner, Roger W McGowan, and Siu Au Lee.  Theoretical and experimental study of the bragg scattering of atoms from a standing light wave.  Physical review A, 52(5):3966, 1995.
  * [67] Christina J Hood, HJ Kimble, and Jun Ye.  Characterization of high-finesse mirrors: Loss, phase shifts, and mode structure in an optical cavity.  Physical Review A, 64(3):033804, 2001.
  * [68] P Münstermann, T Fischer, P Maunz, PWH Pinkse, and G Rempe.  Dynamics of single-atom motion observed in a high-finesse cavity.  Physical review letters, 82(19):3791, 1999.
  * [69] P Münstermann, T Fischer, PWH Pinkse, and G Rempe.  Single slow atoms from an atomic fountain observed in a high-finesse optical cavity.  Optics communications, 159(1-3):63–67, 1999.
  * [70] Thomas Puppe, Peter Maunz, Thomas Fischer, Pepijn WH Pinkse, and Gerhard Rempe.  Single-atom trajectories in higher-order transverse modes of a high-finesse optical cavity.  Physica Scripta, 2004(T112):7, 2004.

## Appendix A:Interaction between the atom in excited state and the cavity in
superposition state

If the atom is in the excited state then the initial state vector will be,

$\ket{\psi(t=0)}=\ket{g,P_{0}}\otimes(\ket{0}+\ket{1})/\sqrt{2}$ (45)

And the proposed state vector can be written as,

$\begin{split}\left|\psi(t)\right>=&e^{-\dot{\iota}\left(\frac{P_{0}^{2}}{2M\hbar}-\frac{\Delta}{2}\right)t}\sum_{l=-\infty}^{\infty}\left[C_{e,0}^{P_{l}}(t)\left|e,0,P_{l}\right>+\right.\\\
&\left.C_{e,1}^{P_{l}}(t)\left|e,1,P_{l}\right>+C_{g,1}^{P_{l}}(t)\left|g,1,P_{l}\right>\right].\end{split}$
(46)

Using the same governing Hamiltonian as presented in Eq. (1), we solve the
Schrodinger equation and after taking projections of probability amplitudes,
we arrive at the following coupled differential equations,

$\begin{split}\dot{\iota}\frac{\partial}{\partial
t}C_{e,0}^{P_{l}}(t)=&\left(\frac{l(l_{0}+l)\hbar
k^{2}}{2M}\right)C_{e,0}^{P_{l}}(t)+\\\
&\frac{\mu}{2}\left[C_{g,1}^{P_{l+1}}(t)+C_{g,1}^{P_{l-1}}(t)\right],\end{split}$
(47)

$\dot{\iota}\frac{\partial}{\partial
t}C_{e,1}^{P_{l}}(t)=\left(\frac{l(l_{0}+l)\hbar
k^{2}}{2M}\right)C_{e,1}^{P_{l}}(t),$ (48)

$\begin{split}\dot{\iota}\frac{\partial}{\partial t}C_{g,1}^{P_{l}}(t)=&\Delta
C_{g,1}^{P_{l}}(t)+\\\
&\frac{\mu}{2}\left[C_{e,0}^{P_{l+1}}(t)+C_{e,0}^{P_{l-1}}(t)\right].\end{split}$
(49)

Putting in the values of $l$ which runs from $-3$ to $1$, we get,

For l=1,

$\dot{\iota}\frac{\partial}{\partial t}C_{g,1}^{P_{1}}(t)=\Delta
C_{g,1}^{P_{1}}(t)+\frac{\mu}{2}\left(C_{e,0}^{P_{2}}(t)+C_{e,0}^{P_{0}}(t)\right),$
(50)

For l=0,

$\dot{\iota}\frac{\partial}{\partial
t}C_{e,0}^{P_{0}}(t)=\frac{\mu}{2}\left(C_{g,1}^{P_{1}}(t)+C_{g,1}^{P_{-1}}(t)\right),$
(51)

For l=-1,

$\dot{\iota}\frac{\partial}{\partial t}C_{g,1}^{P_{-1}}(t)=\Delta
C_{g,1}^{P_{-1}}(t)+\frac{\mu}{2}\left(C_{e,0}^{P_{0}}(t)+C_{e,0}^{P_{-2}}(t)\right),$
(52)

For l=-2,

$\dot{\iota}\frac{\partial}{\partial
t}C_{e,0}^{P_{-}2}(t)=\frac{\mu}{2}\left(C_{g,1}^{P_{-1}}(t)+C_{g,1}^{P_{-3}}(t)\right),$
(53)

For l=-3,

$\dot{\iota}\frac{\partial}{\partial t}C_{g,1}^{P_{-3}}(t)=\Delta
C_{g,1}^{P_{-3}}(t)+\frac{\mu}{2}\left(C_{e,0}^{P_{-2}}(t)+C_{e,0}^{P_{-4}}(t)\right).$
(54)

Applying adiabatic approximation and back substitutions, we reach,

$\frac{\partial}{\partial
t}C_{e,0}^{P_{0}}(t)=\frac{\dot{\iota}\mu^{2}}{4\Delta}\left[2C_{e,0}^{P_{0}}(t)+C_{e,0}^{P_{-}2}(t)\right],$
(55)

$\frac{\partial}{\partial
t}C_{e,0}^{P_{-2}}(t)=\frac{\dot{\iota}\mu^{2}}{4\Delta}\left[C_{e,0}^{P_{-}2}(t)+C_{e,0}^{P_{0}}(t)\right].$
(56)

Solving the above coupled differential equation, we finally arrive at the
governing equations for this system,

$\begin{split}C_{e,0}^{P_{0}}(t)=&e^{\frac{2\dot{\iota}\mu^{2}t}{4\Delta}}\left[C_{e,0}^{P_{0}}(0)\cos\left(\frac{\mu^{2}}{4\Delta}t\right)\right.\\\
&\left.+\dot{\iota}C_{e,0}^{P_{-}2}(0)\sin\left(\frac{\mu^{2}}{4\Delta}t\right)\right],\end{split}$
(57)

$\begin{split}C_{e,0}^{P_{-}2}(t)=&e^{\frac{2\dot{\iota}\mu^{2}t}{4\Delta}}\left[C_{e,0}^{P_{-}2}(0)\cos\left(\frac{\mu^{2}}{4\Delta}t\right)\right.\\\
&\left.+\dot{\iota}C_{e,0}^{P_{0}}(0)\sin\left(\frac{\mu^{2}}{4\Delta}t\right)\right].\end{split}$
(58)

With initial condition, $C_{e,0}^{P_{-2}}(0)=1/\sqrt{2}$ and setting
interaction time as $t=2\pi\Delta/\mu^{2}$,

$\left|\psi\right>_{1}=\frac{1}{\sqrt{2}}\left[\left|1,e,P_{-2}\right>-\dot{\iota}\left|0,e,P_{0}\right>\right].$
(59)