# Fisher information analysis on post-selection involved quantum precision
measurements using optical coherent states

Yingxin Liu  Center for Joint Quantum Studies and Department of Physics,
School of Science,
Tianjin University, Tianjin 300072, China  Lupei Qin<EMAIL_ADDRESS>Center for Joint Quantum Studies and Department of Physics, School of Science,
Tianjin University, Tianjin 300072, China  Xin-Qi Li<EMAIL_ADDRESS>Center for Joint Quantum Studies and Department of Physics, School of Science,
Tianjin University, Tianjin 300072, China

###### Abstract

The weak-value-amplification (WVA)

technique has been extensively considered and debated in the field of quantum
precision measurement, largely owing to the reduced Fisher information caused
by the low probability of successful post-selection. In this work we show
that, rather than the Gaussian meter state as typically considered, using the
optical coherent state as a meter, the WVA measurement can definitely
outperform the conventional measurement not involving the strategy of post-
selection. We also show that the post-selection procedure involved in the WVA
scheme can make a mixture of coherent states work better than a pure coherent
state with identical average photon numbers. This is in sharp contrast to the
claim proved in the absence of post-selection. The post-selection strategy can
also result in the precision of Heisenberg (or even “super-Heisenberg”)
scaling with the photon numbers, but without using any expensive quantum
resources. The present work may stimulate further investigations for the
potential of the post-selection strategy in quantum precision measurements.

##  I Introduction

The concept

of quantum weak values AAV88 ; AV90  has been found useful in quantum
metrology. A novel scheme termed as weak-value amplification (WVA) has been
developed for detecting tiny effects such as the spin Hall effect of light
Kwi08  and measuring weak signals How09a ; How09b ; How10a ; How10b ; Guo13 ,
owing to the advantages of improving the precision of weak measurements by
post-selection and protecting the weak measurements against technical noises
Sim10 ; Ste11 ; Nish12 ; Ked12 ; Jor14 ; Li20 ; Bru15 ; Bru16 ; How17 .

To be more specific, for the measurement of small transverse deflections of an
optical beam How09a ; How09b ; How10a ; How10b , the WVA technique can lead to
experimental sensitivity beyond the detector’s resolution. In this context, it
allowed to use high power lasers with low power detectors while maintaining
the optimal signal-to-noise ratio (SNR), and obtained the ultimate limit in
deflection measurement with a large beam radius. Actually, theoretical
analysis has pointed out that the WVA technique can outperform conventional
measurement in the presence of detector saturation Lun17 . This advantage of
the WVA technique was recently illustrated by achieving a precision of 6 times
higher than that of the conventional measurement ZLJ20 . Other improvements
along the line of the WVA technique include such as strengthening the WVA via
photon recycling Jor13 ; Sim15 ; Jor21  and applying a technique called dual
WVA (DWVA) which allows for a precision metrology with sensitivity several
orders of magnitude higher than the standard approach Zen19 . Also, it has
been pointed out that the WVA technique can reduce the technical noise in some
circumstances Kwi08 ; How09a ; How09b ; Sim10 ; Ste11 ; Nish12 ; Ked12 ; Jor14
; Li20 , and can even utilize it to outperform standard measurement by several
orders of magnitude by means of the imaginary weak-value measurements Sim10 ;
Ste11 ; Ked12 ; Jor14 ; Li20 .

The WVA technique involves an essential procedure of post-selection to the
state of the measured system. It is true that the meter state conditioned on
the successful post-selection contains more information about the parameter to
be estimated than the meter state in conventional measurement for parameter
estimation. However, since the probability of successful post-selection is
low, the practically attainable Fisher information will be greatly reduced. It
is largely this balance consideration that has caused controversial debates in
literature Nish12 ; Ked12 ; Jor14 ; Li20 ; Tana13 ; FC14 ; Kne14 ; ZLJ15 ;
Aha15 . In Ref. Jor14 , using the Fisher-information (FI) metric, it was
proved that WVA technique can put all of the Fisher information about the
parameter to be estimated into a small portion of the events and show how this
fact alone gives technical advantages. Meanwhile, there existed diverse
opinions about the WVA advantages Tana13 ; FC14 ; Kne14 ; ZLJ15 ; Aha15 . For
instance, in Ref. ZLJ15 , it was concluded that both the weak measurement and
post-selection cannot enhance the metrological precision. The analysis was
based on a consideration that the Fisher information contained in the WVA
technique is only a part of the total Fisher information associated with the
post-selection measurement and the subsequent measurement on the meter state,
while the reference standard is the quantum Fisher information (QFI) of the
joint entangled state of the entire system-plus-meter after coupling
interaction (with the strength parameter to be estimated). Obviously, such a
reference standard is too stringent, which is not the standard for comparison
with the conventional precision measurement, such as considered in Refs. Ked12
; Jor14 ; Li20  and in many others.

In precision metrology, another important criterion is the scaling of the
metrological precision with the resource (e.g. photon numbers), aiming to
excess the standard quantum limit (SQL) and to achieve the Heisenberg limit
(HL). The general viewpoint is that, in order to achieve the HL, quantum
resources such as the NOON states Dow08 ; Mac11 ; Ste14  are typically
required. However, it was found in Ref. ZLJ15  and further analyzed in Aha15 ,
that it is possible to achieve the HL with weak measurement using the optical
coherent state, which is a classical resource, as a meter to coupled to a
qubit system. In this case, the information encoded in the distribution of
post-selection process (but not in the post-selected meter state) can be
scaled at the Heisenberg limit with the average photon numbers of the coherent
state.

For employing the optical coherent state as a meter, we become aware of an
interesting conclusion from an earlier study Lui10 , which states that for any
linear coupling interaction, metrological precision better than using the
optical coherent state must come from non-classical effect. Noting that this
statement was proved in the absence of post-selection procedure, we may thus
raise a question: can the post-selection involved in the WVA technique violate
the conclusion proved in Ref. Lui10 ? Of great interest is that, as we will
see in this work, the answer is YES.

In this work, following the line of analysis in Refs. Ked12 ; Jor14 ; Li20 ,
where the Gaussian meter state (i.e. the transverse spatial wavefunction of a
light beam) was considered, we further consider the probe meter as an optical
coherent state and explore its possible advantages in the WVA metrology. This
type of meter state is particularly relevant to the optical cavity-QED or the
solid-state circuit-QED set-up realization. Also, as in Ref. Li20 , our
analysis will go beyond the Aharonov-Albert-Vaidman (AAV) limit by considering
arbitrary measurement-coupling strengths. By means of the Fisher-information
metric, the first important result is that using the optical coherent state as
a probe meter, which is cheap and not a “quantum resource”, the WVA
measurement can definitely outperform the conventional measurement in the
absence of the post-selection strategy. This result is very different from the
one of using the Gaussian meter state, where the Fisher information contained
in the WVA measurement can, at most, reach the same as in the conventional
measurement Jor14 ; Li20 . In this context we will show that, for the
measurement of coherent state in the absence of post-selection, we must
consider quadrature measurement, since the usual photon-number measurement
cannot extract the quantum Fisher information (QFI) encoded in the phase-
shifted coherent state. The second important result is that, in the presence
of the post-selection strategy, the WVA measurement using a mixture of
coherent states can work better than using a single pure coherent state. This
result is of great interest when making a connection with the claim in Ref.
Lui10 , as mentioned above. Since our analysis covers the range of arbitrary
coupling strengths, being somewhat surprising, we find that it is possible to
achieve a scaling behavior with the average photon numbers even better than
the HL, when the measurement-coupling strength is properly increased to
violate the AAV limit.

##  II Quantum Fisher Information Contained in the WVA Measurement

Let us consider

a two-state quantum system (qubit) with states denoted by $|g\rangle$ and
$|e\rangle$, which is coupled to an optical meter prepared in coherent state
$|\alpha\rangle$, while the meter state will be finally monitored by a
classical apparatus via photon number or field quadrature measurement, as
schematically shown in Fig. 1. In dispersive regime, the coupling interaction
Hamiltonian can be assumed as $H_{\rm int}=\chi\hat{\sigma}_{z}\hat{n}$, where
$\hat{\sigma}_{z}$ is the atomic Pauli operator
$\hat{\sigma}_{z}=|e\rangle\langle e|-|g\rangle\langle g|$ and
$\hat{n}=\hat{a}^{\dagger}\hat{a}$ is the photon number operator (with
$\hat{a}^{\dagger}$ and $\hat{a}$ the creation and annihilation operators of a
photon). Over some time, the coupling interaction will generate a unitary
operation $U=e^{i\lambda\hat{\sigma}_{z}\hat{n}}$ on the system-plus-meter
joint state, where $\lambda$ is the integrated interaction strength, which is
the right parameter to be metrologically estimated.

Figure 1:  Sketch for precision metrology of the coupling-interaction strength
parameter ($\lambda$) between a two-state (qubit) system and the meter of
optical coherent state $|\alpha\rangle\equiv|\Phi_{i}\rangle$. The qubit
system is initially prepared in state $|\psi_{i}\rangle$ and post-selected
with $|\psi_{f}\rangle$ after the coupling interaction. The meter state after
post-selection, $|\Phi_{f}\rangle$, is finally monitored by a classical
apparatus via photon number or field quadrature measurement.

Assuming the system initially prepared in a superposition state characterized
by $|\psi_{i}\rangle={\rm cos}\frac{\theta_{i}}{2}|g\rangle+{\rm
sin}\frac{\theta_{i}}{2}e^{i\phi_{i}}|e\rangle$, the coupling interaction will
entangle the joint state of the system-plus-meter as

$\displaystyle|\Psi_{J}\rangle={\rm cos}\frac{\theta_{i}}{2}|g\rangle|\alpha
e^{-i\lambda}\rangle+{\rm
sin}\frac{\theta_{i}}{2}e^{i\phi_{i}}|e\rangle|\alpha e^{i\lambda}\rangle\,.$
(1)

In the WVA strategy, a post-selection procedure is involved by performing a
projective measurement on the system, before finally measuring the meter state
for the parameter-$\lambda$ estimation. Let us assume the post-selection with
state $|\psi_{f}\rangle$, which has the same form of $|\psi_{i}\rangle$ by
replacing only the Bloch-vector polar angles with $\theta_{f}$ and $\phi_{f}$.
After post-selection for the system state, the meter state reads as

$\displaystyle|\tilde{\Phi}_{f}\rangle=\langle\psi_{f}|\Psi_{J}\rangle$
$\displaystyle={\rm cos}\frac{\theta_{i}}{2}{\rm
cos}\frac{\theta_{f}}{2}|\alpha e^{-i\lambda}\rangle+{\rm
sin}\frac{\theta_{i}}{2}{\rm sin}\frac{\theta_{f}}{2}e^{i\phi_{0}}|\alpha
e^{i\lambda}\rangle\,.$ (2)

Here we introduced $\phi_{0}=\phi_{i}-\phi_{f}$. The state
$|\tilde{\Phi}_{f}\rangle$ is not normalized. Let us denote the normalized
meter state as $|\Phi_{f}\rangle=|\tilde{\Phi}_{f}\rangle/\sqrt{p_{a}}$, where
the normalization factor is given by
$p_{a}=\langle\tilde{\Phi}_{f}|\tilde{\Phi}_{f}\rangle$, which is also the
probability of successful post-selection, while the subscript “a” simply
indicates the accepted case of post-selection. This notation will be used
hereafter for other quantities. In practice, after post-selection for the
system state, subsequent measurement to the meter state $|\Phi_{f}\rangle$
should be done, e.g., performing the photon-number measurement and obtaining
the distribution probability $P_{f}(n)$. This probability encodes the
information of the parameter $\lambda$; one can thus calculate the Fisher
information $F_{a}$, which characterizes the metrological precision for the
parameter $\lambda$ via the Cramér-Rao inequality WM09 . In the WVA scheme,
the key quantity to be compared with the conventional approach is the reduced
Fisher information $p_{a}F_{a}$ (to be termed as WVA-FI in this work).

If we do not specify the concrete scheme of measurement on the meter state, we
can talk about and calculate the QFI $Q_{a}$ encoded in the meter state
$|\Phi_{f}\rangle$, using the formula

$\displaystyle
Q_{a}=4\left[(\frac{d\langle\Phi_{f}|}{d\lambda})(\frac{d|\Phi_{f}\rangle}{d\lambda})-|\langle\Phi_{f}|(\frac{d|\Phi_{f}\rangle}{d\lambda})|^{2}\right]\,.$
(3)

Actually, the reduced QFI (or, WVA-QFI) $p_{a}Q_{a}$ represents the maximum
information attainable by the WVA measurement. In Fig. 2 we display the result
of $p_{a}Q_{a}$ as a function of the parameter $\lambda$. This
$\lambda$-dependence will be used later to discuss an important conclusion,
when the WVA scheme is compared with the conventional approach.

Figure 2:  Dependence of the QFI encoded in the WVA scheme on the coupling
interaction strength $\lambda$. The Bloch-vector-angles of the initial and
post-selected states of the qubit are assumed as: $\theta_{i}=\frac{\pi}{2}$,
$\phi_{0}=\pi$, and $\theta_{f}=\frac{3\pi}{2}$ (solid blue) and
$\frac{\pi}{2}$ (dashed red). The average photon number
$\bar{n}=|\alpha|^{2}=4$.

##  III Comparison between the WVA and Conventional Measurements

###  III.1 Photon-number measurement

Now let us consider

the specific scheme of photon-number measurement on the meter state
$|\Phi_{f}\rangle$. The photon-number distribution probability can be
predicted as

$\displaystyle P_{f}(n)={\lvert\langle n|\Phi_{f}\rangle\rvert}^{2}$
$\displaystyle=\frac{e^{-|\alpha|^{2}}}{p_{a}}\left(\frac{|\alpha|^{2n}}{n!}[\,A+B\,{\rm
cos}(2\lambda n+\phi_{0})\,]\right)\,.$ (4)

Here, for simplicity of the expression, we introduced $A={\rm
cos}^{2}\frac{\theta_{i}}{2}{\rm cos}^{2}\frac{\theta_{f}}{2}+{\rm
sin}^{2}\frac{\theta_{i}}{2}{\rm sin}^{2}\frac{\theta_{f}}{2}$, and
$B=\frac{1}{2}\sin\theta_{i}\sin\theta_{f}$. We see clearly that this
probability function encodes the information of the parameter $\lambda$, which
can be quantified by the Fisher information

$\displaystyle
F_{a}^{(n)}=\sum_{n}\frac{1}{P_{f}(n)}\left(\frac{dP_{f}(n)}{d\lambda}\right)^{2}\,.$
(5)

We would like hereafter to denote the FI associated with the photon-number
measurement as $F_{a}^{(n)}$, while the FI based on the $x$-quadrature
measurement is denoted as $F_{a}^{(x)}$ – which will be considered below in
detail. In Fig. 3 (to be discussed later in detail for comparison with other
schemes), we display the numerical results of $p_{a}F_{a}^{(n)}$ as a function
of the post-selection polar angle $\theta_{f}$. We find that via optimal
choice of $\theta_{f}$ the FI $p_{a}F_{a}^{(n)}$ can reach the QFI
$p_{a}Q_{a}$. Otherwise, the FI is less than the QFI, as expected.

###  III.2 Field-quadrature measurement

In the absence of post-selection,

one can prove that performing photon-number measurement on the join-state
$|\Psi_{J}\rangle$ of Eq. (1) does not allow us to extract the information of
$\lambda$. That is, the FI is zero. Therefore, if we consider the conventional
approach as considered in Refs. Ked12 ; Jor14 ; Li20 ; Lui10 , where the
system is prepared in one of the basis states (i.e. in $|g\rangle$ or
$|e\rangle$) and no post-selection procedure is involved, we are unable to
estimate the parameter $\lambda$ via photon-number measurement, despite that
the QFI is nonzero. For instance, consider the system prepared in $|g\rangle$.
After interaction with the meter field, the parameter $\lambda$ is encoded in
the meter state as $|\Phi_{\rm cm}\rangle=|\alpha e^{-i\lambda}\rangle$. The
subscript in $\Phi_{\rm cm}$ and also in the following QFI $Q_{\rm cm}$
indicates “conventional measurement”. Using the formula of Eq. (3), one can
easily calculate the QFI encoded in $|\Phi_{\rm cm}\rangle$ as

$\displaystyle Q_{\rm cm}=4|\alpha|^{2}=4\bar{n}\,.$ (6)

In this work, which is in parallel to the analysis for the case of the
Gaussian meter state, we will use this QFI to set up the best precision
achievable by conventional measurement, when compared with the WVA scheme.

In order to extract out the information of the parameter $\lambda$ encoded in
the coherent state $|\alpha e^{-i\lambda}\rangle$, rather than the photon-
number measurement, one can perform the field-quadrature measurement WM09 .
The quadrature measurement can be fulfilled by means of the homodyne
measurement scheme, in which the signal field (here, the meter field) is mixed
with a strong local oscillator of reference field with phase $\varphi$. Then,
the homodyne readout results correspond to the eigenvalues of the quadrature
operator

$\displaystyle\hat{x}=(\hat{a}e^{-i\varphi}+\hat{a}^{\dagger}e^{i\varphi})/\sqrt{2}\,.$
(7)

This measurement scheme actually defines a representation for us to construct
the wavefunction of the optical coherent state we are interested in, i.e., the
wavefunction $\alpha(x)=\langle x|\alpha\rangle$. In the special case of
$\varphi=0$, the $x$-quadrature operator is reduced as
$\hat{x}=(\hat{a}+\hat{a}^{\dagger})/\sqrt{2}$. This enables us to understand
the wavefunction more easily as the one of a coherent state of the well known
mechanical harmonic oscillator in the coordinate representation. With respect
to the above $x$-quadrature operator, its conjugated $p$-quadrature operator
is given by

$\displaystyle\hat{p}=-i(\hat{a}e^{-i\varphi}-\hat{a}^{\dagger}e^{i\varphi})/\sqrt{2}\,.$
(8)

One can easily check that $[\hat{x},\hat{p}]=i$, i.e., the Heisenberg
equation. Now, let us consider the eigenstate equation
$\hat{a}|\alpha\rangle=\alpha|\alpha\rangle$, which means that the coherent
state $|\alpha\rangle$ is the eigenstate of the field operator $\hat{a}$, with
eigenvalue $\alpha$. Solving $\hat{a}$ from the two quadrature operators
$\hat{x}$ and $\hat{p}$ and substituting into the eigenstate equation, we
obtain

$\displaystyle\frac{1}{\sqrt{2}}(\hat{x}+i\hat{p})=\alpha
e^{-i\varphi}|\alpha\rangle\,.$ (9)

Then, in the $x$-representation, this equation is converted as

$\displaystyle\left[\frac{d}{dx}+(x-\sqrt{2}\alpha
e^{-i\varphi})\right]\alpha(x)=0\,.$ (10)

The solution of this differential equation reads as

$\alpha(x)=(\frac{1}{\pi})^{\frac{1}{4}}\exp\left(\frac{1}{2}\left[\widetilde{\alpha}^{2}-|\widetilde{\alpha}|^{2}-(x-\sqrt{2}\widetilde{\alpha})^{2}\right]\right)\,,$
(11)

where $\widetilde{\alpha}=\alpha e^{-i\varphi}$. In this solution, the phase
factor has been adjusted in order to satisfy the property of coherent states,
$\langle\alpha_{1}|\alpha_{2}\rangle=\int
dx\alpha^{*}_{1}(x)\alpha_{2}(x)=e^{-\frac{1}{2}(|\alpha_{1}|^{2}+|\alpha_{2}|^{2})+\alpha^{*}_{1}\alpha_{2}}$.
Then, straightforwardly, the probability distribution function of the
$x$-quadrature measurement on the coherent state $|\alpha
e^{-i\lambda}\rangle$ is obtained as

$P(x)={\lvert\langle x|\alpha
e^{-i\lambda}\rangle\rvert}^{2}=\frac{1}{\sqrt{\pi}}e^{-[x-\sqrt{2}\,{\rm
Re}(\widetilde{\alpha}e^{-i\lambda})]^{2}}\,.$ (12)

The FI contained in this distribution function can be simply calculated as

$F_{\rm cm}=\int dx\frac{1}{P(x)}\left(\frac{\partial
P(x)}{\partial\lambda}\right)^{2}=4{\rm
Im}^{2}(\widetilde{\alpha}e^{-i\lambda})\,.$ (13)

With no loss of generality, we assume a real $\alpha$ and obtain $F_{\rm
cm}=4|\alpha|^{2}\sin^{2}(\varphi+\lambda)$. Then, we see that through proper
choice for the local oscillator’s phase $\varphi$, one can reach the QFI
$Q_{\rm cm}=4|\alpha|^{2}$.

Let us now come back to the WVA scheme. For the purpose of making a
comprehensive comparison, in addition to the photon-number measurement, we
further consider the $x$-quadrature measurement to the meter field. After the
post-selection, the meter state is a superposition of $|\alpha
e^{-i\lambda}\rangle$ and $|\alpha e^{i\lambda}\rangle$, i.e., the state
$|\Phi_{f}\rangle$ given by Eq. (II). The probability distribution of the
$x$-quadrature measurement is given by $P_{f}(x)=|\langle
x|\Phi_{f}\rangle|^{2}$, which yields

$\displaystyle P_{f}(x)=|u_{1}\alpha_{1}(x)+u_{2}\alpha_{2}(x)|^{2}/p_{a}\,.$
(14)

The two coefficients introduced here read as
$u_{1}=\cos\frac{\theta_{i}}{2}\cos\frac{\theta_{f}}{2}$ and
$u_{2}=\sin\frac{\theta_{i}}{2}\sin\frac{\theta_{f}}{2}e^{i\phi_{0}}$. The two
wavefunctions of the coherent states in this result are given by
$\alpha_{1,2}(x)=\langle x|\alpha e^{\mp i\lambda}\rangle$, which can be
explicitly obtained using the result of Eq. (11). Substituting $P_{f}(x)$ into
the formula used above in Eq. (13), one can numerically compute the Fisher
information associated with the $x$-quadrature measurement, for arbitrary
post-selection options and for arbitrary interaction strengths.

###  III.3 Numerical results

In Fig. 3,

we compare the WVA-FI associated with the photon-number and the $x$-quadrature
measurements, $p_{a}F^{(n)}_{a}$ and $p_{a}F^{(x)}_{a}$, and also the WVA-QFI
$p_{a}Q_{a}$. First, we find that both measurement schemes can reach similar
precision, provided that the local oscillator’s phase $\varphi$ in the
$x$-quadrature measurement is properly chosen. The detailed behaviors of the
phase dependence are shown in Fig. 4. We emphasize that the comparison between
the photon-number and the $x$-quadrature measurements is possible only within
the WVA scheme, since the photon-number measurement does not make sense in the
absence of post-selection. As expected, we find that both $p_{a}F^{(n)}_{a}$
and $p_{a}F^{(x)}_{a}$ are in general smaller than the QFI $p_{a}Q_{a}$. But
at some special post-selection angles, e.g., at $\theta_{f}=3\pi/2$, both can
reach the maximum value of the QFI. In this context, we also mention that the
FI and QFI are periodic functions of $\theta_{f}$ with period of $2\pi$. Here
we only show the results of a single period.

Figure 3:  The WVA-FI $p_{a}F_{a}^{(n,x)}$ (solid purple and dotted blue),
associated with the photon-number and $x$-quadrature measurements, are
compared with the WVA-QFI $p_{a}Q_{a}$ (dash-dotted red) and the QFI $Q_{\rm
cm}$ of conventional metrology (dashed green). The displayed $\theta_{f}$
dependence is a periodic function with period of $2\pi$. Within a single
period, the changing behaviors are shown for a few interaction strengths:
$\lambda=0.01$ (a); 0.05 (b); 0.1 (c); and 1 (d). Parameters used for the
numerical calculations are: $\theta_{i}=\frac{\pi}{2}$, $\phi_{0}=\pi$, and
$\alpha=2$ (thus the average photon number $\bar{n}=|\alpha|^{2}=4$).  Figure
4:  Dependence of the WVA-FI on the local oscillator’s phase $\varphi$ in the
homodyne $x$-quadrature measurement, after post-selection with a few angles:
$\theta_{f}=\frac{\pi}{2}$ (a); $\pi$ (b); and $\frac{3\pi}{2}$ (c). The inset
in (a) is an enlarged display for the regime of small FI. Results for
interaction strengths $\lambda=0.05$, 0.1, and 1 are shown respectively by the
dashed red, solid green, and dash-dotted blue curves. Other parameters used in
the calculation are: $\theta_{i}=\frac{\pi}{2}$, $\phi_{0}=\pi$, and
$\bar{n}=|\alpha|^{2}=4$.

Second, we notice that the post-selection ($\theta_{f}$) dependence behavior
varies drastically with the change of the interaction strength parameter
$\lambda$. Actually, this is true only in the small $\lambda$ regime. For
larger $\lambda$, i.e., the plateau area in Fig. 2, the $\lambda$ dependence
will become insensitive. Very importantly, one finds that, with the increase
of the interaction strength $\lambda$, both the QFI $p_{a}Q_{a}$ and the FI
$p_{a}F_{a}^{(n,x)}$ associated with the WVA scheme can exceed the QFI $Q_{\rm
cm}$ of the conventional metrology scheme, as clearly observed in Fig. 3(d).
For broader range of parameters, this conclusion can be anticipated as well,
based on the result in Fig. 2.

This important observation is beyond the claim debated for long time among the
WVA community Nish12 ; Ked12 ; Jor14 ; Li20 ; Tana13 ; FC14 ; Kne14 ; ZLJ15 ;
Aha15 . That is, it was generally believed that, from either the perspective
of Fisher information or the signal-to-noise ratio, the intrinsic overall
estimate precision cannot be enhanced by the WVA technique. The basic reason
is that, despite the enhanced signal after post-selection, the procedure
itself of post-selection will discard a large number of measurement data and
thus result in a larger uncertainty fluctuations. Therefore, the WVA technique
was conceived of having, at least, no theoretical advantages Tana13 ; FC14 ;
Kne14 ; ZLJ15 , but only having some technical advantages in practice such as
not being bounded by the saturation limit of photo-detectors Lun17 ; ZLJ20 .

We notice that the analysis in Ref. Jor14  (and in many other references) was
based on a Gaussian meter state, i.e., the transverse spatial wavefunction of
the light beam in the optical realization. Here, our analysis above is fully
along the line in Ref. Jor14 . The only difference is that here the meter is
an optical coherent state. Therefore, the claim that the WVA scheme cannot
exceed the conventional approach does not hold in general, which is not a
result imposed by any fundamental principle. The present observation in Fig.
3(d) is an important exception: it clearly shows that the WVA metrology scheme
can reach a precision much better than the conventional approach in the
absence of post-selection.

##  IV Mixed Coherent States

In the above analysis,

we employed the tool of Fisher information, which is connected with the
parameter estimate precision via the well known Cramér-Rao inequality, to
discuss the WVA scheme in comparison with the conventional approach (without
post-selection). In contrast to the conclusion based on the Gaussian meter
state as typically analyzed in literature Nish12 ; Ked12 ; Jor14 ; Li20 ;
Tana13 ; FC14 ; Kne14 , we revealed an important metrological advantage of the
optical coherent state if the strategy of post-selection is involved.

In quantum precision metrology, another important criterion is in concern with
resource consuming, e.g., by considering the (average) photon-number
($\bar{n}=|\alpha|^{2}$) dependence to be standard-quantum-limit (SQL
$\propto\bar{n}$) or Heisenberg-limit (HL $\propto\bar{n}^{2}$) scaling
behavior. Usually, the HL scaling can be achieved by exploiting the quantum
entanglement of photon pairs Dow08 ; Mac11 ; Ste14 . Therefore, an interesting
question is: can the HL precision of the $\bar{n}^{2}$-scaling be achieved by
the post-selection strategy, without using any entangled photon pairs?

For optical coherent states, unfortunately, it was found that the FI gained by
the post-selected results of measurement scales with $\bar{n}$ linearly ZLJ15
. This means that only the precision of the SQL can be achieved by the WVA
technique. However, when applying the procedure of post-selection, it was
found that the FI contained in the distribution probability of the post-
selection results is possible to achieve the HL of $\bar{n}^{2}$-scaling ZLJ15
. This issue was further highlighted and analyzed in detail by Jordan and
Aharonov et al in Ref. Aha15 .

Remarkably, rather than using the single (pure) coherent state as the meter
state, it was found that, if using a statistical mixture of coherent states,
the HL of $\bar{n}^{2}$-scaling can be achieved by the WVA measurement LCF18 .
In the remainder of our present work, we would like to extend the study from
the AAV limit of extremely weak measurement, as assumed in Ref. LCF18 , to
arbitrary strength of measurement interaction. We will show that, with the
increase of the measurement interaction strength, the HL precision (and even
better result) can be achieved in the large $\bar{n}$ regime. Moreover, for
using classical mixed states, we will make a connection with the contradictory
statement proved in Ref. Lui10 , showing that the strategy of post-selection
may cause remarkable impact on some existing conclusions and make important
addition to wider range of problems Lui08 ; Sil10 .

The statistical mixture of multiple coherent states for the optical meter can
be expressed in general as $\rho_{\rm
class}=\sum_{j}w_{j}|\alpha_{j}\rangle\langle\alpha_{j}|$, with
$|\alpha_{j}\rangle$ the individual coherent states and $w_{j}$ the weighting
factors. In parallel to the formulation for the case of the pure state, the
measurement interaction between the system and the meter is described by a
unitary transformation to the initial product state,
$\rho_{T}=U(\rho_{s}\otimes\rho_{\rm class})U^{\dagger}$, where
$U=e^{i\lambda\hat{\sigma}_{z}\hat{n}}$. After post-selecting the system state
with $|\psi_{f}\rangle$, the meter state reads
$\rho_{f}=\langle\psi_{f}|\rho_{T}|\psi_{f}\rangle/\tilde{p}_{a}\equiv\tilde{\rho}_{f}/\tilde{p}_{a}$.
Now the probability of successful post-selection is given by
$\tilde{p}_{a}={\rm Tr}_{M}[\tilde{\rho}_{f}]$, where ${\rm Tr}_{M}(\cdots)$
means the trace over a complete set of the meter states. Then, the photon-
number measurement on the meter state gives the distribution function
$P_{f}(n)=\langle n|\rho_{f}|n\rangle$, and the Fisher information can be
computed straightforwardly.

To be specific, let us consider a simple example of mixed state as

$\displaystyle\rho_{\rm
class}=\frac{1}{3}(|\alpha\rangle\langle\alpha|+|{\alpha+\delta}\rangle\langle{\alpha+\delta}|+|{\alpha-\delta}\rangle\langle{\alpha-\delta}|)$
(15)

Straightforwardly, one can compute the average photon number as $\bar{n}={\rm
Tr}_{M}(\hat{n}\rho_{\rm class})=|\alpha|^{2}+\frac{2}{3}|\delta|^{2}$, and
the variance of the number fluctuation as

$\displaystyle\left(\Delta
n\right)^{2}=\overline{n^{2}}-\left(\bar{n}\right)^{2}$
$\displaystyle=\frac{2}{9}{\lvert\delta\rvert}^{4}+\frac{4}{3}\left[|\alpha\delta|^{2}+{\rm
Re}(\alpha^{2}\delta^{\ast 2})\right]+\bar{n}\,.$ (16)

To gain a preliminary insight, let us assume $|\delta|>>|\alpha|$. Then, we
have $\bar{n}\simeq\frac{2}{3}|\delta|^{2}$, and $(\delta
n)^{2}\simeq\frac{2}{9}|\delta|^{4}\sim\bar{n}^{2}$. In Ref. LCF18 , being
restricted in the AAV limit, it was proved that the FI is directly related
with the variance of the photon numbers of the meter state initially prepared,
which simply reads $F_{a}^{(n)}=4({\rm Im}\sigma^{w}_{z})^{2}(\Delta n)^{2}$,
where
$\sigma^{w}_{z}=\langle\psi_{f}|\sigma_{z}|\psi_{i}\rangle/\langle\psi_{f}|\psi_{i}\rangle$
is the famous expression of the AAV weak values. One obtains thus a
metrological precision of the Heisenberg $\bar{n}^{2}$-scaling.

For an arbitrary strength of measurement interaction beyond the AAV limit,
after post-selection with $|\psi_{f}\rangle$ to the system state, the photon-
number measurement on the meter state would yield the probability distribution

$\displaystyle
P_{f}(n)=\frac{K_{n}}{3\tilde{p}_{a}}\,[p_{n}(\alpha)+p_{n}(\alpha+\delta)+p_{n}(\alpha-\delta)]\,.$
(17)

Here we have introduced the function $p_{n}(y)=e^{-|y|^{2}}|y|^{2n}/n!$ and
the pre-factor

$\displaystyle
K_{n}=[1+\cos\theta_{i}\cos\theta_{f}+\sin\theta_{i}\sin\theta_{f}\cos(2\lambda
n+\phi_{0})]/2\,.$

The normalization factor is given by
$\tilde{p}_{a}=\frac{1}{3}\sum^{3}_{j=1}p_{a}(\bar{n}_{j})$, where
$\bar{n}_{1}=|\alpha|^{2}$ and $\bar{n}_{2,3}=|\alpha\pm\delta|^{2}$, while
the $p_{a}$ function reads as

$\displaystyle
p_{a}(\bar{n}_{j})=\frac{1}{2}[\,1+\cos\theta_{i}\cos\theta_{f}$
$\displaystyle~{}~{}~{}~{}+\sin\theta_{i}\sin\theta_{f}\,e^{-2\bar{n}_{j}\sin^{2}\lambda}\cos(\bar{n}_{j}\sin
2\lambda+\phi_{0})\,]\,.$

Using the probability distribution given by Eq. (17), the FI can be
straightforwardly calculated.

Figure 5:  The WVA-FI $p_{a}F_{a}^{(n)}$ associated with photon-number
measurement of mixed meter states. The exact numerical results are compared
with the approximated ones under the AAV limit, for a few interaction
strengths: $\lambda=10^{-4}$ (a); $10^{-3}$ (b); and $10^{-2}$ (c). The QFI
$p_{a}Q_{a}$ for a pure state with the same average photon number $\bar{n}$ is
displayed, showing that the classical mixed state can work better than the
quantum pure state if employing the strategy of post-selection. Parameters
used for calculation: $\theta_{i}=\frac{\pi}{2}$, $\phi_{0}=\frac{\pi}{2}$,
$\alpha=0.1$ and $\bar{n}=3$.  Figure 6:  $\bar{n}$-scaling behavior of the
WVA-FI associated with the photon-number measurement. Results for the
interaction strengths $\lambda=10^{-3}$ and $10^{-2}$ are shown in (a) and
(b), and a comparison between the results of the mixed and pure states is
plotted. In the large $\bar{n}$ regime, the HL scaling (and even beyond) can
be achieved by using the mixed state, while using a pure state can largely
result in the SQL scaling behavior. Parameters used for calculation:
$\theta_{i}=\frac{\pi}{2}$, $\theta_{f}=\frac{\pi}{2}$,
$\phi_{0}=\frac{\pi}{2}$, and $\alpha=0.1$.

In Fig. 5 we compare the WVA-FI between utilizing mixed and pure states.
Results for the former are computed by considering the photon-number
measurement, while results for the latter are the optimal QFI $p_{a}Q_{a}$ for
a pure state with identical average photon numbers $\bar{n}$. We see clearly
that the classical mixed state can work better than the quantum pure state
after applying the post-selection strategy. To demonstrate the effect of
generalization beyond the AAV limit, we compare the exact numerical results
with the ones from the approximated expression under the AAV limit, for a few
interaction strengths. With the increase of the strength parameter $\lambda$,
greater deviation will occur (not shown in the figure).

The important message got here is that, in the large $\bar{n}$ regime, the
mixed state can work much better than a pure state with the same average
photon numbers. However, in Ref. Lui10 , it was proved that, within the
linear-coupling detection schemes (just as considered in this work), using any
classical states $\rho_{\rm class}$ (i.e. statistical mixture of coherent
states) cannot surpass the metrology precision by using a quantum pure state
$|\alpha_{\rho}\rangle$ with the same $\bar{n}$. This significant difference
should originate from the procedure of post-selection in the WVA scheme, since
in Ref. Lui10  only the standard measurement scheme (i.e. without post-
selection) was considered. Therefore, we can newly claim that, by means of
involving the post-selection strategy as involved in the WVA scheme, rather
than what was claimed in Ref. Lui10 , properly designed classical mixed states
can work much better than the quantum pure states with the same resource (i.e.
the same average photon numbers).

Making a connection between the result revealed in Fig. 5 and the statement
proved in Ref. Lui10  is quite constructive, which is also relevant to a wider
range of problems Lui08 ; Sil10 , i.e., achieving high visibility and super-
resolution in the absence of quantum resources (e.g. quantum entanglement). It
seems that the simple procedure of post-selection involved in the WVA scheme
can make an important addition to some conclusions in the field of quantum
metrology, and deeper understanding needs further investigations.

In Fig. 6 we reveal the $\bar{n}$-scaling behavior of the Fisher information
associated with photon-number measurement within the WVA scheme. Results for
the interaction strengths $\lambda=10^{-3}$ and $10^{-2}$ are shown in Fig.
6(a) and (b), and a comparison between the results from the mixed and pure
states is plotted. In the large $\bar{n}$ regime, as indicated by the slope
$k=2$ or 1 of the fitting lines, the HL scaling (and beyond) can be achieved
by using the mixed state, while using a pure state can largely result in the
SQL scaling behavior. We emphasize that the HL scaling behavior revealed here
for the mixed meter state is also induced by the post-selection procedure in
the WVA scheme, since it was proved in Ref. Lui10  that, for linear coupling
interaction, any classical mixture of the coherent states cannot work better
than a single pure coherent state (under the condition with identical average
photon numbers), while the pure state can only result in the SQL scaling (as
indicated in Fig. 6 by the slope $k=1$).

For the “super-Heisenberg” scaling behavior observed in Fig. 6(b), i.e., the
scaling better than $k>2$, we may provide a simple understanding as follows,
by attributing to violating the AAV limit. Let us denote the photon-number
distribution function for the mixed state as
$\bar{p}(n)=\frac{1}{J}\sum^{J}_{j=1}p_{n}(\alpha_{j})$, with
$p_{n}(\alpha_{j})$ the same function as defined in Eq. (17). In the small
$\lambda$ limit, the probability function of photon-numbers measured after
post-selection can be approximately expressed as

$\displaystyle
P_{f}(n)=\frac{K_{n}}{\tilde{p}}_{a}\,\bar{p}_{(}n)\simeq\left[1-2\lambda{\rm
Im}\sigma_{z}^{w}(n-\bar{n})\right]\,\bar{p}_{n}\,.$ (18)

Then, the FI associated with this distribution function can be simply obtained
as

$\displaystyle F_{a}^{(n)}=4({\rm
Im}\sigma_{z}^{w})^{2}\sum_{n}\frac{(n-\bar{n})^{2}}{1-2\lambda{\rm
Im}\sigma_{z}^{w}(n-\bar{n})}\bar{p}_{n}\,.$ (19)

Based on this expression, if neglecting the small $\lambda$ term in the
denominator, we obtain $F_{a}^{(n)}\simeq 4({\rm
Im}\sigma_{z}^{w})^{2}\langle(n-\bar{n})^{2}\rangle$, where the statistical
average $\langle\cdots\rangle=\sum_{n}(\cdots)\bar{p}_{n}$ is introduced. This
result means that the photon-number scaling behavior is determined by the
initial state of the meter field, and the HL scaling will be achieved as
pointed out below Eq. (IV). However, if we keep the series expansion of
$(1-z)^{-1}=1+z+z^{2}/2!+\cdots$ to higher orders, e.g., keeping the linear
term, we will have an additional term of contribution
$\propto\langle(n-\bar{n})^{3}\rangle$, which makes the scaling better than
the HL as observed in Fig. 6(b). Moreover, we have checked the $\bar{n}$
dependent behavior for the probability $\tilde{p}_{a}$ of successful post-
selection, which can be approximately fitted as $\tilde{p}_{a}=d-b\bar{n}$,
with the slope $b>0$. We find that $b$ is very small for $\lambda=10^{-3}$,
and becomes larger for $\lambda=10^{-2}$. This additional observation also
supports the above explanation for the “super-Heisenberg” scaling behavior
shown in Fig. 6(b). It is indeed owing to the violation of the AAV limit.
Violation of the AAV limit also explains the scaling result better than the
SQL (with slope $k=1$), for the pure state shown in Fig. 6(b).

##  V Summary and Discussion

We have considered

the WVA precision metrology in the case of using the optical coherent state as
a probe meter. We calculated the WVA-FI associated with two schemes of
measuring the meter’s field, after post-selection to the system state. The
first scheme is the photon-number measurement as usually considered; and the
second scheme is the field-quadrature measurement by means of homodyne
detection, which seems rarely discussed in the WVA community. The field-
quadrature measurement is necessarily required by considering the conventional
scheme (not involving the strategy of post-selection), since the photon-number
measurement cannot extract the quantum Fisher information encoded in the
phase-shifted coherent state. The main important results can be briefly
summarized as: (i) With the increase of the measurement coupling strength to
violate the AAV limit, the WVA scheme can definitely outperform the
conventional approach; (ii) the WVA scheme can make a mixture of coherent
states work better than a pure coherent state with identical average photon
numbers; (iii) the WVA scheme can result in the precision of Heisenberg (or
even “super-Heisenberg”) scaling with the photon numbers, but without using
any expensive quantum resources.

We notice that all the above unexpected results are rooted in both the post-
selection procedure in the WVA scheme and the use of optical coherent states
as probe meter. For the above result (i), when considering Gaussian meter
states, the conclusion was very different, which has caused extensively
controversial debates in literature. The Gaussian meter state corresponds to
such as the transverse spatial wavefunction in the Stern-Garlach setup, or the
one of a light beam in experiments. In the case of optical coherent states,
which are quite relevant to the optical cavity-QED system or solid-state
circuit-QED architecture, the above concluding result (i) should be much more
decisive. For the results (ii) and (iii), making a connection with the
statement proved in Ref. Lui10  is quite constructive, which is also relevant
to a wider range of problems Lui08 ; Sil10 , i.e., achieving super-resolution
in the absence of quantum resources. Noting that the different conclusion in
Ref. Lui10  was proved in the absence of post-selection procedure, we may
expect that the simple procedure of post-selection involved in the WVA scheme
can drastically change some conclusions in the field of quantum metrology, and
deeper investigations are needed in future studies.

Acknowledgements.

— This work was supported by the National Key Research and Development Program
of China (No. 2017YFA0303304) and the NNSF of China (Nos. 11675016, 11974011 &
61905174).

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# Open Source Prover in the Attic

Zoltán Kovács The Private University College of Education of the Diocese of
Linz, Austria<EMAIL_ADDRESS>The Private University College of
Education of the Diocese of Linz, Austria Alexander Vujic The Private
University College of Education of the Diocese of Linz, Austria
<EMAIL_ADDRESS>
###### Abstract

The well known JGEX program became open source a few years ago, but seemingly,
further development of the program can only be done without the original
authors. In our project, we are looking at whether it is possible to continue
such a large project as a newcomer without the involvement of the original
authors. Is there a way to internationalize, fix bugs, improve the code base,
add new features? In other words, to save a relic found in the attic and
polish it into a useful everyday tool.

Keywords: JGEX, Java development, open source, education, Dynamics Geometry
Systems.

##  1 Introduction

Wide access to computers in education opens a new horizon in applying
automated reasoning tools in the classroom. Many software tools exist already,
that are of different levels of maturity, and several issues can be found that
prevent direct use of them in education. One of the most mature tools is Java
Geometry Expert (JGEX, [14, 12, 13]), which has its roots in Wichita State
University and was developed by the automated geometry reasoning school, led
by Chou and Gao, and programmed (mostly) by Ye.

JGEX is a remarkable summary of several decades of pioneering work [3]. It
consists of more than 108,000 lines of Java code in more than 230 files. It
has been open-sourced, however, just a couple of years ago (in 2016), and it
was published on GitHub.111https://github.com/yezheng1981/Java-Geometry-Expert
Since then, more than 80 forks were created and the project was starred by
more than 400 users. These numbers clearly show the popularity of JGEX, even
if the original authors discontinued their work: Ye’s latest (very minor)
change was committed in July 2018. Therefore, in our work, we raise the
question, if it is still possible to contribute to this project significantly.

Why are further contributions necessary? First of all, the software tool
supports only a couple of languages. For educational use, a translation for
native speakers seems unavoidable: young learners may have difficulties with
languages. Second, a modern user interface may be more appealing for the new
generation of users. Third, science is clearly at a more mature state-of-the-
art now, so some updates in the used algorithms seem important after a while.
Fourth, it turned out that some minor, annoying bugs still exist in the
program – they should be fixed someday: it is very important for a tool that
is expected to be error-free in all matters when it is about theorem proving.

In our paper we discuss the possibility of continuing an existing tool for
automated proving without the possibility to contact the original authors. Our
discussion focuses on translating JGEX into German and Serbian, and on
extending its capabilities to provide a better user experience. Also, we
discuss how difficult it is to build the program with the newest development
tools available, and what changes have to be done to be somewhat up-to-date in
this concern.

We think this research is important for a whole community, in particular, to
save the achieved algorithms and implementations for the next generation. In
other words, our research is a survival guide which aims at supporting the
community for automated geometry reasoning.

##  2 Expectations

In this section we give an overview of the expected updates and further
improvements.

###  2.1 Translations

JGEX’s final mainstream version supports the Chinese, English, Italian and
Persian languages. Our research team, after forking the mainstream version in
the repository,222https://github.com/kovzol/Java-Geometry-Expert decided to
add further translations: German, Portuguese and Serbian.

It turned out that the translation, technically speaking, is a simple process
of creating a text file, similar to the CSV (comma separated values) format,
that consists of a text ID integer, the English keyword (because in most cases
it is used for the lookup, instead of the ID), the translated text, and
(optionally) a tooltip. With minor edits, a new text file can easily be added
to the source files and it can be inserted in the menu system.

Unfortunately, it also turned out that the tooltip entries are completely
missing for all languages (except for English, but those translations are
already given in the Java source code). On the other hand, some translations
are completely missing by design: such expressions are short phrases like “by
HYP” (which means: “by assumption”, “by hypothesis”) and “because”. We had to
add such phrases in the CSV databases, and modify the Java source code to
search for the phrases also in the CSV files.

Also, two additional lists of phrases are hardcoded in the original source
code. They belong to the Geometry Deduction Database (GDD, [13, 4]) and the
rule database for the Full Angle method [3]. These are given only in English.
It is an ongoing work to move these phrases in the CSV database, instead of
using the Java files to store the English version, and, in our fork, the
German and Serbian translations as well.

Why are all translations stored in one CSV database? There would be a more
efficient way of translating the user interface into different languages.
Nevertheless, a variety of languages have already been translated in the CSV
database, therefore it was more accessible for the team to continue working
with CSV. In our case, the most extensive work was dedicated to the Serbian
translation. Our translation team, led by the second author, started to make a
copy of the German translation (because of the better knowledge of German than
English). After finishing translation of the phrases in the CSV file, it
turned out that, in addition, both rule databases need to be translated. Then,
some additional phrases were found in the Java source code that had to be
translated too. To make a quality work, the translated phrases had to be
double-checked by native speakers who were also experts in geometry.
Unfortunately, by doing the translation in three different steps, every
participant had to work three times: once for the CSV entries, once for the
rule databases, and again for the unexpected phrases shown up randomly in the
source code. This made the work quite inefficient and unexpectedly long.

Overall, it would have been better to unify the translation system first, and
only after then start a concentrated work on doing the real effort in
translating the phrases. But this was, actually, not really possible: JGEX has
no full documentation, and it cannot be therefore assumed that any users
(either newcomers or experienced ones) have a complete overview of all phrases
used in the program.

We also learned that in many cases, unfortunately, the English translation
contains spelling errors. As a consequence, the translation keys have spelling
problems too. This makes it difficult to modify all erroneous translation keys
in all CSV files and the Java source code at the same time.

In fact, there are sophisticated translation systems like gettext. We will
consider changing the translation system in a next round of possible updates.

###  2.2 Modern Interface

First author invited a group of prospective mathematics teachers at the
Private University College of Education of the Diocese of Linz (PHDL) to give
some feedback on the usability of JGEX. While most feedback was positive,
several students mentioned that JGEX had an unusual user interface, and an
old-fashioned look. Many features were difficult to find. Among others, to
obtain a proof that is easy to understand in the classroom, the user needed to
find an appropriate example in the database of showcases. The general method
of showing the proof is to fix the objects and to select the relevant one from
them, and use the right mouse click to access the proof. In certain cases,
such as “08_9point.gex” the students may access the proof by right clicking on
“SHOW: CYCLIC D G E F” and selecting “Prove”.

To solve such difficulties, several ideas were suggested by the students, and
the authors of this paper. One simple possibility is to create tutorials in
both textual and video format. On the other hand, a simplified view of the
most useful capabilities for classroom use could be helpful. Here we refer to
another software package, OK Geometry [8], which comes in different editions
(Easy, Basic, Plus modes). Also, a more extensive use of tooltips could tear
down some barriers. Another option is to force the user into a workflow:
first, she had to create or load a construction, second, point to the searched
properties, and third, to obtain the proof.

Today’s users are more familiar with web applications and mobile phone apps
than native applications. In fact, the original version of JGEX comes with no
installer: the user needs to download the source code, and it is her own task
to compile and run it with an appropriate Java Development Kit (JDK) and Java
Runtime Environment (JRE). First author maintains a ZIP package for Windows
and Mac systems (based on the packr utility333https://github.com/libgdx/packr)
by providing all required files to run JGEX without any further
preparation.444https://github.com/kovzol/Java-Geometry-Expert/releases Also, a
Linux Snap version of JGEX is also available.555https://snapcraft.io/jgex As
of May 2023 there are 175 installations world-wide registered from 48
territories. Anyway, these packages are just one step towards simplifying the
access to JGEX – we think it is unavoidable to make JGEX available as an
embeddable web application by compiling its codebase to JavaScript.

In fact, Ye’s version of JGEX does not even compile automatically when the
newest Java version is used. Some minor updates are required on the source
code: Either the Java version must be downgraded to 8, or some small changes
need to be done to avoid compiler errors. These requirements have been,
luckily, also identified by other contributors who forked JGEX. On the other
hand, the Java technology, selected by the Chinese experts, has proven to be a
good choice, because with just minor modifications, it is easy to import the
project into today’s favorite Java Integrated Desktop Environment (IDE),
IntelliJ IDEA.

Finally, we think it is unavoidable to make it possible to import GeoGebra [5]
figures in JGEX, and to export them in GeoGebra format, since GeoGebra became
the de facto standard of dynamic geometry during the last decade [11]. Even if
JGEX supports a wide set of drawing tools, for newcomers, it can be difficult
to learn its toolset quickly enough.

###  2.3 State-of-the-art Mathematics

JGEX implements several mathematical algorithms. One of the supported
algorithms is the Gröbner basis method [6] that is known to be slower than the
other methods, in a substantial set of input cases. Meanwhile, however, major
speedups have been reported in some implementations of computing Gröbner
bases. One of the successful implementations is included in GeoGebra’s
embedded computer algebra system Giac [7].

In fact, the Gröbner basis method, when using elimination, is known to provide
better non-degeneracy conditions than the faster algebraic method, Wu’s
approach. This result could also be incorporated into JGEX. It’s important to
note that in many cases, the degeneracy conditions output by Wu’s method are
necessary for the theorems to hold, and they help distinguish the cases when
the theorem is true and when it is false. This result could also be
incorporated into JGEX.

###  2.4 Fixing Bugs and Adding Improvements

Even for mature software tools, there is always room to improve minor
problems. Among other minor issues, the proof protocol shown for the GDD
method, could be improved by reducing the number of output lines and showing
the hierarchy of the proof in addition. We show this concept below.

We take the example 1_TOP_TEN/08_9point: It constructs a figure to illustrate
the nine-point circle theorem (Figure 1). We assume a classroom situation to
get a readable proof of the fact that the three midpoints (E, F, G) of an
arbitrary triangle ABC and a perpendicular foot point D (of vertex A,
projected on side BC) are concyclic.

Figure 1: JGEX proves a part of the nine-point theorem by using the GDD
method. Its user interface is set to German.

Now, by issuing some improvements on the source code (further
details666https://github.com/kovzol/Java-Geometry-
Expert/commit/bc91b9ec916f97e38a100c08bec5bfda0c49de8d), we can communicate
the proof in a simpler way (Figure 2).

Figure 2: The output of the GDD method is visualized as a tree.

In fact, the change being performed to get this improvement is quite simple.
Of course, one needs to have a deeper knowledge in the Java language and
eventually in the mathematical background to achieve such changes in a
feasible time. But, we can report that it is possible, and it does not require
a high amount of time. Thus, a follow-up and continuation of the stopped work
seems more than possible

Here we highlight that a tree structure for the obtained GDD proofs cannot
always be accomplished. For example, Figure 3 shows the example
1_TOP_TEN/10_5cir which states the concyclicity of points M1, M2, M3 and M4 in
the given figure. Here we can learn that node 20 is used in nodes 13 and 15
too, so the whole subtree of node 20 is repeated in the subtrees of nodes 13
and 15. The presence of this duplication may be inconvenient for some users.
Therefore, an option to switch forcing the structure off, could be a solution,
or, even better, a different way of visualization might be applied. One
possibility to do that is the embedding of the GraphViz
library,777https://graphviz.org/ to provide a professional look of the outline
of the proof structure (Figure 4).

Figure 3: The GDD method finds a proof that is not a tree. The interface is
set to German and the connecting edges of the graph are not shown.

Structured proofs may be beneficial when explaining the proof steps directly
in the classroom. Of course, proofs that consist of a large number of steps,
may be inappropriate for most students, but rather for gifted learners, mostly
as preparations or training exercises for mathematical contests. Even so,
providing a structured view of the proofs obtained by the GDD algorithm seems
to be a substantial improvement for many classroom situations.

Figure 4: A better visualization of the GDD proof by using GraphViz.

##  3 Conclusion

JGEX is an eminent software tool that summarizes several decades of pioneer
work. Unfortunately, its main authors no longer maintain the code base. In our
research we studied the question if it is possible to continue their work in
some sense, and forward their legacy to the new generations.

According to our case study, we think that such a follow-up is possible,
however, it is not straightforward. Several problems, mostly technical ones,
can occur. The free availability of the source code is, however, a great help.
With enough time and patience, the original version can be extended in a
direction that seems fruitful for the long term.

We need to mention the quality of the code base. Unfortunately, the number of
comments is very low. On the other hand, the variable names and the naming
system for the Java methods are quite straightforward. That is, with little
work, it is possible to learn the internals of the Java source code. Some
parts of the code, however, require a major restructuring. Among others, the
way of how the translations are handled, needs to be improved significantly.
Also, some Java coding standards like capitalization of the source files, seem
to be ignored. Nevertheless, by using modern refactoring tools, these issues
could be solved with minimal efforts.

We remark that a general inspection process in IntelliJ IDEA 2023.1.2 found
269 errors, 9645 warnings, 430 weak warnings, 181 grammar errors and 16449
typos in JGEX’s code base. Even if many of the items of such reports can be of
a matter of taste and coding style, a detailed study of these messages seems
unavoidable for the long term.

Some recent work [2, 9, 10] confirm that continuing the pioneer work being put
in JGEX is an important step towards the more general use of automated
reasoning in the classroom. One possibility is to copy the algorithms and
modify them accordingly (this way was chosen by Baeta and Quaresma, using
C++), but another option (we chose this) is to use the original source code
and do the modifications on it directly.

As a final conclusion, we think it is possible to save the heritage of the
Chinese experts, and continue the hard work of popularizing JGEX and extending
the user community of automated deduction in geometry.

##  4 Acknowledgments

The first author was partially supported by a grant PID2020-113192GB-I00
(Mathematical Visualization: Foundations, Algorithms and Applications) from
the Spanish MICINN.

We acknowledge the kind support of translators Benedek Kovács, Alexander
Thaller, Engelbert Zeintl (German), Jorge Cassio (Portuguese), Amela Hota,
Predrag Janičić and Jelena Marković (Serbian).

We sincerely appreciate Amela Hota for her support in crafting a presentation
of a preliminary version of this paper.

## References

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  * [2] N. Baeta & P. Quaresma (2023): _Towards a Geometry Deductive Database Prover_.  Annals of Mathematics and Artificial Intelligence, 10.1007/s10472-023-09839-0.
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  * [4] S.C. Chou, X.S. Gao & J.Z. Zhang (2000): _A Deductive Database Approach to Automated Geometry Theorem Proving and Discovering_.  Journal of Automated Reasoning 25, pp. 219–246, 10.1023/A:1006171315513.
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  * [12] Z. Ye, S.C. Chou & X.S. Gao (2010): _Visually Dynamic Presentation of Proofs in Plane Geometry, Part 1: Basic Features and the Manual Input Method_.  Journal of Automated Reasoning 45, pp. 213–241, 10.1007/s10817-009-9162-5.
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# An atomic clock with ${10^{-18}}$ instability

N. Hinkley,1,2 J. A. Sherman,1 N. B. Phillips,1 M. Schioppo,1
N. D. Lemke,1 K. Beloy,1 M. Pizzocaro,1,3,4 C. W. Oates,1 A. D. Ludlow1∗

1National Institute of Standards and Technology, 325 Broadway, Boulder,
Colorado 80305, USA
2University of Colorado, Department of Physics, Boulder, Colorado 80309, USA
3Instituto Nazionale di Ricerca Metrologica, Str. delle Cacce 91, 10135
Torino, Italy
4Politecnico di Torino, Corso duca degli Abruzzi 24, 10125 Torino, Italy
∗Corresponding author. E-mail<EMAIL_ADDRESS>
> Atomic clocks have been transformational in science and technology, leading
> to innovations such as global positioning, advanced communications, and
> tests of fundamental constant variation. Next-generation optical atomic
> clocks can extend the capability of these timekeepers, where researchers
> have long aspired toward measurement precision at 1 part in $\bm{10^{18}}$.
> This milestone will enable a second revolution of new timing applications
> such as relativistic geodesy, enhanced Earth- and space-based navigation and
> telescopy, and new tests on physics beyond the Standard Model. Here, we
> describe the development and operation of two optical lattice clocks, both
> utilizing spin-polarized, ultracold atomic ytterbium. A measurement
> comparing these systems demonstrates an unprecedented atomic clock
> instability of $\bm{1.6\times 10^{-18}}$ after only $\bm{7}$ hours of
> averaging.

Periodic physical systems are the basis with which we establish standards of
time and frequency. Quantum mechanical absorbers like atoms serve as the best
available time and frequency references: they are isolatable, possess well-
defined transition frequencies, and exist in abundant identical copies. With
over 50 years of development, clocks based on microwave oscillators matched to
atomic transitions now define the SI _second_ and play central roles in
network synchronization, global positioning systems, and tests of fundamental
physics (?, ?). Under development worldwide, optical clocks oscillate $10^{5}$
times faster than their microwave predecessors, dividing time into finer
intervals (?). While microwave clocks, like Cs fountains, have demonstrated
time and frequency measurements of a few parts in $10^{16}$ (?, ?), optical
clocks are now measuring 1 part in $10^{17}$ (?, ?, ?, ?) and have long
aspired towards levels of 1 part in $10^{18}$. Here we demonstrate an optical
clock with measurement precision of $1.6$ parts in $10^{18}$.

A measurement at the $10^{-18}$ fractional level is equivalent to specifying
the age of the known universe to a precision of less than one second or
Earth’s diameter to less than the width of an atom. To illustrate the power of
this measurement capability, we first consider the gravitational redshift, a
consequence of general relativity dictating that clocks ‘tick’ more slowly in
stronger gravitational fields. This phenomenon has long been accounted for
when remotely comparing atomic clocks with gravitational elevations differing
by many meters or km. Instead, a clock with $10^{-18}$ performance can discern
changes in the gravitational geoid corresponding to 1 cm of elevation at
Earth’s surface. It can thus be exploited as a unique and sensitive probe of
spatial and temporal variations in the gravitational field (?, ?, ?, ?),
revolutionizing geodesic measurements and potentially impacting areas like
hydrology, geology, and climate change studies. Furthermore, alternative
gravitational theories predict deviations to the gravitational redshift
described by general relativity. Optical clocks at the $10^{-18}$ measurement
level, proposed for future space missions, can explore these deviations with
more than three orders of magnitude higher precision than presently possible
(?), as well as stringently test the universality of the gravitational
redshift (a consequence of local positional invariance) (?). Additionally,
some unification theories employing non-metric couplings of matter predict
variations of the fundamental constants of nature throughout space and time.
While these variations are thought to be small, the sensitivity of atomic
clocks to these constants (e.g., the fine structure constant) already provides
a useful constraint on possible variations with time or gravitational field
(?, ?, ?). A clock with $10^{-18}$ measurement capability offers up to two
orders of magnitude tighter constraint on these variations and the theories
predicting them, and may even provide insight into dark energy (?). Finally,
improved timekeepers directly benefit other applications such as navigation
systems, synchronization of telescope arrays (e.g., VLBI), secure
communication, interferometry, and possible redefinition of the SI second (?).

To illustrate an essential property of time and frequency standards, imagine a
pendulum clock. The periodicity of the pendulum swing provides the clock time
base. How this period (or the inverse, frequency) fluctuates over time is a
measure of the clock’s _instability_ , a characteristic of an oscillator
generally quantified by the Allan deviation (?). No time or frequency standard
can make a measurement better than the statistical precision set by its
instability. Further, the systematic uncertainty of an oscillator’s frequency
is often constrained by its long term instability (e.g., slow changes in
temperature and air pressure which perturb the pendulum swing). For these
reasons, and because many timing applications (including most mentioned above)
require only exquisite instability, the instability represents one—if not the
most—important property of an atomic standard.

In the pursuit of lower instability, one approach is particularly promising:
the optical lattice clock (?). In such a clock, a stabilized laser is
referenced to ultracold alkaline-earth (or similar) atoms confined in an
optical standing wave (i.e., lattice) created by a retro-reflected high-power
laser beam. Alignment of the clock interrogation laser along the direction of
tight lattice confinement eliminates most Doppler and motional effects while
probing the ultra-narrowband electronic ‘clock’ transition. Although the
optical lattice induces a Stark shift on the atoms’ electronic states, the net
effect can be nearly canceled by operation at the so-called ‘magic’
wavelength, $\lambda_{m}$, where both electronic states of the clock
transition are shifted equally (?, ?). A key advantage of the optical lattice
is that many ($10^{3}$ to $10^{6}$) atoms are confined in the lattice
potential. All of these atoms are interrogated simultaneously, thereby
improving the atomic detection signal-to-noise and thus the instability, which
is limited fundamentally by quantum projection noise (QPN) (?). Because
lattice clocks combine a large atom number with the ultra-narrow optical
transition, it has long been hoped that they could realize $10^{-18}$ time and
frequency measurement.

The first lattice clocks achieved $10^{-14}$ to $10^{-15}$ level performance
(?, ?, ?), and work has since progressed rapidly worldwide. An important
milestone was the demonstration of a lattice clock with short-term (1 s)
frequency instability in the low $10^{-15}$ levels, averaging down to low
$10^{-16}$ levels at many hundreds of seconds (?)—competitive with the best
clock results over similar timescales. Achievement of the lattice clock’s full
potential was hindered by a technical noise source known as the Dick effect,
which arises when an oscillator is only periodically observed (?, ?). (In the
pendulum clock analogy, imagine trying to discern the pendulum frequency while
the room lights flash on and off.) One approach to overcoming this
limitation—improving the ultra-stabilized laser used to interrogate the clock
transition—demonstrated clock instability well below $10^{-15}$ at short times
(?). Alternatively, synchronized interrogation of two lattice clocks with the
same stabilized laser exploited rejection of the Dick effect in common mode to
probe performance beyond the Dick limit. This technique produced a correlated
measurement instability of $4\times 10^{-16}$ at short times, approaching
$1\times 10^{-17}$ for averaging times greater than 1,000 s (?). Recently, an
uncorrelated comparison of two strontium lattice clocks revealed clock
instability of $3\times 10^{-16}$ at short times, averaging down to $1\times
10^{-17}$ in 1,000 s (?) or, in another case, reaching the $10^{-17}$ level in
20,000 s (?). Here, by comparing two independent optical lattice clocks using
ultracold ${}^{171}\mathrm{Yb}$, we demonstrate an unprecedented clock
instability of $1.6\times 10^{-18}$ in 25,000 s, a key milestone for
applications that have long sought this level of performance.

Both of the Yb lattice clock systems, referred to here as Yb-1 and Yb-2, cool
and collect ${}^{171}\mathrm{Yb}$ atoms from a thermal beam into magneto-
optical traps (see Fig. 1). Two stages of laser cooling, first on the strong
${}^{1}\\!S_{0}$-${}^{1}\\!P_{1}$ cycling transition at 399 nm, followed by
the weaker ${}^{1}\\!S_{0}$-${}^{3}\\!P_{1}$ intercombination transition at
556 nm, reduce the atomic temperature from 800 K to 10 $\mu$K. Each cold atom
sample is then loaded into an optical lattice with $\sim\\!\\!300~{}E_{r}$
trap depth (recoil energy $E_{r}/k_{B}=100$ nK) formed by retro-reflecting
approximately 600 mW of laser power, fixed at $\lambda_{m}\approx 759$ nm (?)
by a reference cavity. For the measurements described here, about 5,000 atoms
captured by each lattice are then optically pumped to one of the $m_{F}=\pm
1/2$ ground states using the ${}^{1}\\!S_{0}$-${}^{3}\\!P_{1}$ transition.
After this state preparation, applying a 140 ms long $\pi$-pulse of 578 nm
light resonant with the ${}^{1}\\!S_{0}$-${}^{3}\\!P_{0}$ clock transition
yields the spectroscopic lineshape shown in Fig. 1C, with a Fourier-limited
linewidth of 6 Hz. Experimental clock cycles alternately interrogate both
$m_{F}$ spin states for cancelation of first order Zeeman and vector Stark
shifts. The optical local oscillator (LO) is an ultra-stable laser source
servo-locked to a high-finesse optical cavity (?) and is shared by both Yb
systems. Light is frequency-shifted into resonance with the clock transition
of each atomic system by independent acousto-optic modulators (AOMs).
Resonance is detected by monitoring the ${}^{1}\\!S_{0}$ ground state
population ($N_{g}$) and ${}^{3}\\!P_{0}$ excited state population ($N_{e}$).
A laser cycles ground state atoms on the ${}^{1}\\!S_{0}$-${}^{1}\\!P_{1}$
transition, and a photomultiplier tube (PMT) collects the resulting
fluorescence, giving a measure of $N_{g}$. After 5-10 ms of cycling, these
atoms are laser-heated out of the lattice. At this point, $N_{e}$ is optically
pumped to the lowest-lying ${}^{3}\\!D_{1}$ state using a 1388 nm laser. These
atoms predominantly decay back to the ground state, where the
${}^{1}\\!S_{0}$-${}^{1}\\!P_{1}$ transition is cycled again, now measuring
$N_{e}$. Combining these measurements yields a normalized excitation for the
atomic ensemble $N_{e}/(N_{e}+N_{g})$. During operation of the clock systems,
special attention was paid to eliminating residual Stark shifts stemming from
amplified spontaneous emission of the lattice lasers, to eliminating residual
Doppler effects from mechanical vibrations of the apparatus correlated with
the experimental cycle, and to controlling the cold collision shift due to
atomic interactions within the lattice (?).

By measuring the normalized excitation while modulating the clock laser
frequency by $\pm 3$ Hz, an error signal is computed for each Yb system.
Subsequently, independent microprocessors provide a digital frequency
correction $f_{1,2}(t)$ to their respective AOMs, thereby maintaining
resonance on the line center. In this way, though derived from the same LO,
the individual laser frequencies for Yb-1 and Yb-2 are decoupled, and are
instead determined by their respective atomic samples (for all but the
shortest time scales). Computers record the frequency correction signals
$f_{1,2}(t)$, as shown in Fig. 2A for a 5,000 s interval. Because the
experimental cycles for each clock system are not synchronized and have
different durations, the recorded correction frequencies are interpolated to a
common time base and then subtracted to compute the frequency difference
between Yb-1 and Yb-2, as shown in Fig. 2B,C,D. Measurements such as these
were repeated several times for intervals of $\sim$15,000 s, demonstrating a
clock instability reaching $4\times 10^{-18}$ at 7,500 s. While collecting
data over a continuous 90,000 s interval, we observed the instability curve in
Fig. 3, shown here as the total Allan deviation for a single Yb clock. Prior
to data analysis, approximately 25% of the attempted measurement time was
excluded due to laser unlocks and auxiliary servo failures. Each servo used to
stabilize the laser to the clock transition had an attack time of a few
seconds, evidenced by the instability bump near 3 s. At $\tau=1-5$ s, the
instability is comparable to previous measurements of the free-running laser
system, and at longer times the instability averages down like white frequency
noise as $\sim\\!\\!3.2\times 10^{-16}/\sqrt{\tau}$ (for averaging times
$\tau$ in seconds), reaching the unprecedented instability of $1.6\times
10^{-18}$ at 25,000 seconds.

Also shown in Fig. 3 is an estimate of the combined instability contribution
(blue dashed) from the Dick effect and QPN, with the shaded region denoting
the uncertainty in these estimates. As can be seen, the observed instability
lies close to the combined contributions. We anticipate that significant
reductions are possible in the QPN limit by simply using higher atom numbers
and longer interrogation times. However, despite earlier reductions in the
Dick effect from improved local oscillators (?), Dick noise continues to be an
important limit in the performance of this clock. Looking to the future, this
limitation must be reduced so that $10^{-18}$ measurement instability can be
realized in $100$ seconds or less. Further stabilization of the optical LO
will continue to reduce the Dick effect, both by lowering the laser frequency
noise, which is down-converted in the Dick process, and by allowing increased
spectroscopy times and thus higher duty cycles. Such laser systems will use
optical cavities exhibiting reduced Brownian thermal-mechanical noise by
exploiting cryogenic operation (?), crystalline optical coatings (?), longer
cavities, or other techniques (?). Fig. 4 demonstrates the benefit of using an
optical LO improved over that used in this work, with four times less laser
frequency noise and with four times longer interrogation time (corresponding
to a short-term laser instability $\sim\\!\\!5\times 10^{-17}$). The red
dotted line gives the Dick instability, while the black dashed line indicates
the QPN limit with the same interrogation time, assuming a moderate atomic
population of 50,000.

Noting that the calculated Dick effect remains several times higher than the
QPN limit, we consider an alternative idea first proposed for microwave ion
clocks: interleaved interrogation of two atomic systems (?). By monitoring the
LO laser frequency at all times with two interleaved atomic systems, the
aliasing problem at the heart of the Dick effect can be highly suppressed. The
solid blue line in Fig. 4 illustrates the potential of a simple interleaved-
clock interrogation using Ramsey spectroscopy, which could be even further
improved using a more selective interleaving scheme. Even with LO noise levels
unimproved from the present work, the Dick effect lies well below a much
improved QPN limit (black dashed line). In this case, spin squeezing of the
atomic sample could reduce the final instability beyond the standard quantum
limit set by QPN (e.g., (?)). The two-system, interleaved technique requires
spectroscopy on each atomic system to last one half or more of the total
experimental cycle. By extending the clock spectroscopy time to $>$250 ms, we
have achieved a 50% duty cycle for each Yb system, demonstrating the
feasibility of this technique. Duty cycles $\geq\\!50\%$ can also be realized
with the aid of nondestructive state detection (?).

Another important property of a clock is its accuracy, which results from
uncertainty in systematic effects that alter the standard’s periodicity from
its natural, unperturbed state. In 2009, we completed a systematic analysis of
Yb-1 at the $3\times 10^{-16}$ uncertainty level (?). Since then, we reduced
the dominant uncertainty due to the blackbody Stark effect by an order of
magnitude (?). With its recent construction, Yb-2 has not yet been
systematically evaluated. The fact that the instability reaches the $10^{-18}$
level indicates that key systematic effects (e.g., blackbody Stark effect,
atomic collisions, lattice light shifts) on each system are being well-
controlled over the relevant timescales. For all measurements described here,
the mean frequency differences $\langle f_{2}(t)-f_{1}(t)\rangle$ were within
the Yb-1 uncertainty at $10^{-16}$. Having demonstrated superior instability,
we can now efficiently characterize systematic effects on each system, and
anticipate $10^{-17}$-level uncertainty in the very near future. Conversely,
because long-term instability is typically limited by systematic drifts,
further reduction and control of our systems’ uncertainties will likely offer
improved instability. With continued progress, we envision that $10^{-18}$
instability at 100 s and long-term instability well below $10^{-18}$ could be
achieved.

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  * 1.

The authors acknowledge DARPA QuASAR, NASA Fundamental Physics, and NIST for
financial support, and T. Fortier and S. Diddams for femtosecond optical
frequency comb measurements.

Figure 1: (A) Laser light at 578 nm is pre-stabilized to an isolated, high-
finesse optical cavity using Pound-Drever-Hall detection and employing
electronic feedback to an acousto-optic modulator (AOM) and laser
piezoelectric-transducer. This stable laser light is then delivered to the
Yb-1 and Yb-2 systems, where it is aligned along the optical lattice axis to
probe the atomic clock transition. Resonance with the atomic transition is
detected by observing atomic fluorescence collected onto a photomultiplier
tube (PMT). The fluorescence signal is digitized and processed by a
microcontroller unit (MCU), which computes a correction frequency,
$f_{1,2}(t)$. This correction frequency is applied to the relevant AOM by way
of a direct digital synthesizer (DDS), and locks the laser frequency onto
resonance with the clock transition. (B) Relevant Yb atomic energy levels and
transitions, including laser cooling transitions (399 and 556 nm), the clock
transition (578 nm), and the optical pumping transition used for excited state
detection (1388 nm). (C) A single-scan, normalized excitation spectrum of the
${}^{1}\\!S_{0}$-${}^{3}\\!P_{0}$ clock transition in 171Yb with 140 ms Rabi
spectroscopy time; the red line is a free-parameter sinc2 function fit. Figure
2: (A) Correction frequencies, $f_{1,2}(t)$, are shown in red and black.
Dominant LO fluctuations are due to the cavity and are thus common to the
atomic systems. (B) Frequency difference between the two Yb clock systems,
$f_{2}(t)-f_{1}(t)$, for a 5,000 s interval. (C) Data set $f_{2}(t)-f_{1}(t)$
over a 90,000 s interval. Gaps represent data rejected before data analysis
due to servo unlocks. (D) A Gaussian fit of the histogram $f_{2}(t)-f_{1}(t)$
for 90,000 s, with a mean frequency of 30 mHz and $\chi^{2}_{r}=0.9996$.
Figure 3: Total Allan deviation of a single Yb clock,
$(f_{2}(t)-f_{1}(t))/\sqrt{2}$ (red circles), and its white-frequency-noise
asymptote of $3.2\times 10^{-16}/\sqrt{\tau}$ (red solid line). The blue
dashed line represents the estimated combined instability contribution from
the Dick effect ($1.4\times 10^{-16}/\sqrt{\tau}$) and QPN ($1\times
10^{-16}/\sqrt{\tau}$), with the shaded region denoting uncertainty in these
estimates. Figure 4: Calculated instability limits for an improved lattice
clock towards the goal of $1\times 10^{-18}$ in 100 s. The Dick limit (red
dotted) is reduced by using a LO which is four times as stable as that used in
this work. The QPN limit is also shown under the same conditions (black
dashed) with a total atom number of 50,000. The inset illustration represents
an interleaved interrogation of two atomic systems, allowing continuous
monitoring of the LO for suppression of the Dick effect. Dead times from
atomic preparation or readout in one system are synchronized with clock
interrogation in the second system. The solid blue line indicates the
suppressed Dick instability in the interleaved-interrogation scheme of two
atomic systems using Ramsey spectroscopy with an unimproved LO.

## Supplementary Materials

### Spectroscopy sequences

Table 1: Spectroscopy sequences for each ytterbium system | Duration (ms)
---|---
Step | Yb-1 | Yb-2
1\. Blue MOT | 60 | 74
2\. Green MOT stages | 80 | 80
3\. State preparation/Pre-spectroscopy | 86 | 58
4\. 578 nm clock Rabi spectroscopy | 140 | 140
5\. State population readout | 39 | 43
Total cycle duration $\bm{T_{1,2}}$ | 405 | 395

Yb-1 and Yb-2 experimental cycles are unsynchronized and have nonidentical but
similar durations. Table 1 details the durations of major operations for both
systems. Initially, a counter-propagating 399 nm beam detuned $\sim\\!\\!120$
MHz below resonance slows ${}^{171}\\!$Yb atoms from a thermal beam. Unlike
many atomic beam magneto-optical traps (MOT)s, we employ no ‘Zeeman slower’.
Slowed atoms are laser-cooled and confined in a blue MOT consisting of three
nearly-orthogonal, retro-reflected 399 nm beams detuned 15 MHz below resonance
and a magnetic field gradient of $\sim\\!\\!3$ mT/cm. The three green MOT
stages, lasting 30 ms, 30 ms, and 20 ms respectively for both systems, use
increasingly resonant 556 nm light and a smaller but varying magnetic field
gradient to further cool the atoms. Notably, doubling the field gradient
during the second stage to $\sim\\!\\!2$ mT/cm compresses the atomic cloud,
ensuring a better overlap with the optical lattice.

After extinguishing the green MOT beams, we apply a $\sim\\!\\!0.5$ mT
magnetic field and spin-polarize trapped atoms by optical pumping on
${}^{1}\\!S_{0}$-${}^{3}\\!P_{1}$. Another $\sim\\!\\!0.1$ mT field lifts the
Zeeman degeneracy during 578 nm clock spectroscopy. Stabilized 578 nm light is
routed to the atomic systems through Doppler-noise-canceled fiber optic paths.
An acousto-optic-modulator (AOM) directs 578 nm light pulses onto the atoms.
To suppress phase chirps from AOM switching, we drive this AOM with $<$10 mW
of radio-frequency power. Rabi spectroscopy with a 140 ms pulse time provides
a convenient combination of narrow lineshape, short clock cycle, and
reliability when locking to the clock transition. Detection of atomic
excitation following spectroscopy is detailed below. During spectroscopy, all
resonant laser beams are extinguished when appropriate by both AOMs and
shutters, with the exception of 578 nm light. The optical lattice beam is
continuously applied with actively stabilized intensity.

### State detection, normalization, and background subtraction

As described in the main text, we apply a series of resonant pulses of 399 nm
light to count the atomic population via scattered fluorescence. The combined
photon collection efficiency is $<~{}\\!\\!\\!1\%$ due to small observational
solid angle, photo-multiplier tube (PMT) quantum efficiency, and transmission
efficiency of a 399 nm bandpass filter. However, by retroreflecting the probe
light to improve the balance of optical forces, each atom scatters $>$$10^{3}$
photons before being laser-heated out of lattice confinement. Unfortunately,
probe light also illuminates the thermal effusive Yb beam, creating a
fluorescence signal that, together with scattered laser light, constitutes an
important background source. Identifying the integrated fluorescence from
trapped ground state atoms as $N_{g}$, and the integrated background signal as
$B$, the first readout in the sequence is recorded as $P_{1}=N_{g}+B$.
Following this, a second identical pulse ideally generates an integrated
signal $P_{2}=B$. Then, application of resonant 1388 nm light optically pumps
nearly all ($>~{}\\!\\!\\!90\%$) trapped atoms from the ${}^{3}\\!P_{0}$ state
into the ground state through the ${}^{3}\\!D_{1}$ state. Finally, a third
identical 399 nm probe pulse yields the integrated signal $P_{3}\simeq
N_{e}+B$.

We synthesize an atomic excitation fraction $\epsilon$, normalized against
total atom number and suppressing background effects, from the three records
$P_{1,2,3}$:

$\epsilon\equiv\frac{N_{e}}{N_{g}+N_{e}}\approx\frac{P_{3}-P_{2}}{P_{1}+P_{3}-2P_{2}}.$
(1)

In order to cleanly implement the atomic state detection, the 399 nm laser is
(1) frequency stabilized to the ${}^{1}\\!S_{0}$-${}^{1}\\!P_{1}$ transition
using a modulation transfer spectrometer, (2) intensity stabilized, (3)
polarization filtered, and (4) sufficiently intense to drive the
${}^{1}\\!S_{0}$-${}^{1}\\!P_{1}$ transition into saturation.

### Clock difference-signal

Each Yb atomic servo consists of a micro-controller unit (MCU) updating a
direct-digital synthesizer (DDS) with a correction signal $f_{1,2}$. System
Yb-$n$ applies discrete corrections $f_{n}(t^{(n)}_{i})$ at unsynchronized
times $t^{(n)}_{i}$. Independent computers record and timestamp each
correction $f_{n}(t^{(n)}_{i})$. During post-processing, we employ a
piecewise-cubic Hermite interpolating polynomial to establish $f_{1}$ and
$f_{2}$ on a common set of timestamps separated by the average of the two
systems’ cycle durations, $\bar{T}=\frac{1}{2}\left(T_{1}+T_{2}\right)$. As
described in the main text, the difference in the correction signals
$f_{2}(t)-f_{1}(t)$ corresponds to the frequency difference between the atomic
systems, and the instability of $f_{2}(t)-f_{1}(t)$ gives the combined
instability measured between the clocks. If systematic variations of each
atomic frequency existed and they were positively correlated between the
systems, then these variations would be reduced in the frequency difference.
In such a case, the measured instability would not reflect the true
instability of the lattice clocks. We note that by design, these atomic
systems are largely independent of each other. Their most significant shared
attribute is co-location in the same laboratory. Thus, while local heat loads
independently influence each atomic apparatus, the ambient laboratory
temperature can drive temperature correlations that influence the blackbody
Stark shift. However, we note the mean laboratory temperature was long-term
stable to $<~{}\\!\\!\\!50$ mK over the measurement period. Most key
systematic effects, like the cold collision shift, Zeeman shifts, probe light
shifts, and other technical effects, are independent for each system. The
optical lattice lasers are independent systems, and their frequencies are
separated by 1 MHz, both held close to the magic wavelength ($\lambda_{m}$) .
We note that control of this optical frequency at the MHz level is routine.

### Lattice laser stabilization

Each Yb system’s lattice is derived from a distinct Titanium Sapphire (Ti:S)
laser that is injection locked by an external cavity diode laser near
$\lambda_{m}\approx 759$ nm. A small fraction of the nominal $\sim$1.5 W
output power from the Yb-2 lattice system is sent via polarization-maintaining
(PM) optical fiber to a reference optical cavity. Feedback to an intra-cavity
piezoelectric transducer (PZT) stabilizes the Yb-2 lattice laser frequency to
the cavity resonance. An AOM upshifts the laser frequency by $80$ MHz,
bringing it to 394,798,295.4 MHz for confinement of Yb-2 atoms near
$\lambda_{m}$.

The frequency difference between the Ti:S of Yb-1 and that of Yb-2 is directly
detected in a heterodyne optical interferometer. This heterodyne beat is
stabilized to $\sim\\!\\!162$ MHz using an RF delay line interferometer, with
Yb-1’s laser frequency chosen higher than that of Yb-2. An AOM downshifts
Yb-1’s laser frequency by $82.3$ MHz, to 394,798,294.3 MHz before the light is
sent via PM fiber to the atomic system. AOMs are also used to intensity
stabilize each lattice beam.