# Quantum Metrology of Absorption and Gain Parameters
using Two-Mode Bright Squeezed Light

Mrunal Kamble  Department of Physics and Astronomy, Texas A&M University,
College Station, Texas 77843, USA  Institute for Quantum Science and
Engineering, Texas A&M University, College Station, Texas 77843, USA  Jiaxuan
Wang  Department of Physics and Astronomy, Texas A&M University, College
Station, Texas 77843, USA  Institute for Quantum Science and Engineering,
Texas A&M University, College Station, Texas 77843, USA  Girish S. Agarwal
Department of Physics and Astronomy, Texas A&M University, College Station,
Texas 77843, USA  Institute for Quantum Science and Engineering, Texas A&M
University, College Station, Texas 77843, USA  Department of Biological and
Agricultural Engineering, Texas A&M University, College Station, Texas 77843,
USA

###### Abstract

Absorption and gain processes are fundamental to any light-matter interaction
and a precise measurement of these parameters is important for various
scientific and technological applications. Quantum probes, specifically the
squeezed states have proved very successful, particularly in the applications
that deal with phase shift and force measurements. In this paper, we focus on
improving the sensitivity of the estimation of the photon loss coefficient of
a weakly absorbing medium as well as the estimation of the gain parameter
using a two-mode bright squeezed state. The generation of this state combines
the advantage of a coherent beam for its large photon number with the quantum
properties of the two-mode squeezing operation in an optical parametric
amplifier. We present two measurement schemes: balanced photodetection and
time-reversed metrology, both utilizing two-mode bright squeezed light. The
maximum quantum advantage we can achieve using two-mode bright squeezed light
is 3.7 times for the absorption parameter $\alpha=0.05$ and 8.4 times for
$\alpha=0.01$ as compared to using only the coherent state. Similarly, the
maximum quantum advantage for the estimation of optical gain is found around
2.81 times for the gain coefficient $G=1.05$ and around 6.28 times for
$G=1.01$. We discuss the significance of using one measurement scheme over the
other under different squeezing conditions. We compare our results with the
Cramér-Rao bound for a two-mode bright squeezed state to assess the quality of
the proposed methodologies.

††preprint: APS/123-QED

##  I  Introduction

Quantum sensing and metrology is one of the emerging fields that demonstrate
the practical utility of quantum mechanical probes in various scenarios. The
sensitivity of the optical systems using the coherent states of light is
limited by the Standard Quantum limit (SQL). In the region where classical
states of light are deemed insufficient to improve the measurement
sensitivity, quantum states step in to push the boundaries PRXQuantum.3.010202
; lawrie2019quantum . Benefits of using quantum probes such as single-mode and
two-mode squeezed states; and entangled states that surpass the SQL have been
shown in phase shift measurements anderson2017phase ; PhysRevLett.111.173601 ;
caves1981quantum ; jianqin , force measurements pooser2020truncated ;
PhysRevA.87.012107 , spectroscopy and imagingli2022quantum ;
casacio2021quantum ; pooser2016plasmonic ; de2020quantum ; triginer2020quantum
, magnetometry li2018quantum  to mention a few.

The standard optical interferometers like the Mach-Zehnder Interferometer play
a crucial role in advancing quantum sensing capabilities by adding quantum
probes as input to the interferometer plick ; gerry2000heisenberg ;
pezze2008mach  as originally suggested by Caves caves1981quantum . An
important goal of such studies with different quantum probes was to find when
the Heisenberg limit can be reached. Yurke, McCall, and Klauder yurke ; zyou ;
caves2020reframing  introduced the SU(1,1) interferometer by replacing the
beam splitters in the Mach-Zehnder interferometer with optical parametric
amplifiers (OPA) and demonstrated the first possibility of reaching the
Heisenberg limit of phase sensitivity. This was an important step forward in
quantum metrology and led to considerable experimental and theoretical work. A
recent experiment jianqin  reports Heisenberg’s limited measurement of phase
using an SU(1,1) interferometer. More recently time-versed metrology is
finding widespread applications with both single and two-mode squeezed vacuum
states of light and even matter agarwal ; vuletic ; burd ; linnemann ;
xin2019optimal ; nanhuo .

The work of Yurke et al. was generalized to add a seed beam to the OPA by
Plick et al. plick . This addition takes advantage of both the coherent
component and the squeezed component in metrology. It leads to what is called
the bright squeezed light and has been extensively used in studies on the
quantum metrology of phase ou2012enhancement ; yuhongliu2 ; anderson2017phase
; liu2018quantum ; PhysRevA.95.063843 . It also turned out that the SU(1,1)
interferometer is more loss-tolerant and thus advantageous in applications
santandrea2023lossy ; tian2023loss ; xin2021phase ; lett ; junxin ; yuhongliu
; manceau2017detection .

While much of the work on sensing with quantum probes has been devoted to
investigations on operations characterized by unitary transformations, studies
of the parameters of the open systems are beginning to appear. The squeezed
vacuum states have been shown to be especially useful gammaT ; wang2024quantum
; nair1 ; PhysRevLett.128.180506 ; moreau ; losero ; monras . The two-mode
squeezed vacuum states are optimum for the measurement of the absorption and
gain parameters. While this is very important, it is so far difficult to
produce squeezed vacuum states with a large number of photons and hence it is
desirable to add a seed to obtain a large number of photons. This is the
object of the present study. As mentioned in the abstract, absorption and gain
are the fundamental processes the study of which yields a wealth of
information on the molecular densities, transition matrix elements, molecular
structure, etc. Thus absorption and gain spectroscopy techniques are very
common tools and the sensitivities of such techniques can be improved
considerably by using quantum light.

Here we explore a readily implementable entangled Gaussian state known as the
Two-Mode Bright Squeezed State (TMBSS), where the squeezing operation is
performed by a non-linear process in an optical parametric amplifier. This
state combines the advantages of a coherent beam, with its high photon count,
together with the noise reduction achieved through two-mode squeezing. It is
to be noted that the goal of this study is to find the level of sensitivity
using a good quantum resource rather than finding when the estimate of the
parameter is optimum. In this study, we present results for quantum advantage
in the measurements of both absorption and gain using TMBSS. The paper is
structured as follows: In Section II, we introduce the characteristics of
TMBSS. Section III delves into studying the Cramér-Rao bound of absorption
estimation with TMBSS. In Section IV, we present measurement approaches for
estimating the absorption parameter using TMBSS. Section V demonstrates the
advantages of these methods by comparing them with the traditional approach
employing a coherent light source. Our analysis encompasses the benefits
provided by intensity differences measurement and the SU(1,1) interferometer
under various squeezing conditions. Furthermore, we compare sensitivity
results with the Cramér-Rao bound, derived in Sec. III. In Section VI, we
present a detailed investigation of the gain using the Cramer-Rao bound and
sensitivities for different types of measurements. Finally, Section VII gives
a comprehensive discussion of our results.

##  II Two-Mode Bright Squeezed Light for Absorption Measurement

In a two-mode bright squeezed state, a coherent field and a vacuum field
interacts non-linearly to produce a quantum state which is such that the
difference in number fluctuations are squeezed. The Hamiltonian of the
squeezing process is given by $H=\Gamma a^{\dagger}b^{\dagger}+h.c$ where
$\Gamma$ indicates the interaction strength of this process. The squeezing
operation is performed by a non-linear process in the Optical Parametric
Amplifier (OPA) where the two input modes, a coherent state, and a vacuum
state are acted upon by the two-mode squeezing operator given by

${S}(r)=e^{r({a}^{\dagger}{b}^{\dagger}-{a}{b})},$ (1)

where $r$ is the squeezing parameter. We take the two-mode squeezing parameter
to be real since absorption estimation does not consider any phase shifts. The
resulting state called the two-mode bright squeezed state is given by

$\displaystyle\left|\psi\right\rangle_{TMBSS}={S}(r)\left|u,0\right\rangle,$
(2)

where $\left|u\right\rangle$ is the coherent state such that
$\left|u\right|^{2}\gg\cosh^{2}r$. The state generated by Eq. (2) is a
Gaussian state.

In the scheme shown in Fig. 1. the part with the OPA describes the generation
of the TMBSS which is used to determine the parameter $\alpha$. It is more
convenient to work with Heisenberg operators in this scenario. The squeezing
transformations, ${a_{1}}={S}^{\dagger}{a_{in}}{S}$ and
${b_{1}}={S}^{\dagger}{b_{in}}{S}$ are given by

$\displaystyle{a_{1}}$ $\displaystyle=$ $\displaystyle{a_{in}}\cosh
r+{b_{in}^{\dagger}}\sinh r,$ $\displaystyle{b_{1}}$ $\displaystyle=$
$\displaystyle{b_{in}}\cosh r+{a_{in}^{\dagger}}\sinh r.$ (3)

Figure 1: The schematics of absorption measurement using two-mode bright
squeezed state. One of the input beams is the coherent beam and the other is
the vacuum field. The two beams at the output of the OPA constitute the two-
mode bright squeezed light. One of the beams called the probe beam passes
through the absorption medium. Detection of the intensity difference of these
beams $\left\langle{N}_{out}\right\rangle=\left\langle
a_{out}^{\dagger}a_{out}\mathbin{-b_{out}^{\dagger}b_{out}}\right\rangle$
forms the signal for the balanced photodetection setup

In the following step, one of the beams is directed into the sample and is
called the probe, and the other beam is called the ancilla. After passing
through the sample the probe carries the information on the absorption
coefficient which is based on the beam splitter transformation model
agarwal2012quantum  given by

$\displaystyle{a_{out}}$ $\displaystyle=$
$\displaystyle{a_{1}}\sqrt{1-\alpha}+{v}\sqrt{\alpha},$
$\displaystyle{b_{out}}$ $\displaystyle=$ $\displaystyle{b_{1}}.$ (4)

When working at the quantum level, we need to introduce the vacuum field given
by annihilation operator ${v}$ to satisfy the commutation relation
$\left[{a_{1}},{a}_{1}^{\dagger}\right]=\left[{a_{out}},{a}_{out}^{\dagger}\right]=1$.
Using Eqs. (3) and (4) we can write

$\displaystyle{a}_{out}$ $\displaystyle=$
$\displaystyle{a_{in}}c_{11}+{b_{in}^{\dagger}}c_{12}+v\,c_{13},$
$\displaystyle{b}_{out}$ $\displaystyle=$
$\displaystyle{a_{in}^{\dagger}}c_{21}+{b_{in}}c_{22},$ (5)

where

$\displaystyle c_{11}=\sqrt{1-\alpha}\cosh r,\hskip
2.84544ptc_{12}=\sqrt{1-\alpha}\sinh r,\hskip 2.84544ptc_{13}=\sqrt{\alpha},$
$\displaystyle c_{21}=\sinh r,\hskip 2.84544ptc_{22}=\cosh r.$

Lastly, the two beams enter the photocurrent mixer to realize the
superposition of the fields which constitutes the signal for the measurement
scheme discussed in section IV subsection A.

##  III Quantum Fisher Information for Two-Mode Bright Squeezed state

With the new emerging quantum technologies for sensing, it is important to
know how far we can push the limits of precisely measuring a physical
parameter. Quantum measurement theory provides us with this limit called the
quantum Cramér-Rao bound which is an extension of its classical counterpart.
We drop the quantum prefix and denote it simply as the Cramér-Rao bound for
the remainder of the paper. It is expressed in terms of the Quantum Fisher
Information (QFI) of the given system that describes the limit on the
distinguishability between two infinitesimally close quantum states.
Typically, the larger the QFI, the better the distinguishability which implies
better precision in estimating the parameter.

In the present research, the precision of the measurement is based on this
Cramér-Rao bound. We first calculate the QFI to determine this bound. As
described in the previous section the input fields to the OPA are Gaussian in
nature. The general approach to computing the quantum Fisher information for a
Gaussian system has been developed pinel2012ultimate ; pinel2013quantum ;
gao2014bounds ; friis2015heisenberg ; vsafranek2015quantum ; banchi2015quantum
; marian2016quantum ; nichols2018multiparameter ; vsafranek2018estimation .
Given that the output field maintains the Gaussian characteristics from the
absorbing medium agarwal1987wigner , we can apply the covariance matrix
method, which is well-formulated in vsafranek2018estimation , to find the QFI
associated with the TMBSS. The calculations are to be done using the explicit
expressions for ${a}_{out}$ and ${b}_{out}$ given by Eq. (5) and using the
fact that ${a}_{in}$ and ${b}_{in}$ represent the coherent and the vacuum
field respectively. We first write the set of the output annihilation and the
creation operators for the given two-mode bosonic system in the form of a
vector of operators given by

${A}=(a_{out},b_{out},a_{out}^{\dagger},b_{out}^{\dagger})^{T}.$ (6)

Introducing the displacement vector $d_{m}=tr[\rho{A}_{m}]$ and its covariance
matrix
$\sigma_{mn}=tr\left[{\rho}\left\\{\Delta{A}_{m},\Delta{A}_{n}^{\dagger}\right\\}\right]$
where $\Delta{A}={A}-d$ and recognizing we only have one parameter to be
estimated ($\alpha$), the QFI is given by

$F_{Q}(\alpha)=\frac{1}{2}vec\left(\frac{\partial\sigma}{\partial\alpha}\right)^{\dagger}\mathfrak{M^{-1}}vec\left(\frac{\partial\sigma}{\partial\alpha}\right)+2\frac{\partial
d^{\dagger}}{\partial\alpha}\sigma^{-1}\frac{\partial d}{\partial\alpha},$ (7)

where, $\mathfrak{M}=\overline{\sigma}\otimes\sigma-\mathit{K\otimes K}$ and
$K=\left[\begin{array}[]{cc}I&0\\\ 0&-I\end{array}\right].$
The required displacement vector and the covariance matrix can be calculated
using Eq. (5) yielding the results:

$d^{(\alpha)}=\left(\begin{array}[]{c}u\cosh r\sqrt{1-\alpha}\\\ u^{\ast}\sinh
r\\\ u^{\ast}\cosh r\sqrt{1-\alpha}\\\ u\sinh r\end{array}\right),$ (8)

and

$\sigma^{(\alpha)}=\begin{bmatrix}\sigma_{11}^{(\alpha)}&0&0&\sigma_{14}^{(\alpha)}\\\
0&\sigma_{22}^{(\alpha)}&\sigma_{23}^{(\alpha)}&0\\\
0&\sigma_{32}^{(\alpha)}&\sigma_{33}^{(\alpha)}&0\\\
\sigma_{41}^{(\alpha)}&0&0&\sigma_{44}^{(\alpha)}\end{bmatrix},$ (9)

where

$\displaystyle\sigma_{11}^{(\alpha)}=\sigma_{33}^{(\alpha)}=1+2(1-\alpha)sinh^{2}r,$
$\displaystyle\sigma_{22}^{(\alpha)}=\sigma_{44}^{(\alpha)}=1+2sinh^{2}r,$
$\displaystyle\sigma_{14}^{(\alpha)}=\sigma_{41}^{(\alpha)}=\sigma_{23}^{(\alpha)}=\sigma_{32}^{(\alpha)}=2\sqrt{1-\alpha}\cosh
r\sinh r.$

Considering the limit $\left|u\right|^{2}\gg\cosh^{2}r$, we can drop the first
term in Eq. (7), which leads to the simplified form given by

$F_{Q}(\alpha)=2\frac{\partial(d^{(\alpha)})^{\dagger}}{\partial\alpha}(\sigma^{(\alpha)})^{-1}\frac{\partial
d^{(\alpha)}}{\partial\alpha}.$ (10)

Thus, using Eqs. (8-10), the QFI of a two-mode bright squeezed state wherein
one of the modes undergoes absorption with a parameter $\alpha$ is expressed
as follows:

${F}_{Q}(\alpha)=\frac{|u|^{2}\cosh^{2}r\cosh 2r}{[1+\alpha(\cosh
2r-1)](1-\alpha)}\,,$ (11)

where $|u|^{2}$ is the total number of photons in the input coherent state
used to generate the two-mode bright squeezed beam and $r$ is the squeezing
parameter. The Camer-Rao bound which is the lowest bound in the estimation of
the given parameter is found by using the inequality

$(\Delta\alpha)_{CR}\geq\frac{1}{\sqrt{\mathcal{N}F_{Q}(\alpha)}},$ (12)

where $F_{Q}(\alpha)$ is the quantum Fisher information Eq. (11) and
$\mathcal{N}$ is the number of independent measurements which we take as 1 for
simplicity. Next, we discuss the practical schemes and assess the possibility
of reaching the Cramér-Rao bound.

##  IV Measurement Schemes

###  IV.1 Balanced Photodetection

The sensitivity of measuring the absorption coefficient ($\alpha$) depends not
only on the state of the light but also on the chosen measurement scheme. Here
we calculate the sensitivity of the setup consisting of three main steps
namely, squeezing, absorption, and detection as shown in Fig. 1. In section II
we discussed the interaction between the probe beam and the absorption sample
as formulated in Eq. (4). The ancilla does not interact with the sample and is
used as a reference beam. In the next step, both the beams enter the balanced
detector to realize the superposition of the fields. Here, the signal is
defined as the intensity difference between the two beams such that,
$\left\langle{N}_{out}\right\rangle=\left\langle
a_{out}^{\dagger}a_{out}\mathbin{-b_{out}^{\dagger}b_{out}}\right\rangle$ and
the associated fluctuations are given by $\Delta N_{out}=\sqrt{\left\langle
N_{out}^{2}\right\rangle-\left\langle N_{out}\right\rangle^{2}}$. Using Eq.
(5) these quantities are found to be

$\left\langle{N}_{out}\right\rangle=|u|^{2}\left(c_{11}^{2}-c_{21}^{2}\right),$
(13)

and

$\displaystyle(\Delta N_{out})^{2}$ $\displaystyle=$
$\displaystyle|u|^{2}[c_{11}^{2}\left(c_{11}^{2}+c_{12}^{2}+c_{13}^{2}\right)+c_{21}^{2}\left(c_{21}^{2}+c_{22}^{2}\right)$
(14) $\displaystyle-$ $\displaystyle
2c_{11}c_{21}\left(c_{11}c_{21}+c_{12}c_{22}\right)].$

The sensitivity of the setup is calculated using

$\Delta\alpha=\frac{\Delta N_{out}}{\mid
d\left\langle{N}_{out}\right\rangle/d\alpha\mid}.$ (15)

From Eqs. (13-15) the sensitivity of the balanced photodetection measurement
scheme could be written in terms of the squeezing parameter r and the
absorption coefficient $\alpha$ as

$(\Delta\alpha)_{BD}=\frac{\sqrt{1+2\alpha^{2}\cosh^{2}r\sinh^{2}r-\alpha\cosh^{2}r}}{|u|\cosh^{2}r}.$
(16)

Next, we calculate the sensitivity of another popular method, namely, the
SU(1,1) interferometer measurement scheme.

###  IV.2 SU(1,1) Interferometer-Time reversed metrology

Figure 2: Experimental SU(1,1) setup for bright squeezed state time-reversal
metrology. The absorption medium, which reduces the output intensity to
$(1-\alpha)$ times the input intensity, is placed between two optical
parametric amplifiers (OPAs), on the upper arm of the interferometer. The
first OPA generates bright squeezed light with coherent light at one of the
input modes. The second OPA reverses the squeezing transformation. Detection
of the number of outgoing photons $\left\langle
a_{out}^{\dagger}a_{out}\right\rangle$ leads to estimating the photon-loss
coefficient $\alpha$.

Here we present the SU(1,1) interferometer scheme which further improves the
signal-to-noise ratio for absorption measurement at certain squeezing
conditions in comparison with the balanced photodetection scheme. Referring to
Fig. 2 this setup has one additional step before detection where the probe
beam and the ancilla both enter another OPA where they undergo the anti-
squeezing process. Thus, the second OPA reverses the squeezing transformation
$r\rightarrow-r$ giving the new output states as

$\displaystyle{a_{out}}$ $\displaystyle=$ $\displaystyle{a_{2}}\cosh
r-{b_{1}^{\dagger}}\sinh r,$ $\displaystyle{b_{out}}$ $\displaystyle=$
$\displaystyle{b_{1}}\cosh r-{a_{2}^{\dagger}}\sinh r.$ (17)

This additional step further improves the sensitivity of the setup as
discussed ahead in the results section. Using Eq. (17) and replacing
${a_{out}}$ in Eq. (4) with ${a_{2}}$ we can write the output operators as

$\displaystyle{a}_{{\rm out}}=$ $\displaystyle{a}_{{\rm
in}}\left(\sqrt{1-\alpha}\cosh^{2}r-\sinh^{2}r\right)$
$\displaystyle-{b}_{{\rm in}}^{\dagger}\left(1-\sqrt{1-\alpha}\right)\sinh
r\cosh r$ $\displaystyle+{v}\cosh r\sqrt{\alpha},$ $\displaystyle{b}_{{\rm
out}}=$ $\displaystyle{b}_{{\rm
in}}\left(\cosh^{2}r-\sqrt{1-\alpha}\sinh^{2}r\right)$ (18)
$\displaystyle+{a}_{{\rm in}}^{\dagger}\left(1-\sqrt{1-\alpha}\right)\sinh
r\cosh r$ $\displaystyle-{v}^{\dagger}\sqrt{\alpha}\sinh r.$

For simplicity, we write ${a}_{out}$ and ${b}_{out}$ as

$\displaystyle{a}_{out}$ $\displaystyle=$
$\displaystyle{a_{in}}d_{11}+{b_{in}^{\dagger}}d_{12}+{v}d_{13},$
$\displaystyle{b}_{out}$ $\displaystyle=$
$\displaystyle{a_{in}^{\dagger}}d_{21}+{b_{in}}d_{22}+{v^{\dagger}}d_{23}.$
(19)

where

$\displaystyle\begin{array}[]{cc}&d_{11}=\sqrt{1-\alpha}\cosh^{2}r-\sinh^{2}r,\\\
&d_{12}=-\left(1-\sqrt{1-\alpha}\right)\sinh r\cosh
r,d_{13}=\sqrt{\alpha}\cosh r,\\\ &d_{21}=\left(1-\sqrt{1-\alpha}\right)\sinh
r\cosh r,\\\
&d_{22}=\cosh^{2}r-\sqrt{1-\alpha}\sinh^{2}r,d_{23}=-\sqrt{\alpha}\sinh r.\\\
\end{array}$

At this point, we need to extract the information on the absorption
coefficient by detecting the mean number of photons and the corresponding
variance. We first consider the case of $\langle{N}_{\rm out}\rangle=\langle
a_{\rm out}^{\dagger}a_{\rm out}+b_{\rm out}^{\dagger}b_{\rm out}\rangle$
which typically constitutes the signal for SU(1,1) technique. The signal and
the fluctuations, in this case, are given by

$\left\langle N_{out}\right\rangle=|u|^{2}\left(d_{11}^{2}+d_{21}^{2}\right),$
(23)

and,

$\displaystyle\Delta^{2}N_{out}$ $\displaystyle=$
$\displaystyle|u|^{2}[d_{11}^{2}\left(1+2d_{12}^{2}\right)+d_{21}^{2}\left(d_{21}^{2}+d_{22}^{2}+d_{23}^{2}\right)$
(24) $\displaystyle+$ $\displaystyle
2d_{11}d_{21}(d_{11}d_{21}+d_{12}d_{22}+d_{13}d_{23})],$

respectively. The sensitivity in Eq. (15) applies to both the measurement
schemes. Thus we get,

$(\Delta\alpha)_{+}=\frac{\sqrt{U}}{V},$ (25)

where

$\displaystyle U$ $\displaystyle=$ $\displaystyle
4-4\zeta-4\alpha\cosh^{2}r+2\zeta\cosh{4r}-2\alpha\eta\sinh^{2}4r,$
$\displaystyle V$ $\displaystyle=$
$\displaystyle\frac{2\cosh^{2}r}{\sqrt{1-\alpha}}(2-\sqrt{1-\alpha}\cosh{2r}).$

We use $\zeta:=1-\sqrt{1-\alpha}$ and $\eta:=\alpha/4+\sqrt{1-\alpha}$ for
simplicity. Notice that V in the denominator approaches zero when
$r\rightarrow 2$ for $\alpha=0.05$ and when $r\rightarrow 3$ for
$\alpha=0.01$. At these points $U\neq 0$, thus the sensitivity ratio has a
singularity at $\sinh^{2}r=\frac{\sqrt{1-\alpha}}{2\zeta}$ as shown in Fig.
3(a). For small $\alpha$, $\sinh r\sim\frac{1}{\sqrt{\alpha}}$. Note that the
singularity can be avoided by detecting the signal only from one of the ports
such that $\langle{N}_{\rm out}\rangle=\langle a_{\rm out}^{\dagger}a_{\rm
out}\rangle$. In this case, we calculate the signal and the associated
fluctuations as

$\left\langle{N}_{out}\right\rangle=|u|^{2}d_{11}^{2},$ (26)
$\Delta^{2}N_{out}=|u|^{2}d_{11}^{2}\left(1+2d_{12}^{2}\right).$ (27)

Using Eqs. (15), (23-24) the sensitivity could be written in terms of r and
$\alpha$ as

$\Delta\alpha=\frac{\sqrt{(1-\alpha)(1+4(1-\sqrt{1-\alpha})^{2}\cosh^{2}r\sinh^{2}r)}}{|u|\cosh^{2}r}.$
(28)

Now that we have calculated the sensitivities from the measurement
perspective, we can compare their quantum advantage and discuss the best
possible schemes under different squeezing conditions.

##  V Quantum Advantage: Sensitivity Enhancement in Measurement Schemes

Figure 3: Sensitivities $\Delta\alpha/\Delta\alpha_{coh}$ (red line - balanced-photodetection, blue line - SU(1,1)) and the Cramér-Rao bound $(\Delta\alpha/\Delta\alpha_{coh})_{min}$ (orange line) using two-mode bright squeezed light, scaled by the sensitivity of using a coherent light without the interferometer, as a function of the squeezing parameter $r$, for weak absorption (a) $\alpha=0.05$ (b) $\alpha=0.01$. In addition, Fig. 3a also shows $\Delta\alpha_{+}$ (purple line). $\alpha$ | r | $QA_{BD}$ | $QA_{SU(1,1)}$ | $QA_{CRB}$
---|---|---|---|---
0.05 | 1.99 | 3.32 | 3.36 | 3.42
2.35 | 2.81 | 3.77 | 3.85
0.01 | 2.82 | 7.39 | 7.53 | 7.63
3.17 | 6.26 | 8.41 | 8.59
Table 1: Quantum Advantage ($QA=\frac{\Delta\alpha_{coh}}{\Delta\alpha}$) for
absorption measurement using: - balanced photodetection (BD), SU(1,1)
interferometer, and the corresponding Cramér-Rao bound (CRB). Values displayed
in bold are the best possible QA attainable using the specified measurement
technique.

We scale the sensitivities derived using the measurement schemes with the
sensitivity using only a coherent state of the same photon number entering the
sample without the interferometer. The ratio for the balanced photodetection
(BD) measurement scheme is expressed as

$\displaystyle\begin{array}[]{cc}&\left(\frac{\Delta\alpha}{\Delta\alpha_{coh}}\right)_{A}=\frac{\sqrt{(1+2\alpha^{2}\cosh^{2}r\sinh^{2}r-\alpha\cosh^{2}r)/(1-\alpha)}}{\cosh
r},\end{array}$ (30)

and, the ratio for SU(1,1) interferometer with signal $\langle a_{\rm
out}^{\dagger}a_{\rm out}\rangle$ is written as

$\displaystyle\begin{array}[]{cc}&\left(\frac{\Delta\alpha}{\Delta\alpha_{coh}}\right)_{B}=\frac{\sqrt{1+4(1-\sqrt{1-\alpha})^{2}\cosh^{2}r\sinh^{2}r}}{\cosh
r}\end{array},$ (32)

where the sensitivity using only a coherent state without an interferometer is
given by

$\Delta\alpha_{coh}=\frac{\sqrt{1-\alpha}}{|u|\cosh r}.$ (33)

Fig. 3 shows the plots of the scaled sensitivities against the squeezing
parameter r for (a) $\alpha=0.05$ and (b) $\alpha=0.01$. Eq. (LABEL:a) and and
Eq. (LABEL:b) are represented by the red line and the blue line respectively.
The Craḿer Rao bound Eq. (12) is represented by the orange line. As displayed
in Table I, both the setups can attain a quantum advantage
($QA=\frac{\Delta\alpha_{coh}}{\Delta\alpha}$) of greater than 3 for
$\alpha=0.05$ and a quantum advantage of greater than 7 for $\alpha=0.01$. For
$\alpha=0.05$ maximum advantage for the BD technique is achieved at
$r=1.99\,(QA=3.32)$, and for the SU(1,1) technique is achieved at
$r=2.35\,(QA=3.77)$. For $\alpha=0.01$ maximum advantage for the BD technique
is achieved at $r=2.82\,(QA=7.39)$ and for the SU(1,1) technique is achieved
at $r=3.17\,(QA=8.41)$. This result is significant in the weak absorption
domain. As we calculate the sensitivity of the measurement it is worth noting
the significance of the large number of photons in the two-mode bright
squeezed state given the $1/\sqrt{N}$ dependence, where N is the total number
of photons passing through the sample at a given time. For example, around
$r=2$, $N$ is approximately $14|u|^{2}$, where $|u|^{2}$ is the total number
of photons in the input coherent field.

Looking at the plots in Fig. 3, we find out that the balanced photodetection
method gives a slightly greater quantum advantage relative to the SU(1,1)
method just before it reaches its maximum around $r=2$ for $\alpha=0.05$ and
$r=2.82$ for $\alpha=0.01$. However, using the full SU(1,1) scheme can provide
a higher quantum advantage for $r\geq 2$ for $\alpha=0.05$ and $r\geq 2.82$
for $\alpha=0.01$. Consequently, it might be advantageous to use the balanced
photodetection method given its benefit of less number of optical components
leading to fewer experimental errors. However, SU(1,1) interferometer can
prove beneficial when technological advancements allow for the higher
squeezing parameters $r\geq 2$.

Next, the scaled sensitivity of both measurement schemes is compared to the
theoretical limit given by the Cramér-Rao bound (orange line) which is
obtained by taking the ratio of sensitivity Eq. (12) calculated using the
quantum Fisher information of the two-mode bright squeezed state and that for
the coherent state again given by
$\Delta\alpha_{coh}=\frac{\sqrt{1-\alpha}}{|u|\cosh r}$. Note that the QFI of
the TMBSS remains unaffected by the anti-squeezing action of the second OPA.
Our results exhibit a close correspondence with the Cramér-Rao bound,
specifically in the region between $1.5\leq r\leq 2.5$ for $\alpha=0.05$ and
$2\leq r\leq 3$ for $\alpha=0.01$. This emphasizes the sensor’s exceptional
performance as measured against the theoretical lower bound for the given
system.

##  VI Two-Mode Bright Squeezed Light for Gain Measurement: Cramér-Rao Bound
and Measurement Schemes

Apart from absorption, gain is another fundamental physical phenomenon that
has broad applications in sensing and imaging technology. For example, gain
sensing proves valuable in detecting Unruh-Hawking radiation using single-mode
probes PhysRevLett.105.151301 . Typically, the gain parameter associated with
these processes is small, as it relies on population inversion within the
medium. Optical amplification mechanisms such as stimulated emission in LASER
media, Stimulated Raman Scattering (SRS), and Stimulated Brillouin Scattering
(SBS) are characterized by this gain parameter. The gain obtained from SBS,
for instance, serves as a signal for measuring the viscoelastic properties of
the medium in imaging and spectroscopic applications. Similarly, the gain from
SRS is utilized to characterize the nonlinear properties of optical fibers,
which are critical components in telecommunication systems. Like absorption,
the gain measurement is sensitive to system noise, necessitating advanced
techniques to achieve precise measurement xu2022quantum ; triginer2020quantum
. Nair et al. PhysRevLett.128.180506  have considered optimal measurement of
gain using quantum probes. However, the advantages offered by adding a
coherent seed while generating the two-mode squeezed state remain unexplored.
Thus here, we again consider the Two-Mode Bright Squeezed State (TMBSS) for
estimating the gain parameter. We first calculate QFI for gain measurement
using the TMBSS and provide the best possible sensitivity for the gain
parameter which is the Cramér-Rao bound. Considering Fig. 1 but for the gain
medium, the annihilation operators after the sample written in terms of the
input operators are given by,

$\displaystyle\begin{array}[]{cc}&{a_{out}}={a_{1}}\sqrt{G}+{v}^{\dagger}\sqrt{G-1},\\\
&{b_{out}}={b_{1}}.\end{array}$ (36)

Figure 4: Sensitivities $\Delta G/\Delta G_{coh}$ (red line - balanced
photodetection, purple line - SU(1,1) with intensity addition) and the Cramér-
Rao Bound $\Delta G_{CRB}/\Delta G_{coh}$ (orange line) using two-mode bright
squeezed light, scaled by the sensitivity of using a coherent light without
the interferometer, as a function of the squeezing parameter $r$, for gain
$G=1.05$.

The Gaussian characteristics of the input state are preserved under the
transformation in a gain medium agarwal1987wigner . Thus, following the same
process as used in Section III, the displacement vector and the covariance
matrix are:

$s^{(G)}=\left(\begin{array}[]{c}u\sqrt{G}\cosh r\\\ u^{\ast}\sinh r\\\
u^{\ast}\sqrt{G}\cosh r\\\ u\sinh r\end{array}\right),$ (37)

and

$\beta^{(G)}=\begin{bmatrix}\beta_{11}^{(G)}&0&0&\beta_{14}^{(G)}\\\
0&\beta_{22}^{(G)}&\beta_{23}^{(G)}&0\\\
0&\beta_{32}^{(G)}&\beta_{33}^{(G)}&0\\\
\beta_{41}^{(G)}&0&0&\beta_{44}^{(G)}\end{bmatrix},$ (38)

where

$\displaystyle\beta_{11}^{(G)}=\beta_{33}^{(G)}=2G\cosh^{2}r-1,$
$\displaystyle\beta_{22}^{(G)}=\beta_{44}^{(G)}=1+2sinh^{2}r,$
$\displaystyle\beta_{14}^{(G)}=\beta_{41}^{(G)}=\beta_{23}^{(G)}=\beta_{32}^{(G)}=2\sqrt{G}\cosh
r\sinh r.$

We obtain the QFI as

$F_{Q}(G)=\frac{|u|^{2}\cosh^{2}r\cosh 2r}{G[G+(G-1)\cosh 2r]},$ (39)

which leads to $(\Delta G)_{CR}=[F_{Q}(G)]^{-1/2}$. Considering the
measurement schemes, we perform similar calculations as done previously to
determine the sensitivity of measuring the gain parameter using the two
measurement schemes. For the balanced photodetection method, the output modes
are given by

$\displaystyle{a}_{out}$ $\displaystyle=$
$\displaystyle{a_{in}}m_{11}+{b_{in}^{\dagger}}m_{12}+{v^{\dagger}}m_{13},$
$\displaystyle{b}_{out}$ $\displaystyle=$
$\displaystyle{a_{in}^{\dagger}}m_{21}+{b_{in}}m_{22},$ (40)

where,

$\displaystyle m_{11}=\sqrt{G}\cosh r,\hskip 2.84544ptm_{12}=\sqrt{G}\sinh
r,\hskip 2.84544ptm_{13}=\sqrt{G-1},$ $\displaystyle m_{21}=\sinh r,\hskip
2.84544ptm_{22}=\cosh r.$

Using the intensity difference $I_{d}$ between two beams as the signal we
calculate the sensitivity for this setup to be

$(\Delta G)_{BD}=\frac{\sqrt{I_{d}(2I_{d}-|u|^{2})}}{|u|\cosh^{2}r},$ (41)

where $I_{d}=|u|^{2}(G\cosh^{2}r-\sinh^{2}r)$.

For the SU(1,1) interferometer scheme with gain medium, the final output
operators are given by

$\displaystyle{a}_{out}$ $\displaystyle=$
$\displaystyle{a_{in}}f_{11}+{b_{in}^{\dagger}}f_{12}+{v^{\dagger}}f_{13},$
$\displaystyle{b}_{out}$ $\displaystyle=$
$\displaystyle{a_{in}^{\dagger}}f_{21}+{b_{in}}f_{22}+{v}f_{23},$ (42)

where

$\displaystyle\begin{array}[]{cc}&f_{11}=\sqrt{G}\cosh^{2}r-\sinh^{2}r,\\\
&f_{12}=\left(\sqrt{G}-1\right)\sinh r\cosh r,f_{13}=\sqrt{G-1}\cosh r,\\\
&f_{21}=-\left(\sqrt{G}-1\right)\sinh r\cosh r,\\\
&f_{22}=\cosh^{2}r-\sqrt{G}\sinh^{2}r,f_{23}=-\sqrt{G-1}\sinh r.\\\
\end{array}$

Considering $\langle{N}_{\rm out}\rangle=\langle a_{\rm out}^{\dagger}a_{\rm
out}+b_{\rm out}^{\dagger}b_{\rm out}\rangle$ as the signal and calculating
the corresponding fluctuations $\Delta N_{out}$, we find the sensitivity as

$(\Delta G)_{+}=\frac{\sqrt{GA}}{B},$ (46)

where

$\displaystyle A$ $\displaystyle=$ $\displaystyle\mu(7G^{3/2}-3G+5\sqrt{G}-1)$
$\displaystyle+$ $\displaystyle 4(3\sqrt{G}\mu-\nu)\mu\nu\cosh 2r$
$\displaystyle+$ $\displaystyle 8\mu^{2}\nu^{2}\cosh 4r+4\mu\nu^{3}\cosh
6r+\nu^{4}\cosh 8r),$ $\displaystyle B$ $\displaystyle=$ $\displaystyle
4\cosh^{2}r(1+\nu\cosh 2r).$

We use $\mu:=\sqrt{G}+1$ and $\nu:=\sqrt{G}-1$ for simplicity.

In Fig. 4 we plot the scaled sensitivities ($\Delta G/\Delta G_{coh}$) for
both the measurement schemes along with the Cramér-Rao bound $(\Delta
G)_{CR}/\Delta G_{coh}$ where

$\Delta G_{coh}=\frac{\sqrt{G}}{|u|\cosh r}.$ (47)

Here, we find out that the SU(1,1) method gives a greater quantum advantage
relative to the balanced photodetection method for any given squeezing
parameter $r$. The best possible quantum advantages for both measurement
schemes are shown in Table II. For $G=1.05$ at $r=2.3$ SU(1,1) is 2.81 times
and the balanced detection is 1.78 times better, and for $G=1.01$ at $r=3.17$,
SU(1,1) is 6.28 times and the balanced detection is 3.93 times better than
using a coherent source. We can also see that both methods, specifically the
SU(1,1) interferometer closely follow the Cramér-Rao bound which is the
theoretical lower limit for the given scheme of measurements.

G | r | $QA_{BD}$ | $QA_{SU(1,1)}$ | $QA_{CRB}$
---|---|---|---|---
1.05 | 2.37 | 1.78 | 2.81 | 3.82
1.01 | 3.17 | 3.93 | 6.28 | 8.58
Table 2: Quantum advantage for the gain measurement schemes: - balanced
photodetection (BD), SU(1,1) interferometer, and the corresponding Cramér-Rao
bound (CRB)

##  VII Discussion

This paper investigates the quantum advantage of using a Two-Mode Bright
Squeezed State (TMBSS) to estimate the absorption parameter in a weakly
absorbing medium as well as the small gain parameter in any optical
amplification process. We present two setups – A) balanced photodetection and
B) SU(1,1) interferometer, and discuss the best possible measurement schemes
under different squeezing conditions. The paper’s main goal is to showcase the
improvement in the sensitivity measurement of the absorption or the gain
parameter of the sample placed in one of the beam paths generated from an
optical parametric amplifier. This study combines the benefits of a large
number of photons in a coherent state with the quantum properties of the two-
mode squeezing operation.

We first calculate the Quantum Fisher Information (QFI) of the two-mode bright
squeezed state which leads us to the theoretical limit called the Cramér-Rao
bound for measuring the absorption sensitivity. We then calculate the
absorption sensitivities of both measurement schemes. We scale both these
sensitivities by using only the coherent source with the same number of
photons but without the interferometer.

The results show these setups can attain more than 7-fold Quantum Advantage
(QA) for $\alpha=0.01$ and 3-fold QA for $\alpha=0.05$. This is an important
result for the weak absorption domain. Plotting the scaled sensitivities
against $r$ reveals the significance of using one measurement scheme over the
other in different squeezing conditions. For example, the balanced
photodetection method gives a slightly greater quantum advantage relative to
the SU(1,1) method just before it reaches its maximum around r = 2 for
$\alpha=0.05$. Thus, given the current technological limitations on the
squeezing parameter $r$, the balanced photodetection method is beneficial
because of its comparatively smaller number of optical components minimizing
the experimental errors. However, the SU(1,1) interferometer gives a higher
quantum advantage as we attain $r\geq 2$ beyond which the QA for balanced
detection goes on decreasing. Thus, the SU(1,1) interferometer holds promise
for future absorption-related applications, particularly as technological
progress enables higher squeezing parameters $r$.

For the gain measurement, we find out that the SU(1,1) method shows a better
quantum advantage than the balanced photodetection method for any given
squeezing parameter $r$. For $G=1.05$ at $r=2.37$, SU(1,1) is 2.81 times and
the balanced detection is 1.78 times better than just using the coherent
source without any interferometer. Comparing the scaled sensitivity of both
the measurement schemes with the Cramér-Rao bound confirms the quantum
advantage of the proposed setups in the estimation of absorption as well as
the gain parameters.

##  VIII ACKNOWLEDGMENTS

The authors are grateful for the support of the Air Force Office of Scientific
Research (Award No. FA-9550-20-1-0366), the Robert A. Welch Foundation
(A-1943-20210327), and the Department of Energy (Grant No. FWP-ERW7011).

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