# High–N00N State Generation: N00N State Output of Floquet Engineering

Yusef Maleki
Institute of Quantum Science and Engineering, Department of Physics and
Astronomy,
College Station, Texas 77843-4242, USA E-mail<EMAIL_ADDRESS>
Here, we review some quantum architectures designed for the engineering of the
N00N state, a bipartite maximally entangled state crucial in quantum metrology
applications. The fundamental concept underlying these schemes is the
transformation of the initial state $|N\rangle\otimes|0\rangle$ to the N00N
state
$\frac{1}{\sqrt{2}}(|N\rangle\otimes|0\rangle+|0\rangle\otimes|N\rangle)$,
where $|N\rangle$ and $|0\rangle$ are the Fock states with $N$ and $0$
excitations. We show that this state can be generated as a superposition of
modes of quantum light, a combination of light and motion, or a superposition
of two spin ensembles. The approach discussed here can generate mesoscopic and
macroscopic entangled states, such as entangled coherent and squeezed states,
as well. We show that a large class of maximally entangled states can be
achieved in such an architecture. The extension of these state engineering
methods to the multi-mode setting is also discussed.

##  I. Introduction

Estimating unknown parameters stands as a pivotal task across diverse
scientific disciplines, playing an important role in technological
advancements and scientific discoveries [1, 2, 3, 4, 5, 6, 7, 8]. The task of
attaining the highest possible accuracy in parameter estimation is a
fundamental objective within the field of metrology [9, 10, 11, 12]. One
effective strategy for enhancing estimation accuracy involves the repetition
of experiments and the collection of additional information on the parameter.
Specifically, utilizing $N$ independent resources for measuring the parameter
$\varphi$, the optimal sensitivity of the parameter is determined by the
central limit theorem, scaling as $\Delta\varphi\propto 1/{\sqrt{N}}$. This
phenomenon is commonly referred to as the shot-noise limit or standard quantum
limit [13, 14, 15].

In the realm of parameter estimation, quantum physics introduces an intriguing
capability that remains beyond the reach of classical methodologies [16, 17,
18, 19]. In fact, quantum physics has the ability to push the boundaries of
precision, enabling measurements that were once deemed impossible within
classical frameworks [20, 21, 22, 23]. Quantum-enhanced precision opens the
door to unprecedented accuracy, revolutionizing fields of science and
technology where extreme accuracy of the measurement is the key [24, 25]. The
exploration of quantum capabilities in parameter estimation has become a focal
point of research, promising breakthroughs that can change our understanding
of measurement precision.

Therefore, quantum resources offer the potential for a substantial enhancement
in precision, surpassing classical limits and approaching the fundamental
bounds dictated by quantum mechanics. This quantum-enhanced precision can go
beyond the classical limit and reach the Heisenberg Limit (HL), where the
estimation error $\Delta\varphi$ scales as $1/{N}$ for a given number of the
resource $N$ [26]. The noteworthy quantum state that can achieve this limit is
the N00N state, represented as
$\frac{1}{\sqrt{2}}(|N\rangle\otimes|0\rangle+|0\rangle\otimes|N\rangle)$ .
This state, renowned for its potential to fulfill Heisenberg limit precision,
holds immense promise for quantum-enhanced metrology [26].

At the heart of this quantum advantage provided by the N00N state is the
quantum entanglement manifested by such a state [26, 27]. Quantum
entanglement, defying classical intuition, showcases the unique properties of
quantum systems, which can serve as the resource for various quantum
applications ranging from quantum computing to quantum communication and
beyond [28, 29]. In recent years, there has been a considerable effort to
utilize entangled states for quantum sensing and metrology.

However, the realization of the N00N state poses a considerable challenge,
restricted by difficulties in generation and susceptibility to decoherence
[26]. Of course, such difficulty is not limited to the N00N state generation
and remains a challenge within quantum science, in general, [30, 31, 32, 33,
34, 35]. The difficulty of generating entangled states in quantum systems is a
challenge that researchers are actively addressing. The inherent difficulty in
achieving the desired entangled states reflects broader challenges in
manipulating quantum states for practical applications. Overcoming these
challenges is essential for harnessing the full potential of quantum-enhanced
precision in parameter estimation scenarios.

This review considers intricate quantum architectures designed to engineer
entangled states, focusing on the N00N state generation. We demonstrate the
generation of this state through a superposition of quantum light modes, a
combination of light and motion modes, or a combination of two spin ensemble
states. The method elucidated in this discussion is capable of producing
mesoscopic and macroscopic entangled states, including entangled coherent and
squeezed states. Hence, we show that a broad spectrum of maximally entangled
states can be realized within this architectural framework. The exploration of
these state engineering techniques is extended to the multi-mode setting and
is also thoroughly examined in our discussion.

##  II. Quantum Circuit for N00N state generation

Figure 1: An scheme of the generating N00N state. The system consists of two
cavities interacting with a qubit. The qubit dispersively interacts with the
cavities.

To generate the N00N state, we suppose that we have an interaction Hamiltonian
of two cavity systems with a two-level qubit given as [17, 36]

$H\textsubscript{eff}=i\kappa(a_{1}a^{\dagger}_{2}-a_{2}a^{\dagger}_{1})\sigma_{z},$
(1)

in which the boson annihilation and creation operators of the first and the
second modes are $a_{1},a_{1}^{\dagger}$ and $a_{2},a_{2}^{\dagger}$,
respectively. These operators can be associated with any harmonic oscillator
system, such as photons in the cavity [37, 68]. The qubit operators in this
equation, $\sigma_{z}=|1\rangle\langle 1|-|0\rangle\langle 0|$,
$\sigma_{+}=|1\rangle\langle 0|$ and $\sigma_{-}=|0\rangle\langle 1|$ are the
Pauli operators, that obey su(2) Lei algebra. Therefore, this Hamiltonian can
be considered as a two-level system (qubit) interacting with two bosonic
modes, such as two modes of the light fields.

As was mentioned earlier, central to our approach for the high-N00N state
generation is the derivation of the effective Hamiltonian in Eq. (1). For
simplicity, one can consider two cavity fields interacting with a single qubit
as in Fig. 1. We will provide a detailed derivation of this effective
Hamiltonian later; however, it is stimulating to see how such a Hamiltonian
results in a generation of N00N state, starting from $N$ photons in one of the
resonators. In fact, we are going to show that, starting from this interaction
Hamiltonian, one can generate the N00N state in the two modes. To this aim, we
consider the qubit to be in the logical superposition state
$\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)$. The first and the second field
modes are prepared in the $|0\rangle_{1}$ and $|N\rangle_{2}$ Fock states,
respectively. Therefore, the initial state of the composed system, including
the atom and the cavities, is thus given by

$|\psi(0)\rangle=\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)|0\rangle_{1}|N\rangle_{2}.$

Starting from this initial state, we consider the time evolution of the entire
system with the effective Hamiltonian above. This implies that the time
evolution operator

$U(t)=e^{-{iH\textsubscript{eff}\Delta t}/{\hbar}},$

should be applied to this initial state, from which we find [17]

$|\psi(t)\rangle=\frac{1}{\sqrt{2}}(e^{-{iH\textsubscript{eff}\Delta
t}/{\hbar}}|0,N\rangle|0\rangle+e^{-{iH\textsubscript{eff}\Delta
t}/{\hbar}}|0,N\rangle|1\rangle),$ (2)

where $|0,N\rangle\equiv|0\rangle_{1}|N\rangle_{2}$.

We let the system evolve for the time interval $\Delta t_{1}=\pi/(4\Omega).$
This entails the time evolved state $|\psi(t)\rangle$ to degenerate into

$|\psi\rangle_{1}=\frac{1}{\sqrt{2^{N}}}\sum_{k=1}^{N}\left(\begin{matrix}N\\\
k\end{matrix}\right)^{\frac{1}{2}}|k,N-k\rangle(|0\rangle+(-1)^{k}|1\rangle).$
(3)

We note that in this state, the two fields and the qubit become entangled. The
measurement of the first resonator in the state $|k\rangle$ projects the qubit
in two orthogonal superposition of the computational basis. Namely, detecting
the first resonator in the state with an even $k$ projects the qubit into the
state $\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)$, and with odd $k$ projects the
qubit into the state $\frac{1}{\sqrt{2}}(|0\rangle-|1\rangle)$. These two
basis are orthogonal. Alternatively, detecting the qubit in these two
orthogonal basis provides information about the $k$. Similar argument can be
made for the second mode.

Next, we apply the Hadamard gate to this state. The Hadamard gate is given by
a $2\times 2$ matrix acting on the Hilbert space of a single qubit which is
given by

$\mathcal{H}=\frac{1}{\sqrt{2}}\left(\begin{matrix}1&1\\\
1&-1\end{matrix}\right).$ (4)

The role of this gate is to rotate the basis of the Hilbert space of the
qubit. Hence, applying this gate translates into a local transformation of the
qubit basis such that
$|0\rangle\rightarrow\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)$, and
$|1\rangle\rightarrow\frac{1}{\sqrt{2}}(|0\rangle-|1\rangle)$. Once the qubit
is subjected to the Hadamard gate, we let the system to undergo the time
evolution again within the time interval $\Delta t_{2}=\pi/(4\Omega)$. With
this time evolution the quantum state of the system reduces to
$|\psi\rangle_{2}$, which is explicitly given by

$|\psi\rangle_{2}=\frac{1}{\sqrt{2^{N}}}\sum_{k=1}^{N}\left(\begin{matrix}N\\\
k\end{matrix}\right)^{\frac{1}{2}}[1+(-1)^{k}]|k,N-k\rangle|0\rangle+\frac{1}{\sqrt{2^{N}}}\sum_{k=1}^{N}\left(\begin{matrix}N\\\
k\end{matrix}\right)^{\frac{1}{2}}[1-(-1)^{k}]|k,N-k\rangle|1\rangle.$ (5)

Next, we measure the qubit in its computational basis, which projects the
field modes into the superposition of the number states. If the outcome of the
measurement performed on the qubit gives $|0\rangle$, the wave function of the
system collapses to

$|\psi\rangle_{3}=\frac{1}{\sqrt{2^{N}}}\sum_{k=1}^{N}\left(\begin{matrix}N\\\
k\end{matrix}\right)^{\frac{1}{2}}[1+(-1)^{k}]|k,N-k\rangle|0\rangle.$ (6)

Clearly, the qubit becomes disentangled from the rest of the system due to the
wave function collapse.

Alternatively, if the outcome of the measurement gives $|1\rangle$, then the
wave function of the system collapses to

$|\psi\rangle_{3}=\frac{1}{\sqrt{2^{N}}}\sum_{k=1}^{N}\left(\begin{matrix}N\\\
k\end{matrix}\right)^{\frac{1}{2}}[1-(-1)^{k}]|k,N-k\rangle|1\rangle$ (7)

Once we measure the state of the qubit, regardless of the outcome of the
measurement, we apply the time evolution operator one more time on the
collapsed wave function of the system. First, let us assume the outcome of the
measurement to be $|0\rangle$. Applying the time evolution operator on the
corresponding collapsed wave function, for a time interval $\Delta
t=\pi/(4\Omega)$ following the measurement step results in the N00N state

$|\textrm{N00N}\rangle=\frac{1}{\sqrt{2}}((-1)^{N}|N\rangle_{1}\otimes|0\rangle_{2}+|0\rangle_{1}\otimes|N\rangle_{2}).$
(8)

Alternatively, if the outcome of the measurement of the qubit gives
$|1\rangle$, then, once we let the system to evolve for a time interval
$3\pi/(4\Omega)$, one arrives at the N00N state of the form

$|\textrm{N00N}\rangle=\frac{1}{\sqrt{2}}(|N\rangle_{1}\otimes|0\rangle_{2}-|0\rangle_{1}\otimes|N\rangle_{2}).$
(9)

It is worth noting that, regardless of what the outcome of the measurement of
the qubit state is, the N00N state can be obtained at the final step of this
protocol. This implies the fact that considering the ideal scenario where no
decoherence is involved and the measurement is perfect, we arrive at the
$100\%$ fidelity of N00N state generation from $|\psi\rangle_{2}$. Therefore,
the N00N state generated with this protocol is deterministic. If the
measurement yields $|0\rangle$, it takes the system less time to evolve to the
N00N state.

It is interesting to note that, once the Fock state $|N\rangle$ is provided,
the number of steps that are needed to arrive at the N00N state is independent
of the number of photons $N$. Most of the N00N state generation schemes start
from $N=1$ N00N state and increase the $N$ in several steps. In these schemes,
the number of the operations is directly related to the number $N$; however,
in the architecture given in this chapter, this may not be the case. This is
important as it can provide a better control of decoherence. In fact, once we
prepare a high-fidelity Fock state $|N\rangle$, in one of the modes, without
getting the rest of the system involved in the process, we can translate that
into a N00N state in just a few operational steps. Thus, this scheme is more
robust to decoherence. On the other hand, contrary to most of the proposed
protocols, this scheme is simpler and requires fewer devices. We only need a
single qubit which interacts with two modes. The architecture described above
is schematically given in Fig. (2).

Figure 2:  N00N state generation circuit: U represents the time evolution of a
quantum state, governed by the time-evolution operator with the effective
Hamiltonian $H\textsubscript{eff}$. $\mathcal{H}$ is the Hadamard gate acting
locally on the qubit. For the time interval $\Delta t_{1}=\pi/(4\Omega)$, the
system evolves from its initial state to $|\psi\rangle_{1}$. The Hadamard gate
yields the state $|\psi\rangle_{2}$. Once a measurement on the state of the
flux qubit is performed, the total wave function of the systems collapses to
$|\psi\rangle_{3}$. finally, time evolution $U(t)$ provides the desired N00N
state.

##  III. Derivation of the effective Hamiltonian

###  1\. Modulating the coupling between the qubit and the resonators:

We consider a setup of two resonators interacting with a same two-level atom,
where the Hamiltonian can be written as a summation of the free part and the
interaction part $H=H_{0}+H_{1}$ [36]. We assume that the coupling parameter
of the cavity fields and the qubit is periodically modulated. Therefore, the
free part of the Hamiltonian is

$H_{0}=\hbar\dfrac{\omega_{0}}{2}\sigma_{z}+\
\hbar\omega_{1}a_{1}^{\dagger}a_{1}+\ \hbar\omega_{2}a_{2}^{\dagger}a_{2},$

and the interaction part reads

$H_{1}=2\hbar g_{0}\sum_{j=0}^{2}\
\cos(\upsilon_{j}t+\varphi_{j})(\sigma^{+}a_{j}+a^{\dagger}_{j}\sigma^{-}).$

Here, $\omega_{0}$ is the frequency of the two-level energy splitting between
its ground and exited levels. $\omega_{1}$ and $\omega_{2}$ are the
frequencies of the first and the second cavities, respectively. $g_{0}$ is the
coupling strength of the resonators and the qubit. $\upsilon_{j}$ is the
modulating frequency and $\varphi_{j}$ is the modulating phase of the
resonator $j$, with $j=1,2$. We consider the resonant case where
$\omega_{1}=\omega_{2}=\omega$ and take $\hbar=1$, for simplicity. Now, with
this Hamiltonian, we are going to extract the effective Hamiltonian of the
N00N state that we already discussed. To this aim we define the unitary
evolution $U_{0}(t)=\exp[-iH_{0}t]$, using the free part of the the
Hamiltonian. Consequently, in the rotating frame the interaction Hamiltonian
can be given as

$H_{I}=U_{0}(t)HU^{-1}_{0}(t)-H_{0}.$

The explicit form of the interaction Hamiltonian, thus, reads

$H_{I}=\delta\sigma_{z}/2+2g_{0}\sum_{j=1}^{2}\
\cos(\upsilon_{j}t+\varphi_{j})(\sigma^{+}a_{j}+a^{\dagger}_{j}\sigma^{-}),$

where we have $\delta=\omega_{0}-\omega$. If we express the
$\cos(\upsilon_{j}t+\varphi_{j})$ term as an exponential form, the Hamiltonian
equivalently can be expressed as

$H_{I}=\delta\sigma_{z}/2+g_{0}\sum_{j=1}^{2}\
[(\sigma^{+}a_{j}+a^{\dagger}_{j}\sigma^{-})e^{i\varphi_{j}}]e^{i\upsilon_{j}t}+[(\sigma^{+}a_{j}+a^{\dagger}_{j}\sigma^{-})e^{-i\varphi_{j}}]e^{-i\upsilon_{j}t}.$

This Hamiltonian has two parts which are related to the time-dependent terms
$e^{i\upsilon_{j}t}$ and $e^{-i\upsilon_{j}t}$, respectively. Therefore, we
define the operator $h_{j}$ such that

$h_{j}=g_{0}(\sigma^{+}a_{j}+a^{\dagger}_{j}\sigma^{-})e^{i\varphi_{j}}.$

With this definition, the Hamiltonian reduces to

$H_{I}=\delta\sigma_{z}/2+\sum_{j=1}^{2}\
h^{\dagger}_{j}e^{i\upsilon_{j}t}+h_{j}e^{-i\upsilon_{j}t}.$

This is a Floquet Hamiltonian [38, 39, 40, 41], which will enable us to
further reduce the Hamiltonian to attain the effective Hamiltonian of the
desire.

Without loss of generality, we consider the same modulation frequencies
$\upsilon_{j}=\upsilon$ in the following. Also we note that $h_{j}$ commutes
with its Hermitian conjugate part, such that $[h_{j},h^{\dagger}_{j}]=0$. With
these considerations, the effective Hamiltonian reduces to

$H\textsubscript{eff}=\delta\sigma_{z}/2+i\dfrac{2g^{2}_{0}}{\upsilon}\sin(\varphi_{1}-\varphi_{2})(a_{1}a^{\dagger}_{2}-a_{2}a^{\dagger}_{1})\sigma_{z}.$

Thus, the coupling strength of the Hamiltonian can be directly controlled by
the modulation phase factor of each cavity. To maximize the coupling strength
in the Hamiltonian we take $\varphi_{1}-\varphi_{2}={\pi}/{2}$, and define
$\kappa=\dfrac{2g^{2}_{0}}{\upsilon}$, where the Hamiltonian can be expressed
as

$H\textsubscript{eff}=\delta\sigma_{z}/2+i\kappa\sigma_{z}(a_{1}a^{\dagger}_{2}-a_{2}a^{\dagger}_{1}).$
(10)

Taking the resonance case $\delta=0$, we attain the desired Hamiltonian of the
system as

$H\textsubscript{eff}=i\kappa\sigma_{z}(a_{1}a^{\dagger}_{2}-a_{2}a^{\dagger}_{1}).$
(11)

It is remarkable that, the term $\sin(\varphi_{1}-\varphi_{2})$ controls the
coupling strength of the scheme, here. Therefore, it is highly important to
have precise control of the modulation of the system in practical scenarios.
It is clear that when the modulation phases are identical, the term
$\sin(\varphi_{1}-\varphi_{2})$ becomes zero, and no effective Hamiltonian can
be attained in this case.

The maximum coupling is determined by the term
$\kappa=\dfrac{2g^{2}_{0}}{\upsilon}$. Therefore, even though the modulation
frequency $\upsilon$, must be larger enough compared to the parameter $g_{0}$,
however, if we choose it to be very large, the coupling constant of the
effective Hamiltonian may be small. Hence, we need to be careful to satisfy
the strong coupling regime when choosing the modulation frequency.
Specifically, if we take $\upsilon=2g_{0}$, the coupling strength of about
$\kappa=g_{0}/2$, can be achieved.

###  2\. Modulating the frequencies of the resonators:

We showed that the effective Hamiltonian of our system can be realized in the
setup of two resonators interacting with the same two-level qubit by a
specific modulation of the coupling of the system. Now, we are going to show
that such a Hamiltonian can be achieved with a slightly different approach. In
this case, we are going to consider the same setup of two resonators
interacting with the same qubit, where, alternatively the resonators are
modulated with the frequencies given by [42, 43]

$\nu_{j}(t)=\nu+\Delta\sin(\nu_{d}t-\varphi_{j}).$

In this scenario, the frequencies of the resonators are modulated, with the
same frequencies but different phases. The frequency of the qubit is constant
and no modulation is applied on that. Therefore, we can express the
Hamiltonian of the system as

$\displaystyle
H=\hbar\dfrac{\omega_{0}}{2}\sigma_{z}+\hbar\nu_{1}(t)a_{1}^{\dagger}a_{1}+\hbar\nu_{2}(t)a_{2}^{\dagger}a_{2}+\hbar
g_{\nu}\
(\sigma^{+}a_{1}+a_{1}^{\dagger}\sigma^{-}+\sigma^{+}a_{2}+a_{2}^{\dagger}\sigma^{-}).$
(12)

Here, $g_{\nu}$ is the coupling constant of the atom to the resonators. Also,
$\nu_{1}(t)$ and $\nu_{2}(t)$ are the frequencies of the resonators that are
time-dependent through the modulation term above. From this Hamiltonian one
can derive the interaction Hamiltonian of the system. Thus, in the framework
of the rotating wave approximate when $\omega_{0}=\nu$ the total Hamiltonian
of the system above can be simplified as

$H_{I}=\hbar
g_{\nu}\sigma^{+}(\hat{a}_{1}e^{i\zeta\cos(\nu_{d}t-\varphi_{1})}+\hat{a}_{2}e^{i\zeta\cos(\nu_{d}t-\varphi_{2})})+h.c.,$

where $\zeta=\Delta/\nu_{d}$.

This Hamiltonian can be reduced to a Floquet Hamiltonian by using the identity
[38, 39]

$e^{i\zeta\cos(\nu_{d}t+\varphi_{j})}=\sum_{n=-\infty}^{\infty}J_{n}(\zeta)e^{in(\nu_{d}t+\varphi_{j})},$

where $J_{n}(\zeta)$ is the $n$th-order Bessel function of the first kind.
Inserting this in the Hamiltonian above enables us to write the Hamiltonian in
the form of a Floquet Hamiltonian which can be expressed as

$H_{I}=H_{0}+\sum_{n=1}^{\infty}H_{n}e^{in\nu_{d}t},$

where the time-independent part of the Hamiltonian is expressed in terms of
the $0$th-order Bessel function of the first kind in the above expansion which
reads

$H_{0}=\hbar
gJ_{0}(\zeta)(\sigma^{+}(\hat{a}_{1}+\hat{a}_{2})+(\hat{a}_{1}^{\dagger}+\hat{a}_{2}^{\dagger})\sigma^{-})$

and the time-dependent part of the Hamiltonian is expressed in terms of the
$n$th-order Bessel function where we can write

$\displaystyle H_{n}=\hbar
gi^{n}J_{n}(\zeta)[(\sigma^{+}\hat{a}_{1}+(-1)^{n}\hat{a}_{1}^{\dagger}\sigma^{-})e^{in\varphi_{1}}+(\sigma^{+}\hat{a}_{2}+(-1)^{n}\hat{a}_{2}^{\dagger}\sigma^{-})e^{in\varphi_{2}}].$

The interaction Hamiltonian of this form can be reduced to the effective
Hamiltonian, for which we have [38, 39, 40, 41]

$H\textsubscript{eff}=H_{0}+\sum_{n=1}^{\infty}[H_{n},H_{-n}]/({n\hbar\nu}),$

Therefore, with a straightforward calculation, the Hamiltonian can be written
as

$H\textsubscript{eff}=\hbar
gJ_{0}(\zeta)(\sigma^{+}(\hat{a}_{1}+\hat{a}_{2})+h.c.)+i\hbar\Omega(\hat{a}_{1}^{\dagger}\hat{a}_{2}-\hat{a}_{1}\hat{a}_{2}^{\dagger})\sigma_{z}.$

In this equation, $\Omega=g^{2}\chi/\nu$ is the coupling coefficient which
depends on $g^{2}$ and $\nu$ similar to the previous scheme. In this relation
the parameter $\chi$ is determined through

$\chi=\sum_{n=1}^{\infty}2J_{n}(\zeta)^{2}\sin(n(\varphi_{1}-\varphi_{2}))/n.$

It is evident that when the modulation phases are identical, this parameter
becomes zero and we cannot attain the desired Hamiltonian, hence the
modulation phases play a crucial role in this scenario. To improve the
coupling of the system, we need to optimize the parameter $\chi$, which
provides a direct strategy to control the overall coupling of the system.

We specifically choose $\zeta=2.40$ ($J_{0}(2.40)=0$) and
$\varphi_{2}\neq\varphi_{1}$ to derive the effective Hamiltonian [36, 17]

$H\textsubscript{eff}=i\hbar\Omega(\hat{a}_{1}^{\dagger}\hat{a}_{2}-\hat{a}_{1}\hat{a}_{2}^{\dagger})\sigma_{z}$
(13)

To maximize the effective coupling coefficient of the Hamiltonian $\Omega$,
one can maximize $\chi$ by controlling the phase difference
$\varphi_{1}-\varphi_{2}$. If we choose $\varphi_{1}-\varphi_{2}=\pi/3$, we
will obtain $\chi\approx 0.628$.

The effective Hamiltonian can be realized in different quantum platforms. In
particular, one can couple two superconductor resonators to the same qubit
[36, 17]. In this case, the coupling strength between the resonators and the
qubit can be tens of MHz [44]. With such a strategy, one is able to strongly
couple the resonators to the qubit and obtain the desired Hamiltonian.
Therefore, in such a setup, one can effectively realize N00N state shared
within the two resonators following the quantum circuit which was already
discussed in a great details. As was already noted such an strategy provides a
robust platform for realizing N00N states, with a high number of photons.
Since Fock state with 15 microwave photons has already been generated in the
lab, and generation of the Fock states with about 20 photons seems quite
feasible, the platform described here can be useful for breaking the existing
limitations of the N00N state generation.

Even though generation of such a N00N state with the method that has already
investigated above, seems quite pleasing, we show that the main Hamiltonian
can be realized in spin systems as well. In particular, we are interested in
generation of the N00N state in nitrogen vacancy centers in diamond. This can
provide many advantages considering the long coherence time of the diamonds,
which enables to overcome the decoherence problem of the N00N state generation
as the main challenge of this state. Furthermore, since NVEs are very good
candidates for quantum memories, generation of such states can be of
particular importance as it enables us to preserve the generated N00N states
for the sufficiently long period of time. Therefore, in the following we are
going to describe this architecture in details.

###  3\. N00N state in nitrogen vacancy centers in diamond

Now, following Ref. [17] we are going to show that N00N states can be
generated in NVEs in diamond. The scheme of the N00N state generation is
following the same N00N state generation circuit. Thus, we are going to sketch
a scenario where a similar effective Hamiltonian can be extracted for NVEs
interacting with a qubit.

In this platform, we investigate a quantum architecture consisting of two
separate noninteracting NVEs in diamond coupled to a common superconducting
qubit. As a part of this architecture, we consider a large and gap-tunable
flux qubit [Fig.3(a)] [45, 46]. Such a hybrid scheme combines the long
coherence time of spins in NV centers, the advantage of controlling NV centers
by microwave and optical fields [47, 48], the tunability property of
superconducting devices, the circuit scalability, and the remarkable
compatibility with the cutting-edge nanotechnologies [45, 46, 49, 50, 51]. It
is notable that, even though microwave photons in superconducting resonators
provide a unique platform for N00N state generation which was mentioned
earlier, unlike microwave photons in superconducting resonators, which have
lifetimes of the order of 1 ms [52], the coherence time of NV centers in
diamond can approach 1 s even at relatively high temperatures [53]. Thus,
combining the advantages from both devices seems to be more desirable for the
quantum state preparation purposes.

Figure 3:  (a) Two ensembles of NV centers coupled to a superconducting flux
qubit consisting of four Josephson junctions forming the main loop and an
$\alpha$ loop. (b,c) Crystal-lattice (b) and energy (c) diagrams of an NV
center in diamond.

Now, we describe the quantum characteristics of the flux qubit. To this aim we
note that we need to choose the biasing in the main loop of the flux qubit
close to half the flux quantum, $\Phi_{0}=h/(2e)$. Therefore, the flux qubit
could be described [46] in terms of a Hamiltonian in a two-dimensional Hilbert
space. The Hamiltonian describing this system could explicitly be given by

$H=-[\epsilon(\Phi_{ext})\sigma_{z}+\Delta(\Phi_{ext}^{\prime})\sigma_{x}]/2$

where $\sigma_{x}$ and $\sigma_{z}$ are the well-known Pauli operators in the
basis of flux qubit. The parameter $\Delta(\Phi_{ext}^{\prime})$ is the flux
qubit tunneling splitting energy, and finally $\epsilon(\Phi_{ext})$ is the
energy bias of the flux qubit. These parameters provide a platform for
controlling the qubit Hamiltonian of the system.

When a static magnetic field of half the flux quantum is applied perpendicular
to the main loop of the flux qubit, the clockwise and counterclockwise
persistent current states in the system become almost degenerate [56, 57].
These clockwise and counterclockwise states provide us with the capability to
define the logical basis states $|0\rangle_{f}$ and $|1\rangle_{f}$. In this
setting, $|0\rangle_{f}$ and $|1\rangle_{f}$ represent the states of the
clockwise and counterclockwise persistent current, respectively. Noting the
fact that $\epsilon(\Phi_{ext})$ and $\Delta(\Phi_{ext}^{\prime})$ can be
adjusted independently by external magnetic fluxes through the main and the
$\alpha$ loops, one can set $\epsilon(\Phi_{ext})=0$. Therefore, by preparing
the system in one of its computational basis $|0\rangle_{f}$ or
$|1\rangle_{f}$, one can perform a single qubit operation on the system
considering the Hamiltonian given above. In the following, we will show that
such a flux qubit can be combined with NVEs to construct an interesting hybrid
architecture for information processing and entangled state generation. To
this aim we first need to consider the NVEs Hamiltonian in details.

For the NV centers in diamond architecture, we only are interested in the
ground state of such a system. Interestingly, the ground state of an NV center
in diamond is a spin triplet state with zero-magnetic-field splitting of about
$D_{gs}\approx$ 2.87 GHz between the $m_{s}=0$ sublevel and the degenerate
$m_{s}=\pm 1$ sublevels [Fig.3(b,c)]. To eliminate the degeneracy in the
$m_{s}=\pm 1$ sublevels, we can apply an external magnetic field to induce
Zeeman splitting. Therefore. we apply the external magnetic field $B_{ext}$
along the [100] direction of the diamond crystal lattice, to remove the
degeneracy of $|m_{s}=\pm 1\rangle$ sublevels [Fig.3(c)]. Hence, one can focus
on the two sublevels $m_{s}=0$ and $m_{s}=-1$, to reduce the system into a
two-level system. This enables us to reduce the Hilbert space of the system
into a two-dimensional space in which one can define the computational basis
for the information processing operations. With this description, the spin
operators of the $j$th NV center can be defined though Pauli operators. Hence,
we explicitly define

$s_{zj}=|-1_{j}\rangle\langle-1_{j}|-|0_{j}\rangle\langle 0_{j}|,$
$s_{+j}=|-1_{j}\rangle\langle 0_{j}|,$ $s_{-}=|0_{j}\rangle\langle-1_{j}|.$

The definition of the Pauli matrices for a single NV, enables one to define a
generic scenario for the situation which includes $N_{0}$ nitrogen vacancy
centers. More precisely, if an NVE contains $N_{0}$ NV spin, collective spin
operators can easily be expressed through summing the spins of the system
which can be written as

$S_{\tau}=\sum_{k=1}^{N_{0}}s_{\tau k}(\tau=z,\pm).$

It is quite notable that qubits based on nitrogen vacancy centers with an
external magnetic field $B_{ext}$ applied to induce splitting between the
$m=1$ and $m=-1$ sublevels have been investigated extensively in the
literature (see, e.g., [54, 55, 56, 57]). One main difference of what we are
going to present here with the earlier works is that we apply additional time-
dependent $B_{j}$ fields, to NVEs beside with $B_{ext}$ to provide a periodic
modulation of the $m=0$ – $m=-1$ splitting. This interestingly provides us
with the capability of generating N00N states which is the main goal of our
discussion in this chapter. To obtain the desired effective Hamiltonian, we
are going to apply the Holstein–Primakoff (HP) transformation [58] to the spin
operators of NV centers. Such a strategy enables mapping of the collective
spin behavior of a quantum system to a harmonics oscillator. Therefore, by
applying HP transformation on the time-dependent $m=0$ and $m=-1$ splitting we
show that the transformation leads to a closed-form Hamiltonian that we are
seeking for.

To be able to apply the HP transformation, we need to restrict ourselves in
the regime of weak excitation, where the spin operators of an NVE with a large
$N_{0}$ can be mapped onto bosonic operators via an HP transformation [58].
Thus, we assume the the the number of the excitation is very small compared to
$N_{0}$. With such an assumption, one can perform the HP transformation
resulting in

$\sum_{k=1}^{N_{0}}s_{+k}^{j}=c_{j}^{{\dagger}}\sqrt{N_{0}-c_{j}^{{\dagger}}c_{j}}\backsimeq\sqrt{N_{0}}c_{j}^{{\dagger}},$

and

$\sum_{k=1}^{N_{0}}s_{-k}^{j}=c_{j}\sqrt{N_{0}-c_{j}^{{\dagger}}c_{j}}\backsimeq\sqrt{N_{0}}c_{j},$

also

$\sum_{k=1}^{N_{0}}s_{zk}^{j}=2c_{j}^{{\dagger}}c_{j}-N_{0}.$

In the relations above $j=1,2$ for the first and the second NVEs,
respectively. In this manner,one has $[c_{j},c_{j}^{{\dagger}}]=1$.

We note that once we have a collective spin system, the coupling of the system
enhances drastically compared to the single spin coupling. More precisely, the
collective coupling strength of an ensemble of $N_{0}$ spins enhances by a
factor of $\sqrt{N_{0}}$ compared to coupling strength of a single spin [59].
Therefore, increasing $N_{0}$ seems to to be favorable from this perspective
as it is much easier to fulfill the requirements of the strong coupling
regime. Furthermore, the weak-excitation requirement mentioned above is
necessary for the validity of HP mapping, could be satisfied via the
conditions $c_{1}^{\dagger}c_{1}<<N_{0}$ and $c_{2}^{\dagger}c_{2}<<N_{0}$.
This in turn limits the $N$ number in attainable N00N states, $N<<N_{0}$.
However, $N_{0}$ can be sufficiently large in a realistic scenario. For
instance, $N_{0}\sim$ $10^{7}$ has been used in the practicle experiments
[60], which is sufficiently higher than what we expect for the N00N state
generations.

The total Hamiltonian of the NVE-flux-qubit hybrid system considered here can
now be written as

$H=-\hbar\frac{\Delta(\Phi_{ext}^{\prime})}{2}\sigma_{x}+\hbar\sum_{j=1}^{2}\omega_{j}c_{j}^{{\dagger}}c_{j}+\hbar\sum_{j=1}^{2}g(c_{j}^{{\dagger}}+c_{j})\sigma_{z},$
(14)

where

$\omega_{j}=D_{gs}-g_{e}\mu_{B}B_{z}-g_{e}\mu_{B}B_{j},$

$g_{e}$ is the ground-state Lande factor and $\mu_{B}$ is the Bohr magneton
and $B_{z}$ is the magnetic field sensed by the spins due to the applied
external magnetic filed $B_{ext}$ and the magnetic field produced by the flux
qubit. Also, the parameter g is the coupling constant between the NVE
esnenbles and the flux qubit.

We Further assume that the sizes of the NVEs are sufficiently small to neglect
spatial variations of the magnetic field induced by the flux qubit.
Furthermore, the magnetic fields are chosen such that

$\omega_{j}=\Delta(\Phi_{ext}^{\prime})/2+\delta\sin(\nu t+\varphi_{j}),$

where $\delta\sin(\nu t+\varphi_{j})$ is controlled by the ac magnetic field.
Choosing large $\nu$ and small $\delta$, so that
$\Delta(\Phi_{ext}^{\prime})\gg\zeta=\delta/\nu$, we apply the rotating-wave
approximation (RWA) and use the basis of flux qubit eigenstates to reduce the
interaction Hamiltonian of the considered system to

$H_{I}=\hbar g\sigma^{+}(\hat{c}_{1}e^{i\zeta\cos(\nu
t+\varphi_{1})}+\hat{c}_{2}e^{i\zeta\cos(\nu t+\varphi_{2})})+H.c.$ (15)

This Hamiltonian can be reduced to a Floquet Hamiltonian. We set $\zeta=2.40$
($J_{0}(2.40)=0$) and $\varphi_{2}\neq\varphi_{1}$ to find the effective
interaction Hamiltonian:

$H\textsubscript{eff}=i\hbar\Omega(\hat{c}_{1}^{\dagger}\hat{c}_{2}-\hat{c}_{2}^{\dagger}\hat{c}_{1})\sigma_{z}.$
(16)

It is straightforward to see now that the coupling coefficient can be tuned by
varying the $B_{j}$ fields. In particular, with
$\varphi_{1}-\varphi_{2}=\pi/3$ and $\nu=5\chi g\simeq 3.14g$, we have
$\Omega/2\pi\approx 14$ MHz. Such coupling strengths have been recently
demonstrated for a system of a flux qubit and an NVE with $N_{0}\approx$
$3\times 10^{7}$ [60]. With the decay rates of NVEs and the flux qubit
estimated as $\gamma_{NV}\sim$ 1 Hz [53] and $\gamma_{FQ}\sim$ 1 MHz [61], we
find that the requirement of strong coupling,
$\Omega\gg\gamma_{FQ},\gamma_{NV}$, is fulfilled.

As one of its interesting features, our protocol of N00N-state generation
requires only local operations on the flux qubit at the second and third
steps, where the state $|\psi\rangle_{1}$ is degenerated into
$|\psi\rangle_{3}$. Consequently, the operation time for each of these two
steps reduces to the single-qubit operation time, $\Delta t_{2}$, $\Delta
t_{3}\sim$ 1 ns [57]. This fact allows the overall time
$\Delta\tau=\sum_{i=1}^{4}\Delta t_{i}$ of N00N-state generation to be
sufficiently shortened. This is useful for avoiding decoherence buildup in the
system. With our earlier estimate of $\Omega/2\pi\simeq 14$ MHz, the system
could be in a strong-coupling regime, for which we have $\Delta t_{1}\approx$
9 ns. Hence, we find for the total time required to create a N00N state
through the considered process to be $\Delta\tau\approx$ 20 ns when the
$|0\rangle_{f}$ state is found at the measurement step and $\Delta\tau\approx$
38 ns if the measurement yields $|1\rangle_{f}$. In both situations, the time
that is needed for N00N-state generation is much shorter than the dephasing
times of both the flux qubit, $\sim 1$ $\mu$s [61] and the NVEs $\sim 350$
$\mu$s [62].

###  4\. N00N state as a superposition of light and motion

We examine a trapped ion featuring an atomic transition between two electronic
states, denoted as $|g\rangle$ and $|e\rangle$, with a transition frequency of
$\omega_{0}$. This ion is coupled to a cavity field with a frequency of
$\omega_{c}$ and is subjected to external stimulation through a classical
laser field operating at the frequency $\omega_{L}$. In a rotating frame with
the laser frequency $\omega_{L}$, the Hamiltonian is expressed as follows [63,
64]

$H=\nu
b^{\dagger}b+\Delta_{c}a^{\dagger}a+\delta\sigma_{z}+\varepsilon_{L}\sigma_{+}+\varepsilon^{*}_{L}\sigma_{-}$
$+g_{0}\ \sin\eta(b^{\dagger}+b)(a^{\dagger}\sigma_{-}+a\sigma_{+}).$

In this context, the symbols $a$, $a^{\dagger}$, $b$, and $b^{\dagger}$
represent the bosonic annihilation and creation operators for the cavity field
and the quantized atomic vibration, respectively. The operators
$\sigma_{z}=|e\rangle\langle e|-|g\rangle\langle g|$,
$\sigma_{+}=|e\rangle\langle g|$, and $\sigma_{-}=|g\rangle\langle e|$ are the
Pauli operators. The Lamb-Dicke parameter corresponding to these operators is
denoted by $\eta$. The detuning parameters, namely $\Delta_{c}$ and $\delta$,
are defined as $\Delta_{c}=\omega_{c}-\omega_{L}$ and
$\delta=\omega_{0}-\omega_{L}$.

The parameter $\varepsilon_{L}=\epsilon_{L}e^{-i\varphi_{L}}$ represents the
amplitude of the laser field. The coupling strength between a single photon,
the atom, and the cavity is denoted by $g_{0}$, and the sinusoidal function
captures the standing wave pattern of the cavity field.

In the Lamb-Dicke regime $(\eta\ll 1)$, we can simplify $\textrm{sin
}\eta(b^{\dagger}+b)\approx\eta(b^{\dagger}+b)$ [63, 64]. Under the resonant
condition $\delta=0$ and in the interaction picture with $\nu=\Delta_{c}$,
after applying the rotating wave approximation (RWA), the reduced Hamiltonian
takes the form

$H^{\prime}=\varepsilon_{L}\sigma_{+}+\varepsilon^{*}_{L}\sigma_{-}+g_{0}\eta(ba^{\dagger}\sigma_{-}+b^{\dagger}a\sigma_{+}).$

We transition to the transformed picture $H_{I}=RH^{\prime}R^{\dagger}$
through the unitary transformation

$R=e^{i\frac{\pi}{4}\sigma_{y}}e^{-i\frac{\varphi_{L}}{2}\sigma_{z}}e^{i\epsilon_{l}\sigma_{z}t}$

Where, following the adiabatic elimination of rapidly oscillating terms using
the Rotating Wave Approximation (RWA) with the condition
$g_{0}\eta\ll\epsilon_{L}$, and introducing
$\varphi_{L}\rightarrow\varphi-\frac{\pi}{2}$, the resulting effective
interaction Hamiltonian is given by [32]

$H_{eff}=i\Omega(e^{-i\varphi}a^{\dagger}b-e^{i\varphi}ab^{\dagger})\sigma_{z},$
(17)

with $\Omega=g_{0}\eta/2$.

##  IV. Entangled state generations beyond the N00N state

Here, we are going to consider generation of entangled state beyond the N00N
state. These entangled states can be very useful for various quantum
information processing tasks [65, 66]. We are going to show that a vast class
of entangled states can be generated with the scheme that will be provided
here. More interestingly, various maximally entangled states can be engineered
with the protocol. We are going to follow [36] in doing so. The starting point
to our system is still the same effective Hamiltonian that we have already
focused on. In this platform, we consider two cavities which interact with a
common qubit. Using the Hamiltonian in Eq. (1) we can derive the dynamics of
the field operators as

$\displaystyle\dot{a}_{1}$ $\displaystyle=\kappa a_{2}(t)\sigma_{z},$
$\displaystyle\dot{a}_{2}$ $\displaystyle=-\kappa a_{1}(t)\sigma_{z}.$

The solution to these equations can be found to be

$\displaystyle a_{1}=a_{1}(0)\cos(\kappa t)-a_{2}(0)\sin(\kappa t)\sigma_{z},$
$\displaystyle a_{2}=a_{2}(0)\cos(\kappa t)+a_{1}(0)\sin(\kappa t)\sigma_{z}.$

At $\kappa t={\pi}/{4}$, we have

$\displaystyle a_{1}=\frac{1}{\sqrt{2}}(a_{1}(0)-a_{2}(0)\sigma_{z}),$
$\displaystyle a_{2}=\frac{1}{\sqrt{2}}(a_{2}(0)+a_{1}(0)\sigma_{z}).$

Next, we let the photons to go through a 50/50 beam splitter given by the
two–mode unitary operation

$U=\text{exp}[-\pi/4(a_{1}^{{\dagger}}(0)a_{2}(0)-a_{2}^{{\dagger}}(0)a_{1}(0))].$

This process results in the following transformation of the field operators

$\displaystyle a_{1}(0)=\frac{1}{\sqrt{2}}(a_{1}(0)+a_{2}(0)),$ $\displaystyle
a_{2}(0)=\frac{1}{\sqrt{2}}(a_{2}(0)-a_{1}(0)).$

Thus, the total transformation, after passing through the beam splitter can be
given by

$\displaystyle a_{1}\rightarrow a_{1}(0)|e\rangle\langle
e|+a_{2}(0)|g\rangle\langle g|,$ (18) $\displaystyle a_{2}\rightarrow
a_{2}(0)|e\rangle\langle e|-a_{1}(0)|g\rangle\langle g|$ (19)

These transformations are central to generation of entangled states which will
become clear soon. Note that these are conditional photon transformation
subjected to the state of the qubit. We will see that such a transformation
maps various initial states into interesting entangled states which could be
useful for various tasks. Now, to see the power of such a transformation, let
us consider an specific example as the input state

$|\psi\rangle=|N\rangle|0\rangle\frac{1}{\sqrt{2}}(|g\rangle+|e\rangle).$

The above transformation, acting upon this initial state, gives rise into

$|\psi\rangle\longrightarrow\frac{1}{\sqrt{2}}(|N\rangle|0\rangle|e\rangle+|0\rangle|N\rangle|g\rangle).$

Figure 4: Architecture of the entangled state generation setup. The input
state first interacts with the system of the two resonators interacting with a
qubit for as long as $\kappa t={\pi}/{4}$. Then the photons in the resonators
are sent to a beam splitter. Finally, by applying a Hadamard gate and
performing a measurement on the qubit we attain a N00N state.

By applying an external $\pi/2$ pulse (Hadamard gate) to the atom in the state
above the overall state of the system degenerates into

$[(|N\rangle|0\rangle+|0\rangle|N\rangle)|e\rangle+(|N\rangle|0\rangle-|0\rangle|N\rangle)|g\rangle]/2.$

Therefore, the measurement on the state of the atom provides different N00N
states. Namely, detection of the atom in the exited state results in the N00N
state

$\frac{1}{\sqrt{2}}(|N\rangle|0\rangle+|0\rangle|N\rangle),$

and detection of the atom in its ground state gives

$\frac{1}{\sqrt{2}}(|N\rangle|0\rangle-|0\rangle|N\rangle).$

Regardless of the sign differences, both of the states give the same result in
the phase estimation scenario, and both are N00N states. We summarize the
transformation presented here in Fig. 4. Accordingly, the input state first
interacts with the resonators–qubit system, then goes through a beam splitter.
Finally, a Hadamard gate followed by a measurement is performed on the qubit
which gives the desired N00N states.

As was mentioned earlier, the usefulness of our architecture can even be
extended beyond the N00N state generation. For instance, let us consider the
case where we prepare the initial state of the first cavity in a coherent
state [67, 68]

$|\alpha\rangle=e^{-\frac{|\alpha|^{2}}{2}}\sum_{n=0}^{\infty}\frac{\alpha^{n}}{\sqrt{n!}}|n\rangle,$

where $\alpha$ is an arbitrary complex number. To this aim, we consider the
initial state of the system to be expressed as

$|\psi\rangle=|\alpha\rangle|0\rangle\frac{1}{\sqrt{2}}(|g\rangle+|e\rangle),$

where we could generate the superposed state of the form

$|\psi\rangle\longrightarrow\frac{1}{\sqrt{2}}(|\alpha\rangle|0\rangle|e\rangle+|0\rangle|\alpha\rangle|g\rangle)$

This is a micro-macro entangled state where an atom is entangled with a shared
coherent state in the cavities. Thus, one can generate entangled state of the
form

$|\varphi^{\pm}\rangle=\frac{1}{\sqrt{2\pm
2e^{-|\alpha|^{2}}}}(|\alpha\rangle|0\rangle\pm|0\rangle|\alpha\rangle).$

The state $|\varphi^{+}\rangle$ is proven to be very usefull in quantum
merology [69]. If we start with a squeezed state at the input stage, we could
entangle the squeezed state with the vacuum.

Our approach can generate states even beyond such a superposition. In general,
we take the quantum states

$|\psi_{1}\rangle=\sum_{n=0}^{d_{1}}b_{n}|n\rangle,$

and

$|\psi_{2}\rangle=\sum_{n=0}^{d_{2}}c_{n}|n\rangle.$

Hence, we prepare the initial state in

$|\psi\rangle=|\psi_{1}\rangle|\psi_{2}\rangle\frac{1}{\sqrt{2}}(|g\rangle+|e\rangle).$

Then, given the detection of the atom in its exited state, we can generate the
symmetric entangled state

$|\psi^{+}\rangle=\dfrac{1}{\sqrt{2+2|\langle\psi_{1}|\psi_{2}\rangle|^{2}}}(|\psi_{1}\rangle|\psi_{2}\rangle+|\psi_{2}\rangle|\psi_{1}\rangle),$
(20)

alternatively, if we detect the atom in its ground state, we attain the anti-
symmetric entangled state

$|\psi^{-}\rangle=\dfrac{1}{\sqrt{2-2|\langle\psi_{1}|\psi_{2}\rangle|^{2}}}(|\psi_{1}\rangle|\psi_{2}\rangle-|\psi_{2}\rangle|\psi_{1}\rangle).$
(21)

Thus, we can generate several interesting states in our architecture such as
superposition of two coherent states $|\alpha\rangle$ and $|\beta\rangle$

$\dfrac{1}{\sqrt{2\pm
2|\langle\alpha|\beta\rangle|^{2}}}(|\alpha\rangle|\beta\rangle\pm|\beta\rangle|\alpha\rangle),$

or superposition of two squeezed states $|\xi_{1}\rangle$ and
$|\xi_{2}\rangle$

$\dfrac{1}{\sqrt{2\pm
2|\langle\xi_{1}|\xi_{2}\rangle|^{2}}}(|\xi_{1}\rangle|\xi_{2}\rangle\pm|\xi_{2}\rangle|\xi_{1}\rangle),$

or even hybrid entangled states of the forms

$\dfrac{1}{\sqrt{2\pm
2|\langle\xi_{1}|\alpha\rangle|^{2}}}(|\alpha\rangle|\xi_{1}\rangle\pm|\xi_{1}\rangle|\alpha\rangle),$
$\dfrac{1}{\sqrt{2\pm 2|\langle
N|\alpha\rangle|^{2}}}(|\alpha\rangle|N\rangle\pm|N\rangle|\alpha\rangle),$
$\dfrac{1}{\sqrt{2\pm 2|\langle
N|\xi_{1}\rangle|^{2}}}(|\xi_{1}\rangle|N\rangle\pm|N\rangle|\xi_{1}\rangle).$

Such states are of paramount importance in beating the shot noise limit in
quantum metrology and have a vast application in quantum information
processing.

To quantify the entanglement degree of the generated states
$|\psi^{\pm}\rangle$ we can use concurrence as the measure of entanglement
[70], which is defined as

$C=|\langle\psi|\sigma_{y}\otimes\sigma_{y}|\psi^{*}\rangle|,$ (22)

where $\sigma_{y}$ is the spin flip operator and $|\psi^{*}\rangle$ is the
complex conjugate of $|\psi\rangle$. With some straightforward calculation, we
find the concurrence as

$C=\dfrac{1-|\langle\psi_{1}|\psi_{2}\rangle|^{2}}{1\pm|\langle\psi_{1}|\psi_{2}\rangle|^{2}}.$
(23)

Where, the plus(minus) sign in the denominator shows the concurrence of the
state $|\psi^{+}\rangle$($|\psi^{-}\rangle$). Interestingly,
$|\psi^{-}\rangle$ is maximally entangled for any input states. However,
$|\psi^{+}\rangle$ is maximally entangled if and only if the input states in
the modes are orthogonal states, i.e., $\langle\psi_{1}|\psi_{2}\rangle=0$.
For instance, considering the input state $|N\rangle|M\rangle$, one can
generate maximally entangled states

$|\psi\rangle\longrightarrow\frac{1}{\sqrt{2}}(|N\rangle|M\rangle+|M\rangle|N\rangle),$

in this class of states. This reduces to the N00N states when $M=0$.

Considering the maximal entanglement of $|\psi^{-}\rangle$, a wast class of
maximally entangled states can be generated in this setup. It is notable that,
having maximally entangled states are extremely important in many quantum
protocols. This scheme can be useful in a vast span of quantum disciplines and
can provide a new platform for information processing tasks.

##  V. Eantangled state generation beyond the two-mode states

As was shown earlier, it is quite possible to generate a large class of
entangled states, such as N00N state and entangled coherent states in the
architecture that we have described earlier. It would be quite interesting to
extend the scheme to a miltimode scenario. Generation of entangled states with
more than two modes could be very important for various applications as well.
However, generation of multimode quantum states is one of the most challenging
tasks in most of the practical applications of quantum science. Thus, what
will be presented here could be found useful in various quantum disciplines.

Figure 5: The network of multi-N00N state generation architecture. The qubit
is sent through the resonator network to interact with each resonator pair.
Then the photons inside the each pair is sent to a beam splitter.

Now, we would like to show how the scheme that we investigated earlier, can be
used to generate a multimode state. To do so. we first prepare an array of
two-cavity system where, the array consists of pairs of cavities and a single
qubit(see Fig. 5). Furthermore, we prepare the initial state of the entire
system in some specific state of the interest. For instance, let us examine
the following initial state as an input of the scheme

$|\psi\rangle=|N,0\rangle\otimes|N,0\rangle\otimes...\otimes|N,0\rangle\frac{1}{\sqrt{2}}(|g\rangle+|e\rangle).$

Then, we send the qubit through the two-resonator network, where it interacts
for the time length of $\kappa t={\pi}/{4}$ with each resonator pair. Once the
qubit is interacted with each pair, the photons go though the beam splitters
after that. Consequently, the initial state of the system degenerates into

$|\psi\rangle\longrightarrow\frac{1}{\sqrt{2}}(|N,0\rangle...|N,0\rangle|e\rangle+|0,N\rangle...|0,N\rangle|g\rangle).$

Now, by applying a ${\pi}/{2}$ pulse on the qubit and performing a measurement
we get the multi-N00N state

$|\psi\rangle\longrightarrow\frac{1}{\sqrt{2}}(|N,0\rangle...|N,0\rangle\pm|0,N\rangle...|0,N\rangle).$

It is interesting that for an array of $M$ resonator pairs, the phase
resolution of the system reduces to $\Delta\varphi\propto\frac{1}{MN}$. To
achieve this value through a N00N state we should be able to increase the
number of phtons to $MN$ in each of the entangled modes which is a very hard
task for relatively large $M$ and $N$.

Recently, a scheme to generate double N00N states was proposed [71]. To
appreciate, the approach we introduce here, we restrict ourselves to $M=2$ and
compare the proposal of our work with the approach of [71]. In their setup
they need $N+2$ operational steps to generate a double N00N state; however, in
our system we can generate a N00N state in just a few operational steps, which
is independent of the number of the photons. This is a big advantage of our
approach. Furthermore, our structure is even capable of generating multi-N00N
states that can facilitate the quantum phase estimation techniques. That being
said, we can generate a vast class of multi-mode entangled state by preparing
each resonator in our specific state of interest.

##  VI. Heisenberg-Limit metrology beyond N00N states

It was already discussed that N00N states have the specific capabilities to
reach the Heisenberg-Limit for the phase estimation sensitivity. Here, we
discuss the HL sensitivity in the su(2) algebra representation and discuss
some of its properties. In fact, what we are going to show is that there is a
large class of Fock state superposition which can reach the HL sensitivity. To
lay out a rather general formalism for such states we will consider the
relation between N00N state and spin coherent state superposition based on
[72]. To this aim, we note that for such a two-mode Fock state representation,
one may introduce the Schwinger realization of the su(2) algebra in terms of
the field operators as [74]

$J_{+}=a^{\dagger}b,\qquad J_{-}=b^{\dagger}a,\qquad
J_{z}=\frac{1}{2}(a^{\dagger}a-b^{\dagger}b).$

Hence, $J_{-}$ and $J_{+}$ are the lowering and raising operators of the spin
states, respectively. The generators of su(2) Lie Algebra $J_{\pm}$ and
$J_{z}$ satisfy the commutation relations

$[J_{+},J_{-}]=2J_{z},\qquad[J_{z},J_{\pm}]=\pm J_{\pm}.$

Taking $j=\frac{n}{2}$, and $m=\frac{n_{a}-n_{b}}{2}$, one can define

$|n_{a}\rangle_{a}|n_{b}\rangle_{b}\equiv|j,m\rangle,\qquad(n=n_{a}+n_{b}).$

This relation maps all the basis of the Hilbert space of the entire system
from Fock state products to the Dicke state basis described by the su(2)
algebra. The two special cases which are interesting to note are given by

$|0\rangle_{a}\otimes|2j\rangle_{b}\equiv|j,-j\rangle,\quad\text{and
}\quad|2j\rangle_{a}\otimes|0\rangle_{b}\equiv|j,j\rangle.$

With this definition, the generators of the spin operators act on the basis as

$J_{\pm}|j,m\rangle=\sqrt{(j\pm m)(j\pm m+1)}|j,m\pm
1\rangle,\quad\text{and}\quad J_{z}|j,m\rangle=m|j,m\rangle.$

Therefore, given $N$ photons in one of the two modes, the two special cases
could be expressed as

$|0\rangle_{a}\otimes|N\rangle_{b}\equiv|j,-j\rangle_{z},\quad\text{and}\quad|N\rangle_{a}\otimes|0\rangle_{b}\equiv|j,j\rangle_{z}.$

These realizations in hand, one can rewrite the N00N state in Dicke basis as
[73]

$|\textrm{N00N}\rangle=\frac{1}{\sqrt{2}}(|j,j\rangle_{z}+|j,-j\rangle_{z}).$

Thus, the N00N state is the superposition of the two north and the south poles
of the Bloch sphere in the Dicke state basis. We note that, the superposition
of the states with other principle numbers, i.e.,

$\frac{1}{\sqrt{2}}(|j,m\rangle_{z}+|j,-m\rangle_{z}),$

cannot give raise to the N00N state since in the Fock basis it is equivalence
to the state

$\frac{1}{\sqrt{2}}(|\frac{n}{2}+m\rangle_{a}|\frac{n}{2}-m\rangle_{b}+|\frac{n}{2}-m\rangle_{a}|\frac{n}{2}+m\rangle_{b}).$

Clearly, such state could not reproduce the HL sensitivity that is achievable
with N00N state. However, considering the fact that N00N state are the two
extreme points on the $z$ direction, one might think of considering the states
in some other extreme points such as points in the $x$ direction. In fact, the
superposition of the two extreme points in $x$ direction could be expressed as

$|j,\pm j\rangle_{x}=\frac{1}{2^{j}}\sum_{m=-j}^{j}\left(\begin{matrix}2j\\\
j+m\end{matrix}\right)^{\frac{1}{2}}(\pm 1)^{j+m}|j,m\rangle_{z},$

Noting that
$|j,m\rangle_{z}=|\frac{n}{2}+m\rangle_{a}|\frac{n}{2}-m\rangle_{b}$, the
superposition of such states is not equal to the N00N state. In fact, the
normalized superposition of the two extreme points in $x$ direction is

$|\psi\rangle=\frac{1}{\sqrt{2}}(|j,j\rangle_{x}+|j,-j\rangle_{x}).$

Now, we show that this state could be equivalent to the N00N state in phase
estimation scenario if we perform an specific strategy that will become clear
soon. To see this, we need to show that this state can indeed give Heisenberg
limit precision. For this purpose, let us apply the unitary operator around
$x$-axis $\exp(i\theta_{x}j_{x})$ on the state $|\psi\rangle$. With this
operator applied, the state $|\psi\rangle$ becomes

$|\psi(\theta_{x})\rangle_{=}\frac{1}{\sqrt{2}}(e^{i\theta_{x}j}|j,j\rangle_{x}+e^{-i\theta_{x}j}|j,-j\rangle_{x}).$

Therefore, we can calculate the phase uncertainty as
$\Delta\theta_{x}=\frac{1}{2j}=\frac{1}{N}$. Thus, the phase estimation is
exactly equal to the HL as the N00N state. From this result one can easily
realize that the superposition of the two lowest and highest states in the $y$
direction, i.e., $|j,\pm j\rangle_{y}$ can reach the HL as well. These
considerations through the extreme points in the $x$ and the $y$ directions
was clearly pointed investigated in Ref. [73].

Figure 6:  Rotation of the $|j,\pm j\rangle_{x}$ about y-axis on the Bloch
sphere.

Now, it is tempting to ask that if there is any other superposed state that
can reach this limit, or are these three states the only ones that can fulfill
the N00N state property for the precise measurement? If there are some other
states what are these states? Also, how we can generate such superposition
states of the Dicke basis say in $x$, $y$ and $z$ directions? We shall address
the generation scheme later; however, we first are going to discuss the first
questions.

It is remarkable that in such a cat state the two superposed points are
antipodals on the Bloch sphere. In general, an element of the Bloch sphere can
be obtained by applying a rotation operator on a reference point (such as
$|j,-j\rangle_{z}$). Such rotation can be given based on the generators of the
su(2) algebra of the spin system. The most general form of the rotation
operator then can be expressed as [74]

$R(\theta,\varphi)e^{i\vartheta
J_{z}}=\exp{\\{\frac{-\theta}{2}(J_{+}e^{-i\varphi}-J_{-}e^{i\varphi})\\}}e^{i\vartheta
J_{z}}.$

On the $z$ direction basis, the term $e^{-i\vartheta J_{z}}$ give rise to an
overall phase, that dose not effect the precision measurement. Since, The
state

$\frac{1}{\sqrt{2}}(e^{i\vartheta_{1}}|N\rangle_{a}\otimes|0\rangle_{b}+e^{-i\vartheta_{1}}|0\rangle_{a}\otimes|N\rangle_{b}),$

is as good as the N00N state, for precise measurement purposes (this extra
phase degree of freedom was introduced in Ref.[73] to facilitate the
generation of N00N state). However, the first term displaces the ground state
of the Dicke basis to a coherent state as

$\displaystyle|\theta,\varphi,j\rangle=\exp{[\frac{\theta}{2}(J_{+}e^{-i\varphi}-J_{-}e^{i\varphi})]}|j,-j\rangle=\frac{1}{(1+|\gamma|^{2})^{j}}\sum_{m=-j}^{j}\left(\begin{matrix}2j\\\
j+m\end{matrix}\right)^{\frac{1}{2}}\gamma^{j+m}|j,m\rangle,$ (24)

where $\gamma=e^{-i\varphi}\tan(\frac{\theta}{2})$. The overlap of two spin
coherent states is given by

$\langle\delta,j|\gamma,j\rangle=\frac{(1+\bar{\delta}\gamma)^{2j}}{(1+|\delta|^{2})^{j}(1+|\gamma|^{2})^{j}}.$

Now, we examine the superposition of two antipodal coherent states on the
Bloch sphere. We consider, with out loss of generality, a superposition given
by [75, 76]

$\frac{1}{\sqrt{2}}(|\theta,\varphi,j\rangle+|\pi-\theta,\pi+\varphi,j\rangle),$

In order for to investigate the phase shift and sensitivity obtained through
this superposition, we consider the following Hermitian operator.

$J_{\gamma}=J_{z}\cos\theta-\frac{1}{2}(J_{+}e^{-i\varphi}+J_{-}e^{i\varphi})\sin\theta.$

In fact, with such a Hermitian operator, we can define the unitary phase shift
operator $U=e^{-i\xi J_{\gamma}}$, such that

$\displaystyle e^{-i\xi
J_{\gamma}}\frac{1}{\sqrt{2}}(|\theta,\varphi,j\rangle+|\pi-\theta,\pi+\varphi,j\rangle)=\frac{1}{\sqrt{2}}(e^{i\xi
j}|\theta,\varphi,j\rangle+e^{-i\xi j}|\pi-\theta,\pi+\varphi,j\rangle).$ (25)

Thus, the phase uncertainty can be obtained as $\Delta\xi=\frac{1}{2j}$.
Therefore, we can get HL phase uncertainty using the superposition of the two
orthogonal spin coherent states as above. This shows that the HL phase
uncertainty is not an exclusive property of the N00N states, and it can indeed
be extracted with a large class of quantum states.

Now, one may ask whether it is possible to get such HL precision via
superposition of two non-orthogonal spin coherent states which are not
antipodal points or is this just limited to the antipodal orthogonal coherent
state superposition? To have a better insight into such superposition we show
the relation between the N00N state and such superposition states. To this aim
we note that the two extreme points of the $x$ direction are related to the
diagonal $z$ component of the spin coherent state in Eq. 24 by the
$\frac{\pi}{2}$ rotation $R_{y}(\frac{\pi}{2})=\exp(-i\frac{\pi}{2}j_{y})$
about the y-axis, i.e.,

$R_{y}(\frac{-\pi}{2})|j,\pm j\rangle_{x}=|j,\pm j\rangle_{z}.$

This is shown in Fig. 6. Thus, by rotating the cat state about y-axis and
noting that $|0\rangle_{a}|N\rangle_{b}\equiv|j,-j\rangle_{z}$, and
$|N\rangle_{a}|0\rangle_{b}\equiv|j,j\rangle_{z}$ we create the N00N state

$|\textrm{N00N}\rangle=\frac{1}{\sqrt{2}}(|2j\rangle_{a}\otimes|0\rangle_{b}+|0\rangle_{a}\otimes|2j\rangle_{b}).$

This result clearly shows that a N00N state, is in fact a typical type of spin
cat state [73]. A coherent state can be expressed as an action of the rotation
operator on the ground state such that
$|\theta,\varphi,j\rangle=R(\theta,\varphi)|j,-j\rangle$. It is also
remarkable that the antipodal point of this coherent state is

$|\pi-\theta,\pi+\varphi,j\rangle=R(\theta,\varphi)R(\pi,\varphi)|j,-j\rangle=R(\theta,\varphi)|j,j\rangle.$

Therefore, by applying the rotation operator
$R^{-1}(\theta,\varphi)=R(-\theta,\varphi)$, we can get the N00N state. Thus,
the spin cat states of antipodal points can be converted to the N00N state by
some global unitary transformation. It is notable that such transformations
change the states entirely and may transform it to a completely different
state. With this description, now consider a superposition of two spin
coherent state as
$|\theta_{1},\varphi_{1},j\rangle+|\theta_{2},\varphi_{2},j\rangle$. We apply
the operator $R(-\theta_{1},\varphi_{1})$ on the state and noting that the
product of two rotation operators gives a rotation operator, up to a phase
factor, we may express the state as

$|j,-j\rangle+R(-\theta_{1},\varphi_{1})R(\theta_{2},\varphi_{2})R(\pi,0)|j,j\rangle.$

This can be fulfilled for
$R(\theta_{2},\varphi_{2})=e^{i\beta}R(\theta_{1},\varphi_{1})R(\pi,\varphi^{\prime})$.
Where we have used the fact that $|j,-j\rangle=R(\pi,0)|j,j\rangle$. Now
considering the fact that multiplication of two rotation operators gives rise
to a rotation operator, up to a phase factor

$R(\gamma_{1})R(\gamma_{2})=R(\gamma_{3})e^{i\Phi(\gamma_{1},\gamma_{2})J_{z}}$

where

$\gamma_{3}=\frac{\gamma_{1}+\gamma_{2}}{1-\gamma_{1}^{*}\gamma_{2}},\qquad\Phi(\gamma_{1},\gamma_{2})=-i\ln(\frac{1-\gamma_{1}^{*}\gamma_{2}}{1-\gamma_{1}\gamma_{2}^{*}})$

with $\gamma_{1}=e^{-i\varphi_{1}}\tan(\frac{\theta_{1}}{2})$, and
$\gamma_{2}=e^{-i\varphi_{2}}\tan(\frac{\theta_{1}}{2})$. Thus, to have a N00N
state one needs to have

$R(-\theta_{1},\varphi_{1})R(\theta_{2},\varphi_{2})R(\pi,0)=Ie^{i\Theta
J_{z}}$

Where, $\Theta$ is some phase determined by the variables of
$R(\theta_{1},\varphi_{1})$ and $R(\theta_{2},\varphi_{2})$. Thus, with this
description the states $|\theta_{1},\varphi_{1},j\rangle$, and
$|\theta_{2},\varphi_{2},j\rangle$ need to be antipodals. In other cases the
action of the operator $e^{i\phi J_{z}}$ shall not result in the HL metrology.
We may express this in Fock basis as

$\frac{1}{(1+|\gamma|^{2})^{\frac{n}{2}}}\sum_{m=0}^{n}\left(\begin{matrix}n\\\
m\end{matrix}\right)^{\frac{1}{2}}\\{\gamma^{m}+e^{i\psi}(-\frac{1}{\bar{\gamma}})^{m}\\}|m\rangle_{a}|n-m\rangle_{b},$
(26)

where $n$ is the number of the total photons. It is quite evident that this
state is completely different from the N00N state, however, it can give the HL
precision as N00N state. Therefore, such a two-mode superposition of the
photon number states can give the set of the states satisfying this limit.
This may allow us to circumvent the difficulty of generating of the N00N state
and permits more general ways to produce the state that may reach HL
precision.

The interaction Hamiltonian of the two-mode field and the two-level atom reads

$H_{1}=i\hbar\Omega(a^{\dagger}b-b^{\dagger}a)\sigma_{z}$ (27)

Now if the atomic state is initially prepared at
$\frac{1}{\sqrt{2}}(|g\rangle+|e\rangle)$ and one of the fields in the Fock
state $|n\rangle_{b}$, and the other is prepared in the vacuum state
$|0\rangle_{a}$, then

$|\psi(0)\rangle=\frac{1}{\sqrt{2}}(|g\rangle+|e\rangle)|0\rangle_{a}|n\rangle_{b}.$

Therefore, defining $\xi(t)=e^{i\varphi}\tan(\Omega t)$ the initial state
evolves to

$|\psi(t)\rangle=\frac{1}{\sqrt{2}}(|e\rangle\otimes|\xi(t),j\rangle+|g\rangle\otimes|-\xi(t),j\rangle).$

This, in fact, is a type of micro-macro entanglement between the electronic
state of the atom and the cavity fields. We apply a $\frac{\pi}{2}$ pulse on
the atomic state which transforms
$|e\rangle\rightarrow\frac{1}{\sqrt{2}}(|g\rangle+|e\rangle)$ then make a
measurement on the ground state of the electronic state that projects the
state to $\frac{1}{\sqrt{2}}(|\xi(t),j\rangle+|-\xi(t),j\rangle),$ which is a
spin cat state. Therefore, there is a large class of spin cat states that can
give HL metrology.

##  VII. SUMMARY

In this review, we considered quantum architectures designed for the
engineering of the N00N state, a bipartite maximally entangled state crucial
in quantum metrology applications. The fundamental concept underlying these
schemes is the transformation of the initial state $|N\rangle\otimes|0\rangle$
to the N00N state
$\frac{1}{\sqrt{2}}(|N\rangle\otimes|0\rangle+|0\rangle\otimes|N\rangle)$. We
showed that this state can be generated as a superposition of modes of quantum
light, a combination of light and motion, or a superposition of two spin
ensembles. The approach discussed here can generate mesoscopic and macroscopic
entangled states, such as entangled coherent and squeezed states, as well. We
show that a large class of maximally entangled states can be achieved in such
an architecture. The extension of these state engineering methods to the
multi-mode setting was also discussed.

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