# Genetics-based deperturbation analysis for the spin-orbit coupled
${\mathrm{A}}^{1}\Sigma^{+}$ and ${\mathrm{b}}^{3}\Pi_{0^{+}}$ states of LiRb

Yide Yin  School of Physics, Dalian University of Technology, 116024 Dalian,
China  Xuhui Bai  School of Physics, Dalian University of Technology, 116024
Dalian, China  Xuechun Li  School of Physics, Dalian University of Technology,
116024 Dalian, China  Xin-Yu Luo  Max Planck Institute of Quantum Optics,
85748 Garching, Germany  Munich Center for Quantum Science and Technology,
80799 München, Germany  Jie Yu<EMAIL_ADDRESS>School of Physics, Dalian
University of Technology, 116024 Dalian, China  Gaoren Wang
<EMAIL_ADDRESS>School of Physics, Dalian University of Technology,
116024 Dalian, China  Yongchang Han  School of Physics, Dalian University of
Technology, 116024 Dalian, China

###### Abstract

We present a deperturbation analysis of the spin-orbit coupled $\rm
A^{1}\Sigma^{+}$ and $\rm b^{3}\Pi_{0^{+}}$ states of LiRb based on the
rovibrational energy levels observed previously by photoassociation
spectroscopy in bosonic 7Li85Rb molecule. Using the genetic algorithm, we fit
the potential energy curves of the $\rm A^{1}\Sigma^{+}$ state and the $\rm
b^{3}\Pi$ state into point-wise form. We then fit these point-wise potentials
along with the spin-orbit coupling into expanded Morse oscillator functional
form and optimise analytical parameters based on the experimental data. From
the fitted results, we calculate the transition dipole moment matrix elements
for transitions from the rovibrational levels of the coupled $\rm
A^{1}\Sigma^{+}$-$\rm b^{3}\Pi_{0^{+}}$ state to the Feshbach state and the
absolute rovibrational ground state for fermionic 6Li87Rb molecule. Based on
the calculated transition dipole moment matrix elements, several levels of the
coupled $\rm A^{1}\Sigma^{+}$-$\rm b^{3}\Pi_{0^{+}}$ state are predicted to be
suitable as the intermediate state for stimulated Raman adiabatic passage
transfer from the Feshbach state to the absolute rovibrational ground state.
In addition, we also provide a similar estimation for ${\rm B}^{1}\Pi$-${\rm
c}^{3}\Sigma_{1}^{+}$-${\rm b}^{3}\Pi_{1}$ state based on available $ab\
initio$ interaction potentials.

††preprint: APS/123-QED

##  I Introduction

Since the turn of the century, the realm of ultracold atoms and molecules has
emerged as one of the most fascinating fields for physicists [1] and chemists
[2]. Compared to ultracold atoms, ultracold molecules offer a richer internal
structure, enhancing application capabilities. The most common techniques used
to prepare ultracold molecules in experiments are photoassociation (PA) [3, 4]
and magnetoassociation [5]. The latter technique prepares Feshbach molecules
[6] that are subsequently transferred to their absolute rovibrational ground
state via stimulated Raman adiabatic passage (STIRAP) [7]. It has been
productively adopted on bialkali polar molecules including 6Li23Na [8], 6Li40K
[9], 23Na39K [10], 23Na40K [11, 12, 13, 14, 15], 23Na87Rb [16, 17], 23Na133Cs
[18, 19], 40K87Rb [20], and 87Rb133Cs [21, 22, 23].

As one of the last missing members of the ultracold bialkali polar molecules
family, LiRb molecule has several unique features in comparison to the species
mentioned above [24, 25]. It exhibits the second largest body-frame dipole
moment among fermionic bialkali molecules [26]. Microwave-shielded LiRb
molecules promise orders of magnitude higher good-to-bad collision rate ratios
compared to existing fermionic polar molecules [27, 28]. As a result, they
have the potential to be cooled to unprecedentedly low temperatures for the
realization of exotic quantum matters [29]. This is made possible by their
large dipole moment and high rotational constant.

LiRb molecule has been investigated experimentally through spectroscopic
techniques. Earlier in 2009, Marzok $et\ al.$ observed Feshbach resonance [30]
in an ultracold 7Li and 87Rb mixture [31]. Ivanova $et\ al.$ observed
spectroscopic data of the $\rm X^{1}\Sigma^{+}$, $\rm a^{3}\Sigma^{+}$, $\rm
B^{1}\Pi$ and $\rm D^{1}\Pi$ states [32, 33]. In 2011, Dutta $et\ al.$ studied
the first 21 vibrational levels of the B${}^{1}\Pi$ via laser-induced
fluorescence spectroscopy [34]. In 2014, Lorenz $et\ al.$ observed the
transitions from the ground electronic state to the $\rm B^{1}\Pi$ and $\rm
D^{1}\Pi$ states via PA and resonantly enhanced multiphoton ionisation
spectroscopy [35]. In 2015, Altar $et\ al.$ probed and identified the high-
lying vibrational levels of $\rm a^{3}\Sigma^{+}$ state and the low-lying
vibrational levels of $(3)^{3}\Pi$ and $(4)^{3}\Sigma^{+}$ [36]. In 2016,
Blasing $et\ al.$ investigated the low-lying vibrational levels of $\rm
a^{3}\Sigma^{+}$ state [37]. Stevenson $et\ al.$ explored the rovibrational
levels in the ${\rm A}^{1}\Sigma^{+}$, ${\rm b}^{3}\Pi$, $\rm C^{1}\Sigma^{+}$
and $\rm d^{3}\Pi$ states [38, 39, 40].

In addition to experimental studies, there have been many theoretical studies
on the underlying interactions in LiRb. Earlier in 2009, Korek $et\ al.$
calculated the potential energy curves (PECs) for the 58 lowest electronic
states of the LiRb molecule using the configuration interaction method, and
the spin-orbit (SO) effect was considered through a semi-empirical SO pseudo-
potential [41]. In 2012, Jendoubi $et\ al.$ calculated the adiabatic and
diabatic PECs, the permanent dipole moments and the transition dipole moments
(TDMs) for 38 electronic states of LiRb [42]. In 2016, You $et\ al.$
investigated the spin-forbidden cooling of LiRb molecule and calculated the
PECs of the $\rm X^{1}\Sigma^{+}$, $\rm a^{3}\Sigma^{+}$, $\rm b^{3}\Pi$ and
$\rm B^{1}\Pi$ states, as well as the SO coupling effects and TDMs using the
multi-reference configuration interaction (MRCI) method [43]. Employing the
quasi-relativistic electronic wave functions obtained by the MRCI method,
Bormotova $et\ al.$ calculated TDMs and SO coupling matrix elements between
the ground and excited states for LiRb [44, 45]. In 2020, Kozlov $et\ al.$
evaluated PECs and SO coupling matrix elements in low-lying electronic states
of LiRb molecules using two different approaches [46].

Although the spectroscopic data of the ${\rm A}^{1}\Sigma^{+}$ and ${\rm
b}^{3}\Pi_{0^{+}}$ states for 7Li85Rb have been observed and assigned in [39],
deperturbation analysis [47] has not been adopted on these electronic states
of LiRb. In the past few years, deperturbation analysis has already been
adopted on the ${\rm A}^{1}\Sigma^{+}$ and ${\rm b}^{3}\Pi_{0^{+}}$ states for
NaRb [48, 49], RbCs [50, 51, 52], Na2 [53], Cs2 [54, 55], NaCs [56], KCs [57,
58], LiCs [59], NaK [60], and KRb [61]. Besides experimental and theoretical
data, efficient optimisation algorithms are essential for deperturbation
analysis. One of the most commonly used algorithms is a modified Levenberg-
Marquardt (LM) algorithm [62]. Following the surge in computational power,
heuristic algorithms like genetic algorithm (GA) have been applied to
optimisation tasks in atomic and molecular physics, such as fitting the
single-channel PEC based on spectroscopic data [63, 64, 65] and optimising
parameters of the laser pulse [66]. Inspired by these works, we present a
genetics-based deperturbation analysis to fit the PECs of ${\rm
A}^{1}\Sigma^{+}$ and ${\rm b}^{3}\Pi$ states and the SO coupling between them
based on the spectroscopic data in [39]. With the fitted PECs and SO coupling,
we predict the rovibrational levels of the coupled ${\rm A}^{1}\Sigma^{+}$ and
${\rm b}^{3}\Pi_{0^{+}}$ states which are suitable as the intermediate state
for STIRAP transfer from the Feshbach state to the absolute rovibrational
ground state. This is vital for preparing the ultracold LiRb molecule in
experiments.

The remaining part of this paper proceeds as follows: the second section
introduces the methodology used in the present work; the third section
discusses the fitted results and future prospects; the last section states the
conclusion of the present work.

##  II Methodologies

###  II.1 model Hamiltonian

Figure 1:  The low-lying electronic states of the LiRb molecule. The PECs of
the $\rm X^{1}\Sigma^{+}$ and $\rm a^{3}\Sigma^{+}$ states are from [32]. The
PECs of the $\rm A^{1}\Sigma^{+}$ and $\rm b^{3}\Pi$ states are from [46]. The
shaded area denotes the observed energy range in [39], which is from 5682 $\rm
cm^{-1}$ to 8943 $\rm cm^{-1}$. The green and dark purple wave functions are
for the Feshbach state and the absolute rovibrational ground state. The red
and light purple wave functions are for the intermediate state chosen from the
SO coupled $\rm A^{1}\Sigma^{+}$ and $\rm b^{3}\Pi$ states. The red and light
purple arrows indicate the transitions induced by the pump and Stokes lasers,
respectively.

The $\rm A^{1}\Sigma^{+}$ and $\rm b^{3}\Pi$ states are two of the lowest
electronic excited states for bialkali polar molecules. Together with $\rm
X^{1}\Sigma^{+}$ and $\rm a^{3}\Sigma^{+}$ states, the PECs of $\rm
A^{1}\Sigma^{+}$ and $\rm b^{3}\Pi$ of LiRb are illustrated in Fig. 1. In
order to establish an effective model Hamiltonian for the SO coupled $\rm
A^{1}\Sigma^{+}$ and $\rm b^{3}\Pi$ states, one should typically consider the
diabatic PECs of $\rm A^{1}\Sigma^{+}$ and $\rm b^{3}\Pi$ states and the SO
splitting of $\rm b^{3}\Pi$ state, as well as the SO coupling between the $\rm
A^{1}\Sigma^{+}$ and $\rm b^{3}\Pi_{0^{+}}$ states and the spin-rotational
interaction between the $\Omega=0^{+},\>1,\>2$ components of $\rm b^{3}\Pi$
state. However, the spectroscopic data in [39] are limited to the
$\Omega=0^{+}$ components of the $\rm b^{3}\Pi$ states, and there are no data
from the $\Omega=1,2$ components. Therefore, we consider only $\rm
A^{1}\Sigma^{+}$ and $\rm b^{3}\Pi_{0^{+}}$ states in the present work.

The relevant coupled-channel (CC) radial Schrödinger equation is

$\left(\begin{array}[]{ll}T+V_{\rm A}&V_{\rm Ab}\\\ V_{\rm
Ab}&T+V_{\rm{b0}}\\\ \end{array}\right)\left(\begin{array}[]{ll}\Phi_{\rm
A}\\\ \Phi_{\rm b0}\\\ \end{array}\right)=E^{\rm
CC}_{v^{\prime}}\left(\begin{array}[]{ll}\Phi_{\rm A}\\\ \Phi_{\rm b0}\\\
\end{array}\right),$ (1)

where $T$ is the kinetic energy term, $V_{\rm A}$, $V_{\rm b0}$ are the
potential energy terms, $V_{\rm Ab}$ is the SO coupling term, $v^{\prime}$
indexes the rovibrational levels of coupled ${\rm A}^{1}\Sigma^{+}$-${\rm
b}^{3}\Pi_{0^{+}}$ state, $E_{v^{\prime}}^{\rm CC}$ is its rovibrational
energy, and $\Phi_{\rm A}$ and $\Phi_{\rm b0}$ are two components of the
eigenstate. The fractional partitions of ${\rm A}^{1}\Sigma^{+}$ state and
${\rm b}^{3}\Pi_{0^{+}}$ state in the eigenstate are $P_{\rm
A}=\left\langle\Phi_{\rm A}|\Phi_{\rm A}\right\rangle$ and $P_{\rm
b0}=\left\langle\Phi_{\rm b0}|\Phi_{\rm b0}\right\rangle$. The eigenstate is
assigned to the electronic state with larger fractional partition.

The potential energy and SO coupling terms in Eq. (1) are

$\begin{array}[]{lll}V_{\rm A}=U_{\rm A}+E^{\rm ROT}_{\rm A},\\\ V_{\rm
b0}=U_{\rm b}-\xi_{\rm bb}+E^{\rm ROT}_{\rm b0},\\\ V_{\rm
Ab}=-\sqrt{2}\xi_{\rm Ab},\\\ \end{array}$ (2)

where $U_{\rm A}$ and $U_{\rm b}$ stand for the diabatic PECs of $\rm
A^{1}\Sigma^{+}$ state and $\rm b^{3}\Pi$ state, respectively, $\xi_{\rm bb}$
is the SO splitting of the $\rm b^{3}\Pi$ state, $\xi_{\rm Ab}$ is the SO
coupling between the $\rm A^{1}\Sigma^{+}$ state and $\rm b^{3}\Pi_{0^{+}}$
state, and $E^{\rm ROT}_{\rm A}$ and $E^{\rm ROT}_{\rm b0}$ are the rotational
Hamiltonians of the $\rm A^{1}\Sigma^{+}$ and $\rm b^{3}\Pi_{0^{+}}$ states,
respectively. Notably, the SO coupling term $\xi_{\rm Ab}$ in $V_{\rm Ab}$ is
multiplied by a sign-consistent multiplicative factor of $-\sqrt{2}$, which
corrects the fine splitting near the dissociation thresholds [45]. The
rotational Hamiltonians [47] in Eq. (2) are derived by

$E^{\rm ROT}_{{}^{2S+1}\Lambda_{\Omega}}=\frac{\hbar^{2}}{2\mu
R^{2}}[J(J+1)-\Omega^{2}+S(S+1)-\Sigma^{2}+L(L+1)-\Lambda^{2}],\\\ $ (3)

where $\hbar$ is the reduced Planck constant, $\mu$ is the reduced mass, $R$
is the internuclear distance, $S$ is the total electron spin, $L$ is the total
electronic orbital angular momentum, $J$ is the total molecular angular
momentum exclusive of nuclear spin, $\Sigma$ and $\Lambda$ are the projection
of $S$ and $L$ along the internuclear axis, and $\Omega=|\Sigma+\Lambda|$.
Therefore,

$\begin{array}[]{lll}E^{\rm ROT}_{\rm A}=E^{\rm ROT}_{\rm
b0}=\frac{\hbar^{2}}{2\mu R^{2}}\left[J(J+1)+2\right].\\\ \end{array}$ (4)

In the present work, the higher-order SO terms and hyperfine coupling are
ignored due to the limitation of spectroscopic data. Within the framework of
our model Hamiltonian, the terms $U_{\rm A}$, $U_{\rm b}$, and $\xi_{\rm Ab}$
are supposed to be optimised. The optimisation algorithm is detailed in the
next two subsections.

###  II.2 Genetic algorithm

GA is a stochastic global optimising method inspired mainly by natural
selection and genetics [67]. In the present work, we modify the algorithm
developed in [65] to optimise the point-wise $U_{\rm A}$ and $U_{\rm b}$,
which has been previously calculated in [46] through the configuration
interaction method with core-valence correlation accounted for using core
polarization potentials (CI-CPP). The flowchart of our modified algorithm is
illustrated in Fig. 2, and each part of it is elaborated in the following.

Figure 2: The flowchart of the genetic algorithm adopted in this work.

####  II.2.1 Initial population

Our algorithm starts with creating an initial population that contains $N$
individuals, each representing a candidate solution to the current problem.
There are two chromosomes in each individual, which represent the diabatic
PECs $U_{\rm A}$ and $U_{\rm b}$, respectively. Both of them are point-wise
PECs along the non-uniform $R$-space grid, as listed in the first column in
Table 1. There are 29 genes in each chromosome, each of which stands for a
potential point. A schematic diagram of the population is shown in Fig. 3.

Figure 3: The schematic diagram of a population in the genetic algorithm of this work. Table 1:  The point-wise PECs of the $\rm A^{1}\Sigma^{+}$ and $\rm b^{3}\Pi$ states of LiRb obtained via $ab\ initio$ calculations in [46] ($U_{\rm A}^{ab}$ and $U_{\rm b}^{ab}$) and GA optimisation ($U_{\rm A}^{\rm GA}$ and $U_{\rm b}^{\rm GA}$). Internuclear distances are given in $\rm\AA$, and energies are in $\rm cm^{-1}$. $R$ | $U_{\rm A}^{ab}$ | $U_{\rm b}^{ab}$ | $U_{\rm A}^{\rm GA}$ | $U_{\rm b}^{\rm GA}$
---|---|---|---|---
2.0 | 26891.94 | 18627.56 | 26812.65 | 18696.66
2.2 | 21646.73 | 13344.37 | 21720.34 | 13399.84
2.4 | 17618.69 | 9853.42 | 17597.78 | 9173.52
2.6 | 14333.90 | 7512.10 | 14320.65 | 6988.95
2.8 | 11691.20 | 5954.89 | 11768.61 | 5625.15
3.0 | 9660.52 | 4992.17 | 9828.51 | 4734.67
3.2 | 8180.83 | 4500.57 | 8134.73 | 4247.30
3.4 | 7157.70 | 4374.61 | 7089.34 | 4123.49
3.6 | 6493.40 | 4523.65 | 6419.08 | 4281.63
3.8 | 6101.02 | 4872.14 | 6010.71 | 4655.91
4.0 | 5908.97 | 5359.15 | 5803.91 | 5183.73
4.2 | 5861.14 | 5936.55 | 5750.32 | 5808.80
4.4 | 5915.47 | 6566.73 | 5811.50 | 6483.36
4.6 | 6042.22 | 7220.42 | 5945.16 | 7169.24
4.8 | 6221.29 | 7875.21 | 6127.08 | 7837.83
5.0 | 6443.57 | 8515.00 | 6357.31 | 8472.52
5.4 | 6983.73 | 9680.68 | 6919.18 | 9619.65
5.8 | 7624.96 | 10639.51 | 7587.32 | 10665.27
6.2 | 8333.23 | 11357.22 | 8296.31 | 11412.43
6.6 | 9063.86 | 11846.76 | 9029.58 | 11911.12
7.0 | 9775.01 | 12159.70 | 9812.40 | 12159.70
8.0 | 11260.91 | 12520.23 | 11295.72 | 12520.23
9.0 | 12125.39 | 12641.24 | 12107.77 | 12641.24
10.0 | 12486.87 | 12688.89 | 12486.87 | 12688.89
11.0 | 12622.40 | 12711.20 | 12622.40 | 12711.20
12.0 | 12678.41 | 12722.59 | 12678.41 | 12722.59
14.0 | 12704.88 | 12729.11 | 12704.88 | 12729.11
17.0 | 12718.63 | 12733.00 | 12718.63 | 12733.00
21.0 | 12726.15 | 12735.30 | 12726.15 | 12735.30

To generate the chromosomes and ensure the physical characteristics of the
point-wise PECs, Wei Hua’s (HW) 4-parameter potential function [68] is
adopted, which is expressed as

$U_{\rm
HW}(R)=T_{e}+D_{e}\left[\frac{1-e^{-b\cdot(R-R_{e})}}{1-Ce^{-b\cdot(R-R_{e})}}\right]^{2},$
(5)

where $T_{e}$ is the potential energy minimum, $D_{e}$ is the potential well
depth, $R_{e}$ is the equilibrium internuclear distance, and $b$ and $C$ are
two parameters controlling the shape of the potential well. $T_{e}+D_{e}$ is
the dissociation threshold of Li 2S + Rb 5P, which is fixed to 12737.35 $\rm
cm^{-1}$ according to the NIST atomic database [69]. Notably, the Li 2S + Rb
5S atomic asymptote serves as the benchmark for energies.

To create the individuals in the initial population, 4 parameters $T_{e}$,
$R_{e}$, $b$ and $C$ are tuned, each of which is uniformly distributed around
a central value with a range. The determination of the central values and
ranges is discussed below. The potential minimum $T_{e}$ is estimated in [39]
by fitting the spectroscopic data to

$T_{v}=T_{e}+\omega_{e}(v+\frac{1}{2})-\omega_{e}\chi_{e}(v+\frac{1}{2})^{2}+\omega_{e}y_{e}(v+\frac{1}{2})^{3},$
(6)

where $\omega_{e}$ is the vibrational harmonic frequency, $\omega_{e}\chi_{e}$
and $\omega_{e}y_{e}$ are first and second anharmonic correction coefficients,
$v$ is the vibrational quantum number of the uncoupled ${\rm A}^{1}\Sigma^{+}$
or ${\rm b}^{3}\Pi_{0^{+}}$ states, and $T_{v}$ is the rovibrational energy.
The parameters $b$ and $C$ are estimated by fitting the $ab\ initio$ PECs in
[46] with HW potential function. The equilibrium distance $R_{e}$ is set to
the value theoretically calculated in [46]. Their central values and ranges
are listed in Table 2. As seen, different to $T_{e}$, $b$ and $C$, all
individuals share the same $R_{e}$ in the initial population. That is because
the PA spectroscopy data in [39] is limited to the rotational level of $J=1$,
the value of $R_{e}$ related to the rotational constant cannot be fitted from
the observed levels. The point-wise $U_{\rm A}$ and $U_{\rm b}$ corresponding
to different $T_{e}$, $R_{e}$, $b$ and $C$, as illustrated in Fig. 4, are
encoded into the population.

Table 2:  The central values and ranges of HW parameters for the $\rm A^{1}\Sigma^{+}$ and $\rm b^{3}\Pi$ states of LiRb. The second and third columns are the central values, and the forth column denotes their ranges. The range is from $1-\Delta$ to $1+\Delta$ multiplied by the central value. Energies are given in $\rm cm^{-1}$, internuclear distances are in $\rm\AA$, coefficient $b$ is in the uint of 1/$\rm\AA$, and $C$ is dimensionless. | $U_{\rm A}$ | $U_{\rm b}$ | $\Delta$
---|---|---|---
$T_{e}$ | 5756.6 | 4112.6 | 2%
$T_{e}+D_{e}$ | 12737.3467 | 12737.3467 | 0
$R_{e}$ | 4.1836 | 3.3802 | 0
$b$ | 0.5269 | 0.8309 | 10%
$C$ | -0.0827 | -0.1979 | 10%
Figure 4: The diagram of encoding point-wise PECs. The subfigures illustrate
potential points in the short-range and long-range regions. The cluster of
light blue lines denotes HW potential with different parameters. The colored
points are the $ab\ initio$ potential points listed in Table 1. The green
points are the reference points mentioned in Appendix A. The red points could
be chosen as crossover point, as mentioned in Section II.2.3. The shaded area
denotes the energy range of the observed rovibrational levels. The dot-dashed
line denotes the upper energy boundary of the reference levels in Appendix A.

####  II.2.2 Fitness function

After the initial population is created, it naturally follows to define a
fitness function that evaluates the performance of individuals. In order to
remedy the limitation of observed energy range in [39], as shown in Fig. 1 and
4, the fitness function in our algorithm should take not only rovibrational
levels but also potential points into account. It contains two terms denoted
as $F_{\alpha}$ and $F_{\beta}$, concerning potential points and rovibrational
levels, respectively, as described in Appendix A.

Considering the difference in the order of magnitude between $F_{\alpha}$ and
$F_{\beta}$, the fitness function in our algorithm is expressed as

$F=\frac{F_{\beta}}{\sigma_{\beta}}+\omega_{F}\cdot\left(\frac{F_{\alpha}}{F_{\beta}}\right)^{2},$
(7)

where $\sigma_{\beta}$ is the experimental uncertainty and $\omega_{F}$ is an
coefficient merging $F_{\alpha}$ and $F_{\beta}$. It needs to set $\omega_{F}$
properly to control the evolution. At the beginning of evolution, the fitting
should primarily focuses on the rovibrational levels. As the evolution goes,
the potential points are also supposed to be fitted together with the
rovibrational levels. The variation of the fitness value $F$ versus
$F_{\alpha}$ and $F_{\beta}$ is shown in Fig. 5. The individuals with smallest
fitness value in the current population, namely the fittest individuals, are
denoted by the circled dots for each generation.

Figure 5: The variation of $F$ versus $F_{\alpha}$ and $F_{\beta}$ with
$\omega_{F}=0.002$. The fittest individuals in each generation are denoted by
circled dots, while the colorbar signifies generations.

The smaller the fitness value of an individual, the better it is. To select
better individuals and speed up the convergence, the weight for each
individual is assigned based on their fitness, as

$W_{n}=\frac{\left(\frac{\bar{F}}{F_{n}}\right)^{\frac{1}{2}}}{\sum_{n}\left(\frac{\bar{F}}{F_{n}}\right)^{\frac{1}{2}}},$
(8)

where $n$ indexes individuals, $F_{n}$ is the fitness value of an individual,
$\bar{F}$ is the median fitness value among all individuals, the exponential
term of $\frac{1}{2}$ is to reduce the effect of extreme individuals [24]. The
weight of an individual stands for its probability of being selected via
roulette selection.

####  II.2.3 Evolutionary operation

The evolutionary operation for GA generally includes selection, crossover and
mutation [70]. In the present work, the evolutionary operation is adapted from
the methods in [65], which works as follows:

  1. 1.

Select 4 individuals from the population via roulette selection and randomly
specify a chromosome to evolve. The chromosomes specified to evolve in these 4
individuals are labelled $\bm{\mathit{x}}$, $\bm{\mathit{y}}$,
$\bm{\mathit{z}}$, $\bm{\mathit{w}}$, respectively.

  2. 2.

Create a new chromosome $\bm{\mathit{u}}$ according to

$\bm{\mathit{u}}=\bm{\mathit{x}}+\lambda(\bm{\mathit{y}}-\bm{\mathit{z}}),$
(9)

where $\lambda$ is a hyperparameter related to the evolutionary intensity.

  3. 3.

Choose a crossover point 111Since the energy range of the observed
rovibrational levels is limited, the crossover points should also be limited
in certain regions. The crossover point is randomly chosen from the red points
shown in Fig. 4.

  4. 4.

Cross $\bm{\mathit{u}}$ with $\bm{\mathit{w}}$ according to

$v_{i}=\left\\{\begin{array}[]{lc}w_{i}&{\rm if}\enspace i\leq I\\\ u_{i}&{\rm
if}\enspace i>I\end{array}\right.,$ (10)

when the crossover point is to the left of the equilibrium distance,
otherwise,

$v_{i}=\left\\{\begin{array}[]{lc}u_{i}&{\rm if}\enspace i\leq I\\\ w_{i}&{\rm
if}\enspace i>I\end{array}\right.,$ (11)

where $i$ indexes genes and $I$ is the index of the crossover point. The
crossover probability is $P_{\rm x}$. If the crossover doesn’t occur, set
$\bm{\mathit{v}}=\bm{\mathit{u}}$ and proceed directly to the next step.

  5. 5.

Assemble $\bm{\mathit{v}}$ with the unevolved chromosome in the old individual
containing $\bm{\mathit{w}}$. Compare the fitness of this new individual to
the old individual containing $\bm{\mathit{w}}$, and add the better one to the
next generation.

The steps above is repeated $N$ times to obtain the next generation
population. After that, elitist preservation is adopted to ensure global
convergence, which directly copies the best individual emerged so far and
replaces the current worst individual [70]. The above operation is iterated to
evolve the population until the $T$-th generation, and the fitted point-wise
PECs $U_{\rm A}$ and $U_{\rm b}$ is obtained.

###  II.3 Analytical potential functions

After the point-wise $U_{\rm A}$ and $U_{\rm b}$ are optimised via GA, we
adopt an optimisation algorithm called sequential least squares programming
(SLSQP) [72] to further optimise $U_{\rm A}$, $U_{\rm b}$ and $\xi_{\rm Ab}$
in the expanded Morse oscillator (EMO) function form. The EMO function is
expressed as

$U_{\rm EMO}(R)=T_{e}+D_{e}[1-e^{-\beta_{\rm EMO}(R)\cdot(R-R_{e})}]^{2},$
(12)

and the coefficient $\beta_{\rm EMO}$ is a polynomial

$\beta_{\rm EMO}(R)=\sum_{i=0}^{N_{\beta}}\beta_{i}\left(\frac{R^{q}-R_{\rm
ref}^{q}}{R^{q}+R_{\rm ref}^{q}}\right)^{i},$ (13)

where $q$ is a small positive integer, and $R_{\rm ref}$ is a positive
reference distance slightly larger than $R_{e}$, both of which are fixed.

Firstly, the point-wise $U_{\rm A}$ and $U_{\rm b}$ obtained by GA and
$\xi_{\rm Ab}$ obtained by $ab\ initio$ calculations in [46] are fitted into
EMO form through betaFIT program [73]. The parameters fitted via betaFIT serve
as initial values of the SLSQP algorithm, and are then optimised to minimise
the loss function

$L=\omega_{L}\sum_{i}(U_{\rm EMO}(R_{i})-U_{\rm
GA}(R_{i}))^{2}+\sum_{j}(E^{j}_{\rm cal}-E^{j}_{\rm obs})^{2},$ (14)

where $i$ indexes the potential points listed in Table 1, $U_{\rm EMO}$ is the
potential energy calculated via EMO function, $U_{\rm GA}$ is the
corresponding GA fitted value listed in Table 1, $j$ indexes the rovibrational
levels observed in [39], $E_{\rm cal}$ is the rovibrational energy calculated
by solving Eq. (1) [74] with $\xi_{\rm bb}$ calculated via CI-CPP in [46],
$E_{\rm obs}$ is the corresponding experimental value in [39], and
$\omega_{L}$ is weight of potential points in the loss function.

The SLSQP algorithm implimented in SciPy 1.13.0 [75] is used in the present
work. As one of the traditional optimisation algorithms, it tends to be
sensitive to the initial value. If the initial value is set inappropriately,
it may fall into a local minimum. In comparison, GA is more efficient to
search larger parameter space and target the global minimum, but costs more
computational resources. Combining these two methods tends to work better,
which is the core strategy of our methodology.

##  III Discussions

###  III.1 Fitted results

Figure 6: The comparison of $U_{\rm A}$, $U_{\rm b}$ (up panel) and $\xi_{\rm Ab}$ (down panel) obtained by $ab\ initio$ calculations (crosses), GA optimisation (circles) and EMO function fitting (solid lines). The shaded area denotes the energy range of observed rovibrational levels in [39]. Table 3:  The parameters of EMO PECs and SO coupling for the $\rm A^{1}\Sigma^{+}$ and $\rm b^{3}\Pi$ states of LiRb. The parameters with superscript f are fixed. The remaining parameters are optimised via the SLSQP algorithm. Energies are given in $\rm cm^{-1}$, internuclear distances are in $\rm\AA$ and coefficients $\beta_{i}$ are in 1/$\rm\AA$. | $U_{\rm A}$ | $U_{\rm b}$ | $\xi_{\rm Ab}$
---|---|---|---
$q$ | 366footnotemark: 6 | 466footnotemark: 6 | 166footnotemark: 6
$R_{\rm ref}$ | 4.266footnotemark: 6 | 3.466footnotemark: 6 | 4.666footnotemark: 6
$T_{e}$ | 5749.9358 | 4124.2063 | 31.4452
$D_{e}$ | 6987.4109 | 8613.1404 | 47.7538
$R_{e}$ | 4.181966footnotemark: 6 | 3.377966footnotemark: 6 | 4.3196
$\beta_{0}$ | 0.4558156100 | 0.6659361156 | 0.2105957306
$\beta_{1}$ | -0.1003385339 | 0.2005301984 | 0.1138233161
$\beta_{2}$ | -0.4728226328 | -0.3765692448 | 0.726857965766footnotemark: 6
$\beta_{3}$ | 1.3235443844 | -1.6712335641 | 2.165308397766footnotemark: 6
$\beta_{4}$ | 6.2113954621 | 4.0219039659 | 2.371627065466footnotemark: 6
$\beta_{5}$ | -12.6227039293 | 15.1945920816 |
$\beta_{6}$ | -30.6398768819 | -20.5050486971 |
$\beta_{7}$ | 60.3403046173 | -54.1544811936 |
$\beta_{8}$ | 76.0445287778 | 55.1534757328 |
$\beta_{9}$ | -138.5647758852 | 81.8727587255 |
$\beta_{10}$ | -92.2212822226 | -71.0566484870 |
$\beta_{11}$ | 152.1359040286 | -45.5314864256 |
$\beta_{12}$ | 43.8793924613 | 35.0427044818 |
$\beta_{13}$ | -63.8511533223 | 2.1812779222 |

We adopt our method on the LiRb molecule. First of all, the $ab\ initio$
point-wise PECs $U_{\rm A}$ and $U_{\rm b}$ are optimised through GA. After
repeated trials, the hyperparameters of GA are tuned to $N=1000$, $T=400$,
$\omega_{F}=0.002$, $\lambda=0.5$, and $P_{\rm x}=0.3$. The procedure is run
10 times under these hyperparameters. The physically best results are picked
from these 10 runs, as listed in Table 1. One can see that the point-wise PECs
near the bottom of potentials have been optimised about hundreds of $\rm
cm^{-1}$. Afterwards, the GA-optimised $U_{\rm A}$ and $U_{\rm b}$ in Table 1
and $\xi_{\rm Ab}$ calculated in [46] are fitted into the EMO function. The
coefficient $\omega_{L}$ in the loss function of SLSQP algorithm is set to
0.0001. The fitted parameters of the EMO functions are listed in Table 3. As
shown in Fig. 6, compared to $ab\ initio$ calculations, the GA-optimised PECs
have deeper potential wells, and so do the EMO PECs. The fitted SO coupling
curve also has a deeper and wider well compared to $ab\ initio$ calculations.

We calculate the rovibrational energies with the PECs and SO terms obtained by
$ab\ initio$ calculations, GA optimisation and EMO function fitting,
respectively, as seen in the supplemental materials 222See Supplemental
Material at [URL will be inserted by publisher].. In [39], the energy spacings
between the neighbor vibrational levels $\Delta E_{v}=E_{v+1}-E_{v}$
calculated with the PECs taken from [41] are compared to the experimental
observed values. There are nearly a half of energy spacings with an error
larger than 1 $\rm cm^{-1}$. Similarly, we also calculate the energy spacings
with the EMO PECs and SO coupling, which agree well with values determined by
the experimental data, as denoted illustrated in Fig. 7. The root mean squared
error between the theoretically calculated energy spacings and their observed
values are less than 0.5 $\rm cm^{-1}$, without considering $\rm
A^{1}\Sigma^{+}$ $v=2$ level and $\rm b^{3}\Pi_{0^{+}}$ $v=8,9$ levels. To fit
these 3 rovibrational levels more accurately, it may need to consider other
components of the $\rm b^{3}\Pi$ state and the spin-rotational coupling
between them.

Figure 7: The comparison of experimental vibrational spacings [39] (red
triangles) to the calculated values with the analytical PECs given in Table 3
(blue circles) for ${\rm A}^{1}\Sigma^{+}$ state (up panel) and ${\rm
b}^{3}\Pi_{0^{+}}$ state (low panel) of 7Li85Rb molecule.

###  III.2 Future prospects

Figure 8: The TDM matrix elements for two legs (up panel) and strengths (down
panel) of STIRAP transfer from the Feshbach state to the absolute
rovibrational ground state for 6Li87Rb, with the rovibrational levels of the
SO coupled ${\rm A}^{1}\Sigma^{+}$ and ${\rm b}^{3}\Pi_{0^{+}}$ states as
intermediate state. The rotational quantum number $J$ equals 1.

The physical model fitted in the present work are useful for the guidance of
preparing ultracold molecules in experiment. As illustrated in Fig. 1, the
rovibrational levels in the SO coupled $\rm A^{1}\Sigma^{+}$ and $\rm
b^{3}\Pi_{0^{+}}$ states could be chosen as the intermediate state for STIRAP.
With the fitted PECs and SO coupling, one could calculate the TDM matrix
elements, which are helpful in predicting feasible intermediate state for
STIRAP transfer.

For fermionic 6Li87Rb, the TDM matrix elements for the transitions from the
rovibrational levels of the coupled ${\rm A}^{1}\Sigma^{+}$-${\rm
b}^{3}\Pi_{0^{+}}$ states to the Feshbach state and to the absolute
rovibrational ground state are calculated as

${\rm TDM}_{v^{\prime}}^{\rm up}=\int\Phi_{\rm F}\cdot{\hat{\mu}_{\rm
ab}}\cdot\Phi_{\rm b0}(v^{\prime},J=1){\rm d}R,$ (15)

and

${\rm TDM}_{v^{\prime}}^{\rm down}=\int\Phi_{\rm X}\cdot{\hat{\mu}_{\rm
AX}}\cdot\Phi_{\rm A}(v^{\prime},J=1){\rm d}R,$ (16)

where $\Phi_{\rm F}$ is the wave function of the Feshbach state, which is
approximated as that for the shallowest bound state of the $\rm
a^{3}\Sigma^{+}$ state [49], $\Phi_{\rm X}$ is the wave function of the
absolute rovibrational ground state, $\Phi_{\rm A}$ and $\Phi_{\rm b0}$ are
components of the eifenfunction calculated by solving Eq. (1),
${\hat{\mu}_{\rm ab}}$ is the $R$-dependent TDM between $\rm a^{3}\Sigma^{+}$
state and $\rm b^{3}\Pi$ state taken from [44], and ${\hat{\mu}_{\rm AX}}$ is
the $R$-dependent TDM between $\rm X^{1}\Sigma^{+}$ state and $\rm
A^{1}\Sigma^{+}$ state taken from [42]. $\Phi_{\rm F}$ and $\Phi_{\rm X}$ are
calculated based on the empirical PECs in [32]. $\Phi_{\rm A}$ and $\Phi_{\rm
b0}$ are calculated based on the PECs and the SO coupling listed in Table 3,
together with $\xi_{\rm bb}$ adopted from [46].

Table 4: The feasible intermediate states for STIRAP predicted in $\rm A^{1}\Sigma^{+}$-$\rm b^{3}\Pi_{0^{+}}$ state. The wavelengths of the pump and Stokes laser are denoted as $\lambda_{\rm p}$ and $\lambda_{\rm S}$, respectively. Energies are given in $\rm cm^{-1}$, wavelengths are in nm, and TDM matrix elements are in Debye. | Feasible intermediate states
---|---
|  ${\rm A}^{1}\Sigma^{+}$ $v=6$ 191919The rovibrational level is assigned to this electronic state. |  ${\rm A}^{1}\Sigma^{+}$ $v=9$ 191919The rovibrational level is assigned to this electronic state.
|  ${\rm b}^{3}\Pi_{0^{+}}$ $v=12$ |  ${\rm b}^{3}\Pi_{0^{+}}$ $v=14$
$|{\rm TDM}_{v^{\prime\prime}}^{\rm up}|$ | 0.0140 | 0.0115
$|{\rm TDM}_{v^{\prime\prime}}^{\rm down}|$ | 2.932 | 2.498
$E_{v^{\prime\prime}}^{\rm CC}$ | 6562.42 | 6908.88
$\lambda_{\rm p}$ | 1523.8 | 1447.4
$\lambda_{\rm S}$ | 807.4 | 785.4

The STIRAP strength is estimated as

$I_{v^{\prime}}=|\rm TDM_{\it v^{\prime}}^{up}|\cdot|TDM_{\it
v^{\prime}}^{down}|.$ (17)

As shown in Fig. 8, the $v^{\prime}=19$ level stands as the best feasible
intermediate state for the STIRAP transfer. It is assigned as $\rm
A^{1}\Sigma^{+}$ $v=6$ level coupled with the $\rm b^{3}\Pi_{0^{+}}$ $v=12$
level. In addition, the $v^{\prime}=23$ level is also recommended as a
feasible intermediate state, which is assigned as $\rm A^{1}\Sigma^{+}$ $v=9$
level coupled with the $\rm b^{3}\Pi_{0^{+}}$ $v=14$ level. Their
rovibrational energies and relevant wavelengths of the pump laser and Stokes
laser are listed in Table 4. More rovibraional energies calculated for 6Li87Rb
are listed in 22footnotemark: 2.

The SO coupled $\rm B^{1}\Pi$, $\rm c^{3}\Sigma^{+}$ and $\rm b^{3}\Pi$ states
have also been used as intermediate states for the STIRAP transfer [14]. Our
methods could be adopted for the deperturbation analysis of the $\rm
B^{1}\Pi$, $\rm c^{3}\Sigma^{+}$ and $\rm b^{3}\Pi$ states. However, there are
no relevant spectroscopic data, especially those of $\rm c^{3}\Sigma^{+}$
state. Based on the $ab\ initio$ PECs and SO couplings in [46], we also
predict the rovibrational levels of the coupled $\rm B^{1}\Pi$, $\rm
c^{3}\Sigma^{+}$ and $\rm b^{3}\Pi$ states that may be suitable as
intermediate state for STIRAP in Appendix B.

##  IV Conclusion

A genetics-based deperturbation analysis on the SO coupled $\rm
A^{1}\Sigma^{+}$ and $\rm b^{3}\Pi$ states of LiRb molecule was performed in
this work. Utilizing experimentally observed rovibrational levels of the $\rm
A^{1}\Sigma^{+}$ and $\rm b^{3}\Pi_{0^{+}}$ states of 7Li85Rb, we optimised
point-wise PECs of $\rm A^{1}\Sigma^{+}$ and $\rm b^{3}\Pi$ states using GA.
To generalise the algorithm in [65] to the CC problem, we introduced the
concept of chromosome to encode each PEC. To address the limitation of the
observed energy range, we employed a nonlinear fitness function considering
both rovibrational levels and potential points. Afterwards, the point-wise
potentials were fitted into the EMO functional form. The parameters of EMO
functions were further optimised using SLSQP algorithm.

The fitted underlying interactions are vital for guiding the preparation of
ultracold molecules in experiments. Based on the fitted PECs and SO coupling,
the TDM matrix elements for transitions from the rovibrational levels of the
coupled $\rm A^{1}\Sigma^{+}$ and $\rm b^{3}\Pi_{0^{+}}$ states to the
Feshbach state and to the absolute rovibrational ground state for fermionic
6Li87Rb molecule were calculated. We recommended a pathway for STIRAP transfer
from the Feshbach state to the absolute rovibrational ground state of 6Li87Rb
with 1523.8 nm pump laser and 807.4 nm Stokes laser. Additionally, we also
estimated the feasible rovibrational levels of the SO coupled $\rm B^{1}\Pi$,
$\rm c^{3}\Sigma^{+}_{1}$ and $\rm b^{3}\Pi_{1}$ states suitable as
intermediate state for STIRAP based on $ab\ initio$ PECs and SO couplings.
Although there is an error within 30 $\rm cm^{-1}$, the estimation is
sufficient to provide a good starting point for the experimental search for a
suitable intermediate state. Moreover, the methodologies developed in this
work can be generalized to other electronic states of LiRb and other molecular
species.

###### Acknowledgements.

This work has been supported by National Natural Science Foundation of China
(Grants No.12104082 and No.12241409), National Key R&D Program of China (Grant
No.2018YFA0306503), Innovation Program for Quantum Science and Technology
(Grant No.2021ZD0302100), and Program of the State Key Laboratory of Quantum
Optics and Quantum Optics Devices (No.KF201814). Xin-Yu Luo is funded by the
European Union (ERC, DiMoBecTe, 101125173). Xin-Yu Luo gratefully acknowledge
support from the Max Planck Society, and the Deutsche Forschungsgemeinschaft
under Germany’s Excellence Strategy – EXC-2111 – 390814868.

##  Appendix A The fitness terms concerning potential points and rovibrational
levels

To define $F_{\alpha}$ concerning potential points, several potential points
in the short-range and long-range regions are taken into consideration, as
illustrated by the green points in Fig. 4. These points are out of the
experimentally observed energy range, and are defined as reference points. For
each individual, $F_{\alpha}$ is defined as

$F_{\alpha}=\sqrt{\sum_{s}\sum_{i}\zeta^{s}_{i}\left(U^{s}_{\rm
GA}(R_{i})-U^{s}_{\rm ref}(R_{i})\right)^{2}},$ (18)

where $i$ indexes reference points, $s$ indexes electronic states, $U^{s}_{\rm
GA}$ is the potential energy encoded in the considered individual, $U_{\rm
ref}^{s}$ is its corresponding theoretical value calculated via CI-CPP in
[46], and $\zeta^{s}_{i}$ is the weight of different reference points.
$\zeta^{s}_{i}$ is defined as

$\zeta^{s}_{i}=\frac{\frac{1}{\sigma^{s}(R_{i})}}{\sum_{s}\sum_{i}\frac{1}{\sigma^{s}(R_{i})}},$
(19)

where $\sigma^{s}(R_{i})$ is the standard deviation of the potential energy at
reference point $R_{i}$ in the initial population. $\sigma^{s}(R_{i})$ is
evaluated as

$\sigma^{s}(R_{i})=\sqrt{\frac{1}{N}\sum_{n=1}^{N}(U^{s}_{n}(R_{i})-{\bar{U}^{s}}(R_{i}))^{2}},$
(20)

where $n$ indexes individuals, $U_{n}^{s}(R_{i})$ is the potential energy of
$s$ electronic state at reference point $R_{i}$ of the $n$-th individual, and
${\bar{U}}^{s}(R_{i})$ is the average among all individuals in the population
for the potential energy at reference point $R_{i}$ of $s$ electronic state.

To define $F_{\beta}$ concerning the rovibraional levels, the positions of
some additional rovibrational levels at the middle and bottom of potentials
are predicted according to Eq. (6), including $\rm A^{1}\Sigma^{+}$
$v=33,34,36,37$ levels and $\rm b^{3}\Pi_{0^{+}}$
$v=0\text{-}4,19,21\text{-}23,25,27,29,31,33$ levels. Together with the
rovibrational levels observed in [39], there are a total of 59 reference
levels taken in account in the fitness function. For each individual,
$F_{\beta}$ is defined as

$F_{\beta}=\frac{1}{2}\sum_{s}\sqrt{\frac{1}{N_{\rm L}^{s}}{\sum_{j=1}^{N_{\rm
L}^{s}}\left(E^{sj}_{\rm GA}-E^{sj}_{\rm ref}\right)^{2}}},$ (21)

where $j$ indexes reference levels, $N_{\rm L}^{s}$ is the number of reference
levels belonging to the $s$ state, $E^{sj}_{\rm GA}$ is the rovibrational
energy calculated by solving Eq. (1) with PECs encoded in current individual
and $\xi_{\rm Ab}$ and $\xi_{\rm bb}$ calculated via CI-CPP in [46], and
$E^{sj}_{\rm ref}$ is the reference value observed experimentally in [39] or
predicted via Eq. (6).

##  Appendix B The estimation of $\rm B^{1}\Pi$, $\rm c^{3}\Sigma^{+}$ and
$\rm b^{3}\Pi$ states as the intermediate state for STIRAP

Figure 9: The TDM matrix elements for two legs (up panel) and strengths (down
panel) for STIRAP transfer from the Feshbach state to the absolute
rovibrational ground state for 6Li87Rb, with the rovibrational levels of the
SO coupled ${\rm B}^{1}\Pi$, ${\rm b}^{3}\Pi$ and ${\rm c}^{3}\Sigma^{+}$
states as intermediate state. The rotational quantum number $J$ equals to 1.

For the SO coupled ${\rm B}^{1}\Pi$, ${\rm c}^{3}\Sigma^{+}$ and ${\rm
b}^{3}\Pi$ states, one should typically consider their components of
$\Omega=1$, which are strongly coupled via SO coupling [18]. The relevant CC
Schrödinger equation is

$\begin{array}[]{ll}\left(\begin{array}[]{lll}T+U_{\rm B}+E^{\rm R}_{\rm
B}&\xi_{\rm Bc}&-\xi_{\rm Bb}\\\ \xi_{\rm Bc}&T+U_{\rm c}+E^{\rm R}_{\rm
c1}&\xi_{\rm bc}\\\ -\xi_{\rm Bb}&\xi_{\rm bc}&T+U_{\rm b}+E^{\rm R}_{\rm
b1}\\\ \end{array}\right)\left(\begin{array}[]{lll}\Phi_{\rm B}\\\ \Phi_{\rm
c1}\\\ \Phi_{\rm b1}\\\ \end{array}\right)=E^{\rm
CC}_{v^{\prime\prime}}\left(\begin{array}[]{ll}\Phi_{\rm B}\\\ \Phi_{\rm
c1}\\\ \Phi_{\rm b1}\\\ \end{array}\right),\\\ \end{array}$ (22)

where $U_{\rm B}$, $U_{\rm c}$ and $U_{\rm b}$ stand for the diabatic PECs of
the $\rm B^{1}\Pi$, $\rm c^{3}\Sigma^{+}$ and $\rm b^{3}\Pi$ states,
respectively, $\xi_{\rm Bc}$, $\xi_{\rm Bb}$ and $\xi_{\rm bc}$ are the SO
coupling terms, $v^{\prime\prime}$ indexes the rovibrational levels of the
coupled state, $E_{v^{\prime\prime}}^{\rm CC}$ is the rovibrational energy,
and $\Phi_{\rm B}$, $\Phi_{\rm c1}$ and $\Phi_{\rm b1}$ are the components of
the eigenstate belonging to the $\rm B^{1}\Pi$, $\rm c^{3}\Sigma^{+}_{1}$ and
$\rm b^{3}\Pi_{1}$ states, respectively. According to Eq. (3), the rotational
Hamiltonians are

$\begin{array}[]{lll}E^{\rm R}_{\rm b1}=E^{\rm R}_{\rm
c1}=\frac{\hbar^{2}}{2\mu R^{2}}\left[J(J+1)+2\right],\\\ E^{\rm R}_{\rm
B}=\frac{\hbar^{2}}{2\mu R^{2}}\left[J(J+1)\right].\\\ \end{array}$ (23)

Similar to the treatment in Section II A, the eigenstate is assigned to ${\rm
B}^{1}\Pi$, $\rm c^{3}\Sigma^{+}_{1}$ or ${\rm b}^{3}\Pi_{1}$ state by
comparing their fractional partitions.

Since the lack of spectroscopic data of $\rm c^{3}\Sigma^{+}_{1}$ and ${\rm
b}^{3}\Pi_{1}$ states, we cannot perform the deperturbation analysis. Instead,
we use PECs and SO couplings calculated by CI-CPP in [46] to predict the
feasible rovibational levels of the coupled $\rm B^{1}\Pi$, $\rm
c^{3}\Sigma^{+}_{1}$ and $\rm b^{3}\Pi_{1}$ states suitable as the
intermediate state for STIRAP. The error of used PECs could be roughly
estimated by comparing with the data of ${\rm B}^{1}\Pi$ rovibrational levels
observed in [33]. Based on the PECs and SO coupling in [46], the rovibrational
energy of ${\rm B}^{1}\Pi$ $v=3$, $J=1$ level for 7Li85Rb is calculated as
11594.3 $\rm cm^{-1}$, and that of ${\rm B}^{1}\Pi$ $v=11$, $J=1$ level for
7Li87Rb is calculated as 12190.9 $\rm cm^{-1}$. According to the spectroscopic
data in [33], their experimentally observed values are 11568.5 $\rm cm^{-1}$
and 12170.6 $\rm cm^{-1}$, respectively. Therefore, the rovibrational energies
calculated based on $ab\ initio$ PECs are higher than the real value, and the
errors are within 30 $\rm cm^{-1}$.

Table 5: The possible intermediate states for STIRAP predicted in $\rm B^{1}\Pi$-$\rm c^{3}\Sigma^{+}_{1}$-$\rm b^{3}\Pi_{1}$ state. The wavelengths of the pump and Stokes laser are denoted as $\lambda_{\rm p}$ and $\lambda_{\rm S}$, respectively. Energies are given in $\rm cm^{-1}$, wavelengths are in nm, and TDM matrix elements are in Debye. | Possible intermediate states
---|---
|  ${\rm b}^{3}\Pi_{1}$ $v=42$ 191919The rovibrational level is assigned to this electronic state. |  ${\rm c}^{3}\Sigma^{+}_{1}$ $v=25$ 191919The rovibrational level is assigned to this electronic state.
|  ${\rm B}^{1}\Pi$ $v=1$ |  ${\rm B}^{1}\Pi$ $v=1$
$|{\rm TDM}_{v^{\prime\prime}}^{\rm up}|$ | 0.2414 | 0.1099
$|{\rm TDM}_{v^{\prime\prime}}^{\rm down}|$ | 0.2562 | 0.4365
$E_{v^{\prime\prime}}^{\rm CC}$ | 11393.4 | 11619.9
$\lambda_{p}$ | 860.6 | 877.7
$\lambda_{S}$ | 573.3 | 580.8

We calculate TDM matrix elements for two legs of the STIRAP transfer, as
follows:

$\begin{array}[]{rr}{\rm TDM}^{\rm up}_{v^{\prime\prime}}=\int\Phi_{\rm
F}\cdot{\hat{\mu}_{\rm ab}}\cdot\Phi_{\rm b1}(v^{\prime\prime},J=1){\rm d}R\\\
+\int\Phi_{\rm F}\cdot{\hat{\mu}_{\rm ac}}\cdot\Phi_{\rm
c1}(v^{\prime\prime},J=1){\rm d}R,\end{array}$ (24) ${\rm TDM}^{\rm
down}_{v^{\prime\prime}}=\int\Phi_{\rm X}\cdot{\hat{\mu}_{\rm
BX}}\cdot\Phi_{\rm B}(v^{\prime\prime},J=1){\rm d}R,$ (25)

where ${\hat{\mu}_{\rm BX}}$ and ${\hat{\mu}_{\rm ac}}$ are $R$-dependent TDMs
from $\rm X^{1}\Sigma^{+}$ state to $\rm B^{1}\Pi$ state and from $\rm
a^{3}\Sigma^{+}$ state to $\rm c^{3}\Sigma^{+}$ state, respectively, both of
which are taken from [44]. The STIRAP strength is estimated as

$I_{v^{\prime\prime}}=\left|{\rm TDM}^{\rm
up}_{v^{\prime\prime}}\right|\cdot\left|{\rm TDM}^{\rm
down}_{v^{\prime\prime}}\right|.$ (26)

The TDM matrix elements and STIRAP strengths are shown in Fig. 9. As seen, one
possible intermediate state for STIRAP transfer is the ${\rm
c}^{3}\Sigma_{1}^{+}$ $v=25$ level, which is coupled to ${\rm B}^{1}\Pi$ $v=3$
level. Another possible intermediate state for STIRAP transfer is the ${\rm
b}^{3}\Pi_{1}$ $v=42$ level coupled to ${\rm B}^{1}\Pi$ $v=1$ level. Their
rovibraional energies and relevant wavelengths of pump laser and Stocks laser
are listed in Table 5. Take the ${\rm c}^{3}\Sigma_{1}^{+}$ $v=25$ level
coupled to ${\rm B}^{1}\Pi$ $v=3$ level as an example, considering there is an
error of 30 $\rm cm^{-1}$ higher than the real value, the rovibrational energy
should be 11589.9 $\rm cm^{-1}$. The corresponding wavelengths of pump laser
and Stocks laser are 862.8 nm and 574.3 nm, respectively. The error here is
less than 3 nm, which is easily covered by tunable wavelength ranges of
semiconductor or Ti:Sappire lasers.

## References

  * Zhai [2021] H. Zhai,  _Ultracold Atomic Physics_ (Cambridge University Press, 2021).
  * Pérez-Ríos [2020] J. Pérez-Ríos,  _An Introduction to Cold and Ultracold Chemistry_ (Springer International Publishing, New York, 2020).
  * Jones  _et al._ [2006] K. M. Jones, E. Tiesinga, P. D. Lett, and P. S. Julienne, REVIEWS OF MODERN PHYSICS 78, 483 (2006).
  * Ulmanis  _et al._ [2012] J. Ulmanis, J. Deiglmayr, M. Repp, R. Wester, and M. Weidemueller, CHEMICAL REVIEWS 112, 4890 (2012).
  * Krems  _et al._ [2009] R. Krems, B. Friedrich, and W. C. Stwalley,  _Cold Molecules: Theory, Experiment, Applications_ (CRC press, 2009).
  * Chin  _et al._ [2010] C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, REVIEWS OF MODERN PHYSICS 82, 1225 (2010).
  * Vitanov  _et al._ [2017] N. V. Vitanov, A. A. Rangelov, B. W. Shore, and K. Bergmann, REVIEWS OF MODERN PHYSICS 89, 015006 (2017).
  * Rvachov  _et al._ [2017] T. M. Rvachov, H. Son, A. T. Sommer, S. Ebadi, J. J. Park, M. W. Zwierlein, W. Ketterle, and A. O. Jamison, Phys. Rev. Lett. 119, 143001 (2017).
  * He  _et al._ [2024] C. He, X. Nie, V. Avalos, S. Botsi, S. Kumar, A. Yang, and K. Dieckmann, arXiv:2310.03300v1 (2024).
  * Voges  _et al._ [2020] K. K. Voges, P. Gersema, M. M. zum Alten Borgloh, T. A. Schulze, T. Hartmann, A. Zenesini, and S. Ospelkaus, PHYSICAL REVIEW LETTERS 125, 083401 (2020).
  * Park  _et al._ [2015] J. W. Park, S. A. Will, and M. W. Zwierlein, Phys. Rev. Lett. 114, 205302 (2015).
  * Seeßelberg  _et al._ [2018] F. Seeßelberg, N. Buchheim, Z.-K. Lu, T. Schneider, X.-Y. Luo, E. Tiemann, I. Bloch, and C. Gohle, Phys. Rev. A 97, 013405 (2018).
  * Liu  _et al._ [2019] L. Liu, D.-C. Zhang, H. Yang, Y.-X. Liu, J. Nan, J. Rui, B. Zhao, and J.-W. Pan, Phys. Rev. Lett. 122, 253201 (2019).
  * Bause  _et al._ [2021] R. Bause, A. Kamijo, X.-Y. Chen, M. Duda, A. Schindewolf, I. Bloch, and X.-Y. Luo, PHYSICAL REVIEW A 104, 043321 (2021).
  * Li  _et al._ [2023] Z.-L. Li, Z.-Y. Gu, P.-J. Wang, and J. Zhang, Science China Physics, Mechanics & Astronomy 66, 293011 (2023).
  * Guo  _et al._ [2016] M. Guo, B. Zhu, B. Lu, X. Ye, F. Wang, R. Vexiau, N. Bouloufa-Maafa, G. Quemener, O. Dulieu, and D. Wang, PHYSICAL REVIEW LETTERS 116, 205303 (2016).
  * Guo  _et al._ [2017] M. Guo, R. Vexiau, B. Zhu, B. Lu, N. Bouloufa-Maafa, O. Dulieu, and D. Wang, PHYSICAL REVIEW A 96, 052505 (2017).
  * Picard  _et al._ [2023] L. R. B. Picard, J. T. Zhang, W. B. Cairncross, K. Wang, G. E. Patenotte, A. J. Park, Y. Yu, L. R. Liu, J. D. Hood, R. González-Férez, and K.-K. Ni, Phys. Rev. Res. 5, 023149 (2023).
  * Stevenson  _et al._ [2023] I. Stevenson, A. Z. Lam, N. Bigagli, C. Warner, W. Yuan, S. Zhang, and S. Will, PHYSICAL REVIEW LETTERS 130, 113002 (2023).
  * Ni  _et al._ [2008] K. K. Ni, S. Ospelkaus, M. H. G. de Miranda, A. Pe’er, B. Neyenhuis, J. J. Zirbel, S. Kotochigova, P. S. Julienne, D. S. Jin, and J. Ye, SCIENCE 322, 231 (2008).
  * Molony  _et al._ [2014] P. K. Molony, P. D. Gregory, Z. Ji, B. Lu, M. P. Köppinger, C. R. Le Sueur, C. L. Blackley, J. M. Hutson, and S. L. Cornish, Phys. Rev. Lett. 113, 255301 (2014).
  * Takekoshi  _et al._ [2014] T. Takekoshi, L. Reichsoellner, A. Schindewolf, J. M. Hutson, C. R. Le Sueur, O. Dulieu, F. Ferlaino, R. Grimm, and H.-C. Naegerl, PHYSICAL REVIEW LETTERS 113, 205301 (2014).
  * Das  _et al._ [2023] A. Das, P. D. Gregory, T. Takekoshi, L. Fernley, M. Landini, J. M. Hutson, S. L. Cornish, and H.-C. Nägerl, SciPost Phys. 15, 220 (2023).
  * Stevenson [2018] I. C. Stevenson,  _Spectroscopy of Ultracold Lithium-Rubidium Molecules_ , Ph.D. thesis, Purdue University (2018).
  * Blasing [2018] D. Blasing,  _Photoassociation in 87RB BECS and in Ultracold 7LI85RB_ , Ph.D. thesis, Purdue University (2018).
  * Li  _et al._ [2020] Z.-A. Li, Z.-W. Wang, G.-R. Wang, and S.-L. Cong, Phys. Rev. A 101, 053409 (2020).
  * Valtolina  _et al._ [2020] G. Valtolina, K. Matsuda, W. G. Tobias, J.-R. Li, L. De Marco, and J. Ye, Nature 588, 239 (2020).
  * Schindewolf  _et al._ [2022] A. Schindewolf, R. Bause, X.-Y. Chen, M. Duda, T. Karman, I. Bloch, and X.-Y. Luo, Nature 607, 677 (2022).
  * Deng  _et al._ [2023] F. Deng, X.-Y. Chen, X.-Y. Luo, W. Zhang, S. Yi, and T. Shi, Phys. Rev. Lett. 130, 183001 (2023).
  * Zhang  _et al._ [2022] R. Zhang, Y.-C. Han, S.-L. Cong, and M. B. Shundalau, CHINESE PHYSICS B 31, 063402 (2022).
  * Marzok  _et al._ [2009] C. Marzok, B. Deh, C. Zimmermann, P. W. Courteille, E. Tiemann, Y. V. Vanne, and A. Saenz, PHYSICAL REVIEW A 79, 012717 (2009).
  * Ivanova  _et al._ [2011] M. Ivanova, A. Stein, A. Pashov, H. Knoeckel, and E. Tiemann, JOURNAL OF CHEMICAL PHYSICS 134, 024321 (2011).
  * Ivanova  _et al._ [2013] M. Ivanova, A. Stein, A. Pashov, H. Knoeckel, and E. Tiemann, JOURNAL OF CHEMICAL PHYSICS 138, 094315 (2013).
  * Dutta  _et al._ [2011] S. Dutta, A. Altaf, D. S. Elliott, and Y. P. Chen, CHEMICAL PHYSICS LETTERS 511, 7 (2011).
  * Lorenz  _et al._ [2014] J. Lorenz, A. Altaf, S. Dutta, Y. P. Chen, and D. S. Elliott, PHYSICAL REVIEW A 90, 062513 (2014).
  * Altaf  _et al._ [2015] A. Altaf, S. Dutta, J. Lorenz, J. Pérez-Ríos, Y. P. Chen, and D. S. Elliott, JOURNAL OF CHEMICAL PHYSICS 142, 114310 (2015).
  * Blasing  _et al._ [2016] D. B. Blasing, I. C. Stevenson, J. Pérez-Ríos, D. S. Elliott, and Y. P. Chen, PHYSICAL REVIEW A 94, 062504 (2016).
  * Stevenson  _et al._ [2016a] I. C. Stevenson, D. B. Blasing, Y. P. Chen, and D. S. Elliott, PHYSICAL REVIEW A 94, 062510 (2016a).
  * Stevenson  _et al._ [2016b] I. C. Stevenson, D. B. Blasing, Y. P. Chen, and D. S. Elliott, PHYSICAL REVIEW A 94, 062503 (2016b).
  * Stevenson  _et al._ [2016c] I. C. Stevenson, D. B. Blasing, A. Altaf, Y. P. Chen, and D. S. Elliott, JOURNAL OF CHEMICAL PHYSICS 145, 224301 (2016c).
  * Korek  _et al._ [2009] M. Korek, G. Younes, and S. Al-Shawa, JOURNAL OF MOLECULAR STRUCTURE-THEOCHEM 899, 25 (2009).
  * Jendoubi  _et al._ [2012] I. Jendoubi, H. Berriche, H. Ben Ouada, and F. X. Gadea, JOURNAL OF PHYSICAL CHEMISTRY A 116, 2945 (2012).
  * You  _et al._ [2016] Y. You, C.-L. Yang, Q.-Q. Zhang, M.-S. Wang, X.-G. Ma, and W.-W. Liu, PHYSICAL CHEMISTRY CHEMICAL PHYSICS 18, 19838 (2016).
  * Bormotova  _et al._ [2018] E. A. Bormotova, S. V. Kozlov, E. A. Pazyuk, and A. V. Stolyarov, Phys. Chem. Chem. Phys. 20, 1889 (2018).
  * Bormotova  _et al._ [2019] E. A. Bormotova, S. V. Kozlov, E. A. Pazyuk, A. V. Stolyarov, W. Skomorowski, I. Majewska, and R. Moszynski, PHYSICAL REVIEW A 99, 012507 (2019).
  * Kozlov  _et al._ [2020] S. Kozlov, V, E. A. Bormotova, A. A. Medvedev, E. A. Pazyuk, A. Stolyarov, V, and A. Zaitsevskii, PHYSICAL CHEMISTRY CHEMICAL PHYSICS 22, 2295 (2020).
  * Lefebvre-Brion and Field [2004] H. Lefebvre-Brion and R. W. Field,  _The Spectra and Dynamics of Diatomic Molecules_ (Elsevier Inc., 2004).
  * Tamanis  _et al._ [2002] M. Tamanis, R. Ferber, A. Zaitsevskii, E. Pazyuk, A. Stolyarov, H. Chen, J. Qi, H. Wang, and W. Stwalley, JOURNAL OF CHEMICAL PHYSICS 117, 7980 (2002).
  * Docenko  _et al._ [2007] O. Docenko, M. Tamanis, R. Ferber, E. A. Pazyuk, A. Zaitsevskii, A. V. Stolyarov, A. Pashov, H. Knoeckel, and E. Tiemann, PHYSICAL REVIEW A 75, 042503 (2007).
  * Bergeman  _et al._ [2003] T. Bergeman, C. Fellows, R. Gutterres, and C. Amiot, PHYSICAL REVIEW A 67, 050501 (2003).
  * Docenko  _et al._ [2010] O. Docenko, M. Tamanis, R. Ferber, T. Bergeman, S. Kotochigova, A. V. Stolyarov, A. d. F. Nogueira, and C. E. Fellows, PHYSICAL REVIEW A 81, 042511 (2010).
  * Kruzins  _et al._ [2014] A. Kruzins, K. Alps, O. Docenko, I. Klincare, M. Tamanis, R. Ferber, E. A. Pazyuk, and A. V. Stolyarov, JOURNAL OF CHEMICAL PHYSICS 141, 184309 (2014).
  * Qi  _et al._ [2007] P. Qi, J. Bai, E. Ahmed, A. M. Lyyra, S. Kotochigova, A. J. Ross, C. Effantin, P. Zalicki, J. Vigue, G. Chawla, R. W. Field, T.-J. Whang, W. C. Stwalley, H. Knoeckel, E. Tiemann, J. Shang, L. Li, and T. Bergeman, JOURNAL OF CHEMICAL PHYSICS 127, 044301 (2007).
  * Bai  _et al._ [2011] J. Bai, E. H. Ahmed, B. Beser, Y. Guan, S. Kotochigova, A. M. Lyyra, S. Ashman, C. M. Wolfe, J. Huennekens, F. Xie, D. Li, L. Li, M. Tamanis, R. Ferber, A. Drozdova, E. Pazyuk, A. V. Stolyarov, J. G. Danzl, H.-C. Naegerl, N. Bouloufa, O. Dulieu, C. Amiot, H. Salami, and T. Bergeman, PHYSICAL REVIEW A 83, 032514 (2011).
  * Znotins  _et al._ [2019] A. Znotins, A. Kruzins, M. Tamanis, R. Ferber, E. A. Pazyuk, A. Stolyarov, V, and A. Zaitsevskii, PHYSICAL REVIEW A 100, 042507 (2019).
  * Zaharova  _et al._ [2009] J. Zaharova, M. Tamanis, R. Ferber, A. N. Drozdova, E. A. Pazyuk, and A. V. Stolyarov, PHYSICAL REVIEW A 79, 012508 (2009).
  * Kruzins  _et al._ [2010] A. Kruzins, I. Klincare, O. Nikolayeva, M. Tamanis, R. Ferber, E. A. Pazyuk, and A. V. Stolyarov, PHYSICAL REVIEW A 81, 042509 (2010).
  * Tamanis  _et al._ [2010] M. Tamanis, I. Klincare, A. Kruzins, O. Nikolayeva, R. Ferber, E. A. Pazyuk, and A. V. Stolyarov, PHYSICAL REVIEW A 82, 032506 (2010).
  * Kowalczyk  _et al._ [2015] P. Kowalczyk, W. Jastrzebski, J. Szczepkowski, E. A. Pazyuk, and A. V. Stolyarov, JOURNAL OF CHEMICAL PHYSICS 142, 234308 (2015).
  * Harker  _et al._ [2015] H. Harker, P. Crozet, A. J. Ross, K. Richter, J. Jones, C. Faust, J. Huennekens, A. V. Stolyarov, H. Salami, and T. Bergeman, PHYSICAL REVIEW A 92, 012506 (2015).
  * Alps  _et al._ [2016] K. Alps, A. Kruzins, M. Tamanis, R. Ferber, E. A. Pazyuk, and A. V. Stolyarov, JOURNAL OF CHEMICAL PHYSICS 144, 144310 (2016).
  * Press  _et al._ [2007] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery,  _Numerical Recipes: The Art of Scientific Computing (3rd Edition)_ (Cambridge University Press, 2007).
  * Marques  _et al._ [2008] J. M. C. Marques, F. V. Prudente, F. B. Pereira, M. M. Almeida, A. M. Maniero, and C. E. Fellows, JOURNAL OF PHYSICS B-ATOMIC MOLECULAR AND OPTICAL PHYSICS 41, 085103 (2008).
  * Almeida  _et al._ [2011] M. M. Almeida, F. V. Prudente, C. E. Fellows, J. M. C. Marques, and F. B. Pereira, JOURNAL OF PHYSICS B-ATOMIC MOLECULAR AND OPTICAL PHYSICS 44, 225102 (2011).
  * Stevenson and Pérez-Ríos [2019] I. C. Stevenson and J. Pérez-Ríos, JOURNAL OF PHYSICS B-ATOMIC MOLECULAR AND OPTICAL PHYSICS 52, 105002 (2019).
  * Londoño and Arce [2023] M. Londoño and J. C. Arce, Phys. Rev. A 108, 013301 (2023).
  * Katoch  _et al._ [2021] S. Katoch, S. S. Chauhan, and V. Kumar, MULTIMEDIA TOOLS AND APPLICATIONS 80, 8091 (2021).
  * Hua [1990] W. Hua, Phys. Rev. A 42, 2524 (1990).
  * [69] Nist: Atomic spectra database, accessed on May 10th, 2024 [URL will be inserted by publisher].
  * Yin  _et al._ [2024] Y. Yin, Z. Li, X. Bai, T. Gong, Z. Ji, Y. Zhao, Y. Han, J. Yu, and G. Wang, Physica Scripta 99, 045003 (2024).
  * Note [1] Since the energy range of the observed rovibrational levels is limited, the crossover points should also be limited in certain regions.
  * Kraft [1988] D. Kraft,  _A software package for sequential quadratic programming_ , Tech. Rep. (DFVLR-FB 88-28, DLR German Aerospace Center – Institute for Flight Mechanics, Koln, Germany., 1988).
  * Le Roy and Pashov [2017] R. J. Le Roy and A. Pashov, JOURNAL OF QUANTITATIVE SPECTROSCOPY & RADIATIVE TRANSFER 186, 210 (2017).
  * MARSTON and BALINTKURTI [1989] C. MARSTON and G. BALINTKURTI, JOURNAL OF CHEMICAL PHYSICS 91, 3571 (1989).
  * [75] Scipy v1.13.0 manual, accessed on May 10th, 2024 [URL will be inserted by publisher].
  * Note [2] See Supplemental Material at [URL will be inserted by publisher].
  * Cairncross  _et al._ [2021] W. B. Cairncross, J. T. Zhang, L. R. B. Picard, Y. Yu, K. Wang, and K.-K. Ni, Phys. Rev. Lett. 126, 123402 (2021).
  * Christakis  _et al._ [2023] L. Christakis, J. S. Rosenberg, R. Raj, S. Chi, A. Morningstar, D. A. Huse, Z. Z. Yan, and W. S. Bakr, Nature 614, 64 (2023).