# Engineering Entangled Schrödinger Cat States of Separated Cavity Modes in
Cavity-QED

Abdul Q. Batin1 Suranjana Ghosh2 Utpal Roy1<EMAIL_ADDRESS>David Vitali3
<EMAIL_ADDRESS>1Indian Institute of Technology Patna, Bihta,
Patna-801103, India
2Indian Institute of Science Education and Research Kolkata, Mohanpur-741246,
India
3School of Science and Technology, Physics Division, University of Camerino,
Camerino, Italy

###### Abstract

We provide a scheme by utilizing a two-cavity setup to generate useful quantum
mechanically entangled states of two cavity fields, which themselves are
prepared in Schrödinger cat states. The underlying atom-field interaction is
considered off-resonant and three atoms are successively sent through the
cavities, initially fed with coherent fields. Analytical solution of the
protocol, followed by conditional measurements on the atoms, produce a family
of eight such entangled states. Entanglement properties of the obtained states
are characterized by the Von Neumann entropy. We reveal the parameter domain
for tuning the entanglement, the prime tuning parameters being the atom-field
interaction time and the field amplitudes. The parameter domains for both
quasi-Bell and non quasi-Bell states are discussed. We also present a Wigner
phase space representation of the reduced state of the cavity, showing
negative values and interference patterns similar to those of a compass state,
used in quantum precision measurements, and despite its large entropy.

##  I Introduction

Two noteworthy concepts fundamentally distinguish quantum mechanics from
classical physics: the ideas of quantum superposition and quantum
entanglement. Distinct quantum states can coexist until being measured and
they can be also made nonseparable with correlated measurement outcomes
Einstein negated the idea of non-local interaction which was initially thought
to occur faster than the speed of light and it was coined as “spooky action at
a distance” [1]. Later, successive experiments have been performed to
demonstrate that quantum entanglement is indeed a genuine occurrence [2, 3, 4,
5, 6]. Combination of quantum entanglement and quantum superposition has
become an essential tool for various emerging applications in quantum
technology. There have been continuous searches for appropriate quantum
superposition states, many of which are found very useful, such as Schrödinger
cat state, compass state, benzene-like state, superposition of two mode
coherent states, quantum tetrachotomous state, hypercube state, superposition
of single or multiple photon added coherent states _etc_ [7, 8]. Their non-
classical features have been intensively studied [9, 10, 11, 12, 13, 14, 15,
16, 17], and they have been shown to be beneficial for a wide range of quantum
technology applications, such as quantum teleportation [18, 19], quantum
sensing [20, 21, 8, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], quantum
communication [32, 33, 34], quantum cryptography [35, 36, 37, 38], and quantum
computing [39, 40, 41, 42, 43]. Highly nontrivial, but interesting properties
are displayed by various entangled states, which include the two-qubit EPR
states [44], the three-qubit GHZ states [45], W-states [46], the four-qubit
cluster-type states [47, 48, 49], each with their unique properties and
manifestations.

The investigation and manipulation of the interaction between light and matter
in systems, such as cavity QED, NMR, molecules, and ultracold atoms provide
favourable platforms to perform tasks in quantum technology and quantum logic
operations. The strong coupling regime, achievable in cavity QED, allows for
the exploration of the light and matter interaction in a confined space [50,
51], making it an intriguing area of study in quantum optics. Particularly,
Haroche and co-workers [52, 53, 54, 55] have demonstrated the production of
mesoscopic superposition states in high quality cavity QED setup with large
dispersive or resonant interaction, which is exploited to generate various
non-classical superposition and entanglement of coherent states [56, 57, 58,
59].

In this work, we exploit the above mentioned two-cavity setup to provide a
quantum circuit, comprising of two microwave cavities, Ramsey zones, atomic
sources and detectors. The ultimate goal is to produce entangled states of two
cavity fields, named here as entangled-field Schrödinger cats (EFSC). A
Schrödinger cat state is a superposition of two distinct coherent states, and
our desired state (EFSC) is an entangled state of two such superposed pairs of
the two cavity fields [60, 61, 62]. Such useful states for quantum information
processing are experimentally generated using various schemes [18, 63, 64].
Here, we consider high-quality cavities, guided by dispersive atom-field
interaction, where conditional measurements of three atoms will leave behind
the desired EFSC for the cavity fields. The correlation between the two cavity
fields is quantified by means of entanglement entropy, that is, the von
Neumann entropy of the reduced state. The variation of the entanglement with
the cavity field parameters reveals a sophisticated engineering of the desired
entanglement of the resulting state. We also riddle out maximally entangled
Schrödinger cat from a variety of obtained EFSC. A systematic Wigner phase
space analysis of the individual reduced state of each cavity manifests
quantum interference patterns typical of Schrodinger cat and compass states,
even though, due to the entanglement, the state is highly mixed state. We
demarcate the parameter regimes for various levels of entanglement, ranging
from un-entangled to maximally-entangled ones and display their phase space
structure to identify appropriate quantum states for quantum sensing, mediated
by entanglement.

The paper is organized as follows. The next section deals with the analytical
derivation from the proposed scheme to obtain a family of EFSC. The
entanglement of the obtained states is characterized in Sec. III and their
variety is delineated through engineering of experimental parameters. Section
IV describes the behavior of the states in phase space, along with their
nonlocal and spatial interference structures. A brief summary and a future
outlook of the work are furnished in the concluding section.

##  II Quantum Circuit of Cavity QED for Generating the EFSC

We consider a two-cavity setup for manipulating three two-level atoms, where
each cavity is initially prepared in a coherent state, which will dispersively
interact with the atoms. The underlying Hamiltonian is written as per the
well-known effective Jaynes Cumming model in the dispersive limit:

$\displaystyle H=\hbar\chi(\sigma_{+}\sigma_{-}+a^{{\dagger}}a\sigma_{3}),$
(1)

where $\chi$ is the dispersive coupling rate, $\sigma_{+}$ and $\sigma_{-}$
are the raising and lowering operators of the two-level atoms, respectively.
Three atoms have been used, consecutively passing through the two cavity
setup, where the atomic Rydberg levels, $|g\big{\rangle}$, $|e\big{\rangle}$
and $|f\big{\rangle}$ are the ones mainly interacting with the cavity modes.
The level $|f\big{\rangle}$ is considered away from the other two levels to
have dispersive interaction in place. The corresponding atomic inversion
operator is given by
$\sigma_{3}=|e\big{\rangle}\big{\langle}e|-|g\big{\rangle}\big{\langle}g|$.
In the dispersive case, the resonant frequency between the two levels is far-
detuned from the cavity fundamental frequency. The cavity fields are prepared
in coherent states, which are initially independent to each other. There are
Ramsey interferometric zones in either sides of the cavities to enable
preparing atoms in desired superposition states, mostly choosing a $\pi$ or a
$\pi/2$ pulse in our case. The proposed experimental setup for the whole
design to manufacture a variety of entangled quantum superposition states is
given in Fig. 1. $C_{1}$ and $C_{2}$ are two microwave cavities, initially
contain coherent fields, $|\alpha_{1}\big{\rangle}$ and
$|\alpha_{2}\big{\rangle}$, respectively. $R_{1}$ to $R_{7}$ are Ramsey
interferometric zones with $\pi/2$ pulses, except $R_{2}$ which is guided by a
$\pi$ pulse. The three boxes in Fig. 1 at the beginning of the channels are
the sources of atoms, $A$, $B$ and $C$, in a selectively prepared initial
state. Using proper Stark shift pulses, atom $B$ can be made noninteracting
with cavity $C_{2}$, while atom $C$ can be made noninteracting with cavity
$C_{1}$ (shown by the faded cavities in Fig. 1). There are detectors for
measuring the atoms in desired combination of tripartite states. In the atom-
cavity dynamics described in the following section, we will neglect any decay
process, both for atoms and for cavities, by exploiting the exceptional long
lifetimes of circular Rydberg states and the fact that the cavities possess a
very high quality factor.

Figure 1: Schematic diagram of a cavity-QED circuit for preparing the EFSC. Three Rydberg atoms, named as $A$, $B$ and $C$, are sent one after another through the two cavity setup. The cavities, labelled as $C_{1}$ and $C_{2}$, are operated under both “ON” and “OFF” (faded) conditions and initially prepared with coherent states, $|\alpha_{1}\big{\rangle}$ and $|\alpha_{2}\big{\rangle}$, respectively. Ramsey zones are placed on either sides of each cavity with $\pi$ or $\pi/2$ pulses. Atomic state | EFSC
---|---
$|g_{1}\big{\rangle}|g_{2}\big{\rangle}|g_{3}\big{\rangle}$ |  $|\psi_{1}\big{\rangle}=\frac{N_{1}}{8}\bigg{[}\Bigr{(}|\alpha_{1}e^{2i\chi_{1}t}\big{\rangle}+e^{-i\chi_{1}t}|\alpha_{1}\big{\rangle}\Bigr{)}\Bigr{(}|\alpha_{2}e^{i\chi_{2}t}\big{\rangle}+e^{-i\chi_{2}t}|\alpha_{2}e^{-i\chi_{2}t}\big{\rangle}\Bigr{)}$
|
$-\Bigr{(}|\alpha_{1}e^{i\chi_{1}t}\big{\rangle}+e^{-i\chi_{1}t}|\alpha_{1}e^{-i\chi_{1}t}\big{\rangle}\Bigr{)}\Bigr{(}|\alpha_{2}e^{2i\chi_{2}t}\big{\rangle}+e^{-i\chi_{2}t}|\alpha_{2}\big{\rangle}\Bigr{)}\bigg{]}$
$|g_{1}\big{\rangle}|e_{2}\big{\rangle}|g_{3}\big{\rangle}$ |  $|\psi_{2}\big{\rangle}=\frac{N_{2}e^{i\delta}}{8}\bigg{[}\Bigr{(}|\alpha_{1}e^{2i\chi_{1}t}\big{\rangle}-e^{-i\chi_{1}t}|\alpha_{1}\big{\rangle}\Bigr{)}\Bigr{(}|\alpha_{2}e^{i\chi_{2}t}\big{\rangle}+e^{-i\chi_{2}t}|\alpha_{2}e^{-i\chi_{2}t}\big{\rangle}\Bigr{)}$
|
$-\Bigr{(}|\alpha_{1}e^{i\chi_{1}t}\big{\rangle}-e^{-i\chi_{1}t}|\alpha_{1}e^{-i\chi_{1}t}\big{\rangle}\Bigr{)}\Bigr{(}|\alpha_{2}e^{2i\chi_{2}t}\big{\rangle}+e^{-i\chi_{2}t}|\alpha_{2}\big{\rangle}\Bigr{)}\bigg{]}$
$|f_{1}\big{\rangle}|g_{2}\big{\rangle}|g_{3}\big{\rangle}$ |  $|\psi_{3}\big{\rangle}=\frac{-N_{3}e^{i\delta}}{8}\bigg{[}\Bigr{(}|\alpha_{1}e^{2i\chi_{1}t}\big{\rangle}+e^{-i\chi_{1}t}|\alpha_{1}\big{\rangle}\Bigr{)}\Bigr{(}|\alpha_{2}e^{i\chi_{2}t}\big{\rangle}+e^{-i\chi_{2}t}|\alpha_{2}e^{-i\chi_{2}t}\big{\rangle}\Bigr{)}$
|
$+\Bigr{(}|\alpha_{1}e^{i\chi_{1}t}\big{\rangle}+e^{-i\chi_{1}t}|\alpha_{1}e^{-i\chi_{1}t}\big{\rangle}\Bigr{)}\Bigr{(}|\alpha_{2}e^{2i\chi_{2}t}\big{\rangle}+e^{-i\chi_{2}t}|\alpha_{2}\big{\rangle}\Bigr{)}\bigg{]}$
$|f_{1}\big{\rangle}|e_{2}\big{\rangle}|g_{3}\big{\rangle}$ |  $|\psi_{4}\big{\rangle}=\frac{-N_{4}e^{i2\delta}}{8}\bigg{[}\Bigr{(}|\alpha_{1}e^{2i\chi_{1}t}\big{\rangle}-e^{-i\chi_{1}t}|\alpha_{1}\big{\rangle}\Bigr{)}\Bigr{(}|\alpha_{2}e^{i\chi_{2}t}\big{\rangle}+e^{-i\chi_{2}t}|\alpha_{2}e^{-i\chi_{2}t}\big{\rangle}\Bigr{)}$
|
$+\Bigr{(}|\alpha_{1}e^{i\chi_{1}t}\big{\rangle}-e^{-i\chi_{1}t}|\alpha_{1}e^{-i\chi_{1}t}\big{\rangle}\Bigr{)}\Bigr{(}|\alpha_{2}e^{2i\chi_{2}t}\big{\rangle}+e^{-i\chi_{2}t}|\alpha_{2}\big{\rangle}\Bigr{)}\bigg{]}$
$|g_{1}\big{\rangle}|g_{2}\big{\rangle}|e_{3}\big{\rangle}$ |  $|\psi_{5}\big{\rangle}=\frac{N_{5}e^{i\delta}}{8}\bigg{[}\Bigr{(}|\alpha_{1}e^{2i\chi_{1}t}\big{\rangle}+e^{-i\chi_{1}t}|\alpha_{1}\big{\rangle}\Bigr{)}\Bigr{(}|\alpha_{2}e^{i\chi_{2}t}\big{\rangle}-e^{-i\chi_{2}t}|\alpha_{2}e^{-i\chi_{2}t}\big{\rangle}\Bigr{)}$
|
$-\Bigr{(}|\alpha_{1}e^{i\chi_{1}t}\big{\rangle}+e^{-i\chi_{1}t}|\alpha_{1}e^{-i\chi_{1}t}\big{\rangle}\Bigr{)}\Bigr{(}|\alpha_{2}e^{2i\chi_{2}t}\big{\rangle}-e^{-i\chi_{2}t}|\alpha_{2}\big{\rangle}\Bigr{)}\bigg{]}$
$|g_{1}\big{\rangle}|e_{2}\big{\rangle}|e_{3}\big{\rangle}$ |  $|\psi_{6}\big{\rangle}=\frac{N_{6}e^{i2\delta}}{8}\bigg{[}\Bigr{(}|\alpha_{1}e^{2i\chi_{1}t}\big{\rangle}-e^{-i\chi_{1}t}|\alpha_{1}\big{\rangle}\Bigr{)}\Bigr{(}|\alpha_{2}e^{i\chi_{2}t}\big{\rangle}-e^{-i\chi_{2}t}|\alpha_{2}e^{-i\chi_{2}t}\big{\rangle}\Bigr{)}$
|
$-\Bigr{(}|\alpha_{1}e^{i\chi_{1}t}\big{\rangle}-e^{-i\chi_{1}t}|\alpha_{1}e^{-i\chi_{1}t}\big{\rangle}\Bigr{)}\Bigr{(}|\alpha_{2}e^{2i\chi_{2}t}\big{\rangle}-e^{-i\chi_{2}t}|\alpha_{2}\big{\rangle}\Bigr{)}\bigg{]}$
$|f_{1}\big{\rangle}|g_{2}\big{\rangle}|e_{3}\big{\rangle}$ |  $|\psi_{7}\big{\rangle}=\frac{-N_{7}e^{i2\delta}}{8}\bigg{[}\Bigr{(}|\alpha_{1}e^{2i\chi_{1}t}\big{\rangle}+e^{-i\chi_{1}t}|\alpha_{1}\big{\rangle}\Bigr{)}\Bigr{(}|\alpha_{2}e^{i\chi_{2}t}\big{\rangle}-e^{-i\chi_{2}t}|\alpha_{2}e^{-i\chi_{2}t}\big{\rangle}\Bigr{)}$
|
$+\Bigr{(}|\alpha_{1}e^{i\chi_{1}t}\big{\rangle}+e^{-i\chi_{1}t}|\alpha_{1}e^{-i\chi_{1}t}\big{\rangle}\Bigr{)}\Bigr{(}|\alpha_{2}e^{2i\chi_{2}t}\big{\rangle}-e^{-i\chi_{2}t}|\alpha_{2}\big{\rangle}\Bigr{)}\bigg{]}$
$|f_{1}\big{\rangle}|e_{2}\big{\rangle}|e_{3}\big{\rangle}$ |  $|\psi_{8}\big{\rangle}=\frac{-N_{8}e^{i3\delta}}{8}\bigg{[}\Bigr{(}|\alpha_{1}e^{2i\chi_{1}t}\big{\rangle}-e^{-i\chi_{1}t}|\alpha_{1}\big{\rangle}\Bigr{)}\Bigr{(}|\alpha_{2}e^{i\chi_{2}t}\big{\rangle}-e^{-i\chi_{2}t}|\alpha_{2}e^{-i\chi_{2}t}\big{\rangle}\Bigr{)}$
|
$+\Bigr{(}|\alpha_{1}e^{i\chi_{1}t}\big{\rangle}-e^{-i\chi_{1}t}|\alpha_{1}e^{-i\chi_{1}t}\big{\rangle}\Bigr{)}\Bigr{(}|\alpha_{2}e^{2i\chi_{2}t}\big{\rangle}-e^{-i\chi_{2}t}|\alpha_{2}\big{\rangle}\Bigr{)}\bigg{]}$
Table 1: Right Column: The obtained entangled mesoscopic superposition state
(EFSC), corresponding to each set of projective measurements on the three
atoms (left Column), total possibilities being eight. $N_{i}$
$(i=1\;\textmd{to}\;8)$ are the normalization constants. $\delta$ contributes
only to the overall phase.

###  II.1 Step-wise Circuit Operation and State Preparation

Here, we provide the detailed analytical steps to evaluate the quantum state
as per the sequence of operations, given in the cavity QED circuit (Fig.1).
The suffixes ($1$, $2$, and $3$), used with the ground state
$|g\big{\rangle}$, exited state $|e\big{\rangle}$ and the auxiliary state
$|f\big{\rangle}$, correspond to atom-A, B and C, respectively. We will also
use the notation, $|\Psi^{ABC}_{R_{6}}\big{\rangle}$, for representing the
resultant quantum state, when atom-A and atom-B have already crossed and
atom-C has just passed Ramsey zone $R_{6}$.
Atom-A: The source (A) emits atom-A upon velocity selection in its ground
state $|g_{1}\big{\rangle}$, which when passes through the first Ramsey zone
becomes a superposition state of $|g_{1}\big{\rangle}$ and an auxiliary state
$|f_{1}\big{\rangle}$, _i.e._ ,

$\displaystyle|\Psi^{A}_{R_{1}}\big{\rangle}=\frac{1}{\sqrt{2}}(|g_{1}\big{\rangle}+e^{i\delta}|f_{1}\big{\rangle}),$
(2)

where $\delta$ is the relative phase, provided by the Ramsey zone, $(R_{1})$.
This linearly superposed atom enters into the cavity $(C_{1})$ and interacts
with its coherent field for which the initial state of atom-cavity system
becomes
$|\Psi^{A}_{C_{1}}(0)\big{\rangle}=|\Psi^{A}_{R_{1}}\big{\rangle}|\alpha_{1}\big{\rangle}$.
The Jaynes- Cummings Hamiltonian in Eq.(1) will apply on atom-A and field
${C_{1}}$ to produce the quantum state at time $t$ just after the cavity
$(C_{1})$:

$\displaystyle|\Psi^{A}_{C_{1}}(t)\big{\rangle}$
$\displaystyle=e^{\frac{-iHt}{\hslash}}|\Psi^{A}_{R_{1}}\big{\rangle}|\alpha_{1}\big{\rangle}$
$\displaystyle=\frac{1}{\sqrt{2}}\Big{(}|g_{1}\big{\rangle}|\alpha_{1}e^{i\chi_{1}t}\big{\rangle}+e^{i\delta}|f_{1}\big{\rangle}|\alpha_{1}\big{\rangle}\Big{)},$

where $\chi_{1}$ is the dispersive coupling rate for cavity-$C_{1}$. We have
used the relations,
$e^{\frac{-iHt}{\hslash}}|g_{1}\big{\rangle}|\alpha_{1}\big{\rangle}=|g_{1}\big{\rangle}|\alpha_{1}e^{i\chi_{1}t}\big{\rangle}$
and
$e^{\frac{-iHt}{\hslash}}|f_{1}\big{\rangle}|\alpha_{1}\big{\rangle}=|f_{1}\big{\rangle}|\alpha_{1}\big{\rangle}$.
It is worth pointing out that the above state in Eq. (II.1) is an entangled
state of atom-A and field ${C_{1}}$. Neither atom-A, nor $C_{1}$ have
distinguishable quantum state anymore, instead, they share a resultant quantum
state of the atom-field system, irrespective of their spatial separation.
Then, atom-A continues to move and interacts with the second Ramsey zone
$(R_{2})$, guided by a $\pi$-pulse, which transforms the component atomic
states as $|g_{1}\big{\rangle}$ $\rightarrow$ $e^{i\delta}|f_{1}\big{\rangle}$
and $|f_{1}\big{\rangle}$ $\rightarrow$ $-e^{-i\delta}|g_{1}\big{\rangle}$.
Such state gets further modified due to the interaction of atom-A and field
${C_{2}}$, which produces the state

$|\Psi^{A}_{C_{2}}(t)=\frac{1}{\sqrt{2}}\Big{(}e^{i\delta}|f_{1}\big{\rangle}|\alpha_{1}e^{i\chi_{1}t}\big{\rangle}|\alpha_{2}\big{\rangle}-|g_{1}\big{\rangle}|\alpha_{1}\big{\rangle}|\alpha_{2}e^{i\chi_{2}t}\big{\rangle}\Big{)}.$
(3)

The final step for Atom-A is a $\pi/2$-pulse in the Ramsey zone $(R_{3})$,
which transforms the state as
$|g_{1}\big{\rangle}\rightarrow\frac{1}{\sqrt{2}}(|g_{1}\big{\rangle}+e^{i\delta}|f_{1}\big{\rangle})$
and
$|f_{1}\big{\rangle}\rightarrow\frac{1}{\sqrt{2}}(|g_{1}\big{\rangle}e^{-i\delta}-|f_{1}\big{\rangle})$.
Hence, the entangled state of Eq. (3) transforms to

$\displaystyle|\Psi^{A}_{R_{3}}\big{\rangle}=$
$\displaystyle\frac{1}{2}\Bigr{[}|g_{1}\big{\rangle}\Big{(}|\alpha_{1}e^{i\chi_{1}t}\big{\rangle}|\alpha_{2}\big{\rangle}-|\alpha_{1}\big{\rangle}|\alpha_{2}e^{i\chi_{2}t}\big{\rangle}\Big{)}-$
(4) $\displaystyle
e^{i\delta}|f_{1}\Big{(}|\alpha_{1}e^{i\chi_{1}t}\big{\rangle}|\alpha_{2}\big{\rangle}+|\alpha_{1}\big{\rangle}|\alpha_{2}e^{i\chi_{2}t}\big{\rangle}\Big{)}\Bigr{]}.$

Atom-B: In the second stage, atom-B is sent through the two-cavity system,
where only cavity-$C_{1}$ interacts with the atom. Prior to that, the atom is
prepared in a superposition state by passing through the Ramsey zone, $R_{4}$,
guided by a $\pi/2$-pulse and the atomic state before entering into the cavity
becomes
$\frac{1}{\sqrt{2}}(|g_{2}\big{\rangle}+e^{i\delta}|e_{2}\big{\rangle})$ with
a relative phase $\delta$. This atom dispersively interacts with field in
${C_{1}}$ (already sharing the state $|\Psi^{A}_{R_{3}}\big{\rangle}$ of Eq.
(4)) through the Jaynes-Cummings Hamiltonian to obtain the state of the system
as

$\displaystyle|\Psi^{AB}_{C_{1}}\big{\rangle}=$
$\displaystyle\frac{1}{2\sqrt{2}}\Bigr{[}|g_{1}\big{\rangle}\tilde{\alpha_{1}}-e^{i\delta}|f_{1}\big{\rangle}\tilde{\alpha_{2}}\Bigr{]}|g_{2}\big{\rangle}+$
$\displaystyle\frac{1}{2\sqrt{2}}e^{i\delta-i\chi_{1}t}\Bigr{[}|g_{1}\big{\rangle}\tilde{\alpha_{3}}-e^{i\delta}|f_{1}\big{\rangle}\tilde{\alpha_{4}}\Bigr{]}|e_{2}\big{\rangle},$

where the cavity field states are collected in the terms $\tilde{\alpha_{1}}$,
$\tilde{\alpha_{2}}$, $\tilde{\alpha_{3}}$ and $\tilde{\alpha_{4}}$, which are
the entangled states of the two cavity fields detailed in Appendix-I.

This state gets further mixed up when atom-B passes through $R_{5}$ with a
$\pi/2$-pulse and the resultant entangled quantum state of the system formed
by atoms A and B and the fields of cavities ${C_{1}}$ and ${C_{2}}$, becomes

$\displaystyle|\Psi^{AB}_{R_{5}}\big{\rangle}=$
$\displaystyle\frac{1}{4}\Bigr{[}|g_{1}\big{\rangle}|g_{2}\big{\rangle}\Bigr{(}\tilde{\alpha_{1}}+e^{-i\chi_{1}t}\tilde{\alpha_{3}}\Bigr{)}+e^{i\delta}|g_{1}\big{\rangle}|e_{2}\big{\rangle}\Bigr{(}\tilde{\alpha_{1}}-e^{-i\chi_{1}t}\tilde{\alpha_{3}}\Bigr{)}$
(5)
$\displaystyle-e^{i\delta}|f_{1}\big{\rangle}|g_{2}\big{\rangle}\Bigr{(}\tilde{\alpha_{2}}+e^{-i\chi_{1}t}\tilde{\alpha_{4}}\Bigr{)}-e^{2i\delta}|f_{1}\big{\rangle}|e_{2}\big{\rangle}\Bigr{(}\tilde{\alpha_{2}}-e^{-i\chi_{1}t}\tilde{\alpha_{4}}\Bigr{)}\Bigr{]}.$

Atom-C: The third atom (C) is made to interact with cavity-$C_{2}$, Ramsey
zones $R_{6}$ and $R_{7}$ (both with a $\pi/2$-pulse). Instead of writing the
lengthy expression here, we would like to write the final state of the whole
setup after Ramsey zone $R_{7}$ as Eq.(12) in Appendix-I. The final state has
$64$ terms in total, where the accumulated field coefficients, different from
the earlier ones, are ${\alpha_{i}}^{\prime}$ and
${\alpha_{i}}^{\prime\prime}$ with $i=1,\;2,\;3,\;4$. These are also entangled
states of the phase-modulated two cavity fields and are given in Appendix-I.
In Eq.(12), for three two-level atoms, there are eight combinations of atomic
states. Each combination has the associated field coefficient as an entangled
state of cavities $C_{1}$ and $C_{2}$, with eight of their tensor product
terms. Conditional measurements on the three atoms generate the desired EFSC
for the state of the two cavities, which does not depend on $\delta$, while
depending upon $\chi_{1}t$, $\chi_{2}t$, and the coherent field amplitudes
$\alpha_{1}$ and $\alpha_{2}$. Hence, it is apparent that Eq.(12) offers a
family of EFSC, which can be produced by tuning these physical parameters in
the experiment. The joint atomic measurements and their corresponding
conditionally generated EFSC states are listed in Table-1.

There are eight measurement outcomes in total, each of which exhibits
entanglement between two quantum superposition states of the cavity fields. We
underline that the cavity field components of the nonseparable state are
individually in the form of a Schrödinger cat state. When one cavity is
measured in a Schrödinger cat state, the other will also be projected onto
another Schrödinger cat state. The separability between the constituent
coherent states of each cat depends on the tuning parameters,
$\chi_{1},\;\chi_{2},\;\alpha_{1}\;\textmd{and}\;\alpha_{2}$.

There are two types of generic mesoscopic superposition: i) superposition
between $|\alpha\big{\rangle}$ and $|e^{2i\chi t}\alpha\big{\rangle}$; and ii)
superposition between $|e^{-i\chi t}\alpha\big{\rangle}$ and $|e^{i\chi
t}\alpha\big{\rangle}$. In both cases, the coherent states are separated in
phase space by $\sqrt{2}$ times the absolute value of the difference of their
magnitudes. Hence, the minimum difference happens for $\chi t=n\pi$ with
integer $n$, which provides an untangled state because it is equivalent to
having no interaction inside the cavities, whereas the maximum separation is
obtained for $\chi t=\pi/2$. The latter case produces an entanglement of two
cat states placed along orthogonal directions in phase space. It would be
interesting to find the quantum states for the maximum separation for the
Schrödinger cats of both the cavities, _i.e.,_ $\chi_{1}t=\chi_{2}t=\pi/2$. We
use this specific case for all the states in Table 1 and obtain eight EFSC
with maximized phase-space distance. We choose this option, corresponding to
compass-like states useful for optimal displacement quantum sensing, and which
are listed in Appendix-I from Eq. (13) to Eq. (20). It is worth noticing that,
out of the eight entangled field states, four states
$\psi_{1},\;\psi_{3},\;\psi_{6}\;\textmd{and}\;\psi_{8}$ take the form of
Bell-like states of the form
$\Bigr{(}|\phi_{1}\rangle|\phi_{2}\rangle\pm|\phi_{2}\rangle|\phi_{1}\rangle\Bigr{)}$.
The remaining four states can be also entangled and any conditional generation
of these states will leave the state of the cavities in a maximally separated
Schrödinger cat state, either in position (real cat) or in momentum (imaginary
cat). However, they do not take the form of a quasi-Bell state. A quasi-Bell
state involving two cat states, one along the real and the other along the
imaginary axis of the phase space, resembles compass states, i.e.,
superpositions of four equally spaced coherent states, whose interference
properties are known to manifest structures with dimension smaller than Planck
constant $(\hbar)$, as first noticed by Zurek [65]. We now study the
entanglement of the conditionally generated states, and we will then study the
phase space properties in Sec. IV.

Figure 2: von Neumann entropy of the quantum state of the first subsystem upon
measuring the atoms in $|g_{1}\rangle|g_{2}\rangle|g_{3}\rangle$ at various
parameter regimes. Upper row corresponds to $\alpha_{1}=\alpha_{2}$ while
$\theta_{1}$ and $\theta_{2}$ are varying. Lower row corresponds to
$\theta_{1}=\theta_{2}$ while $\alpha_{1}$ and $\alpha_{2}$ are varying. The
region shaded in dark-red corresponds to $E\approx 1$, representing the region
of maximal entanglement, and the region in pink corresponds to no entanglement
$E\approx 0$. We note that when $\alpha_{1}=\alpha_{2}$ and
$\theta_{1}=\theta_{2}$ (Figs. (a) and (e)) we have maximum entanglement for
all the parameter values. Figure 3: von Neumann entropy of the quantum state
of the first subsystem upon measuring the atoms in
$|f_{1}\rangle|g_{2}\rangle|e_{3}\rangle$ at various parameter regimes. Upper
row corresponds to $\alpha_{1}=\alpha_{2}$ while $\theta_{1}$ and $\theta_{2}$
are varying. Lower row corresponds to $\theta_{1}=\theta_{2}$ while
$\alpha_{1}$ and $\alpha_{2}$ are varying. The region shaded in dark-red
corresponds to $E\approx 1$, representing the region of maximal entanglement
and the region in pink corresponds to no entanglement $E\approx 0$. Fig (a)
and (e) depict the maximally entangled region for all parameter values, but
satisfying $\alpha_{1}=\alpha_{2}$ and $\theta_{1}=\theta_{2}$. Figure 4: von
Neumann entropy of the eight quantum states in the most nonlocal situation,
$\chi_{1}t=\chi_{2}t=\pi/2$ and $\alpha_{1}=\alpha_{2}=\alpha$. Solid
horizontal line corresponds to states, $|\psi_{j}\big{\rangle}$ : {$j=1,3,6$
and $8$}, having one ebit of entanglement and suitable for teleportation.
Dashed curve refers to the entanglement for the states
$|\psi_{j}\big{\rangle}$ : {$j=2,4,5$ and $7$}, which saturates to unity only
for large enough values of $\alpha$. Only for large amplitudes the two single
mode states becomes quasi-orthogonal and the von-Neumann entropy achieves its
maximum value.

##  III Quantitative Estimate of the Entanglement

As we have seen in the previous section, the eight conditionally generated
quantum states are entangled Schrödinger cat states of the two cavities. Four
of them match the mathematical form of Bell states [75, 76, 77, 78, 79, 80,
81]. If the involved states are orthogonal, then the quasi-Bell states can be
seen as maximally entangled Bell states.

Now, it is worthwhile to quantify the entanglement of these states, which we
do here by using the von-Neumann entropy of the reduced state, which is a
proper entanglement measure for bipartite pure states [82]. The von-Neumann
entropy of the $j$-th cavity field is given by
$E=-Tr_{3-j}[\rho_{j}\log\rho_{j}]$, which can be rewritten as
$E=-\sum_{i}\lambda_{i}\log\lambda_{i}$, where $\lambda_{i}$ are the nonzero
eigenvalues of the reduced density matrix $\rho_{j}$ of the $j$-th subsystem.
We have analytically evaluated this for all the obtained quantum states, but
we show that here for $|\psi_{1}\big{\rangle}$ only. The other entropy
calculations can be performed in a similar manner. The selected state
$|\psi_{1}\big{\rangle}$ is the state conditionally generated when we detect
$|g_{1}\rangle|g_{2}\rangle|g_{3}\rangle$, and is rewritten as

$\displaystyle|\psi_{1}\big{\rangle}$
$\displaystyle=N_{1}\bigg{[}\Bigr{(}|\alpha_{1}e^{2i\chi_{1}t}\big{\rangle}+e^{-i\chi_{1}t}|\alpha_{1}\big{\rangle}\Bigr{)}\Bigr{(}|\alpha_{2}e^{i\chi_{2}t}\big{\rangle}+e^{-i\chi_{2}t}|\alpha_{2}e^{-i\chi_{2}t}\big{\rangle}\Bigr{)}$
(6)
$\displaystyle-\Bigr{(}|\alpha_{1}e^{i\chi_{1}t}\big{\rangle}+e^{-i\chi_{1}t}|\alpha_{1}e^{-i\chi_{1}t}\big{\rangle}\Bigr{)}\Bigr{(}|\alpha_{2}e^{2i\chi_{2}t}\big{\rangle}+e^{-i\chi_{2}t}|\alpha_{2}\big{\rangle}\Bigr{)}\bigg{]},$

Writing
$|\psi_{1}\big{\rangle}=N_{1}\big{[}|\mu_{1}\rangle|\nu_{2}\rangle-|\nu_{1}\rangle|\mu_{2}\rangle\big{]}$,
the corresponding bases modulo normalization become

$\displaystyle|\mu_{1}\rangle=$
$\displaystyle|\alpha_{1}e^{2i\chi_{1}t}\big{\rangle}+e^{-i\chi_{1}t}|\alpha_{1}\big{\rangle},$
$\displaystyle|\nu_{2}\rangle=$
$\displaystyle|\alpha_{2}e^{i\chi_{2}t}\big{\rangle}+e^{-i\chi_{2}t}|\alpha_{2}e^{-i\chi_{2}t}\big{\rangle},$
$\displaystyle|\nu_{1}\rangle=$
$\displaystyle|\alpha_{1}e^{i\chi_{1}t}\big{\rangle}+e^{-i\chi_{1}t}|\alpha_{1}e^{-i\chi_{1}t}\big{\rangle},$
$\displaystyle|\mu_{2}\rangle=$
$\displaystyle|\alpha_{2}e^{2i\chi_{2}t}\big{\rangle}+e^{-i\chi_{2}t}|\alpha_{2}\big{\rangle}.$

The transformation from the non-orthogonal basis to orthogonal basis is done
for both the cavity modes:

$\displaystyle{|0\rangle}_{1}=$ $\displaystyle|\mu_{1}\rangle,$
$\displaystyle{|1\rangle}_{1}=$ $\displaystyle\Big{(}|\nu_{1}\rangle-
p_{1}{|0\rangle}_{1}\Big{)}/q_{1},$

and

$\displaystyle{|0\rangle}_{2}=$ $\displaystyle|\mu_{2}\rangle,$
$\displaystyle{|1\rangle}_{2}=$ $\displaystyle\Big{(}|\nu_{2}\rangle-
p_{2}{|0\rangle}_{2}\Big{)}/q_{2},$

respectively. Here, $p_{1}=\langle\mu_{1}|\nu_{1}\rangle$,
$q_{1}=\sqrt{1-|p_{1}|^{2}}$, $p_{2}=\langle\mu_{2}|\nu_{2}\rangle$, and
$q_{2}=\sqrt{1-|p_{2}|^{2}}$. The quantum state can be rewritten as

$|\psi_{1}\big{\rangle}=N_{1}\big{[}(p_{2}-p_{1}){|0\rangle}_{1}{|0\rangle}_{2}+q_{2}{|0\rangle}_{1}{|1\rangle}_{2}-q_{1}{|1\rangle}_{1}{|0\rangle}_{2}\big{]},$
(7)

where $N_{1}$ is the normalization factor guaranteeing that
$|N_{1}|^{2}\big{[}|p_{2}-p_{1}|^{2}+|q_{2}|^{2}+|q_{1}|^{2}\big{]}=1$. As we
want to find the entropy of mode 1, the corresponding reduced density matrix
is evaluated by tracing out mode 2:
$\rho_{1}=Tr_{2}|\psi_{1}\big{\rangle}\big{\langle}\psi_{1}|$. The reduced
density matrix of mode 1 is written as

$\rho_{1}=\begin{bmatrix}|N_{1}|^{2}\big{[}|p_{2}-p_{1}|^{2}+|q_{2}|^{2}\big{]}&&-|N_{1}|^{2}(p_{2}-p_{1})q^{\ast}_{1}\\\
\\\
-|N_{1}|^{2}(p_{2}-p_{1})^{\ast}q_{1}&&|N_{1}|^{2}|q_{1}|^{2}\end{bmatrix},$

the determinant of which is
$\textmd{det}(\rho_{1})=|N_{1}|^{4}|q_{1}|^{2}|q_{2}|^{2}$ and the eigenvalues
are

$\displaystyle\lambda_{\pm}=\frac{1}{2}\Big{[}1\pm\sqrt{1-4\;\textmd{det}(\rho_{1})}\Big{]}.$
(8)

A similar approach can be used for the other pure conditional states of the
two cavities. We now illustrate the behavior of the von Neumann entropy for
the measurement outcomes of $|g_{1}\rangle|g_{2}\rangle|g_{3}\rangle$ (quasi-
Bell state $|\psi_{1}\rangle$) and $|f_{1}\rangle|g_{2}\rangle|e_{3}\rangle$
(non quasi-Bell state $|\psi_{7}\rangle$) in Fig.2 and Fig.3, respectively,
versus the system parameters. We have defined $\chi t=\theta$ for both modes
with corresponding suffixes. The entropy variations are displayed for two
tuning parameters of the first cavity mode, $\theta_{1}$ and $\alpha_{1}$.
However, this is affected by the second cavity mode parameters, $\theta_{2}$
and $\alpha_{2}$ due to entanglement. It is quite fascinating to observe that,
when the parameters of the two cavity modes are the same,
$\theta_{1}=\theta_{2}$ and $\alpha_{1}=\alpha_{2}$, in the case of a quasi-
Bell state, $|\psi_{1}\big{\rangle}$, one has maximum entanglement $E=1$ for
all physical parameters of both modes (see Fig.2(a) and (e)). This can be
easily understood from the fact that in this case one has $p_{1}=p_{2}$
implying $q_{1}=q_{2}$, and from Eq. (7) one can easily see that the state of
the cavities becomes just a Bell state involving the two orthogonal states
$|0\rangle$ and $|1\rangle$.

On the contrary, deviating from the condition of equal parameters results in a
nontrivial variation of the entanglement and even to its disappearance
($E\approx 0$) at some parameter values (see Fig.2(b)-(d) and (f)-(h)). The
upper row of Fig.2 refers to the variation of the ratio
$\theta_{2}/\theta_{1}$ while keeping $\alpha_{1}=\alpha_{2}$, while the lower
row refers to the variation of the ratio $\alpha_{2}/\alpha_{1}$, by keeping
$\theta_{1}=\theta_{2}$. The contours are used in such a way that the dark-red
color corresponds to the region of very large entanglement with entropy $>0.9$
and pink color shows the region of almost no-entanglement with entropy $<0.1$.
The region of no-entanglement gets gradually inhabited at the cost of
reduction of the area for maximal entanglement when the ratio,
$\theta_{2}/\theta_{1}$ or $\alpha_{2}/\alpha_{1}$, gradually decreases. These
plots clearly reveal the possibility of choosing a desired amount of
entanglement between the cavity modes, by tuning the above interaction
parameters and the amplitude of the initial coherent states in the cavities.
We now study the behavior of the entanglement entropy for a non quasi-Bell
state, $|\psi_{7}\big{\rangle}$, and we show the results in Fig.3. Unlike the
previous case, identical interaction and amplitude parameters for the modes,
$\theta_{1}=\theta_{2}$ and $\alpha_{1}=\alpha_{2}$, does not lead to a
maximally entangled scenario for all parameters (see Fig.3(a) and (e)), but
only for a designated parameter domain, showed by dark-red color. At the same
time, similarly to the previous case of quasi-Bell state, the parameter region
with zero entanglement broadens when the ratio, $\theta_{2}/\theta_{1}$ or
$\alpha_{2}/\alpha_{1}$, decreases from $1.0$ to $0.2$.

We now study the particular situation when $\chi_{1}t=\chi_{2}t=\pi/2$, for
which a maximum separation of the two coherent states of the superposition is
obtained, and the two cat states are directed along two orthogonal directions
in phase space. The entanglement measure through the von Neumann entropy of
the obtained eight quantum states in this case are shown in Fig.4, where two
groups are clearly visible. One group is formed by $|\psi_{j}\big{\rangle}$ :
{$j=1,3,6$ and $8$} with nonzero eigenvalues of the reduced density matrix
${\lambda^{j}_{1}}={\lambda^{j}_{2}}=\frac{1}{2}$. The corresponding
entanglement is $E(|\psi_{j}\big{\rangle})=1$, which implies that these states
are always the maximally entangled state with $1$-ebit of entanglement. These
are represented by the solid horizontal line in Fig.4. On the other hand, two
non-zero eigenvalues of the reduced density matrices of the remaining four
quantum states, $|\psi_{j}\big{\rangle}$: {$j=2,4,5$ and $7$} are given by

${\lambda^{j}_{1}}=\frac{(1-e^{-|\alpha_{1}|^{2}})^{2}}{2(1+e^{-2|\alpha_{1}|^{2}})},\;\;\;\;{\lambda^{j}_{2}}=\frac{(1+e^{-|\alpha_{2}|^{2}})^{2}}{2(1+e^{-2|\alpha_{2}|^{2}})}.$
(9)

When these values of ${\lambda^{j}_{1,2}}$ are used within the von-Neumann
entropy expression, we obtain $E(|\psi_{j}\big{\rangle})\leq 1$, which is
represented by the dashed curve in Fig.4. However, for a larger amplitude
$\alpha$ ($>2$), all the eight obtained states approach one ebit of
entanglement, $E=1$. The above observations can also be seen from the proper
cuts of the general entropy contour plots of Fig. 2(a) or (e) and Fig. 3(a) or
(e), for $\theta_{1}=\theta_{2}=\pi/2$.

Figure 5: Contour plot of the Wigner function for the state
$|\psi_{1}\big{\rangle}$ in both the phase spaces. For ($x_{1},\;p_{1}$), we
integrate the calculated Wigner function over ($x_{2},\;p_{2}$) and vice-
versa. Here, we have taken equal phase-space support for both the fields:
$\alpha_{1}=\alpha_{2}=4$. Left column ((a) and (c)) shows the reduced Wigner
functions of the individual cavity field and the right column ((b) and (d))
display their central interference region. Figure 6: Wigner function in the
first phase space ($x_{1},\;p_{1}$) of the entangled field state, obtained
from the coefficient
$|g_{1}\big{\rangle}|g_{2}\big{\rangle}|g_{3}\big{\rangle}$ in Eq. (12). Here,
$\chi_{1}t$ and $\chi_{2}t$ both are equal and vary from $90^{\circ}$ to
$0.5^{\circ}$. The sub-Planck structures undergo deformation as the
constituent coherent states, that make up the system, are rotated and tend to
blend together. We consider $\alpha_{1}=\alpha_{2}=3$. Figure 7: Wigner
function in the first phase space ($x_{1},\;p_{1}$) of the entangled field
state, obtained from the coefficient
$|g_{1}\big{\rangle}|e_{2}\big{\rangle}|g_{3}\big{\rangle}$ in Eq. (12). Here,
$\chi_{1}t$ and $\chi_{2}t$ both are equal and vary from $90^{\circ}$ to
$0.5^{\circ}$. The sub-Planck structures undergo deformation as the
constituent coherent states, that make up the system, are rotated and tend to
blend together. We consider $\alpha_{1}=\alpha_{2}=3$.

##  IV Phase Space Representation of the EFSC

In the above sections, we have discussed about the quantum state preparation
and their entanglement properties. The generated states possess both the
distinct nonclassical features: _a_) entanglement and _b_) mesoscopic quantum
superposition. Hence, it is interesting to study their coexistence through
phase space Wigner distribution, which has the merit to manifest quantum
interference structures of the correlated state.

However, visualizing the phase space of both cavities will requires a four
dimensional phase space, which is not possible. Thus, we will visualize one of
the field modes, at a time, in its corresponding sub-phase-space by taking the
reduced Wigner function upon integrating over the position and momentum
variables of the other mode. In this way, we are able to see one cavity field
while the two cavities still remain correlated.

Below, we will calculate the Wigner function of a general bipartite state
($|\Psi_{gen}\big{\rangle}$), which can be projected to any one of the eight
listed states of Table 1 modulo normalization:

$\displaystyle|\Psi_{gen}\big{\rangle}$
$\displaystyle=\frac{1}{8}\bigg{[}A\Bigr{(}|\alpha_{1}e^{i\theta_{1}}\big{\rangle}+e^{i\xi_{1}}|-\alpha_{1}e^{i\theta_{1}}\big{\rangle}\Bigr{)}\Bigr{(}|\alpha_{2}e^{i\phi_{2}}\big{\rangle}+e^{i\zeta_{2}}|-\alpha_{2}e^{i\phi_{2}}\big{\rangle}\Bigr{)}$
(10)
$\displaystyle+B\Bigr{(}|\alpha_{1}e^{i\phi_{1}}\big{\rangle}+e^{i\zeta_{1}}|-\alpha_{1}e^{i\phi_{1}}\big{\rangle}\Bigr{)}\Bigr{(}|\alpha_{1}e^{i\theta_{2}}\big{\rangle}+e^{i\xi_{2}}|-\alpha_{2}e^{i\theta_{2}}\big{\rangle}\Bigr{)}\bigg{]},$

where $\theta_{1}$, $\theta_{2}$, $\phi_{1}$, $\phi_{2}$, $\xi_{1}$,
$\xi_{2}$, $\zeta_{1}$ and $\zeta_{2}$ are the phases with the coherent states
and the relative phases between them. $A$ and $B$ are complex coefficients of
the constituent entangled pairs. The corresponding two mode Wigner
distribution function in combined space with variables, $x_{1}$, $p_{1}$;
$x_{2}$, $p_{2}$, can be written as

$\displaystyle W(x_{1},p_{1};x_{2},p_{2})$
$\displaystyle=\frac{1}{\pi^{2}}\int_{-\infty}^{\infty}\Psi_{gen}\big{(}x_{1}+q,x_{2}+s\big{)}$
$\displaystyle\Psi^{{\dagger}}_{gen}\big{(}x_{1}-q,x_{2}-s\big{)}e^{-2i(p_{1}q+p_{2}s)}dqds.$

The quantum state of Eq. (10) is substituted in this expression and a lengthy
but straightforward calculation results

$\displaystyle
W(x_{1},p_{1};x_{2},p_{2})=\frac{1}{64{\pi}^{2}}\exp\\{-2\left(x_{1}^{2}+x_{2}^{2}+p_{1}^{2}+p_{2}^{2}\right)\\}$
$\displaystyle\times\bigg{[}W_{D1}+W_{D2}+W_{OD1}+W_{OD2}\bigg{]}.$ (11)

Here, $W_{D1}$, $W_{D2}$ are diagonal and $W_{OD1}$, $W_{OD2}$ are off-
diagonal components of the obtained Wigner function. These components take big
expressions and are provided in Appendix-II. This analytical expression of the
Wigner function will enable us to visualize the whole family of states with
appropriate experimental parameters, subject to its reduction to one of the
spaces upon integrating over variables of other space. Below, we will show
some of them to reveal their merit towards parameter engineering for
entanglement and quantum sensing applications.

###  IV.1 Wigner Function for a specific case of Largest EFSC

While visualizing the above Wigner function, one needs to consider specific
parameters for the chosen quantum state. The quantum states obtained by
Eqs.(13 to 20) demand specific choices of $\theta_{1}$, $\theta_{2}$,
$\phi_{1}$, $\phi_{2}$, $\xi_{1}$, $\xi_{2}$, $\zeta_{1}$, $\zeta_{2}$, $A$
and $B$. In this subsection, we will demonstrate the state obtained in Eq.(13)
for instance, which maps the general entangled state by putting
$\theta_{1}=0$, $\theta_{2}=0$, $\phi_{1}=\pi/2$, $\phi_{2}=\pi/2$,
$\xi_{1}=\pi/2$, $\xi_{2}=\pi/2$, $\zeta_{1}=3\pi/2$, $\zeta_{2}=3\pi/2$,
$A=-i$ and $B=i$. This is a Bell state of two Schrödinger cats of two
independent cavity modes.

Figure 5 depicts the Wigner function of both the cavity fields separately for
$\chi_{1}t=\chi_{2}t=\pi/2$. For first cavity-field, defined by the phase
space variables ($x_{1},\;p_{1}$), the corresponding Wigner function is
obtained by numerically integrating the total Wigner function over
($x_{2},\;p_{2}$) and vice-versa. The left column of Fig. 5 displays the
states of cavity-$C_{1}$ (up) and cavity-$C_{2}$ (down). In this case of equal
parameters, the pure state of the two cavities is maximally entangled and it
is a Bell state involving the two states $|0\rangle$ and $|1\rangle$
introduced in Sec. III. However, since in this figure we have chosen
$\alpha_{1}=\alpha_{2}=4$, the overlaps $p_{1}=p_{2}\sim 0$ and the reduced
state of each cavity is in practice an equal mixture of two cat states
directed along the real and imaginary axis of the phase space. In this way the
state resembles a compass-state, which is the superposition of four equally
spaced coherent states, and show similar interference patterns at the center
of the phase space, despite being highly mixed and with large entropy.

Figure 8: Wigner function in the first phase space ($x_{1},\;p_{1}$) of the
entangled field state, obtained from the coefficient
$|f_{1}\big{\rangle}|e_{2}\big{\rangle}|g_{3}\big{\rangle}$ in Eq. (12). Here,
$\chi_{1}t$ and $\chi_{2}t$ both are equal and vary from $90^{\circ}$ to
$0.5^{\circ}$. The sub-Planck structures undergo deformation as the
constituent coherent states, that make up the system, are rotated and tend to
blend together. We consider $\alpha_{1}=\alpha_{2}=3$. Figure 9: Wigner
function in the first phase space ($x_{1},\;p_{1}$) of the entangled field
state, obtained from the coefficient
$|g_{1}\big{\rangle}|g_{2}\big{\rangle}|e_{3}\big{\rangle}$ in Eq. (12). Here,
$\chi_{1}t$ and $\chi_{2}t$ both are equal and vary from $90^{\circ}$ to
$0.5^{\circ}$. The sub-Planck structures undergo deformation as the
constituent coherent states, that make up the system, are rotated and tend to
blend together. We consider $\alpha_{1}=\alpha_{2}=3$.

The right column of Fig. 5 manifests their corresponding interference
structures as a zoomed version of the central region of the phase space. It is
quite fascinating to observe such interference structure in the reduced Wigner
function of entangled sub-systems, even in a mixed state. These sub-Planck
structures, resembling a nice checkerboard pattern, have been extensively
studied for quantum sensing. Its sensitivity to environmental decoherence was
also reported [66, 67, 68, 21]. The employment in the measurement of
Heisenberg limited sensitivity and quantum parameter estimation in general
have been investigated [69, 70] and compass state in an arrangement with
cavity QED has also been proposed [71]. Such structures have also become
evident in diverse physical contexts: fractional revivals in molecular wave
packet [20], entangled cat state [72], optical fibers [73], Kirkwood-Rihaczek
distribution [74] _etc_. It is remarkable that one can find interference
fringes and negative values of the Wigner function even in a highly mixed
state, making them also useful for continuous variable quantum computation
[83, 84].

###  IV.2 Wigner Function of Parameter-engineered EFSC

Here, we investigate the phase space corresponding to the general quantum
state in Eq. (12), so that we can incorporate desired tuning to implement
entangled state engineering. In all the figures, we have depicted the reduced
phase space of the first cavity field, where the parameters of the second
cavity also control the state due to their nonseparable mixture. The same
analysis is straightforward for the second field in the similar manner. The
tunability is incorporated here by changing the coherent field amplitudes;
$\chi_{1}t,\;\chi_{2}t$; and all of their asymmetric combinations for both
real and imaginary domains of the coherent state parameters. We will
demonstrate one of the tunability conditions for quantum state engineering,
i.e., by changing $\chi_{1}t$ and $\chi_{2}t$, keeping the field amplitudes
fixed ($\alpha_{1}=\alpha_{2}=3$) in our demonstrations. However, it is worth
mentioning that, changing $\alpha_{1}$ and $\alpha_{2}$ will alter the phase
space support.

Here, we will examine how the constituent coherent states’ orientations in one
phase space (the first phase space) are changed by decreasing the values of
$\chi_{1}t$ and $\chi_{2}t$ from $90^{\circ}$ to $0.5^{\circ}$. The squeezing
of the entangled state is reported along with the deformation of their
corresponding sub-Planck structures. We have arbitrarily chosen four states
from Eq. (12), which are the measurement outcomes of the atomic states,
$|g_{1}\big{\rangle}|g_{2}\big{\rangle}|g_{3}\big{\rangle}$,
$|g_{1}\big{\rangle}|e_{2}\big{\rangle}|g_{3}\big{\rangle}$,
$|f_{1}\big{\rangle}|e_{2}\big{\rangle}|g_{3}\big{\rangle}$, and
$|g_{1}\big{\rangle}|g_{2}\big{\rangle}|e_{3}\big{\rangle}$ and are depicted
in Fig.6, Fig.7, Fig.8 and Fig 9, respectively.

Figure 6 displays the maximally entangled state of real and imaginary cats in
first phase space, where $\chi_{1}t$ and $\chi_{2}t$ vary from $90^{\circ}$ to
$0.5^{\circ}$. The state gets gradually deformed from a compass-like state to
a Gaussian-like state. However, the four constituent coherent states always
remain on a circle of radius $\sqrt{2}|\alpha_{1}|$, shown by the dotted
circle and this is valid for all the figures 6-9. The state evolution in Fig.
7 is quite different from the previous plot. The final plot for
$\chi_{1}t=\chi_{2}t=0.5^{\circ}$ is non-Gaussian, but no negative Wigner
value. On the contrary, Fig.8 shows a changing nature of the state from
$\chi_{1}t=\chi_{2}t=20^{\circ}$ (dip at the middle) to
$\chi_{1}t=\chi_{2}t=0.5^{\circ}$ (a Gaussian-like state). The last figure,
Fig 9, manifests somewhat an opposite nature from Fig.7 for the lower
$\chi_{1}t,\;\chi_{2}t$ values.

In all the above state evolutions, the common features with decreasing
$\chi_{1}t,\;\chi_{2}t$ are i) the reduction in the negative phase space
region, signifying less non-classical field state, ii) the lowering of the
classical action, which is given by the effective phase space support, iii)
prominent sub-Planck structures for higher $\chi_{1}t,\;\chi_{2}t$,
particularly near to $\chi_{1}t=\chi_{2}t=\pi/2$, and iv) non-negative Wigner
function for small $\chi_{1}t,\;\chi_{2}t$, signifying a classical-like state.
For $\alpha_{1}=\alpha_{2}=3$, reduced Wigner function of all the eight field
states with $\chi_{1}t=\chi_{2}t=\pi/2$ possess maximum entropy (see Fig. 4)
and take the shape of a compass-like state (see the first phase space plots of
Figs. 6-9), where the central interference pattern associated to the entangled
cat states along two orthogonal directions, persist despite the presence of
quantum correlations between them and responsible for the maximum von-Neumann
entropy.

##  V Conclusion

We provide a scheme to produce entanglement between two field Schrödinger cats
in a cavity QED system. We considered a two cavity setup, where the
correlation between the high-Q cavities is established by injecting three
Rydberg atoms into them and manipulating their state. A family of entangled
states is produced by the scheme, by using the dispersive Jaynes-Cummings
model. Entanglement is then quantified by means of the entanglement entropy,
that is, the von-Neumann entropy of the reduced state. The generated states
show simultaneously two relevant quantum features, Bell-like nonlocal
correlations, and also phase-space interference and Wigner negativities, due
to the properties of the entangled cat states. These quantum features can be
tuned and engineered by changing the dispersive coupling and the amplitude of
the coherent states in the cavities. Quantum phase space is thoroughly
analyzed through Wigner distribution function, including their state
evolutions to address deformation, squeezing and mixing of the entangled
states. The entangled cat state generated here could be potentially used for
quantum technology applications such as quantum sensing of displacements, or
continuous variable quantum computation.

##  VI Acknowledgement

UR acknowledges the support from the project (CRG/2022/007467, SERB, India);
DV acknowledges the support of PNRR MUR project PE0000023-NQSTI (Italy).

## References

  * [1] A. Einstein, B. Podolsky, N. Rosen, Physical Review 47 777 (1935).
  * [2] J. F. Clauser, A. Shimony, Reports on Progress in Physics 41 1881 (1978).
  * [3] A. Aspect, P. Grangier, G. Roger, Physical Review Letters 49 91 (1982).
  * [4] A. Mair, A. Vaziri, G. Weihs, A. Zeilinger, Nature 412 313–316 (2001).
  * [5] D. M. Greenberger, M. A. Horne, A. Shimony, A. Zeilinger, American Journal of Physics 58 1131–1143 (1990).
  * [6] M. Erhard, M. Krenn, A. Zeilinger, Nature Reviews Physics 2 365–381 (2020).
  * [7] N. Shukla, N. Akhtar, B. C. Sanders, Physical Review A 99 063813 (2019).
  * [8] U. Roy, S. Ghosh, P. K. Panigrahi, D. Vitali, Physical Review A 80 052115 (2009).
  * [9] V. Buvzek, A. Vidiella-Barranco, P. L. Knight, Physical Review A 45 6570 (1992).
  * [10] B. C. Sanders, Journal of Physics A: Mathematical and Theoretical 45 244002 (2012).
  * [11] J. Sun, J. Wang, C. Wang, Physical Review A 46 1700 (1992).
  * [12] B. Yurke, D. Stoler, Physical Review Letters 57 13 (1986).
  * [13] G. Agarwal, R. Simon, Optics Communications 92 105–107 (1992).
  * [14] A. I. Solomon, Physics Letters A 196 29–34 (1994).
  * [15] E. P. Wigner, in: Part I: Physical Chemistry. Part II: Solid State Physics, Springer, pp. 110–120 (1997).
  * [16] M. Hillery, R. F. O’Connell, M. O. Scully, E. P. Wigner, Physics Reports 106 121–167 (1984).
  * [17] A. Kenfack, K. Zyczkowski, Journal of Optics B: Quantum and Semiclassical Optics 6 396 (2004).
  * [18] S. J. van Enk, O. Hirota, Physical Review A 64 022313 (2001).
  * [19] D. Horoshko, G. Patera, M. Kolobov, Optics Communications 447 67–73 (2019).
  * [20] S. Ghosh, A. Chiruvelli, J. Banerji, P. Panigrahi, Physical Review A 73 013411 (2006).
  * [21] S. Ghosh, U. Roy, C. Genes, D. Vitali, Physical Review A 79 052104 (2009).
  * [22] S. Ghosh, U. Roy, Physical Review A 90 022113 (2014).
  * [23] S. Ghosh, J. Bera, P. K. Panigrahi, U. Roy, International Journal of Quantum Information 17 1950019 (2019).
  * [24] N. Akhtar, B. C. Sanders, C. Navarrete-Benlloch, Physical Review A 103 053711 (2021).
  * [25] G. Agarwal, L. Davidovich, Physical Review Research 4 L012014 (2022).
  * [26] J. Bera, S. Ghosh, L. Salasnich, U. Roy, Physical Review A 102 063323 (2020).
  * [27] J. Bera, B. Halder, S. Ghosh, R.-K. Lee, U. Roy, Physics Letters A 453 128484 (2022).
  * [28] J. Joo, W. J. Munro, T. P. Spiller, Physical Review Letters 107 083601 (2011).
  * [29] P. Cappellaro, J. Emerson, N. Boulant, C. Ramanathan, S. Lloyd, D. G. Cory, Physical Review Letters 94 020502 (2005).
  * [30] R. Demkowicz-Dobrza´nski, L. Maccone, Physical Review Letters 113 250801 (2014).
  * [31] Z. Huang, C. Macchiavello, L. Maccone, Physical Review A 94 012101 (2016).
  * [32] R. Cleve, H. Buhrman, Physical Review A 56 1201 (1997).
  * [33] K. Bostr¨om, T. Felbinger, Physical Review Letters 89 187902 (2002).
  * [34] M. B. Hastings, Nature Physics 5 255–257 (2009).
  * [35] K. N. Wilkinson, P. Papanastasiou, C. Ottaviani, T. Gehring, S. Pirandola, Physical Review Research 2 033424 (2020).
  * [36] W. Tittel, J. Brendel, H. Zbinden, N. Gisin, Physical Review Letters 84 4737 (2000).
  * [37] J. Yin, Y.-H. Li, S.-K. Liao, M. Yang, Y. Cao, L. Zhang, J.-G. Ren, W.-Q. Cai, W.-Y. Liu, S.-L. Li, et al., Nature 582 501–505 (2020).
  * [38] T. Jennewein, C. Simon, G. Weihs, H. Weinfurter, A. Zeilinger, Physical Review Letters 84 4729 (2000).
  * [39] M. Zidan, Modern Physics Letters B 34 2050401 (2020).
  * [40] Q. Li, W. H. Chan, C. Wu, Z. Wen, Physical Review A 89 040302 (2014).
  * [41] S. Ding, Z. Jin, Chinese Science Bulletin 52 2161–2166 (2007).
  * [42] C. Ottaviani and D. Vitali, Phys. Rev. A 82, 012319 (2010).
  * [43] J. Guillaud, J. Cohen, and M. Mirrahimi, SciPost Phys. Lect. Notes 72 (2023)
  * [44] H.-K. Lo, S. Popescu, Physical Review A 63 022301 (2001).
  * [45] W. Dur, G. Vidal, J. I. Cirac, Physical Review A 62 062314 (2000).
  * [46] A. Acın, D. Bruß, M. Lewenstein, A. Sanpera, Physical Review Letters 87 040401 (2001).
  * [47] M. A. Nielsen, Reports on Mathematical Physics 57 147–161 (2006).
  * [48] P. Walther, M. Aspelmeyer, K. J. Resch, A. Zeilinger, Physical Review Letters 95 020403 (2005) .
  * [49] H. J. Briegel, R. Raussendorf, Physical Review Letters 86 910 (2001).
  * [50] S. Haroche, D. Kleppner, Phys. Today 42 24–30 (1989).
  * [51] D. Wang, H. Kelkar, D. Martin-Cano, D. Rattenbacher, A. Shkarin, T. Utikal, S. G¨otzinger, V. Sandoghdar, Nature Physics 15 483–489 (2019).
  * [52] L. Davidovich, M. Brune, J. Raimond, S. Haroche, Physical Review A 53 1295 (1996).
  * [53] M. Brune, E. Hagley, J. Dreyer, X. Maitre, A. Maali, C. Wunderlich, J. Raimond, S. Haroche, Physical Review Letters 77 4887 (1996).
  * [54] J. Raimond, M. Brune, S. Haroche, Physical Review Letters 79 1964 (1997).
  * [55] J.-M. Raimond, M. Brune, S. Haroche, Reviews of Modern Physics 73 565 (2001).
  * [56] G. S. Agarwal, R. Puri, R. Singh, Physical Review A 56 2249 (1997).
  * [57] C. C. Gerry, Physical Review A 53 3818 (1996).
  * [58] A. Montina, F. Arecchi, Physical Review A 66 013605 (2002) .
  * [59] E. Solano, G. S. Agarwal, H. Walther, Optics and Spectroscopy 94 805–807 (2003).
  * [60] C.-L. Chai, Physical Review A 46 7187 (1992).
  * [61] D. Mogilevtsev, S. Y. Kilin, Optics Communications 132 452–456 (1996).
  * [62] D. Vitali, M. Fortunat, P. Tombesi, F. De Martini, Fortschritte der Physik: Progress of Physics 48 437–446 (2000).
  * [63] L. A. Kanari-Naish, J. Clarke, S. Qvarfort, M. R. Vanner, Quantum Science and Technology 7 035012 (2022).
  * [64] J. Huang, Y.-H. Liu, J.-F. Huang, J.-Q. Liao, Physical Review A 101 043841 (2020).
  * [65] W. H. Zurek, Nature 412 712–717 (2001).
  * [66] P. Jacquod, I. Adagideli, C. W. Beenakker, Physical Review Letters 89 154103 (2002).
  * [67] D. A. Wisniacki, Physical Review E 67 016205 (2003).
  * [68] P. Pathak, G. Agarwal, Physical Review A 71 043823 (2005).
  * [69] F. Toscano, D. A. Dalvit, L. Davidovich, W. H. Zurek, Physical Review A 73 023803 (2006).
  * [70] D. Dalvit, R. de Matos Filho, F. Toscano, New Journal of Physics 8 276 (2006).
  * [71] G. Agarwal, P. Pathak, Physical Review A 70 053813 (2004).
  * [72] J. R. Bhatt, P. K. Panigrahi, M. Vyas, Physical Review A 78 034101 (2008).
  * [73] M. Stobinska, G. Milburn, K. W´odkiewicz, Physical Review A 78 013810 (2008).
  * [74] J. Banerji, Contemporary Physics 48 157–171 (2007).
  * [75] O. Hirota, S. J. Van Enk, K. Nakamura, M. Sohma, K. Kato, arXiv preprint quant-ph/0101096 (2001).
  * [76] O. Hirota, M. Sasaki, Quantum Communication, Computing, and Measurement 3 359–366 (2002).
  * [77] M. A. Ahmad, L. Shu-Tian, Chinese Physics Letters 23 2964 (2006).
  * [78] R. Zeng, M. A. Ahmad, S. Liu, Optics Communications 271 162–168 (2007).
  * [79] S. H. Bukhari, S. N. Khan, M. A. Ahmad, Acta Phys. Pol. B 42 2077–2086 (2011).
  * [80] O. Hirota, Quantum Measurements and Quantum Metrology 4 70–73 (2017).
  * [81] K. Kato, in: Quantum Communications and Quantum Imaging XIII, volume 9615, SPIE, pp. 65–74
  * [82] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Rev. Mod. Phys. 81, 865 (2009).
  * [83] A. Mari, and J. Eisert, Phys. Rev. Lett. 109, 230503 (2012).
  * [84] M. Walschaers, Quantum 7, 1038 (2023).

##  VII Appendix-I

As mentioned, atom-B after passing through $R_{5}$ (a $\pi/2$-pulse) generates
a combined entangled state of
$atom_{A}$-$atom_{B}$-$field_{C_{1}}$-$field_{C_{2}}$, where none of the atoms
and fields are separable:

$\displaystyle|\Psi^{AB}_{R_{5}}\big{\rangle}=$
$\displaystyle\frac{1}{4}\Bigr{[}|g_{1}\big{\rangle}|g_{2}\big{\rangle}\Bigr{(}\tilde{\alpha_{1}}+e^{-i\chi_{1}t}\tilde{\alpha_{3}}\Bigr{)}+e^{i\delta}|g_{1}\big{\rangle}|e_{2}\big{\rangle}\Bigr{(}\tilde{\alpha_{1}}-e^{-i\chi_{1}t}\tilde{\alpha_{3}}\Bigr{)}$
$\displaystyle-e^{i\delta}|f_{1}\big{\rangle}|g_{2}\big{\rangle}\Bigr{(}\tilde{\alpha_{2}}+e^{-i\chi_{1}t}\tilde{\alpha_{4}}\Bigr{)}-e^{2i\delta}|f_{1}\big{\rangle}|e_{2}\big{\rangle}\Bigr{(}\tilde{\alpha_{2}}-e^{-i\chi_{1}t}\tilde{\alpha_{4}}\Bigr{)}\Bigr{]}.$

Here, the field entangled pairs are collected in the following terms:

$\displaystyle\tilde{\alpha_{1}}$
$\displaystyle=\Bigr{\\{}|\alpha_{1}e^{2i\chi_{1}t}\big{\rangle}|\alpha_{2}\big{\rangle}-|\alpha_{1}e^{i\chi_{1}t}\big{\rangle}|\alpha_{2}e^{i\chi_{2}t}\big{\rangle}\Bigr{\\}},$
$\displaystyle\tilde{\alpha_{2}}$
$\displaystyle=\Bigr{\\{}|\alpha_{1}e^{2i\chi_{1}t}\big{\rangle}|\alpha_{2}\big{\rangle}+|\alpha_{1}e^{i\chi_{1}t}\big{\rangle}|\alpha_{2}e^{i\chi_{2}t}\big{\rangle}\Bigr{\\}},$
$\displaystyle\tilde{\alpha_{3}}$
$\displaystyle=\Bigr{\\{}|\alpha_{1}\big{\rangle}|\alpha_{2}\big{\rangle}-|\alpha_{1}e^{-i\chi_{1}t}\big{\rangle}|\alpha_{2}e^{i\chi_{2}t}\big{\rangle}\Bigr{\\}},$
$\displaystyle\tilde{\alpha_{4}}$
$\displaystyle=\Bigr{\\{}|\alpha_{1}\big{\rangle}|\alpha_{2}\big{\rangle}-|\alpha_{1}e^{-i\chi_{1}t}\big{\rangle}|\alpha_{2}e^{i\chi_{2}t}\big{\rangle}\Bigr{\\}}.$

Cavity-$C_{1}$ does not interact with atom-C, but cavity-$C_{2}$, Ramsey zones
$R_{6}$ and $R_{7}$ (both as $\pi/2$-pulse) operate. Using the standard
operations described earlier, one will be able to derive the final quantum
state of the whole setup as follows

$\displaystyle|\Psi_{Final}\big{\rangle}$
$\displaystyle=\frac{1}{8}\Bigg{[}|g_{1}\big{\rangle}|g_{2}\big{\rangle}|g_{3}\big{\rangle}\Bigr{\\{}{\alpha_{1}}^{\prime}+e^{-i\chi_{1}t}{\alpha_{3}}^{\prime}+{\alpha_{1}}^{\prime\prime}+e^{-i\chi_{1}t}{\alpha_{3}}^{\prime\prime}\Bigr{\\}}$
(12)
$\displaystyle+e^{i\delta}|g_{1}\big{\rangle}|e_{2}\big{\rangle}|g_{3}\big{\rangle}\Bigr{\\{}{\alpha_{1}}^{\prime}-e^{-i\chi_{1}t}{\alpha_{3}}^{\prime}+{\alpha_{1}}^{\prime\prime}-e^{-i\chi_{1}t}{\alpha_{3}}^{\prime\prime}\Bigr{\\}}$
$\displaystyle-e^{i\delta}|f_{1}\big{\rangle}|g_{2}\big{\rangle}|g_{3}\big{\rangle}\Bigr{\\{}{\alpha_{2}}^{\prime}+e^{-i\chi_{1}t}{\alpha_{4}}^{\prime}+{\alpha_{2}}^{\prime\prime}+e^{-i\chi_{1}t}{\alpha_{4}}^{\prime\prime}\Bigr{\\}}$
$\displaystyle-e^{2i\delta}|f_{1}\big{\rangle}|e_{2}\big{\rangle}|g_{3}\big{\rangle}\Bigr{\\{}{\alpha_{2}}^{\prime}-e^{-i\chi_{1}t}{\alpha_{4}}^{\prime}+{\alpha_{2}}^{\prime\prime}-e^{-i\chi_{1}t}{\alpha_{4}}^{\prime\prime}\Bigr{\\}}$
$\displaystyle+e^{i\delta}|g_{1}\big{\rangle}|g_{2}\big{\rangle}|e_{3}\big{\rangle}\Bigr{\\{}{\alpha_{1}}^{\prime}+e^{-i\chi_{1}t}{\alpha_{3}}^{\prime}-{\alpha_{1}}^{\prime\prime}-e^{-i\chi_{1}t}{\alpha_{3}}^{\prime\prime}\Bigr{\\}}$
$\displaystyle+e^{2i\delta}|g_{1}\big{\rangle}|e_{2}\big{\rangle}|e_{3}\big{\rangle}\Bigr{\\{}{\alpha_{1}}^{\prime}-e^{-i\chi_{1}t}{\alpha_{3}}^{\prime}-{\alpha_{1}}^{\prime\prime}+e^{-i\chi_{1}t}{\alpha_{3}}^{\prime\prime}\Bigr{\\}}$
$\displaystyle-e^{2i\delta}|f_{1}\big{\rangle}|g_{2}\big{\rangle}|e_{3}\big{\rangle}\Bigr{\\{}{\alpha_{2}}^{\prime}+e^{-i\chi_{1}t}{\alpha_{4}}^{\prime}-{\alpha_{2}}^{\prime\prime}-e^{-i\chi_{1}t}{\alpha_{4}}^{\prime\prime}\Bigr{\\}}$
$\displaystyle-e^{3i\delta}|f_{1}\big{\rangle}|e_{2}\big{\rangle}|e_{3}\big{\rangle}\Bigr{\\{}{\alpha_{2}}^{\prime}-e^{-i\chi_{1}t}{\alpha_{4}}^{\prime}-{\alpha_{2}}^{\prime\prime}+e^{-i\chi_{1}t}{\alpha_{4}}^{\prime\prime}\Bigr{\\}}\Bigg{]},$

which has total $64$ terms, each being the tensor product of five states
(three atoms and two fields) with the following coefficients of the entangled
fields:

$\displaystyle{\alpha_{1}}^{\prime}$ $\displaystyle=$
$\displaystyle\Bigr{\\{}|\alpha_{1}e^{2i\chi_{1}t}\big{\rangle}|\alpha_{2}e^{2i\chi_{2}t}\big{\rangle}-|\alpha_{1}e^{i\chi_{1}t}\big{\rangle}|\alpha_{2}e^{i\chi_{2}t}\big{\rangle}\Bigr{\\}},$
$\displaystyle{\alpha_{2}}^{\prime}$ $\displaystyle=$
$\displaystyle\Bigr{\\{}|\alpha_{1}e^{2i\chi_{1}t}\big{\rangle}|\alpha_{2}e^{i\chi_{2}t}\big{\rangle}+|\alpha_{1}e^{i\chi_{1}t}\big{\rangle}|\alpha_{2}e^{2i\chi_{2}t}\big{\rangle}\Bigr{\\}},$
$\displaystyle{\alpha_{3}}^{\prime}$ $\displaystyle=$
$\displaystyle\Bigr{\\{}|\alpha_{1}\big{\rangle}|\alpha_{2}e^{i\chi_{2}t}\big{\rangle}-|\alpha_{1}e^{-i\chi_{1}t}\big{\rangle}|\alpha_{2}e^{2i\chi_{2}t}\big{\rangle}\Bigr{\\}},$
$\displaystyle{\alpha_{4}}^{\prime}$ $\displaystyle=$
$\displaystyle\Bigr{\\{}|\alpha_{1}\big{\rangle}|\alpha_{2}e^{i\chi_{2}t}\big{\rangle}+|\alpha_{1}e^{-i\chi_{1}t}\big{\rangle}|\alpha_{2}e^{2i\chi_{2}t}\big{\rangle}\Bigr{\\}},$
$\displaystyle{\alpha_{1}}^{\prime\prime}$ $\displaystyle=$ $\displaystyle
e^{-i\chi_{2}t}\Bigr{\\{}|\alpha_{1}e^{2i\chi_{1}t}\big{\rangle}|\alpha_{2}e^{-i\chi_{2}t}\big{\rangle}-|\alpha_{1}e^{i\chi_{1}t}\big{\rangle}|\alpha_{2}\big{\rangle}\Bigr{\\}},$
$\displaystyle{\alpha_{2}}^{\prime\prime}$ $\displaystyle=$ $\displaystyle
e^{-i\chi_{2}t}\Bigr{\\{}|\alpha_{1}e^{2i\chi_{1}t}\big{\rangle}|\alpha_{2}e^{-i\chi_{2}t}\big{\rangle}+|\alpha_{1}e^{i\chi_{1}t}\big{\rangle}|\alpha_{2}\big{\rangle}\Bigr{\\}},$
$\displaystyle{\alpha_{3}}^{\prime\prime}$ $\displaystyle=$ $\displaystyle
e^{-i\chi_{2}t}\Bigr{\\{}|\alpha_{1}\big{\rangle}|\alpha_{2}e^{-i\chi_{2}t}\big{\rangle}-|\alpha_{1}e^{-i\chi_{1}t}\big{\rangle}|\alpha_{2}\big{\rangle}\Bigr{\\}},$
$\displaystyle{\alpha_{4}}^{\prime\prime}$ $\displaystyle=$ $\displaystyle
e^{-i\chi_{2}t}\Bigr{\\{}|\alpha_{1}\big{\rangle}|\alpha_{2}e^{-i\chi_{2}t}\big{\rangle}+|\alpha_{1}e^{-i\chi_{1}t}\big{\rangle}|\alpha_{2}\big{\rangle}\Bigr{\\}}.$

Table 1 lists the atomic state and field state separately, which will offer
the entangled superposed field states based on a chosen atomic measurement
from eight combinations. From these states, we obtain the entangled states of
maximally nonlocal Schrödinger cats of both the cavities with
$\chi_{1}t=\chi_{1}t=\pi/2$ from Table 1. We use these specific case and
obtain eight entangled states of horizontal and vertical Schrödinger cats:

$\displaystyle|\psi_{1}\big{\rangle}$
$\displaystyle=(-i)\frac{N_{1}}{8}\bigg{[}\Bigr{(}|\alpha_{1}\big{\rangle}+i|-\alpha_{1}\big{\rangle}\Bigr{)}\Bigr{(}|i\alpha_{2}\big{\rangle}-i|-i\alpha_{2}\big{\rangle}\Bigr{)}$
(13)
$\displaystyle-\Bigr{(}|i\alpha_{1}\big{\rangle}-i|-i\alpha_{1}\big{\rangle}\Bigr{)}\Bigr{(}|\alpha_{2}\big{\rangle}+i|-\alpha_{2}\big{\rangle}\Bigr{)}\bigg{]},$
$\displaystyle|\psi_{2}\big{\rangle}$
$\displaystyle=(i)\frac{N_{2}}{8}\bigg{[}\Bigr{(}|\alpha_{1}\big{\rangle}-i|-\alpha_{1}\big{\rangle}\Bigr{)}\Bigr{(}|i\alpha_{2}\big{\rangle}-i|-i\alpha_{2}\big{\rangle}\Bigr{)}$
(14)
$\displaystyle+\Bigr{(}|i\alpha_{1}\big{\rangle}+i|-i\alpha_{1}\big{\rangle}\Bigr{)}\Bigr{(}|\alpha_{2}\big{\rangle}+i|-\alpha_{2}\big{\rangle}\Bigr{)}\bigg{]},$
$\displaystyle|\psi_{3}\big{\rangle}$
$\displaystyle=(i)\frac{N_{3}}{8}\bigg{[}\Bigr{(}|\alpha_{1}\big{\rangle}+i|-\alpha_{1}\big{\rangle}\Bigr{)}\Bigr{(}|i\alpha_{2}\big{\rangle}-i|-i\alpha_{2}\big{\rangle}\Bigr{)}$
(15)
$\displaystyle+\Bigr{(}|i\alpha_{1}\big{\rangle}-i|-i\alpha_{1}\big{\rangle}\Bigr{)}\Bigr{(}|\alpha_{2}\big{\rangle}+i|-\alpha_{2}\big{\rangle}\Bigr{)}\bigg{]},$
$\displaystyle|\psi_{4}\big{\rangle}$
$\displaystyle=(-i)\frac{N_{4}}{8}\bigg{[}\Bigr{(}|\alpha_{1}\big{\rangle}-i|-\alpha_{1}\big{\rangle}\Bigr{)}\Bigr{(}|i\alpha_{2}\big{\rangle}-i|-i\alpha_{2}\big{\rangle}\Bigr{)}$
(16)
$\displaystyle-\Bigr{(}|i\alpha_{1}\big{\rangle}+i|-i\alpha_{1}\big{\rangle}\Bigr{)}\Bigr{(}|\alpha_{2}\big{\rangle}+i|-\alpha_{2}\big{\rangle}\Bigr{)}\bigg{]},$
$\displaystyle|\psi_{5}\big{\rangle}$
$\displaystyle=(-i)\frac{N_{5}}{8}\bigg{[}\Bigr{(}|\alpha_{1}\big{\rangle}+i|-\alpha_{1}\big{\rangle}\Bigr{)}\Bigr{(}|i\alpha_{2}\big{\rangle}+i|-i\alpha_{2}\big{\rangle}\Bigr{)}$
(17)
$\displaystyle+\Bigr{(}|i\alpha_{1}\big{\rangle}-i|-i\alpha_{1}\big{\rangle}\Bigr{)}\Bigr{(}|\alpha_{2}\big{\rangle}-i|-\alpha_{2}\big{\rangle}\Bigr{)}\bigg{]},$
$\displaystyle|\psi_{6}\big{\rangle}$
$\displaystyle=(i)\frac{N_{6}}{8}\bigg{[}\Bigr{(}|\alpha_{1}\big{\rangle}-i|-\alpha_{1}\big{\rangle}\Bigr{)}\Bigr{(}|i\alpha_{2}\big{\rangle}+i|-i\alpha_{2}\big{\rangle}\Bigr{)}$
(18)
$\displaystyle-\Bigr{(}|i\alpha_{1}\big{\rangle}+i|-i\alpha_{1}\big{\rangle}\Bigr{)}\Bigr{(}|\alpha_{2}\big{\rangle}-i|-\alpha_{2}\big{\rangle}\Bigr{)}\bigg{]},$
$\displaystyle|\psi_{7}\big{\rangle}$
$\displaystyle=(i)\frac{N_{7}}{8}\bigg{[}\Bigr{(}|\alpha_{1}\big{\rangle}+i|-\alpha_{1}\big{\rangle}\Bigr{)}\Bigr{(}|i\alpha_{2}\big{\rangle}+i|-i\alpha_{2}\big{\rangle}\Bigr{)}$
(19)
$\displaystyle-\Bigr{(}|i\alpha_{1}\big{\rangle}-i|-i\alpha_{1}\big{\rangle}\Bigr{)}\Bigr{(}|\alpha_{2}\big{\rangle}-i|-\alpha_{2}\big{\rangle}\Bigr{)}\bigg{]},$
$\displaystyle|\psi_{8}\big{\rangle}$
$\displaystyle=(-i)\frac{N_{8}}{8}\bigg{[}\Bigr{(}|\alpha_{1}\big{\rangle}-i|-\alpha_{1}\big{\rangle}\Bigr{)}\Bigr{(}|i\alpha_{2}\big{\rangle}+i|-i\alpha_{2}\big{\rangle}\Bigr{)}$
(20)
$\displaystyle+\Bigr{(}|i\alpha_{1}\big{\rangle}+i|-i\alpha_{1}\big{\rangle}\Bigr{)}\Bigr{(}|\alpha_{2}\big{\rangle}-i|-\alpha_{2}\big{\rangle}\Bigr{)}\bigg{]}.$

All of these states involve the entanglement between a position cat and a
momentum cat.

##  VIII Appendix-II

The Wigner function for the quantum state as given in Eq. (10) is written as

$\displaystyle
W(x_{1},p_{1};x_{2},p_{2})=\frac{e^{-\big{(}|\alpha_{1}|^{2}+|\alpha_{2}|^{2}+x_{1}^{2}+x_{2}^{2}+p_{1}^{2}+p_{2}^{2}\big{)}}}{64{\pi}^{2}}\times\bigg{[}W_{D1}+W_{D2}+W_{OD1}+W_{OD2}\bigg{]}.$

where $W_{D1}$, $W_{D2}$ are diagonal terms and $W_{OD1}$, $W_{OD2}$ are off-
diagonal components, signifying interferences. Their complete expressions are
derived to give the terms as follows.

$\displaystyle W_{D1}=$
$\displaystyle\frac{|A|^{2}}{(1+e^{-2|\alpha_{1}|^{2}}\cos{\xi_{1}})(1+e^{-2|\alpha_{2}|^{2}}\cos{\zeta_{2}})}$
$\displaystyle\bigg{[}e^{-|\alpha_{1}|^{2}}\cos\big{\\{}\sqrt{2}ix_{1}\big{(}\alpha_{1}e^{i\theta_{1}}+\alpha^{\ast}_{1}e^{-i\theta_{1}}\big{)}+\sqrt{2}p_{1}\big{(}\alpha_{1}e^{i\theta_{1}}-\alpha^{\ast}_{1}e^{-i\theta_{1}}\big{)}\big{\\}}$
$\displaystyle+e^{|\alpha_{1}|^{2}}\cos\big{\\{}\xi_{1}+\sqrt{2}ix_{1}\big{(}\alpha_{1}e^{i\theta_{1}}-\alpha^{\ast}_{1}e^{-i\theta_{1}}\big{)}+\sqrt{2}p_{1}\big{(}\alpha_{1}e^{i\theta_{1}}+\alpha^{\ast}_{1}e^{-i\theta_{1}}\big{)}\big{\\}}\bigg{]}$
$\displaystyle\bigg{[}e^{-|\alpha_{2}|^{2}}\cos\big{\\{}\sqrt{2}ix_{2}\big{(}\alpha_{2}e^{i\phi_{2}}+\alpha^{\ast}_{2}e^{-i\phi_{2}}\big{)}+\sqrt{2}p_{2}\big{(}\alpha_{2}e^{i\phi_{2}}-\alpha^{\ast}_{2}e^{-i\phi_{2}}\big{)}\big{\\}}$
$\displaystyle+e^{|\alpha_{2}|^{2}}\cos\big{\\{}\zeta_{2}+\sqrt{2}ix_{2}\big{(}\alpha_{2}e^{i\phi_{2}}-\alpha^{\ast}_{2}e^{-i\phi_{2}}\big{)}+\sqrt{2}p_{2}\big{(}\alpha_{2}e^{i\phi_{2}}+\alpha^{\ast}_{2}e^{-i\phi_{2}}\big{)}\big{\\}}\bigg{]},$
$\displaystyle W_{D2}=$
$\displaystyle\frac{|B|^{2}}{(1+e^{-2|\alpha_{1}|^{2}}\cos{\zeta_{1}})(1+e^{-2|\alpha_{2}|^{2}}\cos{\xi_{2}})}$
$\displaystyle\bigg{[}e^{-|\alpha_{1}|^{2}}\cos\big{\\{}\sqrt{2}ix_{1}\big{(}\alpha_{1}e^{i\phi_{1}}+\alpha^{\ast}_{1}e^{-i\phi_{1}}\big{)}+\sqrt{2}p_{1}\big{(}\alpha_{1}e^{i\phi_{1}}-\alpha^{\ast}_{1}e^{-i\phi_{1}}\big{)}\big{\\}}$
$\displaystyle+e^{|\alpha_{1}|^{2}}\cos\big{\\{}\zeta_{1}+\sqrt{2}ix_{1}\big{(}\alpha_{1}e^{i\phi_{1}}-\alpha^{\ast}_{1}e^{-i\phi_{1}}\big{)}+\sqrt{2}p_{1}\big{(}\alpha_{1}e^{i\phi_{1}}+\alpha^{\ast}_{1}e^{-i\phi_{1}}\big{)}\big{\\}}\bigg{]}$
$\displaystyle\bigg{[}e^{-|\alpha_{2}|^{2}}\cos\big{\\{}\sqrt{2}ix_{2}\big{(}\alpha_{2}e^{i\theta_{2}}+\alpha^{\ast}_{2}e^{-i\theta_{2}}\big{)}+\sqrt{2}p_{2}\big{(}\alpha_{2}e^{i\theta_{2}}-\alpha^{\ast}_{2}e^{-i\theta_{2}}\big{)}\big{\\}}$
$\displaystyle+e^{|\alpha_{2}|^{2}}\cos\big{\\{}\xi_{2}+\sqrt{2}ix_{2}\big{(}\alpha_{2}e^{i\theta_{2}}-\alpha^{\ast}_{2}e^{-i\theta_{2}}\big{)}+\sqrt{2}p_{2}\big{(}\alpha_{2}e^{i\theta_{2}}+\alpha^{\ast}_{2}e^{-i\theta_{2}}\big{)}\big{\\}}\bigg{]},$
$\displaystyle W_{OD1}=$ $\displaystyle\frac{(A\times
B^{\ast})exp[\frac{i((\xi_{1}-\xi_{2})-(\zeta_{1}-\zeta_{2}))}{2}]}{\sqrt{(1+e^{-2|\alpha_{1}|^{2}}\cos{\xi_{1}})(1+e^{-2|\alpha_{2}|^{2}}\cos{\zeta_{2}})(1+e^{-2|\alpha_{1}|^{2}}\cos{\zeta_{1}})(1+e^{-2|\alpha_{2}|^{2}}\cos{\xi_{2}})}}$
$\displaystyle\bigg{[}e^{-|\alpha_{1}|^{2}e^{i(\theta_{1}-\phi_{1})}}\cos\big{\\{}\frac{(\zeta_{1}-\xi_{1})}{2}-\sqrt{2}ix_{1}\big{(}\alpha_{1}e^{i\theta_{1}}+\alpha^{\ast}_{1}e^{-i\phi_{1}}\big{)}-\sqrt{2}p_{1}\big{(}\alpha_{1}e^{i\theta_{1}}-\alpha^{\ast}_{1}e^{-i\phi_{1}}\big{)}\big{\\}}$
$\displaystyle+e^{|\alpha_{1}|^{2}e^{i(\theta_{1}-\phi_{1})}}\cos\big{\\{}\frac{(\zeta_{1}+\xi_{1})}{2}+\sqrt{2}ix_{1}\big{(}\alpha_{1}e^{i\theta_{1}}-\alpha^{\ast}_{1}e^{-i\phi_{1}}\big{)}+\sqrt{2}p_{1}\big{(}\alpha_{1}e^{i\theta_{1}}+\alpha^{\ast}_{1}e^{-i\phi_{1}}\big{)}\big{\\}}\bigg{]}$
$\displaystyle\bigg{[}e^{-|\alpha_{2}|^{2}e^{i(\phi_{2}-\theta_{2})}}\cos\big{\\{}\frac{(\zeta_{2}-\xi_{2})}{2}+\sqrt{2}ix_{2}\big{(}\alpha_{2}e^{i\phi_{2}}+\alpha^{\ast}_{2}e^{-i\theta_{2}}\big{)}+\sqrt{2}p_{2}\big{(}\alpha_{2}e^{i\phi_{2}}-\alpha^{\ast}_{2}e^{-i\theta_{2}}\big{)}\big{\\}}$
$\displaystyle+e^{|\alpha_{2}|^{2}e^{i(\phi_{2}-\theta_{2})}}\cos\big{\\{}\frac{(\zeta_{2}+\xi_{2})}{2}+\sqrt{2}ix_{2}\big{(}\alpha_{2}e^{i\phi_{2}}-\alpha^{\ast}_{2}e^{-i\theta_{2}}\big{)}+\sqrt{2}p_{2}\big{(}\alpha_{2}e^{i\phi_{2}}+\alpha^{\ast}_{2}e^{-i\theta_{2}}\big{)}\big{\\}}\bigg{]},$

and

$\displaystyle W_{OD2}=$ $\displaystyle\frac{(A^{\ast}\times
B)exp[\frac{i((\zeta_{1}-\zeta_{2})-(\xi_{1}-\xi_{2}))}{2}]}{\sqrt{(1+e^{-2|\alpha_{1}|^{2}}\cos{\xi_{1}})(1+e^{-2|\alpha_{2}|^{2}}\cos{\zeta_{2}})(1+e^{-2|\alpha_{1}|^{2}}\cos{\zeta_{1}})(1+e^{-2|\alpha_{2}|^{2}}\cos{\xi_{2}})}}$
$\displaystyle\bigg{[}e^{-|\alpha_{1}|^{2}e^{i(\phi_{1}-\theta_{1})}}\cos\big{\\{}\frac{(\xi_{1}-\zeta_{1})}{2}-\sqrt{2}ix_{1}\big{(}\alpha_{1}e^{i\phi_{1}}+\alpha^{\ast}_{1}e^{-i\theta_{1}}\big{)}-\sqrt{2}p_{1}\big{(}\alpha_{1}e^{i\phi_{1}}-\alpha^{\ast}_{1}e^{-i\theta_{1}}\big{)}\big{\\}}$
$\displaystyle+e^{|\alpha_{1}|^{2}e^{i(\phi_{1}-\theta_{1})}}\cos\big{\\{}\frac{(\xi_{1}+\zeta_{1})}{2}+\sqrt{2}ix_{1}\big{(}\alpha_{1}e^{i\phi_{1}}-\alpha^{\ast}_{1}e^{-i\theta_{1}}\big{)}+\sqrt{2}p_{1}\big{(}\alpha_{1}e^{i\phi_{1}}+\alpha^{\ast}_{1}e^{-i\theta_{1}}\big{)}\big{\\}}\bigg{]}$
$\displaystyle\bigg{[}e^{-|\alpha_{2}|^{2}e^{i(\theta_{2}-\phi_{2})}}\cos\big{\\{}\frac{(\xi_{2}-\zeta_{2})}{2}+\sqrt{2}ix_{2}\big{(}\alpha_{2}e^{i\theta_{2}}+\alpha^{\ast}_{2}e^{-i\phi_{2}}\big{)}+\sqrt{2}p_{2}\big{(}\alpha_{2}e^{i\theta_{2}}-\alpha^{\ast}_{2}e^{-i\phi_{2}}\big{)}\big{\\}}$
$\displaystyle+e^{|\alpha_{2}|^{2}e^{i(\theta_{2}-\phi_{2})}}\cos\big{\\{}\frac{(\xi_{2}+\zeta_{2})}{2}+\sqrt{2}ix_{2}\big{(}\alpha_{2}e^{i\theta_{2}}-\alpha^{\ast}_{2}e^{-i\phi_{2}}\big{)}+\sqrt{2}p_{2}\big{(}\alpha_{2}e^{i\theta_{2}}+\alpha^{\ast}_{2}e^{-i\phi_{2}}\big{)}\big{\\}}\bigg{]}.$