# Poissonian twin beam states and the effect of symmetrical photon subtraction
in loss estimations

N. Samantaray  Email<EMAIL_ADDRESS>J. C. F. Matthews  J. G. Rarity
Quantum Engineering Technology Labs, H. H. Wills Physics Laboratory and
Department of Electrical and Electronic Engineering, University of Bristol,
BS8 1FD, UK

###### Abstract

We have devised an experimentally realizable model generating twin beam states
whose individual beam photon statistics are varied from thermal to Poissonian
keeping the non-classical mode correlation intact. We have studied the
usefulness of these states for loss measurement by considering three different
estimators, comparing with the correlated thermal twin beam states generated
from spontaneous parametric down conversion or four wavemixing . We then
incorporated the photon subtraction operation into the model and demonstrate
their advantage in loss estimations with respect to un-subtracted states at
both fixed squeezing and per photon exposure of the absorbing sample. For
instance, at fixed squeezing, for two photon subtraction, up to three times
advantage is found. In the latter case, albeit the advantage due to photon
subtraction mostly subsides in standard regime, an unexpected result is that
in some operating regimes the photon subtraction scheme can also give up to
20$\%$ advantage over the correlated Poisson beam result. We have also made a
comparative study of these estimators for finding the best measurement for
loss estimations. We present results for all the values of the model
parameters changing the statistics of twin beam states from thermal to
Poissonian.

##  I Introduction

Absorption based measurement underpins many approaches to spectroscopy and
imaging. It finds application in all branches of science from chemistry and
biology to physics and material science. However, the best sensitivity in loss
estimation reached so far using classical light probes is limited by photon
shot noise. In last years, non classical resources such as non-Gaussian states
(by de-Gaussification of Gaussian states) have shown to reach sub-shot noise
(SSN) limit in loss estimations in terms of Fisher information Olivares:2003 .
De-Gaussified single mode squeezed vacuum has been reported for theoretical
quantum enhancement in loss estimation Gerardo:2009 . Two basic operations
that can lead to non-Gaussian states are photon addition to, or photon
subtraction from Gaussian light states kim:2008 ; kim1:2008 . Another
important feature of bi-partite quantum states to reach sub-shot noise limit
(SSNL) are non-classical correlations Rarity:1991 .

It is known that the twin beam state (TBS) generated by spontaneous parametric
down conversion (SPDC) or four wave mixing (FWM) process has thermal photon
statistics in the individual modes, but its perfect photon number non
classical mode correlation allows surpassing the shot-noise limit (SNL)
reaching SSN sensitivity in the realistic scenario of loss estimations
Chekova:2015 ; Brida:2009 ; Lopaeva:2013 ; Marco:2017 ; Moreau:2017 ;
Javier:2019 ; Javier:2017 . More recently, unbiased estimations of optical
losses Losero:2018  (losses are estimated in an absolute way without pre-
calibration of the apparatus) at ultimate quantum limit have been reported
exploiting the quantum correlations in TBS. In the laboratory context, these
correlated beams usually appear Poissonian due to temporal (or spatial)
averaging of thermal statistics.

De-Gaussification by symmetrical photon subtraction on both of the mode of TBS
has not only been shown to improve the individual mode photon statistics from
thermal to sub-Poissonian chekova:2016 , but it also increases the
entanglement between them Carlos:2012 ; Tim:2013 ; ourjoumtsev:2007 . In the
last years, the resulting TBS states after photon subtraction have been
theoretically investigated reporting their advantage over TBS for target
detection in the presence of noise, the so called "quantum illumination"
Zhang:2014 . Their advantage over TBS have also been demonstrated in single
interferometry with parity measurements raul:2012  and more recently for
probing the Plank scale physics Nigam:2019  and distillation of squeezing
Thomas:2019 . Looking at all these advantages of symmetrical photon subtracted
TBS (SPSTBS) over TBS because of their improved photon statistics and non-
classical correlation, we proliferated our interest for using SPTBS for loss
estimations. However, a question on fundamental grounds naturally arises: does
photon subtraction have any advantage in noise suppression if the individual
mode photon statistics of TBS becomes Poissonian (due to averaging of thermal
statistics). Keeping this motivation in mind, we have devised a theoretical
but experimental realizable model (accounting detection losses), where
changing the value of a parameter of the models changes the TBS individual
mode photon statistics from thermal to Poissonian.

For a null value of the model parameter, i.e, when the statistics of
individual beams are Poissonian, the resulting state becomes correlated
Poissonian TBS (CPTBS) keeping the initial TBS non-classical mode correlation
intact. We then incorporated the symmetrical photon subtraction into the model
of absorption measurement. We replace the conventional approach of obtaining
the photon subtraction(placing high transmittance beam splitters on the
individual beam paths ) by an alternate way of seeding the photon number
super-position state Nigam:2019  to the squeezer as shown in the Fig.1.

Figure 1:  An equivalent ways of getting photon subtracted state: (a) The left
hand side image represents the conventional approach in which two high
transmittance beam splitters are placed in the each path of TBS and a
simultaneous photon clicks on the single photon detectors confirms the
implementation of symmetrical photon subtraction, this subtraction operation
can be implemented equivalently by seeding a superposition state to the non-
linear crystal (NL) together with the pump beam as shown in the right image
(b).

One of the important goals of this letter is to answer the following question.
To what extent does photon subtraction bring advantage for absorption
measurement compared to TBS at both fixed squeezing and per photon exposure to
the absorbing sample and furthermore does photon subtraction provide any
advantage particularly when the individual TBS mode statistics turns to
Poissonian. Apart from a few recent works which consider phase measurement
Nigam:2019 ; raul:2012 , most of the quantum optics and information protocols
demonstrate the advantage of photon subtraction at a fixed squeezing parameter
Olivares:2003 ; Genoni:2010 ; Optarny:2000 ; Cochrane:2002 .

This paper is organized in the following way. In section II, we shall briefly
describe the absorption measurement and various types of estimators for loss
estimation. Importance of photon statistics and non-classical correlation in
measuring these estimators will also be addressed. Section III details this
model and a way to incorporate photon subtraction operation. We shall also
present results for different types of absorption estimators up to two photon
subtraction and discuss the usefulness of our model. All the values of the
model parameter that change the statistics of TBS from thermal to Poisonian
and in between have also been considered in the result. We conclude the paper
with a summary in section IV.

##  II Absorption measurement:

Absorption is measured by probing the sample with known light intensity and
then measuring the light intensity at the detection stage as shown in the
fig.2, where $\gamma$ is the absorption coefficient, $\eta$ is the detection
loss, $N_{P}$ and $N^{{}^{\prime}}_{P}$ are the number of detected photons
before and after placing the sample respectively.

Figure 2: Absorption measurement: (a) direct one path imaging of a sample
object of absorption coefficient $\gamma$ and $\eta$ is the detection loss,
(b) beam splitter equivalence of reflection and detection losses in direct one
path imaging, (c) photon number difference measurement in the presence of
quantum correlation generated by pumping the non-linear crystal in SPDC
process.

The losses due to the presence of sample and the detection losses are modelled
by using a single beam splitter with transmittance
$\tau=\eta\left(1-\gamma\right)$. For applications where low light
illumination is required, the uncertainty in measuring $\gamma$ is dominated
by photon shot noise (SN). The uncertainty in absorption measurement due to
photon shot noise can be improved by considering two beams which are
correlated in the photon number basis as in TBS states generated by SPDC
process. The first beam passes though the sample, whereas the second beam acts
as reference thus partially cancelling the SN going below shot noise limit in
realistic scenarios. We shall consider three different absorption estimators
including a new one based on ratio measurement, the TBS state as input source,
and balanced detection (quantum efficiency $\eta$ remains the same for both
beams) throughout this work. The principal point here is to show the
dependence of uncertainties in measuring different absorption estimators on
parameters that characterize photon statistics and correlation of the input
probe states such as Fano factor (F), and noise reduction factor denoted by
the symbol $\sigma$, and finding the best absorption measurement for loss
estimations. Fano factor is defined as the variance of photon number of a
state normalized to its mean value. In terms of statistics, it represents how
a state is different from a coherent state. $F>1$, $F=1$, and $F<1$ refers to
super-Poissonian, Poissonian and sub-Poissonian statistics of the light state
respectively. Analogously, for any bi-partite state, $\sigma$ is defined as
variance of the photon number difference normalized to their mean. One can
easily check $\sigma<1$ refers to non-classical photon number correlation. It
is worth checking the change in these two parameters when individual TBS mode
statistics change from thermal (super-Poissonian) to Poissonian. We have
devised a phenomenological model in this context and we shall detail it in the
next sections. Furthermore, it is interesting for both fundamental
perspectives and applications to see the change in photon statistics and
correlation by subtracting photons symmetrically from each mode of TBS, and to
investigate up-to what extent the photon subtraction operation is advantageous
in this scheme.

###  II.1 Number difference measurement:

In this measurement, the observable under consideration
$\hat{o}=\hat{N_{R}}-\hat{N^{{}^{\prime}}_{P}}$ is photon number difference of
two beams after placing the object as shown in Fig.2 (b). As per theory of
error propagation, the uncertainty in measuring $\gamma$ Gerry:2005  is

$\Delta\gamma_{Diff}=\frac{\sqrt{\Delta^{2}\hat{o}}}{|\frac{\partial\langle\hat{o}\rangle}{\partial\gamma}|}=\sqrt{\frac{\gamma^{2}[F-1]+\gamma+2\sigma\left(1-\gamma\right)}{\langle
N_{P}\rangle}},$ (1)

where $\langle N_{P}\rangle=\eta\sinh^{2}r=\eta\lambda$ is the detected mean
number of photons per mode of the probe TBS state, r being the squeezing
parameter. r carries necessary information about the pump intensity and phase
matching function of the SPDC process. It is easy to check for classical
states, i.e, $F=1$ and $\sigma=1$, a limit
$\Delta\gamma_{Diff}=\sqrt{(2-\gamma)/\langle N_{P}\rangle}$ known as the shot
noise limit (SNL) in differential absorption measurement. It is paramount to
note for no absorption ($\gamma=0$), this limit is twice the standard shot
noise limit in direct one path imaging $1/\sqrt{\langle N_{P}\rangle}$. Each
beam carries one unit of shot noise although for $\gamma=1$, standard SNL is
reached. It can be checked that for low values of of absorption, i.e,
$\gamma<<1$, $F=1$ and $\sigma<1/2$ allows beating SNL, where as for
relatively high $\gamma$, the probe state with $\sigma<1/2$ and $F<1$ is
required for reaching SSN limit.

###  II.2 Optimized balanced absorption estimator:

In the last couple of years, a different absorption estimator of the following
form Moreau:2017  has been considered

$\gamma_{Opt}=1-\frac{\hat{N^{{}^{\prime}}_{P}-k\Delta N_{R}+\delta
E}}{\langle N_{P}\rangle},$ (2)

where k is a factor to be experimentally determined in order to minimize the
uncertainty and $\delta E$ is a correction factor for making the estimator
unbiased. Exploiting theory of error propagation, we worked out the
uncertainty of this estimator

$\Delta\gamma_{Opt}=\sqrt{\frac{\gamma\left(1-\gamma\right)}{\langle
N_{P}\rangle}+\frac{\left(1-\gamma\right)^{2}\sigma}{\langle
N_{P}\rangle}\left(2-\frac{\sigma}{F}\right)}$ (3)

A clear advantage of this estimator is seen as it is $\sqrt{2}$ times
advantageous compared to the number difference at low absorption
($\gamma\rightarrow 0$) for $\sigma=1$ and $F=1$. Another interesting point
about this estimator is the requirement of lower quantum correlation, i.e
$\sigma<1$ ( $\sigma<1/2$ for the number difference case ) and $F<1$ for
reaching SSN limit in absorption measurement. It can be easily checked for
both $\sigma=0,F=1$ (perfect photon number correlation and Poissonian
individual statistics) and $\sigma=1,F=1/2$ (classical photon number
correlation and sub-poissonian individual statistics), the uncertainty in eq.3
simplifies to an ultimate quantum limit (UQL)Losero:2018 ,

$\Delta\gamma_{Opt}=\sqrt{\frac{\gamma\left(1-\gamma\right)}{\langle
N_{P}\rangle}}$ (4)

###  II.3 Ratio measurement:

We have considered a new type of estimator based on ratio measurement. In this
measurement the observable we consider is
$\hat{o(\gamma)}=N^{{}^{\prime}}_{P}/N_{R}$, or
$\hat{o(\gamma)}=N_{R}/N^{{}^{\prime}}_{P}$. Propagating the errors, we obtain
the expression of uncertainty in ratio measurement as:

$\Delta\gamma_{Ratio}=\sqrt{\frac{\gamma\left(1-\gamma\right)}{\langle
N_{P}\rangle}+\frac{\left(1-\gamma\right)^{2}2\sigma}{\langle N_{P}\rangle}}.$
(5)

Unlike other two estimators, it shows dependence of the measured absorption
uncertainty only on correlation of the probe state $\sigma$. It can be checked
for $\sigma=1/2$ and $\sigma=0$ (perfect quantum correlation), the
corresponding uncertainty becomes SNL (direct one path imaging) and UQL (eq.4)
respectively. Thus, similar to the number difference measurement, ratio
estimator beats direct one path imaging when less than 50$\%$ of the photon
number correlation is lost.

##  III Model for correlated TWB state and symmetrical photon subtraction:

Figure 3: Scheme for generating correlated TBS states: Coherent beam pumps a
non linear crystal (NL) for generating correlated twin photon states at the
exit face of the crystal.

Considering the quantum picture of the parametric down conversion, we modelled
the mode operators before and the after the non-linear crystal (see fig. 3) in
the following way

$\hat{c_{1}}(\lambda,\beta)\approx\hat{a_{1}}\sqrt{1+\beta\lambda}+\hat{a^{\dagger}_{2}}\sqrt{\lambda},$
(6)
$\hat{c_{2}}(\lambda,\beta)\approx\hat{a_{2}}\sqrt{1+\beta\lambda}+\hat{a^{\dagger}_{1}}\sqrt{\lambda},$
(7)

where $\beta=1/M$ is the model parameter, $M$ being the number of modes. For
$\beta=1$, these set of equations look like the usual Bougolibov
transformation when vacuum state turns to two mode squeezed state by an action
of two mode squeezed operator $\hat{S}_{1,2}(\lambda)$. In experimental
situation, significance of $\beta$ can be seen as follows: for single temporal
mode in the time window of picoseconds which is the coherence time
$\tau_{coh}$ of the SPDC, $\beta=1$, whereas for more time exposure, many
temporal modes are collected and in that case $\beta\approx 0$. The last case
is realised for high temporal band width pump beam so that many more modes
$M=\tau_{p}/\tau_{coh}$ are generated. This situation is usually considered
experimentally for alleviating the excess noise from the individual TBS
demonstrating its usefulness for SSN absorption measurement Bambrilla:2008 .
In the last years, similar situation of CW pump has been considered
demonstrating experimentally the SSN advantage in reconstructing the
absorption profile of an object Moreau:2017 , and more recently in the
construction of SSN raster scanning microscopeJavier:2019 . Another important
point is the effect of pump photon statistics on the generated twin photon
statistics via SPDC. Intuitively, $0\leq\beta\leq 1$ accounts all of the last
considered experimental situations. Therefore it is intriguing to consider the
parameter $\beta$ in this model. Detected photon statistics and correlation
are calculated using the above set of transformation equations and vacuum
state as follows:

$\displaystyle\langle N_{P}\rangle=\langle N_{R}\rangle=\eta\lambda,$
$\displaystyle\langle\Delta^{2}N_{P}\rangle=\langle\Delta^{2}N_{R}\rangle=\eta\lambda+\beta\eta^{2}\lambda^{2}$
(8)
$\langle\Delta(N_{P},N_{R})\rangle=\eta^{2}\left(\lambda+\beta\lambda^{2}\right),$
$\displaystyle\sigma=\frac{\langle\Delta^{2}\left(N_{P}-N_{R}\right)\rangle}{\langle
N_{P}\rangle+\langle N_{R}\rangle}=1-\eta,$

where $\langle N\rangle$ is the detected mean numbers of photons, the
subscripts $P$ (1), $R$ (2) correspond to probe (signal), reference (idler)
respectively, and $\langle\Delta^{2}N\rangle=\langle N^{2}\rangle-\langle
N\rangle^{2}$, $\langle\Delta(N_{P},N_{R})\rangle=\langle
N_{P}N_{R}\rangle-\langle N_{P}\rangle\langle N_{R}\rangle$ are the respective
variance and covariance. An important remark here is the fact that $\sigma$ is
independent of $\beta$ and it is limited by the detection losses $\eta$. It
implies the photon number correlation remain intact regardless of the values
of $\beta$. Looking at the expression of variance, when $\beta=0$,
$\langle\Delta N^{2}_{j}\rangle=\langle N_{j}\rangle$ (j=P,R), i.e, the
individual beam noise has Poissonian statistics, whereas for $\beta=1$, it is
easy to see the noise of the individual beam has dominant thermal noise
contribution. Thus, the model shows a way to switch from thermal to Poissonian
statistics by varying the model parameter $\beta$ from one to zero.

###  III.1 Symmetrical photon subtraction

In this section, we shall see how to incorporate symmetrical photon
subtraction taking into account the model parameter $\beta$. Theoretically
photon subtraction is a non unitary operation, so a normalization factor is
required for getting symmetrical photon subtracted squeezed vacuum state
(thermal case)

$|\Psi\rangle_{m}=N^{-}_{m}\left(\lambda\right)(\hat{a_{1}})^{m}(\hat{a_{2}})^{m}\hat{S_{1,2}}\left(\lambda\right)|0,0\rangle_{1,2},$
(9)

where $N^{-}_{m}$ is the normalization constant of the form

$N^{-}_{m}(\lambda)=m!(-i\sqrt{\lambda})^{m}P_{m}(i\sqrt{\lambda}),$ (10)

$P_{m}$ being the $m$th order Legendre’s polynomial and $m$ is the number of
subtracted photons. In an experimental scenario, two high transmittance beam
splitter are placed in the paths of the two beams of the TWB. Two simultaneous
clicks at the single photon detectors (SPDS) confirms the probabilistic
generation of subtracted states as shown in the left image of fig 1.
Alternatively by injecting m+1 component superposition state Nigam:2019  to
the NL in place of vacuum, equivalently executes deterministically the m
photon subtraction operation as shown in the right image of fig 1. Such
superposition states can be experimentally generated Lee:2012 .Thus photon
subtraction is incorporated by these $m+1$ component superposition state and
the set of transformation equation defined in eq.6 and eq. 7. For subtracted
states, statistics of the transformed operator are obtained by using this
input superposition state in place of vacuum. We calculated the Fano factors
of the photon subtracted states and they are plotted in the fig 4.

Figure 4: (Color Online) Plots of measured Fano Factor as a function of
$\lambda$ with $\eta=0.98$ ($\gamma=0$) for different values of $m$: $m=0$
(solid red line), $m=1$ (solid blue line), and $m=2$ ( solid green line). We
set $\beta=1$ (top) and $\beta=0$ (bottom).

It is evident that for $\beta=1$ (two individual beams are thermal for $m=0$),
photon subtraction changes the statistics from thermal to sub-Poissonian for
low values of the mean number of photon. On the other hand for $\beta=0$ (two
individual beams are Poissonian as expected), photon subtraction further
improves the statistics from Poissonian to sub-Poissonian and the improvement
increases with the increasing $m$. This is an interesting result. Moreover in
this case, the states have higher threshold in terms of $\lambda$ before they
become thermal compared to the case of $\beta=1$. Our calculation shows that
the noise reduction factor $\sigma$ remains the same regardless of the number
of photon subtraction $m$. This is quite expected as for the balanced case,
$\sigma$ is independent of the statistics of the state and only depends on the
detection losses.

###  III.2 Results

Before interpreting the result, we would like to show the dependence of the
mean number of photons per mode with the model parameter $\beta$ as $m$
changes from $0-2$.

Figure 5: (Color Online) 3D plots of detected mean number of photons with
$\eta=0.98$ gated by m detections: $m=0$ (bottom red sheet), $m=1$ (middle
blue sheet), and $m=2$ (top green sheet).

Fig.5 shows the non-linear rise of the mean number of photons with increasing
$m$, and the increment is monotonous with respect to $\beta$, i.e, minimum for
$\beta=0$ and reaches maximum for $\beta=1$. Furthermore, for the last case,
we checked in the low $\lambda$ limit, the rate of increment is maximum, i.e,
four times for $m=1$ and nine times for $m=2$ with respect to $m=0$.
Substituting the expression of $F$ and $\sigma$ in the uncertainty equations
for different values of $m$ $(m=0-2)$, we worked out the uncertainties for
above described three absorption estimators. The expressions are too
cumbersome to present here, so we shall only depict the results graphically
with relevant parameters of interest in the limiting cases.

####  III.2.1 Fixed squeezing parameter

The analysis of comparing uncertainties for different $m$ at fixed squeezing
has been carried at the same mean energy (photon number exposure) as the un-
subtracted ($m=0$) state. In this way, r is fixed as mean energy $\lambda$
(for $m=0$)=$\sinh^{2}r$. Uncertainties for number difference measurement is
shown in fig 6.

Figure 6: (Color online) Plots of Uncertainty in the number difference
measurement versus absorption coefficient $\gamma$ with $\eta=0.98$ for
different values of m: $m=0$ (solid red line), $m=1$ (solid blue line), and
$m=2$ (solid green line). We set $\lambda=0.05$, $\beta=1$ (top),
$\lambda=0.05$, $\beta=0$ (middle), and $\lambda=2$, $\beta=1$ (bottom).Dotted
lines are the uncertainties evaluated using classical resources with average
energies of $m$ photon subtracted state.

It shows SSN for different values of m (0-2) in this measurement. Albeit,
photon subtraction show the advantage for all values of $\gamma$, particularly
for low $\gamma$ and low $\lambda$, $m=2$ shows maximum advantage of almost
three times over $m=0$. There is little change in the uncertainty as we vary
$\beta$. As $\lambda$ increases, the thermal noise contribution in terms of
$F$ in the uncertainty increases as per eq 1, as a result of which the SSN
advantage for different m values is lost for relative high values of $\gamma$
as shown in figure. For the limit $\gamma\rightarrow 0$ (low absorption) and
$\lambda\rightarrow 0$, the uncertainties in the number difference measurement
for different m values scale as

$\Delta\gamma^{m=0}_{Diff}\approx\frac{\sqrt{2(1-\eta)}}{\sqrt{\eta\lambda}},$
$\Delta\gamma^{m=1}_{Diff}\approx\frac{\sqrt{(1-\eta)}}{\sqrt{2\eta\lambda}}$
(11)
$\Delta\gamma^{m=2}_{Diff}\approx\frac{\sqrt{2(1-\eta)}}{3\sqrt{\eta\lambda}}.$

On the other limiting case of complete absorption ($\gamma\rightarrow 1$) and
for $\lambda\rightarrow 0$, the uncertainty scales as

$\Delta\gamma^{m=0}_{Diff}\approx\frac{1}{\sqrt{\eta\lambda}},$
$\Delta\gamma^{m=1}_{Diff}\approx\frac{1}{2\sqrt{\eta\lambda}},$ (12)
$\Delta\gamma^{m=2}_{Diff}\approx\frac{1}{3\sqrt{\eta\lambda}}.$

This set of equation confirms the standard SNL for the limit
$\gamma\rightarrow 1$ as per the discussion in II.1. The improvement in the
standard SNL for the limit $\gamma\rightarrow 0$ comes from a loss dependent
factor $1-\eta$ in the numerator. The factor of improvements in the same limit
due to different number of photon subtraction are also clear.

Normalized uncertainty for the optimized balanced absorption estimator is
plotted in fig 7.

Figure 7: (Color online) Plots of uncertainty in the optimized balanced
estimator versus absorption coefficient $\gamma$ with $\eta=0.98$ for
different values of m: $m=0$ (solid red line), $m=1$ (solid blue line), and
$m=2$ (solid green line). We set $\lambda=0.05$, $\beta=1$ (top),
$\lambda=0.05$, $\beta=0$ (middle), and $\lambda=2$, $\beta=1$ (bottom).
Dotted lines are the uncertainties evaluated using classical resources with
average energies of $m$ photon subtracted state.

Unlike the number difference estimator, regardless of the values of $\lambda$
for $\beta=1$ (thermal case), the uncertainty in the measurement for the
optimized balanced absorption estimator is SSN enhanced for $\gamma<1$. This
can be explained from the fact that the contribution of the thermal noise in
the limit $\lambda\rightarrow\infty$ in the uncertainty of the measurement
defined in eq 3 of II.2 is less significant. We checked that the advantage of
this optimized estimator is slightly better for $\beta=0$ (thermal case)
compared to $\beta=1$ (Poissonian case) for both $\gamma\approx 1$, and for
$\gamma\approx 0$. This advantage is further improved by the number of
subtracted photon $m$ as shown in the figure. In the limiting case, the
uncertainty of the new absorption estimator for different number of subtracted
number $m$ can be expressed as follows: for $\gamma\rightarrow 0$ (low
absorption) and $\lambda\rightarrow 0$, the uncertainties in the new
absorption estimator for different m values scale as

$\Delta\gamma^{m=0}_{Opt}\approx\frac{\sqrt{1-\eta^{2}}}{\sqrt{\eta\lambda}},$
$\Delta\gamma^{m=1}_{Opt}\approx\frac{\sqrt{1-\eta^{2}}}{2\sqrt{\eta\lambda}},$
(13)
$\Delta\gamma^{m=2}_{Opt}\approx\frac{\sqrt{1-\eta^{2}}}{3\sqrt{\eta\lambda}}.$

On the other limiting case of complete absorption ($\gamma\rightarrow 1$) and
for $\lambda\rightarrow 0$, the uncertainty scales as

$\Delta\gamma^{m=0}_{Opt}\approx\frac{\sqrt{1-\gamma}}{\sqrt{\eta\lambda}},$
$\Delta\gamma^{m=1}_{Opt}\approx\frac{\sqrt{1-\gamma}}{2\sqrt{\eta\lambda}},$
(14)
$\Delta\gamma^{m=2}_{Opt}\approx\frac{\sqrt{1-\gamma}}{3\sqrt{\eta\lambda}}.$

This set of equation says, unlike the number difference measurement, for
$\gamma\rightarrow 1$, the uncertainty scales much better than the standard
SNL and it improves further with the number of subtracted photon $m$. On the
other side of the limit $\gamma\rightarrow 0$, apart from a factor $\sqrt{2}$,
the improvement in the SNL compared to number difference measurement comes
from a factor $1-\eta^{2}$ instead of $1-\eta$. Also in this case, there are
factors of improvement due to photon subtraction, i.e $m+1$ times improvement
in m photon subtraction.

A plot of uncertainty in the ratio measurement is shown in fig 8.

Figure 8: (Color online) Plots of uncertainty in the ratio measurement versus
absorption coefficient $\gamma$ with $\eta=0.98$ for different values of m:
$m=0$ (solid red line), $m=1$ (solid blue line), and $m=2$ (solid green line).
We set $\lambda=0.05$, $\beta=1$ (top), $\lambda=0.05$, $\beta=0$ (middle),
and $\lambda=2$, $\beta=1$ (bottom). Dotted lines are the uncertainties
evaluated using classical resources with average energies of $m$ photon
subtracted state.

It shows SSN limit for any values of mean number of photons per mode $\lambda$
as the uncertainty is unaffected by the thermal noise contribution and only
relies on the photon number non-classical correlation as evident from eq 5.
Again for fixed $\lambda$, we found the uncertainty for subtracted states
(m=0-2) does not change much as $\beta$ changes from zero to one (maximum
advantage is observed for $\beta=0$). Another notable thing is for fixed value
of $\beta$, i.e zero, one or in-between, the uncertainty reduction is maximum
for low values of $\lambda$, which is similar to the case for all our
considered absorption estimators. We checked the uncertainty of the ratio
measurement in the limiting case for different number of subtracted photons
$m$ resembles to the uncertainty of optimized balanced estimator for
$\lambda\rightarrow 0,\gamma\rightarrow 1$ except for $\lambda\rightarrow
0,\gamma\rightarrow 0$, where the uncertainty matches to the uncertainty of
the number difference measurement.

We have plotted the uncertainties showing a comparison among them in fig 9.

Figure 9: (Color online) Comparision of uncertainties among absorption
estimators: number difference (dotted), ratio (dashed), and optimized balanced
(Solid) versus $\gamma$ for $\eta=0.7$, $\beta=0$ and $\lambda=0.05$.
Different colours correspond to different number of photon subtraction.

The optimized estimator outperforms the number difference for the full range
of absorption. Regardless of the values of $\gamma$, the optimized estimator
and ratio performs equally well for low losses (high $\eta$). Nevertheless, in
the limit of low $\gamma$ and high loss (low $\eta$) of about $30\%$, the
optimized estimator performs better than ratio measurement because of the
detection loss dependent factor $\sqrt{1-\eta^{2}}$ instead of $\sqrt{1-\eta}$
as per eq 13. Photon subtraction shows improvement of more than three times
compared to the case of previously considered $2\%$ of detection loss in all
these three estimators. Furthermore, the overall magnitude of the uncertainty
reduction for these three estimators decreases at this high detection loss as
they vary inversely with $\eta$ as evident from the uncertainty equations at
their asymptotic limits.

####  III.2.2 Fixed per photon exposure

Fixed per photon exposure analysis has been carried by balancing numerically
the mean energies of subtracted states so that mean energies for different $m$
are equal to energy of ($m=0$) un-subtracted state. Since there is a $\beta$
dependence on mean energy, we further consider the energy balancing at a fixed
model parameter. Before presenting the uncertainty result, we show the
behaviour of Fano factor (F) in this energy balancing scenario.

Figure 10: (Color online) Plots of Fano factor as a function of $\lambda$ with
$\eta=0.98$ (and $\gamma=0$) for different values of $m$: $m=0$ (solid red
line), $m=1$ (solid blue line), and $m=2$ ( solid green line). We set
$\beta=1$ (top) and $\beta=0$ (bottom).

For $\beta=1$ (thermal) case, $m=2$ remaining sub-poissonian until $\lambda=1$
while for $m=1$ and $m=0$ Fano factor remains super-Poissonian (fig 10). On
the other hand, for $\beta=0$ (Poissonian case), sub-poissonian feature
increases with $m$ other than $m=0$ and becoming maximal between
$\lambda\simeq 1-2$. This shows a shift in $\lambda$ to higher number compared
to the fixed squeezing case.

Figure 11: (Color online) Plots of uncertainty in the number difference
estimator versus efficiency in fixed per photon exposure with $\gamma=0.5$ and
$\lambda=2$ for different values of m: $m=0$ (solid red line), $m=1$ (solid
blue line), and $m=2$ (solid green line). We set $\beta=1$ (top) and $\beta=0$
(bottom). Dotted lines are the SNL.

Uncertainties in the absorption coefficient $\gamma$ for number difference
measurement is shown in the fig 11. Unlike the result at fixed squeezing, we
checked the advantage due to photon subtraction is almost lost in the regime
of low $\lambda$ and $\gamma$ for $\beta=1$ (thermal). Nevertheless, some
advantage still remains at relatively higher $\lambda$ and $\gamma$ compared
to fixed squeezing. This advantage at high $\lambda$ can be related to F, and
looking at the uncertainty expression in eq 1, the advantage at high $\gamma$
value can be understood from the fact that there must exist a value of
$\gamma$ high enough ( loss sufficiently low) to reduce the uncertainty below
the SNL. For instance, for value of $\gamma=0.5$ and $\eta\approx 1$,
uncertainty using $m=0$ is below SNL and $m=2$ provides almost 20$\%$
advantage compared to $m=0$ although the advantage decreases at higher loss as
evident from the fig 11 (top). At 50$\%$ of detection loss, $m=2$ provides
almost 10$\%$ advantage compared to $m=0$. For $\beta=0$ (Poissonian) there is
little advantage for $m=2$ due to limited improvement in F. The magnitude of
uncertainty reduction for different $m$ comes closer to $m=2$ as shown in fig
11 (bottom).

Figure 12: (Color online) Plots of uncertainty in the optimized balanced
estimator versus efficiency in fixed per photon exposure with $\gamma=0.01$
and $\lambda=2$ for different values of m: $m=0$ (solid red line), $m=1$
(solid blue line), and $m=2$ (solid green line). We set $\beta=1$ (top) and
$\beta=0$ (bottom). Dotted lines are the SNL.

Uncertainty for optimized balanced estimator is plotted in fig 12. Like the
number difference estimator, the advantage is almost lost for low $\lambda$,
but still photon subtraction gives a small advantage in the uncertainty
reduction for high $\lambda$ and low $\gamma$, and the advantage is more at
high detection losses unlike number difference measurement for $\beta=1$. For
instance at 50$\%$ detection loss, about 10$\%$ advantage can be obtained for
$m=2$ compared to $m=0$. This can be inferred from the uncertainty in eq 3 as
the uncertainty reduction is more at high losses. The advantage due to
different $m$ is very little and it is not very different compared to $m=0$
for $\beta=0$. Similar to number difference estimator, in this case, the
magnitude of the uncertainty reduction for $m=0,1$ comes closer to $m=2$.
Since the uncerianty in ratio estimator in eq 5 does not depend on the photon
statistics, we check that for both $\beta=1$ and $\beta=0$, unlike the last
two estimators, photon subtraction does not provide any advantage in the fixed
per photon exposure to the absorption sample.

##  IV Conclusions

In summary, for the first time, we have successfully generated TBS whose
individual beam photon statistics vary between thermal and Poissonian
controlled by a ’modal averaging parameter’ $\beta$, and demonstrated their
usefulness for loss estimations. We have considered three different ways of
measuring the absorption and found the best estimator among them in terms of
measured uncertainty reduction. We established a clear connection between the
uncertainty reduction for the three estimators with photon statics of
individual mode of the TBS and their correlation by two factors namely Fano
factor (F) and noise reduction factor ($\sigma$); non-classicality in these
two factors allows sub-shot noise loss estimations. Furthermore, uncertainties
from using number-difference and optimized balanced estimator depends on Fano
factor and $\sigma$, while in the ratio measurement, it only depends on the
mode correlation $\sigma$.

We have then incorporated photon subtraction operation into the model which
brings further improvement in photon statistics. Correlation remains the same
and independent of $\beta$, and for unit value of the model parameter only F
changes from initial super-Poissonian to sub-poissonian with increase in the
number of subtracted photons $m$ which further improves in magnitude for null
value of the model parameter (initial statistics of the individual states are
Poissonian before subtraction). Thus, we inferred that photon subtraction is
advantageous for further improving the individual photon statistics of
correlated TBS with Poissonian statistics.

All the improvements in photon statistics in terms of Fano factor are
reflected in the measured uncertainties of the respective estimators. We have
analysed them with respect to different number of subtracted photons $m$ by
fixing both squeezing parameter and per photon exposure. At fixed squeezing,
uncertainties in the three absorption estimators scale SSN and maximum
uncertainty reduction advantage of three times is obtained for $m=2$ with
respect to $m=0$ at very low absorption. The optimized balanced and ratio
estimator outperform the number difference estimator. Although the ratio and
optimized balanced estimators perform equally well at low detection losses, at
higher losses the latter estimator is slightly better compared to the former.
We notice no significant changes in the uncertainties for null and
intermediate values of the model parameter. Furthermore we have explicitly
computed the uncertainties at the asymptotic absorption limits ($\gamma=0,1$)
which confirm all the discussed quantum enhancements in loss estimations due
to photon subtraction and shows best absorption estimator as well.

In the per photon exposure analysis, i.e when the average photons for
different $m$ is held fixed before entering the sample, advantage due to
photon subtraction in the first model almost subsides in the low $\gamma$ and
low $\lambda$. Nevertheless, for high $\lambda$, some advantage of about
$20\%$ is preserved for the number-difference at low detection losses, and
nearly $10\%$ advantage is retained for the optimized balanced estimator at
$50\%$ detection losses. Ratio measurement does not give any advantage in the
context of photon subtraction at per photon exposure. Therefore, the
improvement in uncertainty reduction due to photon subtraction in overall can
be attributed to the improvement in mean number of photons and photon
statistics.

This work assumes balanced detection loss in two beams of the TBS, however for
the unbalancing scenario, the improvement in correlation and statistics may
vary and finding the best estimator becomes a potentially challenging task
that we will address in future work.

##  V Acknowledgements

The authors acknowledge support from EPSRC through the QUANTIC Hub
EP/T00097X/1

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