# Enhancing phase sensitivity in Mach-Zehnder interferometer with various
detection schemes using SU(1,1) coherent states

N.-E. Abouelkhir  0000-0002-6164-0525 LPHE-Modeling and Simulation, Faculty of
Sciences, Mohammed V University in Rabat, Rabat, Morocco.  A. Slaoui
0000-0002-5284-3240 Corresponding author<EMAIL_ADDRESS>LPHE-
Modeling and Simulation, Faculty of Sciences, Mohammed V University in Rabat,
Rabat, Morocco.  Centre of Physics and Mathematics, CPM, Faculty of Sciences,
Mohammed V University in Rabat, Rabat, Morocco.  E. H. Saidi  LPHE-Modeling
and Simulation, Faculty of Sciences, Mohammed V University in Rabat, Rabat,
Morocco.  Centre of Physics and Mathematics, CPM, Faculty of Sciences,
Mohammed V University in Rabat, Rabat, Morocco.  College of Physical and
Chemical Sciences, Hassan II Academy of Sciences and Technology, Rabat,
Morocco.  R. Ahl Laamara  LPHE-Modeling and Simulation, Faculty of Sciences,
Mohammed V University in Rabat, Rabat, Morocco.  Centre of Physics and
Mathematics, CPM, Faculty of Sciences, Mohammed V University in Rabat, Rabat,
Morocco.  H. El Hadfi  LPHE-Modeling and Simulation, Faculty of Sciences,
Mohammed V University in Rabat, Rabat, Morocco.

###### Abstract

Improving interferometric phase sensitivity is crucial for high-precision
measurements in rapidly developing quantum technologies. The Mach-Zehnder
interferometer (MZI) is a versatile tool for analyzing this phenomenon. By
splitting and recombining a light beam using beam splitters, MZIs allow for
precise phase sensitivity analysis using tools like the quantum Cramér-Rao
bound (QCRB) and the quantum Fisher information (QFI). This paper analyzes the
phase sensitivity of a MZI in various scenarios using different detection
schemes and input states. We compare the single- and two-parameter quantum
estimation and their associated QCRB for three phase-shift situations: in both
arms, only in the upper arm (asymmetric), and in both arms symmetrically. We
then investigate the phase sensitivity under three detection schemes:
difference intensity, single-mode intensity, and balanced homodyne.
Additionally, we explore the use of Perelomov and Barut-Girardello coherent
states, two types of SU(1,1) coherent states, in all scenarios. Notably, we
demonstrate that under optimal conditions, all detection schemes can achieve
the QCRB by utilizing entangled SU(1,1) coherent states as input states.

Keywords: Interferometric phase sensitivity, Barut-Girardello and Perelomov
coherent states, Mach-Zehnder interferometer.

##  I Introduction

Interferometry, exploiting the interaction of superimposed waves, is essential
in precision measurement, quantum metrology, and sensing applications [1, 2,
3, 4]. This technique also contributes significantly to the understanding of
fundamental physics concepts [5]. Notably, many physical quantities, including
distance, local gravity fields, and magnetic fields, are related to the phase
differences of the interfering waves, highlighting the interferometer’s high
sensitivity to phase changes and its wide applicability in precise measurement
and metrology [6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. Phase estimation, a
cornerstone of quantum metrology, is extensively studied due to its crucial
role in diverse precision applications like environmental sensing,
gravitational wave detection [20, 21, 22, 23, 24, 25, 26, 27, 28, 29], and
gyroscopes [30, 31, 32, 33]. This vital role stems from the high sensitivity
of optical interferometry.

In the conventional classical setting with standard resources, the sensitivity
reaches a limit called the shot-noise limit (SNL) or the standard quantum
limit (SQL), scaling as $1/\sqrt{\mathcal{N}}$, where $\mathcal{N}$ is the
number of input photons [34, 10]. To surpass this limit, Caves [34] proposed
the squeezed-state technique, which addresses vacuum fluctuations at the
unused input port. Subsequently, various quantum resources such as entangled
coherent states [35, 36, 37], NOON states [38, 39], number-squeezed states
[40], and two-mode squeezed states [41], have been explored to further enhance
measurement precision, potentially reaching the ultimate quantum limit of
$1/\mathcal{N}$, known as the Heisenberg limit (HL) [42, 43, 44, 41]. In
quantum interferometry, the theoretical limits on phase sensitivity are
derived by applying the quantum Fisher information (QFI) and its corresponding
fundamental limit on precision, the quantum Cramér-Rao bound (QCRB) [45, 46,
47]. Beyond their theoretical significance, these limits prove highly valuable
for assessing the optimality of practical detection schemes. A novel type of
interferometer, referred to as an SU(1,1) interferometer, is configured
similarly to a Mach-Zehnder interferometer (MZI). Yurke et al.[48] first
proposed this concept theoretically, wherein linear beam splitters (BS) are
replaced by non-linear beam splitters (NBS) for the coherent splitting and
mixing of two input fields to achieve precise phase estimation. The term
SU(1,1) is derived from the interaction type utilized in parametric processes
associated with nonlinear wave mixing, distinct from the SU(2) interaction
linked to linear wave mixing through a beam splitter. This nomenclature
reflects the specific nature of the interaction in these two interferometer
types. In this paper, our focus centers on the phase sensitivity of a MZI. It
is recognized that most interferometers can be transformed into an MZI,
enabling the optimization of their phase sensitivity for different input
states and detection schemes.

A critical goal in interferometry is realizing the theoretically optimal phase
sensitivity, which requires optimizing all potential estimators and detection
strategies. The QFI ($\mathcal{H}$) plays a pivotal role in this context [49,
50, 51, 52, 53, 54, 55], being directly linked to the QCRB, expressed as
$\Delta\theta_{QCRB}=1/\sqrt{\mathcal{H}}$. Therefore, finding ways to
increase the QFI becomes an important issue in quantum estimation theory. The
phase sensitivity ($\Delta\theta_{det}$) in any practical detection scheme
always equals or exceeds the QCRB ($\Delta\theta_{QCRB}$), i.e.,
$\Delta\theta_{det}\geq\Delta\theta_{QCRB}$. In a MZI, phase sensitivity
depends on factors like input states and detection schemes. This work explores
entangled SU(1,1) coherent states (CS) for input. The entangled SU(1,1) and
SU(2) coherent states share similarities due to the close relationship between
their Lie algebras. However, SU(1,1) has two relevant types: Perelomov
coherent states (PCS) and Barut-Girardello coherent states (BGCS). PCS,
introduced by Perelomov [56], are analogues of harmonic oscillator coherent
states, achieved by displacing the vacuum state with a displacement operator.
BGCS, introduced by Barut and Girardello [57], are defined as right
eigenstates of the SU(1,1) lowering operator. This work investigates both
types of entangled SU(1,1) CS as input states, alongside three detection
schemes: difference intensity, balanced homodyne detection and single-mode
intensity.

This paper is arranged as follows: Section (II) offers a concise review of
SU(1,1) coherent states. Section (III) introduces conventions and a two-
parameter QFI approach followed by discussion of single-parameter QFI for both
asymmetric and symmetric phase shifts scenarios. In Section (IV), we provide
expressions for the QFIs in all three considered scenarios for input Perelomov
and Barut-Girardello coherent states combined with a vacuum state. The three
detection schemes are described in detail in Section (V), while Section (VI)
analyzes their performances with input SU(1,1) coherent states. Finally,
Section (VII) summarizes the work.

##  II SU(1,1) coherent states

We begin with a brief introduction to the SU(1,1) Lie algebra. This algebra is
spanned by three generators, $\hat{A}_{z}$, $\hat{A}_{+}$, and $\hat{A}_{-}$,
satisfying the commutation relations

$[\hat{A}_{+},\hat{A}_{-}]=-2\hat{A}_{z},\hskip
28.45274pt[\hat{A}_{z},\hat{A}_{\pm}]=\pm\hat{A}_{\pm}$ (1)

In this work, we focus on input states in the context of optical fields,
particularly the entangled SU(1,1) coherent states (CSs). To examine these
states, we employ the Holstein-Primakoff realization (HPR), a potent
theoretical tool in quantum optics. The SU(1,1) CSs can be characterized by a
set of single-mode Bose annihilation and creation operators, aligning with the
HPR representation of the SU(1,1) Lie algebra. This HPR form is given by the
operators

$\displaystyle\hat{A}_{+}=\hat{b}^{\dagger}(\hat{b}^{\dagger}\hat{b}+2a)^{1/2},$
$\displaystyle\hat{A}_{-}=(\hat{b}^{\dagger}\hat{b}+2a)^{1/2}\hat{b},\hskip
28.45274pt\hat{A}_{z}=a+\hat{b}^{\dagger}\hat{b},$ (2)

where the operators $\hat{b}$ and $\hat{b}^{\dagger}$ satisfy the Bose algebra
$[\hat{b},\hat{b}^{\dagger}]=\openone$ and the action of the operators
$\hat{A}_{z}$, $\hat{A}_{+}$, and $\hat{A}_{-}$ on the Fock space states
$|a,g\rangle$ (with $g=0,1,2,...$) is given by the following

$\displaystyle\hat{A}_{+}|a,g\rangle=$
$\displaystyle\left((g+1)(2a+g)\right)^{1/2}|a,g\rangle,$ (3)
$\displaystyle\hat{A}_{-}|a,g\rangle=$
$\displaystyle\left(g(2a+g)\right)^{1/2}|a,g\rangle,$ (4)
$\displaystyle\hat{A}_{z}|a,g\rangle=$ $\displaystyle(g+a)|a,g\rangle.$ (5)

The parameter $a$ called the Bargmann index, is associated with the eigenvalue
determination of the SU(1,1) Casimir operator. This operator is given by
$\hat{C}=\hat{A}^{2}_{z}-(\hat{A}_{+}\hat{A}_{-}+\hat{A}_{-}\hat{A}_{+})/2$,
and by evaluating the eigenvalues of $\hat{C}$, we can obtain the form
$a(a-1)$. Here, we confine our attention solely to the discrete series, which
$a$ is greater than zero, taking values such as $a=1/2,1,3/2,...$.

Now we can move on to the construction of the SU(1,1) CSs. As this group is
non-compact, all its unitary representations must be infinite-dimensional.
Therefore, there are two principal types of coherent states. The first type
consists of displacing the vacuum state by the displacement operator, which
generates the Perelomov coherent states (PCSs). The second type of SU(1,1)
CSs, namely, the Barut-Girardello coherent states (BGCSs), are defined as a
right eigenstate of the SU(1,1) lowering operator.

###  II.1 SU(1, 1) Perelomov coherent states

Following Perelomov’s work [56], the standard PCSs are defined as

$|\xi,a\rangle=D(\mu)|0,a\rangle,$ (6)

with $D(\mu)$ is the displacement operator for this group, defined as

$D(\mu)=e^{\mu\hat{A}_{+}-\mu^{\ast}\hat{A}_{-}},$ (7)

with $\mu$ is a complex number. Using the property
$\hat{A}_{+}^{\dagger}=\hat{A}_{-}$, we can show the following property of the
displacement operator: $D^{+}(\mu)=D(-\mu)$. This operator $D(\mu)$ can be
rewritten as

$D(\mu)=e^{z\hat{A}_{+}}e^{\eta\hat{A}_{z}}e^{-\xi^{\ast}\hat{A}_{-}},$ (8)

where $\xi=e^{-i\varphi}\tanh|\mu|$, $\mu=e^{-i\varphi}\vartheta/2$ and
$\eta=\ln(1-|\xi|^{2})$. The parameter $\vartheta$ is a hyperbolic angle with
$0\leq\vartheta<\infty$ and the angle $\varphi$ is azimuthal with
$0\leq\varphi\leq 2\pi$. By using this equation, we can directly obtain the
PCSs in the form of

$|\xi,a\rangle=(1-|\xi|^{2})^{l}\sum_{g=0}^{\infty}\sqrt{\frac{\Gamma(g+2a)}{g!\Gamma(2a)}}\xi^{g}|a,g\rangle,$
(9)

where $\Gamma(x)$ is the gamma function.

###  II.2 Barut-Girardello Coherent States

Here, we would like to construct the Barut-Girardello coherent state. This
state is defined as the solution to the eigenvalue equation for the
annihilation operator $\hat{A}_{-}$ [57]:

$\hat{A}_{-}|\xi,a\rangle=\xi|\xi,a\rangle,\hskip 28.45274pta>0,$ (10)

where $\xi$ is an arbitrary complex number. In addition, we can decompose the
eigenstates $|\xi,a\rangle$ as a superposition of the complete orthonormal
basis $\\{|\nu,a\rangle\\}$

$|\xi,a\rangle=\sum_{g=0}^{\infty}\langle\nu,l|\xi,a\rangle|\nu,a\rangle.$
(11)

Let the annihilation operator $\hat{A}_{-}$ act on Eq.(11). Then, utilizing
Eqs.(10) and (3) along with the following orthonormality relation

$\langle g,a|g^{\prime},a\rangle=\delta_{\nu\nu^{\prime}},\hskip
28.45274pt\sum_{g=0}^{\infty}|g,a\rangle\langle g,a|=\openone,$ (12)

we can obtain

$\langle g,a|\xi,a\rangle=\frac{z}{\sqrt{n\Gamma(g+2a)}}\langle
g-1,a|\xi,a\rangle.$ (13)

After the recurrence procedure, this equation above transforms into the
following

$\langle g,a|\xi,a\rangle=\xi^{g}\frac{\Gamma(2a)}{g!\Gamma(g+2a)}\langle
0,a|\xi,a\rangle.$ (14)

When we normalize the states $|\xi,a\rangle$ to unity, we have

$|\xi,a\rangle=\sqrt{\frac{|\xi|^{2a-1}}{I_{2a-1}(2|\xi|)}}\sum_{g=0}^{\infty}\frac{\xi^{g}}{\sqrt{g!\Gamma(g+2a)}}|g,a\rangle,$
(15)

where $I_{m}$ is the modified Bessel function of order $m$, defined as

$I_{m}(x)=\sum_{m=0}^{\infty}\frac{1}{m!\Gamma(m+g+1)}\left(x/2\right)^{2m+g}.$
(16)

##  III Quantum Fisher information in Mach-Zehnder Interferometer

We are considering the standard Mach-Zehnder (MZ) interferometric setup
illustrated in Figure (1). In this setup, the two beam splitters, BS1 and BS2,
have transmission (reflection) coefficients $\alpha$ ($\beta$) and
$\alpha^{\prime}$ ($\beta^{\prime}$), respectively. Throughout this work, the
input state is assumed to be pure, with no losses. In general, the precision
of phase estimation in quantum interferometry is bounded by the QFI, which
depends on the way the interferometer phase delay is modeled: (a) two
independent phase shifts, $\theta_{1}$ ($\theta_{2}$) in the upper (lower)
arm; (b) single phase shift in the lower arm; (c) two phase shifts distributed
symmetrically, $\pm\theta/2$.

Figure 1: Experimental setup of a Mach-Zehnder interferometer with three
realistic detection schemes: difference-intensity detection, single-mode
intensity detection and balanced homodyne detection. In these schemes, the
input state $|\psi_{in}\rangle$ undergoes a unitary transformation, resulting
in the output state $|\psi_{out}\rangle$. The primary objective is to estimate
the phase difference $\theta$ between the arms of the MZI using an appropriate
observable.

We first consider the most general scenario, in which the upper and lower arms
of an interferometer contain a phase shift, denoted by $\theta_{1}$ and
$\theta_{2}$, respectively. According to the literature [58, 59, 60, 61], a
two-parameter estimation technique can be used to avoid the problem of
counting supplementary resources, such as an external phase reference, that
are not available. In case where no external phase reference is available, we
are only interested in the phase shift difference
$\theta_{dif}=\theta_{1}-\theta_{2}$. Therefore, it is more convenient to
express the QFIM in the basis $\theta_{s/d}=\theta_{1}\pm\theta_{2}$. To
estimate the values of $\theta_{s}$ and $\theta_{dif}$, we utilize the QFIM,
represented by a $2\times 2$ matrix [62, 63]:

$\mathcal{H}=\left(\begin{array}[]{ll}\mathcal{H}_{dd}&\mathcal{H}_{sd}\\\
\mathcal{H}_{ds}&\mathcal{H}_{ss}\end{array}\right),$ (17)

with

$\mathcal{H}_{ij}=4\mathfrak{R}_{e}\\{\langle\partial_{i}\psi|\partial_{j}\psi\rangle-\langle\partial_{i}\psi|\psi\rangle\langle\psi|\partial_{j}\psi\rangle\\},$
(18)

where $Re$ is the real part, and the subscripts $i$ and $j$ correspond to
$\theta_{s}$ and $\theta_{dif}$, respectively. We consider the wavevector
$|\psi\rangle$, which is expressed as

$|\psi\rangle=\exp\\{-i\frac{\hat{g}_{2}-\hat{g}_{3}}{2}\theta_{dif}\\}\exp\\{-i\frac{\hat{g}_{2}+\hat{g}_{3}}{2}\theta_{s}\\}|\psi^{\prime}\rangle.$
(19)

Here, $\hat{g}_{l}=\hat{b}^{\dagger}_{l}\hat{b}_{l}$ denotes the number
operator corresponding to port $l$. By applying the field operator
transformations above to the input state $|\psi\rangle$, we can obtain the
state $|\psi^{\prime}\rangle$ as

$\hat{b}_{2}=\alpha\hat{b}_{0}+\beta\hat{b}_{1},\hskip
28.45274pt\hat{b}_{3}=\beta\hat{b}_{0}+\alpha\hat{b}_{1}.$ (20)

We have the equation $|\alpha|^{2}+|\beta|^{2}=1$, and
$\alpha\beta^{\ast}=-\alpha^{\ast}\beta$. This relationship implies that
$\alpha^{\ast}\beta=\pm i\alpha\beta$. Without loss of generality, throughout
this work we will adopt the convention $\alpha^{\ast}\beta=i|\alpha\beta|$.
The QCRB is a lower bound on the variance of any unbiased estimator of a
parameter and provides the minimum limit for estimating the parameter. In the
case of multiparameter estimation, the QCRB is given as

$Cov(\hat{\theta})\geq\frac{1}{N}\mathcal{H}^{-1},$ (21)

with $N$ repeated experiments, $\mathcal{H}$ is the QFIM given in Eq.(17) and
$Cov(\hat{\theta})$ stands for the covariance matrix of the estimator,
including both $\theta_{dif}$ and $\theta_{s}$, whose elements are
$Cov(\hat{\theta})_{ij}=E(\hat{\theta}_{i}\hat{\theta}_{j})-E(\hat{\theta}_{i})E(\hat{\theta}_{j})$
with $E$ being a mathematical expectation. Here, we set $N=1$, and when
examining phase difference sensitivity, we have

$(\Delta\theta_{dif})^{2}\geq(\mathcal{H}^{-1})_{dd}.$ (22)

The expression for the first diagonal element of the inverse matrix
$\mathcal{H}^{-1}$ is as follows

$\mathcal{H}^{(a)}=\frac{1}{(\mathcal{H}^{-1})_{dd}}=\mathcal{\mathcal{H}}_{dd}-\frac{(F_{sd})^{2}}{\mathcal{H}_{ss}},$
(23)

thus, inequality (21) can be saturated, which implies the two-parameter QCRB
is reduced to a simpler form of

$\Delta\theta_{QCRB}^{(a)}=\frac{1}{\sqrt{\mathcal{H}^{(a)}}}.$ (24)

According to the definition of $\mathcal{H}_{ij}$ in equation (18), the
elements of the QFIM given in equation (17), namely $\mathcal{H}_{ss}$,
$\mathcal{H}_{dd}$, and $\mathcal{H}_{sd}$, can be determined as follows

$\mathcal{H}_{ss}=\Delta^{2}\hat{g}_{0}+\Delta^{2}\hat{g}_{1},$ (25)
$\displaystyle\mathcal{H}_{dd}=$
$\displaystyle\left(2|\alpha|^{2}-1\right)^{2}\left(\Delta^{2}\hat{g}_{0}+\Delta^{2}\hat{g}_{1}\right)+8|\alpha\beta|^{2}\left(\langle\hat{g}_{0}\rangle\langle\hat{g}_{1}\rangle-|\langle\hat{b}_{0}\rangle|^{2}|\langle\hat{b}_{1}\rangle|^{2}-\mathfrak{R}_{e}\left\\{\langle(\hat{b}_{0}^{\dagger})^{2}\rangle\langle\hat{b}_{1}^{2}\rangle-\langle\hat{b}_{0}^{\dagger}\rangle^{2}\langle\hat{b}_{1}\rangle^{2}\right\\}\right)$
$\displaystyle+4|\alpha\beta|^{2}\left(\langle\hat{g}_{0}\rangle+\langle\hat{g}_{1}\rangle\right)-8|\alpha\beta|\left(2|\alpha|^{2}-1\right)\left(\mathfrak{I}_{m}\left\\{\left(\langle\hat{b}_{0}^{\dagger}\hat{g}_{0}\rangle-\langle\hat{b}_{0}^{\dagger}\rangle\langle\hat{g}_{0}\rangle\right)\langle\hat{b}_{1}\rangle+\langle\hat{b}_{0}\rangle\left(\langle\hat{b}_{1}^{\dagger}\hat{g}_{1}\rangle-\langle\hat{g}_{1}\rangle\langle\hat{b}_{1}^{\dagger}\rangle\right)\right\\}\right),$
(26) $\displaystyle\mathcal{H}_{sd}=$
$\displaystyle\mathcal{H}_{ds}=\left(2|\alpha|^{2}-1\right)\left(\Delta^{2}\hat{g}_{0}-\Delta^{2}\hat{g}_{1}\right)+4|\alpha\beta|\mathfrak{I}_{m}\left\\{\langle\hat{b}_{0}\rangle\langle\hat{b}^{\dagger}_{1}\rangle-\left(\langle\hat{g}_{0}\hat{b}_{0}\rangle-\langle\hat{g}_{0}\rangle\langle\hat{b}_{0}\rangle\right)\langle\hat{b}^{\dagger}_{1}\rangle+\langle\hat{b}_{0}\rangle\left(\langle\hat{b}_{1}^{\dagger}\hat{g}_{1}\rangle-\langle\hat{b}^{\dagger}_{1}\rangle\langle\hat{g}_{1}\rangle\right)\right\\},$
(27)

where $\Delta^{2}\hat{g}$ represents the variance of the number operator
$\hat{g}$, defined as
$\Delta^{2}\hat{g}=\langle\hat{g}^{2}\rangle-\langle\hat{g}\rangle^{2}$.

In the case of a phase shift in one arm, assuming it occurs in output $3$ of
BS1, and using the notations from Fig.(1), we have the state transformed as
$|\psi\rangle=e^{-i\theta\hat{g}_{3}}|\psi^{\prime}\rangle$. Using definition
(18), the single-parameter QFI, denoted as $\mathcal{H}^{(b)}$, is given by

$\mathcal{H}^{(b)}=4\Delta^{2}\hat{g}_{3},$ (28)

and the QCRB, which provides the optimal phase estimation, is given by

$\Delta\theta_{QCRB}^{(b)}=\frac{1}{\sqrt{4\Delta^{2}\hat{g}_{3}}}.$ (29)

By applying the field operator transformations (20), we obtain the following
expression

$\displaystyle\mathcal{H}^{(b)}=$ $\displaystyle
4|\beta|^{4}\Delta^{2}\hat{g}_{0}+4|\alpha|^{4}\Delta^{2}\hat{g}_{1}$ (30)
$\displaystyle+4|\alpha\beta|^{4}\left(\langle\hat{g}_{0}\rangle+\langle\hat{g}_{1}\rangle+2\left(\langle\hat{g}_{0}\rangle\langle\hat{g}_{1}\rangle-|\langle\hat{b}_{0}\rangle|^{2}|\langle\hat{b}_{1}\rangle|^{2}\right)\right)$
$\displaystyle-8|\alpha\beta|^{2}\mathfrak{R}_{e}\left\\{\langle\hat{b}^{2}_{0}\rangle\langle(\hat{b}^{\dagger}_{1})^{2}\rangle-\langle\hat{b}_{0}\rangle\langle\hat{b}^{\dagger}_{1}\rangle\right\\}-8|\alpha\beta|\mathfrak{I}_{m}\left\\{\langle\hat{b}_{0}\rangle\langle\hat{b}^{\dagger}_{1}\rangle\right\\}$
$\displaystyle-16|\alpha\beta||\beta|^{2}\mathfrak{I}_{m}\left\\{\left(\langle\hat{g}_{0}\hat{b}_{0}\rangle-\langle\hat{g}_{0}\rangle\langle\hat{b}_{0}\rangle\right)\langle\hat{b}^{\dagger}_{1}\rangle\right\\}$
$\displaystyle-16|\alpha\beta||\alpha|^{2}\mathfrak{I}_{m}\left\\{\langle\hat{b}_{0}\rangle\left(\langle\hat{b}^{\dagger}_{0}\hat{g}_{1}\rangle-\langle\hat{g}_{1}\rangle\langle\hat{b}^{\dagger}_{1}\rangle\right)\right\\}.$

Comparing equation (30) with the elements of the QFIM derived from the first
estimation scenario, we can see that the single-parameter QFI,
$\mathcal{H}^{(b)}$, can be expressed in terms of the coefficients of the QFIM
as follows

$\mathcal{H}^{(b)}=\mathcal{H}_{dd}+\mathcal{H}_{ss}-2\mathcal{H}_{sd}.$ (31)

If $\mathcal{H}_{sd}=\mathcal{H}_{ss}$, then the above equation and the two-
parameter QFI (23) are equal, $\mathcal{H}^{(a)}=\mathcal{H}^{(b)}$. In this
case, we can prove that

$\mathcal{H}^{(b)}\geq\mathcal{H}^{(a)}.$ (32)

In the last case, which is essentially a single-parameter estimation problem
similar to the second scenario, it is modeled by the unitary operation as
$U(\theta)=e^{i\frac{\theta}{2}(\hat{g}_{2}-\hat{g}_{3})}$. Consequently, the
QFI is given by

$\mathcal{H}^{(c)}=\Delta^{2}\hat{g}_{2}+\Delta^{2}\hat{g}_{3}.$ (33)

Similarly in equation (30), we obtain

$\displaystyle\mathcal{H}^{(c)}=\left(|\alpha|^{4}+|\beta|^{4}\right)\left(\Delta^{2}\hat{g}_{0}+\Delta^{2}\hat{g}_{1}\right)$
(34)
$\displaystyle+2|\alpha\beta|^{2}\left(\langle\hat{g}_{0}\rangle+\langle\hat{g}_{1}\rangle+2\left(\langle\hat{g}_{0}\rangle\langle\hat{g}_{1}\rangle-|\langle\hat{b}_{0}\rangle|^{2}|\langle\hat{b}_{1}\rangle|^{2}\right)\right)$
$\displaystyle-2|\alpha\beta|^{2}\left(\langle\hat{b}^{2}_{0}\rangle\langle(\hat{b}^{\dagger}_{1})^{2}\rangle+\langle(\hat{b}^{\dagger}_{0})^{2}\rangle\langle\hat{b}^{2}_{1}\rangle-\langle\hat{b}_{0}\rangle^{2}\langle\hat{b}^{\dagger}_{1}\rangle^{2}-\langle\hat{b}^{\dagger}_{0}\rangle^{2}\langle\hat{b}_{1}\rangle^{2}\right)$
$\displaystyle+2\alpha^{\ast}\beta\left(2|\alpha|^{2}-1\right)\left(\langle\hat{b}^{\dagger}_{0}\hat{g}_{0}\rangle-\langle\hat{b}^{\dagger}_{0}\rangle\langle\hat{g}_{0}\rangle\right)\langle\hat{b}_{1}\rangle$
$\displaystyle-2\alpha^{\ast}\beta\left(2|\alpha|^{2}-1\right)\left(\langle\hat{g}_{0}\hat{b}_{0}\rangle-\langle\hat{g}_{0}\rangle\langle\hat{b}_{0}\rangle\right)\langle\hat{b}^{\dagger}_{1}\rangle$
$\displaystyle+2\alpha^{\ast}\beta\left(2|\alpha|^{2}-1\right)\langle\hat{b}_{0}\rangle\left(\langle\hat{b}^{\dagger}_{1}\hat{g}_{1}\rangle-\langle\hat{b}^{\dagger}_{1}\rangle\langle\hat{g}_{1}\rangle\right)$
$\displaystyle-2\alpha^{\ast}\beta\left(2|\alpha|^{2}-1\right)\langle\hat{b}^{\dagger}_{0}\rangle\left(\langle\hat{g}_{1}\hat{b}_{1}\rangle-\langle\hat{g}_{1}\rangle\langle\hat{b}_{1}\rangle\right),$

and this implies that the corresponding QCRB becomes

$\Delta\theta_{QCRB}^{(c)}=\frac{1}{\sqrt{\mathcal{H}^{(c)}}}.$ (35)

##  IV Phase estimation with a vacuum state in the first input and SU(1,1)
coherent states in the second input

In this section, our focus lies on input states characterized by entangled
SU(1,1) coherent states combined with a vacuum state. We have identified two
principal types of SU(1,1) CSs: The first type, termed as the Perelomov
coherent states (PCSs; see Eq.(9)), and the second type, the Barut-Girardello
coherent states (BGCSs; see Eq.(15)). The input state is represented as
follows

$|\psi_{in}\rangle=|\xi_{i},a\rangle_{1}\otimes|0\rangle_{0},$ (36)

where the subscript $i=P$ or $B$ corresponds to entangled PCS (Eq.9) and
entangled BGCS (Eq.15), respectively. We also denote the QFI as
$\mathcal{H}_{i}$. Using the results of the three QFIs reported in the
previous section, i.e., $\mathcal{H}^{(a)}$, $\mathcal{H}^{(b)}$, and
$\mathcal{H}^{(c)}$, given by equations (23), (30, and (34), it is easy to
verify that the QFIs in our input states are given by

$\displaystyle\mathcal{H}^{(a)}_{i}=$ $\displaystyle
4|\alpha\beta|^{2}\langle\hat{g}_{1}\rangle_{i},$ (37)
$\displaystyle\mathcal{H}^{(b)}_{i}=$ $\displaystyle
4|\alpha|^{4}\Delta^{2}\hat{g}_{1}+4|\alpha\beta|^{2}\langle\hat{g}_{1}\rangle_{i},$
(38) $\displaystyle\mathcal{H}^{(c)}_{i}=$
$\displaystyle(|\alpha|^{4}+|\beta|^{4})\Delta^{2}\hat{g}_{1}+2|\alpha\beta|^{2}\langle\hat{g}_{1}\rangle_{i}.$
(39)

Using the equations derived above, we can calculate the QFI for each type of
SU(1,1) coherent state. We denote the QFI for the first type PCSs as
$\mathcal{H}_{P}$ and the QFI for the second type BGCSs as $\mathcal{H}_{B}$.
For PCSs, the analytical expressions for the QFIs simplify to

$\displaystyle\mathcal{H}^{(a)}_{P}=$ $\displaystyle
4a|\alpha\beta|^{2}\left(\cosh(v)-1\right),$ (40)
$\displaystyle\mathcal{H}^{(b)}_{P}=$ $\displaystyle
4a|\alpha|^{2}\left(\frac{1}{2}|\alpha|^{2}\sinh^{2}(v)+|\beta|^{2}\left(\cosh(v)-1\right)\right),$
(41) $\displaystyle\mathcal{H}^{(c)}_{P}=$
$\displaystyle\frac{a}{2}\left(|\alpha|^{4}+|\beta|^{4}\right)\sinh^{2}(v)+2a|\alpha\beta|^{2}\left(\cosh(v)-1\right).$
(42)

Therefore, the corresponding QCRBs are

$\displaystyle\Delta\theta^{(a)}_{QCRB,P}=$
$\displaystyle\frac{1}{2|\alpha\beta|\sqrt{a\left(\cosh(v)-1\right)}},$ (43)
$\displaystyle\Delta\theta^{(b)}_{QCRB,P}=$
$\displaystyle\frac{1}{2|\alpha|\sqrt{\frac{a}{2}|\alpha|^{2}\sinh^{2}(v)+a|\beta|^{2}\left(\cosh(v)-1\right)}},$
(44) $\displaystyle\Delta\theta^{(c)}_{QCRB,P}=$
$\displaystyle\frac{1}{\sqrt{\frac{a}{2}\left(|\alpha|^{4}+|\beta|^{4}\right)\sinh^{2}(v)+2a|\alpha\beta|^{2}\left(\cosh(v)-1\right)}},$
(45)

In the case of BGCSs, the QFIs take the following analytical form

$\displaystyle\mathcal{H}^{(a)}_{B}=$ $\displaystyle
4|\xi||\alpha\beta|^{2}\frac{I_{2a}(2|\xi|)}{I_{2a-1}(2|\xi|)},$ (46)
$\displaystyle\mathcal{H}^{(b)}_{B}=$
$\displaystyle\frac{4|\xi|}{I_{2a-1}^{2}(2|\xi|)}\left[|\alpha|^{4}X+|\alpha\beta|^{2}I_{2a-1}(2|\xi|)I_{2a}(2|\xi|)\right],$
(47) $\displaystyle\mathcal{H}^{(c)}_{B}=$
$\displaystyle\frac{|\xi|}{I_{2a-1}^{2}(2|\xi|)}\left[\left(|\alpha|^{4}+|\beta|^{4}\right)X+2|\alpha\beta|^{2}I_{2a-1}(2|\xi|)I_{2a}(2|\xi|)\right],$
(48)

where

$X=I_{2a-1}(2|\xi|)\left[|\xi|I_{2a+1}(2|\xi|)+I_{2a}(2|\xi|)\right]-|\xi|I_{2a}^{2}(2|\xi|),$
(49)

and

$\displaystyle\Delta\theta^{(a)}_{QCRB,B}=$
$\displaystyle\frac{1}{2|\alpha\beta|}\sqrt{\frac{I_{2a-1}(2|\xi|)}{|\xi|I_{2a}(2|\xi|)}},$
(50) $\displaystyle\Delta\theta^{(b)}_{QCRB,B}=$
$\displaystyle\frac{I_{2a-1}(2|\xi|)}{2|\alpha|\sqrt{|\xi|\left(|\alpha|^{2}X+|\beta|^{2}I_{2a-1}(2|\xi|)I_{2a}(2|\xi|)\right)}},$
(51) $\displaystyle\Delta\theta^{(c)}_{QCRB,B}=$
$\displaystyle\frac{I_{2a-1}(2|\xi|)}{\sqrt{|\xi|\left[\left(|\alpha|^{4}+|\beta|^{4}\right)X+2|\alpha\beta|^{2}I_{2a-1}(2|\xi|)I_{2a}(2|\xi|)\right]}}.$
(52)

In the balanced scenario (i.e., $|\alpha|=|\beta|=1/\sqrt{2}$), the QFIs
associated with PCSs takes the form

$\displaystyle\mathcal{H}^{(a)}_{P}=$ $\displaystyle
a\left(\cosh(v)-1\right),$ (53) $\displaystyle\mathcal{H}^{(b)}_{P}=$
$\displaystyle a\left(\frac{1}{2}\sinh^{2}(v)+\cosh(v)-1\right),$ (54)
$\displaystyle\mathcal{H}^{(c)}_{P}=$
$\displaystyle\frac{1}{4}a\sinh^{2}(v)+\frac{1}{2}\left(\cosh(v)-1\right).$
(55)

The corresponding QCRBs are given by

$\displaystyle\Delta\theta^{(a)}_{QCRB,P}=$
$\displaystyle\frac{1}{\sqrt{a\left(\cosh(v)-1\right)}},$ (56)
$\displaystyle\Delta\theta^{(b)}_{QCRB,P}=$
$\displaystyle\frac{1}{\sqrt{a\left(\frac{1}{2}\sinh^{2}(v)+\cosh(v)-1\right)}},$
(57)

and

$\Delta\theta^{(c)}_{QCRB,P}=\frac{1}{\sqrt{\frac{a}{4}\sinh^{2}(v)+\frac{a}{2}(\cosh(v)-1)}}.$
(58)

Similarly for the BGCSs, we can write the analytical expressions as

$\displaystyle\mathcal{H}^{(a)}_{B}=$
$\displaystyle|\xi|\frac{I_{2a}(2|\xi|)}{I_{2a-1}(2|\xi|)},$ (59)
$\displaystyle\mathcal{H}^{(b)}_{B}=$
$\displaystyle\frac{|\xi|}{I_{2a-1}^{2}(2|\xi|)}\left[X+I_{2a-1}(2|\xi|)I_{2a}(2|\xi|)\right],$
(60) $\displaystyle\mathcal{H}^{(c)}_{B}=$
$\displaystyle\frac{|\xi|}{2I_{2a-1}^{2}(2|\xi|)}\left[X+I_{2a-1}(2|\xi|)I_{2a}(2|\xi|)\right].$
(61)

and the corresponding QCRBs take the form

$\displaystyle\Delta\theta^{(a)}_{QCRB,B}=$
$\displaystyle\sqrt{\frac{I_{2a-1}(2|\xi|)}{|\xi|I_{2a}(2|\xi|)}},$ (62)
$\displaystyle\Delta\theta^{(b)}_{QCRB,B}=$
$\displaystyle\frac{I_{2a-1}(2|\xi|)}{\sqrt{|\xi|\left[X+I_{2a-1}(2|\xi|)I_{2a}(2|\xi|)\right]}},$
(63)

and

$\Delta\theta^{(c)}_{QCRB,B}=\frac{\sqrt{2}I_{2a-1}(2|\xi|)}{\sqrt{|\xi|\left[X+I_{2a-1}(2|\xi|)I_{2a}(2|\xi|)\right]}}.$
(64)

$\left(a\right)$

$\left(b\right)$

Figure 2: The dynamics of the three quantum Fisher information metrics,
denoted as $\mathcal{H}^{(a)}$, $\mathcal{H}^{(b)}$, and $\mathcal{H}^{(c)}$,
with respect to the transmission coefficient $|\alpha|^{2}$ of the first beam
splitter for a single input state; Panel ($a$) for the Perelomov coherent
input state with parameters $a=1$ and $v=1$, and panel ($b$) for the Barut-
Girardello coherent input state with parameters $a=1$ and $v=1$. The graphical
representation distinguishes $\mathcal{H}^{(a)}$, $\mathcal{H}^{(b)}$ and
$\mathcal{H}^{(c)}$: the blue line for $\mathcal{H}^{(a)}$, the red line for
$\mathcal{H}^{(b)}$, and the long black line for $\mathcal{H}^{(c)}$.

In Figure (2), we illustrate the dynamic behavior of the three QFIs metrics
considered, $\mathcal{H}^{(a)}$, $\mathcal{H}^{(b)}$, and $\mathcal{H}^{(c)}$,
with respect to the transmission coefficient $|\alpha|^{2}$ of the first beam
splitter for both the Perelomov coherent states (PCS) and the Barut-Girardello
coherent states (BGCS) with parameters $a=1$ and $v=1$. In Fig.2($a$),
focusing on the scenario where the single input state is PCS, the plots show
that the QFI $\mathcal{H}^{(c)}_{P}$ remains nearly constant for different
values of the transmission coefficient $\alpha$, while the QFI
$\mathcal{H}^{(b)}_{P}$ exhibits a linear variation, increasing steadily from
$0$ for $|\alpha|=0$ to $2a\sinh^{2}(v)$ for $|\alpha|=1$. Moreover, the two-
parameter QFI, represented by $\mathcal{H}^{(a)}_{P}$, reaches its maximum
value in the balanced case, i.e., when $|\alpha|=|\beta|=1/\sqrt{2}$. In the
extreme case where $|\alpha|$ equals $0$ or $1$, the QFI reaches its minimum
value, $\mathcal{H}^{(a)}_{P}=0$. Moving to Fig.2($b$), the QFI is plotted for
the single input state BGCS. Here, $\mathcal{H}^{(c)}$ remains constant, while
$\mathcal{H}^{(b)}_{P}$ shows a linear progression, consistently ranging from
its minimum for $|\alpha|=0$ to its maximum for $|\alpha|=1$, which
corresponds to $4|\xi|/I_{2a-1}^{2}(2|\xi|)$. Furthermore, for the two-
parameter QFI, $\mathcal{H}^{(a)}{P}$ reaches its maximum value when
$|\alpha|=|\beta|=1/\sqrt{2}$. The QFI vanishes in the extreme cases where
$|\alpha|$ equals $0$ or $1$. More importantly, comparing Fig.2($a$) with
Fig.2($b$), we observe that the values of the phase estimate for PCSs are
larger than those for BGCSs in all the scenarios considered.

##  V Phase sensitivity in a Mach-Zehnder interferometer

We will now proceed to close the Mach-Zehnder interferometer (MZI) using BS2,
which is characterized by its transmission (reflection) coefficient
$\alpha^{\prime}$ ($\beta^{\prime}$). We will then examine the performance of
three realistic detection schemes: namely, the difference intensity, the
single-mode intensity, and the balanced homodyne detection. To delve deeper
into the analysis, we explore the quantum parameter estimation problem. In
this context, we consider an experimentally accessible Hermitian operator,
denoted by $\hat{S}$ , that depends on a parameter $\theta$. In our specific
scenario, $\theta$ corresponds to the phase shift in a MZI and may or may not
be observable. Its average is given by

$\langle\hat{S}(\theta)\rangle=\langle\psi|\hat{S}(\theta)|\psi\rangle,$ (65)

where $|\psi\rangle$ represents the wave function of the system. When a small
variation $\delta\theta$ is applied to the parameter $\theta$, it induces a
change described as

$\langle\hat{S}(\theta+\delta\theta)\rangle\approx\langle\hat{S}(\theta)\rangle+\frac{\partial\langle\hat{S}(\theta)\rangle}{\partial\theta}\delta\theta.$
(66)

The experimental detectability of the difference between
$\langle\hat{S}(\theta+\delta\theta)\rangle$ and
$\langle\hat{S}(\theta)\rangle$ depends on satisfying the condition

$\langle\hat{S}(\theta+\delta\theta)\rangle-\langle\hat{S}(\theta)\rangle\geq\Delta\hat{S}(\theta),$
(67)

where $\Delta\hat{S}$ is the standard deviation of $\hat{S}$ and is defined as
the square root of the variance $\Delta^{2}\hat{S}$, which is expressed as
$\Delta\hat{S}=\sqrt{\langle\hat{S}^{2}\rangle-\langle\hat{S}\rangle^{2}}$. If
the inequality (67) is saturated by the value of $\delta\theta$, then this
variation $\delta\theta$ is called the sensitivity, denoted by $\Delta\theta$
[64]

$\Delta\theta=\frac{\Delta\hat{S}}{|\frac{\partial}{\partial\theta}\langle\hat{S}\rangle|}.$
(68)

In the following, $\theta$ represents the total phase shift inside the
interferometer, which is divided into two parts: the first part, denoted as
$\theta_{i}$, represents the quantity we want to measure, and the second part
is $\theta_{exp}$, which is experimentally controllable. We express this
relationship as $\theta=\theta_{i}+\theta_{exp}$. In interferometry, the
condition $|\theta_{i}|\ll|\theta|$ is crucial because it indicates that the
unknown phase shift $\theta_{i}$ has a limited effect on the total phase shift
$\theta$. Therefore, the experimenter must adjust $\theta_{exp}$ to approach
the optimal phase shift, denoted as $\theta_{opt}$, to achieve the best
performance. In the above description, we will focus on the phase sensitivity
for each of the considered detection schemes, i.e., difference intensity
detection, single-mode intensity detection, and balanced homodyne detection.

###  V.1 Difference intensity detection scheme

In the difference intensity detection scheme, which is only sensitive to the
difference between the phase shifts $\theta_{1}$ and $\theta_{2}$, We compute
the disparity in the output photocurrents, labeled as $\hat{G}_{dif}$,
specifically those detected at ${\rm D}_{4}$ and ${\rm D}_{5}$, as illustrated
in Fig.(1). Thus, the output operator $\hat{G}_{dif}$ is defined as

$\hat{G}_{dif}=\hat{b}^{\dagger}_{4}\hat{b}_{4}-\hat{b}^{\dagger}_{5}\hat{b}_{5}.$
(69)

To express the operator $\hat{G}_{dif}$ in terms of the input field operators,
we need the field operator transformations,

$\hat{b}_{4}=\alpha^{\prime}\hat{b}_{2}+\beta^{\prime}\hat{b}_{3},\quad\hat{b}_{5}=\beta^{\prime}\hat{b}_{2}+\alpha^{\prime}\hat{b}_{3},$
(70)

where $\alpha^{\prime}$ ($\beta^{\prime}$) represents the transmission
(reflection) coefficients of the second beam splitter (BS2). Using the field
operator transformations (20), we obtain

$\displaystyle\hat{b}_{4}=[(\alpha\alpha^{\prime}e^{-i\theta_{2}}+\beta\beta^{\prime}e^{-i\theta_{1}})\hat{b}_{0}+(\alpha\beta^{\prime}e^{-i\theta_{1}}+\beta\alpha^{\prime}e^{-i\theta_{2}})\hat{b}_{1}],$
$\displaystyle\hat{b}_{5}=[(\alpha\beta^{\prime}e^{-i\theta_{2}}+\beta\alpha^{\prime}e^{-i\theta_{1}})\hat{b}_{0}+(\alpha\alpha^{\prime}e^{-i\theta_{1}}+\beta\beta^{\prime}e^{-i\theta_{2}})\hat{b}_{1}],$
(71)

where $\theta=\theta_{1}-\theta_{2}$. Substituting the field operator
transformations into the definition of $\hat{G}_{dif}$ yields

$\displaystyle\hat{G}_{dif}$
$\displaystyle=\left[(2|\alpha|^{2}-1)(2|\alpha^{\prime}|^{2}-1)-4|\alpha\alpha^{\prime}\beta\beta^{\prime}|\cos\theta\right](\hat{g}_{0}-\hat{g}_{1})$
$\displaystyle+4\mathfrak{R}_{e}\left\\{\left((|\beta|^{2}e^{-i\theta}-|\alpha|^{2}e^{i\theta})\alpha^{\prime\ast}\beta^{\prime}+\alpha^{\ast}\beta(1-2|\alpha^{\prime}|^{2})\right)\hat{g}_{0}\hat{g}_{1}^{\dagger}\right\\}.$

To quantify the phase sensitivity here, we can define it as

$\Delta\theta_{dif}=\frac{\Delta\hat{G}_{dif}}{|\frac{\partial}{\partial\theta}\langle\hat{G}_{dif}\rangle|},$
(73)

where the derivative of $\hat{G}_{dif}$ with respect to $\theta$ is given by

$\displaystyle\frac{\partial}{\partial\theta}\langle\hat{G}_{dif}\rangle$
$\displaystyle=4|\alpha\alpha^{\prime}\beta\beta^{\prime}|\sin\theta(\langle\hat{g}_{0}\rangle-\langle\hat{g}_{1}\rangle)$
$\displaystyle+4|\alpha^{\prime}\beta^{\prime}|\mathfrak{R}_{e}\\{(|\beta|^{2}e^{-i\theta}+|\alpha|^{2}e^{i\theta})\langle\hat{b}_{0}\rangle\langle\hat{g}^{\dagger}_{1}\rangle\\}.$
(74)

Thus, the variance of the operator $\hat{G}_{dif}$ can be derived as the
following form

$\displaystyle\Delta^{2}\hat{G}_{dif}=$
$\displaystyle\delta_{A}^{2}(\Delta^{2}\hat{g}_{0}+\Delta^{2}\hat{g}_{1})+|\delta_{B}|^{2}(\langle\hat{g}_{0}\rangle+\langle\hat{g}_{1}\rangle)$
$\displaystyle+2|\delta_{B}|^{2}\left(\langle\hat{g}_{0}\rangle\langle\hat{g}_{1}\rangle-|\langle\hat{b}_{0}\rangle|^{2}||\langle\hat{b}_{1}\rangle|^{2}\right)$
$\displaystyle+2\mathfrak{R}_{e}\left\\{\delta_{B}^{2}(\langle\hat{b}_{0}^{2}\rangle\langle(\hat{b}_{1}^{\dagger})^{2}\rangle)-\langle\hat{b}_{0}\rangle^{2}\langle\hat{b}_{1}^{\dagger}\rangle^{2}\right\\}$
$\displaystyle+4\delta_{A}\mathfrak{R}_{e}\left\\{\delta_{B}\left((\langle\hat{g}_{0}\hat{b}_{0}\rangle-\langle\hat{g}_{0}\rangle\langle\hat{b}_{0}\rangle)\langle\hat{b}_{1}^{\dagger}\rangle\right.\right.$
$\displaystyle\qquad\left.\left.-\langle\hat{b}_{0}\rangle(\langle\hat{b}_{1}^{\dagger}\hat{g}_{1}\rangle-\langle\hat{g}_{1}\rangle\langle\hat{b}_{1}^{\dagger}\rangle)\right)\right\\},$
(75)

where

$\displaystyle\delta_{A}=$ $\displaystyle
1-2\left(|\alpha||\beta^{\prime}|+|\beta||\alpha^{\prime}|\right)^{2}+4|\alpha\beta||\alpha^{\prime}\beta^{\prime}|(1-\cos\theta),$
(76) $\displaystyle\delta_{B}=$ $\displaystyle
2i\left(|\alpha\beta|(1-2|\alpha^{\prime}|^{2})+(1-2|\alpha|^{2})|\alpha^{\prime}\beta^{\prime}|\cos\theta\right)$
(77) $\displaystyle+2|\alpha^{\prime}\beta^{\prime}|\sin\theta,$

satisfy the following condition

$\delta_{A}^{2}+|\delta_{B}|^{2}=1.$ (78)

In this detection scheme, the phase sensitivity remains the same for both
scenarios ($b$) and ($c$). For simplicity, we will use the notation
$\Delta\theta_{dif}$ to collectively represent the phase sensitivity in all
detection schemes.

###  V.2 Single-mode intensity detection scheme

In the single-mode intensity detection scheme, we focus on a single
photocurrent at output port $4$ (see Fig.(1)), represented by its associated
operator $\hat{g}_{4}=\hat{b}_{4}^{\dagger}\hat{b}_{4}$. The phase sensitivity
in this scenario is defined as

$\Delta\theta_{sing}=\frac{\Delta\hat{g}_{4}}{|\frac{\partial}{\partial\theta}\langle\hat{g}_{4}\rangle|}.$
(79)

From equation (V.1), we can determine the average number of photons with
respect to the input field operator as

$\displaystyle\langle\hat{g}_{4}\rangle$
$\displaystyle=\left(|\alpha\alpha^{\prime}|^{2}+|\beta\beta^{\prime}|^{2}-2|\alpha\alpha^{\prime}\beta\beta^{\prime}|\cos\theta\right)\langle\hat{g}_{0}\rangle$
(80)
$\displaystyle+\left(|\alpha\beta^{\prime}|^{2}+|\alpha^{\prime}\beta|^{2}+2|\alpha\alpha^{\prime}\beta\beta^{\prime}|\cos\theta\right)\langle\hat{g}_{1}\rangle$
$\displaystyle+2\mathfrak{R}_{e}\left\\{\left(\alpha^{\ast}\beta(2|\alpha^{\prime}|^{2}-1)+\alpha^{\prime\ast}\beta^{\prime}e^{-i\theta}(|\alpha|^{2}-|\beta|^{2}e^{2i\theta})\right)\langle\hat{b}_{0}^{\dagger}\rangle\langle\hat{b}_{1}\rangle\right\\}.$

Using the above equation, we immediately get

$\displaystyle\frac{\partial\langle\hat{g}_{4}\rangle}{\partial\theta}=$
$\displaystyle
2|\alpha\alpha^{\prime}\beta\beta^{\prime}|\sin\theta(\langle\hat{g}_{0}\rangle-\langle\hat{g}_{1}\rangle)$
(81)
$\displaystyle+2||\alpha^{\prime\ast}\beta^{\prime}|\mathfrak{R}_{e}\left\\{\left(|\alpha|^{2}e^{-i\theta}+|\beta|^{2}e^{i\theta}\right)\langle\hat{b}_{0}^{\dagger}\rangle\langle\hat{b}_{1}\rangle\right\\}.$

To find $\Delta^{2}\hat{g}_{4}$, we first calculate the square of the operator
$\hat{g}_{4}$. Then we get the final expression for $\Delta^{2}\hat{g}_{4}$ as

$\displaystyle\Delta^{2}\hat{g}_{4}$
$\displaystyle=|\delta_{3}|^{2}\left(\langle\hat{g}_{0}\rangle+\langle\hat{g}_{1}\rangle+2\langle\hat{g}_{0}\rangle\langle\hat{g}_{1}\rangle-2|\langle\hat{b}_{0}\rangle|^{2}|\langle\hat{b}_{1}\rangle|^{2}\right)$
$\displaystyle+2\delta_{0}\mathfrak{R}_{e}\left\\{\delta_{3}(\langle\hat{g}_{0}\hat{b}_{0}^{\dagger}\rangle+\langle\hat{b}_{0}^{\dagger}\hat{g}_{0}\rangle-2\langle\hat{g}_{0}\rangle\langle\hat{b}_{0}^{\dagger}\rangle)\langle\hat{b}_{1}\rangle\right\\}$
$\displaystyle+2\delta_{1}\mathfrak{R}_{e}\left\\{\delta_{3}\langle\hat{b}_{0}^{\dagger}\rangle(\langle\hat{g}_{1}\hat{b}_{1}\rangle+\langle\hat{b}_{1}\hat{g}_{1}\rangle-2\langle\hat{g}_{1}\rangle\langle\hat{b}_{1}\rangle)\right\\}$
$\displaystyle+2|\delta_{3}^{2}|\mathfrak{R}_{e}\left\\{\langle\hat{b}_{0}^{2}\rangle\langle(\hat{b}_{1}^{\dagger})^{2}\rangle-\langle\hat{b}_{0}\rangle^{2}\langle\hat{b}_{1}^{\dagger}\rangle^{2}\right\\}+\delta_{0}^{2}\Delta^{2}\hat{g}_{0}$
$\displaystyle+\delta_{1}^{2}\Delta^{2}\hat{g}_{1},$ (82)

where

$\displaystyle\delta_{0}=$
$\displaystyle|\alpha\alpha^{\prime}|^{2}+|\beta\beta^{\prime}|^{2}-2|\alpha\alpha^{\prime}\beta\beta^{\prime}|\cos\theta,$
(83) $\displaystyle\delta_{1}=$
$\displaystyle|\alpha\beta^{\prime}|^{2}+|\alpha^{\prime}\beta|^{2}+2|\alpha\alpha^{\prime}\beta\beta^{\prime}|\cos\theta,$
(84) $\displaystyle\delta_{3}=$
$\displaystyle\alpha^{\ast}\beta(2|\alpha^{\prime}|^{2}-1)+\alpha^{\prime\ast}\beta^{\prime}(|\alpha|^{2}e^{-i\theta}-|\beta|^{2}e^{i\theta}).$
(85)

###  V.3 Balanced homodyne detection scheme

We now turn to the balanced homodyne detection scheme at output port 4 (see
Fig.(1)). The operator of interest for modeling this detection scheme is given
by

$\hat{X}_{\theta_{L}}=\mathfrak{R}_{e}\left\\{e^{-i\theta_{L}}\hat{b}_{4}\right\\},$
(86)

where $\theta_{L}$ is the phase of the local coherent source $|\gamma\rangle$,
where $|\gamma\rangle=|\gamma|e^{i\theta_{L}}$ and $\gamma$ is a complex
number. In this detection scheme, we define the phase sensitivity as follows

$\Delta\theta_{hom}=\frac{\sqrt{\Delta^{2}\hat{X}_{\theta_{L}}}}{|\frac{\partial\langle\hat{X}_{\theta_{L}}\rangle}{\partial\theta}|}.$
(87)

Using the field operator transformations (V.1), we arrive at the final
expression for $\langle\hat{X}_{\theta_{L}}\rangle$ as

$\displaystyle\langle\hat{X}_{\theta_{L}}\rangle=$
$\displaystyle\mathfrak{R}_{e}\left\\{e^{-i\theta_{L}}\left((\alpha\alpha^{\prime}e^{-i\theta_{2}}+\beta\beta^{\prime}e^{-i\theta_{1}})\langle\hat{b}_{0}\rangle\right.\right.$
(88)
$\displaystyle\left.\left.+(\alpha\beta^{\prime}e^{-i\theta_{1}}+\beta\alpha^{\prime}e^{-i\theta_{2}})\langle\hat{b}_{1}\rangle\right)\right\\},$

and the variance of the above operator is given by

$\displaystyle\Delta^{2}\hat{X}_{\theta_{L}}=$
$\displaystyle\frac{1}{4}+2\mathfrak{R}_{e}\left\\{x^{2}\Delta^{2}\hat{b}_{0}+y^{2}\Delta^{2}\hat{b}_{1}\right\\}$
(89)
$\displaystyle+2|x|^{2}(\langle\hat{g}_{0}\rangle-|\langle\hat{b}_{0}\rangle|^{2})+2|y|^{2}(\langle\hat{g}_{1}\rangle-|\langle\hat{b}_{1}\rangle|^{2}),$

where the coefficients $A$ and $B$ are given by

$\displaystyle x=$
$\displaystyle\frac{1}{2}e^{-i(\theta_{L}+\theta_{2})}\left(\alpha\alpha^{\prime}+\beta\beta^{\prime}e^{-i\theta}\right),$
(90) $\displaystyle y=$
$\displaystyle\frac{1}{2}e^{-i(\theta_{L}+\theta_{2})}\left(\alpha\beta^{\prime}e^{-i\theta}+t^{\prime}r\right).$
(91)

For scenario ($b$) in Fig.(1), where $\theta_{1}=\theta$ and $\theta_{2}=0$,
the absolute value of the derivative of $\langle\hat{X}_{\theta_{L}}\rangle$
with respect to $\theta$ is given by

$|\frac{\partial\langle\hat{X}_{\theta_{L}}\rangle}{\partial\theta}|=|\mathfrak{R}_{e}\left\\{e^{-i(\theta_{L}+\theta)}\left(\beta\langle\hat{b}_{0}\rangle+\alpha\langle\hat{b}_{1}\rangle\right)\right\\}||\beta^{\prime}|$
(92)

and for scenario ($c$), where $\theta_{1}=-\theta_{2}=\theta/2$, we obtain

$\displaystyle|\frac{\partial\langle\hat{X}_{\theta_{L}}\rangle}{\partial\theta}|=$
$\displaystyle\frac{1}{2}|\mathfrak{R}_{e}\left\\{ie^{-i\theta_{L}}\left(\left(\alpha\alpha^{\prime}e^{i\theta/2}-\beta\beta^{\prime}e^{-i\theta/2}\right)\langle\hat{b}_{0}\rangle\right.\right.$
$\displaystyle\left.\left.+\left(\beta\alpha^{\prime}e^{i\theta/2}-\alpha\beta^{\prime}e^{-i\theta/2}\right)\right)\langle\hat{b}_{1}\rangle\right\\}|$
(93)

##  VI Phase sensitivity with SU(1,1) coherent states in the input

In this section, we compare the phase sensitivities achievable by the three
considered detection schemes, as presented in Section IV, for input SU(1,1)
coherent states with the QCRBs implied by the various QFIs discussed in
Section III. Using the results of the phase sensitivities reported in the
previous section, i.e., $\Delta\theta_{dif}$, $\Delta\theta_{sing}$, and
$\Delta\theta_{hom}$, and considering the input state (36), it is easy to
verify that the phase sensitivities in our input states are as follows:

For a difference-intensity detection scheme, we get the final analytical
expression of the phase sensitivity for the two types of SU(1,1) CSs, as

$\Delta\theta_{dif}^{i}=\frac{\Delta_{i}\hat{G}_{dif}}{4|\alpha\alpha^{\prime}\beta\beta^{\prime}||\sin\theta\langle\hat{g}_{1}\rangle_{i}|},$
(94)

where the subscript $i=P$ or $B$ corresponds to entangled PCS (Eq.9) and
entangled BGCS (Eq.15), respectively. Then, from the above expression of the
phase sensitivity, we calculate its final analytical expression for the two
types of SU(1,1) CSs, as

$\Delta\theta_{dif}^{P}=\frac{\sqrt{\frac{1}{2}\delta_{A}^{2}\sinh^{2}v+|\delta_{B}|^{2}(\cosh
v-1)}}{4\sqrt{a}|\alpha\alpha^{\prime}\beta\beta^{\prime}||(\cosh
v-1)\sin\theta|},$ (95) $\displaystyle\Delta\theta_{dif}^{B}=$
$\displaystyle\frac{\sqrt{\delta_{A}^{2}|\xi|\left[I_{2a-1}I_{2a+1}-I_{2a}^{2}\right]+I_{2a-1}I_{2a}}}{4|\alpha\alpha^{\prime}\beta\beta^{\prime}||\sin\theta|\sqrt{|\xi|}I_{2a}},$
(96)

where $\delta_{A}$ and $\delta_{B}$ were defined in equation (76). To simplify
notation, we define $I_{2a}\equiv I_{2a}(2|\xi|)$. Interestingly, for both
scenarios ($b$) and ($c$), this detection scheme yields the same phase
sensitivity result.

For a single-mode intensity detection scheme, we obtain the phase sensitivity
in all considered scenarios as

$\Delta\theta_{sing}^{i}=\frac{\Delta_{i}\hat{g}_{4}}{2|\alpha\alpha^{\prime}\beta\beta^{\prime}||\sin\theta\langle\hat{g}_{1}\rangle_{i}|},$
(97)

and the analytical expression of the phase sensitivity for our input SU(1,1)
coherent states takes the form

$\Delta\theta_{sing}^{P}=\frac{\sqrt{\frac{1}{2}\delta_{1}^{2}\sinh^{2}v+|\delta_{3}|^{2}(\cos
v-1)}}{2\sqrt{a}|\alpha\alpha^{\prime}\beta\beta^{\prime}||\sin\theta(\cosh
v-1)|},$ (98)
$\displaystyle\Delta\theta_{sing}^{B}=\frac{\sqrt{\delta_{1}^{2}|\xi|\left[I_{2a-1}I_{2a+1}-I_{2a}^{2}\right]+(\delta_{1}^{2}+|\delta_{3}|^{2})^{2}I_{2a-1}I_{2a}}}{2\sqrt{|\xi|}|\alpha\alpha^{\prime}\beta\beta^{\prime}||\sin\theta|I_{2a}},$
(99)

where $\delta_{1}$ and $\delta_{3}$ were defined in equation (83).Finally, for
a balanced homodyne detection scheme, we have

$\Delta^{2}_{i}\hat{X}_{\theta_{L}}=\frac{1}{4}+2\mathfrak{R}_{e}\left\\{y^{2}\Delta^{2}_{i}\hat{b}_{1}\right\\}+2|y|^{2}(\langle\hat{g}_{1}\rangle_{i}-|\langle\hat{b}_{1}\rangle_{i}|^{2}),$
(100)

In the case of scenario ($b$), where $\theta_{1}=\theta$ and $\theta_{2}=0$,
and assuming $\theta_{L}=\varphi$, the variance of the operator
$\hat{X}_{\theta_{L}}$ is given by

$\displaystyle\Delta_{i}^{2}\hat{X}_{\theta_{L}}=$
$\displaystyle\frac{1}{4}-\frac{1}{2}\left(|\alpha\beta^{\prime}|^{2}\cos
2\theta+|\alpha^{\prime}\beta|^{2}+2|\alpha\alpha^{\prime}\beta\beta^{\prime}|\cos\theta\right)\mu_{i}$
$\displaystyle+\frac{1}{2}\left(|\alpha\beta^{\prime}|^{2}+|\alpha^{\prime}\beta|^{2}+2|\alpha\alpha^{\prime}\beta\beta^{\prime}|\cos\theta\right)\left(\bar{g}_{i}-|\nu_{i}|^{2}\right),$
(101)

where

$\displaystyle\mu_{P}=\frac{\cosh^{-4a}(v/2)}{\tanh^{2}(v/2)}\left[\sum_{g=0}^{\infty}\frac{\sqrt{\Gamma(g+2a)\Gamma(g+2a-2)}}{\Gamma(2a)(g-2)!}\tanh^{2g}(v/2)-\frac{1}{\cosh^{4a}(v/2)}\left(\sum_{g=0}^{\infty}\frac{\sqrt{\Gamma(g+2a)\Gamma(g+2a-1)}}{\Gamma(2a)(g-1)!}\tanh^{2g}(v/2)\right)^{2}\right],$
(102)
$\displaystyle\mu_{B}=\frac{\tanh^{2a-3}(v/2)}{I_{2a-1}}\left[\sum_{g=0}^{\infty}\frac{\tanh^{2g}(v/2)}{(g-2)!\sqrt{\Gamma(g+2a)\Gamma(g+2a-2)}}-\frac{\tanh^{2a-1}(v/2)}{I_{2a-1}}\left(\sum_{g=0}^{\infty}\frac{\tanh^{2g}(v/2)}{(g-1)!\sqrt{\Gamma(g+2a)\Gamma(g+2a-1)}}\right)^{2}\right],$
(103)
$\displaystyle\nu_{P}=\left(1-\tanh^{2}(v/2)\right)^{2a}\sum_{g=0}^{\infty}\frac{\sqrt{\Gamma(g+2a)\Gamma(g+2a-1)}}{\Gamma(2a)(g-1)!}\tanh^{(2g-1)}(v/2)$
(104)
$\displaystyle\nu_{B}=\frac{\tanh^{2(a-2)}(v/2)}{I_{2a-1}}\sum_{g=0}^{\infty}\frac{\tanh^{2g}(v/2)}{(g-1)!\sqrt{\Gamma(g+2a)\Gamma(g+2a-1)}},$
(105) $\displaystyle\bar{g}_{P}=a(\cosh
v-1),\quad\mbox{and}\quad\bar{g}_{B}=\frac{I_{2a}}{I_{2a-1}}\tanh(v/2).$ (106)

From equation (92), we get

$|\frac{\partial\langle\hat{X}_{\theta_{L}}\rangle}{\partial\theta}|=\frac{1}{|\tan(\frac{v}{2})|}|\alpha\beta^{\prime}||\cos(\theta)\nu_{i}|.$
(107)

Based on these results, the phase sensitivity is given by

$\displaystyle\Delta\theta_{hom}^{(b)}=$
$\displaystyle|\tan(\frac{v}{2})|\frac{\sqrt{\Delta_{i}^{2}\hat{X}_{\theta_{L}}}}{|\alpha\beta^{\prime}||\cos(\theta)\nu_{P}|}.$
(108)

In the case of scenario ($c$), where $\theta_{1}=-\theta_{2}=\theta/2$, the
above variance is given by

$\displaystyle\Delta_{i}^{2}\hat{X}_{\theta_{L}}$
$\displaystyle=\frac{1}{4}-\frac{1}{2}\left((|\alpha\beta^{\prime}|^{2}+|\alpha^{\prime}\beta|^{2})\cos\theta+2|\alpha\alpha^{\prime}\beta\beta^{\prime}|\right)\mu_{i}$
$\displaystyle+\frac{1}{2}\left(|\alpha\beta^{\prime}|^{2}+|\alpha^{\prime}\beta|^{2}+2|\alpha\alpha^{\prime}\beta\beta^{\prime}|\cos\theta\right)\left(\bar{g}_{i}-|\nu_{i}|^{2}\right).$
(109)

Based on the same reasoning, from equation (V.3) we obtain

$|\frac{\partial\langle\hat{X}_{\theta_{L}}\rangle}{\partial\theta}|=\frac{1}{2|\tan(\frac{v}{2})|}||\alpha^{\prime}\beta|-|\alpha\beta^{\prime}|||\cos(\frac{\theta}{2})\nu_{i}|,$
(110)

From this, we can calculate the phase sensitivity in this last scenario as

$\displaystyle\Delta\theta_{hom}^{(c)}=$
$\displaystyle|2\tan(\frac{v}{2})|\frac{\sqrt{\Delta_{i}^{2}\hat{X}_{\theta_{L}}}}{||\alpha^{\prime}\beta|-|\alpha\beta^{\prime}|||\cos(\frac{\theta}{2})\nu_{i}|}.$
(111)

Figure 3: The phase sensitivity $\Delta\theta$ with respect to the phase shift
in the difference-intensity detection, single-mode detection, and balanced
homodyne detection schemes, denoted as $\Delta\theta_{dif}$,
$\Delta\theta_{sing}$, $\Delta\theta_{hom}^{(b)}$, and
$\Delta\theta_{hom}^{(c)}$ respectively, together with the three QCRBs
$\Delta\theta_{QCRB}^{(a)}$, $\Delta\theta_{QCRB}^{(b)}$, and
$\Delta\theta_{QCRB}^{(c)}$. In panels ($a$) and ($c$), for the Perelomov
coherent input state with parameters $a=1$ and $v=1$, and in panels ($b$) and
($d$), for the Barut-Girardello coherent input state with parameters $a=1$ and
$v=1$, the graphical representation includes a gray curve for
$\Delta\theta_{dif}$, the green curve for $\Delta\theta_{sing}$, the blue
curve for $\Delta\theta_{hom}^{(b)}$, the orange curve for
$\Delta\theta_{hom}^{(c)}$, the red line for $\Delta\theta_{QCRB}^{(a)}$, the
green line for $\Delta\theta_{QCRB}^{(b)}$, and the black line for
$\Delta\theta_{QCRB}^{(c)}$. Figure 4: The performance ratio
$R=\Delta\theta_{i}/\Delta\theta_{j}$, with $i$ representing the Perelomov
coherent input state and $j$ representing the Barut-Girardello coherent input
state, of optimal phase estimation in different detection schemes for the two
types of SU(1,1) CSs with $v=1$

Fig.(3) presents the variation of the four phase sensitivities as a function
of the MZI’s phase shift for both the Perelomov and Barut-Girardello coherent
input states with parameters $a=1$ and $v=1$. The figure shows these phase
sensitivities for the three detection schemes discussed previously, alongside
the corresponding three QCRBs.

We plot the difference-intensity detection and single-mode detection schemes
for the case of a balanced beam splitter, i.e., $\alpha=1/\sqrt{2}$ and
$\beta=i/\sqrt{2}$, considering its optimal setup. For the balanced homodyne
detection scheme, we consider transmission coefficients $\alpha=1$ for the
first beam splitter and $\alpha^{\prime}=0$ for the second beam splitter,
implying its optimal performance. The gray curve and the green curve in
Fig.3(a) and Fig.3(b) correspond to the difference intensity detection scheme
and the single-mode detection scheme, respectively. As can be seen in this
figure, both curves have an optimum that achieves the QCRB, which is implied
by the two-parameter QFI. The blue and orange lines are for two balanced
homodyne detection schemes. As observed in this figure, these phase
sensitivities reach the QCRBs. For the chosen values, the phase sensitivity
$\Delta\theta_{hom}^{(b)}$ shows significantly better performance than the
phase sensitivity $\Delta\theta_{dif}$. This suggests an advantage to using an
external phase reference for both the Perelomov coherent input state and the
Barut-Girardello coherent input state. The input state, including the local
oscillator, can be expressed as

$|\psi\rangle=|\psi_{in}\rangle\otimes|\gamma\rangle=|\xi_{i},a\rangle\otimes||\gamma|e^{i\theta_{L}}\rangle.$
(112)

Our interferometer consists of two arms: one involving input port 1, passing
through the first beam splitter with total transmission, phase shift
$\theta_{1}$, the second beam splitter with total reflection, and reaching the
beam splitter (BSL). The other arm represents the local oscillator fed into
the balanced beam splitter of the homodyne setup.

To compare the performance of optimal phase estimation in different detection
schemes for the two types of SU(1,1) CSs, we use a technique where we
introduce the ratio between the phase sensitivities of these two states in
different detection schemes as $R=\Delta\theta_{i}/\Delta\theta_{j}$, with $i$
representing the Perelomov coherent input state and $j$ representing the
Barut-Girardello coherent input state. As a result, when the ratio $R<1$, the
error limit of the phase sensitivity for the Perelomov coherent input state in
different detection schemes is smaller and offers an advantage over that of
the Barut-Girardello coherent input state. As shown in Fig.(4), we observe
that the values of the phase sensitivities for the two states satisfy the
inequality $\Delta\theta_{P}<\Delta\theta_{B}$, indicating that the
performance of the phase sensitivities for the Perelomov coherent input state
is better and would provide a more precise result than that of the Barut-
Girardello coherent input state.

##  VII CONCLUSION

Optimizing the sensitivity of a MZI requires careful consideration of both the
input state and the detection scheme. QFI serves as a valuable tool to
identify the optimal operating points that achieve the highest possible
sensitivity. This paper presents theoretical calculations of QCRBs for both
two-parameter and single-parameter estimation in quantum interferometry,
considering two input scenarios. We explore the performance of Perelomov and
Barut-Girardello coherent input states within the SU(1,1) Lie algebra. We
investigate their phase sensitivity across various detection schemes,
including difference-intensity, single-mode, and balanced homodyne detection.
Furthermore, we analyze the QCRB associated with the QFI obtained for these
states in all the aforementioned scenarios.

Our results demonstrate that the phase sensitivities for the Perelomov
coherent input state in different detection schemes are better and would
provide a more precise result than that of the Barut-Girardello coherent input
state. The use of balanced homodyne detection techniques has been studied. The
availability of an external phase reference can significantly enhance the
performance of these input states, particularly in the unphysical limits where
the transmission coefficients of the beam splitters approach
$|\alpha|\rightarrow 1$ and $|\alpha^{\prime}|\rightarrow 0$. This suggests
that an external phase reference can improve the performance of these input
states in quantum interferometry, potentially leading to improved measurement
precision.

Data Availability: No data associated in the manuscript.

Disclosures: The authors declare no conflicts of interest

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