# Improving the precision of weak-value-amplification with two cascaded
Michelson interferometers based on Vernier-effect

Jing-Hui Huang1<EMAIL_ADDRESS>Fei-Fan He1
<EMAIL_ADDRESS>Xue-Ying Duan2,3,4<EMAIL_ADDRESS>Guang-
Jun Wang2,3,4<EMAIL_ADDRESS>Xiang-Yun Hu1<EMAIL_ADDRESS>1
School of Institute of Geophysics and Geomatics, China University of
Geosciences, Lumo Road 388, 430074 Wuhan, China.  2 School of Automation,
China University of Geosciences, Lumo Road 388, 430074 Wuhan, China.  3 Hubei
Key Laboratory of Advanced Control and Intelligent Automation for Complex
Systems, China  4 Engineering Research Center of Intelligent Technology for
Geo-Exploration, Ministry of Education, China

###### Abstract

A modified-weak-value-amplification(MWVA) technique of measuring the mirror’s
velocity based on the Vernier-effect has been proposed. We have demonstrated
with sensitivity-enhanced and the higher signal-to-noise ratio(${\rm SNR}$) by
using two cascaded Michelson interferometers. These two interferometers are
composed of similar optical structures. One interferometer with a fixed mirror
acts as a fixed part of the Vernier-scale, while the other with a moving
mirror acts as a sliding part of the Vernier-scale for velocity sensing. The
envelope of the cascaded interferometers shifts much more than a single one
with a certain enhancement factor, which is related to the free space range
difference between these two interferometers. In addition, we calculate the
${\rm SNR}$ based on the Fisher information with both the MWVA technique and
the traditional-weak-value-amplification(TMVA) technique. The results show
that the ${\rm SNR}$ with our MWVA technique is larger than the the ${\rm
SNR}$ with the TWVA technique within the range of our time measurement window.
Our numerical analysis proved that our MWVA technique is more efficient than
the TWVA technique. And by using the principles of the Vernier-effect, it is
applicative and convenient to ulteriorly improving the sensitivity and ${\rm
SNR}$ in measuring other quantities with the MWVA technique.

##  I Introduction

A weak measurement is a new tool for characterizing post-selected quantum
systems in the recent development of quantum mechanicsLecocq  _et al._ (2021);
Monroe  _et al._ (2021). Historically speaking, the idea of this measurement
was originally proposed by Aharonov, Albert and VaidmanAharonov  _et al._
(1988) as an extension to the standard("projective") von-Neumann model of
measurement. In weak measurement, information is gained by weakly probing the
system, while approximately preserving its initial state. The average shift of
the final pointer state can go far beyond the eigenvalue spectrum of the
system observable in sharp contrast to the standard quantum measurement. This
average shift is called the weak value, which is usually complexPeres (1989);
Leggett (1989). These features have allowed a wide range of applicability in
understanding many counter-intuitive quantum results. For example, weak value
can offer new insights into quantum paradox, such as Hardy’s paradoxLundeen
and Steinberg (2009); Yokota  _et al._ (2009), the three-box paradoxResch  _et
al._ (2004), apparent superluminal travelRohrlich and Aharonov (2002), direct
measuring the real and imaginary components of the wave-function Weissberger
and Rossiter (2011).

In addition to the fundamental physics interest in weak values, one of the
major goals in weak measurement is to enhance the sensitivity of estimating
weak signalsSzorkovszky  _et al._ (2011); Magaña Loaiza  _et al._ (2014); Li
_et al._ (2014); Luo  _et al._ (2020); Huang.Jing-Hui  _et al._ (2021). For
example, Dixon et al. amplified very small transverse deflections of an
optical beam, then measured the angular deflection of a mirror down to 400
$\pm$ 200 frad and the linear travel of a piezo actuator down to 14 $\pm$7 fm
Dixon  _et al._ (2009). Boyd et al. demonstrated the first realization of
weak-value amplification in the azimuthal degree of freedom and had achieved
effective amplification factors as large as 100Magaña Loaiza  _et al._ (2014).
Viza et al. achieved a velocity measurement of 400 fm/s by measuring one
Doppler shifted due to a moving mirror in a Michelson interferometerViza  _et
al._ (2013). Pati et al. even proposed an alternative method to measure the
temperature of a bath using the weak measurement scheme with a finite-
dimensional meterPati  _et al._ (2020). Therefore, these applications have
been termed as the weak-value-amplification(WVA) technique in literatureRen
_et al._ (2020).

It is worth noting that weak-value-amplification cannot be arbitrarily large
in the cost of decreasing output laser intensityPang and Brun (2015). The
analysis of Koike et al.Koike and Tanaka (2011) showed that the measured
displacement and the amplification factor are not proportional to the weak
value and rather vanish in the limit of infinitesimal output intensity. The
work of Pang et al.Pang  _et al._ (2014) investigated the limiting case of
continuous-variable systems to demonstrate the influence of system dimension
on the amplification limit. Therefore, it is still the focus in weak
measurement to enhance the sensitivity of detecting small signals. The
experiment of Starling et al.Starling  _et al._ (2009) and the proposal of
Feizpour et al. Feizpour  _et al._ (2011) showed that post-selection can
significantly raise the signal-to-noise ratio (${\rm SNR}$) of weak
measurement. Huang et alHuang  _et al._ (2019) proposed an approach called
dual weak-value amplification (DWVA), with sensitivity several orders of
magnitude higher than the standard approach without losing signal intensity.
Krafczyk et alKrafczyk  _et al._ (2021). experimentally demonstrated recycled
weak-value measurements. By using photon counting detectors, they demonstrated
a signal improvement by a factor of $4.4\pm 0.2$ and a signal-to-noise ratio
improvement of $2.1\pm 0.06$, compared to a single-pass weak-value experiment.

Recently, Vernier-effect has been an effective tool to enhance the sensitivity
of photonic devicesDai (2009); Claes  _et al._ (2010); Zhu  _et al._ (2017);
Liu  _et al._ (2017). The Vernier-effect is an efficient method to enhance the
accuracy of measurement instruments, which consists of two/three scales with
different periods. In Ref.Dai (2009) a sensor is introduced that consists of
two cascaded ring resonators, and it is shown theoretically that it can obtain
very high sensitivities thanks to the Vernier principle. In Zhu  _et al._
(2017) a high sensitivity of 1892dB/RIU of the optical sensor was achieved for
the intensity interrogation based on the cascaded reflective Mach–Zehnder
interferometer(MZI)s and the micro-ring resonators. Further, in Ref.Liu  _et
al._ (2017) a three cascaded micro-ring resonators based refractive index
optical sensor with a high sensitivity of 5866 nm/RIU was demonstrated, the
measurement range of which is significantly improved 24.7 times compared with
the traditional two cascaded micro-ring resonators.

In this letter, we put forward a modified-weak-value-amplification(MWVA) with
two cascaded Michelson interferometers to improve the sensitivity of
traditional-weak-value-amplification(TMVA) based on the Vernier-effect. More
specifically, in the framework of quantum weak measurements, we observe the
temporal shift induced by a small spectral shift with two cascaded Michelson
interferometers. The spectral shift in our scheme is a Doppler frequency shift
produced by a mirror in one of the interferometers, while the other
interferometer has the same optical structure with no movement of the mirror.
By numerical simulation, We obtain the temporal shift dependence of the
velocity, the sensitivity, and the signal-to-noise ratio (${\rm SNR}$)
calculated from the Fisher informationHoyle (2010). Besides, our MWVA
technique is comparable to the TWVA technique, but allows us to reach the
higher sensitivity and ${\rm SNR}$ under the same conditions of measuring
time.

The paper is organized as follows. In Sec II.A, we briefly review the TWVA
technique for measurement velocity in a single Michelson interferometer. Then,
in Sec II.B, we first derive the MWVA technique for the velocity with two
cascaded Michelson interferometers based on Vernier-effect. In Sec III, by
ensuring the same time measurement window and determine the respective meter
in the two techniques, we obtained analytic results of the sensitivity and the
${\rm SNR}$ from the Fisher information. Sec IV is devoted to the summary and
the discussions.

##  II The TWVA technique and the MWVA technique

In this article, we propose and numerically demonstrate a weak measurement
scheme, in which the amplification of the phase shifts in a Michelson
interferometer can be effectively enhanced by introducing another cascaded
interferometer based on the Vernier-effect. In order to compare our new
technique to the traditional ones, we briefly review the traditional WVA
technique to measuring velocity with a Michelson interferometer by following
the Ref.Viza  _et al._ (2013) in the following subsection.

###  II.1 The WVA technique for velocity measurements

In this section, we briefly review an interferometer scheme in Ref.Viza  _et
al._ (2013) combined WVA technique to measure longitudinal velocities with the
temporal meter. The temporal shift is proportional to the weak value and can
be amplified in the measurement of $v$, which is accompanied by a decrease in
the measured intensity due to the nature of the weak measurement. The temporal
shift is induced by the spectral shift, which is a Doppler frequency shift
produced by a moving mirror in a Michelson interferometer. The weak
measurement is characterized by state preparation, a weak perturbation, and
post-selection. The detail of the WVA technique for velocity measurements is
shown in Fig. 1.

The TWVA technique of weak measurement can be characterized into three parts:
state preparation, weak interaction, and post-selection. The initial state
$|{\Phi_{i}}\rangle$ of the system and $|{\Psi_{i}}\rangle$ of the meter are
prepared with the polarizer and AO Modulator. The initial polarization state
of the system can be described by the polarization qubit:

$\displaystyle|{\Phi_{i}}\rangle=sin(\alpha)|{H}\rangle+cos(\alpha)|{V}\rangle$
(1)

where $\alpha=\pi/4$ is the angle between the horizontal line and the
transmission axis of line polarizer(pre-selection); The horizontal
polarization state $|{H}\rangle$ correspond to the arm with the mirror which
the light goes through, while the vertical polarization state $|{V}\rangle$
correspond to the arm with piezo-driven mirror in the Michelson
interferometer. The preparation of the meter consists of the generation of the
temporal mode

$\displaystyle
I_{single}^{i}=|\langle{p}|{\Psi_{i}}\rangle|^{2}(t)=\frac{1}{\sqrt{2\pi\tau^{2}}}e^{-(t+t_{0})^{2}/2\tau^{2}}$
(2)

where $\tau$ is the length of the Gaussian pulse and $t_{0}$ is the center of
the pulse. The advantage of that temporal meter and the non-Fourier limited
Gaussian-shaped pulse has been studied in Ref.Viza  _et al._ (2013). The pulse
is injected into a Michelson interferometer, where the horizontally polarized
component of the pulse goes through the arm with a slowly moving mirror at
speed $v$, and the vertically one goes through the arm with a piezo-Driven
mirror. The weak interaction in Fig. 1 can be expressed as

$\displaystyle U=e^{-ig\hat{A}\otimes\hat{p}}=e^{-i\omega_{d}\hat{A}t/2}$ (3)

where $\omega_{d}=2\pi f_{d}=2\pi\frac{2v}{\lambda}$. Note that the spectral
shift $f_{d}=2v/\lambda$ is proportional to velocity $v$. The observable
$\hat{A}$ satisfies:
$\hat{A}=\frac{1}{2}(|{H}\rangle\langle{H}|-|{V}\rangle\langle{V}|)$.

Figure 1: Schematic of the WVA technique for velocity measurements in the
single Michelson. AOM is the Acousto-Optic Modulator, which can modulate the
light into a temporal Gaussian shaped pulse. PBS is the polarizer beam
splitter. The arrival time of single photons is measured with an avalanche
photodiode(APD). Figure 2: The time shift of meter in the scheme of the WVA
technique for measuring velocity $v=30$ nm/s with the length of the Gaussian
pulse $\tau=$ 0.2 ms and the phase offset $\phi$= 0.3 rad of the post-
selection. The vertical dot line represents the corresponding the center of
the Gaussian function.

Finally, the post-selection of the weak measurement is controlled by inducing
a phase offset $2\phi$ by the piezo-Driven mirror. Thus, the final post-
selection of the system is given by:

$\displaystyle|{\Phi_{f}}\rangle=\frac{1}{\sqrt{2}}(|{H}\rangle-e^{i2\phi}|{V}\rangle)$
(4)

and the weak value can be calculated by:

$\displaystyle
A_{w}=\frac{\langle{\Phi_{f}}|\hat{A}|{\Phi_{i}}\rangle}{\langle{\Phi_{f}}|{\Phi_{i}}\rangle}=-i\rm
cot\phi\approx\frac{-i}{\phi},$ (5)

By following the original work in Ref.Aharonov  _et al._ (1988), the meter
state of the temporal mode after the post-selection becomes:

$\displaystyle\langle{p}|{\Psi_{f}}\rangle$ $\displaystyle=$
$\displaystyle\langle{\Phi_{f}}|U|{\Phi_{i}}\rangle\langle{p}|{\Psi_{i}}\rangle$
(6) $\displaystyle=$
$\displaystyle\langle{\Phi_{f}}|e^{-ig\hat{A}\otimes\hat{p}}|{\Phi_{i}}\rangle\langle{p}|{\Psi_{i}}\rangle$
$\displaystyle=$
$\displaystyle\langle{\Phi_{f}}|{\Phi_{i}}\rangle[1-igA_{w}\hat{p}]\langle{p}|{\Psi_{i}}\rangle+O(g^{2})$
$\displaystyle\approx$ $\displaystyle\langle{\Phi_{f}}|{\Phi_{i}}\rangle
e^{-igA_{w}\hat{p}}\langle{p}|{\Psi_{i}}\rangle$

and therefore, the absolute value squared of the meter state (6) is given by:

$\displaystyle I_{single}^{f}$ $\displaystyle=$
$\displaystyle|\langle{p}|{\Psi_{f}}\rangle|^{2}$ (7) $\displaystyle=$
$\displaystyle(sin\phi)^{2}e^{-\frac{2\pi
vt}{\lambda\phi}}|\langle{p}|{\Psi_{i}}\rangle|^{2}$ $\displaystyle=$
$\displaystyle\frac{(sin\phi)^{2}}{\sqrt{2\pi\tau^{2}}}e^{-\frac{2\pi
vt}{\lambda\phi}}e^{-(t+t_{0})^{2}/2\tau^{2}}$ $\displaystyle\approx$
$\displaystyle\frac{(sin\phi)^{2}}{\sqrt{2\pi\tau^{2}}}e^{-(t+t_{0}+\frac{4\pi\tau^{2}v}{\lambda\phi})^{2}/2\tau^{2}}$

The final step approximation of Eq.(7) condition is $4\pi
v\tau^{2}\ll\lambda\phi$. In the scheme of the traditional WVA technique for
measuring velocity, the time shift $\delta
t=\frac{4\pi\tau^{2}v}{\lambda\phi}$ in Eq.(7) is amplified in the measurement
of $v$. Note that the work in Ref.Viza  _et al._ (2013) and our derivation
show that the spectral shift $f_{d}=2v/\lambda$ can translate into the time
shift $\delta
t=\frac{2\pi\tau^{2}f_{d}}{\phi}=\frac{4\pi\tau^{2}v}{\lambda\phi}$.

In addition, we show the time shift of meter in the scheme of the TWVA
technique for measuring velocity $v=30$ nm/s with the length $\tau=$ 0.2 ms
and central wavelength $\lambda=$ 800 nm of the Gaussian pulse in Fig. 2. Note
that the work in Ref.Brunner and Simon (2010) studied an interferometric
scheme based on a purely imaginary weak value, combined with a frequency-
domain analysis, which may have the potential to outperform standard
interferometry by several orders of magnitude. However, the amplification of
the TWVA technique can not be arbitrarily large due to the limit of the
resolution of the detector. Therefore, the MWVA technique based on Vernier-
effect with two cascaded Michelson interferometers for measuring velocity is
proposed in the next subsection.

###  II.2 The MWVA technique based on Vernier-effect

Figure 3: The scheme of the MWVA technique with two cascaded Michelson
interferometers based on Vernier-effect.

In this section, we propose a sensitivity-enhanced scheme of the MWVA
technique with two cascaded Michelson interferometers based on the Vernier-
effect. In addition, through numerical simulation experiments, we verify the
effectiveness and feasibility of the MWVA technique.

The schematic diagram of the MWVA technique for measuring velocity is shown in
Fig. 3. It includes two Michelson interferometers based on the TWVA technique,
and each Michelson interferometer is the same as the Michelson interferometer
with the TWVA technique in Fig. 1. The upper interferometer with moving mirror
1 serves as the sliding part of the Vernier-scalar, as the velocity changes
will cause a shift of the final meter. The lower interferometer with statical
mirror 2 serves as the fixed part of the Vernier-scalar. The Vernier-scale is
an efficient method to enhance the accuracy of measurement instruments, which
consists of two scales with different periodsShao  _et al._ (2015).

In our work, the upper interferometer and the lower interferometer are
prepared with different free spectrum range(FSR). The total transmission meter
of the cascaded Michelson interferometer is the superposition of the optical
power output of two interferometers, which exhibits peaks at times where two
interference peaks of the respective interferometers partially overlap. The
envelope period is given by

$\displaystyle\frac{FSR_{sliding}\times
FSR_{fixed}}{|FSR_{sliding}-FSR_{fixed}|}$ (8)

When the velocity change, the transmission of the single(upper) interferometer
will shift. Then the shift of cascaded Michelson interferometers envelope is
magnified by a certain factor $Enh$. The shift of the transmission peak is
proportional to the enhanced factor

$\displaystyle{Enh}=\frac{FSR_{fixed}}{|FSR_{sliding}-FSR_{fixed}|}$ (9)

In order to produce the Vernier-effect, the parameters of the main optical
components in our scheme in Fig. 3 shall meet the following requirements:

(i) Light: high-power light source is suitable for our scheme. Note that the
pulses for light 1 and light 2 don’t have to be coherent, because the laser
detected at APD is the superposition of the optical power output of two
interferometers.

(ii) Piezo -Driven Mirror: the post-selection of the TWVA technique is
controlled by inducing a phase offset $2\phi$ with the Piezo-Driven Mirror. In
other words, the intensity of the outgoing light is proportional to the phase
offset $2\phi$, which is shown in Eq.(7). To obtain the overlap with easily
discernible peaks, the intensity of the outgoing light of each interferometer,
namely the phase offset $2\phi$ should keep the same value.

(iii) AOM: AOM is the most critical device in our scheme. According to the
researches in Ref. Shao  _et al._ (2015); Zhu  _et al._ (2017); Liu  _et al._
(2017), the pulse, which serves as the sliding part or the fixed part of the
Vernier-scale, ought to be produced with equally spaced Gaussian mode rather
than individual pulse (2). At the same time, AOM is mainly used outside the
laser cavity, using the electric signal of the driving source to modulate the
laserZeng  _et al._ (2007). In our work, AOM 1 modulates the laser with
$FRS_{sliding}$ as the pre-selection meter
$|\langle{p}|{\Psi_{i}}\rangle|_{sliding}^{2}(t)$ $=I_{u}^{i}$ in upper
interferometer, while AOM 2 modulates the laser with $FRS_{fixed}$ as the pre-
selection meter $|\langle{p}|{\Psi_{i}}\rangle|_{fixed}^{2}(t)$ $=I_{d}^{i}$
in lower interferometer. Then $I_{u}^{i}$ and $I_{d}^{i}$ are given by:

$\displaystyle
I_{u}^{i}=\sum_{m=0}^{N}\frac{1}{\sqrt{2\pi\tau^{2}}}e^{-(t+t_{0}-m\times
FRS_{sliding})^{2}/2\tau^{2}}$ (10) $\displaystyle
I_{d}^{i}=\sum_{m=0}^{N}\frac{1}{\sqrt{2\pi\tau^{2}}}e^{-(t+t_{0}-m\times
FRS_{fixed})^{2}/2\tau^{2}}$ (11)

The final meter $|\langle{p}|{\Psi_{f}}\rangle|_{fixed}^{2}(t)$ =$I_{d}^{f}$
of the lower interferometer becomes:

$\displaystyle
I_{d}^{f}=\sum_{m=0}^{N}\frac{(sin\phi)^{2}}{\sqrt{2\pi\tau^{2}}}e^{-(t+t_{0}-m\times
FRS_{fixed})^{2}/2\tau^{2}}$ (12)

The final meter $|\langle{p}|{\Psi_{f}}\rangle|_{sliding}^{2}(t)$=$I_{u}^{f}$
of the upper interferometer with moving mirror 1 at velocity $v$ becomes:

$\displaystyle
I_{u}^{f}=\sum_{m=0}^{N}\frac{(sin\phi)^{2}}{\sqrt{2\pi\tau^{2}}}e^{-(t+t_{0}-m\times
FRS_{sliding}+\frac{4\pi\tau^{2}v}{\lambda\phi})^{2}/2\tau^{2}}$ (13)

The intensity $I_{sum}^{f}=I_{u}^{f}+I_{d}^{f}$ of the total laser detected at
APD is the sum of the light power $I_{u}^{f}$ from the upper interferometer
and the light power $I_{d}^{f}$ from the lower interferometer.

Figure 4:  Numerical simulation of the transmission temporal shift of (a)
single Michelson interferometer and (b) two cascaded Michelson
interferometers.

In the end, we fix a specific experimental setup, that is, the given chosen
post-selection phase $\phi=0.3$ rad, the length $\tau=$ 0.2 ms and central
wavelength $\lambda=$ 800 nm, the parameter $t_{0}$=0.0 of the pulse
$I_{u}^{i}$ (10) and the pulse $I_{d}^{i}$ (11), the free spectrum range
$FSR_{fixed}$=2.0 ms of the lower interferometer and the free spectrum range
$FSR_{sliding}$=1.96 ms of the upper interferometer in Fig. 3, so that
numerical simulation of the transmission temporal shift of single Michelson
interferometer is shown in Fig. 4(a), and numerical simulation of the
transmission temporal shift of two cascaded Michelson interferometers is shown
in Fig. 4(b). Then the sensitivity-enhanced factor $Enh$=
$\frac{3\;ms}{0.06\;ms}=50$ due to the Vernier-effect, which is the same value
as the value $Enh$= $\frac{2.00\;ms}{2.00\;ms-1.96\;ms}=50$ calculated from
Eq. (9). The results verify that the sensitivity for measuring velocity can be
enhanced much more according to the calculated sensitivity-enhanced factor (9)
by choosing two interferometers with smaller differences in FRS. However, in
practice, there is a compromise between the sensitivity and measurement
accuracy. Because the smaller FRS difference will increase the FRS of the
envelope, which introduces the difficulty in tracking the temporal shift of
the envelope peak Shao  _et al._ (2015).

The numerical results show that the MWVA technique based on Vernier-effect is
more sensitive than the TWVA technique despite how much time is used in the
two techniques. In the scheme of the TWVA technique, Eq.(7) indicates that
time shift $\delta t=\frac{4\pi\tau^{2}v}{\lambda\phi}$ can be effectively
enhanced by increasing the measuring time. Meanwhile, the Fig. 4. show that
our scheme costs more time than the scheme of the single Michelson
interferometer. Therefore, our work leads to a new issue: is the MWVA
technique based on Vernier-effect more sensitive than the TWVA technique at
the same measurement time? And further discussion is shown in the next
section.

Figure 5:  Numerical simulation of the transmission temporal shift of (a)
single Michelson interferometer and (b) two cascaded Michelson interferometers
for measuring velocity $v$= 30 nm/s(left plane) and velocity $v$= 10
nm/s(right plane). The unit of the y-coordinate of each graph is
$(2\pi\tau^{2})^{-1/2}$.

##  III Further discussions and Numerical Results

###  III.1 The sensitivity-enhanced factor at the same measurement time

In this subsection, we will study the sensitivity-enhanced factor in the two
schemes mentioned above at the same measurement time. Note that we can obtain
the temporal shift in a certain and effective time window, and the rest time
window is useless in the measurement. For example, When time is greater than 2
ms of the window in Fig. 4(a), the measurement is invalid.

In the next step, by choosing the appropriate meter in the TWVA technique and
the MWVA technique, we compare the sensitivity of these techniques at the same
measurement time. Note that we can obtain the temporal shift in a certain and
effective time window. The simulation is shown in Fig. 5. The $I_{u}^{i}$ and
$I_{d}^{i}$ are chosen with chosen post-selection phase $\phi=0.3$ rad, the
length $\tau$=0.2 ms of the individual Gaussian pulse, the parameter
$t_{0}$=46.0 of the pulse $I_{u}^{i}$ (10) and the pulse $I_{d}^{i}$ (11), the
free spectrum range $FSR_{fixed}$=2.0 ms of the lower interferometer and the
free spectrum range $FSR_{sliding}$=1.96 ms of the upper interferometer in
Fig. 3. Then $I_{u}^{i}$ and $I_{d}^{i}$ are given by

$I_{d}^{i}=\left\\{\begin{array}[]{rcl}\sum\limits_{m=0}^{N}\frac{1}{\sqrt{2\pi\times
0.2^{2}}}e^{-(t+46-2.00m)^{2}/(2\times 0.2^{2}})&&{t<4ms}\\\
0&&{t>4ms}\end{array}\right.$
$I_{d}^{i}=\left\\{\begin{array}[]{rcl}\sum\limits_{m=0}^{N}\frac{1}{\sqrt{2\pi\times
0.2^{2}}}e^{-(t+46-1.96m)^{2}/(2\times 0.2^{2}})&&{t<4ms}\\\
0&&{t>4ms}\end{array}\right.$

In order to keep the same measurement time, the meter mode $I_{single}^{i}$
(2) of the schematic of the TWVA technique with the single Michelson in Fig. 1
should be modulated with the length $\tau$=0.4 ms and the length $t_{0}$=-2.0
of the Gaussian pulse. The $I_{single}^{i}$ is given by:

$I_{d}^{i}=\left\\{\begin{array}[]{rcl}\frac{1}{\sqrt{2\pi\times
0.4^{2}}}e^{-(t-2.0)^{2}/(2\times 0.4^{2})}&&{t<4ms}\\\
0&&{t>4ms}\end{array}\right.$

With the input of the $I_{single}^{i}$ of the single Michelson in Fig. 1, we
can obtain the final meter $I_{single}^{f}$ (7) of measuring velocity $v$= 30
nm/s and $v$= 10 nm/s in Fig. 5(a) and Fig. 5(c). Under the same window of
measurement time, the results of the detected intensity $I_{sum}^{f}$ of the
two cascaded Michelson interferometers measuring velocity $v$= 30 nm/s and
$v$= 10 nm/s based on Vernier-effect are shown in Fig. 5(b) and Fig. 5(d).

The method of how to chose the window of measurement time is explained as
follows. The selected goal is to obtain the envelope through movement of
$I_{sum}^{f}$ within the smallest possible measurement time window. In our
work, we find that the intersection of straight lines determined by the
interval troughs can track the shift of the envelope trough. More
specifically, the shift of $I_{sum}^{f}$ can be calculated by tracking the
shift of the trough, which is the nearest wave to the right of the
intersection. Therefore, we call the above method the intersecting-point-
tracing(IPT) method. By the IPT method, we obtain the shifts of the envelope
$I_{sum}^{f}$ at measuring velocity $v$= 30 nm/s and $v$= 10 nm/s based on
Vernier-effect are shown in Fig. 5(b) and Fig. 5(d). And the results show that
the IPT method is effective and convenient for tracking the shifts.

We can obtain the effective sensitivity-enhanced factor $eEnh$ of measuring
velocity $v$= 30 nm/s from Fig. 5(a) and Fig. 5(c). $eEnh$ is calculated by
$eEnh=\frac{3.00\;ms}{0.25\;ms}=12$. In addition, the temporal shifts of
measuring velocity $v$= 10 nm/s in Fig. 5(b) and Fig. 5(d) conclude the same
result $eEnh=\frac{1.00\;ms}{0.08\;ms}=12$. Though the effective sensitivity-
enhanced factor $eEnh=12$ is smaller than the sensitivity-enhanced factor
$Enh=50$ calculated from Eq. (9), the sensitivity of the TWVA technique for
velocity measurements is still enhanced by several orders of magnitude thanks
to the Vernier-effect.

In the actual measurement process, for the temporal shift to be measurable, we
require that $\delta t>\Delta t$, where $\Delta t$ is the resolution of the
detector. Assuming that the shift $\delta t=0.08\;ms$ in Fig. 5(c) has reached
the resolution $\Delta t$ and the $\delta t$ can not be measured. However the
shift $\delta t=1.0ms>\Delta t$ in Fig. 5(d) can be effectively measured due
to the sensitivity-enhanced of Vernier-effect. Therefore, the MWVA technique
based on Vernier-effect can break the resolution limit of the detector of the
TWVA technique for velocity measurements.

###  III.2 The analysis of the signal-to noise ratio based on the Fisher
information

Figure 6:  The Fisher information (a) and the signal-to-noise (b) at different
velocity with the traditional WVA technique and the modified WVA technique
based on Vernier-effect.

In this subsection, by calculation the cram$\rm\acute{e}$r-Rao
bound($CRB$)Frieden and Gatenby (2013); Pfister  _et al._ (2011), we consider
the fundamental limitation of velocity measurements with both the traditional
WVA technique and the modified WVA technique based on the Vernier-effect. The
$CRB$ is the fundamental limit in the minimum uncertainty for parameter
estimation. And the $CRB$ is equal to the inverse of the Fisher information.
In information theory, for a parameter $\Omega$ dependent probability
distribution of a random variable $x$, $\textbf{P}(x|\Omega)$, the Fisher
informationHoyle (2010) is defined as

$\displaystyle\textbf{F}(\Omega)=N\int
dx\;\textbf{P}(x|\Omega)[\frac{d}{d{\Omega}}\;{\rm
ln}\;\textbf{P}(x|\Omega)]^{2}$ (14)

where $N$ represents that $N$ photons are sent through the interferometer.
Then the $CRB$, which is the minimum variance $\Delta^{2}({\Omega})$ of an
unbiased of ${\Omega}$ can be obtained as $CRB$=$1/\textbf{F}(\Omega)$. Next,
we calculate $CRB$ and ${\rm SNR}$ in the TWVA technique and the MWVA
technique based on Vernier-effect at the same measurement time
$0<t<4\;ms$(discussed in Sec. III.1). In our work, the parameter $\Omega$ is
corresponding to the temporal shift $\delta t$.

In the scheme of Fig. 1 with the traditional WVA technique, we chose the same
pulse in Sec. III.1 and obtain the probability distribution of a random
variable $t$:

$\textbf{P}_{s}=\left\\{\begin{array}[]{rcl}\frac{(sin\phi)^{2}}{\sqrt{0.32\pi}}e^{-(t-2.0+\delta
t)^{2}/0.32}&&{t<4ms}\\\ 0&&{t>4ms}\end{array}\right.$

and the Fisher information can be computed by

$\displaystyle\textbf{F}_{s}=N\int dt\;\textbf{P}(t|\delta
t)[\frac{d}{d{\delta t}}\;{\rm ln}\;\textbf{P}_{s}(t|\delta t)]^{2}$ (15)

The sensitivity in determination of $\delta
t=\frac{4\pi\tau^{2}v}{\lambda\phi}$ is bounded by the square root of the
minimum variance $\Delta(\delta t)=\sqrt{1/\textbf{F}_{s}}$. And the smallest
resolvable velocity is determined by the signal-to-noise

$\displaystyle SNR_{s}=\frac{\delta t_{s}}{\Delta(\delta t)}=\frac{v}{\Delta
v}=\frac{4v\pi\tau^{2}\sqrt{\textbf{F}_{s}}}{\lambda\phi}$ (16)

where $\delta t_{s}=\delta t$.

Meanwhile the Fisher information $\textbf{F}_{c}$ with the MWVA technique
based on Vernier-effect in Fig. 3. To keep the same measurement times with the
TWVA technique, we choose the input $I_{u}^{i}$ (10) and $I_{d}^{i}$ (12) with
the same parameters in Sec. III.1, then the Fisher information
$\textbf{F}_{c}$ can be calculated by

$\displaystyle\textbf{F}_{c}=N\int dt\;\textbf{P}(t|\delta
t)[\frac{d}{d{\delta t}}\;{\rm ln}\;\textbf{P}_{c}(t|\delta t)]^{2}$ (17)

where the probability distribution $\textbf{P}_{c}$ satisfies:

$\textbf{P}_{c}=\left\\{\begin{array}[]{rcl}\sum\limits_{m=0}^{N}\frac{1}{\sqrt{0.08\pi}}[e^{-(t+46-2.00m+\delta
t)^{2}/0.08}&&\\\ +e^{-(t+46-1.96m+\delta t)^{2}/0.08}]&&{t<4ms}\\\
0&&{t>4ms}\end{array}\right.$

Therefore, the $SNR_{c}$ is given by:

$\displaystyle SNR_{c}=\frac{\delta t_{c}}{\Delta(\delta t)}=\frac{12v}{\Delta
v}=\frac{48v\pi\tau^{2}\sqrt{\textbf{F}_{s}}}{\lambda\phi}$ (18)

where $\delta t_{c}=12\delta t$. Note that the $SNR_{c}$ with the TWVA
technique will be enhanced with the factor $eEnh=12$ due to the Vernier-
effect.

The complexity of the probability distribution $\textbf{P}_{s}$ and
$\textbf{P}_{c}$ make it hard to analytically calculate the Fisher
information. In our work, we numerically solve the integration (15) and the
integration (17) for $N=\times 10^{6}$, and the results are shown in Fig.
6(a). And on this basis, we calculate the signal-to-noise at different
velocities with the TWVA technique and the MWVA technique in Fig. 6(b).

In our work, the cram$\rm\acute{e}$r-Rao bound($CRB$)Frieden and Gatenby
(2013); Pfister  _et al._ (2011) states that the inverse of the Fisher
information is a lower bound on the variance of any unbiased estimator of
$\delta t$, by noting that $\Delta^{2}(\delta t)$ characterizes the extent of
the shot noise of the quantum measurementRen  _et al._ (2020). Firstly, the
intuitive explanation for the Fisher Information is that the Fisher
Information reflects the accuracy of our parameter estimation. The larger it
is, the higher the accuracy of parameter estimation, that is, it represents
more information. Fig. 6(a) shows that the Fisher Information $\textbf{F}_{c}$
with our MWVA technique is larger than $\textbf{F}_{s}$ with the traditional
TWVA technique, when our measured velocity is less than 32 nm/s (read from
Fig. 6(a)). However, with further increase of velocity $v$, our MWVA technique
will become less efficient than the TWVA technique. As we have pointed out,
our MWVA technique is an effective alternative to the TWVA technique, when the
measurement is beyond the limits of resolution of the detector in Fig. 1.

Let us continue the discussion related to the signal-to-noise in Fig. 6(b).
The results show that the signal-to-noise $SNR_{c}$ is larger than $SNR_{s}$
with the traditional WVA technique When our measured velocity is less than 58
nm/s (read from Fig. 6(b)). Note that an interesting conclusion can be drawn
from Fig. 6. The Fisher information characterization is not fully equivalent
to the ${\rm SNR}$ characterization with the increase of measurement velocity.
And the similar conclusion is obtained by analyzing The WVA technique beyond
the Aharonov-Albert-Vaidman limit in Ref Ren  _et al._ (2020). In addition, as
the measurement velocity is smaller, the signal-to-noise ratio of our MWVA
technique is improved more.

We notice that the time shift $\delta t_{c}$ in our MWVA technique is less
than 3.5 ms thanks to ensuring the same time measurement window. And the upper
limit of the corresponding velocity measurement is 40 nm/s. Therefore, we can
conclude that our MWVA technique is more efficient than the TWVA technique,
because $SNR_{c}$ is larger than $SNR_{s}$ as $v<$ 40 nm/s in Fig. 6(b).

##  IV Summary and Discussions

In summary, using the feature of sensitivity-enhanced in Vernier-effect, we
have proposed a modified-weak-value-amplification(MWVA) technique of measuring
the mirror’s velocity. Compared with the traditional-weak-value-
amplification(TWVA) technique, we demonstrated sensitivity-enhanced and higher
${\rm SNR}$ by using two cascaded Michelson interferometers. In our work, the
two cascaded interferometers are composed of the similar optical structures.
One interferometer with a fixed mirror acts as a fixed part of the Vernier-
scale, while the other with a moving mirror acts as a sliding part of the
Vernier-scale is for velocity sensing. By choosing the appropriate meter in
our MWVA technique and in the TWVA technique to ensuring the same measurement
time, we obtained numerical results of the sensitivity and the ${\rm SNR}$ in
the two techniques.

Our numerical results show that our MWVA technique is more efficient than the
traditional one with 12 times sensitivity-enhanced and higher ${\rm SNR}$. In
addition, as the measurement velocity is smaller, the ${\rm SNR}$ of our MWVA
technique is improved more. Note that our MWVA technique can propose a viable
and effective alternative for the measurement out of the limit of the
resolution in the TWVA technique. In addition, by using the principles of the
Vernier-effect, it is applicative and convenient to further improve the
sensitivity and ${\rm SNR}$ in measuring other physical quantities with the
WVA technique.

A few remarks are in order. We have proposed an intersecting-point-
tracing(IPT) method for tracking the shift of the meter with cascaded
Michelson interferometers, the IPT method indicates that the intersection of
straight lines which is determined by interval troughs can track the shift of
the envelope trough. It is remarked that we have found that the fisher
information characterization is not fully equivalent to the ${\rm SNR}$
characterization with the increase of measurement velocity. And the conclusion
is the response to the attention in Ref Ren  _et al._ (2020) from a purely
theoretical interest.

In addition, our MWVA technique can be realized beyond the Gaussian meter wave
function. Note that the workTurek  _et al._ (2015a) shows that the
Hermite–Gaussian and Laguerre–Gaussian pointer states for a given coupling
direction have advantages and disadvantages over the fundamental Gaussian mode
in improving the ${\rm SNR}$. And the workTurek  _et al._ (2015b) indicates
that the post-selected weak measurement scheme for non-classical pointer
states(coherent, squeezed vacuum, and Schrodinger cat states) to be superior
to semi-classical ones. Therefore, a MWVA technique with a non-Gaussian meter
wave function based on the Vernier-effect will be studied in our future works.
Meanwhile, the relevant experiments are also being carried out gradually.

###### Acknowledgements.

This study is financially supported by the National Key Research and
Development Program of China (Grant No. 2018YFC1503705) and the Fundamental
Research Funds for National Universities, China University of
Geosciences(Wuhan) (Grant No. G1323519204).

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