††thanks: These authors contributed equally to this work.††thanks: These
authors contributed equally to this work.

# Lifetime Determination of the $5s5p$ ${}^{3}P^{\rm{o}}_{0}$ Metastable State
in 87Sr from the Electric Dipole Matrix Element

Xiao-Tong Lu  National Time Service Center, Chinese Academy of Sciences, Xi’an
710600, China  Feng Guo  National Time Service Center, Chinese Academy of
Sciences, Xi’an 710600, China  Yan-Yan Liu  National Time Service Center,
Chinese Academy of Sciences, Xi’an 710600, China  Jing-Jing Xia  National Time
Service Center, Chinese Academy of Sciences, Xi’an 710600, China  School of
Astronomy and Space Science, University of Chinese Academy of Sciences,
Beijing 100049, China  Guo-Dong Zhao  National Time Service Center, Chinese
Academy of Sciences, Xi’an 710600, China  School of Astronomy and Space
Science, University of Chinese Academy of Sciences, Beijing 100049, China
Ying-Xin Chen  National Time Service Center, Chinese Academy of Sciences,
Xi’an 710600, China  School of Astronomy and Space Science, University of
Chinese Academy of Sciences, Beijing 100049, China  Ye-Bing Wang  National
Time Service Center, Chinese Academy of Sciences, Xi’an 710600, China  Ben-
Quan Lu<EMAIL_ADDRESS>National Time Service Center, Chinese Academy of
Sciences, Xi’an 710600, China  Hong Chang<EMAIL_ADDRESS>National Time
Service Center, Chinese Academy of Sciences, Xi’an 710600, China  School of
Astronomy and Space Science, University of Chinese Academy of Sciences,
Beijing 100049, China

###### Abstract

We report a measurement of the radiative lifetime of the
$5s5p\;{}^{\rm{3}}P^{\rm{o}}_{\rm{0}}$ metastable state in 87Sr, which is
coupled to the 5$s^{\rm{2}}\;$ ${}^{\rm{1}}S_{\rm{0}}$ ground state via a
hyperfine-induced electric dipole transition. The radiative lifetime is
determined to be 151.4(48) s, in good agreement with theoretical results. Our
approach relies on accurate measurements of laser intensity and free-space
Rabi frequency, enabling lifetime measurements of any excited state and
particularly suitable for long-lived states.

##  I Introduction

Precisely determining the lifetime of electronic states serves as a crucial
reference point for understanding atomic structure. As such, it plays a
critical role in improving the performance of various quantum techniques
Bothwell2022[1] ; Sanner2019[2] ; Takamoto2022[3] ; Drscher2021[4] ;
Golovizin2019[5] ; Grotti2018[6] ; Lu2021[7] ; Daley2008[8] . For example, the
understanding of atomic structure can help to reduce the uncertainty of
dynamical black-body-radiation shift, which limits the accuracy of the state-
of-the-art 87Sr Bothwell2022[1]  and 171Yb McGrew2018[31]  optical lattice
clocks and the corresponding correction relies on the ab-initio calculations
Porsev2006 ; Safronova2013 . It is relatively straightforward to precisely
measure the lifetime by directly observing the spontaneous decay Beloy2012[9]
; Olmschenk2009[10] . However, when the lifetime of an excited state exceeds
seconds, such as in the case of metastable states, the direct observation of
spontaneous decay faces challenges due to competing processes. In such
scenarios, careful consideration must be given to the evaluation of decaying
rates arising from collisions and off-resonant laser radiation. Additionally,
the long measurement period is necessary to ensure adequately small
statistical uncertainty Yasuda2004[11] ; Dorscher2018[12] .

The natural lifetime of the metastable state could be precisely inferred from
the cavity-enhanced dispersion measurements, where a transition with a well-
known spontaneous emission rate provides a reference for the target level.
This complicated method has been demonstrated for lifetime measurements of the
$\left|5s5p\;{}^{3}P^{\rm{o}}_{0}\right\rangle$ state in 87Sr Muniz2021[13] ,
achieving a measurement uncertainty of 2.5%. However, it is worth noting that
the result of 118(3) s obtained in their experiment is inconsistent with the
prior measurement of 330(170) s Dorscher2018[12] . This discrepancy highlights
the necessity for additional investigation into the lifetime of the
$\left|5s5p\;{}^{3}P^{\rm{o}}_{0}\right\rangle$ state in 87Sr. Alternatively,
combining measurements of Rabi frequency and Stark shifts from the
interrogation laser can extract the lifetime from the transition matrix
element Hettrich2015[14] . Recently, this method was used to measure the
lifetime of the ${}^{2}F_{7/2}$ level in $\rm{{}^{171}Yb^{+}}$ with record
precision for such a long lifetime in the $10^{7}$ s Lange2021[15] . It
appears that this technique reduces the challenging requirement for precise
determination of the laser intensity. However, it is still necessary to
determine the differential polarizability between states, and in certain
cases, accurately measuring the laser intensity remains crucial
Huntemann2016[23] .

In this paper, we have developed a technique that enables accurate
reconstruction of the two-dimensional laser intensity profile. This
advancement allows for precise determination of the natural lifetime of
metastable states through the measurement of the transition matrix element. By
employing cold-atom ensemble as a sensor, we can effectively detect the
distribution of laser intensity and accurately determine the central position
of the laser beam profile by systematically adjusting its location. Combining
this information with the result of the free-space Rabi frequency, the
corresponding transition matrix element can be accurately determined. To
demonstrate this technique, we investigate the hyperfine-induced E1 transition
from the
$\left|5s5p\;{}^{3}P^{\rm{o}}_{0},F^{\rm{e}}=I,M^{\rm{e}}_{\rm{F}}=+9/2\right\rangle$
excited state to the
$\left|5s^{2}\;{}^{1}S_{0},F^{\rm{g}}=I,M^{\rm{g}}_{\rm{F}}=+9/2\right\rangle$
ground state in 87Sr (with a nuclear spin $I=9/2$). The natural lifetime of
the excited $\left|5s5p\;{}^{3}P^{\rm{o}}_{0}\right\rangle$ state has been
determined with 3.2% uncertainty.

##  II Method

The transition rate $A(M^{\rm{g}}_{\rm{F}},M^{\rm{e}}_{\rm{F}})$ from an
excited state $\left|\Gamma
IJ^{\rm{e}}F^{\rm{e}}M^{\rm{e}}_{\rm{F}}\right\rangle$ to a lower state
(ground state) $\left|\Gamma
IJ^{\rm{g}}F^{\rm{g}}M^{\rm{g}}_{\rm{F}}\right\rangle$ can be written as
Grumer2014[16]

$\begin{split}&A(M^{\rm{g}}_{\rm{F}},M^{\rm{e}}_{\rm{F}})\\\
&=\frac{C^{\rm{\rho k}}}{\lambda^{\rm{2k+1}}}\sum_{\rm{q}}|\left\langle\Gamma
IJ^{\rm{g}}F^{\rm{g}}M^{\rm{g}}_{\rm{F}}|O^{\rm{\rho k}}_{\rm{q}}|\Gamma
IJ^{\rm{e}}F^{\rm{e}}M^{\rm{e}}_{\rm{F}}\right\rangle|^{2}.\end{split}$ (1)

$\left\langle\Gamma IJ^{\rm{g}}F^{\rm{g}}M^{\rm{g}}_{\rm{F}}|O^{\rm{\rho
k}}_{\rm{q}}|\Gamma IJ^{\rm{e}}F^{\rm{e}}M^{\rm{e}}_{\rm{F}}\right\rangle$ is
the transition matrix element between the ground state and the excited state.
$O^{\rm{\rho k}}_{\rm{q}}$ indicates the multipole radiation-field tensor
operator and is typically determined by the rank $k$, which corresponds to the
angular momentum of the photon relative to atoms. $\rho=(-1)^{\rm{k}}$ or
$(-1)^{\rm{k+1}}$ represents the parity for the electric (E$k$) or magnetic
(M$k$) multipole transitions, respectively. $C^{\rm{\rho k}}$ is a constant
depending on the transition type and $\lambda$ is the transition wavelength.
In relation to the $\left|\Gamma
IJ^{\rm{e}}F^{\rm{e}}M^{\rm{e}}_{\rm{F}}\right\rangle$ $\rightarrow$
$\left|\Gamma IJ^{\rm{g}}F^{\rm{g}}M^{\rm{g}}_{\rm{F}}\right\rangle$
transition, the free-space Rabi frequency, which indicates the strength of the
interaction between atoms and the electromagnetic field used for
interrogation, can be expressed by Scar1981[17]

$\Omega_{\rm{M^{eg}_{F}}}=|\left\langle\Gamma
IJ^{\rm{g}}F^{\rm{g}}M^{\rm{g}}_{\rm{F}}|O^{\rm{\rho k}}|\Gamma
IJ^{\rm{e}}F^{\rm{e}}M^{\rm{e}}_{\rm{F}}\right\rangle|\cdot\vec{E}/\hbar.$ (2)

$\hbar$ represents the reduced Planck constant, and $\vec{E}$ represents the
electric field strength, which is directly related to the intensity ($I_{0}$)
of the interrogation laser by
$|\vec{E}|^{2}=2I_{0}n_{\rm{r}}/c_{0}n_{0}\varepsilon_{0}$. Here, $n_{\rm{r}}$
and $n_{0}$ denote the refractive index of the atom ensemble and the
surrounding space, respectively. $c_{0}$ stands for the speed of light in a
vacuum, while $\varepsilon_{0}$ represents the vacuum permittivity. Therefore,
the transition rate $A(M^{\rm{g}}_{\rm{F}},M^{\rm{e}}_{\rm{F}})$ can be
directly determined based on the measurements of $\Omega_{\rm{M^{eg}_{F}}}$
and $I_{0}$.

Figure 1:  Experimental setup for optical lattice and clock transition
detection. The lattice is created by overlapping the incident 813 nm laser
beam with its retroreflected beam using a concave mirror (CR). The 698 nm
interrogation laser beam is aligned with the lattice laser using a dichroic
mirror (DM). Both lasers are polarized along the Y direction, and their
polarizations are ensured using the linear polarizer (LP). The position of the
698 nm laser can be finely adjusted using a three-dimensional translation
stage. The inset shows that the waist of the 698 nm laser is much larger than
the lattice size, and atoms can only interact with a small portion of the 698
nm laser.

##  III Experiment setup

We present our method with the
$\left|5s5p\;{}^{3}P^{\rm{o}}_{0},F^{\rm{e}}=9/2,M^{\rm{e}}_{\rm{F}}=+9/2\right\rangle$
$\rightarrow$
$\left|5s^{2}\;{}^{1}S_{0},F^{\rm{g}}=9/2,M^{\rm{g}}_{\rm{F}}=+9/2\right\rangle$
transition of 87Sr atoms. The experimental setup is based on our 87Sr
transportable optical lattice clock Kong2020[18] ; Kong2021[19] . Fig. 1 shows
the setup for the optical lattice and the clock transition detection. After
undergoing two stages of laser cooling [see Appendix A for details], cold
atoms are trapped within a horizontal one-dimensional optical lattice. The
initial trap depth is set at 120 $E_{\rm{R}}$ ($E_{\rm{R}}$ is the recoil
energy from a lattice photon). Subsequently, we employ an optical repumping
technique to prepare the atoms in
$\left|{}^{1}S_{0},F^{\rm{g}}=I,M^{\rm{g}}_{\rm{F}}=+9/2\right\rangle$ ground
state. In order to remove hotter atoms, we employ an energy-filtering method.
This involves linearly reducing the lattice trap depth to 60 $E_{\rm{R}}$ over
a period of 20 ms. We then wait for 10 ms before increasing the trap depth
back up to 176 $E_{\rm{R}}$ within another 20 ms. Through this process,
approximately 1000 atoms are retained within the lattice, while the hotter
atoms are effectively removed. Following the filtering procedure, the axial
temperature is measured to be 2.7 $\rm{\mu K}$ and the radial temperature is
determined to be 3.9 $\rm{\mu K}$ using the sideband spectra Blatt2009[20] .

Figure 2:  The measurements of the square of the normalized free-space Rabi
frequency $\Omega^{2}_{0}$ as a function of axis position. (a)
$\Omega^{2}_{0}$ as a function of the X. (b) $\Omega^{2}_{0}$ as a function of
the Y. The continuous lines show the Gaussian fittings.

After preparing the quantum reference system, we proceed to interrogate the
$\left|{}^{3}P^{\rm{o}}_{0},F^{\rm{e}}=9/2,M^{\rm{e}}_{\rm{F}}=+9/2\right\rangle$
$\rightarrow$
$\left|{}^{1}S_{0},F^{\rm{g}}=9/2,M^{\rm{g}}_{\rm{F}}=+9/2\right\rangle$
transition using a 698 nm laser. The 698 nm laser is stabilized to an
ultralow-expansion (ULE) cavity (with a fineness of 300,000) using the Pound-
Drever-Hall (PDH) technique. To minimize frequency drift caused by changes in
the cavity length, we implement a linear sweeping of the driving frequency of
the acoustic optical modulator (AOM) and, effectively reduce the drift to
below 1 mHz/s. The excitation fraction is determined using the normalized
electronic shelving method Nagourney1986[21] . After interrogating the 698 nm
transition, the number of atoms in the $\left|{}^{1}S_{0}\right\rangle$ state
($N_{\rm{g}}$) is probed using a 461 nm laser pulse, while concurrently
expelling these atoms from the lattice. Next, the atoms in the
$\left|{}^{3}P^{\rm{o}}_{0}\right\rangle$ state are repumped to the
$\left|{}^{1}S_{0}\right\rangle$ state, and the number of atoms in this state
($N_{\rm{e}}$) probed. Additionally, a third probe light pulse measures the
background noise ($N_{\rm{b}}$). The excitation fraction ($p_{\rm{e}}$) is
then determined by
$(N_{\rm{e}}-N_{\rm{b}})/(N_{\rm{e}}+N_{\rm{g}}-2N_{\rm{b}})$ (see Appendix A
for more detalis). The Rabi oscillation is measured by maintaining resonance
on the transition and observing the maximum excitation fraction at various
interrogation times. By taking into account the thermal distribution of the
cold ensemble, the free-space Rabi frequency can be accurately extracted from
the Rabi oscillation Blatt2009[20] .

Figure 3:  Measurements of the free-space Rabi frequency $\Omega_{0}$ and
laser intensity. (a) The measurements of $\Omega_{0}$. The orange area
represents the average value and its corresponding $\pm 1$ standard deviation.
The inset shows a typical Rabi oscillation. (b) presents the reconstruction of
the laser intensity. Its center is the expected intensity of interaction with
atoms.

##  IV ${}^{3}P^{\rm{o}}_{0}$ state lifetime measurement

Due to the significantly larger size of the interrogation laser compared to
the lattice, it is essential to secure the lattice at the center of the
Gaussian interrogation laser beam. In this way, we can not only accurately
ascertain the intensity of the laser’s interaction with atoms but also
significantly minimize the uncertainty in intensity attributed to errors in
position measurement. To pinpoint the center position, we measure the Rabi
frequency as a function of the position of the interrogation laser along the X
and Y directions, as illustrated in Fig. 2. By adjusting the position of the
translation stage, we initially determine the position of peak intensity in
the X-direction ($X_{\rm{p}}$=6.69(5) mm). With the X scale fixed at
$X_{\rm{p}}$, we proceed to measure the peak point in the Y-direction
($Y_{\rm{p}}$=7.13(4) mm). Gaussian fitting analysis reveals a $e^{-2}$ width
of 6.77(16) mm for the X-direction and 6.66(22) mm for the Y-direction.
Remarkably, these widths are consistent in the measurements obtained from the
laser beam profiler (Thorlabs BP209-VIS/M) at the estimated lattice position,
which recorded values of 6.87(7) mm and 6.74(7) mm for the X and Y directions,
respectively. This agreement further validates our methodology and
demonstrates the consistency of our experimental results.

Figure 3(a) shows measurements of the free-space Rabi frequency at
(X=$X_{\rm{p}}$, Y=$Y_{\rm{p}}$). By analyzing the measured laser beam profile
at the inferred distance of the lattice, we determine the ratio $R_{\rm{p}}$
of the measured total laser power to the total flux $C_{\rm{t}}$ recorded in
the beam profile. Here $C_{\rm{t}}$ is the sum of the brightness level signals
$C_{\rm{ij}}$ of all pixels in the digitally generated beam profile image.
Thus, we can express the optical intensity of pixel (i,j) as
$R_{\rm{p}}C_{\rm{ij}}/A_{\rm{p}}$, where $A_{\rm{p}}$ denotes the pixel area.

As described in Ref. Lu2023[22] , in terms of 87Sr, the hyperfine interaction
can induce the $\left|{}^{3}P^{\rm{o}}_{1}\right\rangle$ and
$\left|{}^{1}P^{\rm{o}}_{1}\right\rangle$ states mixing with the
$\left|{}^{3}P^{\rm{o}}_{0}\right\rangle$ state, which opened the transition
channel between the ground state and the
$\left|{}^{3}P^{\rm{o}}_{0}\right\rangle$ state. According to Eq.(1), the
transition rate $A(M^{\rm{g}}_{\rm{F}},M^{\rm{e}}_{\rm{F}})$ between the
excited Zeeman state
$\left|5s5p\;{}^{3}P^{\rm{o}}_{0},F^{\rm{e}}=9/2,M^{\rm{e}}_{\rm{F}}=+9/2\right\rangle$
and the ground Zeeman state
$\left|5s^{2}\;{}^{1}S_{0},F^{\rm{g}}=9/2,M^{\rm{g}}_{\rm{F}}=+9/2\right\rangle$
is Lu2023[22]

$\begin{split}&A(M^{\rm{g}}_{\rm{F}},M^{\rm{e}}_{\rm{F}})=\frac{2.02613\times
10^{18}}{\lambda^{\rm{3}}}\sum_{\rm{q}}\\\
&|\left\langle"IJ^{\rm{g}}F^{\rm{g}}=IM^{\rm{g}}_{\rm{F}}"|O^{(1)}_{\rm{q}}|"IJ^{\rm{e}}F^{\rm{e}}=IM^{\rm{e}}_{\rm{F}}"\right\rangle|^{2},\end{split}$
(3)

where the state within the quotation marks describes the dominant component of
the eigenvector and $O^{(1)}_{\rm{q}}$ is the $q$th component of the electric
dipole transition operator $O^{(1)}$. For the
$\left|5s5p\;{}^{3}P^{\rm{o}}_{0},F^{\rm{e}}=9/2,M^{\rm{e}}_{\rm{F}}=+9/2\right\rangle$
$\rightarrow$
$\left|5s^{2}\;{}^{1}S_{0},F^{\rm{g}}=9/2,M^{\rm{g}}_{\rm{F}}=+9/2\right\rangle$
($\rm{\pi}$ transition, $q=0$) in 87Sr, $\lambda=6984.457096$ $\rm{\AA}$ and
the Rabi frequency is

$\Omega_{\rm{M^{eg}_{F}}}=|\left\langle"IJ^{\rm{g}}F^{\rm{g}}=IM^{\rm{g}}_{\rm{F}}"|O^{(1)}_{\rm{q}}|"IJ^{\rm{e}}F^{\rm{e}}=IM^{\rm{e}}_{\rm{F}}"\right\rangle|\cdot\vec{E}/\hbar.$
(4)

Figure 4:  Comparison of experimental (blue) and theoretical (red) results of
the 87Sr state natural lifetime. Previous experimental results were based on
rate equation analysis Dorscher2018[12]  and the cavity-enhanced dispersion
measurements Muniz2021[13] . Theoretical values are given in Refs. Lu2023[22]
; Boyd2007[24] ; Santra2004[25] ; Porsev2004[26] ; Palchikov2022[27] . The
purple line and shadow area show the arithmetic average value and the
corresponding 1 standard error of theoretical results.

In our experiment, taking into consideration the low atomic density (below
$10^{12}\;\rm{cm^{-3}}$) and the spontaneous radiation rate of the
$\left|5s5p\;{}^{3}P^{\rm{o}}_{0}\right\rangle$ state, we can estimate that
the value of $n_{\rm{r}}$ is approximately 1 with an error below $10^{-5}$
McCutcheon2022[add1] . Additionally, given the ultra-low pressure of $10^{-9}$
Pa in the science chamber, we can safely estimate $n_{\rm{0}}$ to be 1.
Combining Eq.(3) and Eq. (4), the transition rate of
$\left|{}^{3}P^{\rm{o}}_{0},F^{\rm{e}}=9/2,M^{\rm{e}}_{\rm{F}}=+9/2\right\rangle$
$\rightarrow$
$\left|{}^{1}S_{0},F^{\rm{g}}=9/2,M^{\rm{g}}_{\rm{F}}=+9/2\right\rangle$
transition is experimentally determined as $5.41(17)\times 10^{-3}$
$\rm{s^{-1}}$, and the corresponding dipole moment of
$|\left\langle"IJ^{\rm{g}}F^{\rm{g}}=IM^{\rm{g}}_{\rm{F}}"|O^{(1)}_{\rm{0}}|"IJ^{\rm{e}}F^{\rm{e}}=IM^{\rm{e}}_{\rm{F}}"\right\rangle|$
is determined to be $3.0(1)\times 10^{-5}$ a.u. (a.u. indicates the atomic
unit). There are two decaying channels from
$\left|{}^{3}P^{\rm{o}}_{0},F^{\rm{e}}=9/2,M^{\rm{e}}_{\rm{F}}=+9/2\right\rangle$
to the ground state, including
$\left|{}^{3}P^{\rm{o}}_{0},M^{\rm{e}}_{\rm{F}}=+9/2\right\rangle$
$\rightarrow$ $\left|{}^{1}S_{0},M^{\rm{g}}_{\rm{F}}=+9/2\right\rangle$
($\rm{\pi}$ transition) and
$\left|{}^{3}P^{\rm{o}}_{0},M^{\rm{e}}_{\rm{F}}=+9/2\right\rangle$
$\rightarrow$ $\left|{}^{1}S_{0},M^{\rm{g}}_{\rm{F}}=+7/2\right\rangle$
($\rm{\sigma}$ transition), and the radiation rate ratio of $\rm{\pi}$
transition to $\rm{\sigma}$ transition is 4.5 [see Appendix C for details].
The total radiation rate of the
$\left|5s5p\;{}^{3}P^{\rm{o}}_{0},F^{\rm{e}}=9/2,M^{\rm{e}}_{\rm{F}}=+9/2\right\rangle$
level is determined to be $6.61(21)\times 10^{-3}\;\rm{s^{-1}}$, corresponding
to a natural lifetime of 151.4(48) s. In this experiment, the magnetic field
strength is 0.4 G, resulting in a change of the Rabi frequency less than
$5\times 10^{-8}$ to the $M^{\rm{e}}_{\rm{F}}=\pm 9/2$ Zeeman sub-levels
Lu2023[22] . Therefore, the magnetic field effect on the transition rate can
be safely neglected. The natural lifetime of the
$\left|5s5p\;{}^{3}P^{\rm{o}}_{0}\right\rangle$ state, which accounts for the
statistical average lifetime of all ten Zeeman sub-levels, is the same as that
of the
$\left|5s5p\;{}^{3}P^{\rm{o}}_{0},F^{\rm{e}}=9/2,M^{\rm{e}}_{\rm{F}}=+9/2\right\rangle$
state.

##  V Correction and uncertainty evaluation for lifetime measurement

The total uncertainty associated with the optical power meter (PM100D) and
sensor (S130C) is 3%, which translates to an uncertainty of 4.5 s in the
lifetime measurement. The laser power interacting with atoms is determined by
averaging the powers before and after passing through the windows. As a
result, influences caused by the windows (such as reflection, absorption, and
potential etalon effects) can be disregarded. The precision of reconstructing
the laser intensity is limited by the knowledge of the precise position of the
atoms. We can only measure the beam profile outside the science chamber at the
calculated position. Measurement uncertainty will occur due to the size of the
lattice and the positional inaccuracies. We have observed beam profiles
deviating by $\pm 40$ mm from the expected position and have determined that
each millimeter of positional error contributes to a 0.188 s difference in the
lifetime measurement. Taking into account the largest deviation of 8 mm (the
radius of the cavity windows), this effect will contribute an uncertainty of
1.2 s to the lifetime measurement. The uncertainty of the Rabi frequency is
determined to be 0.23%, which corresponds to an uncertainty of 0.74 s. This
uncertainty includes contributions from the precision of atomic temperature
measurements (0.3%) and the fitting error at 1 standard error (0.15%). The
center position of the laser beam is determined with an uncertainty of 0.22 mm
as shown in Fig. 2, and the $e^{-2}$ waist diameter of the lattice is 0.1 mm.
Therefore, we calculate the $\left|{}^{3}P^{\rm{o}}_{0}\right\rangle$ state
lifetime using the average intensity around the measured center point with a
radius of 0.3 mm. We conservatively estimate that this method may introduce a
maximum error of 0.23 s based on the difference between the average and peak
intensities.

We observed that a small portion of the probe light extends beyond the beam
profiler (with an effective diameter of 9 mm), resulting in a considerable
systematic error. Through two-dimensional Gaussian fitting to deduce the
complete intensity distribution, we determined that the power collected by the
beam profiler is 1.81% smaller than the total power. Additionally, the power
meter sensor (with a diameter of 9.5 mm) will also leak a small amount of
power, estimated at 1.17%. Therefore, the systematic error due to the finite
active aperture of the beam profiler and power meter is -0.97(52) s, with
uncertainty estimated considering a perceived position error of the power
meter of 0.5 mm. This systematic error can be suppressed by using an aperture
close to the fiber collimator to restrict the size of the probe light. The
presence of noise sidebands in the probe light introduces a systematic error
as these sidebands contribute to the light power measured by the power meter
but are not reflected in the Rabi oscillation.

In the PDH scheme, the feedback loop tends to amplify noise beyond the
feedback bandwidth, resulting in what is commonly referred to as ’servo-bumps’
Akerman2015 ; Li2022 ; Chao2023 . These servo-bumps can lead to an
overestimation of the measured light power since they do not contribute to the
transition detection but instead occupy a small portion of the light power. By
analyzing the beat signal between the probe light and a reference laser
operating at 698 nm, we can determine the relative amplitudes of the servo-
bumps in relation to the carrier signal. To prevent potential overlap of
servo-bumps between the two lasers, the cavity transmission light of the
reference laser is used to cancel out the servo-bumps Akerman2015 .
Furthermore, we also observe spectral sidebands possibly created by a coherent
source at $\pm 19.2$ kHz and $\pm 38.4$ kHz (these sidebands are also observed
in the atomic response to an excitation of the 698 nm transition). Considering
the contributions of both the servo-bumps and spectral sidebands, we estimate
that they will introduce a correction of -1.24(26) s to the lifetime [see
Appendix D for further details]. The uncertainty here reflects the variation
in the calculated lifetime when we increase the PDH gain by 20% from its
typical operational point, where the amplitude of PDH error is minimized. The
impact of servo-bumps can be strongly suppressed through techniques such as
spectral filtering and post-correction methods Akerman2015 ; Li2022 ; Chao2023
.

Table 1 presents a summary of the evaluation results that contribute to the
error in lifetime measurements. The total measurement uncertainty is
determined to be 4.8 s (corresponding to a relative uncertainty of 3.2%),
which is mainly dominated by the power meter uncertainty. If the power meter
uncertainty is reduced to 0.5% Huntemann2016[23]  and a larger beam waist is
used, this method has great potential to determine the lifetime with an
uncertainty below 1%. Figure 4 summarizes the experimental and theoretical
results of the $\left|{}^{3}P^{\rm{o}}_{0}\right\rangle$ state lifetime in
${}^{87}\rm{Sr}$. Our result not only agrees with our previous theoretical
results using the multiconfiguration Dirac-Hartree-Fock method, but also
accords with the arithmetic average value of all theoretical calculations.
Thus, our result provides a clear point of reference for atomic structure
calculations.

Table 1:  Corrections of the lifetime measurement of the $5s5p$ ${}^{3}P^{\rm{o}}_{0}$ state. Source | Correction (s) | Uncertainty (s)
---|---|---
Optical power meter | - | 4.5
Lattice position | - | 1.2
Rabi frequency | - | 0.74
Averaging optical intensity | - | 0.23
Finite active aperture | -0.97 | 0.52
servo-bumps and |  |
spectral sidebands | -1.24 | 0.26
Total | -2.21 | 4.8

##  VI CONCLUSION

We use a straightforward yet effective technique for determining the lifetime
of the $5s5p$ ${}^{3}P^{\rm{o}}_{0}$ in ${}^{87}\rm{Sr}$. By combining
measurements of the free-space Rabi frequency and reconstructing the laser
intensity to determine the transition matrix element, we infer the lifetime to
be 151.4(48) seconds. This method explained here holds potential for broad
application in other atomic species, including other alkaline-earth atoms and
ions Dube2013[28] ; Barrett2019[29] ; Huang2022[30] ; McGrew2018[31] ;
Filzinger2023[32] , ${}^{175}\rm{Lu}^{+}$Zhang2023[33] , $\rm{Pb}^{2+}$
Beloy2021[34]  and highly charged ions Steven2022[35] . Accurate lifetime
measurements contribute to a deeper understanding of the atomic structure,
enable testing of theoretical models, and support quantitative experimental
investigations of the hyperfine quenching effect Johnson2010[36]  by measuring
the lifetimes of all stable isotopes (such as in Yb Roman2021[37] ).
Additionally, the precise determination of laser intensity can also have a
positive impact on other measurements, such as the measurement of the
differential polarizability Huntemann2016[23] .

##  VII ACKNOWLEDGMENTS

We thank Lin-Xiang He from the Innovation Academy for Precision Measurement
Science and Technology, CAS for helpful discussions, and Luis A. Orozco from
the University of Maryland and Yan-Ting Zhao from Shanxi University for his
careful reading of the manuscript. This work is supported by the National
Natural Science Foundation of China (Grant No. 12203057), and the Strategic
Priority Research Program of the Chinese Academy of Sciences (Grant No.
XDB35010202).

##  Appendix A Preparation of the cold 87Sr ensemble

Figure 5(a) shows the energy levels of 87Sr relevant to this experiment. After
preliminary cooling by passing the Zeeman slower, atoms from atomic oven are
trapped and cooled by the blue magneto-optical trap (MOT). The bule MOT is
based on the $\left|{}^{1}P^{\rm{o}}_{1},F\rm{{}^{\prime}}=11/2\right\rangle$
$\rightarrow$ $\left|{}^{1}S_{0},F^{\rm{g}}=9/2\right\rangle$ transition and
can reduce atomic temperature to $1\sim 5$ mK. The
$\left|{}^{3}P^{\rm{o}}_{0}\right\rangle$ $\rightarrow$
$\left|{}^{3}S_{1}\right\rangle$ and $\left|{}^{3}P^{\rm{o}}_{2}\right\rangle$
$\rightarrow$ $\left|{}^{3}S_{1}\right\rangle$ transitions can repump atoms
populated in ${}^{3}P^{\rm{o}}_{0}$ and ${}^{3}P^{\rm{o}}_{2}$ states back to
${}^{1}S_{0}$ state, increasing the number of trapped atoms. Following the
bule MOT, the $\left|{}^{3}P^{\rm{o}}_{1},F\rm{{}^{\prime}}=11/2\right\rangle$
$\rightarrow$ $\left|{}^{1}S_{0},F^{\rm{g}}=9/2\right\rangle$ and
$\left|{}^{3}P^{\rm{o}}_{1},F\rm{{}^{\prime}}=9/2\right\rangle$ $\rightarrow$
$\left|{}^{1}S_{0},F^{\rm{g}}=9/2\right\rangle$ transitions are simultaneously
used to further decrease the thermodynamic temperature of atoms to 3 $\rm{\mu
K}$. Then, atoms are loaded into the optical lattice by turning off MOT lasers
and quadrupole magnetic field. The lattice laser frequency of 368 554 485.0(1)
MHz is stabilized to an ultra-low-expansion (ULE) cavity with a fineness of
20,000. The lattice AC Stark shift is almost zero at this ”magic frequency”.
Before the energy filtering process, atoms are optically repumped into the
$\left|{}^{1}S_{0},F^{\rm{g}}=9/2,M^{\rm{g}}_{\rm{F}}=+9/2\right\rangle$ state
using a right-handed circularly polarized light resonant on the
$\left|{}^{3}P^{\rm{o}}_{1},F^{\rm{e}}=9/2\right\rangle$ $\rightarrow$
$\left|{}^{1}S_{0},F^{\rm{g}}=9/2\right\rangle$ transition.

Figure 5(b) illustrates the timing scheme for excitation fraction detection.
Following the interrogation of the 698 nm transition, three 461 nm pulses
(each with a duration of 2 ms) are employed to determine the excitation
fraction. The first 461 nm pulse is used to ascertain the number of atoms
remaining in the $\left|{}^{1}S_{0}\right\rangle$ state. Strong radiation
pressure compels atoms out of the lattice. Prior to applying the second 461 nm
pulse to measure atoms in the $\left|{}^{3}P_{0}\right\rangle$ state, 679 nm
and 707 nm lasers are used (with pulse durations of 10 ms) to repump atoms
populated in the $\left|{}^{3}P_{0}\right\rangle$ state back to the
$\left|{}^{1}S_{0}\right\rangle$ state. The third 461 nm pulse is utilized for
assessing background noise.

Figure 5:  (a) Energy levels of 87Sr. (b) Timing scheme of excitation fraction
detection.

##  Appendix B Determination of the free-space Rabi frequency

As atoms have the Boltzmann distribution in the external states, the
excitation fraction with Rabi detection at zero frequency detuning can be
expressed by Blatt2009[20]

$\begin{split}P_{carrier}(t_{\rm{p}})&=\sum_{\rm{n_{x}=1}}^{\rm{N_{x}}}\sum_{\rm{n_{z}=1}}^{\rm{N_{z}}}Q_{\rm{n_{x}}}(T_{\rm{r}})\\\
&Q_{\rm{n_{z}}}(T_{\rm{z}})\sin^{2}(0.5t_{\rm{p}}\Omega_{\rm{n_{x}},\rm{n_{z}}}).\end{split}$
(5)

In Eq.(5), $t_{\rm{p}}$ indicates the interrogation time.
$\Omega_{\rm{n_{x}},\rm{n_{z}}}=\Omega_{0}e^{-\eta^{2}_{\rm{x}}/2}e^{-\eta^{2}_{\rm{z}}/2}L_{\rm{n_{x}}}(\rm{\eta^{2}_{x}})\it{L}_{\rm{n_{z}}}(\rm{\eta^{2}_{z}})$
represents the effective Rabi frequency in the quantum number of external
states ($\rm{n_{x}}$, $\rm{n_{z}}$), where $L_{\rm{n}}$ is the $n$th order
Laguerre polynomial, and
$\eta_{\rm{x}}=\delta\theta/\lambda_{\rm{p}}\sqrt{h/2m_{\rm{a}}v_{\rm{r}}}$
and $\eta_{\rm{z}}=1/\lambda_{\rm{p}}\sqrt{h/2m_{\rm{a}}v_{\rm{z}}}$ are the
Lamb-Dick parameters, in the transverse and longitudinal directions,
respectively. Herein, $\lambda_{\rm{p}}$ is the wavelength of interrogation,
$v_{\rm{z}}$ is the longitudinal trap frequency, $v_{\rm{r}}$ indicates the
transverse trap frequency, $m_{\rm{a}}$ is the mass of 87Sr and $\delta\theta$
represents effective misalignment between lattice and interrogation laser
beams.
$Q_{\rm{n_{x}}}(T_{\rm{r}})=(1-e^{-hv_{\rm{r}}/k_{\rm{b}}T_{\rm{r}}})e^{-n_{\rm{x}}hv_{\rm{r}}/k_{\rm{b}}T_{\rm{r}}}$
and
$Q_{\rm{n_{z}}}(T_{\rm{z}})=(1-e^{-hv_{\rm{z}}/k_{\rm{b}}T_{\rm{z}}})e^{-n_{\rm{z}}hv_{\rm{z}}/k_{\rm{b}}T_{\rm{z}}}$
are the normalized Boltzmann weights. $N_{\rm{x}}$ and $N_{\rm{z}}$ are the
number of states in the trap at the transverse and longitudinal directions,
respectively. According to the resolved sideband spectrum in longitudinal and
transverse directions, parameters of $v_{\rm{z}}$=91.44 kHz, $v_{\rm{r}}$= 340
Hz, $T_{\rm{z}}$= 2.7(3) $\rm{\mu K}$, $T_{\rm{r}}$= 3.9(4) $\rm{\mu K}$,
$N_{\rm{z}}$=7, $N_{\rm{x}}$=1787 can be determined. The free-space Rabi
frequency $\Omega_{0}$ can be extracted by fitting the data of Rabi
oscillation using Eq. (5) with free parameters of $\delta\theta$ and
$\Omega_{0}$.

##  Appendix C The transition rate ratio of $\rm{\pi}$ transition to
$\rm{\sigma}$ transition

In 87Sr, the hyperfine interaction can induce the ${}^{3}P^{\rm{o}}_{1}$ and
${}^{1}P^{\rm{o}}_{1}$ states mixing with the ${}^{3}P^{\rm{o}}_{0}$ state,
which opened the transition channel between the ground state and the
${}^{3}P^{\rm{o}}_{0}$ state. According to the theory of the unexpected
transition Brage2009 ; Andersson2009 ; Li2011 ; Grumer2014[16] , the wave
function of the ${}^{3}P^{\rm{o}}_{0}$ state can be written as linear
combination of the ${}^{3}P^{\rm{o}}_{1}$ and ${}^{1}P^{\rm{o}}_{1}$
perturbing states Lu2023[22]

$\begin{split}&\left|"5s5p\;{}^{3}P^{\rm{o}}_{0},F^{\rm{e}}=IM^{\rm{e}}_{\rm{F}}"\right\rangle=\\\
&\left|5s5p\;{}^{3}P^{\rm{o}}_{0},F^{\rm{e}}=IM^{\rm{e}}_{\rm{F}}\right\rangle+\\\
&\sum_{\rm{s=1,3}}d_{\rm{s}}\left|"5s5p\;{}^{\rm{s}}P^{\rm{o}}_{1},F^{\prime}=IM^{\rm{e}}_{\rm{F}}"\right\rangle,\end{split}$
(6)

where the state within the quotation marks describes the dominant component of
the eigenvector. Other interactions from different configurations can be
neglected due to their large energy separations and weak hyperfine
interactions. In first-order perturbation theory, the mixing coefficients are
given by

$d_{\rm{s}}=\frac{\left\langle
5s5p\;{}^{\rm{s}}P_{1}F^{\prime}M_{\rm{F}}|H_{\rm{hfs}}|5s5p\;{}^{\rm{3}}P_{0}FM_{\rm{F}}\right\rangle}{E(5s5p\;{}^{\rm{3}}P_{0}FM_{\rm{F}})-E(5s5p\;{}^{\rm{s}}P_{1}F^{\prime}M_{\rm{F}})}.$
(7)

The ground state is separate from other states, its wave function is given as

$\left|"5s^{2}\;{}^{1}S_{0},F^{\rm{g}}=IM^{\rm{g}}_{\rm{F}}"\right\rangle=\left|5s^{2}\;{}^{1}S_{0},F^{\rm{g}}=IM^{\rm{g}}_{\rm{F}}\right\rangle.$
(8)

Therefore, the hyperfine induced transition rate
$A(M^{\rm{g}}_{\rm{F}},M^{\rm{e}}_{\rm{F}})$ between the excited Zeeman state
$\left|5s5p\;{}^{3}P^{\rm{o}}_{0},F^{\rm{e}}=IM^{\rm{e}}_{\rm{F}}\right\rangle$
and the ground Zeeman state
$\left|5s^{2}\;{}^{1}S_{0},F^{\rm{g}}=IM^{\rm{g}}_{\rm{F}}\right\rangle$ is
presented as

$\begin{split}&A(M^{\rm{g}}_{\rm{F}},M^{\rm{e}}_{\rm{F}})=\frac{2.02613\times
10^{18}}{\lambda^{\rm{3}}}\sum_{\rm{q}}\\\
&|\left\langle"5s^{2}{}^{1}S_{0}\;,F^{\rm{g}}=IM^{\rm{g}}_{\rm{F}}"|O^{(1)}_{\rm{q}}|"5s5p\;{}^{3}P^{\rm{o}}_{0},F^{\rm{e}}=IM^{\rm{e}}_{\rm{F}}"\right\rangle|^{2}.\end{split}$
(9)

The E1 transition matrix element is in the atomic unit (a.u.), but the units
of $\Omega_{\rm{M^{eg}_{F}}}$ and $I_{0}$ are Hz and $\rm{W/m^{2}}$,
respectively. Thus, the measured value of the transition matrix element should
be divided by $4\rm{\pi}\varepsilon_{0}\hbar^{2}/m_{\rm{e}}e_{0}$ for
conversion to the atomic unit, where $m_{\rm{e}}$ represents the mass of the
electron, and $e_{0}$ represents the charge of the electron. By submitting
Eqs. (6) and (8) into the above equation, the transition rate
$A(M^{\rm{g}}_{\rm{F}},M^{\rm{e}}_{\rm{F}})$ is then given as

$\begin{split}&A(M^{\rm{g}}_{\rm{F}},M^{\rm{e}}_{\rm{F}})=\frac{2.02613\times
10^{18}}{\lambda^{\rm{3}}}\times\\\
&\sum_{\rm{q}}|\sum_{\rm{s=1,3}}d_{\rm{s}}\sqrt{2F^{\rm{g}}+1}\sqrt{2F^{\rm{e}}+1}\\\
&{\begin{pmatrix}F^{g}&1&F^{\rm{e}}\\\
-M^{g}_{F}&q&M^{e}_{F}\end{pmatrix}}{\begin{Bmatrix}J^{g}&F^{g}&I\\\
F^{\rm{e}}&J^{e}&1\end{Bmatrix}\langle{5s^{2}\
{}^{1}\\!S_{0}}||O^{(1)}_{\rm{q}}||{5s5p\;{}^{\rm{s}}\\!P^{\rm{o}}_{1}}\rangle{|^{2}}}.\end{split}$
(10)

According to Eq. (10), the ratio of the E1 transition matrix elements between
the
$\left|5s5p\;{}^{3}P^{\rm{o}}_{0},F^{\rm{e}}=9/2,M^{\rm{e}}_{\rm{F}}=+9/2\right\rangle$
$\rightarrow$
$\left|5s^{2}\;{}^{1}S_{0},F^{\rm{g}}=9/2,M^{\rm{g}}_{\rm{F}}=+9/2\right\rangle$
and
$\left|5s5p\;{}^{3}P^{\rm{o}}_{0},F^{\rm{e}}=9/2,M^{\rm{e}}_{\rm{F}}=+9/2\right\rangle$
$\rightarrow$
$\left|5s^{2}\;{}^{1}S_{0},F^{\rm{g}}=9/2,M^{\rm{g}}_{\rm{F}}=+7/2\right\rangle$
transitions is ${\begin{pmatrix}9/2&1&9/2\\\
-9/2&0&9/2\end{pmatrix}}/{\begin{pmatrix}9/2&1&9/2\\\
-7/2&-1&9/2\end{pmatrix}}=-3/\sqrt{2}$. The corresponging ratio of transition
rate is its square and equals 9/2.

##  Appendix D Measurements of the servo-bumps and spectral peaks

The probe laser utilized in this study is compared with another 698 nm
reference laser. The reference laser employs the cavity transmission light
(with a power of approximately 19.8 $\rm{\mu W}$) to filter its servo-bumps. A
photodetector (FPD310-FS-VIS, MenloSystems) with a frequency response range
from 1 MHz to 1.5 GHz is employed to detect the beat signal. In Figure 6(a),
the power spectral densities of the beat signal are displayed through Fourier
analysis using a spectrum analyzer (N9030A, KEYSIGHT). By simultaneously
examining the Fourier analysis of the PDH error, we validate that the observed
servo-bumps originate from the laser used in this experiment Akerman2015 . In
both cases, the peak frequencies of the servo-bumps are around $\pm 0.352(4)$
MHz. To quantify the contributions of the servo-bumps and spectral peaks to
the total light power, we integrate the regions excluding the carrier in the
linear coordinates presented in Fig. 6(b). It is estimated that the servo-
bumps and spectral peaks contribute 0.815(8)% to the total power, with the
uncertainty indicating the 1$\sigma$ standard error of 10 measurements.

Figure 6:  Spectra of the 698 nm probe laser. (a) The servo-bumps and spectral
peaks within the logarithmic Y-axis. The inset shows the Fourier analysis of
the PDH error signal of the probe light used in this work. (b) The same data
in (a) but shown in the linear Y-axis. The inset shows the details of the
small servo-bumps and spectral peaks.

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Report on the 61st Annual International Mathematical Olympiad

BÉLA BAJNOK

Gettysburg College
Gettysburg, PA 17325
<EMAIL_ADDRESS>
EVAN CHEN

Massachusetts Institute of Technology
Cambridge, MA 02139
<EMAIL_ADDRESS>
The International Mathematical Olympiad (IMO) is the world’s leading
mathematics competition for high school students, and is organized annually by
different host countries. The competition consists of three problems each on
two consecutive days, with an allowed time of four and a half hours both days.
In recent years, more than one hundred countries send teams of up to six
students to compete.

The 61st IMO was to take place in July of 2020 in St. Petersburg, Russia. Due
to the COVID-19 pandemic, however, the competition was postponed and moved
remotely where students could take the exam in their home countries. The
competition was ultimately given on September 21 and 22, 2020, at universally
coordinated times (which meant that students in the Americas had to start
their work in the middle of the night).

The members of the US team are chosen during the Math Olympiad Program (MOP)
each year, a year-long endeavor organized by the MAA’s American Mathematics
Competitions (AMC) program. Students gain admittance to MOP based on their
performance on a series of examinations, culminating in the USA Mathematical
Olympiad (USAMO). A report on the 2020 USAMO can be found in the April 2021
issue of this Magazine; more information on the American Mathematics
Competitions program can be found on the site https://www.maa.org/math-
competitions.

The members of the 2020 US team were Quanlin Chen (11th grade, Princeton
International School of Mathematics and Science, Princeton, NJ); Gopal Krishna
Goel (11th grade, Krishna Homeschool, Portland, OR); Tianze Jiang (11th grade,
Princeton International School of Mathematics and Science, Princeton, NJ);
Jeffrey Kwan (12th grade, Harker Upper School, San Jose, CA); Luke Robitaille
(10th grade, Robitaille Homeschool, Euless, Texas); and William Wang (West
Windsor-Plainsboro High School North, Plainsboro Township, NJ). Chen,
Robitaille, and Wang each earned Gold Medals, and Goel, Jiang, and Kwan each
earned Silver Medals. In the unofficial ranking of countries, the United
States finished third after China (first) and Russia (second).

Below we present the problems and solutions of the 61st IMO. The solutions we
feature here are those of the indicated students, as edited by the two authors
of this report.

Problem 1 proposed by Dominik Burek, Poland. Consider the convex quadrilateral
$ABCD$. The point $P$ is in the interior of $ABCD$. The following ratio
equalities hold:

$\angle PAD:\angle PBA:\angle DPA=1:2:3=\angle CBP:\angle BAP:\angle BPC.$

Prove that the following three lines meet in a point: the internal bisectors
of angles $\angle ADP$ and $\angle PCB$ and the perpendicular bisector of
segment $AB$.

Solution by Isabella Quan, 10th grade, Westlake High School, Austin, TX. Let
$O$ be the circumcenter of $\triangle PAB$. We will prove that each of the
three lines specified in the problem go through $O$. Since $AO=BO$, this
clearly holds for the perpendicular bisector of $AB$. Below we prove that $O$
lies on the internal bisector of $\angle PCB$; the claim about the internal
bisector of $\angle ADP$ can be verified similarly.

Note that

$\angle PCB=\pi-(\angle CBP+\angle BPC)$

and, by the given condition,

$\angle CBP+\angle BPC=2\angle BAP.$

But by the inscribed angle theorem, $\angle BOP=2\angle BAP$ as well, and thus
$\angle PCB$ and $\angle BOP$ are supplementary, which implies that
quadrilateral $BOPC$ is cyclic. Because $OP=OB$, the inscribed angles $\angle
PCO$ and $\angle OCB$ in the circumcircle of quadrilateral $BOPC$ are equal,
and hence $O$ lies on the angle bisector of $\angle PCB$, as claimed.

Problem 2 proposed by Stijn Cambie, Netherlands. The real numbers $a,b,c,d$
are such that $a\geq b\geq c\geq d>0$ and $a+b+c+d=1$. Prove that

$(a+2b+3c+4d)a^{a}b^{b}c^{c}d^{d}<1.$

Solution by Luke Robitaille, 10th grade, Robitaille Homeschool, Euless, TX. We
will prove that, for real numbers $a$, $b$, $c$, and $d$ satisfying the
conditions of the problem, we have:

$\displaystyle a^{a}b^{b}c^{c}d^{d}$ $\displaystyle\leq$ $\displaystyle
a^{2}+b^{2}+c^{2}+d^{2}$ (1)

and

$\displaystyle(a+2b+3c+4d)(a^{2}+b^{2}+c^{2}+d^{2})$ $\displaystyle<$
$\displaystyle 1;$ (2)

these two inequalities together clearly imply the claim of the problem.

Our first inequality follows from the Weighted AM-GM Inequality, as we now
explain. Let us recall that for a positive integer $n$ and nonnegative real
numbers $x_{1},\dots,x_{n},$ $w_{1},\dots,w_{n}$ with
$w=w_{1}+\cdots+w_{n}>0$, the weighted arithmetic mean of $x_{1},\dots,x_{n}$
with corresponding weights $w_{1},\dots,w_{n}$ is defined as
$(w_{1}x_{1}+\cdots+w_{n}x_{n})/w,$ and their weighted geometric mean is
defined as $\sqrt[w]{{x_{1}}^{w_{1}}\cdot\cdots\cdot{x_{n}}^{w_{n}}}.$ The
Weighted AM-GM Inequality then states that the weighted arithmetic mean is
always at least as much as the weighted geometric mean; that is:

$\displaystyle\sqrt[w]{{x_{1}}^{w_{1}}\dots x_{n}^{w_{n}}}$
$\displaystyle\leq$ $\displaystyle(w_{1}x_{1}+\cdot\cdots\cdot+w_{n}x_{n})/w.$

(The case when all weights equal 1 yields the regular AM-GM Inequality.)
Applying the Weighted AM-GM Inequality for $n=4$, $x_{1}=w_{1}=a$,
$x_{2}=w_{2}=b$, $x_{3}=w_{3}=c$, and $x_{4}=w_{4}=d$, and noting that
$a+b+c+d=1$, readily gives (1).

To establish (2), we prove more generally that for real numbers $a$, $b$, $c$,
and $d$ with $a\geq b\geq c\geq d>0$, we have

$\displaystyle(a+2b+3c+4d)(a^{2}+b^{2}+c^{2}+d^{2})$ $\displaystyle<$
$\displaystyle(a+b+c+d)^{3}.$ (3)

In order to do so, we transform to new variables $r,s,t,u$, defined as
$r=a-b$, $s=b-c$, $t=c-d$, and $u=d$; our assumptions imply that $r,s,t\geq 0$
and $u>0$. Since $a=r+s+t+u$, $b=s+t+u$, $c=t+u$, and $d=u$, we have

$\displaystyle a+2b+3c+4d$ $\displaystyle=$ $\displaystyle r+3s+6t+10u$

and

$\displaystyle a^{2}+b^{2}+c^{2}+d^{2}$ $\displaystyle=$
$\displaystyle(r+s+t+u)^{2}+(s+t+u)^{2}+(t+u)^{2}+u^{2}$ $\displaystyle=$
$\displaystyle r^{2}+2s^{2}+3t^{2}+4u^{2}+2rs+2rt+2ru+4st+4su+6tu,$

and thus we get

$\displaystyle(a+2b+3c+4d)(a^{2}+b^{2}+c^{2}+d^{2})$ $\displaystyle=$
$\displaystyle r^{3}+6s^{3}+18t^{3}+40u^{3}+$ $\displaystyle
5r^{2}s+8r^{2}t+12r^{2}u+8rs^{2}+24s^{2}t+32s^{2}u+$ $\displaystyle
15rt^{2}+33st^{2}+66t^{2}u+24ru^{2}+52su^{2}+84tu^{2}+$ $\displaystyle
22rst+30rsu+38rtu+82stu.$

Furthermore,

$\displaystyle(a+b+c+d)^{3}$ $\displaystyle=$ $\displaystyle(r+2s+3t+4u)^{3}$
$\displaystyle=$ $\displaystyle r^{3}+8s^{3}+27t^{3}+64u^{3}+$ $\displaystyle
6r^{2}s+9r^{2}t+12r^{2}u+12rs^{2}+36s^{2}t+48s^{2}u+$ $\displaystyle
27rt^{2}+54st^{2}+108t^{2}u+48ru^{2}+96su^{2}+144tu^{2}+$ $\displaystyle
36rst+48rsu+72rtu+144stu.$

Making term-wise comparisons, we can verify that

$(a+2b+3c+4d)(a^{2}+b^{2}+c^{2}+d^{2})\leq(a+b+c+d)^{3},$

and since $u>0$ implies that $40u^{3}<64u^{3}$, we see that, in fact, strict
inequality holds, which proves (3). This completes our proof.

Problem 3 proposed by Milan Haiman (Hungary) and Carl Schildkraut (USA). There
are $4n$ pebbles of weights $1,2,3,\dots,4n$. Each pebble is colored in one of
$n$ colors, and there are four pebbles of each color. Show that we can arrange
the pebbles into two piles, so that the total weights of both piles are the
same and each pile contains two pebbles of each color.

Solution by Alex Zhao, 8th grade, Kamiakin Middle School, Kirkland, WA. We
begin by pairing off the pebbles so that each pair sums to the same value;
namely, pairing $1$ and $4n$, $2$ and $4n-1$, $3$ and $4n-2$, …, so that each
pair adds up to $4n+1$. Then, it is sufficient to split these $2n$ _pairs_
into two groups of $n$ pairs each, such that each pile contains two pebbles of
each color.

We separate the pebbles by color, and put pebbles of the same color into a box
of their own, and then draw a line between the two pebbles of each pair. (Some
of the lines could start and end in the same box.) The problem can then be
rephrased as follows: we wish to color each of the lines either blue or green,
such that each box has two pebbles which are endpoints of blue lines, and two
pebbles which are endpoints of green lines. (These new colors are not related
to the colors in the original problem. They correspond to whether we put the
pair in one pile or the other.) An example with $n=5$ is illustrated below.

We can capture this information in a single multigraph $G$. (A multigraph is
like a graph, but edges may be repeated; we also allow an edge to join a
vertex to itself.) Each box is a vertex of $G$, and each line is an edge of
the multigraph. Hence, this graph will have $n$ vertices and degree $4$ at
every vertex. Shown below is the multigraph $G$ corresponding to the example
we drew before. (The circle that passes through $A$ denotes the edge from $A$
to itself.)

Consider any connected component of this multigraph. Since the degree of every
vertex is even, the component has an _Eulerian cycle_ — a cycle passing
through every edge. (In the above figure, one example of an Eulerian cycle is
$AABCDEBECDA$.)

So, consider any connected component and one of its Eulerian cycles. Since the
degree of each vertex is 4, the sum of all degrees is a multiple of 4, and so
the number of edges (half the sum of the degrees) is even. Therefore we can
alternately color the edges in the Eulerian cycle blue and green. Shown below
is an alternating coloring of $AABCDEBECDA$ in this way.

We can now easily see that this coloring works; indeed:

  * •

If the vertex has one self-loop, the two outgoing edges must be colored the
same, and differently from the self-loop.

  * •

If the vertex has no self-loops, the cycle visits the vertex twice, and the
two edges going in and out of each visit are colored differently from each
other.

  * •

If the vertex has two self-loops (which can only happen if the connected
component has only a single vertex) then those two self-loops are colored
differently.

Having found the desired coloring, the problem is solved. The figure below
shows the translation of our coloring example into a valid partition of the
pairs.

Problem 4 proposed by Tejaswi Navrinarekallu, India. Let $n>1$ be an integer.
There are $n^{2}$ stations on a slope of a mountain, all at different
altitudes. Each of two cable car companies, $A$ and $B$, operates $k$ cable
cars; each cable car provides a transfer from one of the stations to a higher
one (with no intermediate stops). The $k$ cable cars of $A$ have $k$ different
starting points and $k$ different finishing points, and a cable car that
starts higher also finishes higher. The same conditions hold for $B$. We say
that two stations are linked by a company if one can start from the lower
station and reach the higher one by using one or more cars of that company (no
other movements between stations are allowed). Determine the smallest positive
integer $k$ for which one can guarantee that there are two stations that are
linked by both companies.

Solution by Ryan Li, 10th grade, Solon High School, Solon, OH. We claim that
the answer is $k=n^{2}-n+1$.

We label the stations by $0$, $1$, $2$, …, $n^{2}-1$ in ascending order of
altitude. Let $G_{A}$ be a simple graph with $n^{2}$ vertices defined as
follows:

  * •

The vertices are the $n^{2}$ stations.

  * •

Two vertices are joined by an edge if and only if they are joined by a cable
car from company $A$.

The graph $G_{B}$ is defined in an analogous way, with the connections from
company $B$ instead of company $A$.

> Lemma. Two stations are linked by $A$ (as defined in the problem statement)
> if and only if they are connected in the graph $G_{A}$.

###### Proof.

Obviously, being linked implies being connected. It remains be verified that
the converse is true.

Suppose we have a path from station $i$ to station $j$ in $G_{A}$, where
$i<j$, given by labels $i=a_{0}\to a_{1}\to\cdots\to a_{m-1}\to a_{m}=j$. We
claim that

$i<a_{1}<a_{2}<\cdots<a_{m-1}<j.$

Indeed, note that for $r=1,2,\dots,m-1$, each $a_{r}$ is connected to
$a_{r-1}$ and $a_{r+1}$ by cable cars . Since the starting points and ending
points of each cable car are unique, $a_{r-1}$ and $a_{r+1}$ cannot be both
higher or lower than $a_{r}$, so either $a_{r-1}<a_{r}<a_{r+1}$ or
$a_{r-1}>a_{r}>a_{r+1}$ for each $r$.

Since $i<j$, this implies $i=a_{0}<a_{1}$, $a_{1}<a_{2}$, …,
$a_{m-1}<a_{m}=j$, as claimed. ∎

Now we prove that $k=n^{2}-n+1$. First, we show that there exists a
construction for $k=n^{2}-n$ such that no two stations are linked by both
company $A$ and company $B$. Consider the situation in which for every integer
$0\leq i\leq n-1$, company $A$ links together $i,i+n,i+2n,\dots,i+n^{2}-n$ in
a series and company $B$ links together stations $ni,ni+1,ni+2,\dots,ni+n-1$.
This gives a total of $n^{2}-n$ cable cars for each company. Depicted below is
the case for $n=4$, with cars for $A$ in bold, and cars for $B$ dashed.

It is clear that no car is linked by both companies: since any two cars linked
by $A$ have indices differing by at least $n$, while any two cars linked by
$B$ have indices differing by at most $n-1$.

Finally, we prove that $k=n^{2}-n+1$ suffices. In the earlier proof of the
lemma, we saw that a path from a vertex $i$ to a vertex $j$ must in fact pass
through strictly increasing indices. This implies that $G_{A}$ and $G_{B}$ are
acyclic — i.e., they are forests.

Now since $G_{A}$ has $n^{2}$ vertices and $k$ edges but no cycles, the number
of connected components in $G_{A}$ is exactly

$n^{2}-k=n^{2}-(n^{2}-n+1)=n-1.$

So by the pigeonhole principle, some connected component $S$ of $G_{A}$ has at
least $\left\lceil\frac{n^{2}}{n-1}\right\rceil=n+1$ vertices. On the other
hand, $G_{B}$ also has exactly $n-1$ connected components. So by the
pigeonhole principle again, there are two vertices in $S$ which lie in the
same connected component of $G_{B}$. These two stations are therefore linked
by both companies.

Problem 5 proposed by Oleg Košik, Estonia. A deck of $n>1$ cards is given. A
positive integer is written on each card. The deck has the property that the
arithmetic mean of the numbers on each pair of cards is also the geometric
mean of the numbers on some collection of one or more cards. For which $n$
does it follow that the numbers on the cards are all equal?

Solution by Pravalika Putalapattu, 10th grade, Thomas Jefferson High School
for Science and Technology, Alexandria, VA. We will prove that this assertion
is true for all $n$.

Proceeding indirectly, let us assume that there exists some positive integer
$n$ and a set of cards labeled $a_{1},a_{2},\ldots,a_{n}$ that satisfy the
above property but are not all equal. We may assume without loss of generality
that our cards are in decreasing order of their labels:

$a_{1}\geq a_{2}\geq\cdots\geq a_{n}.$

Note that multiplying all card values by some constant will multiply all
arithmetic and geometric means by the same value. Thus, the property is
preserved if we replace each label $a_{i}$ with $a_{i}/d$, where
$d=\gcd(a_{1},a_{2},\ldots,a_{n})$. Therefore, we may assume without loss of
generality that the $n$ labels are relatively prime.

If $a_{1}=1$, then all $a_{i}$ must equal 1, violating our assumption that not
all $a_{i}$ are equal, thus, $a_{1}\geq 2$. Let $p$ be a prime divisor of
$a_{1}$. Since $\gcd(a_{1},a_{2},\ldots,a_{n})=1$, there must be some $a_{i}$
not divisible by $p$. Let $k$ be the smallest index such that $p$ does not
divide $a_{k}$. Note that $a_{k}<a_{1}$.

Now, consider the arithmetic mean of $a_{1}$ and $a_{k}$, which we know is
greater than $a_{k}$. This must equal some geometric mean, and thus

$\frac{a_{1}+a_{k}}{2}=\sqrt[m]{a_{i_{1}}\cdots a_{i_{m}}}$

for some positive integer $m$ and indices $i_{1},\dots,i_{m}$. In particular,

$\sqrt[m]{a_{i_{1}}\cdots a_{i_{m}}}>a_{k}.$

Next, observe that, since $a_{k}$ is not divisible by $p$, none of
$a_{i_{1}},\dots,a_{i_{m}}$ can be divisible by $p$ either. Since $k$ is the
smallest index such that $p$ does not divide $a_{k}$, each of
$i_{1},\ldots,i_{m}$ must be at least $k$, and therefore each of
$a_{i_{1}},\ldots,a_{i_{m}}$ must be at most $a_{k}$. But then their geometric
mean is also at most $a_{k}$, resulting in a contradiction, as desired.

Problem 6 proposed by Ting-Feng Lin and Hung-Hsun Hans Yu, Taiwan. Consider an
integer $n>1$ and a set $\mathcal{S}$ of $n$ points in the plane such that the
distance between any two different points in $\mathcal{S}$ is at least $1$.
Prove that there is a line $\ell$ separating $\mathcal{S}$ such that the
distance from any point of $\mathcal{S}$ to $\ell$ is at least $cn^{-1/3}$ for
some positive constant $c$. (A line $\ell$ separates a set of points $S$ if
some segment joining two points in $\mathcal{S}$ crosses $\ell$.)

Solution by Gopal Krishna Goel, 11th grade, Krishna Homeschool, Portland, OR,
and Jaedon Whyte, 10th grade, Archimedean Upper Conservatory, Miami, FL. We
will show that there exists a line $\ell$ separating $\mathcal{S}$ such that
the distance from any point in $\mathcal{S}$ to $\ell$ is at least $0.01\cdot
n^{-1/3}$.

Let $[\mathcal{P}]$ denote the area of a region $\mathcal{P}$ in the plane. We
will need the following lemma.

> Lemma. Let $\mathcal{R}$ be an $a\times b$ rectangle in the plane, where
> $a,b\geq\tfrac{1}{2}$. Then $|\mathcal{S}\cap\mathcal{R}|\leq
> 20[\mathcal{R}]$.

###### Proof.

For each point $P$, let $\mathcal{D}_{P}$ denote the open disk of radius
$\tfrac{1}{2}$ centered at $P$. Since $PQ\geq 1$ for all distinct
$P,Q\in\mathcal{S}$, their respective disks $\mathcal{D}_{P}$ and
$\mathcal{D}_{Q}$ cannot intersect.

Now, for any $P\in\mathcal{S}\cap\mathcal{R}$, we see that
$\mathcal{D}_{P}\subseteq\mathcal{R}^{\prime}$, where $\mathcal{R}^{\prime}$
is an $(a+1)\times(b+1)$ rectangle with each side of $\mathcal{R}^{\prime}$ a
distance exactly $1/2$ from a side of $\mathcal{R}$. For the total area
covered by the disks we have:

$\sum_{P\in\mathcal{S}\cap\mathcal{R}}[\mathcal{D}_{P}]\leq[\mathcal{R}^{\prime}]=|\mathcal{S}\cap\mathcal{R}|\cdot\frac{\pi}{4}=(a+1)(b+1).$

Since $a,b\geq\tfrac{1}{2}$, we see that
$\frac{(a+1)(b+1)}{ab}=\left(1+\frac{1}{a}\right)\left(1+\frac{1}{b}\right)\leq
3\cdot 3=9$, so:

$|\mathcal{S}\cap\mathcal{R}|\leq\frac{4}{\pi}(a+1)(b+1)\leq\frac{36}{\pi}ab<20[\mathcal{R}],$

as desired. ∎

Let $A$ and $B$ denote the two points in $\mathcal{S}$ such that $AB$ is
maximized (if there are multiple pairs, pick one arbitrarily), and let $r=AB$
be this maximum value. Let $\mathcal{T}$ denote the set of points $P$ in the
plane such that $\max(AP,BP)\leq r$. We see that $\mathcal{T}$ is simply the
intersection of the disk centered at $A$ with radius $r$ and the disk centered
at $B$ with radius $r$, as shown below. By definition, we must have
$\mathcal{S}\subseteq\mathcal{T}$.

We will first resolve the case $r\geq n^{2/3}$. Project all the points of
$\mathcal{S}$ onto $AB$, and suppose they are
$A=P_{1},P_{2},P_{3},\ldots,P_{n}=B$, appearing in that order (it is possible
that we may have $P_{i}=P_{i+1}$ for some values of $i$). Now, since

$P_{1}P_{2}+P_{2}P_{3}+\cdots+P_{n-1}P_{n}=r,$

we see that there exists some $i$ such that

$P_{i}P_{i+1}\geq\frac{r}{n-1}>r/n\geq n^{-1/3}.$

Take $\ell$ to be the perpendicular bisector of $P_{i}P_{i+1}$. We see that
the distance from any point $P\in\mathcal{S}$ to $\ell$ is simply the distance
from the projection of $P$ onto $AB$ to the midpoint of $P_{i}P_{i+1}$, which
is at least

$\frac{1}{2}P_{i}P_{i+1}>\frac{1}{2}n^{-1/3},$

as desired. This resolves the case $r\geq n^{2/3}$.

We now turn our attention to the case $r<n^{2/3}$. Let $XY$ be a chord of the
circle centered at $B$ with radius $r$ such that $XY\perp AB$ and the distance
from $A$ to $XY$ is $\tfrac{1}{2}$. Let $M$ be its midpoint. Since $r=AB\geq
1$, we see that $X$ and $Y$ are on the arc of the circle centered at $B$ with
radius $r$ that is a part of $\mathcal{T}$. The line $XY$ divides
$\mathcal{T}$ into two parts; let $\mathcal{T}^{\prime}$ be the part
containing $A$.

Note that $\mathcal{T}^{\prime}$ can be contained in a rectangle of size
$\tfrac{1}{2}\times XY$. By the Pythagorean theorem on $\triangle{XMB}$, we
have

$XY=2\sqrt{r^{2}-\left(r-\frac{1}{2}\right)^{2}}=2\sqrt{r-\frac{1}{4}}<2\sqrt{r}<2n^{1/3},$

so by the lemma above, we see that

$|\mathcal{T}^{\prime}\cap\mathcal{S}|\leq
20\cdot\frac{1}{2}\cdot|XY|<20n^{1/3}.$

Similar to before, let
$Q_{1}=A,Q_{2},Q_{3},\ldots,Q_{|\mathcal{T}^{\prime}\cap\mathcal{S}|}$ denote
the projections of the points of $\mathcal{T}^{\prime}\cap\mathcal{S}$ onto
segment $AB$, in that order.

If $A$ is the only point of $S$ in $\mathcal{T}^{\prime}$, then we take our
line as the perpendicular bisector of line $AM$. So assume
$|\mathcal{T^{\prime}}\cap\mathcal{S}|\geq 2$. Since all points $Q_{i}$ lie in
a segment of length $\tfrac{1}{2}$, so there must be some $i$ such that

$Q_{i}Q_{i+1}\geq\frac{\tfrac{1}{2}}{|\mathcal{T}^{\prime}\cap\mathcal{S}|-1}>\frac{1}{2|\mathcal{T}^{\prime}\cap\mathcal{S}|}>\frac{1}{40}n^{-1/3}.$

We may now take $\ell$ to be the perpendicular bisector of $Q_{i}Q_{i+1}$, and
as before, the distance from any point of $\mathcal{S}$ to $\ell$ is at least
$\tfrac{1}{80}n^{-1/3}>0.01n^{-1/3}$, as desired. This completes the solution.

BÉLA BAJNOK (MR Author ID: 314851) is a Professor of Mathematics at Gettysburg
College and the Director of the American Mathematics Competitions program of
the MAA.

EVAN CHEN (MR Author ID: 1158569) is a graduate student at the Massachusetts
Institute of Technology and the Assistant Academic Director of the Math
Olympiad Program of the MAA.