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

# Preparation of a quantum degenerate mixture of 23Na40K molecules and 40K
atoms

Jin Cao  Hefei National Research Center for Physical Sciences at the
Microscale and School of Physical Sciences, University of Science and
Technology of China, Hefei 230026, China  Shanghai Research Center for Quantum
Science and CAS Center for Excellence in Quantum Information and Quantum
Physics, University of Science and Technology of China, Shanghai 201315, China
Huan Yang  Hefei National Research Center for Physical Sciences at the
Microscale and School of Physical Sciences, University of Science and
Technology of China, Hefei 230026, China  Shanghai Research Center for Quantum
Science and CAS Center for Excellence in Quantum Information and Quantum
Physics, University of Science and Technology of China, Shanghai 201315, China
Zhen Su  Hefei National Research Center for Physical Sciences at the
Microscale and School of Physical Sciences, University of Science and
Technology of China, Hefei 230026, China  Shanghai Research Center for Quantum
Science and CAS Center for Excellence in Quantum Information and Quantum
Physics, University of Science and Technology of China, Shanghai 201315, China
Xin-Yao Wang  Hefei National Research Center for Physical Sciences at the
Microscale and School of Physical Sciences, University of Science and
Technology of China, Hefei 230026, China  Shanghai Research Center for Quantum
Science and CAS Center for Excellence in Quantum Information and Quantum
Physics, University of Science and Technology of China, Shanghai 201315, China
Jun Rui  Hefei National Research Center for Physical Sciences at the
Microscale and School of Physical Sciences, University of Science and
Technology of China, Hefei 230026, China  Shanghai Research Center for Quantum
Science and CAS Center for Excellence in Quantum Information and Quantum
Physics, University of Science and Technology of China, Shanghai 201315, China
Hefei National Laboratory, University of Science and Technology of China,
Hefei 230088, China  Bo Zhao  Hefei National Research Center for Physical
Sciences at the Microscale and School of Physical Sciences, University of
Science and Technology of China, Hefei 230026, China  Shanghai Research Center
for Quantum Science and CAS Center for Excellence in Quantum Information and
Quantum Physics, University of Science and Technology of China, Shanghai
201315, China  Hefei National Laboratory, University of Science and Technology
of China, Hefei 230088, China  Jian-Wei Pan  Hefei National Research Center
for Physical Sciences at the Microscale and School of Physical Sciences,
University of Science and Technology of China, Hefei 230026, China  Shanghai
Research Center for Quantum Science and CAS Center for Excellence in Quantum
Information and Quantum Physics, University of Science and Technology of
China, Shanghai 201315, China  Hefei National Laboratory, University of
Science and Technology of China, Hefei 230088, China

###### Abstract

We report on the preparation of a quantum degenerate mixture of 23Na40K
molecules and 40K atoms. A deeply degenerate atomic mixture of 23Na and 40K
atoms with a large number ratio ($N_{F}/N_{B}\approx 6$) is prepared by the
mode-matching loading of atoms from a cloverleaf-type magnetic trap into a
large-volume horizontal optical dipole trap and evaporative cooling in a
large-volume three-beam optical dipole trap. About $3.0\times 10^{4}$ 23Na40K
ground-state molecules are created through magneto-association followed by
stimulated adiabatic Raman passage. The 2D density distribution of the
molecules is fit to the Fermi-Dirac distribution with $T/T_{F}\approx
0.4-0.5$. In the atom-molecule mixture, the elastic collisions provide a
thermalization mechanism for the molecules. In a few tens of milliseconds
which are larger than the typical thermalization time, the degeneracy of the
molecules is maintained, which may be due to the Pauli exclusion principle.
The quantum degenerate mixture of 23Na40K molecules and 40K atoms can be used
to study strongly interacting atom-molecule mixtures and to prepare ultracold
triatomic molecular gases.

Ultracold molecules have attracted great interest due to their potential
applications in the study of chemical reactions at nearly absolute zero
temperatures, quantum simulation of exotic models in condense matter physics,
and precision measurements Krems (2008); Quéméner and Julienne (2012); Carr et
al. (2009). Recently, magnetic tunable atom-molecule Feshbach resonances have
been observed in elastic and inelastic collisions between 23Na40K molecules
and 40K atoms Yang et al. (2019, 2022); Wang et al. (2021); Su et al. (2022),
and reactive collisions between 23Na6Li molecules and 23Na atoms Son et al.
(2022). The analysis in Ref.Wang et al. (2021) suggests that these resonances
are a common feature of atom-molecule systems and thus provide a powerful tool
to control atom-molecule interactions Chin et al. (2010). The ultracold atom-
molecule mixtures with tunable interactions open up many new research
opportunities, such as the adiabatic magneto-association of triatomic
molecules Köhler et al. (2006); Hermsmeier et al. (2021), the Efimov
resonances involving molecules Kraemer et al. (2006), the study of the angulon
problem Schmidt and Lemeshko (2015, 2016) and the superfluid involving atom-
molecule pairs. However, many of these applications require the preparation of
a quantum degenerate atom-molecule mixture, which remains elusive.

Although the techniques of preparing quantum degenerate gas of atoms are well
established, the creation of quantum degenerate gas of molecules is very
difficult. Recently, various diatomic molecules have been created via
ultracold association Ni et al. (2008); Molony et al. (2014); Takekoshi et al.
(2014); Park et al. (2015); Guo et al. (2016); Rvachov et al. (2017);
Seeßelberg et al. (2018); Liu et al. (2019); Voges et al. (2020) and direct
laser cooling Barry et al. (2014); Cheuk et al. (2018); Caldwell et al.
(2019); Ding et al. (2020). However, only two groups can create quantum
degenerate gas of molecules DeMarco et al. (2019); Valtolina et al. (2020);
Duda et al. (2021); Schindewolf et al. (2022). In the experiment at the JILA
group DeMarco et al. (2019), a deeply degenerate Bose-Fermi mixture of 87Rb
and 40K atoms with a large number imbalance is prepared to form the 87Rb40K
molecules in a conventional optical dipole trap. For a large number ratio of
$N_{F}/N_{B}\approx 7$ with $N_{B}$ ($N_{F}$) the number of bosonic
(fermionic) atoms, the spatial mismatch of density between Bose-Einstein
condensates (BEC) and degenerate Fermi gases (DFG) is mitigated, and thus a
high efficiency of association of molecules can be achieved. In the experiment
at the MPQ group Duda et al. (2021), a deeply degenerate Bose-Fermi mixture of
23Na and 40K atoms with a small number imbalance ($N_{F}/N_{B}\approx 3$) is
prepared. To form molecules with a high efficiency, a species-dependent dipole
trap is used to achieve density-matching between BEC and DFG.

To prepare a quantum degenerate mixture of 23Na40K molecules and 40K atoms,
the formation of molecules in a deeply degenerate mixture of 23Na and 40K
atoms with a large number imbalance is desired, since it does not require the
complicated species-dependent dipole trap, and the Feshbach molecules can be
quickly transferred to the ground state once formed. However, it is
challenging to prepare a deeply degenerate mixture of 23Na and 40K atoms with
a large number imbalance, because the three-body loss rate between 23Na and
40K is about 10 times larger than that between 87Rb and 40K Duda et al.
(2021). Another difficulty in the study of atom-molecule mixtures is that the
molecules usually suffer from losses due to collisions with the atoms.
Different mechanisms may contribute to the losses, such as photoexcitation by
trap lasers, inelastic collisions, and sticky collisions Christianen et al.
(2019); Liu et al. (2020); Gregory et al. (2020); Nichols et al. (2022); Mayle
et al. (2012); Frye and Hutson (2021). In the mixtures, the elastic collisions
must be faster than the inelastic collisions, so that the collisional
interactions can be studied before notable losses take place.

In this Letter, we report on the preparation of a quantum degenerate mixture
of 23Na40K molecules and 40K atoms. We first prepare a deeply degenerate
mixture of 23Na and 40K atoms with a large number imbalance
($N_{F}/N_{B}\approx 6$). This is achieved by the mode-matching loading of
atoms from a cloverleaf-type magnetic trap into a large-volume horizontal
optical dipole trap and performing evaporative cooling in a large-volume
three-beam optical dipole trap. We typically create approximately $3.9\times
10^{5}$ 40K atoms with $T/T_{F}\approx 0.2-0.25$ at a temperature of about 120
nK coexisting with approximately $6.2\times 10^{4}$ 23Na atoms with a BEC
fraction of about 80%. About $4.3\times 10^{4}$ 23Na40K Feshbach molecules can
be formed by adiabatic magneto-association. For a typical STIRAP efficiency of
70%, about $3.0\times 10^{4}$ 23Na40K ground-state molecules can be prepared.
The 2D column density distribution of the molecules is fit to the Fermi-Dirac
distribution with $T/T_{F}\approx 0.4-0.5$. By selectively removing 23Na
atoms, we obtain a quantum degenerate mixture of 23Na40K molecules and 40K
atoms. The elastic collisions between them provide the thermalization
mechanism for the molecules. After a few tens of milliseconds which is larger
than the typical thermalization time, the momentum distribution of the
molecules can still be fit to the Fermi-Dirac distribution with
$T/T_{F}\approx 0.5-0.6$. This indicates that the low entropy of the molecular
gas can be maintained in the mixture, which may be caused by the suppression
of collisions due to Pauli exclusion principle. The preparation of a quantum
degenerate atom-molecule mixture paves the way toward studying strongly
interacting atom-molecule mixtures and preparing ultracold triatomic molecular
gases.

Our experiment starts with the preparation of a deeply degenerate mixture of
23Na and 40K atoms. The experimental setup has been introduced in previous
works Yang et al. (2022); Wang et al. (2022). We first load 23Na and 40K atoms
into a two-species dark-spot magneto-optical trap. The atoms are optically
pumped to the $|f,m_{f}\rangle_{\rm{Na}}=|2,2\rangle$ state and the
$|f,m_{f}\rangle_{\rm{K}}=|9/2,9/2\rangle$ state and are then loaded into a
cloverleaf-type magnetic trap to perform evaporative cooling of 23Na atoms and
the 40K atoms are sympathetically cooled. After that, the atomic mixtures are
loaded into a horizontal dipole trap formed by two elliptical laser beams
($\lambda=1064$ nm) with beam waists of about $38\times 152$ $\mu$m crossed at
an angle of about 5 degrees. The axial direction of the horizontal dipole trap
is along the axial direction of the magnetic trap to ensure mode-matching with
the cigar-shaped magnetic trap. In this horizontal dipole trap, 23Na atoms are
transferred to the $|1,1\rangle$ state. A vertical dipole trap ($\lambda=1064$
nm) with a beam waist of approximately $235$ $\mu$m is adiabatically ramped up
to load the 23Na and 40K atoms into a large-volume dipole trap formed by the
three laser beams. Optical evaporative cooling is performed in the large-
volume three-beam optical dipole trap. At the end of the optical evaporation,
we typically create $6.2\times 10^{4}$ 23Na atoms with a BEC fraction of about
80% and about $3.9\times 10^{5}$ 40K atoms with a temperature of about 120 nK
and $T/T_{F}\approx 0.2-0.25$. The absorption images of the quantum degenerate
atomic mixture are shown in Fig. 1. The trap frequencies for 40K atoms are
$(\omega_{x},\omega_{y},\omega_{z})=2\pi\times(240,69,24)$ Hz.

A key improvement in the current work is that the interference between the two
laser beams forming the horizontal dipole trap is largely suppressed. We find
that the residual interference between these two laser beams causes additional
heating. Although in our previous works a quantum degenerate atomic mixture
can still be produced Wang et al. (2022), a deeply degenerate atomic mixture
with a large number imbalance was difficult. In the current work, the
interference between the two beams is largely suppressed by carefully
optimizing the polarizations and frequencies of the laser beams. This
improvement and further optimization of the optical evaporative cooling allow
us to create a deeply degenerate atomic mixture of 23Na and 40K atoms with a
large number ratio ($N_{F}/N_{B}\approx 6$), which serves as an ideal starting
point for the creation of quantum degenerate gas of molecules.

It is worth mentioning that in our work, we do not implement gray molasses on
atoms. In Refs. DeMarco et al. (2019); Duda et al. (2021), both the JILA and
MPQ group implemented the gray molasses to improve the initial phase space
density of the atoms. It was one of the key techniques for the preparation of
a deeply degenerate mixture of 87Rb atoms and 40K atoms with a large number
imbalance at the JILA group DeMarco et al. (2019). Our work demonstrates that
mode-matching loading from the magnetic trap into an optical dipole trap and
evaporative cooling in a large-volume dipole trap can be used to prepare a
deeply degenerate Bose-Fermi mixture with a large number ratio, even if the
mixture has a large three-body loss rate.

Figure 1:  Absorption images (average of 5 images) of a deeply degenerate
Bose-Fermi mixture of 23Na and 40K atoms with a large number imbalance after a
time-of-flight expansion of 13 ms. The mixture contains about $6.2\times
10^{4}$ 23Na atoms with a Bose-Einstein condensate fraction of about 80% and
about $3.9\times 10^{5}$ 40K atoms with $T/T_{F}=0.24$.

After the preparation of the degenerate atomic mixture, we use the Feshbach
resonance between the $|1,1\rangle$ and $|9/2,-7/2\rangle$ states at about
89.8 G to form 23Na40K Feshbach molecules. To this end, we first transfer the
40K atoms to the $|9/2,-5/2\rangle$ state by a Landau-Zener passage at 32 G.
We then ramp the magnetic field to 93 G and transfer the 40K atoms to the
$|9/2,-7/2\rangle$ state by a $\pi$-pulse. The 23Na40K Feshbach molecules are
created by ramping the magnetic field from 93 G to 89.5 G at a rate of about 3
G/ms. We typically create about $4.3\times 10^{4}$ 23Na40K Feshbach molecules.
The number of Feshbach molecules is determined by dissociating the Feshbach
molecules into the $|1,1\rangle$+$|9/2,-9/2\rangle$ state, and the 40K atoms
in the $|9/2,-9/2\rangle$ state are detected by absorption imaging. The
efficiency of molecule creation for 23Na atoms is about 70%. Note that this
high association efficiency may be overestimated, because the thermal
components of the 23Na atoms are difficult to discern for a nearly pure BEC
and thus the number of 23Na atoms may be underestimated.

In the experiment at the MPQ group Duda et al. (2021), high efficiency
creation of 23Na40K Feshbach molecules can only be achieved by using a
species-dependent optical dipole trap. However, we can achieve a high molecule
creation efficiency in a conventional optical dipole trap, which is similar to
the formation of 87Rb40K molecules at the JILA group. This may be because the
large number ratio ($N_{F}/N_{B}\approx 6$) in our work is close to that used
in the JILA group ($N_{F}/N_{B}\approx 7$) DeMarco et al. (2019). As pointed
out in Refs. Cumby et al. (2013); DeMarco et al. (2019), a large ratio of the
number of fermionic atoms to bosonic atoms can help to mitigate the spatial
density mismatch and suppress the three-body losses. In addition, the large
volume dipole trap in our experiment also reduces the density of the atoms and
thus can further suppress the three-body losses.

After the Feshbach molecules are created, we transfer the molecules to the
rovibrational ground state by stimulated Raman adiabatic passage (STIRAP). We
prepare the 23Na40K molecules in the hyperfine level
$|v,N,m_{i_{\rm{Na}}},m_{i_{\rm{K}}}\rangle=|0,0,-3/2,-3\rangle$ of the ground
state, where $v$ and $N$ represent the vibrational and rotational quantum
numbers, and $m_{i_{\rm{Na}}}$ and $m_{i_{\rm{K}}}$ denote the nuclear spin
projections of 23Na and 40K, respectively. The $\sigma^{-}$ polarized pump
laser and the $\sigma^{+}$ polarized Stokes laser propagate along the
direction of the magnetic field. For a typical STIRAP efficiency of 70%, we
create about $3.0\times 10^{4}$ ground state molecules. To measure the
degeneracy of the 23Na40K molecules, we remove the remaining 23Na atoms and
40K atoms by resonant light pulses. After a short hold time, we transfer the
ground-state molecules back to the Feshbach state for detection. The Feshbach
molecules are detected by direct absorption imaging.

Figure 2: (a) The degeneracy parameter $T/T_{F}$ of the 23Na40K molecules as a
function of hold time. The 2D column density distribution (average of $10-12$
absorption images) is fit to the Fermi-Dirac distribution and $T/T_{F}$ is
determined from the fitted fugacity. The fluctuations of $T/T_{F}$ for a hold
time less than about 25 ms are mainly affected by the breathing modes along
the x direction. The collective modes of the molecules induced by STIRAP are
shown in the inset, where the center-of-mass position of x direction and the
widths of the molecular clouds along the x and y directions oscillate as a
function of hold time. The $1/e$ time constants of the damping of the dipole
oscillations and the breathing modes along the x direction are $46(12)$ ms and
$19(4)$ ms, respectively. No notable damping is observed for the breathing
mode along the y direction. The azimuthal averaged momentum distribution for a
hold time of $3.5$ ms and $56.5$ ms are shown in (b) and (c), respectively.
The azimuthal averaging is applied after the fitting of the 2D distribution.
The solid lines are the fit to the Fermi-Dirac distribution (black curve) and
the Gaussian distribution (red curve). The inset shows the absorption images
(average of (b)12 and (c)10 images) after a time-of-flight expansion of 9 ms.
Error bars represent the standard error.

The 2D column density distribution of the molecules is fit to the Fermi-Dirac
distribution. The degeneracy parameter $T/T_{F}$ is obtained from the fitted
fugacity and is shown in Fig. 2. We find that STIRAP excites the collective
modes due to the misalignment and the differences of the trap frequencies. The
dominant collective modes are the dipole modes along the x direction and the
breathing modes in both the x and y directions. The degeneracy parameter
$T/T_{F}$ changes significantly for a hold time of less than 25 ms, which is
mainly affected by the breathing mode along the x direction. Due to the lack
of a thermalization process, the collective oscillations can only be damped by
the anharmonicity of the dipole trap. After a hold time of about 50 ms, the
breathing mode along the x direction is damped while the breathing mode along
the y direction remains. In this case, the amplitude that the degeneracy
parameter fluctuates is small, and we obtain $T/T_{F}\approx 0.4-0.5$. Since
STIRAP conserves the shape of the molecule cloud, this value reflects the
degeneracy of the Feshbach molecules. This value is smaller than the
degeneracy parameter of $T/T_{F}\approx 0.6$ calculated from the number of
Feshbach molecules and the trap frequencies. This may be because the 23Na
atoms are localized at the center of the 40K atoms, and the molecules are
dominantly created from the low-entropy part of the deeply degenerate Fermi
gas DeMarco et al. (2019). Therefore, the Feshbach molecules inherit the low
entropy of the 40K atoms. STIRAP introduces additional holes in the particle
distribution due to the limited efficiency and thus reduces the degeneracy of
the molecules. The effective degeneracy parameter of the ground-state
molecules may be estimated from the modified peak occupancy Tobias et al.
(2020). For $T/T_{F}\approx 0.4-0.5$ and a STIRAP efficiency of 70%, the
effective degeneracy parameter is $0.5-0.7$. However, a thermalization process
is required for the ground-state molecules to reach thermal equilibrium.

After selectively removing the 23Na atoms, we obtain a quantum degenerate
mixture of approximately $3\times 10^{4}$ 23Na40K molecules and approximately
$3\times 10^{5}$ 40K atoms at a temperature of about 120 nK. The elastic
collisions between 23Na40K molecules and 40K atoms provide a thermalization
mechanism for the molecules. We measure the elastic scattering cross sections
using the method in Ref. Su et al. (2022), by monitoring the cross-species
thermalization in a classical atom-molecule mixture. The measured elastic
scattering cross section is $\sigma_{\rm{el}}=6.4(12)\times 10^{-11}$ cm2. The
thermalization time may be calculated using the formula
$\tau_{\rm{th}}=(3/\xi)/\Gamma_{\rm{coll}}$, where
$\Gamma_{\rm{coll}}=n_{\rm{ov}}\sigma_{\rm{el}}v_{\rm{rel}}$ is the elastic
collision rate Mosk et al. (2001); Son et al. (2020); Tobias et al. (2020).
Here $n_{\rm{ov}}$ is the overlap density, $v_{\rm{rel}}$ is the relative mean
velocity, and the parameter
$\xi=4m_{\rm{NaK}}m_{\rm{K}}/(m_{\rm{NaK}}+m_{\rm{K}})^{2}$ represents the
mass effect. For $T/T_{F}>0.2$, the Gaussian distribution is a good
approximation for calculating $n_{\rm{ov}}v_{\rm{rel}}$ DeMarco et al. (2001).
Therefore, we have $n_{\rm{ov}}=(N_{\rm{NaK}}+N_{\rm{K}})[(\frac{2\pi
k_{\rm{B}}T_{\rm{K}}}{m_{\rm{K}}\bar{\omega}_{\rm{K}}^{2}})(1+\frac{m_{\rm{K}}T_{\rm{NaK}}}{m_{\rm{NaK}}T_{\rm{K}}\gamma_{t}^{2}})]^{-\frac{3}{2}}$
with $\gamma_{t}\approx 0.77$ being the ratio between the trap frequencies for
23Na40K molecules and 40K atoms and
$v_{\rm{rel}}=\sqrt{(8k_{\rm{B}}/\pi)(T_{\rm{K}}/m_{\rm{K}}+T_{\rm{NaK}}/m_{\rm{NaK}})}$.
For $T_{\rm{NaK}}=T_{\rm{K}}=120$ nK, $N_{\rm{K}}=3.0\times 10^{5}$ and
$N_{\rm{NaK}}=3.0\times 10^{4}$, we obtain $\tau_{\rm{th}}\approx 7$ ms.

These calculations do not consider the suppression of the elastic collisions
due to the Pauli exclusion principle in the quantum degenerate regime, which
is difficult to consider in the theoretical model. The effective elastic
collision cross sections in the quantum degenerate regime may be obtained from
the damping of the dipole oscillation in the mixture Tobias et al. (2020). The
damping rate of the dipole oscillations may be expressed in terms of the
elastic collision rate as
$1/\tau_{\rm{d}}-1/\tau_{0}=(2m_{\rm{K}}N_{\rm{K}})/[3(m_{\rm{K}}+m_{\rm{NaK}})(N_{\rm{K}}+N_{\rm{NaK}})\Gamma_{\rm{coll}}]$,
where $\tau_{\rm{d}}=17$ ms and $\tau_{0}=46$ ms are the measured $1/e$ time
constants of damping of the dipole oscillations with and without atoms,
respectively. We obtain an effective elastic scattering cross section
$2.7(5)\times 10^{-11}$ cm${}^{2}/$. This value is smaller than that measured
using the thermalization method in a classical mixture. We attribute this
reduction to the suppression of the elastic collisions due to the Pauli
exclusion principle DeMarco et al. (2001); DeMarco (2001). The thermalization
time in the quantum degenerate regime may be increased by a similar factor,
and thus we infer a thermalization time of about 17 ms, which is still much
shorter than the lifetime of the molecules of about 85 ms in the mixtures.

Figure 3:  The degeneracy parameter $T/T_{F}$ of the 23Na40K molecules in the
atom-molecule mixture as a function of hold time. The 2D column density
distribution (average of $20-30$ images) is fit to the Fermi-Dirac
distribution and the $T/T_{F}$ is determined from the fitted fugacity. The
damping of the collective oscillations of the molecules by the atom-molecule
elastic collisions is shown in the inset. The $1/e$ time constants of the
damping of the dipole oscillations and the breathing modes along the x and y
directions are $17(2)$ ms, $8(2)$ ms, and $28(6)$ ms respectively. Error bars
represent the standard error.

In the mixtures, the breathing modes are also damped due to the elastic
collisions. The damping time of the breathing mode is similar to that of the
dipole oscillation. As shown in Fig. 3, we measure the degeneracy parameter of
the molecules in the atom-molecule mixture for a hold time of $30-50$ ms,
which is larger than the damping time of the collective oscillations and the
estimated thermalization time. We find that the 2D column density distribution
can still be fit to the Fermi-Dirac distribution with $T/T_{F}\approx
0.5-0.6$. These degeneracy parameters are close to the effective degeneracy
parameters estimated from the modified peak occupancy. However, they are
smaller than the degeneracy parameters $T/T_{F}\approx 0.8-1$ calculated from
the trap frequencies and ground-state molecule numbers. A possible explanation
is that increasing the degeneracy parameter $T/T_{F}$ requires absorbing
energy from the atoms and creating more holes in the peak occupancy. However,
absorbing energy from the more degenerate atoms is suppressed due to the Pauli
exclusion principle DeMarco et al. (2001) and thus a complete thermalization
process may need a longer time than the thermalization time that the simple
thermalization model predicts. On the timescale of a few tens of milliseconds,
the degeneracy of the molecules can be maintained, which is favorable for the
study of the quantum degenerate atom-molecule mixtures.

In summary, we have prepared a quantum degenerate mixture of 23Na40K molecules
and 40K atoms. The method of preparing the degenerate molecules in our work
may be useful to other atomic mixtures that also have large three-body loss
rates. The collective oscillations induced by the STIRAP lasers can be
suppressed by using 1D optical lattices DeMarco et al. (2019) and optimizing
the alignment. Further increasing the degeneracy may be achieved by
selectively removing the high-energy 40K atoms, and using the atom-molecule
elastic collisions to perform evaporative cooling. There are many Feshbach
resonances between 23Na40K molecules and 40K atoms. Therefore, the preparation
of the quantum degenerate mixture with tunable interactions paves the way
toward studying strongly interacting atom-molecule mixtures, quantum
simulation of fermionic angulon Schmidt and Lemeshko (2015, 2016), and
preparation of triatomic molecular gases Trenkwalder et al. (2011); Köhler et
al. (2006); Hermsmeier et al. (2021). The mixture may also be used to study
the BCS pairing of atoms and molecules, similar to the ultracold mixture of
mass-imbalanced fermionic atoms Ravensbergen et al. (2020); Green et al.
(2020).

###### Acknowledgements.

This work was supported by the National Key R&D Program of China (under Grant
No. 2018YFA0306502), the National Natural Science Foundation of China (under
Grant Nos. 11521063, 11904355), the Chinese Academy of Sciences, the Anhui
Initiative in Quantum Information Technologies, the Shanghai Municipal Science
and Technology Major Project (Grant No.2019SHZDZX01), the Shanghai Rising-Star
Program (Grant No. 20QA1410000), the Innovation Program for Quantum Science
and Technology (Grant No. 2021ZD0302101).

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a\sigma_{3}}S_{k}^{\infty}e^{\pi a\sigma_{3}},$ (165)

We get that for every integer $k$,

$S_{k}^{\infty}=e^{2\pi a\sigma_{3}}S_{k}^{\infty}e^{-2\pi a\sigma_{3}}.$

Each Stokes matrix is known to have the triangular structure with units on the
diagonal and one (generally nontrivial) off-diagonal element, $s_{k}$, called
the Stokes multiplier. The last equation implies for the Stokes multipliers
the following equation,

$s_{k}=e^{4\pi a}s_{k}.$

Therefore, we arrive at the conclusion that for $a\neq in$ and $a\neq i/2+in$,
$n\in\mathbb{Z}$, the meromorphic solution of Equation (1) vanishing at the
origin has vanishing Stokes multipliers,

$s_{0}^{\infty}=s_{1}^{\infty}=0.$ (166)

We have to cope with the two remaining cases of $a$: $a=in$ and $a=i/2+in$.
Theorem 2.1 says, that for $a=i/2+in$ there exists the unique odd meromorphic
solution vanishing at $\tau=0$. Sure, by the continuity argument (monodromy
data depends analytically on $a$) we can prove that Equation (166) holds also
for this case. As for the case $a=in$, we know that for $a=0$, $\pm 1$,…,$\pm
5$ (see formulae for $u_{2n}$ underneath Remark 2.2) the meromorphic solution
vanishing at $\tau=0$ does not exists. Sure explicit calculations can be
continued further and the nonexistence can be confirmed for the larger values
of $n$. Since Section 5 says that the nontrivial function generating residues
at $a=in$ can be constructed for any $n$, such solutions do not exists for all
$n\in\mathbb{Z}$. The proof of Lemma 2.1 shows that if holomorphic solution at
$\tau=0$ exists for some $a=in_{0}$, $n_{0}\in\mathbb{Z}$, it is odd and not
unique. The continuity argument, analogous to the one given above for the case
$a=i/2+in$, shows that Equation (166) should hold for at least one limiting
case as $a\to in_{0}$. However, it contradicts to the last equation of System
(170) defining the monodromy manifold.

Before going further, let us consider the symmetry for System (12) of [1]
related with the odd solutions $u(\tau)$ and reproduce Condition (166) for all
such solutions. This symmetry is considered in Subsection 6.2 (Item 6.2.1 page
1199 of [1]) and requires correction.

So, we assume that for some solution $u(\tau e^{i\pi})=-u(\tau)$ in a
neighborhood of $\tau=0$. Then obviously, $u(\tau)=-u(\tau e^{-i\pi})$. The
coefficients of System (12) of [1] are denoted as $A(\tau)$, $B(\tau)$,
$C(\tau)$, and $D(\tau)$. Equations in Proposition 1.2 of [1] shows that,

$A(\tau)=A(-\tau),\quad B(\tau)=B(-\tau),\quad C(\tau)=-C(-\tau),\quad
D(\tau)=-D(-\tau).$

These equations implies the following relations for the canonical solutions:

$\displaystyle
Y_{k}^{\infty}(e^{i\pi}\mu,e^{-i\pi}\tau)=e^{-\frac{i\pi}{4}\sigma_{3}}Y_{k}^{\infty}(\mu,\tau)e^{\pi
a\sigma_{3}},$ (167) $\displaystyle
Y_{k+2}^{\infty}(e^{i\pi}\mu,e^{i\pi}\tau)=e^{-\frac{i\pi}{4}\sigma_{3}}Y_{k}^{\infty}(\mu,\tau),$
(168)

Equations (167) and (168) imply for the following relations Stokes matrices

$\displaystyle S_{k}^{\infty}=e^{-\pi a\sigma_{3}}S_{k}^{\infty}e^{\pi
a\sigma_{3}},\quad S_{k+2}^{\infty}=S_{k}^{\infty},\qquad k=0,\pm 1,\ldots,$

respectively. With the help of Equation (165) we see that both above equations
are equivalent to the following equation for the Stokes multipliers,

$s_{k}^{\infty}=s_{k}^{\infty}e^{2\pi a}.$ (169)

Equation (169) implies Condition (166) for all odd solutions of Equation (1)
in the neighborhood of $\tau=0$ and $a\neq in$, $n\in\mathbb{Z}$.

Analogous reasoning does not work for the canonical solutions $X_{k}(\mu)$, in
the neighbourhood of $\mu=0$, because their asymptotics contains terms
$\tau^{1/2}$ and $\tau^{1/4}$. Actually, the first equation of System (158)
implies,

$s_{0}^{0}=2i\cosh(\pi a)=ie^{\pi a}+ie^{-\pi a}.$

Moreover, Equation (44) of [1], together with the restrictions (43) of [1],
implies $\rho=\pm\frac{ai}{2}$. Again we can choose here any sign of $\rho$
because Asymptotic Formula (45) of [1] is symmetric with respect $\rho$.
Calculations slightly simpler with $\rho=-\frac{ai}{2}$. After that a simple
analysis of System (158) allows us to prove the following

######  Proposition 7.1.

For $a\notin i\mathbb{Z}/2$ there exists the only one solution of Equation (1)
such that both functions $u(\tau)$ and $e^{i\varphi(\tau)}$ are meromorphic.
It is the odd function of $\tau$. For $a=i/2+in$ with $n\in\mathbb{Z}$ there
exists the unique odd solution of Equation (1) such that both functions
$u(\tau)$ and $e^{i\varphi(\tau)}$ are meromorphic. Their monodromy parameters
are as follows:

$\begin{gathered}s_{0}^{\infty}=s_{1}^{\infty}=0,\qquad s_{0}^{0}=2i\cosh(\pi
a),\\\ g_{21}=-ie^{\pi a}g_{11},\quad g_{12}=ie^{\pi a}g_{22},\qquad
g_{11}g_{22}\left(1-e^{2\pi a}\right)=1.\end{gathered}$ (170)

###### Proof.

The main part of the proof is given before Proposition. The last equation of
System (170) implies $a\notin i\mathbb{Z}$, which is consistent with Equation
(24).

To finish the proof we have to notice that the monodromy data given by System
(170) contains one parameter, say, $g_{12}$ or $g_{21}$. As soon as this
parameter is fixed the others are uniquely defined. This parameter defines the
constant of integration in Equation (157) and does not effect on the function
$u(\tau)$. The function $u(\tau)$ is uniquely determined by the bilinear
combinations of the monodromy data: $g_{11}g_{22}$ and $g_{11}g_{12}$. In our
case these combinations are uniquely defined as long as the parameter $a$ is
fixed. ∎

######  Remark 7.1.

Note, that System (170) for $a=\pm i/2$ coincides with the corresponding
System (163), therefore the latter systems are proved.

######  Remark 7.2.

The functions $u(\tau)$ and $\varphi^{\prime}(\tau)$ are odd and meromorphic.
Note that function $\varphi(\tau)$ is not a meromorphic function.

######  Remark 7.3.

The solution $u(\tau)$ is not the only meromorphic solution of Equation (1).
For the the other meromorphic solutions the corresponding functions
$e^{i\varphi(\tau)}$ are not single-valued.

##  8 Asymptotics as $\tau\to+\infty$

Here we apply the results obtained in the previous section and [1] to get
asymptotics of $u(\tau)$ for the large values of $\tau$. In this section we
assume the following restrictions on the coefficients of Equation (1)

$|{\rm Im}a|<1,\qquad b>0.$

First of all we have to check the conditions on the monodromy data for
applicability of Theoreom 3.1 of [1]. There are two such conditions:

$\displaystyle g_{11}g_{12}g_{21}g_{22}\neq 0,$ (171) $\displaystyle\left|{\rm
Re}\left(\frac{i}{2\pi}\ln(g_{11}g_{22})\right)\right|<1/6.$ (172)

Inequality (171) is an obvious consequence of the second line of equations of
System (170). The second condition should be examined more carefully.
Substituting the last equation of System (170) into Condition (172) we find,

$\left|{\rm Re}\left(\frac{i}{2\pi}\ln(1-e^{2\pi a})\right)\right|<1/6.$ (173)

######  Remark 8.1.

The leading term of asymptotics $u(\tau)$ obtained in [1] contains the
$\cosh$-function which is by definition can be written as the half-sum of two
exponents. Restriction (172) obtained in [1] guarantee that both exponents are
greater than the correction term. In Appendix $B$ of our subsequent Paper [2]
we have corrected the phase-shift in the $cosh$-function obtained in [1] and
also pointed out that in case we require that only the largest exponent of the
$cosh$-function is greater than the correction term, then Restriction (172) is
weaker, namely, $1/6$ in the r.h.s. should be changed by $1/2$. For a better
numerical correspondence of the leading term of asymptotics with the exact
solution outside Restriction (173) one should use the result obtained in [2].
From the point of view of the complete asymptotic expansions, which are not
yet considered, it means rearrangement of the corresponding series.

######  Remark 8.2.

The function $\ln$ is multivalued: the sense of Equation (173) is that there
should exists the branch of $\ln$-function such that the restriction holds. If
such branch of $\ln(\cdot)$ exists, then it is fixed uniquely and it is the
branch which should be used in the corresponding asymptotic formula. This
remark applies also for the weaker condition discussed in Remark 8.1.

$\displaystyle
u(\tau)\underset{\tau\to+\infty}{=}u_{as}(\tau)+o(\tau^{-\delta}),\quad\delta>0,$
(174) $\displaystyle
u_{as}(\tau)=\frac{b^{1/2}}{3^{1/4}}\left(\sqrt{\frac{\vartheta(\tau)}{12}}+\sqrt{\nu+1}\,e^{\frac{3\pi
i}{4}}\cosh(i\vartheta(\tau)+(\nu+1)\ln\vartheta(\tau)+z)\right),$ (175)
$\displaystyle\vartheta(\tau)=3^{3/2}b^{1/3}\tau^{2/3},\qquad\nu+1=\frac{i}{2\pi}\ln(g_{11}g_{22}),$
(176) $\displaystyle z=\frac{\ln(2\pi)}{2}-\frac{\pi i}{2}-\frac{3\pi
i}{2}(\nu+1)+ia\ln(2+\sqrt{3})+(\nu+1)\ln 12$ (177)
$\displaystyle-\ln\left(\omega\sqrt{\nu+1}\Gamma(\nu+1)\right),\qquad\omega=g_{11}g_{12}.$
(178)

The error estimate is written in Equation (174) in a different form comparing
with Theorem 3.1 of [1] because here it is convenient to introduce notation
$u_{as}(\tau)$ for the leading term of asymptotics. Since $\delta>0$ does not
fixed the present formulation is equivalent to the original one in [1]. In
Equations (174)-(178), we put $\varepsilon_{1}=\varepsilon_{2}=0$ and
$\varepsilon=1$. The very last equation in (178) is corrected accordingly
Appendix B of [2] (see Equation B.6 at page 53 of [2]).

Consider Condition (173) and the weaker one mentioned in Remark 8.1 more
explicitly, note that we have to take into account also Equation (159). Here,
we address two specific cases of the parameter $a$, namely, $a$ is purely
imaginary and $a$ is real:

###  8.1 $a=i\alpha$, $\alpha\in\mathbb{R}$

In this case Condition (173) reads, $1/6<|\alpha|<5/6$, the corresponding
condition from Remark 8.1 is $0<|\alpha|<1$. As we mentioned above in both
cases the leading term of asymptotics is given by Equations (174)–(178).
However, the leading term numerically describes the solution better when
$\alpha$ is closer to $1/2$. Below I present the real and imaginary part of
the leading term of asymptotics of the function $u(x)$. To make it more
explicit we consider the case

$\alpha\in(0,1/2)\cup(1/2,1).$ $\displaystyle{\rm Re}\,u_{as}$
$\displaystyle=\frac{b^{2/3}}{2}\tau^{1/3}+\frac{b^{1/2}}{3^{1/4}}\sqrt{\nu_{1}}\left(\sin(\pi/4+\psi)\cosh(\chi)\cos(\phi)+\cos(\pi/4+\psi)\sinh(\chi)\sin(\phi)\right)\\!,$
$\displaystyle{\rm Im}\,u_{as}$
$\displaystyle=\frac{b^{1/2}}{3^{1/4}}\sqrt{\nu_{1}}\left(\sin(\pi/4+\psi)\sinh(\chi)\sin(\phi)-\cos(\pi/4+\psi)\cosh(\chi)\cos(\phi)\right),$

where

$\displaystyle\nu_{1}$
$\displaystyle=\frac{1}{2}\sqrt{(\alpha-1/2)^{2}+(\ln(2\sin(\pi\alpha)))^{2}/\pi^{2}},$
$\displaystyle\psi$
$\displaystyle=-\frac{1}{2}\arctan\left(\frac{\ln(2\sin(\pi\alpha))}{\pi(\alpha-1/2)}\right)+\frac{\pi}{4}({\rm
sign}(\alpha-1/2)-1),$ $\displaystyle\chi$
$\displaystyle=\frac{1}{2}(\alpha-1/2)\ln\left(3\sqrt{3}\,b^{1/3}\tau^{2/3}\right)+\chi_{0},$
$\displaystyle\chi_{0}$
$\displaystyle=\frac{\ln(2\pi)}{2}+\frac{\ln(2\sin(\pi\alpha))}{4}-\alpha\ln(2+\sqrt{3})+\frac{\ln(12)}{2}(\alpha-1/2)$
$\displaystyle-\ln\left(\sqrt{\nu_{1}}\left|\Gamma\left(\frac{1}{2}(\alpha-1/2)-\frac{i}{2\pi}\ln(2\sin(\pi\alpha))\right)\right|\right),$
$\displaystyle\phi$
$\displaystyle=3\sqrt{3}\,b^{1/3}\tau^{2/3}-\frac{\ln(2\sin(\pi\alpha))}{2\pi}\ln\left(3\sqrt{3}\,b^{1/3}\tau^{2/3}\right)+\phi_{0},$
$\displaystyle\phi_{0}$
$\displaystyle=\frac{\pi}{2}-\psi-\frac{3\pi}{4}(\alpha-1/2)-\frac{\ln(12)}{2\pi}\ln(2\sin(\pi\alpha)))$
$\displaystyle-\arg\left(\Gamma\left(\frac{1}{2}(\alpha-1/2)-\frac{i}{2\pi}\ln(2\sin(\pi\alpha)\right)\right).$

In the formulae above all roots and fractional powers of positive numbers are
positive, the function $\ln$ with positive argument is positive, $\arctan$ is
the principle branch of the corresponding inverse function with the values in
$(-\pi/2,\pi/2)$, and ${\rm sign(z)}=1$ for $z>0$ and $-1$ for $z<0$,
respectively.

Below we present a comparison of the solution with its asymptotic
approximation, with the help of the above asymptotic formulae and MAPLE 16. As
it often happens for the Painlevé equations the leading term of asymptotics
for large values of independent variable gives quite good numerical
approximation of the solution when the independent variable takes relatively
small values. For illustrative purposes I especially show the plots (see
Figures 1 and 2) for a small value of the parameter $b$, at such values the
difference between the solution and its asymptotics can be easily observed.
The increase of $b$ means a rescaling of the argument and we, in some sense,
observe how the same solution would behave for the larger values of the
argument although we keep the same interval for our plots (see Figures 3 and
4). Of course, when $\alpha$ approaches to the boundaries of the validity of
Asymptotics (174) the numerical correspondence with the solution for the small
and finite values of $\tau$ becomes worse, however by increasing $b$, one can
observe that asymptotics is working on the whole interval $(0,1)$.

Figure 1: The plots of ${\rm Re}\,u(\tau)$ and ${\rm Re}\,u_{as}(\tau)$ for
$\alpha=2/7$ and $b=1/80$. The first one has higher corresponding extremums.
Figure 2: The plots of ${\rm Im}\,u(\tau)$ and ${\rm Im}\,u_{as}(\tau)$ for
$\alpha=2/7$ and $b=1/80$. The first one has higher corresponding extremums.
Figure 3: The plots of ${\rm Re}\,u(\tau)$ and ${\rm Re}\,u_{as}(\tau)$ for
$\alpha=5/7$ and $b=1$. The corresponding extremums are higher for ${\rm
Re}\,u(\tau)$. Figure 4: The plots of ${\rm Im}\,u(\tau)$ and ${\rm
Im}\,u_{as}(\tau)$ for $\alpha=5/7$ and $b=1$. The orresponding extremums are
higher for ${\rm Im}\,u(\tau)$.

Now consider the Suleimanov solution, i.e., the odd meromorphic solution for
$\alpha=1/2$. In this case, the formulae presented above can be considerably
simplified:

$\displaystyle{\rm Im}\,u_{as}(\tau)$ $\displaystyle=-\sqrt{\frac{b\sqrt{3}\ln
2}{4\pi}}\,\cos{\phi}$ $\displaystyle{\rm Re}\,u_{as}(\tau)$
$\displaystyle=\frac{b^{2/3}\tau^{1/3}}{2}-\sqrt{\frac{b\ln
2}{4\pi\sqrt{3}}}\,\sin{\phi},$

where

$\displaystyle\phi$ $\displaystyle=3\sqrt{3}\,b^{1/3}\tau^{2/3}-\frac{\ln
2}{2\pi}\ln(3\sqrt{3}\,b^{1/3}\tau^{2/3})+\phi_{0},$ $\displaystyle\phi_{0}$
$\displaystyle=\frac{3}{4}\pi-\frac{\ln
2\ln{12}}{2\pi}-\arg\Gamma\left(-i\frac{\ln 2}{2\pi}\right).$

Qualitatively the plots of the Suleimanov solution and its asymptotics look
similar to the ones presented on Figures 1–4, with the same comment concerning
dependence on the parameter $b$.

######  Remark 8.3.

The asymptotics of Suleimanov’s solution for $\alpha=-1/2$ has the same real
part as for the case $\alpha=1/2$ presented above, whilst its imaginary part
differs by sign, namely, ${\rm Im}\,u_{as}(\tau)=\sqrt{\frac{b\sqrt{3}\ln
2}{4\pi}}\,\cos{\phi}$.

###  8.2 $a<0$

Assume $a\in\mathbb{R}$. Both conditions, (173) and the weaker one discussed
in Remark 8.1, give the same result $a<0$, which is assumed below. In this
case the solution $u(\tau)$ is real. After some calculation one proves that
Asymptotics (175)–(178) is also, as it should be, real and can be rewritten as
follows,

$\displaystyle u_{as}$ $\displaystyle=\frac{b^{2/3}}{2}\tau^{1/3}$ (179)
$\displaystyle-\frac{b^{1/2}}{3^{1/4}}\sqrt{\frac{-\ln(1-e^{2\pi
a})}{2\pi}}\cos\left(3^{3/2}b^{1/3}\tau^{2/3}-\frac{\ln(1-e^{2\pi
a})}{2\pi}\ln\left(3^{3/2}b^{1/3}\tau^{2/3}\right)+\phi_{0}\right),$
$\displaystyle\phi_{0}$
$\displaystyle=a\ln\left(2+\sqrt{3}\right)-\ln(12)\frac{\ln(1-e^{2\pi
a})}{2\pi}-\frac{\pi}{4}-\arg\left(\Gamma\left(-i\frac{\ln(1-e^{2\pi
a})}{2\pi}\right)\right).$

On Figures 5 and 6 we present examples of Asymptotics (179). The smaller
values of $a$ the better approximation of $u(\tau)$ by $u_{as}(\tau)$, it can
be seen comparing the scale of the $y$-axes and also taking into account that,
as mentioned above, by enlarging $b$ we in a sense enlarging $\tau$ therefore
approximation of $u(\tau)$ by $u_{as}(\tau)$ becomes better. In making
numerics this, however, means that if we consider plots on the same
$\tau$-segment, then the larger $b$ the higher accuracy of calculations are
required, because, it is equivalent to consideration of the solution on a
longer interval. We also see that for the large negative values of $a$, say,
$a=-3$ is already ”large”, the amplitude of oscillation becomes so small that
the plot visually looks like the cubic parabola. We have chosen the parameters
$a$ and $b$ such that on one hand one can see the difference between $u(\tau)$
and $u_{as}(\tau)$ and on the other hand the fact the convergence of the
asymptotics to solution.

Figure 5: The plots of $u(\tau)$ and $u_{as}(\tau)$ for $a=-2/3$ and $b=1/8$.
The second plot is higher. Figure 6: The plots of $u(\tau)$ and $u_{as}(\tau)$
for $a=-3$ and $b=10$. The second plot is higher.

##  9 Positiveness of ${\rm Re}\,u(\tau)$

Looking on Figures 5 and 6 one arrives at the following

######  Proposition 9.1.

The function $u(\tau)$ is bounded for $\tau\in\mathbb{R}$ and $\tau\in
i\mathbb{R}$. Moreover, $u(\tau)$ is the odd function and $u(\tau)>0$ for
$\tau>0$.

As follows from Expansion (23) the solution possesses the following
symmetries:

$u(\tau)=-u(-\tau),\qquad u(\tau,a)=-iu(i\tau,-a).$ (180)

###### Proof.

In the last equation we indicated dependence of $u(\tau)$ on parameter $a$.
Therefore, it is enough to prove boundedness of our solution for $\tau>0$.
Sure, the proof should be independent of the sign of the parameter $a$.
Expansion (23) implies (recall $a<0$) $u(\tau)>0$ for $0<\tau<\delta$ at least
for some rather small $\delta>0$. Assume that $\tau_{z}>0$ is a zero of
$u(\tau)$, then substituting the Taylor expansion of $u(\tau)$ at $\tau_{z}$
with the leading term $A(\tau-\tau_{z})^{n}$ with $n>0$ into Equation (1) one
proves that $n=1$ and $b^{2}/A^{2}=-1$. Therefore, real solutions of Equation
(1) have no zeroes on the real line. It means that our solution is positive on
the positive semiaxis Analogously, substituting the Laurent expansion of
$u(\tau)$ at pole $\tau_{p}$ with the leading term $A_{1}/(\tau-\tau_{p})^{m}$
with $m>0$ one finds $m=2$ and $2\tau_{p}=-8A_{1}$, If we assume $\tau_{p}>0$
we find that
$u(\tau)\underset{\tau\to\infty}{\sim}-\tfrac{\tau_{p}}{4(\tau-\tau_{p})^{2}}<0$.
Thus, our positive solution does not have poles on positive semiaxis. ∎

######  Lemma 9.1.

If a solution of Equation (1) has positive real part for $\tau>0$, then it
does not have poles for $\tau>0$.

###### Proof.

The Laurent expansion of solutions of Equation (1) at the pole $\tau_{p}$
reads

$-\frac{\tau_{p}}{4(\tau-\tau_{p})^{2}}+O(1)\quad\tau\to\tau_{p}$

∎

Since $\tau$ and $\tau_{p}$ are assumed real, then the imaginary part of any
solution does not have poles on the real axis except possibly $\tau_{p}=0$.
Moreover, if we assume that $\tau_{p}>0$, then real part of any solution with
a pole is negative in some its neighborhood, however we assume that the real
part is positive.

######  Proposition 9.2.

For ${\rm Re}\;a<0$ there exists $\varepsilon>0$ such that for $|{\rm
Im}\;a|<\varepsilon$ and $\tau>0$ ${\rm Re}\;u(\tau)>0$.

Assume there is a sequence $a_{n}$ with ${\rm Re}\;a_{n}\equiv\beta<0$ and
${\rm Im}\;a_{n}\to 0$, such that the functions $u(\tau)=u(\tau,a_{n})$ has a
sequence of zeroes $\tau_{n}$. If the sequence $\tau_{n}$ has a bounded
subsequence, then it leads to immediate contradiction with Proposition 9.1,
since in this case $u(\tau_{0})=u(\tau_{0},0)=0$ where $\tau_{0}$ is a finite
limiting point of the sequence $\tau_{n}$.
The case when the sequence $\tau_{n}$ is unbounded also cannot happen. It
follows from the general property of local uniformness of asymptotics of the
Painlevé equations. This property means the following: assume that an
asymptotic formula is valid for some open subset in the space of the monodromy
parameters, as it happens for Asymptotics (174)–(178), which is valid in the
open strip, $a=\beta+i\alpha$ with $\beta<0$ and $|\alpha|<1$. We consider any
compact subset of this strip, for our purposes it is enough to fix
$\beta=\beta_{0}<0$ and a segment $|\alpha|\leq 1-\epsilon$ for some positive
$\epsilon<1$. Then there exist $\Upsilon=\Upsilon(\beta,\epsilon)$ such that
the error estimate in Equation (174) holds for all $\tau>\Upsilon$ and all $a$
from the above compact subset. Thus, if we choose the error estimate in (174)
small enough by fixing $\Upsilon$, then the zeroes $\tau_{n}>\Upsilon$ cannot
exist because $u(\tau)$ is close to the first term in Asymptotics (174) and,
therefore, cannot vanish for any values of $\tau>\Upsilon$. The following
conjecture is the extension of Proposition 9.2.

######  Conjecture 9.1.

If ${\rm Re}\;a\leq 0$ and $0<|{\rm Im}\;a|<1$, then ${\rm Re}\;u(\tau)>0$ for
$\tau>0$.

######  Remark 9.1.

Solutions for the parameter $a$ from Conjecture 9.1 have neither zeroes nor
poles on the positive semiaxis. To prove the absence of poles one can surely
apply Zhou’s vanishing lemma [11]. In our case the Riemann-Hilbert problem
corresponding to the monodromy data defined in Section 7 can be formulated on
the positive real semiaxis and the circle centered at $0$. The jump matrices
are:

$\begin{pmatrix}1&s_{0}\\\ s_{0}&1+s_{0}^{2}\end{pmatrix},\qquad
e^{2\pi(a-i/2)\sigma_{3}},\quad{\rm and}\quad G$

inside, outside, and on the circle, respectively. There are singular points at
zero and infinity with the proper behavior, which we do not discuss here. The
vanishing lemma implies that the jump matrices on the real axis are positively
defined; we get the following condition for the trace of the diagonal matrix,

$\cosh(2\pi(a-i/2))+\cosh(2\pi(\bar{a}+i/2))>0\;\;\Rightarrow\;\;\cos(2\pi{\rm
Im}\,a)<0\;\;\Rightarrow\;\;\frac{1}{4}<{\rm Im}\,a<\frac{3}{4},$

and arbitrary ${\rm Re}\,a$. The trace of the nondiagonal matrix is
$2+s_{0}^{2}$ its positiveness exactly coincide with the above condition on
$a$. There is one more condition that demand the vanishing lemma,
$G^{\dagger}=G^{-1}$. I recall that matrix $G$ (see the second line of
equations in System (170)), contains one free complex parameter, which is
related with the constant of integration in Equation (157), so that for a
given $u(\tau)$ it can be chosen arbitrarily. If we choose it to satisfy the
condition on matrix $G$ we get ${\rm Im}\,a=\pm 1$ with arbitrary ${\rm
Re}\,a$.

Thus, in particular, the Suleimanov solution is regular on the positive
semiaxis, in fact, on the coordinate cross, because of the symmetries (180).
Unfortunately, this remark does not pour any light on the proof of Conjecture
9.1.

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