SAPPHIRES collaboration

# Search for sub-eV axion-like particles in a stimulated resonant photon-
photon collider with two laser beams based on a novel method to discriminate
pressure-independent components

Yuri Kirita  Graduate School of Advanced Science and Engineering, Hiroshima
University, Kagamiyama, Higashi-Hiroshima, Hiroshima 739-8526, Japan  Takumi
Hasada  Graduate School of Advanced Science and Engineering, Hiroshima
University, Kagamiyama, Higashi-Hiroshima, Hiroshima 739-8526, Japan  Masaki
Hashida  Institute for Chemical Research, Kyoto University Uji, Kyoto
611-0011, Japan  Research Institute of Science and Technology, Tokai
University, 4-1-1 Kitakaname, Hiratsuka, Kanagawa 259-1292, Japan  Yusuke
Hirahara  Graduate School of Advanced Science and Engineering, Hiroshima
University, Kagamiyama, Higashi-Hiroshima, Hiroshima 739-8526, Japan  Kensuke
Homma111corresponding author Graduate School of Advanced Science and
Engineering, Hiroshima University, Kagamiyama, Higashi-Hiroshima, Hiroshima
739-8526, Japan  Shunsuke Inoue  Institute for Chemical Research, Kyoto
University Uji, Kyoto 611-0011, Japan  Graduate School of Science, Kyoto
University, Sakyouku, Kyoto 606-8502, Japan  Fumiya Ishibashi  Graduate School
of Advanced Science and Engineering, Hiroshima University, Kagamiyama,
Higashi-Hiroshima, Hiroshima 739-8526, Japan  Yoshihide Nakamiya  Institute
for Chemical Research, Kyoto University Uji, Kyoto 611-0011, Japan  Extreme
Light Infrastructure-Nuclear Physics (ELI-NP)/Horia Hulubei National Institute
for R&D in Physics and Nuclear Engineering (IFIN-HH), 30 Reactorului St., P.O.
Box MG-6, Bucharest-Magurele, Judetul Ilfov, RO-077125, Romania  Liviu Neagu
Extreme Light Infrastructure-Nuclear Physics (ELI-NP)/Horia Hulubei National
Institute for R&D in Physics and Nuclear Engineering (IFIN-HH), 30 Reactorului
St., P.O. Box MG-6, Bucharest-Magurele, Judetul Ilfov, RO-077125, Romania
National Institute for Laser, Plasma and Radiation Physics, 409 Atomistilor PO
Box MG-36, 077125, Magurele, Jud. Ilfov, Romania  Akihide Nobuhiro  Graduate
School of Advanced Science and Engineering, Hiroshima University, Kagamiyama,
Higashi-Hiroshima, Hiroshima 739-8526, Japan  Takaya Ozaki  Graduate School of
Advanced Science and Engineering, Hiroshima University, Kagamiyama, Higashi-
Hiroshima, Hiroshima 739-8526, Japan  Madalin-Mihai Rosu  Extreme Light
Infrastructure-Nuclear Physics (ELI-NP)/Horia Hulubei National Institute for
R&D in Physics and Nuclear Engineering (IFIN-HH), 30 Reactorului St., P.O. Box
MG-6, Bucharest-Magurele, Judetul Ilfov, RO-077125, Romania  Shuji Sakabe
Institute for Chemical Research, Kyoto University Uji, Kyoto 611-0011, Japan
Graduate School of Science, Kyoto University, Sakyouku, Kyoto 606-8502, Japan
Ovidiu Tesileanu  Extreme Light Infrastructure-Nuclear Physics (ELI-NP)/Horia
Hulubei National Institute for R&D in Physics and Nuclear Engineering (IFIN-
HH), 30 Reactorului St., P.O. Box MG-6, Bucharest-Magurele, Judetul Ilfov,
RO-077125, Romania

###### Abstract

Sub-eV axion-like particles (ALPs) have been searched for by focusing two-
color near-infrared pulse lasers into a vacuum along a common optical axis.
Within the focused quasi-parallel collision system created by combining a
creation field ($2.5\,\mathrm{mJ}/47\,\mathrm{fs}$ Ti:Sapphire laser) and a
background inducing field ($1.5\,\mathrm{mJ}/9\,\mathrm{ns}$ Nd:YAG laser),
the detection of signal photons via stimulated resonant photon-photon
scattering by exchanging ALPs was attempted in a vacuum chamber. The signal
wavelength can be determined via energy-momentum conservation in the vacuum,
and it coincides with that determined from the atomic four-wave-mixing (aFWM)
process. In this work, the pulse energies were one order of magnitude higher
than those in the previous search, allowing aFWM from optical elements to be
observed as a pressure-independent background for the first time, in addition
to the residual-gas-originating aFWM following a quadratic pressure
dependence. In principle the four-wave-mixing process in vacuum via ALP
exchanges (vFWM) must also be pressure-independent, so the development of a
new method for discriminating the optical-element aFWM is indispensable for
increasing the pulse energies to the values needed for future upgraded
searches. In this paper, we will present the established method for
quantifying the yield from the optical-element aFWM process based on the beam
cross-section dependence. With the new method, the number of signal photons
was found to be consistent with zero. We then successfully obtained a new
exclusion region in the relation between ALP-photon coupling, $g/M$, and the
ALP mass $m$, reaching the most sensitive point $g/M=1.14\times
10^{-5}\,\mathrm{GeV^{-1}}$ at $m=0.18\,\mathrm{eV}$.

##  I Introduction

Axion and axion-like particles (ALPs) can be generic candidates for the dark
components of the Universe if their couplings to matter are reasonably weak
with the proper masses. Axion is a pseudoscalar boson associated with the
breaking of the Peccei-Quinn symmetry which is introduced to solve the strong
CP problem PQ , and could be cold dark matter. Additionally, a miracle
scenario miracle  which explains both dark matter and inflation predicts the
existence of an ALP having a mass range that overlaps with the axion models.
Scalar fields, such as dilaton dilaton ; DEptp  and chameleon chameleon ;
HuntingDE , have been widely explored in attempts to explain the dark energy
in the Universe.

In the low mass region, many experiments pvlas ; alps ; osqar ; PTEP-EXP00
have been conducted focusing on ALP-photon coupling as a method of searching
for ALPs. In this paper, we focus on the coupling between sub-eV ALPs and two
photons. For scalar ($\phi$)- and pseudoscalar ($\sigma$)-type ALPs, the
following effective Lagrangians are assumed:

$-L_{\phi}=gM^{-1}\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\phi,\hskip 10.0pt-
L_{\sigma}=gM^{-1}\frac{1}{4}F_{\mu\nu}\tilde{F}^{\mu\nu}\sigma,$ (1)

where $g$ is a dimensionless constant for a given energy scale $M$ at which a
relevant global symmetry is broken.
$F^{\mu\nu}=\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}$ is the
electromagnetic field strength tensor, and its dual tensor is defined 222We
explicitly note that the factor $1/2$ is newly multiplied to the definition
used in the previous search SAPPHIRES00 .  as
$\tilde{F}^{\mu\nu}\equiv\frac{1}{2}\varepsilon^{\mu\nu\alpha\beta}F_{\alpha\beta}$
with the Levi-Civita symbol $\varepsilon^{ijkl}$.

---
Figure 1: Concept of a stimulated resonant photon-photon collision (vFWM
process) via scalar ($\phi$) or pseudoscalar ($\sigma$) fields. The coaxially
combined green and red lasers, which are, respectively, used to create ALPs
and to simultaneously induce those decays, are focused with a parabolic mirror
in a vacuum. The wavy arrows represent two-body photon-photon scattering via
ALP exchange in the laser fields. The induced signal photons with wavelength
determined by energy-momentum conservation (blue arrow) are produced primarily
around the focal point.

The SAPPHIRES collaboration searches for sub-eV ALPs via stimulated resonant
photon-photon collisions (SRPC) by focusing two lasers in the manner
illustrated in Fig. 1. A collision between two photons with a shallow incident
angle in a creation field (Ti:Sapphire laser, the green beam in Fig. 1)
produces an ALP resonance state while a background field (Nd:YAG laser, the
red beam in Fig. 1) simultaneously induces the ALP decay. The wavelength of
the signal photon (the wavy blue line) in the final state can differ from
those of the lasers due to energy-momentum conservation. In the terminology of
laser physics, this process can be interpreted as four-wave-mixing in vacuum
(vFWM).

Figure 2 illustrates possible background sources with respect to vFWM. The
dominant background process is four-wave mixing originating from atoms (aFWM).
Both vFWM and aFWM are kinematically similar and indistinguishable with the
exception of polarization states of signal photons, which depend on the types
of exchanged ALPs. The dominant background source of aFWM is residual gas
around the focal spot in the vacuum chamber that interacts with the high-
intensity [$\mathrm{W/m^{2}}$] laser fields. However, a new background
component originating from optical elements along the propagation path of the
laser fields is expected to be significant for much higher intensities, even
when the beams have not been focused. In addition to aFWM, other background
sources include the plasma generated by focusing lasers and the thermal noise
in the photo-detection device.

---
Figure 2: Possible background sources originating from atoms and plasma in the
quasi-parallel collision system. The atomic four-wave mixing (aFWM) photons
can be produced from residual gas and optical elements, with the same
wavelength as the signal photons. Plasma can be generated by focusing high-
intensity laser fields at the focal point and ionizing the residual gas around
it.

In the SAPPHIRES00 search SAPPHIRES00 , we looked for ALPs with sub-mJ-level
pulse lasers. We provided a way to evaluate the background yields from
residual-gas-originating aFWM, plasma, and other noise sources. In this work,
we used a $2.5\,\mathrm{mJ}/47\,\mathrm{fs}$ Ti:Sapphire laser and a
$1.5\,\mathrm{mJ}/9\,\mathrm{ns}$ Nd:YAG laser, whose energies are one order
of magnitude higher than those in SAPPHIRES00. Since the yield from optical-
element aFWM was observed for the first time with mJ-level laser pulses, we
have established a new method for discriminating between the new background
component and signals from vFWM. In the next section we summarize the
principles behind the new method, which is based on the beam cross-section
dependence. We then present the new search results obtained using this method
and provide new exclusion regions for the relation between ALP coupling with
photons and ALP mass. Finally, we conclude with prospects for the future work.

##  II Method for discriminating against optical-element atomic four-wave-
mixing

To extract the yield from ALP exchanges, that is, vacuum-originating FWM
$n_{vFWM}$, we first summarize the summation rules for the total number of
signal-like FWM photons $n_{FWM}$:

$n_{FWM}\equiv n_{vFWM}+n_{aFWM}$ (2)

with

$n_{aFWM}\equiv n_{gas}+n_{opt},$ (3)

where $n_{gas}$ and $n_{opt}$ are the numbers of photons from residual-gas-
originating aFWM and optical-element aFWM, respectively. Another expression
for $n_{FWM}$ based on the observed numbers of signal-like photons, denoted
with capital characters as $N_{S,C,I,P}$, is given by

$n_{FWM}=N_{S}-(N_{C}-N_{P})-(N_{I}-N_{P})-N_{P}$ (4)

with

$\displaystyle N_{S}$ $\displaystyle=$ $\displaystyle
n_{FWM}+n_{cPLS}+n_{iPLS}+n_{PED}$ (5) $\displaystyle N_{C}$ $\displaystyle=$
$\displaystyle n_{cPLS}+n_{PED}$ (6) $\displaystyle N_{I}$ $\displaystyle=$
$\displaystyle n_{iPLS}+n_{PED}$ (7) $\displaystyle N_{P}$ $\displaystyle=$
$\displaystyle n_{PED},$ (8)

where S, C, I, and P denote combinations of laser beams (see Fig. 6 for
details): S for a signal-expected case with both creation and inducing beams,
C for a case with only a creation laser, I for a case with only an inducing
laser, and P for a case without beams, hence, a pedestal of the detection
system. $n_{cPLS}$ and $n_{iPLS}$ are the expected numbers of photons due to
plasma formation with only creation and inducing beams, respectively.

As had already been performed in past searches PTEP2015 ; PTEP2020 ;
SAPPHIRES00 , $n_{FWM}$ in Eq.(2) can be determined from Eq.(4) based on
measurements in the different beam combinations above. Past searches have also
confirmed that $n_{gas}$ follows a square dependence on the residual gas
pressure. Thus, we first take a look at the pressure dependence of $n_{FWM}$
by changing vacuum pressure values. If $n_{FWM}$ exhibits pressure
independence, then it implies that

$n_{FWM}=n_{vFWM}+n_{opt}$ (9)

in a low-pressure environment because vFWM and optical-element aFWM are, in
principle, pressure independent. Therefore, we need a method for evaluating
$n_{vFWM}$ by discriminating against $n_{opt}$.

---
Figure 3: Relations between beam cross sections for creation and inducing
lasers: (a) at planes in optical elements before focusing and (b) at the focal
plane after focusing. (a) The creation and inducing laser cross sections are
drawn with dotted and dashed green lines and a solid red line, respectively.
The optical-element aFWM photons are produced at the overlapping areas between
the cross sections of the creation and inducing beams. (b) Behavior of the two
beams at a focal plane after focusing is illustrated. Since the beam waist is
proportional to the inverse of the beam diameter at the focal point, a large
(small) creation beam diameter makes the beam waist small (large) and the beam
intensity higher (lower). Photons from vFWM and residual-gas-originating aFWM
can be generated from the overlapping areas between the two beam cross
sections, while optical-element aFWM photons are never produced.

The principle for discrimination is illustrated in Fig. 3. In any FWM process,
$n_{FWM}$ originates from the space-time overlapping region of the two laser
pulses. Figure 3(a) shows cross sections of a creation laser (dotted and
dashed lines) with different beam diameters and an inducing laser (solid line)
with a fixed diameter measured at one of the optical elements before focusing
the two laser beams. Figure 3(b) shows the corresponding cross sections at a
focal plane after focusing. In Fig. 3(a), the peripheral ring beyond the
inducing diameter (the area between the dotted and solid lines) will never be
able to produce aFWM because the inducing field does not exit there.
Therefore, when the creation beam diameter is beyond the range of the inducing
beam diameter, as shown by the dotted line, optical-element aFWM yields must
be constant even if the creation beam diameter is increased. On the other
hand, in Fig. 3(b), since the beam waist $w_{0}$ in an ideal Gaussian beam at
a focal plane is expressed as Yariv

$w_{0}\equiv\frac{f\lambda}{\pi(d/2)}$ (10)

with laser photon wavelength $\lambda$, focal length $f$, and beam diameter
$d$, the dotted area is fully contained within the range of the inducing focal
spot. Increasing the creation beam diameter introduces a higher photon density
at the focal point, that is, a higher beam intensity in $\mathrm{W/m^{2}}$.
Thus, both $n_{vFWM}$ and $n_{gas}$ are expected to be strongly dependent on
the creation beam diameter. Therefore, observing how $n_{FWM}$ increases when
the creation beam diameter is increased, especially in the case where the
creation beam diameter is larger than that of the inducing beam, enables
discrimination against the optical-element aFWM. If $n_{FWM}$ behaves in a
pressure-independent fashion in a low-pressure environment, then we can test
the dependence on beam cross sections to investigate whether the vFWM process
is included or not on top of the constant behavior from optical-element aFWM.
We must therefore know in advance how $n_{vFWM}$ behaves as a function of the
beam cross sections.

(a) $m=0.1\,\mathrm{eV}$

(b) $m=0.01\,\mathrm{eV}$

Figure 4: Behavior of individual factors in Eq. (11) as a function of creation
beam cross sections $A_{c}$ with respect to a fixed inducing beam cross
section $A_{i}=200\,\mathrm{mm}^{2}$ (yellow dashed line) obtained by assuming
an ALP mass of (a) $m=0.1\,\mathrm{eV}$ and (b) $m=0.01\,\mathrm{eV}$. The
red, black, blue, and green curves indicate $\mathcal{Y}_{c+i}$,
$\mathcal{N}_{c+i}\equiv\frac{1}{4}N_{c}^{2}N_{i}$, $\mathcal{D}_{I}$, and
$\overline{\Sigma}_{I}$, normalized by their values at
$A_{c}=1000\mathrm{mm}^{2}$, respectively. Since the creation beam cross
section is cut by an iris (A1 in Fig. 5), the number of photons in a creation
laser pulse $N_{c}$ is proportional to $A_{c}$. $\overline{\Sigma}_{I}$, which
describes ALP-photon interactions, has the characteristic beam cross-section
dependence, because ALP resonance masses depend on the incident angles of two
stochastically selected photons from among those in the focused creation beam.

The vFWM yield, which, more specifically from the particle physics point of
view, is the signal yield per pulse collision from stimulated resonant photon-
photon scattering, ${\mathcal{Y}}_{c+i}$, is factorized as JHEP2020 ;
SAPPHIRES00

$\displaystyle{\mathcal{Y}}_{c+i}[1]\equiv\frac{1}{4}N^{2}_{c}N_{i}{\mathcal{D}}_{I}\left[s/L^{3}\right]\overline{\Sigma}_{I}\left[L^{3}/s\right].\mbox{\hskip
65.44142pt}$ (11)

$N_{c}$ and $N_{i}$ are the average number of photons in the two beams, with
the subscripts $c$ and $i$ denoting the creation and inducing beams,
respectively. ${\mathcal{D}}_{I}$ is a space-time overlapping factor between
the incident beams, and $\overline{\Sigma}_{I}$ is the interaction volume
rate, not the interaction cross section JHEP2020 ; SAPPHIRES00 , specified in
the units within [ ], which are expressed in terms of length $L$ and seconds
$s$.

Figures 4 (a) and (b) demonstrate how individual factors in Eq. (11) behave as
a function of the creation beam cross sections $A_{c}$ with respect to a fixed
inducing beam cross section $A_{i}=200\,\mathrm{mm}^{2}$, represented by the
yellow dashed line. The assumed ALP mass was $m=0.1$ eV and $m=0.01$ eV,
respectively. Individual factors are explained with different colors inside
the figures and have individual normalizations of $A_{c}=1000$ mm2, which is
close to the actual value of the maximum creation beam cross sections used in
this work. What is important is the behavior of $\mathcal{Y}_{c+i}$ in the
region $A_{c}>A_{i}$, shown with the red curves. Depending on the ALP mass,
$\overline{\Sigma}_{I}$ behaves differently, and the $\mathcal{Y}_{c+i}$
dependence on $A_{c}$ obviously deviates from the constant behavior seen when
optical-element aFWM contributes as the sole background source. Therefore, if
$\mathcal{Y}_{c+i}$ exhibits a significant non-constant behavior for
$A_{c}>A_{i}$ while in a low-pressure environment, then, in principle, we can
determine the mass scale of ALPs from the creation beam cross-section
dependence JHEP2020  in addition to testing the existence of ALPs.

##  III Experimental setup

---
Figure 5: Experimental setup for this work. The basic design is the same as
that in the previous search SAPPHIRES00 . The propagations of individual
lasers (creation laser, inducing laser, and calibration laser) are painted
with green, red, and blue stripes, respectively. Ti:Sapphire and Nd:YAG lasers
were used as the creation and inducing lasers. The calibration laser was a
He:Ne laser, and it was used to obtain the acceptance of signal photons
propagating from the Interaction point (IP) to the photomultiplier tube (PMT).
The two irises (A1 and A2) were introduced in order to change the creation
beam diameter and to fix the inducing beam diameter for the measurement of the
beam cross-section dependence. Both beam diameters were determined using the
knife-edge mechanism (KE). We intentionally introduced the window (W) not only
to keep a low pressure value in the interaction chamber but also to enhance
the background from optical-element aFWM, and to establish the method for
discriminating against optical-element aFWM.

Figure 5 shows the experimental setup. It is almost identical to the setup
used in the previous search SAPPHIRES00 . For the creation and inducing fields
we used a 47 fs-pulsed Ti:Sapphire laser, located at the Institute for
Chemical Research in Kyoto University ($\mathrm{T}^{6}$ system), and a 9 ns-
pulsed Nd:YAG laser. The central wavelengths of these lasers were
$816\,\mathrm{nm}$ and $1064\,\mathrm{nm}$, respectively. The signal central
wavelength was expected to be $662\,\mathrm{nm}$ via FWM:
$2\omega_{c}-\omega_{i}$ with creation photon energy $\omega_{c}$ and inducing
photon energy $\omega_{i}$. A linearly polarized (P-pol) creation beam and a
circularly polarized (left-handed) inducing beam were combined at a beam
combiner using a dichroic mirror (DM2) in the transport chamber, which was
maintained at $\sim 10^{-2}\,\mathrm{Pa}$. The combined laser fields were
focused coaxially into the interaction point (IP) in the interaction chamber,
which was designed to reach $\sim 10^{-8}\,\mathrm{Pa}$ but was operated at
$\sim 10^{-5}\,\mathrm{Pa}$ for this search. Since the pressure values in the
two chambers were so different, we introduced a window (W) between them. To
reduce the background amount in the upstream so that the photo-detection
device could be operated without saturation, a set of long pass filters (LPF)
for cutting the signal wavelength was inserted between the window and the
focusing off-axis parabolic mirror (OAP1). A second off-axis parabolic mirror
(OAP2), identical to OAP1, was placed at the symmetric position with respect
to the IP to collect signal photons. The signal photon selection was made
through a set of five identical dichroic mirrors (DM3 through DM7), and the
collimated signal photons were detected by a photomultiplier (PMT) through a
set of tight band-pass filters (BPFs) that rejected remnant beam photons. The
voltage-time information was recorded with a waveform digitizer, and the peak
finding algorithm was applied to obtain arrival times and the number of
photons through amplitudes of waveforms. The detailed specifications for these
devices and for the peak analysis method can be found in PTEP2020 ;
SAPPHIRES00 . To discuss the beam cross-section dependence of optical-element
aFWM, we introduced two irises (A1 and A2) with circular apertures along the
individual beam propagations: the c-Iris for changing the beam diameter of the
creation laser and the i-Iris for fixing the beam diameter of the inducing
laser. To accurately determine the beam diameters before focusing, the knife-
edge mechanism (KE) was installed after the DM2.

##  IV Measurements

###  IV.1 Arrival time distributions in four beam combinations

To evaluate $n_{FWM}$ in Eq. (4), we configured four beam combinations with
the two laser pulses. The repetition rates of the creation and inducing laser
pulses are 5Hz at equal and irregular intervals, respectively, as illustrated
in Figure 6.

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Figure 6: Four beam combinations with the two laser pulses. The green and red pulses are creation and inducing laser pulses, and the repetition rates are 5Hz at equal and irregular intervals, respectively. We denote four pulse combinations: S for both laser pulses, C for only the creation laser pulses, I for only the inducing laser pulses, and P for a pedestal measure without laser pulses.  Table 1: Sources of photons included in individual beam combination patterns. The characters $N_{S,C,I,P}$ denote the numbers of observed photons in the individual combination patterns expressed by the subscripts.  Four beam combinations |  S pattern |  C pattern |  I pattern |  P pattern
---|---|---|---|---
Characters | $N_{S}$ | $N_{C}$ | $N_{I}$ | $N_{P}$
Vacuum FWM ($n_{vFWM}$) | $\bigcirc$ | - | - | -
Atomic FWM ($n_{aFWM}$) | $\bigcirc$ | - | - | -
Plasma in the creation laser ($n_{cPLS}$) | $\bigcirc$ | $\bigcirc$ | - | -
Plasma in the inducing laser ($n_{iPLS}$) | $\bigcirc$ | - | $\bigcirc$ | -
Noise ($n_{PED}$) | $\bigcirc$ | $\bigcirc$ | $\bigcirc$ | $\bigcirc$

We then measured the arrival time distributions of photon-like signals at the
photomultiplier using peak finding on waveforms for four pulse combinations:
two laser pulses (Signal pattern, S), only the creation laser pulses (Creation
pattern, C), only the inducing laser pulses (Inducing pattern, I), and without
laser pulses (Pedestal pattern, P). The expected sources of photons contained
in individual patterns are summarized in table 1. The observed photons in the
P pattern are pedestal components from all kinds of ambient noises, primarily
from the thermal noise of the photo-detection device. The observed photons in
the C and I patterns contain plasma-originating photons from the individual
lasers and pedestals. The observed photons in the S pattern include signal
photons and photons from all possible background sources.

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Figure 7: Arrival-time distributions of observed photons in individual beam
combinations at $4.32\times 10^{-5}\,\mathrm{Pa}$. The red lines represent a
time window within which signal photons are expected to arrive. The upper-
left, upper-right, lower-left, and lower-right histograms correspond to the S,
C, I, and P beam combinations, respectively.

Figure 7 shows the arrival-time distributions of observed photons in
individual combinations of laser beams. The histograms for the S, C, I, and P
patterns are shown in the upper left (Blue), upper right (Green), lower left
(Purple) and lower right (Gray), respectively. The area between the two red
lines represents the expected arrival time window for signal photons.

###  IV.2 Pressure dependence of the number of FWM photons

---
Figure 8: Pressure dependence of the number of observed FWM photons $n_{FWM}$
per shot within the time window for expected signal arrival. The number of
observed FWM photons follows the square of the pressure when the chamber is
above $10\,\mathrm{Pa}$. This means that residual-gas-originating FWM
backgrounds are dominant in high pressure environments. Below
$1\,\mathrm{Pa}$, a nonzero constant component was observed for the first
time.

Using subtractions between the four beam combinations based on Eq. (4), we
measured the number of FWM photons in the interaction chamber. It is known
that the amount of residual-gas-originating aFWM scales with the square of the
pressure values PTEP2015 ; PTEP2020 ; SAPPHIRES00 . Figure 8 shows the
pressure dependence of the number of FWM photons per shot $\mathcal{N}$, fit
with the function

$\mathcal{N}=a+b\mathcal{P}^{c},$ (12)

where $\mathcal{P}$ is the pressure with fitting parameters $a$, $b$, and $c$.
The fit result, $c=2.05\pm 0.09$, is consistent with the square scaling. From
this result, we confirmed that the number of FWM photons via the residual gas
$n_{gas}$ was negligibly small at $10^{-5}\,\mathrm{Pa}$. Additionally, the
constant component was finally observed as $a=0.35\pm 0.02$ for the first
time.

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Figure 9: Arrival-time distribution of the number of observed photons after
subtracting background photons from plasma and thermal noise, based on Eq. (2)
with total S-pattern statistics of 15,000 shots at $2.87\times
10^{-5}\,\mathrm{Pa}$. The region between the two red lines is the time window
within which signal photons are expected to arrive.

More explicitly, Fig. 9 shows the arrival-time distribution after the
subtraction between the four beam combinations, based on Eq. (4) with total
S-pattern statistics of 15,000 shots at $2.87\times 10^{-5}\,\mathrm{Pa}$.
This pressure-independent component is considered to be a possible mixture of
$n_{vFWM}$ and $n_{opt}$, as introduced in Eq. (9) of section II.

###  IV.3 Beam cross-section dependence of the number of observed photons

---
Figure 10: Creation beam cross-section dependence of the number of observed
FWM photons $n_{FWM}$ per shot, drawn on a double-logarithmic scale. The
yellow dashed line is a fixed inducing pulse cross section
$A_{i}=217\,\mathrm{mm}^{2}$. Four datapoints from the right-most point, where
creation pulse diameters are sufficiently larger than the inducing pulse
diameter with considering errors, are shown enlarged in the insert with a
linear scale. These were fitted with Eq. (13), resulting in the slope
$e=(-1.5\pm 2.6)\times 10^{-4}$ represented by the red line. Since the number
of observed FWM photons are independent of the creation pulse cross section in
the $A_{c}>A_{i}$ region, the optical-element aFWM accounts for the observed
photons.

Figure 10 shows the beam cross-section dependence of the number of FWM photons
at $10^{-5}$ Pa, where the beam cross sections of the creation laser $A_{c}$
were changed with the variable size iris, c-Iris, while that of the inducing
laser $A_{i}$ was fixed at $A_{i}=217\,\mathrm{mm}^{2}$ (yellow dashed line in
Fig. 10). As discussed in section II, we only need to consider the dependence
on creation beam cross sections above the inducing beam cross section. We thus
focused on the last four cross section points, and fitting was performed using
the following linear function of the creation beam cross sections $A_{c}$ (red
line in Fig. 10):

$\mathcal{N}=d+eA_{c}.$ (13)

The slope $e$ was obtained as $e=(-1.5\pm 2.6)\times 10^{-4}$. Therefore, we
conclude that the observed number of FWM photons is consistent with being
constant as a function of the creation beam cross sections, indicating that
the number of optical-element aFWM photons $n_{opt}$ occupies the $n_{FWM}$ in
Eq. (9).

##  V Exclusion regions in ALP coupling-mass relations

We obtained new exclusion regions in ALP coupling-mass relations by following
the same procedure as in the previous publicationSAPPHIRES00 . With a set of
laser beam parameters $P$, the number of ALP-originating signal photons, that
is, vacuum-originating FWM $n_{vFWM}$, as a function of mass $m$ and coupling
$g/M$ is expressed as

$n_{vFWM}={\cal Y}_{c+i}(m,g/M;P)t_{a}r\epsilon,$ (14)

where $t_{a}$ is the data acquisition time, $r$ is the repetition rate of the
pulsed beams, and $\epsilon$ is the efficiency of detecting signal photons.
For a set of $m$ values and an $n_{vFWM}$, a set of coupling $g/M$ is
evaluated by solving this equation numerically. The laser parameters are
summarized in Table 2.

Table 2: Experimental parameters for numerical calculation of the exclusion limits in the $m-g/M$ parameter spaces.  Parameter | Value
---|---
Creation pulse laser |
Centeral wavelength $\lambda_{c}$ | 816 nm
Relative linewidth, $\delta\omega_{c}/<\omega_{c}>$ | $1.2\times 10^{-2}$
Duration time of pulse, $\tau_{c}$ | 40 fs
Measured pulse energy per $\tau_{c}$, $E_{c}$ | (2.53 $\pm$ 0.05) mJ
Pulse energy fraction within 3 $\sigma_{xy}$ focal spot, $f_{c}$ | 0.78
Effective pulse energy per $\tau_{c}$ within 3 $\sigma_{xy}$ focal spot |  $E_{c}f_{c}=1.97$ mJ
Effective number of creation photons, $N_{c}$ |  $8.1\times 10^{15}$ photons
Beam diameter of pulse, $d_{c}$ | (36.4 $\pm$ 2.2 $\pm$ 3.1) mm
Polarization | linear (P-polarized state)
Inducing pulse laser |
Central wavelength, $\lambda_{i}$ | 1064 nm
Relative linewidth, $\delta\omega_{i}/<\omega_{i}>$ | $1.0\times 10^{-4}$
Duration time of pulse, $\tau_{ibeam}$ | 9 ns
Measured pulse energy per $\tau_{ibeam}$, $E_{i}$ |  $(1.51\pm 0.02)$ mJ
Linewidth-based duration time of pulse, $\tau_{i}/2$ |  $\hbar/(2\delta\omega_{i})=2.8$ ps
Pulse energy fraction within 3 $\sigma_{xy}$ focal spot, $f_{i}$ | 0.80
Effective pulse energy per $\tau_{i}$ within 3 $\sigma_{xy}$ focal spot |  $E_{i}(\tau_{i}/\tau_{ibeam})f_{i}=0.75$ $\mu$J
Effective number of inducing photons, $N_{i}$ |  $4.0\times 10^{12}$ photons
Beam diameter of pulse, $d_{i}$ |  $(16.63\pm 0.03\pm 0.84)$ mm
Polarization | circular (left-handed state)
Focal length of off-axis parabolic mirror, $f$ | 279.1 mm
Single-photon detection efficiency, $\epsilon_{det}$ | 1.4%
Efficiency of optical path from IP to PMT, $\epsilon_{opt}$ | 33%
Total number of shots in S-pattern, $W_{S}$ |  $15,000$ shots
$\delta{n}_{opt}$ | 427.7

Since we conclude that the contribution from $n_{vFWM}$ is consistent with
null from the creation beam cross-section dependence, we can evaluate new
exclusion regions in the coupling-mass relation based on the fact that the
background distribution is only from the optical-element aFWM process. We do
this by assuming the Gaussian distribution on $n_{opt}$ because $n_{opt}$ is
obtained from subtractions between different trigger patterns whose baseline
fluctuations are expected to individually follow Gaussian distributions. We
focus on the result from the highest ALP mass range, which corresponds to
$n_{opt}$ at the maximum creation beam cross section, because a sensible mass
$m$ is expressed as $m=2\omega_{c}\sin\vartheta$ with half the incident angles
$\vartheta$ between two incident photons varying in the range
$0<\vartheta<\sin^{-1}(f/(d/2))$. A confidence level is thus defined with
respect to the null hypothesis by stating that the observed $n_{FWM}$ is all
from the optical-element aFWM process.

A confidence level $1-\alpha$ for excluding the null hypothesis is expressed
as

$1-\alpha=\frac{1}{\sqrt{2\pi}\sigma}\int^{\mu+\delta}_{\mu-\delta}e^{-(x-\mu)^{2}/(2\sigma^{2})}dx=\mbox{erf}\left(\frac{\delta}{\sqrt{2}\sigma}\right),$
(15)

where $\mu$ is the expected value of an estimator $x$ corresponding to
$n_{opt}$, and $\sigma$ is one standard deviation $\delta n_{opt}$ on the
measurement of $n_{opt}$. We assign the acceptance-uncorrected uncertainty for
$\delta n_{opt}$ from the quadratic sum of all error components in the
following result by multiplying the total statistics of 15,000 shots in the S
pattern:

$\displaystyle n_{opt}=(8.6613\pm 0.0427\mathrm{(stat.)}\pm
0.0023\mathrm{(syst.I)}\pm 0.0009\mathrm{(syst.II)})\times
10^{4}\,\mathrm{photons}.$ (16)

Systematic error I originates from baseline fluctuations measured before the
signal time window defined by the two red lines in Fig. 9, reflecting the
possibility of different pedestal distributions depending on the different
time domains. Systematic error II comes from the threshold value used to
define the single photon waveform. We note that the uncertainty of the focal
spot overlap factor evaluated in SAPPHIRES00  is omitted in this analysis
because the optical-element aFWM process is insensitive to it, due to the the
background only being produced when two parallel beams are propagating.

To obtain a confidence level of 95%, $2\alpha=0.05$ with $\delta=2.24\sigma$
was assigned, where a one-sided upper limit by excluding above $x+\delta$
PDGstatistics  was applied. For the evaluation of the upper limits on the
coupling–mass relation, we numerically solve

$2.24\delta n_{opt}={\cal Y}_{c+i}(m,g/M;P)t_{a}r\epsilon,$ (17)

where $t_{a}r=W_{S}=15,000$, and the overall efficiency
$\epsilon\equiv\epsilon_{opt}\epsilon_{det}$, with $\epsilon_{opt}$ being the
optical path acceptance to the signal detector position and $\epsilon_{det}$
the single photon detection efficiency.

---
Figure 11:  Upper limit in the $m_{\phi}-g/M$ parameter space for scalar field
exchanges achieved in this work, SAPPHIRES01, represented by the red shaded
area. The red filled area is from the pilot search, SAPPHIRES00 SAPPHIRES00 .
Limits from the LSW experiments (ALPS alps  and OSQAR osqar ) and the VMB
experiment (PVLAS pvlas ) are shown in the blue and gray areas, respectively.
The green lines are exclusion limits from the non-Newtonian force searches
(Stanford 1 Stanford1 ) and the Casimir force measurement (Lamoreaux Stanford1
).
---
Figure 12:  Upper limit in the $m_{\sigma}-g/M$ parameter space for
pseudoscalar field exchanges achieved in this work, SAPPHIRES01, represented
by the red shaded area. The red filled area is from the pilot search,
SAPPHIRES00 SAPPHIRES00 . Limits from the LSW experiments (ALPSalps  and
OSQARosqar ), the VMB experiment (PVLASpvlas ), and a Helioscope experiment
(CASTcast ) are shown with the blue, gray, and green filled areas,
respectively. The yellow band and the black dashed lines represent QCD axion
predictions from the KSVZ model KSVZ  with $0.07<|E/N-1.95|<7$ and $E/N=0$,
respectively. The solid line is the prediction from the DFSZ DFSZ  model with
$E/N=8/3$.

Figures 11 and 12 provide the evaluated upper limits of the coupling–mass
relations for scalar and pseudoscalar fields, respectively, at a 95%
confidence level. The red shaded areas are from this work, while the red
filled areas are from previous searches SAPPHIRES00  333 As for Fig. 12, we
note that the limit from the previous search SAPPHIRES00  is rescaled due to
the factor 1/2 in the new definition for the dual electromagnetic tensor in
Eq. (1).. The ALPS alps  and OSQAR osqar  experiments are “Light Shining
through Wall (LSW)” experiments and are filled with blue. The individual
sinusoidal terms in the sensitivities from ALPS and OSQAR are simplified to 1
above $5.0\,\mathrm{meV}$ and $2.5\,\mathrm{meV}$, respectively. The gray area
is the region excluded by the vacuum magnetic birefringence (VMB) experiment
(PVLAS pvlas ). In Fig. 11, the lines with green are excluded by a non-
Newtonian force search (Stanford 1 Stanford1 ) and a Casimir force measurement
(Lamoreaux Lamoreaux ). In Fig. 12, the green area is the region excluded by a
Helioscope experiment (CAST cast ). The yellow band is a region corresponding
to the benchmark QCD axion model (KSVZ KSVZ ) with $0.07<|E/N-1.95|<7$. The
black dashed and solid lines are predicted by the KSVZ model with $E/N=0$ and
the DFSZ DFSZ  model with $E/N=8/3$, respectively.

##  VI Conclusions

In this paper, we performed a stimulated resonant photon-photon collider
experiment using two lasers with pulse energies around one order of magnitude
higher than those in previous searches SAPPHIRES00 . The generation of
background photons via the atomic four-wave mixing (aFWM) process in optical
elements was observed for the first time. By measuring the creation beam cross
section dependence of the number of FWM photons, we observed a constant
behavior, indicating that the observed photons originated only from the
optical-element aFWM background. The expected value for the number of observed
photons from resonant scattering via ALP exchanges was consistent with null.
We then obtained the upper limit, which reached its maximum sensitivity in the
coupling $g/M=1.14\times 10^{-5}\,\mathrm{GeV^{-1}}$ at $m=0.18\,\mathrm{eV}$.

In this search the window material between the interaction chamber and the
transport chamber was introduced to keep a high vacuum condition in the
interaction chamber and to investigate how to handle optical-element aFWM
background in real data analysis. We then succeeded in establishing the method
for discussing ALP exchanges under a nonzero pressure-independent background,
based on the beam cross-section dependence of the FWM yield. In future
searches a specialized intermediate chamber without the window material will
be inserted, which will reduce the optical-element background even if we
increase the intensities of both beams. Building off of the established
method, we expect the upgrade to higher laser intensities to drastically
improve the sensitivity to ALP exchanges.

## Acknowledgments

The $\mathrm{T}^{6}$ system was financially supported by the MEXT Quantum Leap
Flagship Program (JPMXS0118070187) and the program for advanced research
equipment platforms (JPMXS0450300521). Y. Kirita acknowledges support from the
JST, the establishment of university fellowships for the creation of science
technology innovation, Grant No. JPMJFS2129, and a Grant-in-Aid for JSPS
fellows No. 22J13756 from the Ministry of Education, Culture, Sports, Science
and Technology (MEXT) of Japan. K. Homma acknowledges the support of the
Collaborative Research Program of the Institute for Chemical Research at Kyoto
University (Grant Nos. 2018–83, 2019–72, 2020–85, 2021–88, and 2022–101) and
Grants-in-Aid for Scientific Research Nos. 17H02897, 18H04354, 19K21880, and
21H04474 from the Ministry of Education, Culture, Sports, Science and
Technology (MEXT) of Japan. The authors in ELI-NP acknowledge the support from
Extreme Light Infrastructure Nuclear Physics Phase II, a project co-financed
by the Romanian Government and the European Union through the European
Regional Development Fund and the Competitiveness Operational Programme (No.
1/07.07.2016, COP, ID 1334).

## References

  * (1) R. D. Peccei and H. R. Quinn, Phys. Rev. Lett 38, 1440 (1977).
  * (2) R. Daido, F. Takahashi and W. Yin, Journal of High Energy Physics, 02, (2018) 104.
  * (3) Y. Fujii and K. Maeda, The Scalar-Tensor Theory of Gravitation Cambridge Univ. Press (2003).
  * (4) Y. Fujii and K. Homma, Prog. Theor. Phys 126, 531 (2011); Prog. Theor. Exp. Phys. 089203 (2014) [erratum].
  * (5) J. Khoury and A. Weltman, Phys. Rev. Lett., 93, 171104 (2004).
  * (6) T. Katsuragawa, S. Matsuzaki, and K. Homma, Phys. Rev. D 106, 044011 (2022).
  * (7) K. Ehret et al. (ALPS), Phys. Lett. B 689, 149 (2010).
  * (8) K. Homma, T. Hasebe, and K.Kume, Prog. Theor. Exp. Phys. 083C01 (2014).
  * (9) R. Ballou et al. (OSQAR), Phys. Rev. D 92, 9, 092002 (2015).
  * (10) A. Ejlli et al., Physics Reports 871 (2020) 1–74.
  * (11) K. Homma, Y. Kirita, M. Hashida, Y. Hirahara, S. Inoue, F. Ishibashi, Y. Nakamiya, L. Neagu, A. Nobuhiro, T. Ozaki, M. Rosu, S. Sakabe and O. Tesileanu (SAPPHIRES), Journal of High Energy Physics, 12, (2021) 108.
  * (12) T. Hasebe, K. Homma, Y. Nakamiya, K. Matsuura, K. Otani, M. Hashida, S. Inoue, S. Sakabe, Prog. Theo. Exp. Phys. 073C01 (2015).
  * (13) A. Nobuhiro, Y. Hirahara, K. Homma, Y. Kirita, T. Ozaki, Y. Nakamiya, M. Hashida, S. Inoue, and S. Sakabe, Prog. Theo. Exp. Phys. 073C01 (2020).
  * (14) A. Yariv, Optical Electronics in Modern Communications Oxford University Press (1997).
  * (15) K. Homma and Y. Kirita, JHEP 09 (2020) 095.
  * (16) J. Chiaverini et al., Phys. Rev. Lett. 90, 151101 (2003).
  * (17) S. K. Lamoreaux, Phys. Rev. Lett. 78, 5 (1997); 81, 5475 (1998).
  * (18) E. Arik et al. (CAST), J. Cosmol. Astropart. Phys. 02, 008 (2009); M. Arik et al. (CAST), Phys. Rev. Lett. 107, 261302 (2011); M. Arik et al. (CAST), Phys. Rev. Lett. 112, 9, 091302 (2014); V. Anastassopoulos et al. (CAST), Nature Phys. 13, 584 (2017).
  * (19) J. E. Kim, Phys. Rev. Lett. 43, 103 (1979); M. Shifman, A. Vainshtein, and V. Zakharov, Nucl. Phys. B166, 493 (1980).
  * (20) M. Dine, W. Fischler, and M. Srednicki, Phys. Lett. 104B, 199 (1981); A. Zhitnitskii, Sov. J. Nucl. Phys. 31, 260 (1980).
  * (21) See Eq.(36.56) in J. Beringer et al. (Particle Data Group), Phys. Rev. D 86, 010001 (2012).


* [7] H. C. Berg and E. M. Purcell, Physics of Chemoreception, Biophys. J. 20, 193 (1977).
  * [8] R. D. Astumian and Z. A. Schelly, Geometric effects of reduction of dimensionality in interfacial reactions, J. Am. Chem. Soc. 106, 304 (1984).
  * [9] P. H. Lee and J. J. Kozak, Lattice theory of reaction efficiency in compartmentalized systems. II. Reduction of dimensionality, J. Chem. Phys. 80, 705 (1984).
  * [10] D. Shoup, G. Lipari and A. Szabo, Diffusion-controlled bimolecular reaction rates. The effect of rotational diffusion and orientation constraints, Biophys. J. 36, 697 (1981).
  * [11] R. Zwanzig and A. Szabo, Time dependent rate of diffusion influenced ligand binding to receptors on cell surfaces, Biophys. J. 60, 671 (1991).
  * [12] D. Axelrod and M. D. Wang, Reduction-of-Dimensionality Kinetics at Reaction-Limited Cell Surface Receptors, Biophys. J. 66, 588 (1994).
  * [13] M. Schmick and P. I. H. Bastiaens, The Interdependence of Membrane Shape and Cellular Signal Processing, Cell 156, 1132 (2014).
  * [14] M. A. McCloskey and M-m Poo, Rates of Membrane-associated Reactions: Reduction of Dimensionality Revisited, J. Cell Biology 102, 88 (1986).
  * [15] O. G. Berg, R. B. Winter and P. H. von Hippel, Diffusion-driven mechanisms of protein translocation on nucleic acids. 1\. Models and theory, Biochemistry 20, 6929 (1981).
  * [16] P. H. von Hippel and O. G. Berg, Facilitated target location in biological systems, J. Biol. Chem. 264, 675 (1989).
  * [17] M. Coppey, O. Bénichou, R. Voituriez and M. Moreau, Kinetics of Target Site Localization of a Protein on DNA: A Stochastic Approach, Biophys. J. 87, 1640 (2004).
  * [18] S. E. Halford and J. F. Marko, How do site-specific DNA-binding proteins find their targets?, Nucleic Acids Res. 32, 3040 (2004).
  * [19] T. Hu, A. Y. Grosberg and B. I. Shklovskii, How Proteins Search for Their Specific Sites on DNA: The Role of DNA Conformation, Biophys. J. 90, 2731 (2006).
  * [20] L. Mirny, Cell commuters avoid delays, Nature Phys. 4, 93 (2008).
  * [21] L. Mirny, M. Slutsky, Z. Wunderlich, A. Tafvizi, J. Leith, and A. Kosmrlj, How a protein searches for its site on DNA: the mechanism of facilitated diffusion, J. Phys. A: Math. Theor. 42, 434013 (2009).
  * [22] M. A. Lomholt, B. van den Broek, S-M. J. Kalisc, G. J. L. Wuite and R. Metzler, Facilitated diffusion with DNA coiling, Proc. Natl. Acad. Sci. USA 106, 8204 (2009).
  * [23] B. van den Broek, M. A. Lomholt, S.-M. J. Kalisch, R. Metzler, and G. J. L. Wuite, _How DNA coiling enhances target localization by proteins_ , Proc. Natl. Acad. Sci. USA 105, 15738 (2008).
  * [24] K. V. Klenin, H. Merlitz, J. Langowski, and C.-X. Wu, _Facilitated Diffusion of DNA-Binding Proteins_ , Phys. Rev. Lett. 96, 018104 (2006).
  * [25] G. Kolesov, Z. Wunderlich, O. N. Laikova, M. S. Gelfand, and L. A. Mirny, _How gene order is influenced by the biophysics of transcription regulation_ , Proc. Natl. Acad. Sci. USA 104, 13948 (2007).
  * [26] O. Pulkkinen and R. Metzler, _Distance matters: the impact of gene proximity in bacterial gene regulation_ , Phys. Rev. Lett. 110, 198101 (2013).
  * [27] M. Slutsky and L. A. Mirny, _Kinetics of Protein-DNA Interaction: Facilitated Target Location in Sequence-Dependent Potential_ , Biophys. J. 87, 4021 (2004).
  * [28] M. Bauer and R. Metzler, _In vivo facilitated diffusion model_ , PLoS ONE 8, e53956 (2013).
  * [29] M. Sheinman, O. Bénichou, Y. Kafri and R. Voituriez, Classes of fast and specific search mechanisms for proteins on DNA, Rep. Prog. Phys. 75, 026601 (2012).
  * [30] R. Peters, Translocation through the nuclear pore complex: Selectivity and speed by reduction-of-dimensionality, Traffic 6, 421 (2005)
Translocation through the nuclear pore: Kaps pave the way, BioEssays 31, 466
(2009).

  * [31] O. Bénichou, M. Coppey, M. Moreau, P-H. Suet, and R. Voituriez, Optimal Search Strategies for Hidden Targets, Phys. Rev. Lett.94, 198101 (2005).
  * [32] O. Bénichou, C. Loverdo, M. Moreau, and R. Voituriez, Intermittent search strategies, Rev. Mod. Phys. 83, 81 (2011).
  * [33] O. Bénichou, D. S. Grebenkov, P. Levitz, C. Loverdo, and R. Voituriez, Optimal Reaction Time for Surface-Mediated Diffusion, Phys. Rev. Lett. 105, 150606 (2010).
  * [34] O. Bénichou, D. S. Grebenkov, P. Levitz, C. Loverdo, and R. Voituriez, Mean First-Passage Time of Surface-Mediated Diffusion in Spherical Domains, J. Stat. Phys. 142, 657-685 (2011).
  * [35] J.-F. Rupprecht, O. Bénichou, D. S. Grebenkov, and R. Voituriez, Kinetics of Active Surface-Mediated Diffusion in Spherically Symmetric Domains, J. Stat. Phys. 147, 891-918 (2012).
  * [36] J.-F. Rupprecht, O. Bénichou, D. S. Grebenkov, and R. Voituriez, Exact mean exit time for surface-mediated diffusion, Phys. Rev. E 86, 041135 (2012).
  * [37] F. Rojo and C. E. Budde, Enhanced diffusion through surface excursion: A master-equation approach to the narrow-escape-time problem, Phys. Rev. E 84, 021117 (2011).
  * [38] F. Rojo, C. E. Budde Jr., H. S. Wio, and C. E. Budde, Enhanced transport through desorption-mediated diffusion, Phys. Rev. E 87, 012115 (2013).
  * [39] O. Bénichou, D. S. Grebenkov, L. Hillairet, L. Phun, R. Voituriez, and M. Zinsmeister, Mean exit time for surface-mediated diffusion: spectral analysis and asymptotic behavior, Anal. Math. Phys. (2015).
  * [40] M. A. Lomholt, T. Koren, R. Metzler, and J. Klafter, _Lévy strategies in intermittent search processes are advantageous_ , Proc. Natl. Acad. Sci. USA 105, 11055 (2008).
  * [41] G. Oshanin, M. Tamm and O. Vasilyev, Narrow-escape times for diffusion in microdomains with a particle-surface affinity: mean-field results, J. Chem. Phys. 132, 06B607 (2010).
  * [42] G. Oshanin, H. S. Wio, K Lindenberg, and S. F. Burlatsky, Intermittent random walks for an optimal search strategy: one-dimensional case, J. Phys.: Condens. Matt. 19, 065142 (2007).
  * [43] G. Oshanin, K. Lindenberg, H. S. Wio, and S. Burlatsky, Efficient search by optimized intermittent random walks, J. Phys. A: Math. and Theor. 42, 434008 (2009).
  * [44] F. Rojo, J. Revelli, C. E. Budde, H. S. Wio, G. Oshanin, and K. Lindenberg, Intermittent search strategies revisited: effect of the jump length and biased motion, J. Phys. A: Math. and Theor. 43, 345001 (2010).
  * [45] V. V. Palyulin, A. V. Chechkin, and R. Metzler, _Lévy flights do not always optimize random blind search for sparse targets_ , Proc. Natl. Acad. Sci. USA 111, 2931 (2014).
  * [46] V. Palyulin, A. V. Chechkin, R. Klages, and R. Metzler, _Search reliability and search efficiency of combined Lévy-Brownian motion: long relocations mingled with thorough local exploration_ , J. Phys. A 49, 394002 (2016).
  * [47] O. Bénichou and R. Voituriez, From first-passage times of random walks in confinement to geometry-controlled kinetics, Phys. Rep. 539, 225 (2014).
  * [48] S. Redner, A Guide to First Passage Processes (Cambridge: Cambridge University Press, 2001).
  * [49] R. Metzler, G. Oshanin and S. Redner, eds., First-passage phenomena and their applications (World Scientific Publishers: Singapore, 2014).
  * [50] D. S. Grebenkov, R. Metzler, and G. Oshanin, Distribution of first-reaction times with target sites on boundaries of shell-like regions, New J. Phys. 23, 123049 (2021).
  * [51] S. R. McGuffee and A. H. Elcock, Diffusion, Crowding and Protein Stability in a Dynamic Molecular Model of the Bacterial Cytoplasm, PLoS ComputBiol 6, e1000694 (2010).
  * [52] S. K. Ghosh, A. G. Cherstvy, D. S. Grebenkov, and R. Metzler, Anomalous, non-Gaussian tracer diffusion in heterogeneously crowded environments, New J. Phys. 18, 013027 (2016).
  * [53] J. Ma, M. Do, M. A. Le Gros, C. S. Peskin, C. A. Larabell, Y. Mori, and S. A. Isaacson, Strong intracellular signal inactivation produces sharper and more robust signaling from cell membrane to nucleus, PLoS Comput. Biol. 16, e1008356 (2020).
  * [54] E. R. Weikum, X. Liu, and E. A. Ortlund, _The nuclear receptor superfamily: A structural perspective_ , Protein Sci. 27, 1876 (2018).
  * [55] G. Antczak and G. Ehrlich, Jump processes in surface diffusion, Surface Science Rep. 62, 39 (2007).
  * [56] M. D. Huber and L. Gerace, The size-wise nucleus: nuclear volume control in eukaryotes, J. Cell Biol. 179, 583 (2007).
  * [57] A. Rizzotto and E. C. Schirmer, Breaking the scale: how disrupting the karyoplasmic ratio gives cancer cells an advantage for metastatic invasion, Biochem. Soc. Trans. 45, 1333 (2017).
  * [58] M. E. Malerba and D. J. Marshall, Larger cells have relatively smaller nuclei across the Tree of Life, Evol. Lett. 5, 306 (2021).
  * [59] E. W. Sinnott and V. V. Trombetta, The cytonuclear ratio in plant cells, Am. J. Botany 23, 602 (1936).
  * [60] C. Mejía-Monasterio, G. Oshanin and G. Schehr, First passages for a search by a swarm of independent random searchers, J. Stat. Mech. P06022 (2011).
  * [61] T. G. Mattos, C. Mejía-Monasterio, R. Metzler, and G. Oshanin, First passages in bounded domains: When is the mean first passage time meaningful? Phys. Rev. E 86 031143 (2012).
  * [62] A. Godec and R. Metzler, Optimization and universality of Brownian search in quenched heterogeneous media, Phys. Rev. E 91, 052134 (2015).
  * [63] A. Godec and R. Metzler, Universal proximity effect in target search kinetics in the few encounter limit, Phys. Rev. X 6, 041037 (2016).
  * [64] D. S. Grebenkov, R. Metzler and G. Oshanin, Strong defocusing of molecular reaction times results from an interplay of geometry and reaction control, Comm. Chem. 1, 96 (2018).
  * [65] D. S. Grebenkov, R. Metzler and G. Oshanin, Towards a full quantitative description of single-molecule reaction kinetics in biological cells, Phys. Chem. Chem. Phys. 20, 16393 (2018).
  * [66] D. S. Grebenkov, R. Metzler and G. Oshanin, Full distribution of first exit times in the narrow escape problem, New J. Phys. 21, 122001 (2019).
  * [67] D. S. Grebenkov, Paradigm Shift in Diffusion-Mediated Surface Phenomena, Phys. Rev. Lett. 125, 078102 (2020).
  * [68] D. S. Grebenkov, Joint distribution of multiple boundary local times and related first-passage time problems with multiple targets, J. Stat. Mech. 103205 (2020)
  * [69] D. S. Grebenkov, An encounter-based approach for restricted diffusion with a gradient drift, J. Phys. A: Math. Theor. 55, 045203 (2022).
  * [70] D. S. Grebenkov and G. Oshanin, Diffusive escape through a narrow opening: new insights into a classic problem, Phys. Chem. Chem. Phys. 19, 2723 (2017).
  * [71] G. Oshanin, M. N. Popescu and S. Dietrich, Active colloids in the context of chemical kinetics, J. Phys. A: Math. Theor. 50, 134001 (2017).
  * [72] D. S. Grebenkov, R. Metzler, and G. Oshanin, Effects of the target aspect ratio and intrinsic reactivity onto diffusive search in bounded domains, New J. Phys. 19, 103025 (2017).
  * [73] D. Purves et al, eds., Neuroscience, 2nd ed. (Sunderland (MA): Sinauer Associates, 2001).
  * [74] B. Alberts et al, Molecular Biology of the Cell, $6$th ed. (New York: Garland, 2015).
  * [75] D. S. Grebenkov, R. Metzler, and G. Oshanin, A molecular relay race: sequential first-passage events to the terminal reaction centre in a cascade of diffusion controlled processes, New J. Phys. 23, 093004 (2021).
  * [76] D. S. Grebenkov, Surface Hopping Propagator: An Alternative Approach to Diffusion-Influenced Reactions, Phys. Rev. E 102, 032125 (2020).
  * [77] N. Chao, S. H. Young and M. Poo, Localization of cell membrane components by surface diffusion into a trap, Biophys. J. 36, 139 (1981).
  * [78] H. Sano and M. Tachiya, Theory of diffusion-controlled reactions on spherical surfaces and its application to reactions on micellar surfaces, J. Chem. Phys. 75, 2870 (1981).
  * [79] T. Prüstel and M. Tachiya, Reversible diffusion-influenced reactions of an isolated pair on some two dimensional surfaces, J. Chem. Phys. 139, 194103 (2013).
  * [80] D. S. Grebenkov, Reversible reactions controlled by surface diffusion on a sphere, J. Chem. Phys. 151, 154103 (2019).
  * [81] S. N. Majumdar and B. Meerson, Statistics of first-passage Brownian functionals, J. Stat. Mech. (2020) 023202; Corrigendum: J. Stat. Mech. (2021) 039801.
  * [82] B. Meerson and N. R. Smith, Geometrical optics of constrained Brownian motion: three short stories, J. Phys. A: Math. Theor. 52, 415001 (2019).
  * [83] C. Antoine and J. Talbot, Enhancing search efficiency through diffusive echo (submitted; E-print: arXiv:2205.09942v1).
  * [84] G. H. Weiss, K. E. Shuler, and K. Lindenberg, Order Statistics for First Passage Times in Diffusion Processes, J. Stat. Phys. 31, 255 (1983).
  * [85] K. Basnayake, A. Hubl, Z. Schuss, and D. Holcman, Extreme narrow escape: Shortest paths for the first particles among n to reach a target window, Phys. Lett. A 382, 3449 (2018).
  * [86] K. Basnayake, Z. Schuss, and D. Holcman, Asymptotic formulas for extreme statistics of escape times in 1, 2 and 3-dimensions, J. Nonlinear Sci. 29, 461-499 (2019).
  * [87] Z. Schuss, K. Basnayake, and D. Holcman, Redundancy principle and the role of extreme statistics in molecular and cellular biology, Phys. Life Rev. 28, 52-79 (2019).
  * [88] D. S. Lawley and J. B. Madrid, A Probabilistic Approach to Extreme Statistics of Brownian Escape Times in Dimensions 1, 2, and 3, J. Nonlinear Sci. 30, 1207-1227 (2020).
  * [89] D. S. Lawley, Distribution of extreme first passage times of diffusion, J. Math. Biol. 80, 2301 (2020).
  * [90] D. S. Grebenkov, R. Metzler, and G. Oshanin, From single-particle stochastic kinetics to macroscopic reaction rates: fastest first-passage time of N random walkers, New J. Phys. 22, 103004 (2020).
  * [91] S. N. Majumdar, A. Pal, and G. Schehr, Extreme value statistics of correlated random variables: a pedagogical review, Phys. Rep. 840, 1 (2020).
  * [92] D. S. Grebenkov, Depletion of Resources by a Population of Diffusing Species, Phys. Rev. E 105, 054402 (2022).
  * [93] J. F. Reverey, J.-H. Jeon, H. Bao, M. Leippe, R. Metzler, and C. Selhuber-Unkel, _Superdiffusion dominates intracellular particle motion in the supercrowded space of pathogenic Acanthamoeba castellanii_ , Sci. Rep. 5, 11690 (2015).
  * [94] S. Thapa, N. Lukat, C. Selhuber-Unkel, A. Cherstvy, and R. Metzler, _Transient superdiffusion of polydisperse vacuoles in highly-motile amoeboid cells_ , J. Chem. Phys. 150, 144901 (2019).
  * [95] P. Witzel, M. Götz, Y. Lanoiselée, T. Franosch, D. S. Grebenkov, and D. Heinrich, _Heterogeneities Shape Passive Intracellular Transport_ , Biophys. J. 117, 203-213 (2019).
  * [96] A. V. Chechkin, F. Seno, R. Metzler, and I. M. Sokolov, Brownian yet Non-Gaussian Diffusion: From Superstatistics to Subordination of Diffusing Diffusivities, Phys. Rev. X 7, 021002 (2017).
  * [97] Y. Lanoiselée, N. Moutal, and D. S. Grebenkov, Diffusion-limited reactions in dynamic heterogeneous media, Nature Commun. 9, 4398 (2018).
  * [98] D. S. Grebenkov, A physicist’s guide to explicit summation formulas involving zeros of Bessel functions and related spectral sums, Rev. Math. Phys. 33, 2130002 (2021).
  * [99] A. Erdélyi, Higher transcendental functions, vol. 1\. (Robert E. Krieger Publishing, Malabar FL, 1953).
  * [100] R. Szmytkowski, The parameter derivatives $[\partial^{2}P_{\nu}(z)/\partial\nu^{2}]_{\nu=0}$ and $[\partial^{3}P_{\nu}(z)/\partial\nu^{3}]_{\nu=0}$, where $P_{\nu}(z)$ is the Legendre function of the first kind, E-print arXiv:1301.6586 (2013).
  * [101] B. J. Laurenzi, Derivatives with respect to the order of the Legendre Polynomials, E-print arXiv:1502.06507v1 (2015).