Science, vol. 298, no. 5594, pp. 824–827, 2002.

[75]
U. Alon, “Network motifs: theory and experimental approaches,” Nature
Reviews Genetics, vol. 8, no. 6, pp. 450–461, 2007.

[76]
J. M. Hofman and C. H. Wiggins, “Bayesian approach to network modularity,”
Physical review letters, vol. 100, no. 25, p. 258701, 2008.

[77]
E. Ziv, M. Middendorf, and C. H. Wiggins, “Information-theoretic approach to
network modularity,” Physical Review E, vol. 71, no. 4, p. 046117,

[78]
R. Velez and J. Clune, “Identifying core functional networks and functional
modules within artificial neural networks via subsets regression,” in
Proceedings of the Genetic and Evolutionary Computation Conference
2016, 2016, pp. 181–188.

[79]
R. G. Winther, “Varieties of modules: kinds, levels, origins, and
behaviors,” Journal of Experimental Zoology, vol. 291, no. 2, pp.
116–129, 2001.

[80]
J. A. Bolker, “Modularity in Development and Why It Matters to Evo-Devo1,”
American Zoologist, vol. 40, no. 5, pp. 770–776, 08 2015. [Online].
Available: <https://doi.org/10.1093/icb/40.5.770>

[81]
M. Parter, N. Kashtan, and U. Alon, “Facilitated variation: How evolution
learns from past environments to generalize to new environments,” PLOS
Computational Biology, vol. 4, no. 11, pp. 1–15, 11 2008. [Online].
Available: <https://doi.org/10.1371/journal.pcbi.1000206>

[82]
N. Kashtan and U. Alon, “Spontaneous evolution of modularity and network
motifs,” Proceedings of the National Academy of Sciences, vol. 102,
no. 39, pp. 13 773–13 778, 2005. [Online]. Available:

[83]
J. Clune, J.-B. Mouret, and H. Lipson, “The evolutionary origins of
modularity,” Proceedings of the Royal Society B: Biological Sciences,
vol. 280, no. 1755, p. 20122863, 2013. [Online]. Available:

[84]
H. Chung, A. P. Asnodkar, and C. Teuscher, “A structural analysis of evolved
complex networks-on-chip,” in Proceedings of the Fifth International
Workshop on Network on Chip Architectures, ser. NoCArc '12.1em plus
0.5em minus 0.4emNew York, NY, USA: Association for Computing
Machinery, 2012, p. 17–22. [Online]. Available:

[85]
D. Melo and G. Marroig, “Directional selection can drive the evolution of
modularity in complex traits,” Proceedings of the National Academy of
Sciences, vol. 112, no. 2, pp. 470–475, 2015. [Online]. Available:

[86]
J. Gunawardena, “Weak linkage and learning,” online lecture notes, 2018.

[87]
M. Conrad, “The geometry of evolution,” Biosystems, vol. 24, no. 1,
pp. 61–81, 1990. [Online]. Available:

[88]
J. Monod, J. Wyman, and J.-P. Changeux, “On the nature of allosteric
transitions: a plausible model,” J Mol Biol, vol. 12, no. 1, pp.
88–118, 1965.

[89]
M. Csete and J. Doyle, “Bow ties, metabolism and disease,” TRENDS in
Biotechnology, vol. 22, no. 9, pp. 446–450, 2004.

[90]
T. Friedlander, A. E. Mayo, T. Tlusty, and U. Alon, “Evolution of bow-tie
architectures in biology,” PLOS Computational Biology, vol. 11,
no. 3, pp. 1–19, 03 2015. [Online]. Available:

[91]
J. M. Whitacre, “Degeneracy: a link between evolvability, robustness and
complexity in biological systems,” Theoretical Biology and Medical
Modelling, vol. 7, no. 1, pp. 1–17, 2010.

[92]
J. A. Draghi, T. L. Parsons, G. P. Wagner, and J. B. Plotkin, “Mutational
robustness can facilitate adaptation,” Nature, vol. 463, no. 7279,
pp. 353–355, 2010.

[93]
G. M. Viswanathan, S. V. Buldyrev, S. Havlin, M. Da Luz, E. Raposo, and H. E.
Stanley, “Optimizing the success of random searches,” nature, vol.
401, no. 6756, pp. 911–914, 1999.

[94]
F. Bartumeus, J. Catalan, U. Fulco, M. Lyra, and G. Viswanathan, “Optimizing
the encounter rate in biological interactions: Lévy versus brownian
strategies,” Physical Review Letters, vol. 88, no. 9, p. 097901,

[95]
B. B. Mandelbrot and B. B. Mandelbrot, The fractal geometry of
nature.1em plus 0.5em minus 0.4emWH freeman New York, 1982,
vol. 1.

[96]
G. M. Edelman and J. A. Gally, “Degeneracy and complexity in biological
systems,” Proceedings of the National Academy of Sciences, vol. 98,
no. 24, pp. 13 763–13 768, 2001.

[97]
P. Mason, J. Domínguez D., B. Winter, and A. Grignolio, “Hidden in plain
view: degeneracy in complex systems,” Biosystems, vol. 128, pp. 1–8,
2015. [Online]. Available:

[98]
L. Seifert, J. Komar, D. Araújo, and K. Davids, “Neurobiological
degeneracy: a key property for functional adaptations of perception and
action to constraints,” Neuroscience & Biobehavioral Reviews,
vol. 69, pp. 159–165, 2016.

[99]
B. Winter, “Spoken language achieves robustness and evolvability by exploiting
degeneracy and neutrality,” BioEssays, vol. 36, no. 10, pp. 960–967,

[100]
G. Tononi, O. Sporns, and G. M. Edelman, “A measure for brain complexity:
relating functional segregation and integration in the nervous system,”
Proceedings of the National Academy of Sciences, vol. 91, no. 11, pp.
5033–5037, 1994.

[101]
——, “A complexity measure for selective matching of signals by the
brain,” Proceedings of the National Academy of Sciences, vol. 93,
no. 8, pp. 3422–3427, 1996.

[102]
——, “Measures of degeneracy and redundancy in biological networks,”
Proceedings of the National Academy of Sciences, vol. 96, no. 6, pp.
3257–3262, 1999.

[103]
S. J. Gould and E. S. Vrba, “Exaptation—a missing term in the science of
form,” Paleobiology, vol. 8, no. 1, pp. 4–15, 1982.

[104]
J. Whitacre and A. Bender, “Degeneracy: a design principle for achieving
robustness and evolvability,” Journal of theoretical biology, vol.
263, no. 1, pp. 143–153, 2010.

[105]
E. L. Berlow, “Strong effects of weak interactions in ecological
communities,” Nature, vol. 398, no. 6725, pp. 330–334, 1999.

[106]
P. Csermely, Weak links: the universal key to the stability of networks
and complex systems.1em plus 0.5em minus 0.4emSpringer Science
& Business Media, 2009.

[107]
A.-L. Barabasi and Z. N. Oltvai, “Network biology: understanding the cell's
functional organization,” Nature reviews genetics, vol. 5, no. 2, pp.
101–113, 2004.

[108]
Y. Bar-Yam and I. R. Epstein, “Response of complex networks to stimuli,”
Proceedings of the National Academy of Sciences, vol. 101, no. 13, pp.
4341–4345, 2004. [Online]. Available:

[109]
G. Byrne, D. Dimitrov, L. Monostori, R. Teti, F. van Houten, and R. Wertheim,
“Biologicalisation: Biological transformation in manufacturing,” CIRP
Journal of Manufacturing Science and Technology, vol. 21, pp. 1–32, 2018.

[110]
O. Berger-Tal, J. Nathan, E. Meron, and D. Saltz, “The
exploration-exploitation dilemma: a multidisciplinary framework,” PloS
one, vol. 9, no. 4, p. e95693, 2014.

[111]
M. Ghavamzadeh, S. Mannor, J. Pineau, A. Tamar et al., “Bayesian
reinforcement learning: A survey,” Foundations and
Trends® in Machine Learning, vol. 8, no. 5-6, pp. 359–483,

[112]
L. F. D. Sotto, Sotto, S. Mayer, and J. Garcke, “The pole balancing problem
from the viewpoint of system flexibility,” in GECCO ’22 Companion,
July 9–13, 2022, Boston, MA, USA, ser. GECCO'22.1em plus 0.5em
minus 0.4emRed Hook, NY, USA: ACM, 2022.

§ EVOLVABILITY OF LOGIC CIRCUITS

One of the striking features of evolution is the appearance of novel structures in organisms. The origin of the ability to generate novelty is one of the main mysteries in evolutionary theory. The molecular mechanisms that enhance the evolution of novelty were integrated by Kirschner and Gerhart in their theory of facilitated variation (FV) [41]. This theory suggests that organisms have a design that makes it more likely that random genetic changes will result in organisms with novel shapes that can survive. The paper [81] demonstrates by means of computer simulations of two well-studied model systems, logic circuits and RNA secondary structure, that FV can spontaneously emerge. We recapitulate their main results on logic circuits using the formalism developed in Section <ref> and discuss which elements of flexibility play a role here.

Before we get to the formalism, let us first describe the essence of the flexibility problem considered in [81]. A Boolean expression is a formula that evaluates to $1$ (true) or $0$ (false). An example for a Boolean expression with four variables $x_1,x_2,x_3,x_4 \in \{0,1\}$ is

\begin{align}
t(x_1,x_2,x_3,x_4) = (x_1\; \mathrm{ AND }\; x_2)\; \mathrm{ OR }\; (x_3\; \mathrm{ EQ }\; x_4).
\end{align}

A Boolean expression including Boolean variables is also called Boolean function.
It is known that every Boolean function can be equivalently constructed from a combination of NAND gates.
The NAND gate is defined by the truth table given in Table <ref> and its standard shape symbol is depicted in Figure <ref>.

(0,0) node
(1.5, 1) node[ieeestd nand port] (mynand) ;

IEEE shape symbol of NAND gate.

$x_1$     $x_2$     $\mathrm{NAND}(x_1,x_2)$

$0$     $0$     $1$

$0$     $1$     $1$

$1$     $0$     $1$

$1$     $1$     $0$

Truth table of NAND gate.

As an example, consider the OR gate. It can be written as

\begin{align}
x_1\; \mathrm{ OR }\; x_2 =  (x_1\; \mathrm{ NAND }\;x_1) \; \mathrm{ NAND }\; (x_2\; \mathrm{ NAND }\;x_2).
\end{align}

The corresponding circuit is depicted in Figure <ref>.

[ocirc,fill=black](split1) at (0.42,2) ;
[ocirc,fill=black](split2) at (0.42,0) ;
(-0.1,2) node (in1) $x_1$
(-0.1,0) node (in2) $x_2$
(1.5,2) node[ieeestd nand port] (mynand1)
(1.5,0) node[ieeestd nand port] (mynand2)
(4,1) node[ieeestd nand port] (mynand3)
(5.3,1) node (out1) $y$
(in1.east) – (split1.west)
(in2.east) – (split2.west)
(split1.north) – (mynand1.in 1)
(split1.south) – (mynand1.in 2)
(split2.north) – (mynand2.in 1)
(split2.south) – (mynand2.in 2)
(mynand1.out) – (mynand3.in 1)
(mynand2.out) – (mynand3.in 2);

NAND construction of the OR gate.

The flexibility problem that is addressed in [81] is essentially the following.

Given Boolean functions $t_1, t_2,\dots,t_n$, find a template Boolean function build solely from NAND gates such that its structure allows for the following:

* For any Boolean function $t_i$, it takes only few modifications to the template to have a construction of $t_i$ based on NAND gates (low reconfiguration cost).

* It is easy to find the necessary modifications (low adaption cost).

Moreover, we wish the reconfiguration and adaption cost to be low for as many different Boolean functions as possible (high task diversity).

Definition <ref> captures the essence of the flexibility problem. A number of details are still missing to have a problem formulation that is completely precise and detailed enough that a computer can solve it. We use the flexibility formalism to guide us through the procedure of filling the gaps. In the end, we will have a precise picture of the exact version of LCFP considered in [81] and the system used to solve it.

§.§ Formal description of the tasks

Reproducing the truth values of a Boolean expression in $d$ Boolean variables $x=(x_1,\dots,x_d)$  can be described as a task

\begin{align}
T = (E,F,t,P_T)
\end{align}

with the following elements. The environment is given by the space of all possible values (or states) a vector of $d$ Boolean variables can take, $E=\{0,1\}^d$. All states are feasible, so $F=E$, and all states are possible initial states, hence $X=E$. The goal is now to map an input state $x \in X$ to a target state in

\begin{align}
Y = \{ (x_1,1,1,1): x_1 \in \{0,1\}  \}.
\end{align}

Note that $Y$ is one of $2^{d-1}$ subspaces in $E$ that are equivalent to $\{0,1\}$. The particular choice of $Y$ is arbitrary and we may simply identify $Y=\{0,1\}$. The goal is formally given by a Boolean function

\begin{align}
t: \{0,1\}^d \to \{0,1\}.
\end{align}

The performance (or fitness) of a given Boolean function $u:  \{0,1\}^d \to \{0,1\}$ in reaching the goal is measured by the fraction of outputs where $u$ coincides with $t$,

\begin{align}
\tilde P (u,t) = \frac{|\{x \in E: u(x) = t(x) \}|}{|E|}.
\end{align}

To be able to read adaptability and reconfigurability numbers from [81], we consider a task to be performed successfully by a Boolean function $u$ if we have $\tilde P(u,t) \geq \varepsilon$, where the specific choice of $\varepsilon \in [0,1]$ is discussed in Section <ref>. Hence, the ultimate performance measure  is given by

\begin{align}
P_T(u) = \begin{cases}
1, \text{ if } \tilde P(u,t) \geq \varepsilon,\\
0, \text{ otherwise}.
\end{cases}
\end{align}

We discuss the specific task context considered in the flexibility problem in Section <ref>.

§.§ Formal description of the system

In [81] a standard genetic algorithm (GA) is used to evolve a population of logic circuits consisting of NAND gates towards varying Boolean target functions $t$. The population size is $N_{\mathrm{pop}} = 5000$. We consider both together—the logic circuit and the GA—as a self-adapting system $S$ where the GA operates as the learning system. To be more precise, the system $S$ consists of the following elements:

* A logic circuit consisting of

* $d$ input gates ($d=4$),

* $M$ NAND gates  ($M=12$),

* one output gate.

The input gates can be connected to more than one non-input gate. A NAND gate can be connected to more than one of the other NAND gates and to the output gate.
This is the phenotype in the language of evolutionary computation.

* A binary encoding $\gamma$ of the circuit wiring. This is the genotype in the language of evolutionary computation and what we consider the system configuration to be adapted.

* A genotype-phenotype-mapping that translates the genotype into an actual wiring of the logic circuit.

* An array of length $N_{\mathrm{pop}}$ that contains alternative system configurations.

* The genetic algorithm.

The genetic algorithm operates on the array of alternative genotypes. Every generation it evaluates the performance of all genotypes/system configurations in the array and selects the best-performing as the current configuration $\gamma$ of the system. The system then realizes an algorithm $S(\gamma)$ that compute the $d$-variate Boolean function specified by the wiring of the NAND gates.

It remains to describe the structure of the binary encoding and the resulting system configuration space $\Gamma(S)$. The encoding consists of $M$ “gate genes” and one “output gene”. A gate gene has a length of

\begin{align}
2m = 2\lceil\log_2(M+d)\rceil=8
\end{align}

bits where the first $m$ bits encode which input or NAND gate is connected to the first input of the gate, and the second $m$ bits encode the wiring of the second input. The output gate has a length of $m=4$ bits. Hence, the binary encoding has a length of $B=4(2M+1)$ bits and the system's configuration space is given by

\begin{align}
\Gamma(S) = \{0,1\}^B.
\end{align}

§.§ Adaption and reconfiguration cost

Given a current system configuration $\gamma \in \Gamma(S)$ and a new task $T$, the adaption cost are the number of generations it takes the genetic algorithm to find a configuration $\gamma' \in \Gamma(S)$ such that $P_T(S(\gamma')) = 1$. The reconfiguration cost are the number of mutations that have to be applied to get from the previous configuration $\gamma$ to the new configuration $\gamma'$ (i.e., the Hamming distance between $\gamma$ and $\gamma'$).

§.§ The flexibility optimization problem in detail

Let us now describe one variant of LCFP that is actually addressed in [81] using the language of the flexibility formalism. The considered task context consists of tasks $T$ that all share the same environment $E$ but have different goals $t: X \to Y$ of the form

\begin{align}
t(x_1,x_2,x_3,x_4) = f(g(x_1,x_2), h(x_3,x_4))
\end{align}

where $g,h \in \{\mathrm{EQ}, \mathrm{XOR}\}$ and $f \in \{\mathrm{AND},\mathrm{OR}\}$. More specifically, we have a training task context

\begin{align}
W^{\mathrm{train}} = ((T_1, T_2, T_3), Q)
\end{align}

and a test task context $W^{\mathrm{test}}$ consisting of twenty equally probable tasks where the goals are such that either $f$, $g$, or $h$ is a Boolean function not present in $T_1, T_2, T_3$. What is now examined by means of a computer experiment in the paper, see <cit.>, is the effect of two different task histories generated from the training context $W^{\mathrm{train}}$ on the adaptability and reconfigurability of the system $S$ in the test context $W^{\mathrm{test}}$. Hence, LCFP is solved for the Boolean expressions given in $W^{\mathrm{test}}$. We discuss their results in the next section

§.§ Adaptability and reconfigurability results

[circle,draw] (T_1) at (10,10) $T_1$;
[circle,draw] (T_2) at (8.5,8.5) $T_2$;
[circle,draw] (T_3) at (11.2,8.5) $T_3$;

(T_1) edge[<->] node[above,left] $1/3$ (T_2);
(T_1) edge[<->] node[above,right] $1/3$ (T_3);
(T_1) edge[loop above] node $1/3$ (C);

Transition probabilities between tasks in the MVG scenario.

In the first case, called fixed goal (FG) scenario in the paper, the system is only presented one task $T_1$ for a certain number of generations $L$ (formally, the probability measure $Q$ is chosen such that $T_1$ has probability $1$). In the second case, called modularly varying goals (MVG) scenario, the system is presented a random task history that results from performing a random walk between three tasks $T_1, T_2, T_3$ according to the transition probabilities in Figure <ref>, where the task is switched every $E=20$ generations. This leads to two task histories $H_n^{\mathrm{FG}}$ and $H_n^{\mathrm{MVG}}$ of length $n=L/E$.

Let $T$ be a random task from $W^{\mathrm{test}}$. Given our choice of the performance threshold $\varepsilon$, see Section <ref>, the numerical results in <cit.> show that

\begin{align}
\mathbb{E}[c^{\textrm{ada}}(S,H_n^{\mathrm{FG}}, T)] > 10,
\end{align}

\begin{align}
\mathbb{E}[c^{\textrm{ada}}(S,H_n^{\mathrm{MVG}}, T)] = 3.
\end{align}

Moreover, the reconfiguration cost in the MVG scenario are on average five times smaller than in the FG scenario, see <cit.>.

Though we do not change the design of the system $S$ or the design of the configuration space $\Gamma(S)$, one may still consider the training phase in the MVG scenario as a system optimization that improves the array of alternative system configurations and the initial configuration in the testing phase by locating them in a beneficial region of the configuration space $\Gamma(S)$. In that sense, the FG and the MVG scenario lead to two different systems $S^{\mathrm{FG}}$ and $S^{\mathrm{MVG}}$, where the latter is at least three times more adaptable,

\begin{align}
\alpha_n(S^{\mathrm{MVG}}, W^{\mathrm{test}}) > 3\, \alpha_n(S^{\mathrm{FG}}, W^{\mathrm{test}}),
\end{align}

and five times easier to reconfigure on average,

\begin{align}
\rho^{\mathrm{avg}}(S^{\mathrm{MVG}}, W^{\mathrm{test}}) > 5\, \rho^{\mathrm{avg}}(S^{\mathrm{FG}}, W^{\mathrm{test}}).
\end{align}

§.§ Elements of flexibility

The reason for the better adaptability and reconfigurability in the MVG scenario is that the switching between the tasks in the history $H_n^{\mathrm{MVG}}$ lets the genetic algorithm search in a region of the configuration space $\Gamma(S)$ where the corresponding wirings between the NAND gates reflect the hierarchical and modular structure of goals, which is not the case in the FG scenario. To see how also weak regulatory linkage appears in the MVG scenario, we refer to the explanations in [81].

§.§ Further details

Choice of threshold $\varepsilon$.
To exclude that the same results could be reached by random guessing, the authors consider a normalized fitness measure given by

\begin{align}
F_N = \frac{F - F_r}{1- F_r},
\end{align}

where $F = \max_{\gamma_1,\dots,\gamma_{N_{\mathrm pop}}} P_T(S(\gamma_i))$ is the maximal fitness in the evolved population and $F_r$ the average maximal fitness in a population of random genotypes of the same size $N_{\mathrm{pop}}$. The results in <cit.> are specified for the normalized fitness. Since $F_r$ is independent of the considered system $S$, we may consider $F_r$ as a constant such that $F_N$ is proportional to $F$. We choose $\varepsilon$ such that

\begin{align}
F = F_r + (1-F_r) F_N > \varepsilon
\end{align}

if $F_N > 0.8$.


# Universal scattering phase shift in the presence of spin-orbit coupling

Cai-Xia Zhang1,3and Shi-Guo Peng2<EMAIL_ADDRESS>1Guangdong Provincial
Key Laboratory of Quantum Engineering and Quantum Materials, GPETR Center for
Quantum Precision Measurement and SPTE, South China Normal University,
Guangzhou 510006, China  2State Key Laboratory of Magnetic Resonance and
Atomic and Molecular Physics, Innovation Academy for Precision Measurement
Science and Technology, Chinese Academy of Sciences, Wuhan 430071, China
3Guangdong-Hong Kong Joint Laboratory of Quantum Matter, Frontier Research
Institute for Physics, South China Normal University, Guangzhou 510006, China

(August 27, 2024)

###### Abstract

Scattering phase shift, as a key parameter in scattering theory, plays an
important role in characterizing low-energy collisions between ultracold
atoms. In this work, we theoretically investigate the universal low-energy
behavior of the scattering phase shifts for cold atoms in the presence of
spin-orbit coupling. We first construct the asymptotic form of the two-body
wave function when two fermions get as close as the interaction range, and
consider perturbatively the correction of the spin-orbit coupling up to the
second order, in which new scattering parameters are introduced. Then for
elastic collisions, the scattering phase shifts are defined according to the
unitary scattering $S$ matrix. We show how the low-energy behavior of the
scattering phase shifts is modified by these new scattering parameters
introduced by spin-orbit coupling. The universality of the scattering phase
shifts is manifested as the independence of the specific form of the
interatomic potential. The explicit forms of the new scattering parameters are
analytically derived within a model of the spherical-square-well potential.
Our method provides a unified description of the low-energy properties of
scattering phase shifts in the presence of spin-orbit coupling.

##  I Introduction

Owing to their versatility, ultracold atomic gases provide ideal platform on
which to study fascinating quantum many-body phenomena in a highly
controllable and tunable way Bloch2008M ; Giorgini2008T ; Chin2010F . As a
building block of interacting many-body systems, the two-body problem is of
fundamental importance in ultracold atomic physics Braaten2006U ; Blume2012F ;
Greene2017U . In one hand, two-body solutions determine the essential
interaction parameter in the many-body Hamiltonian. In the other hand, the
two-body physics even gives rise to a set of universal relations that
characterize various properties of many-body systems, ranging from macroscopic
thermodynamics to microscopic correlation functions Tan2008E ; Tan2008L ;
Tan2008G ; Zwerger2011T . It then opens up a new direction of studying many-
body problems based on the two-body physics Braaten2008E ; Zhang2009U ;
Stewart2010V ; Kuhnle2010U ; Yu2015U ; Yoshida2015U ; He2016C ; Luciuk2016E ;
Peng2016L ; Peng2019U . An important feature of ultracold atomic systems is
that the mean distance between atoms is usually much larger than the length
scale associated with interatomic potentials. Therefore, the two-body
scattering properties outside the interatomic potential become independent of
the short-range detail of the potential between atoms, and are universally
characterized by the so-called scattering phase shift Landau1999Q . Moreover,
microscopic two-body scattering parameters, such as the scattering length and
effective range, can be defined based on the low-energy expansion of the
scattering phase shift Mott1965T .

Recently, the successful attainment of the spin-orbit (SO) coupling in cold
atomic gases is one of remarkable breakthroughs Lin2011S ; Cheuk2012S ;
Wang2012S ; Wu2016R ; Huang2016E ; Chen2018S ; Chen2018R ; Zhang2019G . Owing
to the high controllability of cold atoms in interatomic interaction, purity,
and geometry Chin2010F ; Kohler2006P ; Zhu2006S ; Peng2014M ; Gross2017Q , SO-
coupled cold atoms have became a manifold platform for further researching and
understanding of novel phenomena in condensed-matter physics, such as
topological insulators and superconductors Qi2011T ; Dalibard2011C ;
Zhang12019T . However, the introducing of SO coupling brings new challenges to
the few-body physics as well as many-body physics. In the presence of SO
coupling, the center-of-mass (c.m.) motions of pairs are coupled to their
relative motions Wang2016T . Besides, all the scattering partial waves are
mixed, since the orbital angular momentum of the relative motion of two atoms
is no longer conserved because of SO coupling Cui2012M . These complications
dramatically affect the theoretical description of interactions between cold
atoms, and especially modify the asymptotic behavior of the many-body wave
function when two atoms get close Zhang2012M . Then new scattering parameters
need to be introduced by SO coupling besides the well-known scattering length
and effective range Peng2018C ; Zhang2020U . Consequently, Tan’s universal
relations, governed by the short-range behavior of the two-body physics, are
also amended by SO-coupling. New contacts are then introduced, and additional
universal relations related to SO-coupling appear Peng2018C ; Zhang2020U ;
Zhang2018U ; Jie2018U ; Qin2020L ; Qin2020U .

Two-body scattering problems in the presence of SO coupling have
comprehensively been studied in the past few years Cui2012M ; Duan2013U ;
Guan2016S ; Guan2016A ; Wang2018U ; Wu2013S . However, a unified description
of the low-energy behavior of the scattering phase shift is still elusive.
Since new scattering parameters need to be introduced by SO coupling Peng2018C
; Zhang2020U , it is of special interest how these new scattering parameters
characterize the low-energy behavior of the scattering phase shift, and
whether it is universal and independent of the specific form of interatomic
potentials. It is the target of the present work to address these issues. In
this work, we consider the two-body scattering problem of Fermi gases in the
presence of three-dimensional (3D) isotropic SO coupling. We perturbatively
take into account of SO coupling up to the second-order correction, and
construct the short-range asymptotic form of the two-body wave function. We
show that additional scattering parameters need to be introduced by SO
coupling, besides the scattering length (volume) and effective range.
Utilizing the unitarity of the elastic collision between two fermions, we
define the scattering phase shifts according to the $S$ matrix, and then
represent the form of the two-body wave function outside the interatomic
potential by the scattering phase shifts. By expanding the two-body wave
function at short distance, and comparing it with that obtained perturbatively
by introducing new scattering parameters, we acquire the low-energy behavior
of the scattering phase shifts. The low-energy expansion of the scattering
phase shifts, characterizing by new scattering parameters, is found universal
and independent of the specific form of interatomic potentials. As a simple
example, we verify our results within a model of the spherical-square-well
potential. The explicit forms of new scattering parameters introduced by SO
coupling are analytically derived, which show excellent agreement with
previous numerical calculations near the $s$-wave resonance Cui2012M .

The paper is arranged as follows. To warm up, we briefly review our method in
Sec.II to construct the short-range asymptotic form of the two-body wave
function in the presence of SO coupling, and take into account of the second-
order corrections of SO coupling. In Sec.III, the scattering phase shifts are
defined according to the unitary $S$ matrix for elastic collisions, based on
the exact two-body solution outside the interatomic potential. Then the
asymptotic form of the two-body wave function represented by the scattering
phase shifts is obtained. By comparing this short-range form of the two-body
wave function with that obtained perturbatively by introducing new scattering
parameters, the low-energy behavior of the scattering phase shifts is then
acquired in Sec.IV. The explicit forms of the new scattering parameters
introduced by SO coupling are derived and verified by using the model of a
spherical-square-well potential in Sec.V. Finally, the remarks and conclusions
are summarized in Sec.VI.

##  II model and two-body wave function

To generalize our previous results Peng2018C ; Zhang2020U , we briefly review
the introducing of new parameters in handling two-body scattering problems
with SO coupling, and construct the short-range form of the two-body wave
function up to the second-order corrections of SO coupling. Let us consider
two spin-$1/2$ fermions in the presence of 3D isotropic SO coupling, and the
single-particle Hamiltonian takes the form of Cui2012M ; Peng2018C ; Wu2013S

$\hat{\mathcal{H}}_{1}=\frac{\hbar^{2}\mathbf{\hat{k}^{\mathrm{2}}}}{2M}+\frac{\hbar^{2}\lambda}{M}\mathbf{\hat{k}}\cdot\hat{\boldsymbol{\sigma}}+\frac{\hbar^{2}\lambda^{2}}{2M},$
(1)

where $\hat{{\bf k}}=-i\nabla$ and $\hat{\boldsymbol{\sigma}}$ are
respectively the single-particle momentum and spin operators, $\lambda>0$
denotes the strength of SO coupling, $M$ is the atomic mass, and $\hbar$ is
the Planck’s constant divided by $2\pi$. For two fermions, by introducing the
c.m. and relative coordinates ${\bf R}=\left({\bf r}_{1}+{\bf r}_{2}\right)/2$
and ${\bf r}={\bf r}_{1}-{\bf r}_{2}$ as usual, the two-body Hamiltonian can
formally be written as
$\hat{\mathcal{H}}_{2}=\hat{\mathcal{H}}_{cm}+\hat{\mathcal{H}}_{r}$ with

$\displaystyle\mathcal{\hat{H}}_{cm}$ $\displaystyle=$
$\displaystyle\frac{\hbar^{2}\hat{\mathbf{K}}^{2}}{4M}+\frac{\hbar^{2}\lambda}{2M}\mathbf{\hat{K}}\cdot\left(\hat{\boldsymbol{\sigma}}_{1}+\hat{\boldsymbol{\sigma}}_{2}\right),$
(2) $\displaystyle\mathcal{\hat{H}}_{r}$ $\displaystyle=$
$\displaystyle\frac{\hbar^{2}\mathbf{\hat{k}}^{2}}{M}+\frac{\hbar^{2}\lambda}{M}\mathbf{\hat{k}}\cdot\left(\hat{\boldsymbol{\sigma}}_{1}-\hat{\boldsymbol{\sigma}}_{2}\right)+\frac{\hbar^{2}\lambda^{2}}{M}+V(r),$
(3)

which describe the c.m. motion with total momentum $\hat{{\bf K}}$ and
relative motion with momentum $\hat{{\bf k}}=\left(\hat{{\bf k}}_{1}-\hat{{\bf
k}}_{2}\right)/2$, respectively. Here, $V\left(r\right)$ is the short-range
interaction potential between two fermions. As discussed in Cui2012M , the
total angular momentum ${\bf J}$ of two fermions as well as their total
momentum ${\bf K}$ is conserved. We may conveniently focus on the scattering
problem in the subspace of $\mathbf{K=0}$ and $\mathbf{J}=0$. In this case,
the two-body Hamiltonian $\hat{\mathcal{H}}_{2}$ is simply reduced to
$\hat{\mathcal{H}}_{r}$, and only $s$\- and $p$-wave scatterings are involved.
Moreover, the subspace of $\mathbf{K=0}$ and $\mathbf{J}=0$ is spanned by two
angular orthogonal basis
$\left\\{\Omega_{0}\left(\mathbf{\hat{r}}\right),\Omega_{1}\left(\mathbf{\hat{r}}\right)\right\\}$
Cui2012M ; Peng2018C ; Zhang2020U

$\displaystyle\Omega_{0}\left(\mathbf{\hat{r}}\right)$ $\displaystyle=$
$\displaystyle Y_{00}\left(\mathbf{\hat{r}}\right)\left|S\right\rangle,$ (4)
$\displaystyle\Omega_{1}\left(\mathbf{\hat{r}}\right)$ $\displaystyle=$
$\displaystyle-\frac{i}{\sqrt{3}}\left[Y_{1-1}\left(\mathbf{\hat{r}}\right)\left|\uparrow\uparrow\right\rangle\right.$
(6)
$\displaystyle\left.+Y_{11}\left(\mathbf{\hat{r}}\right)\left|\downarrow\downarrow\right\rangle-
Y_{10}\left(\mathbf{\hat{r}}\right)\left|T\right\rangle\right],$

where $Y_{lm}\left(\hat{{\bf r}}\right)$ with angular variable $\hat{{\bf
r}}=\left(\theta,\varphi\right)$ for the relative motion of two fermions is
the spherical harmonics, and
$\left|S\right\rangle=\left(\left|\uparrow\downarrow\right\rangle-\left|\downarrow\uparrow\right\rangle\right)/\sqrt{2}$
and
$\left\\{\left|\uparrow\uparrow\right\rangle,\left|\downarrow\downarrow\right\rangle,\left|T\right\rangle=\left(\left|\uparrow\downarrow\right\rangle+\left|\downarrow\uparrow\right\rangle\right)/\sqrt{2}\right\\}$
are the singlet and triplet spin states for two fermions, respectively. Then
the two-body wave function in this subspace can generally be written in the
form of

$\Psi\left(\mathbf{r}\right)=\psi_{0}\left(r\right)\Omega_{0}\left(\mathbf{\hat{r}}\right)+\psi_{1}\left(r\right)\Omega_{1}\left(\mathbf{\hat{r}}\right),$
(7)

and $\psi_{i}\left(r\right)\,\left(i=0,1\right)$ denotes the radial part of
the wave function.

The existence of SO coupling dramatically changes the short-range behavior of
the two-body wave function Zhang2012M . However, for a realistic interaction
potential $V\left(r\right)$ with a short range $\epsilon$, the SO-coupling
strength $\lambda$ as well as the relative momentum $k$ between two fermions
is usually much smaller than $\epsilon^{-1}$ in current experiments of cold
atoms Cheuk2012S ; Wang2012S . In this case, we may perturbatively construct
the asymptotic form of the two-body wave function when two fermions approach
as close as $\epsilon$, i.e.,

$\Psi\left(\mathbf{r}\right)\approx\phi\left(\mathbf{r}\right)+k^{2}F(\mathbf{r})-\lambda
G(\mathbf{r})\\\ +k^{4}X(\mathbf{r})-\lambda
k^{2}Y\left(\mathbf{r}\right)-\lambda^{2}Z\left(\mathbf{r}\right).$ (8)

To generalize our previous results Peng2018C ; Zhang2020U , here we take into
account of the second-order corrections of $\left(k^{2},\lambda\right)$. Later
we will see that these second-order corrections modify the low-energy
expansion of the scattering phase shifts.

Substituting Eq.(8) into the two-body Schrödinger equation
$\mathcal{\hat{H}}_{2}\Psi\left(\mathbf{r}\right)=E\Psi\left(\mathbf{r}\right)$,
following the similar route as that in Peng2018C ; Zhang2020U , and after some
straightforward algebra, we obtain the general form of the two-body wave
function at short distance (see appendix for detail)

$\displaystyle\Psi\left(\mathbf{r}\right)$ $\displaystyle=$
$\displaystyle\alpha_{0}\left[\frac{1}{r}+\left(-\frac{1}{a_{0}}+\frac{b_{0}k^{2}}{2}+\frac{\alpha_{1}}{\alpha_{0}}u\lambda-\frac{\alpha_{1}}{\alpha_{0}}h\lambda
k^{2}-c_{0}\lambda^{2}\right)-\left(\frac{k^{2}}{2}+\frac{\alpha_{1}}{\alpha_{0}}\lambda
k^{2}+\frac{3\lambda^{2}}{2}\right)r\right]\Omega_{0}\left(\mathbf{\hat{r}}\right)$
(9)
$\displaystyle+\alpha_{1}\left[\frac{1}{r^{2}}+\frac{1}{2}\left(k^{2}+2\lambda\frac{\alpha_{0}}{\alpha_{1}}-\lambda^{2}\right)+\left(-\frac{1}{3a_{1}}+\frac{b_{1}k^{2}}{6}+\frac{\alpha_{0}}{\alpha_{1}}v\lambda-\frac{\alpha_{0}}{\alpha_{1}}q\lambda
k^{2}-c_{1}\lambda^{2}\right)r\right]\Omega_{1}\left(\mathbf{\hat{r}}\right)+\mathcal{O}\left(r^{2}\right)$

for $r\gtrsim\epsilon$, where $a_{0},b_{0}$ ($a_{1},b_{1}$) are the well-known
$s$-wave ($p$-wave) scattering length (volume) and effective range, $u,v$ are
the new scattering parameters resulted from the first-order correction of SO
coupling, and $h,q,c_{0},c_{1}$ are those introduced by the second-order
correction of SO coupling. Here, $\alpha_{0}$ and $\alpha_{1}$ are two complex
superposition coefficients. Near the $s$-wave resonance, the contribution of
the $p$-wave scattering could be ignored and we have $\alpha_{1}\approx 0$.
The short-range form of the two-body wave function reduces to (up to a
constant $\alpha_{0}$)

$\Phi_{0}\left(\mathbf{r}\right)=\left[\frac{1}{r}+\left(-\frac{1}{a_{0}}+\frac{b_{0}k^{2}}{2}-c_{0}\lambda^{2}\right)\right.\\\
\left.-\frac{1}{2}\left(k^{2}+3\lambda^{2}\right)r\right]\Omega_{0}\left(\hat{\mathbf{r}}\right)\\\
+\lambda\left[1+\left(v-qk^{2}\right)r\right]\Omega_{1}\left(\hat{\mathbf{r}}\right)+\mathcal{O}\left(r^{2}\right),$
(10)

which recovers the modified Bethe-Peierls boundary condition of Zhang2012M ,
by noticing that we expand the wave function up to the order of $r$ at short
distance and the second-order terms of $\lambda$ are retained. We can see that
a considerable $p$-wave component is involved because of SO coupling, even
near the $s$-wave resonance. In like manner, near the $p$-wave resonance, the
contribution of the $s$-wave scattering is small and could be ignored
($\alpha_{0}\approx 0$). Subsequently, the form of the two-body wave function
at short distance reduces to (up to a constant $\alpha_{1}$)

$\Phi_{1}\left(\mathbf{r}\right)=\lambda\left[\left(u-hk^{2}\right)-k^{2}r\right]\Omega_{0}\left(\hat{\mathbf{r}}\right)+\left[\frac{1}{r^{2}}+\frac{1}{2}\left(k^{2}-\lambda^{2}\right)\right.\\\
\left.+\left(-\frac{1}{3a_{1}}+\frac{b_{1}k^{2}}{6}-c_{1}\lambda^{2}\right)r\right]\Omega_{1}\left(\hat{\mathbf{r}}\right)+\mathcal{O}\left(r^{2}\right).$
(11)

Similarly, an $s$-wave component is induced by SO coupling even near the
$p$-wave resonance.

##  III The $S$ matrix and scattering phase shifts

The range $\epsilon$ of interaction potentials between neutral atoms is
usually much smaller than the inverse of the relative momentum $k$ as well as
that of SO-coupling strength $\lambda$. The two-body wave function outside the
potential in the presence of SO coupling can easily be obtained by solving the
Schrödinger equation with $V\left(r\right)=0$ for $r>\epsilon$ Wu2013S ,

$\Psi\left({\bf r}\right)=A\Psi_{-}^{(in)}\left({\bf
r}\right)+B\Psi_{+}^{(in)}\left({\bf r}\right)\\\ +C\Psi_{-}^{(out)}\left({\bf
r}\right)+D\Psi_{+}^{(out)}\left({\bf r}\right)$ (12)

with

$\Psi_{\pm}^{(in)}\left({\bf
r}\right)=k_{\pm}\left[h_{0}^{(2)}\left(k_{\pm}r\right)\Omega_{0}\left(\hat{\mathbf{r}}\right)\pm
h_{1}^{(2)}\left(k_{\pm}r\right)\Omega_{1}\left(\hat{\mathbf{r}}\right)\right],$
(13) $\Psi_{\pm}^{(out)}\left({\bf
r}\right)=k_{\pm}\left[h_{0}^{(1)}\left(k_{\pm}r\right)\Omega_{0}\left(\hat{\mathbf{r}}\right)\pm
h_{1}^{(1)}\left(k_{\pm}r\right)\Omega_{1}\left(\hat{\mathbf{r}}\right)\right],$
(14)

$k_{\pm}=k\pm\lambda$, and $k=\sqrt{ME/\hbar^{2}}$. Here,
$h_{\nu}^{\left(1\right)}$ and $h_{\nu}^{\left(2\right)}$ denote the $\nu$th-
order spherical Hankel functions of the first and second kinds, respectively,
and $A,B,C,D$ are superposition coefficients. The physical meaning of the
solution (12) is apparent: due to unique properties of the single-particle
dispersion relation Cui2012M ; Wu2013S , the incident wave with energy $E$ is
an arbitrary superposition of two spherical waves with two different
magnitudes of momenta $k_{\pm}$ (corresponding to the spherical Hankel
function of the second kind $h_{\nu}^{(2)}$); scattered elastically by the
interatomic potential $V\left(r\right)$, the outgoing wave becomes a different
superposition of the same two spherical waves (corresponding to the spherical
Hankel function of the first kind $h_{\nu}^{(1)}$).

Using the time-dependent Schrödinger equation $i\hbar\partial\Psi\left({\bf
r},t\right)/\partial t=\hat{\mathcal{H}}_{2}\Psi\left({\bf r},t\right)$, and
noticing $\Psi\left({\bf
r},t\right)=\psi_{0}\left(r,t\right)\Omega_{0}\left(\mathbf{\hat{r}}\right)+\psi_{1}\left(r,t\right)\Omega_{1}\left(\mathbf{\hat{r}}\right)$,
we easily obtain the following continuity equation

$\frac{\partial\rho}{\partial t}+\frac{dj}{dr}=0,$ (15)

where
$\rho=r^{2}\left(\left|\psi_{0}\right|^{2}+\left|\psi_{1}\right|^{2}\right)$
is the radial probability density, and

$j\left(r\right)=\\\
\frac{i\hbar}{M}r^{2}\left[\sum_{i=0}^{1}\left(\psi_{i}\frac{d\psi_{i}^{*}}{dr}-\psi_{i}^{*}\frac{d\psi_{i}}{dr}\right)-2\lambda\left(\psi_{0}^{*}\psi_{1}-\psi_{0}\psi_{1}^{*}\right)\right]$
(16)

is the radial probability current density. For the stationary state of the
system, the radial probability density $\rho$ is obviously independent of
time, and thus the radial probability current density $j\left(r\right)$ is a
constant. Moreover, we have $j\left(r\right)=0$ at $r=0$, which in turn
indicates that $j\left(r\right)$ should be zero everywhere. Inserting the
scattering solution Eq.(12) into Eq.(16) and using $j\left(r\right)=0$, we
easily obtain

$\left|A\right|^{2}+\left|B\right|^{2}=\left|C\right|^{2}+\left|D\right|^{2}.$
(17)

This means that the outgoing wave is different from the incident wave only up
to a unitary transformation, i.e.

$\left[\begin{array}[]{c}C\\\
D\end{array}\right]=\mathcal{S}\left[\begin{array}[]{c}A\\\
B\end{array}\right],$ (18)

and $\mathcal{S}$ is a unitary $2\times 2$ matrix determined by the specific
form of the short-range interaction potential. The unitarity of the matrix
$\mathcal{S}$ was confirmed by using a spherical-square-well model in Wu2013S
. Here, we emphasize that this property is obviously universal for any short-
range interaction potential in the presence of SO coupling, which is a natural
consequence of the probability conservation during elastic collisions.

The unitary $\mathcal{S}$ matrix can formally be diagonalized as

$W^{\dagger}\mathcal{S}W=\left[\begin{array}[]{cc}e^{i2\delta_{0}}&0\\\
0&e^{i2\delta_{1}}\end{array}\right]$ (19)

by

$W=\left[\begin{array}[]{cc}\omega_{0-}&\omega_{1-}\\\
\omega_{0+}&\omega_{1+}\end{array}\right],$ (20)

where $\left[\omega_{0-},\omega_{0+}\right]^{T}$ and
$\left[\omega_{1-},\omega_{1+}\right]^{T}$ are two eigenvectors of
$\mathcal{S}$ corresponding to different eigenvalues $e^{i2\delta_{0}}$ and
$e^{i2\delta_{1}}$. We can see that $\delta_{0,1}$ are the new scattering
phase shifts characterizing the scattering effect in the presence of SO
coupling. Inserting Eq.(18) into Eq.(12), we may represent the solution
$\Psi\left({\bf r}\right)$ outside the interatomic potential by using
eigenvectors and eigenvalues of the $\mathcal{S}$ matrix, i.e.,

$\Psi\left({\bf r}\right)=\alpha_{0}\Phi_{0}\left({\bf
r}\right)+\alpha_{1}\Phi_{1}\left({\bf r}\right)$ (21)

with

$\Phi_{i}\left({\bf
r}\right)=\omega_{i-}\Psi_{-}^{(in)}+\omega_{i+}\Psi_{+}^{(in)}\\\
+e^{i2\delta_{i}}\left(\omega_{i-}\Psi_{-}^{(out)}+\omega_{i+}\Psi_{+}^{(out)}\right),$
(22)

and $\alpha_{i}=\omega_{i-}^{*}A+\omega_{i+}^{*}B$ for $i=0,1$. Substituting
Eqs. (13) and (14) into Eq.(22), we arrive at

$\Phi_{i}\left({\bf r}\right)=2e^{i\delta_{i}}\cos\delta_{i}\\\
\times\sum_{\nu=0}^{1}\sum_{\eta=\pm}\eta^{\nu}\omega_{i\eta}k_{\eta}\left[j_{\nu}\left(k_{\eta}r\right)-n_{\nu}\left(k_{\eta}r\right)\tan\delta_{i}\right]\Omega_{\nu}\left(\hat{{\bf
r}}\right)$ (23)

for $i=0,1$, where $j_{\nu}$ and $n_{\nu}$ are the $\nu$th-order spherical
Bessel functions of the first and second kinds, respectively.

##  IV low energy expansion of scattering phase shifts

For ultracold atoms, the low-energy behavior of the scattering phase shift
gives rise to some key microscopic scattering parameters, such as the
scattering length (volume) and effective range. In the presence of SO
coupling, the low-energy expansion of the scattering phase shift is expected
to be modified. Then we may consider such modification perturbatively, since
the energy scale of the SO-coupling strength is usually much smaller than that
corresponding to the range of interatomic potentials. This can be done by
simply comparing the short-range expansion of Eq.(23) with those of Eqs.(10)
and (11).

Near $s$-wave resonances, we may expand Eq.(23) with $i=0$ at small $r$ and
obtain

$\Phi_{0}\left(\mathbf{r}\right)=\left(\frac{1}{r}+\frac{k+\omega_{0}\lambda}{\tan\delta_{0}}-\frac{k^{2}+\lambda^{2}+2\omega_{0}\lambda
k}{2}r\right)\Omega_{0}\left(\hat{\mathbf{r}}\right)\\\
+\left[\frac{\omega_{0}k-\lambda}{k^{2}-\lambda^{2}}\frac{1}{r^{2}}+\frac{\omega_{0}k+\lambda}{2}\right.\\\
\left.+\frac{\omega_{0}\left(k^{2}+\lambda^{2}\right)+2\lambda
k}{3\tan\delta_{0}}r\right]\Omega_{1}\left(\hat{\mathbf{r}}\right)+\mathcal{O}\left(r^{2}\right)$
(24)

for $r\sim\epsilon^{+}$, where we have omitted an overall factor
$2\left(\omega_{0+}+\omega_{0-}\right)e^{i\delta_{0}}\sin\delta_{0}$, and
introduced
$\omega_{0}\equiv\left(\omega_{0+}-\omega_{0-}\right)/\left(\omega_{0+}+\omega_{0-}\right)$.
Comparing the corresponding terms between the Eq.(24) and Eq.(10), we find

$k\cot\delta_{0}\approx\frac{k^{2}}{k^{2}+\lambda^{2}}\left(-\frac{1}{a_{0}}+\frac{b_{0}}{2}k^{2}-c_{0}\lambda^{2}\right),$
(25)

which recovers that of Cui2012M  if we keep the terms in $\tan\delta_{0}$ up
to $\lambda^{2}$ and notice $b_{0}\approx 0$ near broad $s$-wave resonances.
It is apparent that $\delta_{0}$ is the counterpart of the $s$-wave scattering
phase shift in the absence of SO coupling, and then Eq.(25) reduces to the
well-known effective-range expansion of the $s$-wave scattering phase shift,
i.e., $k\cot\delta_{0}=-1/a_{0}+b_{0}k^{2}/2+\mathcal{O}\left(k^{4}\right)$.
In the presence of SO coupling, the new scattering parameter $c_{0}$ is
involved, which modifies the low-energy behavior of the $s$-wave scattering
phase shift.

Near $p$-wave resonances, we again expand Eq.(23) with $i=1$ at small $r$ and
obtain (up to an overall constant
$2\left(\omega_{1+}/k_{+}-\omega_{1-}/k_{-}\right)e^{i\delta_{1}}\sin\delta_{1}$)

$\Phi_{1}\left(\mathbf{r}\right)=\left[\frac{\omega_{1}\left(k^{2}-\lambda^{2}\right)}{k-\omega_{1}\lambda}\frac{1}{r}+\frac{\left(k^{2}-\lambda^{2}\right)\left(\omega_{1}k+\lambda\right)}{\left(k-\omega_{1}\lambda\right)\tan\delta_{1}}\right.\\\
\left.-\frac{\omega_{1}\left(k^{4}-\lambda^{4}\right)+2\lambda
k^{3}-2\lambda^{3}k}{2\left(k-\omega_{1}\lambda\right)}r\right]\Omega_{0}\left(\hat{\mathbf{r}}\right)\\\
+\left[\frac{1}{r^{2}}+\frac{\left(k^{2}-\lambda^{2}\right)\left(k+\omega_{1}\lambda\right)}{2\left(k-\omega_{1}\lambda\right)}+\frac{k^{2}-\lambda^{2}}{3\left(k-\omega_{1}\lambda\right)}\right.\\\
\left.\times\frac{k^{2}+\lambda^{2}+2\omega_{1}\lambda
k}{\tan\delta_{1}}r\right]\Omega_{1}\left(\hat{\mathbf{r}}\right)+\mathcal{O}(r^{2})$
(26)

for $r\sim\epsilon^{+}$, and
$\omega_{1}\equiv\left(\omega_{1+}+\omega_{1-}\right)/\left(\omega_{1+}-\omega_{1-}\right)$.
In like manner, comparing the corresponding terms of Eq.(26) with Eq.(11), we
easily obtain

$k^{3}\cot\delta_{1}\approx\frac{k^{4}}{k^{4}-\lambda^{4}}\left(-\frac{1}{a_{1}}+\frac{b_{1}}{2}k^{2}-3c_{1}\lambda^{2}\right).$
(27)

It recovers that of Cui2012M  near $p$-wave resonances NOTE-PWave . We can see
that the SO-coupling induced scattering parameter $c_{1}$ is involved in
characterizing the scattering phase shift $\delta_{1}$. Obviously, the
scattering phase shift $\delta_{1}$ is the counterpart of the $p$-wave
scattering phase shift in the absence of SO coupling. It reduces to the well-
known effective-range expansion of the $p$-wave scattering phase shift without
SO coupling, i.e.,
$k^{3}\cot\delta_{1}=-1/a_{1}+b_{1}k^{2}/2+\mathcal{O}\left(k^{4}\right)$.

##  V a spherical-square-well potential model

Till now, we have discussed the introducing of new scattering parameters in
the short-range asymptotic behavior of the two-body wave function, and
considered how these new scattering parameters characterize and modify the
scattering phase shifts at low energy. In order to demonstrate that these new
scattering parameters $c_{0}$ and $c_{1}$ are not artificially introduced by
our perturbation method and are in fact physically determined by real
interatomic potentials, in the follows, we consider a two-body scattering
problem within a model of the spherical-square-well potential, i.e.,

$V\left(r\right)=\begin{cases}-V_{0},&0\leq r\leq\epsilon,\\\
0,&r>\epsilon\end{cases}$ (28)

with the depth $V_{0}>0$. By solving the Schrödinger equation inside and
outside the potential, respectively, and utilizing the continuity of the two-
body wave function at $r=\epsilon$ as well as its first-order derivative, we
obtain (see appendix)

$\displaystyle\frac{c_{0}}{\epsilon}$ $\displaystyle=$
$\displaystyle-1+\frac{\widetilde{V}_{0}}{3\left[\sqrt{\widetilde{V}_{0}}-\tan\left(\sqrt{\widetilde{V}_{0}}\right)\right]^{2}}+\frac{1}{\widetilde{V}_{0}-\sqrt{\widetilde{V}_{0}}\tan\left(\sqrt{\widetilde{V}_{0}}\right)},$
(29) $\displaystyle c_{1}\epsilon$ $\displaystyle=$
$\displaystyle-\frac{\widetilde{V}_{0}\left[\left(-15+\widetilde{V}_{0}\right)\sqrt{\widetilde{V}_{0}}\cos\left(\sqrt{\widetilde{V}_{0}}\right)+3\left(5-2\widetilde{V}_{0}\right)\sin\left(\sqrt{\widetilde{V}_{0}}\right)\right]}{5\left[-3+\widetilde{V}_{0}+3\sqrt{\widetilde{V}_{0}}\cot\left(\sqrt{\widetilde{V}_{0}}\right)\right]^{2}\left[\sqrt{\widetilde{V}_{0}}\cos\left(\sqrt{\widetilde{V}_{0}}\right)-\sin\left(\sqrt{\widetilde{V}_{0}}\right)\right]},$
(30)

and $\tilde{V}_{0}=M\epsilon^{2}V_{0}/\hbar^{2}$.

Figure 1: (Color online) The new scattering parameter $c_{0}/\epsilon$,
characterizing the second-order correction of $\lambda$ to the scattering
phase shift $\delta_{0}$, evolving with the depth of the potential
$\tilde{V}_{0}=M\epsilon^{2}V_{0}/\hbar^{2}$. The red dashed line indicates
the $s$-wave resonance in the absence of spin-orbit coupling, and the blue
numerical data is from Cui2012M . The inset shows the position of $s$-wave
scattering resonance changing with the spin-orbit-coupling strength.

We present $c_{0}$ as a function of the reduced depth $\tilde{V}_{0}$ of the
potential near the $s$-wave resonance in Fig.1. The new parameter $c_{0}$
introduced by SO coupling characterizes the second-order correction of
$\lambda$ to the scattering phase shift $\delta_{0}$, and has numerically been
evaluated in Cui2012M  near the $s$-wave resonance. Our analytical result,
i.e., Eq.(29), shows accurate agreement with the numerical calculation of
Cui2012M . The scattering phase shift $\delta_{0}$ involves all short-range
information of the interatomic potential, and governs the $s$-wave scattering
properties outside the potential $r>\epsilon$. The resonance position is
determined by $\delta_{0}=\pi/2$, which yields the well depth
$\tilde{V}_{0}=\pi^{2}/4$ at the resonance in the absence of SO coupling as
indicated by the red dashed line in Fig.1. When the SO coupling is gradually
turned on, the resonance position is dramatically shifted by SO coupling
because of considerable value of $c_{0}$, as shown in the inset of Fig.1. For
the same consideration, we plot $c_{1}$ as a function of the reduced depth
$\tilde{V}_{0}$ near the $p$-wave resonance in Fig.2, which characterizes the
second-order correction of $\lambda$ to the $p$-wave scattering phase shift
$\delta_{1}$. Unlike the case of the $s$-wave scattering, we can see that
$c_{1}$ is extremely small near the $p$-wave resonance without SO coupling,
which means the correction from the SO coupling to the $p$-wave scattering
phase shift is negligibly small. Consequently, the resonance position is
nearly unchanged when the SO coupling is gradually turned on as shown in the
inset of Fig.2.

Figure 2: (Color online) The new scattering parameter $c_{1}\epsilon$,
characterizing the second-order correction of $\lambda$ to the scattering
phase shift $\delta_{1}$, evolving with the depth potential
$\tilde{V}_{0}=M\epsilon^{2}V_{0}/\hbar^{2}$. The red dashed line indicates
the $p$-wave resonance in the absence of spin-orbit coupling. The inset shows
the position of $p$-wave scattering resonance changing with the spin-orbit-
coupling strength.

##  VI Conclusions

In this work, we investigate the two-body scattering problems of Fermi gases
in the presence of the three-dimensional isotropic spin-orbit coupling. Since
the energy scale corresponding to the spin-orbit-coupling strength is much
smaller than that corresponding to the range of interatomic potentials, we
perturbatively construct the asymptotic behavior of two-body wave function
when two fermions approach as close as the interaction range. To generalize
our previous results Peng2018C ; Zhang2020U , we consider up to the second-
order correction of the spin-orbit-coupling strength to the two-body wave
function, and additional new scattering parameters are introduced. Further,
the scattering phase shifts are defined based on the unitary $S$ matrix for an
elastic collision. The low-energy behavior of the scattering phase shifts is
discussed, which is modified by the new scattering parameters in the presence
of spin-orbit coupling. Our results naturally reduce to the well-known
effective-range expansions of the scattering phase shifts without spin-orbit
coupling. Within the model of a spherical-square-well potential, our
analytical results are verified and agree well with previous numerical
calculations near the $s$-wave resonance Cui2012M . To simplify the
presentation of this work, we focus our discussions on the subspace of zero
center-of-mass momentum and zero total angular momentum of two fermions, and
then only $s$\- and $p$-wave scatterings are involved. The advantage of our
method is that the correction of the spin-orbit coupling to the two-body wave
function at short distance may perturbatively be considered order by order, as
well as to the scattering phase shifts at low energy. Therefore, it is
straightforward for the generalization to the case of non-zero center-of-mass
momentum and non-zero total angular momentum. Then more scattering partial
waves should be involved, and additional scattering parameters would be
introduced to characterize the short-range behavior of the two-body wave
function as well as the low-energy expansion of the scattering phase shifts.
Our method provides a unified description of the low-energy properties of
scattering phase shifts in the presence of spin-orbit coupling.

###### Acknowledgements.

C.-X.Z. is supported by NSFC under Grant No.11947097. S.-G.P. is supported by
NSFC under Grants No. 11974384, and NKRDP under Grant No. 2016YFA0301503.

##  Appendix A Scattering under a spherical-square-well potential

In this appendix, we present the derivation detail of the short-range behavior
of the two-body wave function (9), and calculate the specific forms of the new
scattering parameters for a spherical-square-well potential. Substituting the
ansatz Eq.(8) into the two-body Schr$\ddot{\mathrm{o}}$dinger equation
$\mathcal{\hat{H}}_{2}\Psi\left(\mathbf{r}\right)=E\Psi\left(\mathbf{r}\right)$,
and comparing the corresponding coefficients of the terms ( $k^{2}$,
$\lambda$, $k^{4}$, $\lambda k^{2}$, $\lambda^{2}$) on both sides, we obtain
the following coupled equations,

$\displaystyle\left[-\nabla^{2}+\frac{MV\left(\mathbf{r}\right)}{\hbar^{2}}\right]\phi\left(\mathbf{r}\right)$
$\displaystyle=$ $\displaystyle 0,$ (31)
$\displaystyle\left[-\nabla^{2}+\frac{MV\left(\mathbf{r}\right)}{\hbar^{2}}\right]F\left(\mathbf{r}\right)$
$\displaystyle=$ $\displaystyle\phi\left(\mathbf{r}\right),$ (32)
$\displaystyle\left[-\nabla^{2}+\frac{MV\left(\mathbf{r}\right)}{\hbar^{2}}\right]G\left(\mathbf{r}\right)$
$\displaystyle=$
$\displaystyle\hat{Q}\left(\mathbf{r}\right)\phi\left(\mathbf{r}\right),$ (33)
$\displaystyle\left[-\nabla^{2}+\frac{MV\left(\mathbf{r}\right)}{\hbar^{2}}\right]Y\left(\mathbf{r}\right)$
$\displaystyle=$
$\displaystyle\hat{Q}\left(\mathbf{r}\right)F\left(\mathbf{r}\right)+G\left(\mathbf{r}\right),$
(34)
$\displaystyle\left[-\nabla^{2}+\frac{MV\left(\mathbf{r}\right)}{\hbar^{2}}\right]Z\left(\mathbf{r}\right)$
$\displaystyle=$
$\displaystyle\phi\left(\mathbf{r}\right)-\hat{Q}\left(\mathbf{r}\right)G\left(\mathbf{r}\right)$
(35)

with
$\hat{Q}\left(\mathbf{r}\right)=\mathbf{\hat{k}}\cdot\left(\hat{\boldsymbol{\sigma}}_{1}-\hat{\boldsymbol{\sigma}}_{2}\right)$.
The corresponding expressions of functions $\phi\left(\mathbf{r}\right)$,
$F(\mathbf{r})$, $G(\mathbf{r})$, $Y\left(\mathbf{r}\right)$ and
$Z\left(\mathbf{r}\right)$ outside the potential for $r\gtrsim\epsilon$ can
easily be obtained by solving these coupled equations

$\displaystyle\phi\left(\mathbf{r}\right)$ $\displaystyle=$
$\displaystyle\alpha_{0}\left(\frac{1}{r}-\frac{1}{a_{0}}\right)\Omega_{0}\left(\mathbf{\hat{r}}\right)+\alpha_{1}\left(\frac{1}{r^{2}}-\frac{1}{3a_{1}}r\right)\Omega_{1}\left(\mathbf{\hat{r}}\right)+\mathcal{O}\left(r^{2}\right),$
(36) $\displaystyle F\left(\mathbf{r}\right)$ $\displaystyle=$
$\displaystyle\alpha_{0}\left(\frac{1}{2}b_{0}-\frac{1}{2}r\right)\Omega_{0}\left(\mathbf{\hat{r}}\right)+\alpha_{1}\left(\frac{1}{2}+\frac{b_{1}}{6}r\right)\Omega_{1}\left(\mathbf{\hat{r}}\right)+\mathcal{O}\left(r^{2}\right),$
(37) $\displaystyle G\left(\mathbf{r}\right)$ $\displaystyle=$
$\displaystyle-\alpha_{1}u\Omega_{0}\left(\mathbf{\hat{r}}\right)-\alpha_{0}\left(1+vr\right)\Omega_{1}\left(\mathbf{\hat{r}}\right)+\mathcal{O}\left(r^{2}\right),$
(38) $\displaystyle Y\left(\mathbf{r}\right)$ $\displaystyle=$
$\displaystyle\alpha_{1}\left(h+r\right)\Omega_{0}\left(\mathbf{\hat{r}}\right)+\alpha_{0}qr\Omega_{1}\left(\mathbf{\hat{r}}\right)+\mathcal{O}\left(r^{2}\right),$
(39) $\displaystyle Z\left(\mathbf{r}\right)$ $\displaystyle=$
$\displaystyle\alpha_{0}\left(c_{0}+\frac{3}{2}r\right)\Omega_{0}\left(\mathbf{\hat{r}}\right)+\alpha_{1}\left(\frac{1}{2}+c_{1}r\right)\Omega_{1}\left(\mathbf{\hat{r}}\right)+\mathcal{O}\left(r^{2}\right).$
(40)

Inserting these functions into the two-body wave function (8), we arrive at
Eq.(9). Obviously, the derivation here is independent of the specific form of
the interatomic potential, and new scattering parameters are introduced in the
short-range form of the two-body wave function.

To determine the new scattering parameters in the two-body wave function, let
us consider a specific model of the spherical-square-well potential. Outside
the potential, i.e., $V\left(r\right)=0$, we have already obtain the form of
the two-body wave function as shown in Eqs.(36) to (40). While inside the
potential, we have $V\left(r\right)=-V_{0}$, and the corresponding specific
forms of these functions inside the potential are obtained by solving the
Schröding equation. By using the continuity of the two-body wave function at
$r=\epsilon$ as well as its first-order derivative, all the scattering
parameters are then be determined. To simplify the presentation, we only show
the specific forms of $c_{0}$ and $c_{1}$ in Eqs.(29)-(30), which characterize
the second-order corrections of SO coupling to the scattering phase shifts.

## References

  * (1) I. Bloch, J. Dalibard, and W. Zwerger, Many-body physics with ultracold gases, Rev. Mod. Phys. $\mathbf{80}$, 885 (2008).
  * (2) S. Giorgini, L. P. Pitaevskii, and S. Stringari, Theory of ultracold atomic Fermi gases, Rev. Mod. Phys. $\mathbf{80}$, 1215 (2008).
  * (3) C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Feshbach resonances in ultracold gases, Rev. Mod. Phys. $\mathbf{82}$, 1225 (2010).
  * (4) E. Braaten and H.-W. Hammer, Universality in few-body systems with large scattering length, Phys. Rep. $\mathbf{428}$, 259 (2006).
  * (5) D. Blume, Few-body physics with ultracold atomic and molecular systems in traps, Rep. Prog. Phys. $\mathbf{75}$, 046401 (2012).
  * (6) C. H. Greene, P. Giannakeas, and J. P$\mathrm{\acute{e}}$rez-R$\mathrm{\acute{i}}$os, Universal few-body physics and cluster formation, Rev. Mod. Phys. $\mathbf{89}$, 035006 (2017).
  * (7) S. Tan, Energetics of a strongly correlated Fermi gas, Ann. Phys. $\mathbf{323}$, 2952 (2008).
  * (8) S. Tan, Large momentum part of a strongly correlated Fermi gas, Ann. Phys. $\mathbf{323}$, 2971 (2008).
  * (9) S. Tan, Generalized virial theorem and pressure relation for a strongly correlated Fermi gas, Ann. Phys. $\mathbf{323}$, 2987 (2008).
  * (10) W. Zwerger, The BCS-BEC Crossover and the Unitary Fermi Gas, Lecture Notes in Physics Vol. 836 (Springer, Berlin, 2011), see Chap. 6 for a brief review.
  * (11) E. Braaten and L. Platter, Exact relations for a strongly interacting Fermi gas from the operator product expansion, Phys. Rev. Lett. $\mathbf{100}$, 205301 (2008).
  * (12) S. Zhang and A. J. Leggett, Universal properties of the ultracold Fermi gas, Phys. Rev. A $\mathbf{79}$, 023601 (2009).
  * (13) J. T. Stewart, J. P. Gaebler, T. E. Drake, and D. S. Jin, Verification of universal relations in a strongly interacting Fermi gas, Phys. Rev. Lett. $\mathbf{104}$, 235301 (2010).
  * (14) E. D. Kuhnle, H. Hu, X.-J. Liu, P. Dyke, M. Mark, P. D. Drummond, P. Hannaford, and C. J. Vale, Universal behavior of pair correlations in a strongly interacting Fermi gas, Phys. Rev. Lett. $\mathbf{105}$, 070402 (2010).
  * (15) Z. H. Yu, J. H. Thywissen, and S. Z. Zhang, Universal relations for a Fermi gas close to a $p$-wave interaction resonance, Phys. Rev. Lett. $\mathbf{115}$, 135304 (2015).
  * (16) S. M. Yoshida and M. Ueda, Universal high-momentum asymptote and thermodynamic relations in a spinless Fermi gas with a resonant $p$-wave interaction, Phys. Rev. Lett. $\mathbf{115}$, 135303 (2015).
  * (17) M. Y. He, S. L. Zhang, H. M. Chan, and Q. Zhou, Concept of a contact spectrum and its applications in atomic quantum Hall states, Phys. Rev. Lett. $\mathbf{116}$, 045301 (2016).
  * (18) C. Luciuk, S. Trotzky, S. Smale, Z. Yu, S. Zhang, and J. H. Thywissen, Evidence for universal relations describing a gas with $p$-wave interactions, Nature Phys. $\mathbf{12}$, 599 (2016).
  * (19) S.-G. Peng, X. J. Liu, and H. Hu, Large-momentum distribution of a polarized Fermi gas and $p$-wave contacts, Phys. Rev. A $\mathbf{94}$, 063651 (2016).
  * (20) S.-G. Peng, Universal relations for a spin-polarized Fermi gas in two dimensions, J. Phys. A: Math. Theor. $\mathbf{52}$, 245302 (2019).
  * (21) P. Zhang, S. Zhang, and Z. Yu, Effective theory and universal relations for Fermi gases near a $d$-wave-interaction resonance, Phys. Rev. A $\mathbf{95}$, 043609 (2017).
  * (22) L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Butterworth-Heinemann, Oxford, 1999).
  * (23) N. F. Mott and H. S. W. Massey, Theory of Atomic Collisions (3rd ed., Oxford, 1965).
  * (24) Y. J. Lin, K. Jiménez-Garcia, and I. B. Spielman, Spin-orbit-coupled Bose-Einstein condensates, Nature $\mathbf{471}$, 83 (2011).
  * (25) L. W. Cheuk, A. T. Sommer, Z. Hadzibabic, T. Yefsah, W. S. Bakr, and M. W. Zwierlein, Spin-injection spectroscopy of a spin-orbit-coupled Fermi gas, Phys. Rev. Lett. $\mathbf{109}$, 095302 (2012).
  * (26) P. J. Wang, Z. Q. Yu, Z. K. Fu, J. Miao, L. H. Huang, S. J. Chai, H. Zhai, and J. Zhang, Spin-orbit-coupled degenerate Fermi gases, Phys. Rev. Lett. $\mathbf{109}$, 095301 (2012).
  * (27) Z. Wu, L. Zhang, W. Sun, X. T. Xu, B. Z. Wang, S. C. Ji, Y. Deng, S. Chen, X. J. Liu, and J. W. Pan, Realization of two-dimensional spin-orbit coupling for Bose-Einstein condensates, Science $\mathbf{354}$, 83 (2016).
  * (28) L. H. Huang, Z. M. Meng, P. J. Wang, P. Peng, S. L. Zhang, L. C. Chen, D. H. Li, Q. Zhou, and J. Zhang, Experimental realization of two-dimensional synthetic spin-orbit coupling in ultracold Fermi gases, Nat. Phys. $\mathbf{12}$, 540 (2016).
  * (29) H.-R. Chen, K.-Y. Lin, P.-K. Chen, N.-C. Chiu, J.-B. Wang, C.-A. Chen, P.-P. Huang, S.-K. Yip, Y. Kawaguchi, and Y.-J. Lin, Spin-orbital-angular-momentum-coupled Bose-Einstein condensates, Phys. Rev. Lett. $\mathbf{121}$, 113204 (2018).
  * (30) P.-K. Chen, L.-R. Liu, M.-J. Tsai, N.-C. Chiu, Y. Kawaguchi, S.-K. Yip, M.-S. Chang, and Y.-J. Lin, Rotating atomic quantum gases with light-induced azimuthal gauge potentials and the observation of the Hess-Fairbank effect, Phys. Rev. Lett. $\mathbf{121}$, 250401 (2018).
  * (31) D. Zhang, T. Gao, P. Zou, L. Kong, R. Li, X. Shen, X.-L. Chen, S.-G. Peng, M. Zhan, H. Pu, and K. Jiang, Ground-state phase diagram of a spin-orbital-angular-momentum-coupled Bose-Einstein condensate, Phys. Rev. Lett. $\mathbf{122}$, 110402 (2019).
  * (32) T. Köhler, K. Góral, and P. S. Julienne, Production of cold molecules via magnetically tunable Feshbach resonances, Rev. Mod. Phys. $\mathbf{78}$, 1311 (2006).
  * (33) S.-L. Zhu, H. Fu, C.-J. Wu, S.-C. Zhang, and L.-M. Duan, Spin Hall effects for cold atoms in a light-induced gauge potential, Phys. Rev. Lett. $\mathbf{97}$, 240401 (2006).
  * (34) S.-G. Peng, S. Tan, and K. Jiang, Manipulation of $p$-wave scattering of cold atoms in low dimensions using the magnetic field vector, Phys. Rev. Lett. $\mathbf{112}$, 250401 (2014).
  * (35) C. Gross and I. Bloch, Quantum simulations with ultracold atoms in optical lattices, Science $\mathbf{357}$, 995 (2017).
  * (36) X. L. Qi and S. C. Zhang, Topological insulators and superconductors, Rev. Mod. Phys. $\mathbf{83}$, 1057 (2011).
  * (37) J. Dalibard, F. Gerbier, G. Juzeliunas, and P. Öhberg, Colloquium: Artificial gauge potentials for neutral atoms, Rev. Mod. Phys. $\mathbf{83}$, 1523 (2011).
  * (38) D. W. Zhang, Y. Q. Zhu, Y. X. Zhao, H. Yan, and S. L. Zhu, Topological quantum matter with cold atoms, Adv. Phys. $\mathbf{67}$, 253 (2019).
  * (39) J. K. Wang, W. Yi, and W. Zhang, Two-body physics in quasi-low-dimensional atomic gases under spin-orbit coupling, Front. Phys. $\mathbf{11}$, 118102 (2016).
  * (40) X. Cui, Mixed-partial-wave scattering with spin-orbit coupling and validity of pseudopotentials, Phys. Rev. A $\mathbf{85}$, 022705 (2012).
  * (41) P. Zhang, L. Zhang, and Y. Deng, Modified Bethe-Peierls boundary condition for ultracold atoms with spin-orbit coupling, Phys. Rev. A $\mathbf{86}$, 053608 (2012).
  * (42) S.-G. Peng, C.-X. Zhang, S. Tan, and K. Jiang, Contact theory for spin-orbit-coupled Fermi gases, Phys. Rev. Lett. $\mathbf{120}$, 060408 (2018).
  * (43) C.-X. Zhang, S.-G. Peng, and K. Jiang, Universal relations for spin-orbit-coupled Fermi gases in two and three dimensions, Phys. Rev. A $\mathbf{101}$, 043616 (2020).
  * (44) P. Zhang and N. Sun, Universal relations for spin-orbit-coupled Fermi gas near an $s$-wave resonance, Phys. Rev. A $\mathbf{97}$, 040701(R) (2018).
  * (45) J. Jie, R. Qi, and P. Zhang, Universal relations of an ultracold Fermi gas with arbitrary spin-orbit coupling, Phys. Rev. A $\mathbf{97}$, 053602 (2018).
  * (46) F. Qin, P. Zhang, and P.-L. Zhao, Large-momentum tail of one-dimensional Fermi gases with spin-orbit coupling, Phys. Rev. A $\mathbf{101}$, 063619 (2020).
  * (47) F. Qin and P. Zhang, Universal relations for hybridized $s$\- and $p$-wave interactions from spin-orbital coupling, arXiv: 2005.04997v1 (2020).
  * (48) H. Duan, L. You, and B. Gao, Ultracold collisions in the presence of synthetic spin-orbit coupling, Phys. Rev. A $\mathbf{87}$, 052708 (2013).
  * (49) Q. Guan and D. Blume, Scattering framework for two particles with isotropic spin-orbit coupling applicable to all energies, Phys. Rev. A $\mathbf{94}$, 022706 (2016).
  * (50) Q. Guan and D. Blume, Analytical coupled-channel treatment of two-body scattering in the presence of three-dimensional isotropic spin-orbit coupling, Phys. Rev. A $\mathbf{95}$, 020702(R) (2017).
  * (51) J. Wang, C. R. Hougaard, B. C. Mulkerin, and X. J. Liu, Ultracold collisions between spin-orbit-coupled dipoles: General formalism and universality, Phys. Rev. A $\mathbf{97}$, 042709 (2018).
  * (52) Y. Wu and Z. Yu, Short-range asymptotic behavior of the wave functions of interacting spin-$1/2$ fermionic atoms with spin-orbit coupling: A model study, Phys. Rev. A $\mathbf{87}$, 032703 (2013).
  * (53) Here, near the $p$-wave resonance, the divergent term $r^{-1}$ in the $s$-wave channel of Cui2012M  could be eliminated by using $\phi\left({\bf r}\right)$. Then our results agree with that in Cui2012M .