# Nonclassical features of the pointer states in the $q$-deformed post-
selected weak measurement

Seyyede Elham Mousavigharalari<EMAIL_ADDRESS>Department of
Physics, Istanbul University, Vezneciler, Istanbul 34134, Türkiye  Azmi Ali
Altintas  Department of Physics, Istanbul University, Vezneciler, Istanbul
34134, Türkiye  Fatih Ozaydin  Institute for International Strategy, Tokyo
International University, 4-42-31 Higashi-Ikebukuro, Toshima-ku, Tokyo
170-0013, Japan

(August 28, 2024)

###### Abstract

We study $q$-deformed coherent states of the Arik-Coon harmonic oscillator as
the quantum resource in the post-selected weak measurement. First, we show how
the precision of weak measurement is improved significantly due to
$q$-deformation. Next, we focus on the role of the interplay between the
deformation parameter and the interaction strength on the nonclassical nature
of light. In particular, we show that sub-Poissonian photon distribution as
characterized by Mandel parameter, photon antibunching effect, and quadrature
squeezing are greatly enhanced by $q$-deformation. Our results not only
advance the understanding of the fundamentals of $q$-deformed quantum
mechanics, but also raise the potential to contribute to quantum technologies.

##  I Introduction

In recent decades, q-deformation has garnered escalating interest across
multiple domains of physics, including but not limited to quantum mechanics
and quantum optics [1, 2, 3, 4, 5, 6, 7, 8], quantum information [9, 10, 11],
nuclear physics [12], quantum thermodynamics [13, 14, 15, 16, 17], and
statistical mechanics [18, 19]. This stems from the fact that it provides a
richer structure for exhibiting nonclassical features and also makes it
possible to obtain a deeper understanding of the physical systems by giving
the system an additional degree of freedom [20, 21, 22, 23]. Considering the
widespread application of quantum optical states in almost every branch of
physics, the study of the generalization of these states has become one of the
most important tasks in physics.

Initial efforts were concentrated on generalization of the coherent states
within the framework of group theory [24]. While mathematical challenges have
impeded the pace of this progression, significant achievements have been made.
The $q$-generalization offers a systematic framework that establishes a valid
quantum optics approach for diverse deformed quantum mechanical models. The
initial generalization of the simple harmonic oscillator using $q$-boson
annihilation and creation operators was introduced by Arik and Coon in 1976,
with $q\rightarrow 1$ corresponding to the non-deformed scenario [25].

Subsequently, numerous authors have proposed diverse forms of $q$-deformations
for the simple harmonic oscillator and investigated their associated novel
quantum states, including $q$-coherent states, $q$-cat states, and
$q$-squeezed states. It has been observed that these states exhibit distinct
behaviors, commonly referred to as quantum or nonclassical effects, which are
contingent upon the deformation parameter $q$ [26, 27, 28, 29, 30, 31, 32, 33,
34, 35, 36, 37, 38, 39, 40, 41]. According to the conventional quantum theory,
a quantum measurement, also known as the strong measurement, is a set of
probabilistic orthogonal projections onto the eigenstates of an observable. In
such a measurement, first a physical property of a system we intend to
measure, i.e., the observable $\hat{A}$ is coupled to the meter, and then the
meter is used to read out information about the state of the observable.

Since the strong measurement is not time-symmetric, in $1988$ Aharonov, Albert
and Vaidman (AAV) proposed a different quantum measurement technique called
the post-selected weak measurement (PSWM) [42]. It implies that when
considering time-symmetric measurement, it becomes necessary to post-select
the measuring system subsequent to the measurement interaction [43]. The
primary assumption underlying their discussion is that the coupling between
the meter’s probe (the pointer) and the physical property of the system that
is being measured (the observable) is sufficiently weak that the entanglement
between them, which typically arises in standard measurements, can be
disregarded. In standard quantum measurement, the output of a quantum
measurement or the shift of the meter is typically within the range of the
minimal and the maximal eigenvalues of $\hat{A}$. Nonetheless, in weak
measurement, the meter shift is smaller than the uncertainty associated with
the observables of the measurement apparatus, rendering it challenging to
interpret. Consequently, achieving a precise estimation of the desired
parameter becomes unfeasible.

However, when appropriate pre- and post-selections are applied to the state of
the observable $\hat{A}$ following the weak coupling, the measurement outcome
is enhanced by the post-selection. The magnitude of this enhancement can be
quantified by a parameter known as the “weak value”, which significantly
surpasses the range of eigenvalues associated with the observable $\hat{A}$.
In strong measurements, the outcome might be random, while for the weak
measurements it has always a definite value and is equal to the real part of
the weak value [44, 45, 46, 47, 48, 49, 50]. An important aspect of weak
measurement is its relation to nonclassicality witnesses and advantages in
quantum metrology [51], where surpassing the classical limit via nonclassical
resources is crucial [52, 53, 54, 55]. Weak measurement has also enabled
direct observation of Hardy’s paradox [56].

Nonclassical light is a potential resource to overcome classical limitations
in high-precision measurements [57]. For instance, the use of squeezed light
which represents prominent nonclassical properties, has shown to enhance
measurement sensitivities beyond the classical limits. This has emerged as a
fundamental concept in quantum metrology [58, 59, 60, 61]. Considering post-
selection and nonclassicality of the probe states as two main resources to
achieve high precision in a weak measurement regime, we will exploit the
advantage of more nonclassical $q$-deformed quantum resources as the probe
states [62, 63, 64, 65, 66, 67]. Numerous authors have shown that by tuning
the deformation parameter $q$, the $q$-deformed coherent states (in brief,
$q$-coherent states) will offer a richer structure, exhibiting nonclassical
properties such as sub-Poissonian photon distribution and antibunching and
quadrature squeezing in comparison to their non-deformed counterparts [68, 69,
70, 71, 72, 73, 74].

In this work, the non-orthogonal $q$-coherent states of the Arik-Coon harmonic
oscillator will be utilized as the probe states in the post-selected weak
measurement. This paper is organized as follows. In Sec.II we will go beyond
the state of the art and will study the weak values of $q$-deformed position
operator in the PSWM by making use of the $q$-coherent states of the Arik-Coon
harmonic oscillator as the pre- and post-selected states of the pointer to
determine the role of $q$-deformation on the weak value amplification effect.
The goal is pursued by investigating the nonclassical properties of the
$q$-coherent state pointers including sub-Poissonian photon statistics, anti-
bunching effect, and amplitude and quadrature squeezing in the PSWM in Sec.
III. Finally, we conclude in Sec. IV.

##  II $q$-generalized post-selection

$q$-deformed nonlinear coherent states were first introduced by Arik-Coon as
the eigenstates of the $q$-deformed annihilation operator, i.e.,
${a_{q}}|z\rangle_{{}_{q}}=z|z\rangle_{{}_{q}}$, where $z$ is a complex
variable with the polar form $z=|z|e^{i\theta}$. Post-selected weak
measurement basically involves three steps:

1) Pre-selection, i.e., the preparation of the measuring system and the
measuring probe in specific initial states $|\psi_{i}\rangle$ and
$|\phi\rangle$, respectively.

2) Weak interaction between the system and the probe through a unitary
transformation which will lead to the updated state
$|\Psi\rangle=\hat{U}|\psi_{i}\rangle$ with the unitary operator
$\hat{U}=e^{{-i\int_{t_{0}}^{t}H_{\text{int}}dt}}$. Here,
$H_{\text{int}}=g(t)\hat{A}\otimes\hat{P}$ is the interaction Hamiltonian
between the system that is being measured and the meter probe (pointer), where
$\hat{A}$ is the observable of the system to be measured and $\hat{P}$ is the
momentum operator relevant to the pointer. Moreover, the coupling
$g(t)=g\delta(t-t_{0})$ is a nonzero function in a finite interaction time
interval $t$ with $g$, a small real coupling constant, and $\delta(t-t_{0})$
implies that the weak interaction occurs over a very brief duration of time.

3) Post-selection, i.e., performing a strong measurement and observing the
system such that if it is found in the specific preferred state
$|\psi_{f}\rangle$ of the probe’s pointer, it will be recorded, and otherwise,
it is ruled out and in this way the final state of the pointer, $|\Phi\rangle$
will be obtained [43].

The main result of AAV’s measurement regime is that for a real coupling
constant $g$, during the interaction time $t$, the weak interaction strength
$gt$ (note that it is different from $g(t)$) is sufficiently weak after the
post-selection. Therefore, the mean shift of the pointer position is given by
$\Delta{x}=gt\text{Re}(A_{w})$, with Re denoting the real part, and $A_{w}$ is
the so-called weak value

${A_{w}}=\frac{\langle\psi_{f}|\hat{A}|\psi_{i}\rangle}{\langle\psi_{f}|\psi_{i}\rangle}.$
(1)

In other words, weak value is the average shift of the final pointer state or
the information obtained about the system between the initial and final
states.

In particular, the weak value can be complex with the real and imaginary parts
that contain distinct information about the system [75]. The real part
represents the value measured and registered by the pointer [76], while the
imaginary part corresponds to the displacement in the momentum-space
distribution of the meter indicating that how the initial state of the system
would be perturbed unitarily [77]. It was found in a recent study by Guo et
al. that the pointer states with nonclassical features result in a larger
average pointer shifts and a wider range of weak values [78], so, the authors
believe that this leads to higher accuracy compared with using ordinary state
as the pointer state.

$q$-deformed quantum field theory, which violates the Pauli exclusion
principle and deviates from Bose statistics [79, 80], has led to the
formulation of $q$-deformed harmonic oscillator algebra by introducing the
deformation parameter $q=e^{\hbar}$ into the canonical commutation relations.
These $q$-deformed simple harmonic oscillators, often referred to as
$q$-oscillators, have garnered significant attention in physics due to two
distinct properties: their significant impact in completely integrable
theories and their connection with a specific form of nonlinearity arising
from the dependence of the oscillation frequency on the energy of the
vibrations [39, 20, 40, 73, 74].

The two primary commutation relations $aa^{\dagger}-qa^{\dagger}a=1$ and
$aa^{\dagger}-qa^{\dagger}a=q^{-N}$, with $0<q<1$ are associated with the
Math-type ($M$-type) and Physics-type ($P$-type) $q$-deformed quantum harmonic
oscillators in the literature introduced by Arik and Coon [25], and Biedenharn
[81] and Macfarlane [82], respectively. According to Katriel and Solomon’s
model [83], $q$-coherent states provide a more accurate description of non-
ideal He-Ne lasers, and it is hypothesized that the deformation parameter $q$
functions as a tuning parameter that defines the degree of deviation of the
implemented device from an ideal one.

$q$-deformed systems exhibit specific characteristics wherein they manifest
more nonclassical properties in comparison to their non-deformed counterparts.
Furthermore, an intriguing aspect is that the extent of nonclassicality is
heavily contingent upon the value of the deformation parameter. Put
differently, the deformation parameter $q$ acts akin to an additional degree
of freedom for the system, thereby enhancing its nonclassical properties.

The exact form of the q-coherent state is

$|z\rangle_{q}=\frac{1}{\sqrt{e_{q}{(|z|^{2})}}}\sum_{n=0}^{\infty}\frac{z^{n}}{\sqrt{{[n]_{q}}!}}|n\rangle,$
(2)

with ${[n]_{q}}=\frac{1-q^{n}}{1-q}$, the relative $q$-number of the M-type
bosons [71]. The $q$-generalized exponential function is of the form

$e_{q}({|z|}^{2})=\sum_{n=0}^{\infty}\frac{{|z|^{2n}}}{{[n]_{q}}!}=\sum_{n=0}^{\infty}\frac{((1-q)\,|z|^{2})^{n}}{(q;q)_{n}},$
(3)

where ${(a;q)_{n}:=(1-a)(1-aq)(1-aq^{2})\cdots(1-aq^{n-1})}$ and
${(a;q)_{0}}=1$. Note that any result regarding the $q$-deformed coherent
states of the probes are required to reduce to their non-deformed counterparts
in ordinary quantum mechanics, by taking the deformation parameter
$q\rightarrow 1$.

Considering the $q$-coherent states of the Arik-Coon harmonic oscillator in
PSWM as the probe is expected to bring about more nonclassicality through
giving the system two extra degrees of freedom, one being the deformation
parameter $q$ and the other, the coupling constant $g$, both playing a key
role in improving the nonclassicality of the system which is necessary to
obtain more precision in quantum measurement.

Let us introduce the $q$-generalized post-selected weak measurement
considering the $q$-coherent states as the initial and final states of the
system and pointer. The probe will be coupled to the system with an
interaction Hamiltonian of the form

$H_{\text{int},q}=g(t)\hat{A_{q}}\otimes\hat{P_{q}},$ (4)

where $\hat{A_{q}}$ and $\hat{P_{q}}$ are the $q$-deformed position and
momentum quadratures of the system and probe, respectively. Hence, the unitary
evolution operator in terms of the $q$-exponential function turns out to be

$\begin{split}\hat{U}_{q}&=e_{q}^{-i\int_{t_{0}}^{t}H_{\text{int.,}q}\,dt}\\\
&=\sum_{n=0}^{\infty}\frac{{\left((1-q)(-ig\hat{A}_{q}\otimes\hat{P}_{q})\right)^{n}}}{{(q;q)_{n}}}.\end{split}$
(5)

Suppose that the system and probe are initially prepared in the states
$|\psi_{i}\rangle_{q}$ and $|\phi\rangle_{q}$, respectively. Then the overall
state of the joint system after interaction will be turn into

$|\Psi\rangle_{q}=\hat{U_{q}}|\psi_{i}\rangle_{q}{|\phi\rangle}_{q}.$ (6)

When the interaction is sufficiently weak, only the lowest order terms of $gt$
are considered. Then $\hat{U}_{q}$ is obtained as

$\hat{U_{q}}=1-ig\hat{A_{q}}\otimes\hat{P_{q}}.$ (7)

Post-selection of the system to the final state ${|\psi\rangle}_{f}$ leads to

${|\Phi\rangle}_{q}={{}_{q}\langle}{\psi_{{}_{f}}}|\Psi\rangle_{q},$ (8)

and therefore the final state of the probe will be

$|\Phi\rangle_{{}_{q}}={\cal{N}}\,\,_{{}_{q}}\langle\psi_{f}|\psi_{i}\rangle_{{}_{q}}\left(|\phi\rangle_{{}_{q}}-ig\langle\hat{A}_{q}\rangle_{{}_{w}}\otimes\hat{P}_{q}|\phi\rangle_{{}_{q}}\right),$
(9)

with
$\langle\hat{A}_{q}\rangle_{{}_{w}}=\frac{{}_{{}_{q}}\langle\psi_{f}|{\hat{A}}_{q}|\psi_{i}\rangle_{{}_{q}}}{{}_{{}_{q}}\langle\psi_{f}|\psi_{i}\rangle_{{}_{q}}}$
being the weak value of the observable ${\hat{A}}_{q}$ to be measured. The
normalization factor for $|\Phi\rangle_{f}$ can be found as

${\cal{N}}(g,q)=\left[{|_{{}_{q}}\langle\psi_{f}|\psi_{i}\rangle_{{}_{q}}|}^{2}\
\left(1+2\sqrt{2}\
\text{Im}(z)\,\text{Im}(g\langle\hat{A}_{q}\rangle_{{}_{w}})+\frac{g^{2}}{2}{|\langle\hat{A}_{q}\rangle_{{}_{w}}|}^{2}\,\left(1+(1+q){|z|}^{2}-2{({\mbox{Re}}(z))}^{2}\,+2{(\mbox{Im}(z))}^{2}\right)\right)\right]^{-\frac{1}{2}}.$
(10)

In this work we consider measuring two $q$-generalized position operators
$\hat{A}_{{}_{1,q}}=\hat{X}_{{}_{1,q}}=\frac{1}{\sqrt{2}}({{a_{q}}^{\dagger}}+{a_{q}})$
and
$\hat{A}_{{}_{2,q}}=\hat{X}_{{}_{2,q}}=\frac{1}{\sqrt{2}}({q^{\frac{N}{2}}}\,{a_{q}}^{\dagger}+\,{a_{q}}\,q^{\frac{N}{2}})$
via a $q$-deformed pointer with momentum quadrature
$\hat{P_{q}}=\frac{i}{\sqrt{2}}({{\hat{a}}_{q}}^{\dagger}-{\hat{a}}_{q})$.

Let us suppose the initial and final states of the system are
$|\psi_{i}\rangle_{{}_{q}}=|\alpha\rangle_{{}_{q}}$ and
$|\psi_{f}\rangle_{{}_{q}}=|\beta\rangle_{{}_{q}}$, respectively. Then, it is
straightforward to calculate the weak values of $\hat{X}_{{}_{1,q}}$ and
$\hat{X}_{{}_{2,q}}$ from Eq. 1, as

$\langle\hat{X}_{{}_{1,q}}\rangle_{{}_{w}}=\frac{(\alpha+{\beta}^{*})}{\sqrt{2}},$
(11)

and

$\langle\hat{X}_{{}_{2,q}}\rangle_{{}_{w}}={\frac{{q^{\frac{1}{2}}}\,{e_{q}}({q^{\frac{1}{2}}}\alpha{\beta}^{*})}{{e_{q}}(\alpha{\beta}^{*})}}{\frac{(\alpha+{\beta}^{*})}{\sqrt{2}}}$
(12)

with $\alpha=|\alpha|e^{i\lambda}$ and $\beta=|\beta|e^{i\varphi}$. As it is
seen, the weak values of both observables depend on the coherency variables
$|\alpha|$ and ${|\beta|}$, and the angles $\lambda$ and $\varphi$. The most
important point is that $\langle\hat{X}_{{}_{1,q}}\rangle_{{}_{w}}$ is
$q$-independent while $\langle\hat{X}_{{}_{2,q}}\rangle_{{}_{w}}$ is
completely $q$-dependent. It should be remarked that the fidelity between the
initial and final states of the system used in the above calculation is

${}_{{}_{q}}\langle\beta|\alpha\rangle_{{}_{q}}=\frac{{e_{q}}({\beta}^{*}\alpha)}{\sqrt{{e_{q}}(|\beta|^{2}){e_{q}}(|\alpha|^{2})}}.$
(13)

In ordinary systems, the desired fidelity between the pre-and post-selection
is a small one, so the weak value can be very large, indicating a large shift
in the pointer position which can be a powerful tool to amplify the weak
signals and detecting the weak effects [84]. Here, the dependency of the
fidelity to the deformation parameter gives the system the opportunity for
obtaining large weak values by tuning the value of $q$.

Fig. 1 demonstrates the amplification of the $q$-dependent complex weak value
with respect to the deformation parameter $q$, showing that both the real and
imaginary parts of the weak value are greater than the eigenvalues for
$q\lessapprox 0.8$, and the weak value amplification due to $q$-deformation is
clearly confirmed. For the range $0.8\lessapprox q<1$, this trend is reversed
and the eigenvalues are greater than the weak values.

Therefore, based on experimental and theoretical evidences [78, 46], this
system can effectively suppress technical noise by amplifying the signal by
tuning the deformation parameter properly. This consequently leads to more
precise measurements and enriches the ability to detect weak physical effects.

Figure 1: Eigenvalue and complex and real parts of the complex weak values of
the observable $\langle\hat{X}_{{}_{2,q}}\rangle_{{}_{w}}$ with respect to
deformation parameter $q$ with $|\lambda|=2$, $|\varphi|=0.5$,
$\theta=\frac{\pi}{8}$, $\lambda=\frac{7\pi}{8}$.

##  III Nonclassical properties of the $q$-coherent state pointers

In this section we will focus on the effects of weak measurement on the
nonclassical features of the coherent states of the $q$-deformed radiation
fields. We will first study the photon statistical properties of these states,
namely the super-(sub-)Poissonian statistics. Deviation of the standard photon
statistics from Poisson distribution is characterized by the Mandel parameter,
$Q$, such that $Q=0$ corresponds to Poissonian statistics, whereas $Q<0$ and
$Q>0$ indicates sub- and super- Poissonian statistics, respectively [85]. Sub-
Poissonian radiation is called nonclassical light, since the number of photons
distribution on a state is narrower than a Poissonian one with the same
intensity [86]. We will first show how the nonclassical nature of light can be
enhanced with $q$-deformation.

Next, we will explore photon antibunching as a pure quantum effect [87, 88].
Second-order correlation function which represents the probability of two
photons arriving simultaneously from a random source can be used to
investigate the photon antibunching, a coherent state of a laser beam and
bunching effect of a classical field. Observation of antibunching trait
reveals whether there is only a single photon emitter present in a sample. We
will show that $q$-deformation leads to stronger photon antibunching.

The last nonclassical characteristics we will study is the quadrature
squeezing corresponding to the case where the variance of at least one of the
quadratures $X$ or $P$ is less than the right hand side of the Weyl-Heisenberg
uncertainty relation,
$\sigma_{XX}\,\sigma_{PP}\geq\frac{1}{4}{|[\hat{X}_{q},\hat{P}_{q}]|}^{2}$
[89]. According to the uncertainty relation, when there is squeezing in one of
the quadratures, the other quadrature is expanded. Here, we will show how the
quadrature squeezing is enhanced due to $q$-deformation.

Figure 2: Photon distribution function for the $q$-coherent state probe for
the weak value $\langle\hat{X}_{{}_{2,q}}\rangle_{{}_{w}}$ with various
instances of $q$, and $g=0.3$, $|z|=1.5$, $|\alpha|=2$, $|\beta|=1$,
$\theta=\lambda=\varphi=\frac{\pi}{2}$.

###  III.1 Sub-Poissonian behavior and photon number squeezing

For an arbitrary non-negative integer $n$, the distribution function or the
conditional probability of detecting $n$ photons in a weak measurement post-
selected with a q-deformed coherent states probe ${|\Phi\rangle}_{{}_{q}}$ can
be found for $\langle\hat{X}_{{}_{1,q}}\rangle_{{}_{w}}$ and
$\langle\hat{X}_{{}_{2,q}}\rangle_{{}_{w}}$ as

$\begin{split}{P_{{\langle\hat{X}_{{}_{1,q}}\rangle}_{{w}}}}(q;n)&\equiv\left|{}_{{}_{q}}\langle
n|\Phi\rangle_{{}_{q}}\right|_{w_{1}}^{2}\\\ &={{|\cal
N|}^{2}}\frac{{e_{{}_{q}}(\alpha\beta^{*})}{e_{{}_{q}}(\alpha^{*}\beta)}}{e_{{}_{q}}(|\alpha|^{2})e_{{}_{q}}(|\beta|^{2})e_{{}_{q}}(|z|^{2})}\frac{|z|^{2n}}{{[n]_{q}}!}\\\
&\crossproduct\left(1-g\mbox{Re}((\alpha+\beta^{*})z)+\frac{g^{2}}{4}|\alpha+\beta^{*}|^{2}|z|^{2}\right),\end{split}$
(14)

and

$\begin{split}{P_{{\langle\hat{X}_{{}_{2,q}}\rangle}_{{w}}}}(q;n)&\equiv\left|{}_{{}_{q}}\langle
n|\Phi\rangle_{{}_{q}}\right|_{w_{2}}^{2}\\\ &={{|\cal
N|}^{2}}\frac{{e_{{}_{q}}(\alpha\beta^{*})}{e_{{}_{q}}(\alpha^{*}\beta)}}{e_{{}_{q}}(|\alpha|^{2})e_{{}_{q}}(|\beta|^{2})e_{{}_{q}}(|z|^{2})}\frac{|z|^{2n}}{{[n]_{q}}!}\\\
&\Bigg{(}1-g\mathrm{Re}\left(\frac{{q^{\frac{1}{2}}}\,{e_{q}{({q^{\frac{1}{2}}}\alpha\beta^{*}})}}{e_{q}{(\alpha\beta^{*})}}{(\alpha+\beta^{*})z}\right)\\\
&+\frac{g^{2}}{4}{|\alpha+\beta^{*}|^{2}\frac{q\,e_{q}{({q^{\frac{1}{2}}}\alpha\beta^{*})}\,e_{q}{({q^{\frac{1}{2}}}\alpha^{*}\beta})}{e_{q}{(\alpha\beta^{*})}\,e_{q}{(\alpha^{*}\beta)}}}|z|^{2}\bigg{)}.\end{split}$
(15)

(a)

(b)

(c)

(d)

Figure 3: $q$-Mandel parameter for the $q$-coherent state probes with (a)
$\langle\hat{X}_{{}_{1,q}}\rangle_{{}_{w}}$ with $g=0.6$, (b)
$\langle\hat{X}_{{}_{2,q}}\rangle_{{}_{w}}$ with $g=0.6$, (c)
$\langle\hat{X}_{{}_{1,q}}\rangle_{{}_{w}}$ with $q=0.8$, and (d)
$\langle\hat{X}_{{}_{2,q}}\rangle_{{}_{w}}$ with $q=0.8$, and all with
$|\alpha|=2$, $|\beta|=0.5$ and $\theta=\frac{\pi}{2}$,
$\lambda=\frac{\pi}{8}$, $\varphi=\frac{7\pi}{8}$.

Variations of the above photon distribution function
${P_{{\langle\hat{X}_{{}_{2,q}}\rangle}_{{w}}}}(q;n)$ are presented in Fig. 2,
while the former function is omitted due to the strong similarity of the two
plots. The height of the $q$-post-selected photon distribution function
decreases with decreasing $q$. Moreover, the peaks slightly shift to the left.
Deviation of the distribution function from the Poissonian reveals that the
deformation parameter $q$ affects the distribution significantly, enhancing
the nonclassical features. Mandel parameter $Q$ is an appropriate measure of
deviation from the Poisson distribution that identifies the nonclassical
nature of photon probability distribution function [85, 90, 91]. Hence, we
consider $Q$ in the $q$-deformed setting as a $q$-Mandel parameter as in [73,
74]

$\begin{split}Q_{{{|\Phi\rangle}_{q}}}^{({\cal
A})}&\equiv\frac{\langle(\Delta\hat{n})^{2}\rangle}{\langle\hat{n}\rangle}-1\\\
&=\frac{{{}_{{}_{q}}\langle\Phi|}({{\cal A}^{\dagger}}{\cal
A})^{2}{{|\Phi\rangle_{{}_{q}}}}-{{{}_{{}_{q}}\langle\Phi|}{{\cal
A}^{\dagger}}{\cal
A}{{|\Phi\rangle_{{}_{q}}}}}^{2}}{{{}_{{}_{q}}\langle\Phi|}{{\cal
A}^{\dagger}}{\cal A}{{|\Phi\rangle_{{}_{q}}}}}-1,\end{split}$ (16)

and derive the $q$-Mandel parameter, $Q$ under weak measurement with
$q$-deformed coherent state probes. Skipping the intermediary steps, we find
$Q$ for a weak value $\langle\hat{A}_{{}_{q}}\rangle_{{}_{w}}$ as

$\begin{split}Q=&\Bigg{\\{}\bigg{\\{}\bigg{[}{{|\cal
N|}^{2}|_{{}_{q}}\langle\beta|\alpha\rangle_{{}_{q}}|^{2}}\bigg{(}{|z|^{2}}+q|z|^{4}+\frac{2g}{\sqrt{2}}\mathrm{\mbox{Re}}\bigg{(}\langle\hat{A}_{{}_{q}}\rangle_{{}_{w}}\left({z^{*}}+(2qz^{*}+q^{2}z^{*}-z){|z|^{2}}+(q^{3}z^{*}-qz)|z|^{4}\right)\bigg{)}\\\
&+\frac{g^{2}}{2}|\langle\hat{A}_{{}_{q}}\rangle_{{}_{w}}|^{2}\bigg{(}1-z^{2}-{z^{*}}^{2}-\left((2q+q^{2}){(z^{2}+{z^{*}}^{2})}-(q^{3}+3q^{2}+3q)\right)|z|^{2}\\\
&-\left(q^{3}({z^{2}}+{z^{*}}^{2})-(q^{5}+2q^{4}+3q^{3})-1\right){|z|^{4}}+({q^{6}}+q){|z|^{6}}\bigg{)}\bigg{)}\bigg{]}\\\
&-\bigg{[}{|\cal
N|}^{2}|_{{}_{q}}\langle\beta|\alpha\rangle_{{}_{q}}|^{2}\bigg{(}{|z|^{2}}+\frac{2g}{\sqrt{2}}\mathrm{\mbox{Re}}\left(\langle\hat{A}_{{}_{q}}\rangle_{{}_{w}}\left({z^{*}}+(qz^{*}-z){|z|^{2}}\right)\right)\\\
&+\frac{g^{2}}{2}|\langle\hat{A}_{{}_{q}}\rangle_{{}_{w}}|^{2}\left(1+(q+2-z^{2}-{{z^{*}}^{2}}){q|z|^{2}}+(1+q^{3}){|z|^{4}}-{z^{2}}-{{z^{*}}^{2}}\right)\bigg{)}\bigg{]}\Bigg{\\}}\\\
&\times\bigg{\\{}\bigg{[}{|\cal
N|}^{2}|_{{}_{q}}\langle\beta|\alpha\rangle_{{}_{q}}|^{2}\bigg{(}{|z|^{2}}+\frac{2g}{\sqrt{2}}\mathrm{\mbox{Re}}\left(\langle\hat{A}_{{}_{q}}\rangle_{{}_{w}}\left({z^{*}}+(qz^{*}-z){|z|^{2}}\right)\right)\\\
&+\frac{g^{2}}{2}|\langle\hat{A}_{{}_{q}}\rangle_{{}_{w}}|^{2}\left(1+(q+2-z^{2}-{{z^{*}}^{2}}){q|z|^{2}}+(1+q^{3}){|z|^{4}}-{z^{2}}-{{z^{*}}^{2}}\right)\bigg{)}\bigg{]}^{-1}\Bigg{\\}}-1.\end{split}$
(17)

In Figs. 3(a) and (b), we present the $q$-Mandel parameter of the $q$-post-
selected coherent state probes for
$\langle\hat{A}_{{}_{q}}\rangle_{{}_{w}}=\langle\hat{X}_{{}_{1,q}}\rangle_{{}_{w}}$
and
$\langle\hat{A}_{{}_{q}}\rangle_{{}_{w}}=\langle\hat{X}_{{}_{2,q}}\rangle_{{}_{w}}$,
respectively, for a range of $q$ with $g=0.6$. In both cases, the distribution
is completely sub-Poissonian and therefore it is nonclassical. Moreover, the
sub-Poissonian character becomes stronger, showing that the nonclassicality is
enhanced with decreasing $q$.

Mandel parameter for $q=0.8$ and a range of interaction parameters $g$ are
presented in Figs. 3(c) and (d) for
$\langle\hat{A}_{{}_{q}}\rangle_{{}_{w}}=\langle\hat{X}_{{}_{1,q}}\rangle_{{}_{w}}$
and
$\langle\hat{A}_{{}_{q}}\rangle_{{}_{w}}=\langle\hat{X}_{{}_{2,q}}\rangle_{{}_{w}}$,
respectively. It can be seen easily by comparing for example the red solid
curve in Fig. 3(a) corresponding to $q=1$ and $g=0.6$, with the dashed green
curve in Fig. 3(c) corresponding to $q=0.8$ and $g=0.6$, that for $q<1$, the
role of the interaction strength in making the distribution sub-Poissonian
becomes more prominent.

Another important point is that a special type of squeezing, the so called
photon number (amplitude) squeezing, exists for the $q$-post-selected
pointers. Photon number squeezing is a behavior exhibited by a sub-Poissonian
photon distribution, where the photon number dispersion is narrower than the
average number of photons in a state, i.e.,
$\langle(\Delta\hat{n})^{2}\rangle<\langle\hat{n}\rangle$ [91].

This property results in optical noise that can be much lower than the
classical shot-noise limit, making sub-Poissonian light highly useful for
quantum information processing [38]. Therefore, the advantage of choosing a
$q$-deformed space is evident in this context. In response to stronger
interaction and smaller $q$, the magnitude of the negative Mandel parameter
becomes greater. This phenomenon is unique to the $q$-deformed states and is
not observed in the ordinary (non-deformed) case presented by the solid red
lines in Figs. 3 (a) and (b).

###  III.2 Photon antibunching effect

The existence of antibunched light is of interest to theoretical physics due
to its compatibility with the quantum theory of light and its variance with
classical theory [92]. Second-order correlation function in zero delay time
$g^{(2)}(0)$ associated with normalized $q$-coherent state pointers in weak
measurement is defined as

$g^{(2)}_{{{|\Phi\rangle}_{q}}}(0)\equiv\frac{{{}_{q}\langle\Phi|}{{{\cal
A}^{\dagger}}}^{2}{{\cal
A}}^{2}{{{{|\Phi\rangle}_{q}}}}}{{{}_{q}\langle\Phi|}{{\cal
A}^{\dagger}}{{\cal A}}{{{{|\Phi\rangle}_{q}}}}^{2}}.$ (18)

For the $q$-post-selected pointer state with every weak value
${\langle\hat{A}_{{}_{q}}\rangle_{{}_{w}}}$, we find

$\displaystyle g^{(2)}_{{|\Phi\rangle}_{{}_{q}}}(0)=$
$\displaystyle\Bigg{\\{}{{{{|\cal
N|}^{2}}}\,|_{{}_{q}}\langle\beta|\alpha\rangle_{{}_{q}}|^{2}}\Bigg{(}{|z|^{4}}+\frac{2g}{\sqrt{2}}\mathrm{\mbox{Re}}\bigg{(}{\langle\hat{A}_{{}_{q}}\rangle_{{}_{w}}}\left(\left(1+q\right){z^{*}}{|z|^{2}}+\left({{q^{2}}{z^{*}}-z}\right){|z|^{4}}\right)\bigg{)}$
(19)
$\displaystyle+\frac{g^{2}}{2}{|{\langle\hat{A}_{{}_{q}}\rangle_{{}_{w}}}|^{2}}\bigg{(}\left({q^{2}}+2q+1-(1+q){({z^{2}}+{z^{*}}^{2})}\right){|z|^{2}}-\left(q^{2}(z^{2}+{z^{*}}^{2})-2q^{3}-2q^{2}-q^{4}\right){|z|^{4}}$
$\displaystyle+\left(1+q^{5}\right){|z|^{6}}\bigg{)}\Bigg{)}\Bigg{\\}}\times\Bigg{\\{}{{{{|\cal
N|}^{2}}}\,|_{{}_{q}}\langle\beta|\alpha\rangle_{{}_{q}}|^{2}}\Bigg{(}{|z|^{2}}+\frac{2g}{\sqrt{2}}\mathrm{\mbox{Re}}\bigg{(}{\langle\hat{A}_{{}_{q}}\rangle_{{}_{w}}}\left(z^{*}+q{z^{*}}{|z|^{2}}-z{|z|^{2}}\right)\bigg{)}$
$\displaystyle+\frac{{g^{2}}}{2}|{\langle\hat{A}_{{}_{q}}\rangle_{{}_{w}}}|^{2}\bigg{(}1-z^{2}-{z^{*}}^{2}-\left(q(z^{2}+{z^{*}}^{2})-2q-q^{2}\right){|z|^{2}}+\left(1+q^{3}\right){|z|^{4}}\bigg{)}\Bigg{)}\Bigg{\\}}^{-2}.$

(a)

(b)

(c)

(d)

Figure 4: Second order $q$-correlation function of the $q$-post-selected
probes with (a) the weak value $\langle\hat{X}_{{}_{1,q}}\rangle_{{}_{w}}$ and
(b) $\langle\hat{X}_{{}_{2,q}}\rangle_{{}_{w}}$ both with $g=0.2$,
$|\alpha|=2$, $|\beta|=0.5$ and $\theta=\frac{\pi}{2}$,
$\lambda=\frac{\pi}{8}$, $\varphi=\frac{7\pi}{8}$. (c) for
$\langle\hat{X}_{{}_{1,q}}\rangle_{{}_{w}}$ and (d) for
$\langle\hat{X}_{{}_{2,q}}\rangle_{{}_{w}}$ both with $q=0.3$, $0<g<1$,
$|\alpha|=2$, $|\beta|=0.5$ and $\theta=\frac{\pi}{2}$,
$\lambda=\frac{\pi}{8}$, $\varphi=\frac{7\pi}{8}$.

Substituting the two weak values
$\langle\hat{A}_{{}_{q}}\rangle_{{}_{w}}=\langle\hat{X}_{{}_{1,q}}\rangle_{{}_{w}}$
and
$\langle\hat{A}_{{}_{q}}\rangle_{{}_{w}}=\langle\hat{X}_{{}_{2,q}}\rangle_{{}_{w}}$
in Eq. 19, we obtain Figs. 4(a)-(d), where classical photon bunching and pure
quantum antibunching effects will be distinguished by
$g^{(2)}_{{|\Phi\rangle}_{{}_{q}}}(0)>1$ and
$g^{(2)}_{{|\Phi\rangle}_{{}_{q}}}(0)<1$, respectively [92]. Fig. 4 shows that
both bunching and antibunching effects are exhibited by the $q$-post-selected
pointer states with a wide range of values of $q$ and $g$.

It is evident from Figs. 4 (a) and (b) that when the interaction parameter is
fixed at $g=0.2$, decreasing the value of the deformation parameter $q$
results in a stronger antibunching effect in a small range of $|z|$. However,
for the remaining values of $|z|$, the bunching effect will be the dominant
mechanism.

Fig. 4(c) and (d) show that for $q=0.3$, increasing the value of $g$, stronger
antibunching effect is exhibited by the pointer state for a limited values of
the coherency parameter. For the remaining values of $|z|$, the bunching
effect prevails, as seen in the case with interaction constant $g=0.2$.

Observing photon antibunching indicates that the distribution of photon
numbers is squeezed in both cases, whether the deformation parameter or
interaction coefficient is fixed [93, 94]. However, when the deformation
parameter is fixed and the interaction constant enhances, the photon number
distribution is even more strongly affected.

###  III.3 Quadrature Squeezing

In this subsection, we consider the squeezing and expansion of position
$\hat{X}=\frac{1}{\sqrt{2}}({{\hat{{a}}}^{\dagger}}_{q}+\hat{a_{q}})$ and
momentum $\hat{P}=\frac{i}{\sqrt{2}}({{\hat{{a}}}^{\dagger}}_{q}-\hat{a_{q}})$
operators of the pointer state using the generalized Weyl-Heisenberg
uncertainty inequality
$\sigma_{XX}\sigma_{PP}\geqslant\frac{1}{4}|\langle[{\hat{X}}_{q},{\hat{P}}_{q}]\rangle|^{2}$
[89].

To compute
$\sigma_{XX}=\langle{{\hat{X}}^{2}}\rangle-{\langle{{\hat{X}}}\rangle}^{2}$
and
$\sigma_{PP}=\langle{{\hat{P}}^{2}}\rangle-{\langle{{\hat{P}}}\rangle}^{2}$,
we obtain the following expectation values in $q$-coherent state pointer

$\begin{split}{{}_{q}\langle\Phi|}\hat{X}_{q}^{2}(\hat{P}_{q}^{2})|\Phi\rangle_{q}&=\frac{|\mathcal{N}|^{2}|_{q}\langle\beta|\alpha\rangle_{q}|^{2}}{2}\bigg{\\{}1\pm
z^{2}\pm{z^{*}}^{2}+(1+q)|z|^{2}\pm\sqrt{2}g\mbox{Re}\Bigg{(}{\langle\hat{A}_{q}\rangle_{w}}\bigg{(}z^{*}-z\pm{z^{*}}^{3}\mp
z^{3}+(1+q)(z^{*}\pm z)\\\ &+(\pm q^{2}z+q(1+q)z^{*}-(1+q)z\mp
z^{*})|z|^{2}\bigg{)}\Bigg{)}\\\
&-\frac{g^{2}}{2}|\langle\hat{A}_{q}\rangle_{w}|^{2}\Bigg{(}\pm
z^{4}\pm{z^{*}}^{4}+\bigg{(}(2\mp 1)+(1\mp 1)q\mp
q^{2}\bigg{)}(z^{2}+{z^{*}}^{2})\\\ &+\bigg{(}\mp q^{3}+q^{2}+q\mp
1\bigg{)}(z^{2}+{z^{*}}^{2})|z|^{2}+\bigg{(}-q^{3}-3q^{2}-(3\mp 2)q\mp(-2\pm
1)\bigg{)}|z|^{2}\\\ &-\bigg{(}q^{4}+q^{3}\mp
2q^{2}+q+1\bigg{)}|z|^{4}-q-2\bigg{)}\Bigg{)}\bigg{\\}},\end{split}$ (20)

$\displaystyle{{}_{q}\langle\Phi|}\hat{X}_{{}_{q}}(\hat{P}_{{}_{q}}){{|\Phi\rangle}_{q}}$
$\displaystyle=\frac{{{|\cal
N|}^{2}\,\,|_{{}_{q}}\langle\beta|\alpha\rangle_{{}_{q}}|^{2}}\,{i^{(\frac{1}{2}\mp\frac{1}{2})}}}{\sqrt{2}}\Bigg{\\{}z^{*}\pm
z\pm{\sqrt{2}\,g\,{i^{(\frac{1}{2}\mp\frac{1}{2})}}}\mbox{Re}(\mbox{Im})\bigg{(}{\langle\hat{A}_{{}_{q}}\rangle_{{}_{w}}}(1\pm{z^{*}}^{2}-z^{2}$
$\displaystyle\quad+(q\mp
1)|z|^{2}\bigg{)}-\frac{g^{2}}{2}{|\langle\hat{A}_{{}_{q}}\rangle_{{}_{w}}}|^{2}\bigg{(}{z^{*}}^{3}\pm
z^{3}-q(z^{*}\pm z)+(1\mp 1)(z-z^{*})$ $\displaystyle\quad+(\mp q^{2}+q\mp
1)z{|z|^{2}}-(q^{2}\mp q+1){z^{*}}|z|^{2}\bigg{)}\Bigg{\\}},$ (21)

and

$\displaystyle\langle[{\hat{X}}_{q},{\hat{P}}_{q}]\rangle$
$\displaystyle={{|\cal
N|}^{2}\,\,|_{{}_{q}}\langle\beta|\alpha\rangle_{{}_{q}}|^{2}}\Bigg{\\{}i(1-|z|^{2}+q|z|^{2})$
$\displaystyle+i\frac{g}{\sqrt{2}}\mbox{Re}\Bigg{(}{\langle\hat{A}_{{}_{q}}\rangle_{{}_{w}}}\bigg{(}z^{3}+{z^{*}}^{3}+q{z^{*}}-(2+q)z+(q^{2}-q-1){z^{*}}{|z|^{2}}-(q^{2}+q-1)z{|z|^{2}}\bigg{)}\Bigg{)}$
$\displaystyle-i\frac{g}{\sqrt{2}}\mbox{Re}\Bigg{(}{\langle\hat{A}_{{}_{q}}\rangle_{{}_{w}}}\bigg{(}z^{3}+{z^{*}}^{3}-q(z+z^{*})-({q^{2}}-q+1)z|z|^{2}-(q^{2}-q+1)z^{*}|z|^{2}\bigg{)}\Bigg{)}$
$\displaystyle-i\frac{g^{2}}{2}\,|{\langle\hat{A}_{{}_{q}}\rangle_{{}_{w}}}|^{2}\mbox{Re}\Bigg{(}\bigg{(}{z^{*}}^{4}-z^{4}-(q^{3}+q^{2}-q+1)|z|^{2}+(q^{2}+2q+1)z^{2}-(1+q^{2}){z^{*}}^{2}$
$\displaystyle-(q^{3}-q^{2}+q+1){{z^{*}}^{2}}{|z|^{2}}+(q^{3}+q^{2}-q+1){z^{2}}{|z|^{2}-(q^{4}-q^{3}+q-1){|z|^{4}}-q}\bigg{)}\Bigg{)}\Bigg{\\}}.$
(22)

It should be noted that in Eqs. 20 and III.3, the upper and lower signs are
for $\hat{X}$ and $\hat{P}$, respectively. Consequently, substituting the
above values in the Weyl-Heisenberg uncertainty inequality, plots in Fig. 5
are obtained. Figs. 5(a) and (b) represent the $q$-generalized uncertainty
relation for the $q$-coherent states of the meter with the weak values of the
observables $\hat{A_{1}}=\hat{{X_{1}}_{q}}$ and
$\hat{A_{2}}=\hat{{X_{2}}_{q}}$, respectively. It is clear that in both cases
the Weyl-Heisenberg inequality is completely satisfied.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 5: Plots of (a) generalized uncertainty relation (GUR) with coherence
parameters $|\alpha|=4$, $|\beta|=2$, $\theta=\frac{\pi}{2}$,
$\lambda=\frac{\pi}{8}$, $\varphi=\frac{7\pi}{8}$ for the observable
$\hat{X}_{1}$ and (b) for the observable $\hat{X}_{2}$. (c) and (d) belong to
the variances of the $q$-generalized position and momentum operators for
various values of $0<q<1$ and $g=0.8$ with $|\alpha|=4$, $|\beta|=0.5$ and
$\theta=\frac{\pi}{2}$, $\lambda=\frac{\pi}{3}$, $\varphi=\frac{2\pi}{3}$. The
solid red, yellow, green and blue curves show the variance in $X$, while the
dashed red, gray, orange and cyan curves show the variance in $P$. (e) and (f)
are related to the variances of the $q$-generalized position and momentum
operators for various values of $0<g<1$ and $q=0.7$ with
$\theta=\frac{\pi}{2}$, $\lambda=\frac{\pi}{3}$, $\varphi=\frac{2\pi}{3}$.

As shown in Figs. 5(c) and (d), the $\hat{X}$ quadrature is expanded and
$\hat{P}$ is squeezed for the fixed value of the interaction parameter $g=0.9$
and various values of the deformation parameter. Furthermore, as the value of
$q$ decreases, the degree of squeezing (expanding) in the $\hat{P}$
($\hat{X}$) quadrature gets stronger (weaker).

Figs. 5(e) and (f) clarify that in the case that the interaction parameter
changes, quadrature squeezing is observed again in quadrature $\hat{P}$ and
its conjugate quadrature, i.e. $\hat{X}$ is expanded. Moreover, the degree of
squeezing gets stronger by decreasing the value of $g$, while the deformation
parameter is fixed at $q=0.8$. These findings confirm that, unlike the
ordinary (non-deformed) pointer coherent states, the uncertainties in the two
quadratures are not equal.

Hence, a compelling aspect of $q$-post-selected systems is that the level of
squeezing can be adjusted through two parameters: $q$ and $g$. This parallels
the idea that such systems possess two distinct supplementary degrees of
freedom, namely the deformation parameter $q$ and the interaction coefficient
$g$. This characteristic significantly contributes to the crucial role played
by these states in quantum metrology.

The ideal squeezed states or intelligent states are special states that show
squeezing in one quadrature and satisfy the uncertainty relation [91]. Here,
squeezing in one quadrature for various values of $q$ and $g$ is observed and
the minimum uncertainty relation is also satisfied. Consequently, it can be
inferred that the $q$-post-selected probe states also qualify as intelligent
states.

##  IV Concluding remarks

We have leveraged a particular kind of post-selected weak measurement in which
the initial and final states of the system, and the state of the pointer are
replaced by the coherent states of a special type of harmonic oscillator, the
$q$-deformed Arik-Coon harmonic oscillator with $0<q<1$. To the best of our
knowledge, $q$-deformed systems have not been studied under the weak
measurement regime.

Our efforts aimed to modify the weak measurement technique by measuring the
$q$-generalized observables with a $q$-deformed coherent state meter, which
best fits the non-ideal He-Ne lasers. We have shown how the strategic choices
of the $q$-deformed initial and final post-selected states can result in large
flexible weak values and optimized probes that are essential in quantum
metrology for obtaining more precise measurements. $q$-dependency of the weak
value makes it more tunable and flexible while setting a limit for its
amplification value. Furthermore, the fidelity between the pre- and post-
selected states was found to be dependent on $q$, varying with changes in the
deformation parameter. Plots of the probability of finding $n$ photons in a
$q$-coherent state of the probe related to each weak value revealed that the
height and position of the peaks for different values of $q$ deviates from the
non-deformed case. $q$ and $g$-dependent antibunching effect and sub-
Poissonian statistics are the additional nonclassical features exhibited
simultaneously by the meter states, which cannot be observed with the non-
deformed states. Number (amplitude) squeezing is another quantum
characteristic of the $q$-post-selected pointer states as well.

It was observed that the generalized Weyl-Heisenberg uncertainty relation is
fully satisfied by the $q$-post-selected coherent pointer states. Quadrature
squeezing, the other quantum phenomena, is shown by the meter states. It was
found that with the weak values of both types of $q$-deformed observables,
both the $P$ and $X$ quadratures were squeezed with $g=cte$ and $q=cte$,
respectively. Existence of both the quadrature and amplitude squeezing in this
weak measurement regime is a special statistical trait shown by the $q$-post-
selected pointers.

The principal finding of this research indicates that through $q$-deformation
of the post-selected weak measurement, enhanced nonclassicality of the probe
state is attained, a pivotal factor in enhancing precision in quantum
metrology.

###### Acknowledgements.

This research is supported by Türkiye Scholarships, the Turkish Government
Scholarship Program, under the Scholarship No. 23IR009840. F.O. acknowledges
Personal Research Fund of Tokyo International University.

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