# General approach of weak-to-strong measurement transition for Fock-state-
based pointer states

Yusuf Turek1<EMAIL_ADDRESS>Janarbek Yuanbek2 Ahmad Abliz2 1School of
Physics Science and Technology, Xinjiang University, Urumqi, Xinjiang 830046,
China  2School of Physics and Electronic Engineering, Xinjiang Normal
University, Urumqi, Xinjiang 830054, China

(August 28, 2024; August 28, 2024)

###### Abstract

The transition from von Neumann’s projective strong measurement to Aharonov’s
weak measurement has recently received large attention, theoretical and
experimental. In this work, we present a general approach to describe the
weak-to-strong measurement transition for Fock-state-based pointer states, and
analyze in some details the case of coherent pointer states. A possible
realization of our measurement scheme using trapped ions is also discussed.

###### pacs:

03.65.-w, 03.65.Ta, 42.50.−p, 03.67.−a

##  I Introduction

Measurements are at the basis of physics, and represent the major tool for the
quantitative understanding of our world. In quantum systems, the measurement
problem has fostered a better understanding of quantum theory itself, and its
applications to technology (Zurek, 2003; Haroche and J.-M.Raymond, 2006;
Briegel  _et al._ , 2009; Taylor  _et al._ , 2013; Wiseman and Milburn, 2014;
Matsuno, 2017; Wasielewski  _et al._ , 2020; Huggins  _et al._ , 2021). In
standard textbooks, quantum measurements are usually introduced using the
projective measurement model formulated by von Neumann (von Neumann J, 1955).
The description of the Stern-Gerlach experiment is a typical example of this
kind of measurement (Sakurai, 1993). This is a strong measurement model where
the strength of the interaction between the pointer (measuring device) system
and the measured one is strong enough to obtain the desired information about
the measured system after a single trial, but the system itself collapses onto
an eigenstate of the measured system observable. On the other hand, there is
another type of measurement modelm proposed in 1988 by Aharonov et al
(Aharonov  _et al._ , 1988). In this scheme, which also follows the basic
requirements of von Neumann measurement theory, the coupling between the
measured and the measuring systems is weak. As a consequence, the initially
prepared system state is not destroyed after a single trial, and we still can
get the required information statistically, after many repetitions (Tollaksen
_et al._ , 2010). In the weak measurement model, the value of the measured
system observable is extracted in the form of weak value caused by the pre-
and post-selection processes. This weak value is, in general, a complex number
and different from the eigenvalue or expectation value of the corresponding
system observable. Another important feature of weak values is that they could
exceed the range of the eigenvalues. This feature is usually referred to as
weak signal amplification and has been found useful in the solution of several
problems in physics and related sciences (Hosten and Kwiat, 2008; Starling
_et al._ , 2009; Dixon  _et al._ , 2009; Lundeen  _et al._ , 2011; Pfeifer and
Fischer, 2011; Lundeen and Bamber, 2012; Salvail  _et al._ , 2013; Boyd, 2013;
Malik  _et al._ , 2013; Zhou  _et al._ , 2013; Di Lorenzo, 2013; Pan  _et al._
, 2019; Hariri  _et al._ , 2019; Denkmayr  _et al._ , 2017). For more details,
the reader is referred to (Kofman  _et al._ , 2012; Malik  _et al._ , 2014)
and the references therein.

In a recent study (Vaidman  _et al._ , 2017), it has been shown that the
nature of weak values is different from that of the expectation values of the
system observable, though weak values describe the interaction in the same way
as the eigenvalue does. In general, a system observable is characterized by
three sets of values: eigenvalues, (conditional) expectation values and weak
values. The system information we want to obtain from the pointer after
measurement is related to the pointer’s shifts. In order to get these values,
we rely on different types of measurement models. The eigenvalues and the
expectation values of the system observable are usually related to the
position shift of the pointer after strong measurement. On the other hand, in
the weak measurement procedures (Jozsa, 2007), the real and imaginary parts of
weak values are obtained from the position and momentum shifts of the pointer,
respectively.

As a matter of fact, a relevant point in von Neumann measurement model is that
the coupling strength between the measured system and the pointer may assume
arbitrary values, so that we can connect the weak and strong measurement
schemes by adjusting it. Recently, the weak-to-strong measurement transition
problems has been investigated experimentally (Pan  _et al._ , 2020) and
theoretically (Araya-Sossa and Orszag, 2021; Turek  _et al._ , 2021) for some
specific pointer states. However, a general approach to the weak-to-strong
measurement transition for arbitrary pointer states has not been yet
investigated.

In this paper, we present a general approach to describe the weak-to-strong
measurement transition problem for infinite-dimensional (pure) pointer states
in the Fock space, i.e. for the so-called Fock-state-based pointer states. We
know that Fock number states $|n\rangle$ are the eigenstates of the number
operator $\hat{n}=\hat{a}^{\dagger}\hat{a}$ , i.e.,
$\hat{a}^{\dagger}\hat{a}|n\rangle=n|n\rangle$. Thus, we can expand any
quantum state $|\phi\rangle$ in the basis of $|n\rangle$ as
$|\phi\rangle=\sum_{n}c_{n}|n\rangle$ with $c_{n}=\langle n|\phi\rangle$. We
consider the internal (e.g. spin or polarization) and external degrees (e.g.
spatial or field distribution) of freedom of the state as the measured system
and pointer (measuring device), respectively, and give the general expressions
of pointer shifts corresponding to the pointer position and momentum operators
after measurement. Since these expressions are derived without any
approximation, they are valid for any coupling, and we can described the
transitions from Aharonov’s weak measurement to von Neumann’s projective
strong measurement by tuning the coupling strength.

By using our general approach, the two extreme cases can be recovered, in
which the shifted values of the pointers are directly associated with the
conditional expectation values (strong measurement regime) and weak values
(weak measurement regime), respectively. In particular, we derive a simple
formula for the position shift of the pointer in the weak measurement regime
and verify its agreement with previous results. We also analyze in some
details the case of coherent state pointers by means of the Husimi-Kano $Q$
function in the phase space. Finally, we discuss the possible implementation
of our transition measurement model in trapped ion systems.

The rest of this paper is organized as follows. In Sec. II, we provide a brief
introduction to the von Neumann measurement by considering the weak and strong
interaction cases. In Sec. IV, we derive the final pointer state of our weak-
to-strong measurement transition model and present the general expressions of
the position and momentum shifts. We also illustrate how to realize the weak-
to-strong measurement transition by tuning the coupling strengths between the
measured system and the pointer. We also verify our results by comparison with
previous studies. In Sec.IV, as an example of our general approach, we
investigate the coherent state based weak-to-strong measurement transition. In
Sec. V, we discuss the possible experimental implementation of our model in
trapped ion systems. In Sec. VI, we provide a brief summary of this paper.
Throughout this paper, we use the unit $\hbar=1$.

##  II  Brief introduction of quantum measurement

The overall Hamiltonian of our system is the sum of three terms
$H=H_{s}+H_{p}+H_{int}$, where $H_{s}$ and $H_{p}$ denote the free
Hamiltonians of the measured system and the pointer (measuring device),
respectively, and $H_{int}$ is the interaction Hamiltonian between the system
and pointer. In the following, since the readout information is related to the
coupling between the system and the pointer, we describe the dynamics in an
interaction picture and only consider the interaction Hamiltonian $H_{int}$.
According to the von Neumann measurement theory, the general form of the
interaction Hamiltonian can be written as (von Neumann J, 1955)

$H_{int}=g(t)\hat{A}\otimes\hat{P},\text{\ \
$\int_{t_{0}}^{t}g(t)dt=g\delta(t-t_{0}).$}$ (1)

Here, $\hat{A}$ is the system observable that we want to measure, whereas
$\hat{P}$ represents the momentum operator of the pointer. The position
operator $\hat{X}$ is its canonically cojugated operator, i.e.
$[\hat{X},\hat{P}]=i$. The function $g(t)$ is nonzero only in a finite
interaction time interval, and $g$ fully characterizes the coupling strength
between the system and the pointer. If we denote by $|a_{i}\rangle$ and
eigenstate of the observable $\hat{A}$ with the eigenvalue $a_{i}$, i.e.,
$\hat{A}|a_{j}\rangle=a_{j}|a_{j}\rangle$, then the observable $\hat{A}$ can
be written as (Sakurai, 1993)

$\hat{A}=\sum_{j}a_{j}|a_{j}\rangle\langle a_{j}|.$ (2)

In general, given an observable $\hat{A}$, there are three sets of relevant
values to be considered: the eigenvalues, the (conditional) expectation
values, and the weak values. According to the measurement theory, these values
can be obtained by a pointer’s shift after measurement. Since each eigenvalue
is a special case of a (conditional) expectation values and weak values of the
observable, we now provide a brief review of the readout procedures to obtain
(conditional) expectation values and weak values.

1. Expectation value. Assume that the initial state of the pointer is $|\Phi\rangle$ with wave function $\Phi(x)$, and the measured system state is prepared in a superposition of eigenstates of $\hat{A}$, i.e., $|\psi_{i}\rangle=\sum_{j}\alpha_{j}|a_{j}\rangle$ with $\sum_{j}|\alpha_{j}|^{2}=\sum_{j}|\langle a_{j}|\psi_{j}\rangle|^{2}=1$. Under the action of the unitary evolution operator $e^{-ig\hat{A}\otimes\hat{P}}$, the total system state evolves into

$\displaystyle|\Psi\rangle$
$\displaystyle=e^{-ig\hat{A}\otimes\hat{P}}|\psi_{i}\rangle\otimes|\Phi(x)\rangle$
$\displaystyle=\sum_{j}\alpha_{j}|a_{j}\rangle\otimes|\Phi(x-ga_{j})\rangle.$
(3)

The Eq. (3) implies that if a strong measurement of the observable $\hat{A}$
is performed with outcome $a_{j}$, then the wave function of the pointer
collapses onto $\Phi(x-ga_{j})$, i.e. it is displaced by the amount $ga_{j}$
compared to the initial wave function $\Phi(x)$. After some algebra, the final
position shift of the pointer for the strong measurement can be obtained as

$\displaystyle\delta x_{st}$
$\displaystyle={\langle\Psi|\hat{X}|\Psi\rangle}-\langle\Phi|\hat{X}|\Phi\rangle$
$\displaystyle=g\langle A\rangle.$ (4)

Here, $\langle A\rangle$ is the expectation value of the observable on the
state $|\psi_{i}\rangle$, i.e.,

$\langle
A\rangle=\langle\psi_{i}|\hat{A}|\psi_{i}\rangle=\sum_{j}a_{j}|\alpha_{j}|^{2}.$
(5)

This result can be esily understood by writing the initial density matrix

$\displaystyle\rho$ $\displaystyle=|\psi_{i}\rangle\langle\psi_{i}|$
$\displaystyle=\sum_{j}|\alpha_{j}|^{2}|a_{j}\rangle\langle
a_{j}|+\sum_{j,k}\alpha_{j}\alpha_{k}^{\ast}|a_{j}\rangle\langle a_{k}|.$ (6)

It is clear that $|\alpha_{i}|^{2}$ is the probability of eigenvalue $a_{i}$
of the observable $A$, corresponding to the eigenstate $|a_{i}\rangle$, if the
system is prepared in $|\psi_{i}\rangle$. In order to obtain a determined
outcome (i.e., an eigenvalue of the measured observable corresponding to a
given eigenstate ), the second term (off-diagonal part which represented the
coherence) of $\rho$ has to vanish after the measurement. This means that,
after the measurement the system is a mixture of the eigenstates of the
measured observable, i.e., $\sum_{j}P_{j}\rho P_{j}$, where
$P_{j}=|a_{j}\rangle\langle a_{j}|$ are the projectors on the different
eigenstates of $\hat{A}$. The disappearance of the off-dagonal terms in the
above mixture is usually referred to as the projection postulate.

2\. Conditional Expectation value. In time-symmetric quantum mechanics
(Aharonov  _et al._ , 1964; Aharonov and Vaidman, 1991, 2008), if we assume
that a postselection onto the state
$|\psi_{f}\rangle=\sum_{j}\beta_{j}|a_{j}\rangle$ of $|\Psi\rangle$ [see Eq.
(3)], is possible, than the unnormalized final state of the total system is
given by

$|\Psi^{\prime}\rangle=\sum_{j}\alpha_{j}\beta_{j}^{\ast}|a_{j}\rangle\otimes|\Phi(x-ga_{i})\rangle\,,$
(7)

where $\sum_{j}|\beta_{j}|^{2}=\sum_{i}|\langle
a_{i}|\text{$\psi_{f}$}\rangle|^{2}=1$. In this case, after the measurement
the position shift of the pointer is proportional to the average value of the
position operator

$\displaystyle\delta x_{st}$
$\displaystyle=\frac{\langle\Psi^{\prime}|\hat{X}|\Psi^{\prime}\rangle}{\langle\Psi^{\prime}|\Psi^{\prime}\rangle}-\langle\Phi|\hat{X}|\Phi\rangle=g\langle
A\rangle_{c},$ (8)

where $\langle A\rangle_{c}$ is the conditional expectation value of the
observable $A$, and defined as (Aharonov  _et al._ , 1964)

$\displaystyle\langle A\rangle_{c}$
$\displaystyle=\frac{\sum_{j}a_{j}|\langle\psi_{f}|a_{j}\rangle\langle
a_{j}|\psi_{i}\rangle|^{2}}{\sum_{k}|\langle\psi_{f}|a_{k}\rangle\langle
a_{k}|\psi_{i}\rangle|^{2}}$
$\displaystyle=\frac{\sum_{j}a_{j}|\alpha_{j}\beta_{j}^{\ast}|^{2}}{\sum_{j}|\alpha_{j}\beta_{j}|^{2}}.$
(9)

This value is also referred to a the postselected strong value of $\hat{A}$.
In the above derivations, we have assumed that the coupling between the
measured system and the pointer is strong enough to make the sub-wave-packets
of the pointer corresponding to different eigenvalues distinguishable, i.e.,
$g\triangle a\gg\sigma$, where $\triangle a=a_{j}-a_{j-1}$ and $\sigma$
represent the differences of neighboring eigenvalues and the width of the sub-
wave-packets, respectively.

3\. Weak value. At variance with the situation considered above, if
$g\triangle a\text{$<\sigma$}$, i.e., if system-pointer coupling is
sufficiently weak, we cannot get the needed information after a single trial.
In this case, we may consider the expansion of the unitary operator up to the
first order term, such that the first line of Eq. (3) may be written as

$|\eta\rangle\approx(1-ig\hat{A}\otimes\hat{P})|\psi_{i}\rangle\otimes|\Phi(x)\rangle.$
(10)

As for the conditional strong measurement process, if we take a postselection
on the state $|\psi_{f}\rangle$, then the (unnormalized) total system state
may be written

$\displaystyle|\eta^{\prime}\rangle$ $\displaystyle\approx e^{-ig\langle
A\rangle_{w}\hat{P}}|\psi_{f}\rangle\otimes|\Phi(x)\rangle$
$\displaystyle\propto|\psi_{f}\rangle\otimes\left|\Phi\left(x-gtRe\left[\langle
A\rangle_{w}\right]\right)\right\rangle.$ (11)

After the measurement, the position and momentum shifts of the pointer are
given by (Jozsa, 2007)

$\delta x_{w}\propto gRe\left[\langle A\rangle_{w}\right],$ (12)

and

$\delta p_{w}=2gIm\left[\langle A\rangle_{w}\right]Var\left(P\right),$ (13)

respectively, where
$Var\left(P\right)=\langle\Phi|P^{2}|\Phi\rangle-\langle\Phi|P|\Phi\rangle^{2}$
is the variance of the momentum operator $\hat{P}$ on the initial pointer
state, and $\langle A\rangle_{w}$ is the weak value, which is defined as

$\displaystyle\langle A\rangle_{w}$
$\displaystyle=\frac{\langle\psi_{i}|\hat{A}|\psi_{f}\rangle}{\langle\psi_{i}|\psi_{f}\rangle}=\frac{\sum_{j}\sum_{k}\alpha_{j}\beta_{k}^{\ast}\langle
a_{k}|\hat{A}|a_{j}\rangle}{\sum_{j}\sum_{k}\alpha_{j}\beta_{k}^{\ast}\langle
a_{k}|a_{j}\rangle}$
$\displaystyle=\frac{\sum_{j}a_{j}\alpha_{j}\beta_{j}^{\ast}}{\sum_{j}\alpha_{j}\beta_{j}^{\ast}}\,,$
(14)

where, $Re[c]$ and $Im[c]$ denote the real and the imaginary parts of the
complex number $c$, respectively. In general, the conditional expectation
value $\langle A\rangle_{c}$ and the weak value $\langle A\rangle_{w}$ are
different [see Eqs. (9) and (14) ], and thus correspond to different
measurement strengths (Ban, 2015). In other words, the (conditional)
expectation value of system observable is related to the (postselected) strong
measurement, while the weak value is caused by the postselected weak
measurement. In can be easily seen that if $\beta_{i}=\alpha_{i},$ the
conditional expectation value and the weak value reducedto the expectation
value given in Eq. (5). As shown above, these values can be obtained from the
position shifts of the pointer after the measurement.

##  III General Approach to the weak-to-strong measurement transition

In some recent studies, the weak-to-strong measurement transition has been
investigated theoretically and experimentally by using some specific pointer
state (Pan  _et al._ , 2020; Araya-Sossa and Orszag, 2021). However, a general
approach to this problem has not been presented yet. In this section, we
provide a general approach based on the measurement scheme illustrated in Fig.
1. We know that the Fock number states are the eigenstates of the number
operator $\hat{n}=\hat{a}^{\dagger}\hat{a}$, i.e.,
$\hat{a}^{\dagger}\hat{a}|n\rangle=n|n\rangle$. One can expand any state in
the basis of $|n\rangle$ since the number operator is Hermitian. In other
words, any state $|\phi\rangle$ can be written as:

$|\phi\rangle=\sum_{n}c_{n}|n\rangle,$ (15)

where the expansion coefficients $c_{n}$ are determined by $c_{n}=\langle
n|\phi\rangle$. Let us assume that the pointer and the measured system are
prepared in the states $|\phi\rangle$ and $|\psi_{i}\rangle,$ respectively.
Then the total initial state can be written as
$|\phi\rangle\otimes|\psi_{i}\rangle$ and the evolution of the total system is
generated by the interaction Hamiltonian given in Eq. (1) as

$\displaystyle|\Psi_{evol}\rangle$
$\displaystyle=\exp\left(-i\int_{0}^{t}H_{int}(\tau)d\tau\right)|\psi_{i}\rangle\otimes|\phi\rangle$
$\displaystyle=\sum_{i}\alpha_{i}D\left(\frac{\Gamma}{2}a_{i}\right)|a_{i}\rangle\otimes|\phi\rangle$
$\displaystyle=\sum_{n,i}\alpha_{i}c_{n}|a_{i}\rangle\otimes\left|\frac{\Gamma}{2}a_{i},n\right\rangle,$
(16)

where

$D\left(\frac{\Gamma}{2}a_{i}\right)=\exp\left(\frac{g}{2\sigma}a_{i}(a^{\dagger}-a)\right),\
\ \ \Gamma=\frac{g}{\sigma}$ (17)

is the displacement operator, and

$\left|\frac{\Gamma}{2}a_{i},n\right\rangle=D\left(\frac{\Gamma}{2}a_{i}\right)|n\rangle$
(18)

is usually referred to as displaced Fock states. In the derivation of Eq.
(16), we used the expression of the pointer momentum operator $\hat{P}$ in
terms of the annihilation and creation operators $\hat{a}$ and
$\hat{a}^{\dagger}$as

$\hat{P}=\frac{i}{2\sigma}\left(\hat{a}^{\dagger}-\hat{a}\right).$ (19)

The parameter $\Gamma=g/\sigma$ characterizes the coupling strength between
the measured system and the pointer. If $\Gamma\gg 1$ ($\Gamma\ll 1$), the
measurement is in the strong (weak) measurement regime. After a postselection
onto the state $|\psi_{f}\rangle$ the normalized final state of the pointer
reads as follows

$|\varTheta\rangle=\mathcal{N}\sum_{i}\sum_{n}\alpha_{i}\beta_{i}^{\ast}c_{n}\left|\frac{\Gamma}{2}a_{i},n\right\rangle,$
(20)

where $\mathcal{N}$ is a normalization coefficient given by

$\displaystyle\mathcal{N}^{-2}$
$\displaystyle=\sum_{i,j}\sum_{n,m}c_{m}^{\ast}c_{m}\alpha_{j}^{\ast}\beta_{j}\alpha_{i}\beta_{i}^{\ast}L_{n}\left(\frac{\Gamma^{2}}{4}|a_{i}-a_{j}|^{2}\right)\times$
$\displaystyle\exp\left[-\frac{\Gamma^{2}}{8}|a_{i}-a_{j}|^{2}\right]$
$\displaystyle=\sum_{i}|\alpha_{i}\beta_{i}^{\ast}|^{2}+\sum_{n}\sum_{i\neq
j}\alpha_{j}^{\ast}\beta_{j}\alpha_{i}\beta_{i}^{\ast}|c_{n}|^{2}L_{n}\left(\frac{\Gamma^{2}}{4}|a_{i}-a_{j}|^{2}\right)\times$
$\displaystyle\exp\left[-\frac{\Gamma^{2}}{8}|a_{i}-a_{j}|^{2}\right],$ (21)

where $L_{n}(x)$ denote the n-th Laguerre polynomials. Eq. (20) contains the
information about the system observable $A$ that we want to measure, and holds
for all Fock-state-based pointer states. In the next subsection, we derive the
general formulas of pointer shifts and discuss the weak-to-strong measurement
transition.

Figure 1: Schematic diagram of our scheme to realize weak-to-strong
measurement transition . Panel (a) shows the postselected von Neumann
measurement model; (b) and (c) show the relationships between the shifts of
the pointer and the values of the associated system observable.

###  III.1 Position shift

Here, we give the general expression of the position shift of the Fock-state-
based pointer states after the postselected von Neumann measurement. The
position operator $\hat{X}$ of the pointer can be written in term of
annihilation and creation operators $\hat{a}$ and $\hat{a}^{\dagger}$as

$\hat{X}=\sigma\left(\hat{a}^{\dagger}+\hat{a}\right).$ (22)

Since the interaction causes the shifts of the pointer, we can read the system
information by comparing the pointer state before and after the measurement.
For our scheme, the position shift of the pointer after the postselected von
Neumann measurement reads as

$\displaystyle\delta x$
$\displaystyle=\langle\varTheta|\hat{X}|\varTheta\rangle-\langle\phi|\hat{X}|\phi\rangle$
$\displaystyle=\mathcal{|N}|^{2}\sigma\sum_{i,j}\sum_{n,m}c_{m}^{\ast}c_{n}\alpha_{j}^{\ast}\beta_{j}\alpha_{i}\beta_{i}^{\ast}$
$\displaystyle\text{$\times$}\\{\sqrt{n+1}\langle
m|D\left[\frac{\Gamma}{2}\left(a_{i}-a_{j}\right)\right]|n+1\rangle$
$\displaystyle+\sqrt{n}\langle
m|D\left[\frac{\Gamma}{2}\left(a_{i}-a_{j}\right)\right]|n-1\rangle+$
$\displaystyle\Gamma a_{i}\langle
m|D\left[\frac{\Gamma}{2}\left(a_{i}-a_{j}\right)\right]|n\rangle\\}-\langle\phi|\hat{X}|\phi\rangle,$
(23)

where

$\langle\phi|\hat{X}|\phi\rangle=2\sigma\sum_{n}\sqrt{n+1}Re\left[c_{n}c_{n+1}^{\ast}\right]$
(24)

is the initial average value of the position operator on the initial pointer
state $|\phi\rangle$. The explicit forms of the matrix elements of
displacement operator in the above formula are given by (de Oliveira  _et al._
, 1990)

$\displaystyle\langle m|D\left(\alpha\right)|n\rangle$
$\displaystyle=e^{-\frac{|\alpha|^{2}}{2}}\begin{cases}\sqrt{\frac{n!}{m!}}L_{n}^{m-n}\left(|\alpha|^{2}\right)\left(\alpha\right)^{m-n},&m\geq
n\\\
\sqrt{\frac{m!}{n!}}L_{m}^{n-m}\left(|\alpha|^{2}\right)\left(\alpha\right)^{n-m}.&m\leq
n\end{cases}$ (25)

where the generalized Laguerre polynomials are defined as

$L_{n}^{(\eta)}(x)=\sum_{k=0}^{n}\left(\begin{array}[]{c}n+\eta\\\
n-k\end{array}\right)\frac{(-1)^{k}}{k!}x^{k}.$ (26)

Eq. (23) is the general expression of the position shift of the Fock- state-
based pointer measurement schemes, and it holds in all measurement regimes.

If one wants to know the position shift of the pointer in the postselected
strong measurement regime, then one can consider larger values of the coupling
$\Gamma$. In order to do that, a limit $\Gamma\rightarrow\infty$ should be
taken to Eq. (23 ) which gives

$\displaystyle\left(\delta x\right)_{st}$
$\displaystyle=\lim_{\Gamma\rightarrow\infty}\delta x$ $\displaystyle=$
$\displaystyle\frac{\text{$\sigma$}}{\sum_{i}|\alpha_{i}\beta_{i}^{\ast}|^{2}}\\{\sum_{i}\sum_{n,m}c_{m}^{\ast}c_{n}|\alpha_{i}\beta_{i}^{\ast}|^{2}\times$
$\displaystyle\left(\sqrt{n+1}\delta_{m,n+1}+\sqrt{n}\delta_{m,n-1}+\Gamma
a_{i}\delta_{m,n}\right)\\}$ $\displaystyle-\langle\phi|\hat{X}|\phi\rangle$
$\displaystyle=g\frac{\sum_{i}a_{i}|\alpha_{i}\beta_{i}^{\ast}|^{2}}{\sum_{i}|\alpha_{i}\beta_{i}^{\ast}|^{2}}=g\langle
A\rangle_{c}.$ (27)

In the above expression, $\delta_{m,n}$ represents the Kronecker delta
function

$\delta_{m,n}=\begin{cases}1,&m=n\\\ 0,&m\neq n\end{cases}$ (28)

Eq. (27) is the position shift formula for the postselected strong von Neumann
measurement introduced in Sec. II. On the other hand, if we are interested in
the postselected weak measurement, $\Gamma$ should be set vanishingly small.
In this case, the position shift of the pointer reads as follows

$\displaystyle\left(\delta x\right)_{w}$
$\displaystyle=\lim_{\Gamma\rightarrow 0}\delta x$
$\displaystyle=gtRe\left[\langle A\rangle_{w}\right]+2gtIm\left[\langle
A\rangle_{w}\right]\times$
$\displaystyle\\{Im\left[\sum_{n}\sqrt{(n+1)(n+2)}c_{n}^{\ast}c_{n+2}\right]$
$\displaystyle-Im\left[\sum_{n}\sqrt{n+1}c_{n}^{\ast}c_{n+1}\right]^{2}\\}$
$\displaystyle=gRe\left[\langle A\rangle_{w}\right]+2gIm\left[\langle
A\rangle_{w}\right]Im\left[\langle a^{2}\rangle-\langle a\rangle^{2}\right].$
(29)

Eq. (29) is the concise form of the expression presented in (Dressel and
Jordan, 2012). It gives the position shift of the pointer in posteselectd weak
measurement, and holds for any pointer state that can be written in Fock state
basis. In the above position shift formula, $\langle a\rangle$ ($\langle
a^{2}\rangle$) denotes the expectation value of the annihilation operator $a$
(squared annihilation operator $a$) in the initial pointer state
$|\phi\rangle$.

Let us now exploit Eq. (29) to investigate the transition for some specific
pointer state, which have been investigated in recent studies cite those
studies here? (1) Coherent state. If we assume that the pointer is initially
prepared in a coherent state, i.e.,
$|\phi\rangle=|\alpha\rangle=\sum_{n}c_{n}|n\rangle$ with
$c_{n}=e^{-\frac{|\alpha|^{2}}{2}}\frac{\alpha^{n}}{\sqrt{n!}}$, then

$\langle a^{2}\rangle_{ch}=\langle\alpha|a^{2}|\alpha\rangle=\alpha^{2},\ \ \
\langle\alpha|a|\text{$\alpha\rangle=\alpha$ .}$ (30)

In this case the position shift, Eq. (29), is given by

$\left(\delta x\right)_{w,coh}=gRe\left[\langle A\rangle_{w}\right].$ (31)

This result coincides with that presented in Ref. (Turek  _et al._ , 2015; Xu
_et al._ , 2022).

(2) Coherent squeezed state. Assuming the pointer initially prepared in a
coherent squeezed state, we recover the results of Ref. (Araya-Sossa and
Orszag, 2021) by using Eq. (29). Squeezed coherent states are defined as
$|\phi\rangle=|\alpha,\xi\rangle=D(\alpha)S(\xi)|0\rangle$ where

$D(\alpha)=\exp\left(\alpha
a^{\dagger}-\alpha^{\ast}a\right),S(\xi)=\exp\left(\frac{1}{2}\xi^{\ast}a^{2}-\frac{1}{2}\xi
a^{\dagger 2}\right).$ (32)

Here, $D(\alpha)$ and $S(\xi)$ are the displacement and squeezed operator,
respectively, with $\alpha=|\alpha|e^{i\phi_{\alpha}}$ and
$\xi=re^{i\phi_{\xi}}$. This state also can be written in terms of Fock state
basis as (Gerry and Knight, 2005)

$|\alpha,\xi\rangle=\sum_{n}c_{n}|n\rangle,$ (33)

where

$\displaystyle c_{n}$ $\displaystyle=\frac{1}{\sqrt{\cosh
r}}\exp\left[-\frac{|\alpha|^{2}}{2}-\frac{1}{2}\alpha^{\ast
2}e^{i\phi_{\xi}}\tanh r\right]$
$\displaystyle\times\sum_{n=0}^{\infty}\frac{\left[\frac{1}{2}e^{i\phi_{\xi}}\tanh
r\right]^{n/2}}{\sqrt{n!}}H_{n}\left[\gamma\left(e^{i\phi_{\xi}}\sinh(2r)\right)^{-\frac{1}{2}}\right]$
(34)

with $\gamma=\alpha\cosh r+\alpha^{\ast}e^{i\phi_{\xi}}\sinh r$, and
$H_{n}(x)$ being the the order $n$ Hermite polynomial. The expectation value
of $\langle a\rangle$ and $\langle a^{2}\rangle$ under the state
$|\alpha,\xi\rangle$ are given as

$\langle a\rangle_{sc}=\alpha,\ \ \ \langle
a^{2}\rangle_{sc}=\alpha^{2}-e^{i\phi_{\xi}}\sinh r\cosh r.$ (35)

In this case, after the postselected weak measurement, the position shift of
the pointer reads as

$\displaystyle\left(\delta x\right)_{w,sc}$ $\displaystyle=gRe\left[\langle
A\rangle_{w}\right]-gIm\left[\langle A\rangle_{w}\right]\sinh
2r\sin\phi_{\xi}.$ (36)

This is the same result derived in Ref. (Araya-Sossa and Orszag, 2021) for
squeezed coherent pointer.

(3) Single-photon-added coherent state(SPAC). The SPAC state is defined as
(Agarwal and Tara, 1991)

$|1,\alpha\rangle=\sum_{n}c_{n}|n+1\rangle$ (37)

where

$c_{n}=\frac{e^{-\frac{|\alpha|^{2}}{2}}}{\sqrt{1+|\alpha|^{2}}}\sqrt{\frac{n+1}{n!}}.$
(38)

The expectation values $\langle a\rangle_{spac}$ and $\langle
a^{2}\rangle_{spac}$ for the state $|1,\alpha\rangle$ are given by

$\langle a\rangle_{spac}=\frac{\alpha(2+|\alpha|^{2})}{1+|\alpha|^{2}},$ (39)

and

$\langle
a^{2}\rangle_{spac}=\frac{\alpha^{2}(3+|\alpha|^{2})}{1+|\alpha|^{2}},\ \ \ \
$ (40)

respectively. Thus, if we take the pointer initially prepared in a SPAC state,
its position shift after the postselected weak measurement would be shifted by

$\left(\delta x\right)_{w,spac}=gRe\left[\langle
A\rangle_{w}\right]-2g\frac{|\alpha|^{2}\sin
2\phi_{\alpha}}{\left(1+|\alpha|^{2}\right)^{2}}Im\left[\langle
A\rangle_{w}\right].$ (41)

This is the same result obtained in Ref. (Xu  _et al._ , 2022; Turek  _et al._
, 2021). From the above examples it can be confirmed that the Eq. ( 29) can be
used for any pointer states.

###  III.2 Momentum shift

The shifts in the momentum of the pointer may be evaluated in a similar way.
After a postselected von Neumann measurement we have

$\delta p$
$\displaystyle=\langle\varTheta|\hat{P}|\varTheta\rangle-\langle\phi|\hat{P}|\phi\rangle$
$\displaystyle=\frac{i}{2\sigma}\mathcal{|N}|^{2}\sum_{i,j}\sum_{n,m}c_{m}^{\ast}c_{n}\alpha_{j}^{\ast}\beta_{j}\alpha_{i}\beta_{i}^{\ast}$
$\displaystyle\times\\{\sqrt{n+1}\langle
m|D\left[\frac{\Gamma}{2}\left(a_{i}-a_{j}\right)\right]|n+1\rangle$
$\displaystyle-\sqrt{n}\langle
m|D\left[\frac{\Gamma}{2}\left(a_{i}-a_{j}\right)\right]|n-1\rangle\\}-\langle\phi|\hat{P}|\phi\rangle\,,$
(42)

where

$\langle\phi|\hat{P}|\phi\rangle=\frac{1}{\sigma}Im\left[\sum_{n}\sqrt{n+1}c_{n+1}^{\ast}c_{n}\right]$
(43)

is the expectation value of the momentum operator in the initial pointer state
$|\phi\rangle.$ No approximation has been used to derive Eq. ( 42), and thus
it provides the momentum shift of the pointer in all measurement regimes.
Again, the two extremes give the momentum shifts corresponding to the strong
and weak measurement, respectively: (i) For a very strong coupling ($\Gamma\gg
1)$, corresponding to the strong measurement regime, the momentum shift
becomes

$\displaystyle\left(\delta p\right)_{st}$
$\displaystyle=\lim_{\Gamma\rightarrow\infty}\delta p$
$\displaystyle=\frac{i}{2\sigma\sum_{j}|\alpha_{i}\beta_{i}|^{2}}\sum_{i,j}\sum_{n,m}c_{m}^{\ast}c_{n}\alpha_{j}^{\ast}\beta_{j}\alpha_{i}\beta_{i}^{\ast}$
$\displaystyle\times\\{\sqrt{n+1}\langle
m|D\left[\frac{\Gamma}{2}\left(a_{i}-a_{j}\right)\right]|n+1\rangle$
$\displaystyle-\sqrt{n}\langle
m|D\left[\frac{\Gamma}{2}\left(a_{i}-a_{j}\right)\right]|n-1\rangle\\}-\langle\phi|\hat{P}|\phi\rangle$
$\displaystyle=\frac{i}{2\sigma}\sum_{n,m}c_{m}^{\ast}c_{n}\\{\sqrt{n+1}\langle
m|n+1\rangle-\sqrt{n}\langle m|n-1\rangle\\}$
$\displaystyle-\frac{1}{\sigma}Im\left[\sum_{n}\sqrt{n+1}c_{n+1}^{\ast}c_{n}\right]=0.$
(44)

(ii) On the contrary, in the weak measurement regime ($\Gamma\ll 1$), we get

$\displaystyle\left(\delta p\right)_{w}$
$\displaystyle=\lim_{\Gamma\rightarrow 0}\delta p$
$\displaystyle=\frac{i}{2\text{$\sigma\sum_{i,j}\alpha_{j}^{\ast}\beta_{j}\alpha_{i}\beta_{i}^{\ast}$}}\sum_{i,j}\sum_{n,m}c_{m}^{\ast}c_{n}\alpha_{j}^{\ast}\beta_{j}\alpha_{i}\beta_{i}^{\ast}$
$\displaystyle\times\\{\sqrt{n+1}\langle
m|D\left[\frac{\Gamma}{2}\left(a_{i}-a_{j}\right)\right]|n+1\rangle$
$\displaystyle-\sqrt{n}\langle
m|D\left[\frac{\Gamma}{2}\left(a_{i}-a_{j}\right)\right]|n-1\rangle\\}-\langle\phi|\hat{P}|\phi\rangle$
$\displaystyle=2gIm\left[\langle
A\rangle_{w}\right]\left(\langle\hat{P}^{2}\rangle-\langle\hat{P}\rangle^{2}\right)$
$\displaystyle=2gIm\left[\langle A\rangle_{w}\right]Var(P).$ (45)

where $Var(P)$ is the variance of the momentum operator $\hat{P}$ in the
initial pointer state $|\phi\rangle$. The final results of Eq. (44) and Eq.
(45) agree with those contained in previous works (Araya-Sossa and Orszag,
2021; Jozsa, 2007; Turek  _et al._ , 2021), and holds for any pointer states.

##  IV Example: coherent state pointer case

###  IV.1 Measurement transition

As an example of application of our general method to described the weak-to-
strong measurement transition, we here consider a pointer state initially
prepared in a coherent state. We assume that the interaction Hamiltonian
between the measured system and the pointer has the form (1), and the system
observable $\hat{A}$ is the Pauli-$x$ operator, i.e.,
$\hat{A}=\hat{\sigma}_{x}=|+\rangle\langle+|+|-\rangle\langle-|$. Here,
$|\pm\rangle\equiv\left(|\uparrow\rangle\pm|\downarrow\rangle\right)/\sqrt{2}$
are the eigenstates of $\hat{\sigma}_{x}$ with eigenvalues $\pm 1$,
respectively. If we assume that the measured system is initially prepared in
the state $|\psi_{ini}\rangle=|\downarrow\rangle$, and pointer is prepared in
coherent state $|\phi(x)\rangle=|\beta\rangle$ with $\beta=re^{i\varphi}$,
then the initial total state of the system can be written as
$|\downarrow\rangle\otimes|\beta\rangle$. Given the unitary operator
$U=e^{-ig\hat{\sigma}_{x}\otimes\hat{P}}$, the total system state given in Eq.
(16) evolves as

$\displaystyle|\Omega\rangle$ $\displaystyle=\frac{1}{\sqrt{2}}\left[|+\rangle
D\left(\frac{\Gamma}{2}\right)|\beta\rangle-|-\rangle
D^{\dagger}\left(\frac{\Gamma}{2}\right)|\beta\rangle\right],$ (46)

with $\Gamma=\frac{g}{\sigma}.$ If we take a postselection onto the state
$|\psi_{f}\rangle=\cos\theta|\uparrow\rangle-\sin\theta|\downarrow\rangle$,
then the final state of the pointer [see Eq. (20] reduces to

$\displaystyle|\Psi^{\prime\prime}\rangle=$
$\displaystyle\frac{\sin\left(\frac{\pi}{4}-\theta\right)e^{-i\frac{\Gamma}{2}r\sin\varphi}|\beta+\frac{\Gamma}{2}\rangle-\cos\left(\frac{\pi}{4}-\theta\right)e^{i\frac{\Gamma}{2}r\sin\varphi}|\beta-\frac{\Gamma}{2}\rangle}{\sqrt{1-\cos\left(2\theta\right)\cos\left(2\Gamma
r\sin\varphi\right)e^{-\frac{1}{2}\Gamma^{2}}}}.$ (47)

As we can see, this is a superposition of two coherent states with different
coherent amplitudes. In the $x$-representation the state
$|\Psi^{\prime}\rangle$ can be expressed as

$\displaystyle\Psi^{\prime\prime}(x)$ $\displaystyle=\langle
x|\Psi^{\prime\prime}\rangle=$
$\displaystyle\frac{\sin\left(\frac{\pi}{4}-\\!\\!\theta\right)e^{-i\frac{\Gamma}{2}r\sin\varphi}\psi_{+}(x)\\!-\\!\cos\left(\frac{\pi}{4}-\\!\\!\beta\right)e^{i\frac{\Gamma}{2}r\sin\varphi}\psi_{-}(x)}{\sqrt{1-\cos\left(2\theta\right)\cos\left(2\Gamma
r\sin\varphi\right)e^{-\frac{1}{2}\Gamma^{2}}}},$ (48)

where

$\displaystyle\psi_{\pm}(x)$ $\displaystyle=\langle
x|\beta\pm\frac{\Gamma}{2}\rangle$
$\displaystyle=\left(\frac{1}{2\pi\sigma^{2}}\right)^{\frac{1}{4}}e^{-|\beta\pm\frac{\Gamma}{2}|^{2}}$
$\displaystyle\times\exp\left(\frac{x^{2}}{4\sigma^{2}}\right)\exp\left[-\frac{\left(x-\sigma(\beta\pm\frac{\Gamma}{2})\right)^{2}}{2\sigma^{2}}\right]$
(49)

are Gaussian wave packets with central position shifts $\pm\frac{\Gamma}{2}$
compared to the initial Gaussian caused by the interaction with the measured
system. On the other hand, for our pre- and postselected states
$|\downarrow\rangle$ and
$\cos\theta|\uparrow\rangle-\sin\theta|\downarrow\rangle$ , the corresponding
conditional expectation value and weak value defined in Eq. (9) and Eq. (14)
can be written as

$\langle\sigma_{x}\rangle_{c}=-\sin(2\theta),$ (50)

and

$\langle\sigma_{x}\rangle_{w}=-\cot(\theta),$ (51)

respectively. As we show in the following, the conditional expectation value
$\langle\sigma_{x}\rangle_{c}$ and weak value $\langle\sigma_{x}\rangle_{w}$
are directly related to the pointer shifts of strong postselected measurement
and postselected weak measurement, respectively.

Position shift. We can check the weak-to-strong measurement transition by
investigating the position and momentum shifts of the pointer after
measurement. As mentioned in Sec. II, the the parameter $\Gamma$ quantifies
the measurement strength and it can takes all values, thus realizing a weak-
to-strong transition. The general expression of the position shift after
postselected von Neumann measurement for coherent pointer state, can be
calculated by using Eq. (48). We hav

$\displaystyle\delta x_{coh}$ $\displaystyle=\langle x\rangle_{fi}-\langle
x\rangle_{ini}$
$\displaystyle=\frac{-\text{$g$$\sin$$(2\theta)$}+\frac{1}{2}gr\cos\varphi\cos\left(2\theta\right)\cos\left(2\Gamma
r\sin\varphi\right)e^{-\frac{1}{2}\Gamma^{2}}}{1-\cos\left(2\theta\right)\cos\left(2\Gamma
r\sin\varphi\right)e^{-\frac{1}{2}\Gamma^{2}}}.$ (52)

It is straightforward to see from this expression that the measurement
transition can be controlled by adjusting $\Gamma$ in the term
$e^{-\frac{1}{2}\Gamma^{2}}$. To get the position shift of the coherent state
pointer after a strong measurement, we take the limit of Eq. (52) for $\Gamma$
goes to infinity, i.e.,

$\displaystyle\left(\delta x\right)_{strong}$
$\displaystyle=\lim_{\Gamma\rightarrow\infty}\delta
x_{coh}=-g\text{$\sin$$(2\theta)$}=g\langle\sigma_{x}\rangle_{c}.$ (53)

This result exactly matches with the Eq. (27). Taking the opposite limit,
i.e., $\Gamma\ll 1$, we go to weak measurement regime and pointer shift
position becomes

$\displaystyle\left(\delta x\right)_{weak}$
$\displaystyle=\lim_{\Gamma\rightarrow 0}\delta
x_{coh}=-g\cot\theta=g\langle\sigma_{x}\rangle_{w}.$ (54)

This result validates again Eq. ( 29). In order to explore the weak-to-strong
measurement transition, as characterized by the position shift of the coherent
pointer state, we have calculated $\delta x_{coh}$ as a function of
postselected system state parameter $\theta$ for various coupling strengths
$\Gamma$. Results are shown in Fig. 2 and agree well with our theoretical
predictions. In panels (a) and (b) we report results for two values of $r$ and
a fixed value of $\varphi$.

Figure 2: Weak-to-strong measurement transition, as characterized by position
shift (see the Eq.(52)) of the coherent pointer state. The $\delta x_{coh}$ is
shown as function of system postselected state parameter $\theta$ for
different values of the coupling strengths $\Gamma$: (a) $r=3$ and (b) $r=0$.
Here, we take $\varphi=\frac{\pi}{6}$.

Momentum shift. The momentum shift of the coherent pointer state also can be
obtained by substituting the form of the state into Eq. (48). Its explicit
expression reads as follows

$\displaystyle\delta p_{coh}$
$\displaystyle=\langle\Psi^{\prime\prime}|\hat{P}|\Psi^{\prime\prime}\rangle-\langle\beta|\hat{P}|\beta\rangle$
$\displaystyle=-\frac{1}{2\sigma\left(1-\cos\left(2\theta\right)\cos\left(2\Gamma
r\sin\varphi\right)e^{-\frac{1}{2}\Gamma^{2}}\right)}$
$\displaystyle\times\\{2iIm(\beta^{\ast})-i\sin\left(2\Gamma
r\sin\varphi\right)\cos(2\theta)e^{-\frac{1}{2}\Gamma^{2}}\Gamma$
$\displaystyle\\!\\!-\\!2iIm(\beta^{\ast})\cos(2\theta)\cos\left(2\Gamma
r\sin\varphi\right)e^{-\frac{1}{2}\Gamma^{2}}\\!\\}\\!+\\!\frac{1}{2\sigma}2Im(\beta^{\ast}).$
(55)

It is obvious from the above expression that the factor
$e^{-\frac{1}{2}\Gamma^{2}}$ can be used to control the measurement regimes in
the same way as the position shift. Similarly, we obtain

$\displaystyle\left(\delta p\right)_{weak}$
$\displaystyle=\lim_{\Gamma\rightarrow 0}\delta p_{coh}=0$ (56)

and

$\displaystyle\left(\delta p\right)_{st}$
$\displaystyle=\lim_{\Gamma\rightarrow\infty}\delta p_{coh}=0,$ (57)

respectively. These results well support our theoretical derivations given in
Sec. II. Notice that in this scheme the weak value is real, whereas in general
the momentum shift of the pointer in postselected weak measurement is
proportional to the imaginary part of the weak value [see the Eq.(45)]. Thus,
in our current case the momentum shift of the pointer for weak measurement
vanishes. In Fig. 3, we show the the momentum shift [see Eq. (55)] as a
function of parameter $\theta$.

Figure 3: Momentum shift of the coherent pointer state after postselected von
Neuamann measurement as function of system postselected state parameter
$\theta$ for various coupling strengths $\Gamma$. Here we take
$\varphi=\frac{\pi}{6}$ , $r=3$.

In Ref. (Pan  _et al._ , 2020), the authors have experimentally studied the
weak-to-strong measurement transition problem in ${}^{40}Ca^{+}$ trapped ion
system by considering a zero mean Gaussian as a pointer. The zero Gaussian
profile corresponds to the ground state ($n=0$) of the coherent pointer state.
Thus, our discussion in this section can be seen as an extension of the
theoretical part of Ref. (Pan  _et al._ , 2020).

###  IV.2 The phase space function—– the Husimi-Kano Q function

In this subsection, in order to better visualize the weak-to-strong transition
for coherent pointer states, we calculate the phase space distribution— $Q$
function. In quantum mechanics, Heisenberg’s uncertainty relations prevent the
notion of a system being characterized by a point in phase space. However,
since coherent states minimize the uncertainty product for two orthogonal
quadrature operators and the two uncertainties are equal, we can write the
phase space distribution of a quantum states by using the coherent state
representation. Notice that the quadrature operators are dimensionless and
scaled versions of the position and momentum operators. There are three main
phase space quasi-probability distributions (Gerry and Knight, 2005), i.e. the
Glauber-Sudarshan P-function, the Husimi-Kano Q function, and the Wigner
function. Among them, the Q function is the expectation value of the density
operator on a coherent state. As shown in (Tombesi and Pike, 1989) the
interference effect in phase space can be described by considering the
properties of Q function

$Q_{\psi}(\alpha)=\frac{1}{\pi}|\langle\alpha|\psi\rangle|^{2}$ (58)

for the state $|\psi\rangle$ under study. This expression may also be
rewritten as

$\langle\alpha|\psi\rangle=\sqrt{\pi
Q_{\psi}(\alpha)}e^{i\phi_{\psi}(\alpha)}\,.$ (59)

If $|\alpha\rangle$ and $|\psi\rangle$ denote two different quantum states,
then their inner product $\langle\alpha|\psi\rangle$ is linked to the overlap
in the phase space. In particular, $\pi Q_{\psi}(\alpha)$ denotes the overlap
and $\phi_{\psi}(\alpha)$ is its phase. If we take
$|\psi\rangle=|\Psi^{\prime\prime}\rangle$, its $Q$ function can be calculated
as

$\displaystyle Q_{\Psi^{\prime\prime}}(\alpha)$
$\displaystyle=\frac{1}{\pi}|\langle\alpha|\Psi^{\prime\prime}\rangle|^{2}$
$\displaystyle=\frac{1}{\pi}\frac{1}{1-\cos\left(2\theta\right)\cos\left(2\Gamma
r\sin\varphi\right)e^{-\frac{1}{2}\Gamma^{2}}}$
$\displaystyle\\!\\!\\!\\!\times\\!\\!\\{\sin^{2}\left(\frac{\pi}{4}-\\!\theta\right)e^{-|\alpha-\beta-\frac{\Gamma}{2}|^{2}}\text{\\!\\!\\!}+\cos^{2}\left(\\!\frac{\pi}{4}-\\!\theta\right)e^{-|\alpha-\beta+\frac{\Gamma}{2}|^{2}}$
$\displaystyle-\cos 2\theta\cos\left(2\Gamma
r\sin\varphi\right)e^{-\frac{1}{2}|\alpha-\beta-\frac{\Gamma}{2}|^{2}}e^{-\frac{1}{2}|\alpha-\beta+\frac{\Gamma}{2}|^{2}}\\}.$
(60)

The $Q$\- function of our initial coherent pointer state $|\beta\rangle$ is
given by

$Q_{coh}(\alpha)=\frac{1}{\pi}|\langle\alpha|\beta\rangle|^{2}=\frac{1}{\pi}e^{-|\alpha-\beta|^{2}}.$
(61)

By comparing the Eqs. (60) and (61), we can see that
$Q_{\Psi^{\prime\prime}}(\alpha)$ contains three terms: the first and second
terms represent the $Q-$functions of two different coherent states with
amplitudes $|\beta+\frac{\Gamma}{2}|$ and $|\beta-\frac{\Gamma}{2}|$,
respectively, whereas the third one describes their interference. The extra
$\pm\frac{\Gamma}{2}$ in the exponential function part of the first and second
terms of Eq.(60) [compare to Eq. (61)] are caused by the interaction between
the measured system and the pointer. After the measurement, the wave-packet
separates into sub-wave-packets corresponding to the two eigenvalues of the
Pauli-$x$ operator. We also notice that in this expression the transition
factor $e^{-\frac{1}{2}\Gamma^{2}}$ is still present. If we fix the coherent
state parameter $\beta$, with the increasing of the coupling parameter
$\Gamma$, the overlap gradually decreases and the third term vanishes. To
describe the measurement transition phenomena in the phase-space, we show, in
Fig. 4, the function $Q_{\Phi}(\alpha)$ function for different coupling
strengths $\Gamma$. In panel (a) we see the $Q$\- function of the coherent
state $|\beta\rangle$, which is a Gaussian located at $\alpha_{r}=\sqrt{3}/2$
and $\alpha_{i}=1/2$. This Gaussian function is symmetric, the corresponding
contour plot denotes the $1/e$ decay of the $Q_{coh}(\alpha)$-function. If the
interaction is weak, there are two overlapping Gaussian wave-packets and thus
interference effects appear, due to the third term in Eq.(60) [see the Fig. 4
(b)-(e)] . However, as it can be seen in Fig. 4(f), if the coupling between
the system and pointer is strong enough, the initially overlapping two sub-
wave-packets are fully separated, and the measurement transition from weak to
strong measurement regimes occured. Here we take
$\beta=\beta_{r}+i\beta_{i}=\sqrt{3}/2+0.5i$, though the $Q$-function of a
coherent state is always a Gaussian fuction, independently on the value of the
parameters $\beta_{r}$ and $\beta_{i}$. By comparing Fig.4(a) to Fig.4(e) and
(f), we find that the radius of the circular contour plot is indeed
independent of $\beta$. Since a coherent state corresponds to a displaced
ground state of a harmonic oscillator, the fluctuations of the electric field
operator in a coherent state are independent of the displacement $\beta$, i.e.
fluctuations are only determined by the properties of the harmonic oscillator,
and not by the displacement amplitude.

Figure 4:  $Q_{\Phi}(\alpha)$ function (see the Eq. (60)) for different
coupling strengths $\Gamma$ : (a) no interaction case ($\Gamma=0$); (b)
$\Gamma=0.5$; (c) $\Gamma=1$; (d) $\Gamma=2$; (e) $\Gamma=3$; (f) $\Gamma=5$.
Here, we take $r=1$, $\varphi=\frac{\pi}{6}$, and $\theta=0.01$.

##  V A possible experimental realization

In this Section, we put forward a possible implementation of our measurement
scheme in a trapped-ion system. As shown in previous studies , [8], trapped
ions subject to laser cooling represent an excellent platform for quantum
state preparation and manipulation (Cirac  _et al._ , 1993a) since they are
characterized by a decoherence time much longer than the typical times
required to perform quantum measurements. Furthermore, the (quantum) motion
state of trapped ions can be fully characterized by quantum tomographic
measurements (Wallentowitz and Vogel, 1995; Leibfried  _et al._ , 1996). Up to
date, a variety of schemes have been proposed and experimentally observed for
the preparation of nonclassical motional states of a trapped ion (Meekhof  _et
al._ , 1996; Zeng and Lin, 1993; Kienzler  _et al._ , 2015; Mccormick  _et
al._ , 2019), such as Fock (Cirac  _et al._ , 1993b), coherent (Hume  _et al._
, 2011; Munro  _et al._ , 2000; Raffa  _et al._ , 2012; Alonso  _et al._ ,
2016), squeezed (Zeng and Lin, 1995; Cirac  _et al._ , 1993a; Drechsler  _et
al._ , 2020), Schrodinger cat (de Matos Filho and Vogel, 1996; Poyatos  _et
al._ , 1996; Gerry, 1997; Haroche and J.-M.Raymond, 2006; Wineland, 2013),
pair coherent (Gou  _et al._ , 1996a), and pair cat (Gou  _et al._ , 1996b)
states. In order to implement our scheme, we consider an ion trapped in a
harmonic potential with frequency $\nu$, and driven by two laser beams
interacting resonantly with the system, tuned to the lower (red) and upper
(blue) vibrational sidebands, respectively. By taking the Lamb-Dicke regime
(Javanainen, 1981) into account, the total system Hamiltonian in the
interaction picture is given by (Zheng, 1998; Wallentowitz and Vogel, 1995)

$\displaystyle H$
$\displaystyle=\eta\Omega\left(\hat{\sigma}_{x}\sin\phi_{+}+\hat{\sigma}_{y}\cos\phi_{+}\right)$
$\displaystyle\otimes\left(\sigma\sin\phi_{-}\hat{P}-\frac{\hbar}{2\sigma}\cos\phi_{-}\hat{X}\right),$
(62)

where $\eta$ is the Lamb-Dcike parameter, $\Omega$ is the Rabi frequency, and
$\phi_{\pm}=\frac{1}{2}\left(\phi_{red}\pm\phi_{blue}\right)$ are phases
depending on the lower and upper sideband lasers phases $\phi_{red}$ and
$\phi_{blue}$. Here, $\hat{X}=\sigma\left(\hat{a}+\hat{a}^{\dagger}\right)$
and $\hat{P}=\frac{i}{2\sigma}\left(\hat{a}^{\dagger}-\hat{a}\right)$ are the
position and momentum operators of the vibrational degree of freedom of the
ion, and $\sigma=\sqrt{1/2\nu m}$ characterizes the size of the motional
state, which itself depends on the mass $m$ and on the vibrational $\nu$ of
the ion. Another important point is that here the ion is considered as a two
level system. In the above Hamiltonian, the $\hat{\sigma}_{x}$ and
$\hat{\sigma}_{y}$ are the Pauli-$x$ and -$y$ operatorsm which can be written
in terms of ion’s ground ($|\downarrow\rangle$) and optically excited
($|\uparrow\rangle$) states as
$\hat{\sigma}_{x}=|\uparrow\rangle\langle\downarrow|+|\downarrow\rangle\langle\uparrow|$
and
$\hat{\sigma}_{y}=i\left(|\uparrow\rangle\langle\downarrow|-|\downarrow\rangle\langle\uparrow|\right)$,
respectively. The Lamb-Dcike parameter $\eta$ is related to the wave vector
$k$, and is given by$\eta=k/\sqrt{2\nu m}$. In the derivation of the above
Hamiltonian we have assumed $\eta\ll 1$.

If we take the external and internal parts of the two level trapped ion as the
pointer and measured system, respectively, Eq. (62) describes the typical von
Neumann type measurement by adjusting some of the parameters. In general, the
system observable $\hat{A}$ satisfies $\hat{A}^{2}=\hat{A}$ or
$\hat{A}^{2}=\hat{I}$, and the $\hat{A}^{2}=\hat{I}$ case is very usual in
measurement problems. If we set $\phi_{-}=\frac{\pi}{2}$ ,
$\phi_{+}=\frac{\pi}{2}$ or $0$, the above interaction Hamiltonian takes the
form

$H=g\hat{\sigma}_{x}\otimes\hat{P},$ (63)

or

$H=g\hat{\sigma}_{y}\otimes\hat{P},$ (64)

respectively, with $g=\eta\sigma\Omega$. These correspond to the von Nuemann
type measurement Hamiltonians introduced in Sec. II. In a recent study (Pan
_et al._ , 2020), the authors have investigated the weak-to-strong measurement
transition problem by taking a zero mean Gaussian state as pointer in
${}^{40}Ca^{+}$ trapped ion system. Since state preparation in trapped ion
system is very realiable, and the generated states are stable and mantain
coherence over a long time, we anticipate that our general approach to weak-
to-strong measurement transition may be implemented by using the Hamiltonians
in Eq. (63) or Eq.(64) using Fock (José and Mizrahi, 2000; Um  _et al._ ,
2016; Wolf  _et al._ , 2019; Simón  _et al._ , 2020), SPACS (Kienzler  _et
al._ , 2015) and Schodinger cat pointer states (Kienzler  _et al._ , 2016).

##  VI Conclusion and remarks

In conclusion, we have given a general expressions of the position and
momentum shifts which holds for any infinite-dimensional pointer states and
any value of coupling strengths between the measured system and the pointer.
At first, we have derived the general formula of the position shift
corresponding to postselected weak measurements [see Eq. (29)], and have
verified that it agrees with previous results. We have also proved that by
adjusting the coupling strength parameter $\Gamma$, we can link the weak value
and the conditional expectation value of system observable obtained by the
shifts of the pointer in weak and strong measurement regimes, respectively. As
a typical example of our general approach, we have illustrated the weak-to-
strong measurement transition for a coherent pointer state, also analyzing the
phenomenon in the phase space by using the $Q$\- function. Finally, we have
discussed a possible experimental realization of our weak-to-strong
measurement transition scheme in trapped ion system.

###### Acknowledgements.

This work was supported by the National Natural Science Foundation of China
(Grants No. 11865017), the Natural Science Foundation of Xinjiang Uyghur
Autonomous Region of China (Grant No. 2020D01A72).

## References

  * Zurek (2003) W. H. Zurek, Rev. Mod. Phys. 75, 715 (2003).
  * Haroche and J.-M.Raymond (2006) S. Haroche and J.-M.Raymond,  _Exploring the Quantum_ (Oxford University Press, Oxford, 2006).
  * Briegel  _et al._ (2009) H. J. Briegel, D. E. Browne, W. Dur, R. Raussendorf, and M. Van den Nest, Nat. Phys 5, 19 (2009).
  * Taylor  _et al._ (2013) M. A. Taylor, J. Janousek, V. Daria, J. Knittel, B. Hage, H.-A. Bachor, and W. P. Bowen, Nat. Photonics 7, 229 (2013).
  * Wiseman and Milburn (2014) H. M. Wiseman and G. J. Milburn,  _Quantum Measurement and Control_ (Cambridge University Press, Cambridge, England, 2014).
  * Matsuno (2017) K. Matsuno, Prog. Biophys. Mol. Bio. 131, 131 (2017).
  * Wasielewski  _et al._ (2020) M. R. Wasielewski, M. D. E. Forbes, N. L. Frank, K. Kowalski, G. D. Scholes, J. Yuen-Zhou, M. A. Baldo, D. E. Freedman, R. H. Goldsmith, T. Goodson, M. L. Kirk, J. K. McCusker, J. P. Ogilvie, D. A. Shultz, S. Stoll, and K. B. Whaley, Nat. Rev. Chem 4, 490 (2020).
  * Huggins  _et al._ (2021) W. J. Huggins, J. R. McClean, N. C. Rubin, Z. Jiang, N. Wiebe, K. B. Whaley, and R. Babbush, npj Quantum Infor 7, 23 (2021).
  * von Neumann J (1955) von Neumann J,  _Mathematical Foundations of Quantum Mechanics_ (Princeton University Press, Princeton, NJ, 1955).
  * Sakurai (1993) J. J. Sakurai,  _Modern Quantum Mechanics, Revised Edition_ (Addison Wesley,, 1993).
  * Aharonov  _et al._ (1988) Y. Aharonov, D. Z. Albert, and L. Vaidman, Phys. Rev. Lett. 60, 1351 (1988).
  * Tollaksen  _et al._ (2010) J. Tollaksen, Y. Aharonov, A. Casher, T. Kaufherr, and S. Nussinov, New. J. Phys 12, 013023 (2010).
  * Hosten and Kwiat (2008) O. Hosten and P. Kwiat, Science 319, 787 (2008).
  * Starling  _et al._ (2009) D. J. Starling, P. B. Dixon, A. N. Jordan, and J. C. Howell, Phys. Rev. A 80, 041803 (2009).
  * Dixon  _et al._ (2009) P. B. Dixon, D. J. Starling, A. N. Jordan, and J. C. Howell, Phys. Rev. Lett. 102, 173601 (2009).
  * Lundeen  _et al._ (2011) J. S. Lundeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber, Nature 474, 188 (2011).
  * Pfeifer and Fischer (2011) M. Pfeifer and P. Fischer, Opt. Express 19, 16508 (2011).
  * Lundeen and Bamber (2012) J. S. Lundeen and C. Bamber, Phys. Rev. Lett. 108, 070402 (2012).
  * Salvail  _et al._ (2013) J. Z. Salvail, M. Agnew, A. S. Johnson, E. Bolduc, J. Leach, and R. W. Boyd, Nat. Photon. 7, 316 (2013).
  * Boyd (2013) R. W. Boyd, Frontiers in Optics (2013).
  * Malik  _et al._ (2013) M. Malik, M. Mirhosseini, M. P. J. Lavery, J. Leach, M. J. Padgett, and R. W. Boyd, in  _Frontiers in Optics 2013_ (Optical Society of America, 2013) p. FTu1C.6.
  * Zhou  _et al._ (2013) L. Zhou, Y. Turek, C. P. Sun, and F. Nori, Phys. Rev. A 88, 053815 (2013).
  * Di Lorenzo (2013) A. Di Lorenzo, Phys. Rev. Lett. 110, 010404 (2013).
  * Pan  _et al._ (2019) W.-W. Pan, X.-Y. Xu, Y. Kedem, Q.-Q. Wang, Z. Chen, M. Jan, K. Sun, J.-S. Xu, Y.-J. Han, C.-F. Li, and G.-C. Guo, Phys. Rev. Lett. 123, 150402 (2019).
  * Hariri  _et al._ (2019) A. Hariri, D. Curic, L. Giner, and J. S. Lundeen, Phys. Rev. A 100, 032119 (2019).
  * Denkmayr  _et al._ (2017) T. Denkmayr, H. Geppert, H. Lemmel, M. Waegell, J. Dressel, Y. Hasegawa, and S. Sponar, Phys. Rev. Lett. 118, 010402 (2017).
  * Kofman  _et al._ (2012) A. G. Kofman, S. Ashhab, and F. Nori, Phys. Rep. 520, 43 (2012).
  * Malik  _et al._ (2014) M. Malik, M. Mirhosseini, M. P. J. Lavery, J. Leach, M. J. Padgett, and R. W. Boyd, Nat. Commun. 5, 3115 (2014).
  * Vaidman  _et al._ (2017) L. Vaidman, A. Ben-Israel, J. Dziewior, L. Knips, M. Weißl, J. Meinecke, C. Schwemmer, R. Ber, and H. Weinfurter, Phys. Rev. A 96, 032114 (2017).
  * Jozsa (2007) R. Jozsa, Phys. Rev. A 76, 044103 (2007).
  * Pan  _et al._ (2020) Y. Pan, J. Zhang, E. Cohen, C.-w. Wu, P.-X. Chen, and N. Davidson, Nat. Phys 16, 1206 (2020).
  * Araya-Sossa and Orszag (2021) K. Araya-Sossa and M. Orszag, Phys. Rev. A 103, 052215 (2021).
  * Turek  _et al._ (2021) Y. Turek, A. Islam, and A. Abliz, “arxiv.2109.10128,”  (2021).
  * Aharonov  _et al._ (1964) Y. Aharonov, P. G. Bergmann, and J. L. Lebowitz, Phys. Rev. 134, B1410 (1964).
  * Aharonov and Vaidman (1991) Y. Aharonov and L. Vaidman, J. Phys. A 24, 2315 (1991).
  * Aharonov and Vaidman (2008) Y. Aharonov and L. Vaidman,  _The two-vector state formalism: an updated review. In: Lecture Notes in Physics_ , Vol. 734 (Springer, Berlin, 2008) pp. 399–447.
  * Ban (2015) M. Ban, Quantum Studies: Math. Found 2, 263 (2015).
  * de Oliveira  _et al._ (1990) F. A. M. de Oliveira, M. S. Kim, P. L. Knight, and V. Buek, Phys. Rev. A 41, 2645 (1990).
  * Dressel and Jordan (2012) J. Dressel and A. N. Jordan, Phys. Rev. A 85, 012107 (2012).
  * Turek  _et al._ (2015) Y. Turek, W. Maimaiti, Y. Shikano, C.-P. Sun, and M. Al-Amri, Phys. Rev. A 92, 022109 (2015).
  * Xu  _et al._ (2022) W. J. Xu, T. Yusufu, and Y. Turek, Phys. Rev. A 105, 022210 (2022).
  * Gerry and Knight (2005) C. Gerry and P. Knight,  _Introductory Quantum Optics_ (Cambridge University Press, Cambridge, England, 2005).
  * Agarwal and Tara (1991) G. S. Agarwal and K. Tara, Phys. Rev. A 43, 492 (1991).
  * Tombesi and Pike (1989) P. Tombesi and E. R. Pike,  _Squeezed and Non-classical light_ (New York: Plenum, 1989).
  * Cirac  _et al._ (1993a) J. I. Cirac, A. S. Parkins, R. Blatt, and P. Zoller, Phys. Rev. Lett. 70, 556 (1993a).
  * Wallentowitz and Vogel (1995) S. Wallentowitz and W. Vogel, Phys. Rev. Lett. 75, 2932 (1995).
  * Leibfried  _et al._ (1996) D. Leibfried, D. M. Meekhof, B. E. King, C. Monroe, W. M. Itano, and D. J. Wineland, Phys. Rev. Lett. 77, 4281 (1996).
  * Meekhof  _et al._ (1996) D. M. Meekhof, C. Monroe, B. E. King, W. M. Itano, and D. J. Wineland, Phys. Rev. Lett. 76, 1796 (1996).
  * Zeng and Lin (1993) H. Zeng and F. Lin, Phys. Rev. A 48, 2393 (1993).
  * Kienzler  _et al._ (2015) D. Kienzler, H.-Y. Lo, B. Keitch, L. de Clercq, F. Leupold, F. Lindenfelser, M. Marinelli, V. Negnevitsky, and J. P. Home, Science 347, 53 (2015).
  * Mccormick  _et al._ (2019) K. C. Mccormick, J. Keller, S. C. Burd, D. J. Wineland, A. C. Wilson, and D. Leibfried, Nature 572, 86 (2019).
  * Cirac  _et al._ (1993b) J. I. Cirac, R. Blatt, A. S. Parkins, and P. Zoller, Phys. Rev. Lett. 70, 762 (1993b).
  * Hume  _et al._ (2011) D. B. Hume, C. W. Chou, D. R. Leibrandt, M. J. Thorpe, D. J. Wineland, and T. Rosenband, Phys. Rev. Lett. 107, 243902 (2011).
  * Munro  _et al._ (2000) W. J. Munro, G. J. Milburn, and B. C. Sanders, Phys. Rev. A 62, 052108 (2000).
  * Raffa  _et al._ (2012) F. Raffa, M. Rasetti, and M. Genovese, Phys. Lett. A 376, 330 (2012).
  * Alonso  _et al._ (2016) J. Alonso, F. M. Leupold, Z. U. Solššr, M. Fadel, and J. P. Home, Nat. Commun 7, 11243 (2016).
  * Zeng and Lin (1995) H. Zeng and F. Lin, Phys. Rev. A 52, 809 (1995).
  * Drechsler  _et al._ (2020) M. Drechsler, M. Belén Farías, N. Freitas, C. T. Schmiegelow, and J. P. Paz, Phys. Rev. A 101, 052331 (2020).
  * de Matos Filho and Vogel (1996) R. L. de Matos Filho and W. Vogel, Phys. Rev. Lett. 76, 608 (1996).
  * Poyatos  _et al._ (1996) J. F. Poyatos, J. I. Cirac, R. Blatt, and P. Zoller, Phys. Rev. A 54, 1532 (1996).
  * Gerry (1997) C. C. Gerry, Phys. Rev. A 55, 2478 (1997).
  * Wineland (2013) D. J. Wineland, Rev. Mod. Phys. 85, 1103 (2013).
  * Gou  _et al._ (1996a) S. C. Gou, J. Steinbach, and P. L. Knight, Phys. Rev. A 54, R1014 (1996a).
  * Gou  _et al._ (1996b) S.-C. Gou, J. Steinbach, and P. L. Knight, Phys. Rev. A 54, 4315 (1996b).
  * Javanainen (1981) S. Javanainen, J.and Stenholm, Appl. phys 24, 151 (1981).
  * Zheng (1998) S.-B. Zheng, Phys. Rev. A 58, 761 (1998).
  * José and Mizrahi (2000) W. D. José and S. S. Mizrahi, J.Opt.B-Quantum. S.O. 2, 306 (2000).
  * Um  _et al._ (2016) M. Um, J. Zhang, D. Lv, Y. Lu, S. An, J.-N. Zhang, H. Nha, M. S. Kim, and K. Kim, Nat. Commun 7, 11410 (2016).
  * Wolf  _et al._ (2019) F. Wolf, C. Shi, J. C. Heip, M. Gessner, L. Pezzšš, A. Smerzi, M. Schulte, K. Hammerer, and P. O. Schmidt, Nat. Commun 10, 2929 (2019).
  * Simón  _et al._ (2020) M. A. Simón, M. Palmero, S. Martínez-Garaot, and J. G. Muga, Phys. Rev. Research 2, 023372 (2020).
  * Kienzler  _et al._ (2016) D. Kienzler, C. Flühmann, V. Negnevitsky, H.-Y. Lo, M. Marinelli, D. Nadlinger, and J. P. Home, Phys. Rev. Lett. 116, 140402 (2016).