# Spin-orbit-coupled spinor gap solitons in Bose-Einstein condensates

Jing Yang  Department of Physics, Shanghai University, Shanghai 200444, China
Yongping Zhang<EMAIL_ADDRESS>Department of Physics, Shanghai
University, Shanghai 200444, China

###### Abstract

Spin-1 spin-orbit-coupled spinor Bose-Einstein condensates have been realized
in experiment. We study spin-orbit-coupled spinor gap solitons in this
experimentally realizable system with an optical lattice. The spin-dependent
parity symmetry of the spin-orbit coupling plays an important role in the
properties of gap solitons. Two families of solitons with opposite spin-
dependent parity are found. Using an approximate model by replacing the
optical lattice with a Harmonic trap, we demonstrate the physical origin of
the two families. For the zero effective quadratic Zeeman shift, we also find
a type of gap soliton that spontaneously breaks the spin-dependent parity
symmetry.

##  I introduction

In nonlinear phenomena, solitons are notably interesting [1, 2].
Nonlinearities are able to balance the dispersion of spatially localized wave
packets to form solitary waves. Depending on the physical origin and nonlinear
backgrounds, the types of solitons are very diverse [3, 4]. Among them, there
is a particular family, called gap solitons [5, 6]. The existence of gap
solitons requires that there must be energy gaps in the dispersion relation of
corresponding linear systems. Nonlinearities excite gap solitons and exactly
situate them into these linear energy gaps. In general, periodic potentials
provide an important approach to open gaps in a linear dispersion relation.
Therefore, nonlinear periodic systems are widely used to explore gap solitons.
In nonlinear optics, periodic potentials can be afforded by waveguide arrays
and optically induced photonic lattices [7]. In atomic Bose-Einstein
condensates (BECs), optical lattices are a powerful experimental means to
control dispersion relation for the gap opening [8].

Besides their specific location in the linear gaps, gap solitons have a
generic feature: they can be classified as fundamental and higher-order modes
[9]. Higher-order modes can be considered as the composites of fundamental
ones [10, 11, 12]. Different from usual bright solitons that are supported by
only attractive nonlinearities, gap solitons can exist in the presence of
attractive [13, 14] or repulsive [15] nonlinearities. This makes it possible
to experimentally observe gap solitons in atomic BECs with repulsive
interactions [16].

The experimentally tunable optical lattices and interactions make the BECs an
ideal platform to investigate gap solitons [17, 18, 19, 20, 21]. Furthermore,
one of the outstanding properties of the BEC systems is the experimentally
possible implementation of multiple components. Interactions and couplings
[22] between multiple components bring novel properties into gap solitons [23,
24, 25, 26, 27, 28]. Recently, gap solitons in spin-orbit-coupled two-
component BECs loaded into optical lattices have demonstrated interesting
features originating from spin-flip symmetries of the spin-orbit coupling [29,
30, 31, 32]. The study of the spin-orbit-coupled gap solitons is encouraged by
the great achievement and rapid development of experiments on spin-orbit-
coupled BECs [33, 34, 35, 36, 37, 38, 39]. The research interest along the
direction of spin-orbit-coupled gap solitons gradually increases [40, 41, 42,
43, 44, 45, 46, 47, 48].

In this paper, we study the existence and properties of gap solitons in a
spin-1 spin-orbit-coupled spinor BEC in the presence of optical lattices. The
study is stimulated by the following aspects. First, a two-component spin-
orbit-coupled BEC has been successfully loaded into optical lattices in the
experiment [49]. Meanwhile, the spin-orbit coupling has been experimentally
synthesized into a spin-1 spinor BEC [50]. These experimental advancements
inspire people to study the physics of a spin-orbit-coupled spinor BEC with
optical lattices. Second, in comparison with two-component BECs, the spin-1
spinor BECs have more degrees of freedom, and spinor interactions include the
spin-spin collision which allows the spin exchange [51]. Many significant
spin-mixing solitonic states have been experimentally observed in spinor BECs
[52, 53]. Spinor gap solitons have already shown interesting spin structures
[26]. The interplay between the symmetries of the spin-orbit coupling and
spinor interactions is expected to give rise to more distinct spin states.
Finally, the spin-orbit-coupled optical lattices can have an alternative
interpretation by the aid of the synthetic dimension concept [54, 55]. Based
on this idea, spin states become discrete sites along synthetic dimensions.
The coupling between them turns out into the tunneling of the sites. The
connection of the spin-orbit coupled optical lattices and the synthetic
dimension has been experimentally examined [56]. Using the synthetic
dimension, the system of spin-1 spin-orbit-coupled spinor optical lattices can
be mapped into a model of three-leg ladders with a tunable magnetic flux [57,
58, 59]. Such mapping enables experimental accessibility to chiral edge states
by the standard spin-resolved atomic measurements [58, 59]. The study on
spatially localized states in the spin-1 spin-orbit-coupled spinor optical
lattices may shed light on the existence of novel quantum many-body edge
states.

Our study completely bases on the spin-1 spin-orbit-coupled spinor BEC
experiment of ${}^{87}\text{Rb}$ atoms in Ref. [50]. The spin-orbit coupling
is synthesized by Raman lasers. It respects a spin-dependent parity symmetry.
We find that this symmetry plays an important role in the existence of the
spin-orbit-coupled spinor gap solitons and brings interesting features into
them. In the experiment of the spin-1 spin-orbit-coupled spinor BECs, the
tunable parameters are the effective quadratic Zeeman shift and the Raman
coupling. We find that the different effective quadratic Zeeman shift leads to
diverse solitons. For the zero effective quadratic Zeeman shift, the
population in the $|m_{\mathrm{F}}=0\rangle$ component is not preferred, and
gap solitons have two different types; one type occupies the
$|m_{\mathrm{F}}=-1\rangle$ and $|m_{\mathrm{F}}=1\rangle$ components and
obeys the spin-dependent parity symmetry, while the other just occupies the
one of $|m_{\mathrm{F}}=-1\rangle$ and $|m_{\mathrm{F}}=1\rangle$ components
and spontaneously breaks the spin-dependent parity symmetry. For a large
effective quadratic Zeeman shift, we uncover two different families of gap
solitons having opposite spin-dependent parity symmetries. For a further
understanding, we develop an approximating model to provide a physical picture
for the origin of these two families. All solitonic solutions can exist in a
broad range of the Raman coupling and are stable. Their features are
identified.

This paper is organized as follows. In Sec. II, we present the theoretical
model for the spin-1 spin-orbit-coupled spinor BEC with optical lattices. In
Sec. III, the linear properties of the spin-orbit-coupled spinor optical
lattice are studies, with a particular attention on the identification of
linear energy gaps which shall accommodate gap solitons. In Sec. IV, two
different types of spin-orbit-coupled spinor gap solitons are demonstrated in
the case of the zero effective quadratic Zeeman shift. In Sec. V, we
demonstrate the existence of two families of solitons which have opposite
spin-dependent parity symmetries for the case of a large effective quadratic
Zeeman shift. An approximating model is developed to explain the origin of
solitons. Finally, the conclusion including a discussion of “edge state
soliton” follows in Sec. VI.

Figure 1:  Scheme for the realization of the spin-1 spin-orbit-coupled spinor
optical lattice. (a) The spin-orbit coupling is implemented by the
interactions between the atom cloud and three Raman lasers represented by the
arrows in the horizontal direction, the two of them propagate in the same
direction, and the third one in the opposite direction. Two additional far
detuning lasers with a $\pi/2$ incident angle represented by the black arrows
generate the optical lattice. (b) The interactions of three Raman lasers and
the unequal splitting hyperfine states of ${}^{87}\text{Rb}$ atoms. Two sets
of two-photon transitions share the same laser represented by the red lines.

##  II Model

Our system is quasi-one-dimensional. The spin-orbit coupling and the optical
lattice are along the longitudinal direction, and the motions along the
transverse direction are completely frozen due to strong traps. Ref. [50] has
experimentally realized a spin-1 spin-orbit-coupled spinor BEC. We follow the
experimental design. The scheme to realize the spin-1 spin-orbit-coupled
spinor optical lattice is shown in Fig. 1. With a large bias magnetic field,
three hyperfine states of ${}^{87}\text{Rb}$ atoms
($|-1\rangle=|\mathrm{F}=1,m_{\mathrm{F}}=-1\rangle$,
$|0\rangle=|\mathrm{F}=1,m_{\mathrm{F}}=0\rangle$, and
$|1\rangle=|\mathrm{F}=1,m_{\mathrm{F}}=1\rangle$) are split unequally between
$|-1\rangle\leftrightarrow|0\rangle$ and $|0\rangle\leftrightarrow|1\rangle$.
Three external Raman lasers (with the wavelength
$\lambda_{\text{Ram}}=790\text{nm}$) are employed to interact with the atom
cloud. Laser frequencies and polarizations are properly chosen such that the
Raman lasers can couple the hyperfine states together via two sets of two-
phonon transitions [see Fig. 1(b)]. Two of the lasers propagate in same
direction and the third one propagates oppositely. Such arrangement of laser
propagating configuration introduces a spin-momentum locking during the
transitions. The resulted single-particle spin-orbit-coupled Hamiltonian is
[50],

$H_{\mathrm{soc}}=\frac{1}{2m}\left(p_{x}+2\hbar
k_{\mathrm{Ram}}F_{z}\right)^{2}+\delta F_{z}^{2}+\sqrt{2}\bar{\Omega}F_{x}.$
(1)

In above, $p_{x}$ is the momentum along the longitudinal direction, and $m$ is
atom mass. $F_{z}$ and $F_{x}$ are the spin-1 Pauli matrices.
$\left(p_{x}+2\hbar k_{\mathrm{Ram}}F_{z}\right)^{2}/2m=p_{x}^{2}/2m+2\hbar
k_{\mathrm{Ram}}p_{x}F_{z}/m+2\hbar^{2}k_{\mathrm{Ram}}^{2}F_{z}^{2}/m$. The
spin-1 spin-orbit coupling is $2\hbar k_{\mathrm{Ram}}p_{x}F_{z}/m$ with the
strength $2\hbar k_{\mathrm{Ram}}/m$, here
$k_{\mathrm{Ram}}=2\pi/\lambda_{\mathrm{Ram}}$. $\delta$ relates to the two-
photon detuning. In the experiment [50], the detunings for the two sets of
two-photon transitions are adjusted to be equal, which results in $\delta
F_{z}^{2}$. Incorporating $2\hbar^{2}k_{\mathrm{Ram}}^{2}F_{z}^{2}/m$ from the
first term in $H_{\mathrm{soc}}$, the effective quadratic Zeeman shift becomes
$\left(2\hbar^{2}k_{\mathrm{Ram}}^{2}/m+\delta\right)F_{z}^{2}$. The last term
in $H_{\mathrm{soc}}$ is the so-called Raman coupling with the strength
$\bar{\Omega}$.

Like the experiment of the two-component spin-orbit-coupled optical lattices
in Ref. [49], we load such spin-1 spin-orbit-coupled system into an optical
lattice by shining two $\lambda_{\mathrm{lat}}=1540\text{nm}$ lattice lasers.
There is a $\pi/2$ incident angle between these two lasers which are
represented by the black arrows in Fig. 1. The generated optical lattice is
spin-independent and becomes $V(x)=-\bar{V}\cos(2k_{\mathrm{lat}}x)/2$ with
$k_{\mathrm{lat}}=2\pi/(\sqrt{2}\lambda_{\mathrm{lat}})$. $\bar{V}$ is the
lattice depth which can be tuned by changing the intensity of the lattice
lasers. Consequently, we end up with the total single-particle Hamiltonian for
the spin-orbit-coupled spinor optical lattice,

$H_{\mathrm{sin}}=H_{\mathrm{soc}}-\frac{\bar{V}}{2}\cos(2k_{\mathrm{lat}}x).$
(2)

The spin-1 BEC with the spin-orbit-coupled optical lattice is described by the
standard mean-field Gross-Pitaevskii (GP) equations with the above single-
particle Hamiltonian. Since the system has three components, the GP equations
are three coupled nonlinear equations as follows,

$\displaystyle i\frac{\partial\psi_{-1}}{\partial
t}=H_{-}\psi_{-1}+\Omega\psi_{0}+\eta\psi_{0}^{2}\psi_{1}^{*},$ (3)
$\displaystyle i\frac{\partial\psi_{0}}{\partial
t}=H_{0}\psi_{0}+\Omega(\psi_{-1}+\psi_{1})+2\eta\psi_{-1}\psi_{0}^{*}\psi_{+1},$
$\displaystyle i\frac{\partial\psi_{1}}{\partial
t}=H_{+}\psi_{1}+\Omega\psi_{0}+\eta\psi_{-1}^{*}\psi_{0}^{2},$

with,

$\displaystyle H_{\mp}=\mathcal{L}+\epsilon\mp i\gamma\frac{\partial}{\partial
x}-2\eta|\psi_{\pm 1}|^{2},$ $\displaystyle
H_{0}=\mathcal{L}-2\eta|\psi_{0}|^{2},$
$\displaystyle\mathcal{L}=-\frac{1}{2}\frac{\partial^{2}}{\partial
x^{2}}-\frac{V}{2}\cos(2x)+(1+\eta)n.$

In above GP equations, the spinor wave functions are
$\psi(x,t)=[\psi_{-1}(x,t),\psi_{0}(x,t),\psi_{1}(x,t)]^{T}$, which describe
the probability amplitudes of corresponding three hyperfine states
$|-1\rangle,|0\rangle,|1\rangle$. The total density $n$ is defined as
$n=|\psi_{-1}|^{2}+|\psi_{0}|^{2}+|\psi_{1}|^{2}$. For the convenience of
numerical calculations, the above GP equations are dimensionless. The units of
length, energy, and time are $1/k_{\mathrm{lat}}$,
$2E_{\mathrm{lat}}=\hbar^{2}k_{\mathrm{lat}}^{2}/m=2\pi\hbar\times
0.97\text{kHz}$, and $\hbar/(2E_{\mathrm{lat}})$ respectively. The wave
functions are also dimensionless with the unit being
$\sqrt{E_{\mathrm{lat}}/(\hbar\omega_{\perp}c_{0})}$, here $\omega_{\perp}$ is
the harmonic trap frequency along the transverse direction and
$c_{0}=(a_{0}+2a_{2})/3$ with $a_{0}$ and $a_{2}$ being s-wave scattering
lengths in the total spin 0 and 2 channels [60, 61]. With these units and
experimental parameters of lasers ($\lambda_{\text{Ram}}=790\text{nm}$ and
$\lambda_{\mathrm{lat}}=1540\text{nm}$) [50, 49], the dimensionless
quantities, the spin-orbit coupling strength $\gamma$ and the effective
quadratic Zeeman shift $\epsilon$, become

$\displaystyle\gamma=2\frac{k_{\mathrm{Ram}}}{k_{\mathrm{lat}}}=2\sqrt{2}\frac{\lambda_{\mathrm{lat}}}{\lambda_{\mathrm{Ram}}}=5.52,$
$\displaystyle\epsilon=2\frac{k_{\mathrm{Ram}}^{2}}{k_{\mathrm{lat}}^{2}}+\frac{\delta}{2E_{\mathrm{lat}}}=15.2+\frac{\delta}{2E_{\mathrm{lat}}},$
(4)

and $V=\bar{V}/(2E_{\mathrm{lat}})$,
$\Omega=\bar{\Omega}/(2E_{\mathrm{lat}})$, and $\eta=c_{2}/c_{0}$ with
$c_{2}=(a_{2}-a_{0})/3$. For ${}^{87}\text{Rb}$ atoms, $a_{0}=101.8a_{B}$ and
$a_{2}=100.4a_{B}$ with $a_{B}$ being the Bohr radius [62], therefore,

$\eta=\frac{c_{2}}{c_{0}}=\frac{a_{2}-a_{0}}{a_{0}+2a_{2}}=-0.005,$ (5)

which represents a ferromagnetic spin-spin interaction. We characterize the
spin-orbit-coupled spinor gap soliton by its atom number

$N=N_{0}\int dx\left[|\psi_{-1}|^{2}+|\psi_{0}|^{2}+|\psi_{1}|^{2}\right],$
(6)

which is measured with $N_{0}=\hbar k_{\mathrm{lat}}/(2m\omega_{\perp}c_{0})$.
A typical experimental transverse trap frequency
$\omega_{\perp}/(2\pi)=300\text{Hz}$ leads to $N_{0}=105$.

The ground states of a spin-1 spinor BEC are polar states when $\eta>0$ and
ferromagnetic states when $\eta<0$ [61]. Meanwhile, the ground states have an
infinite spin degeneracy; all spinors associated with spin rotations are
degenerate [61]. The spin-1 spinor BEC can support stable bright solitons [63,
64, 65], which may have the spin degeneracy [64]. This may be understood from
the single-mode approximation [66]. Three components share a same spatially
localized profile which is decoupled from the spin degrees of freedom.
Therefore, the spin rotation invariance of the spinor interactions is not
affected under the single-mode approximation. In the presence of optical
lattices, spinor bright solitons exist inside the linear energy gaps,
converting to gap solitons. Ref. [26] has found a family of spin-1 spinor gap
solitons which have the same spatial profile in three components and have the
spin degeneracy. Different from the ground states, the stable polar-like and
ferromagnetic-like spinor gap solitons can exist no matter the sign of $\eta$.
Moreover, Ref. [26] has found the existence of stable gap solitons that do not
satisfy the single-mode approximation.

The coupling between the spin and orbit degrees of freedom leads to that the
single-mode approximation generally cannot apply to spin-orbit-coupled spinor
BECs. The ground states of a spin-1 spin-orbit-coupled spinor BEC have
intriguing phases including stripe, plane-wave and zero-momentum states [67,
68, 69, 70, 71]. Spin textures and spin dynamics [72] of spin-orbit-coupled
spinor bright solitons become important due to the lacking of the spin
degeneracy [73, 74, 75, 76, 77, 78, 79, 80].

Here, we study spin-1 spin-orbit-coupled spinor gap solitons by numerically
solving the Eq. (3). In the experiment [50], the Raman coupling $\Omega$ and
the two-photon detuning $\delta$ are tunable parameters. In our study, we keep
$\Omega$ as a free parameter and choose two typical values for the detuning
$\delta=-30.4E_{\mathrm{lat}}$ and $\delta=0$. This leads to that the
effective quadratic Zeeman term of the GP equations in Eq. (3) and Eq. (4)
becomes $\epsilon=0$ and $\epsilon=15.2$ correspondingly. Due to the locations
of gap solitons, it is important to identify linear energy gaps firstly.

Figure 2:  Linear Bloch spectrum of the spin-1 spin-orbit-coupled spinor
optical lattice and corresponding Bloch waves. The dimensionless parameters
are $\gamma=5.52$ and $\Omega=1$. In (a)-(d), the effective quadratic Zeeman
shift is $\epsilon=0$. (a): The Bloch spectrum as a function of the optical
lattice depth $V$. The shadow areas indicate Bloch energy bands and the white
areas are energy gaps. The solid lines are the maximum and minimum of each
Bloch band. (b): The Bloch spectrum as a function of the quasimomentum $k$ for
a fixed depth $V=3$ [indicated by the vertical dashed line in (a)]. (c) and
(d): The density distributions of the Bloch waves labeled by the solid circles
in (b). The shadow areas indicate the optical lattice regimes with
$-V\cos(2x)/2>0$. ($|\psi_{-1}|^{2},|\psi_{0}|^{2},|\psi_{1}|^{2}$) are
labeled by the red, olive and blue lines respectively. In (c),
$|\psi_{-1}|^{2}=|\psi_{1}|^{2}$, and $|\psi_{0}|^{2}=0$. In (d),
$|\psi_{-1}|^{2}=|\psi_{0}|^{2}=0$. (e)-(h) show same quantities as in
(a)-(d), but the effective quadratic Zeeman shift is $\epsilon=15.2$.

##  III Linear spectrum

The optical lattice is periodic, so the single-particle Hamiltonian
$H_{\mathrm{sin}}$ in Eq. (2) possesses Bloch band gap spectrum. The
corresponding Bloch waves are defined as $\psi(x,t)=\exp(-i\mu t+ikx)\phi(x)$.
Here $\phi(x)=[\phi_{-1}(x),\phi_{0}(x),\phi_{1}(x)]^{T}$ are periodic
functions having the same period as the optical lattice, $k$ is the
quasimomentum, and $\mu$ is the chemical potential. The linear Bloch spectrum
$\mu(k)$ can be calculated by a plane-wave expansion of $\phi(x)$.

Two typical linear Bloch spectra for the effective quadratic Zeeman shift
$\epsilon=0$ and $\epsilon=15.2$ are shown in Fig. 2. For the case of
$\epsilon=0$ in Fig. 2(a), the lowest two bands mix together and there is an
energy gap between the second and third bands. The size of this gap increases
as the increase of the optical lattice depth. The lowest four bands of the
linear spectrum at $V=3$ are shown in Fig. 2(b) as a function of the
quasimomentum. Clearly the gap between the second and third bands is big
enough in comparison with the band widths of the lowest two bands. It may
provide an accommodation to support spinor gap solitons. The density
distributions of the Bloch waves at $k=0$ and at the maximum of the second
band [labeled by the solid circles in Fig. 2(b)] are demonstrated in Figs.
2(c) and 2(d) respectively. It is interesting to see that these Bloch waves
have specific spin populations, i.e., $|\psi_{-1}|^{2}=|\psi_{1}|^{2}$ and
$|\psi_{0}|^{2}=0$ in Fig. 2(c), and $|\psi_{-1}|^{2}=|\psi_{0}|^{2}=0$ in
Fig. 2(d).

We depict the linear spectrum for a large effective quadratic Zeeman shift
$\epsilon=15.2$ in Figs. 2(e)-(h). As shown in Fig. 2(e), the lowest three
bands mix together and there is no energy gap between them when the depth $V$
is small. For a sufficient large $V$, gaps are weakly opened between the
lowest three bands. There always is an energy gap opening between the third
and fourth bands. Its size also increases as a function of $V$. Therefore, for
the existence of spinor gap solitons in this gap, it is reasonable to choose a
large $V$. Here we choose $V=3$. The linear spectrum as a function of the
quasimomentum for $V=3$ is demonstrated in Fig. 2(f). As expected, the energy
gap between the third and fourth bands is a proper accommodation for spinor
gap solitons. Furthermore, two typical Bloch waves are described in Figs. 2(g)
and 2(h). They are the Bloch waves at the Brillouin zone center and edge in
the third band. The outstanding feature of these Bloch waves is spin
occupations; $|\psi_{0}|^{2}\ll|\psi_{-1}|^{2}=|\psi_{1}|^{2}$ in Fig. 2(g),
and $|\psi_{0}|^{2}>|\psi_{-1}|^{2}=|\psi_{1}|^{2}$ in Fig. 2(h).

Once the energy gaps are identified, we numerically find gap solitons located
inside them. In the following, we shall focus on the energy gaps shown in
Figs. 2(b) and 2(f). The profiles of linear Bloch waves shown in Fig. 2 are
instructive for the structures of gap solitons since they may bifurcate from
these linear waves.

Figure 3: Two types of spin-1 spin-orbit-coupled spinor gap solitons in the
case of the zero effective quadratic Zeeman shift $\epsilon=0$. The other
parameters are the optical lattice depth $V=3$ and the Raman coupling
$\Omega=1$. (a) The existence of spinor gap solitons is reflected by the
dependence of the solitonic atom number $N$ on the chemical potential $\mu$.
Two types are degenerate in the plane of ($N,\mu$). The shadow areas
correspond to linear bands. The density profiles of the labeled point are
shown in (b)-(d). (b) A density profile of the first type,
$|\psi_{-1}|^{2}=|\psi_{1}|^{2}\neq 0$ and $|\psi_{0}|^{2}=0$, which is polar-
like. The inset zooms in the density oscillating tail. (c) Density profile of
the second type, $|\psi_{-1}|^{2}\neq 0$ (red line) and
$|\psi_{0}|^{2}=|\psi_{1}|^{2}=0$. (d) Density profile of the second type,
$|\psi_{-1}|^{2}=|\psi_{0}|^{2}=0$ and $|\psi_{1}|^{2}\neq 0$ (Blue dashed
line). In (b)-(d), the red, olive, and blue-dashed lines correspond to
$|\psi_{-1}|^{2}$, $|\psi_{0}|^{2}$ and $|\psi_{1}|^{2}$, respectively. The
shadow areas indicate the regimes for the lattices $-V\cos(2x)/2>0$. (e)-(g)
The momentum-space distributions corresponding to (b)-(d). The $|-1\rangle$
($|1\rangle$) component has a momentum peak centered at $k=-\gamma$
($\gamma$). (h) The stable time evolution of a gap soliton. The initial state
is $\psi(1+0.1\mathrm{Rand})$ with $\psi$ being the first-typed gap soliton
labeled by the dot in (a) and $10\%$ Gaussian distributed random noise is
considered.

##  IV Spin-orbit-coupled spinor gap soliton with the zero effective quadratic
Zeeman shift

Spin-orbit-coupled spinor gap solitons are stationary solutions of the Eq.
(3). $\psi(x,t)=\exp(-i\mu t)\psi(x)$ with the chemical potential $\mu$. Then,
$\psi(x)$ satisfy the stationary GP equations. We use the Newton relaxation
method to solve the spatially discretized stationary GP equations. The
chemical potential is fixed to be the values inside the linear energy gaps
during the calculations. We first consider the case of the zero effective
quadratic Zeeman shift $\epsilon=0$. The solitons will locate inside the
linear energy gap demonstrated in Fig. 2(b).

The results of gap solitons are shown in Fig. 3. We find that the fundamental
solitons have two different types. The dependence of solitonic atom number $N$
[defined in Eq. (6)] on the chemical potential $\mu$ is shown in Fig. 3(a).
Two types are completely degenerate in the ($N,\mu$) plane. From this figure,
we can know their existence; they exist inside the linear energy gap except a
very small regime close to the second band where the wave functions go to
zero. Without the spin-orbit coupling, the wave functions of spinor gap
solitons can be scaled into real numbers [26]. However, the presence of the
spin-orbit coupling makes the scaling impossible, and the wave functions are
in general complex numbers. The first type is
$|\psi_{-1}|^{2}=|\psi_{1}|^{2}\neq 0$ and $|\psi_{0}|^{2}=0$, whose profile
is demonstrated in Fig. 3(b). The second type has two configurations:
$|\psi_{-1}|^{2}\neq 0$ and $|\psi_{0}|^{2}=|\psi_{1}|^{2}=0$ as shown in Fig.
3(c), and $|\psi_{-1}|^{2}=|\psi_{0}|^{2}=0$ and $|\psi_{1}|^{2}\neq 0$ as
shown in Fig. 3(d). Since the degeneracy of these two types, the density
amplitude of the second type is double that of the first. This is because that
the second type has only one-component occupation, while the first type
includes two components. No matter which type the soliton is, the common
feature is the zero population of the $|0\rangle$ component,
$|\psi_{0}|^{2}=0$.

In the following, we provide a physical insight for these numerical solutions
of Eq. (3). The effective quadratic Zeeman shift consists of two parts: one is
due to the spin-dependent momentum displacement
$2\hbar^{2}k_{\mathrm{Ram}}^{2}/m$ and the other is the two-photon detuning
$\delta$. $\epsilon=0$ means $\delta=-30.4E_{\mathrm{lat}}$, which is a large
negative value. The momentum displacement does not affect the energy of
system. The energy functional of the $\delta$ term is,
$E_{\text{qua}}=\delta(|\psi_{-1}|^{2}+|\psi_{1}|^{2})$. Since $\delta\ll 0$,
minimizing this energy functional requires $\psi_{0}=0$. Physically, when the
two-photon detuning $\delta$ is a large negative value we can adiabatically
eliminate the $|0\rangle$ state. Under the condition of $\psi_{0}=0$ and
$\epsilon=0$, the stationary GP equations in Eq. (3) are simplified as,

$\displaystyle\mu\psi_{-1}=\mathcal{L^{\prime}}\psi_{-1}-i\gamma\frac{\partial\psi_{-1}}{\partial
x}+\eta(|\psi_{-1}|^{2}-|\psi_{1}|^{2})\psi_{-1},$
$\displaystyle\mu\psi_{1}=\mathcal{L^{\prime}}\psi_{1}+i\gamma\frac{\partial\psi_{1}}{\partial
x}-\eta(|\psi_{-1}|^{2}-|\psi_{1}|^{2})\psi_{1},$ (7)

with

$\mathcal{L^{\prime}}=-\frac{1}{2}\frac{\partial^{2}}{\partial
x^{2}}-\frac{V}{2}\cos(2x)+(|\psi_{-1}|^{2}+|\psi_{1}|^{2}).$ (8)

Note that the Raman coupling $\Omega$ disappears in the above reduced
equations due to $\psi_{0}=0$. In the absence of $\Omega$, the spin-orbit
coupling can be gauged out from the equations. After applying a unitary
transformation $\psi_{-1}=\exp(-i\gamma x)\phi_{-1}$ and
$\psi_{1}=\exp(i\gamma x)\phi_{1}$, the above equations become,

$\displaystyle\mu\phi_{-1}=\mathcal{L^{\prime}}\phi_{-1}+\eta(|\phi_{-1}|^{2}-|\phi_{1}|^{2})\phi_{-1},$
$\displaystyle\mu\phi_{1}=\mathcal{L^{\prime}}\phi_{1}-\eta(|\phi_{-1}|^{2}-|\phi_{1}|^{2})\phi_{1}.$
(9)

An irrelevant constant $\gamma^{2}$ has been dropped off. The energy
functional of the nonlinear terms in the above equations is

$E_{\text{non}}=\frac{1}{2}(1+\eta)(|\phi_{-1}|^{4}+|\phi_{1}|^{4})+(1-\eta)|\phi_{-1}|^{2}|\phi_{1}|^{2}.$
(10)

Since $\eta=-0.005<0$, the above interactions are immiscible. In the absence
of the optical lattice, to minimize the immiscible interactions, the ground
state has two configurations: $\phi_{-1}\neq 0$, $\phi_{1}=0$ and
$\phi_{-1}=0$, $\phi_{1}\neq 0$. The ground state chooses the one of two
configurations spontaneously. Therefore, considering the zero $|0\rangle$
component $\psi_{0}=0$, the ground state of the spin-orbit coupled spinor BEC
without the optical lattice is ferromagnetic and is a plane wave as
$\psi_{-1}=\exp(-i\gamma x)\phi_{-1}$, $\psi_{1}=\exp(i\gamma x)\phi_{1}$ for
the situation of the zero effective quadratic Zeeman shift. This result is
consistent with that in Ref. [50].

In the presence of the optical lattice, gap soliton solutions of the Eq. (9)
can be numerically found as $\phi_{-1}^{(gs)}(x)$ and $\phi_{1}^{(gs)}(x)$.
Since all quantities in Eq. (9) are real-valued, the wave functions of gap
solitons $\phi_{-1}^{(gs)}$ and $\phi_{1}^{(gs)}$ can be scaled into real
numbers. On the other hand, the conservation of the total density
$|\phi_{-1}|^{2}+|\phi_{1}|^{2}$ and the relative density
$|\phi_{-1}|^{2}-|\phi_{1}|^{2}$ in the time-dependent version of Eq. (9)
brings two free degrees of freedoms into the gap soliton solutions; they are
the global phase $\alpha$ and relative phase $\theta$. Finally, going back to
the basis of $\psi$, we get the solutions of gap solitons as

$\begin{pmatrix}\psi_{-1}\\\ \psi_{0}\\\
\psi_{1}\end{pmatrix}_{(gs)}=\begin{pmatrix}e^{-i\gamma
x}\phi_{-1}^{(gs)}(x)\\\ 0\\\ e^{i\gamma
x}\phi_{1}^{(gs)}(x)e^{i\theta}\end{pmatrix}e^{i\alpha}.$ (11)

The numerical results demonstrated in Fig. 3 belongs to the solutions in Eq.
(11). In order to show this, we plot the momentum-space density distributions
of corresponding gap solitons in Figs. 3(e)-(g). The $|-1\rangle$ component of
solitons always has a momentum peak locating at $k=-\gamma$, while it sits at
$k=\gamma$ in the $|1\rangle$ component. Results of momentum peaks are
consistent with solutions in Eq. (11). It is noticed that there is a small
amplitude splitting in the each peak. The splitting originates from the
existence of very small oscillating tails in gap solitons which are shown in
the zooming-in inset in Fig. 3(b).

The spin-orbit-coupled Hamiltonian in Eq. (1) obeys a spin-dependent parity
symmetry $\mathcal{O}$,

$\mathcal{O}=-\mathcal{P}e^{i\pi F_{x}}=\mathcal{P}\begin{pmatrix}0&0&1\\\
0&1&0\\\ 1&0&0\end{pmatrix}.$ (12)

Here, $\mathcal{P}$ is the parity operator, $\mathcal{P}x\mathcal{P}^{-1}=-x$,
and $e^{i\pi F_{x}}$ is the operator to rotate spins along the $F_{x}$ by an
angle of $\pi$. The first type of spin-orbit-coupled spinor gap solitons
satisfies this symmetry, which gives rise to $\psi_{-1}(x)=\pm\psi_{1}(-x)$.
This restriction in the end leads to $\phi_{-1}^{(gs)}(x)=\phi_{1}^{(gs)}(x)$
and $\theta$ is fixed to $0$ or $\pi$. While the global phase in the Eq. (11)
still cannot be fixed and $\alpha$ can be arbitrary real constants since the
GP equations have the U(1) symmetry. The second type wants the one of
$\phi_{-1}^{(gs)}$ and $\phi_{1}^{(gs)}$ to disappear. Therefore, it does not
obey the $\mathcal{O}$ symmetry.

Finally, we calculate the spin average value for the gap solitons. It is
defined as,

$\langle\mathbf{F}\rangle=\int dx\psi^{T*}(x)\mathbf{F}\psi(x).$ (13)

The first type satisfies $\langle\mathbf{F}\rangle=0$, which is reminiscent of
the spinor polar ground state in the case of $\eta>0$ [61]. However, for spin-
orbit-coupled spinor gap solitons, the first type is polar-like but can exist
when $\eta<0$. While, the second type is ferromagnetic-like, $|\langle
F_{z}\rangle|\neq 0$ and $\langle F_{x}\rangle=\langle F_{y}\rangle=0$.

The stability of gap solitons found in Fig. 3(a) is examined by the nonlinear
evolution of the GP equations. The nonlinear evolution is implemented by using
the initial states as $\psi(1+0.1\mathrm{Rand})$ with $\psi$ being gap
solitons and $\mathrm{Rand}$ being the randomly distributed noise. The stable
solutions shall evolve without changing density profiles. The results are that
all gap soliton solutions in Fig. 3(a) are stable except for a very small
regime that the chemical potential is close to the second band. In this
regime, both types of solitons have a weak oscillation. In Fig. 3(h), we
demonstrate a typically stable evolution of the first-typed soliton [which is
labeled by the dot in Fig. 3(a)]. It shows that the corresponding soliton
evolves stably up to $t=1000$ [which corresponds to 165ms considering the time
unit] in the presence of noise.

Figure 4: Two types of spin-1 spin-orbit-coupled spinor gap solitons with the
zero effective quadratic effect $\epsilon=0$ can exist in a broad range of the
Raman coupling $\Omega$. The optical lattice depth is $V=3$. (a) Linear Bloch
spectrum as a function of $\Omega$. The shadow areas represent energy bands.
(b) The dependence of the solitonic atom number on $\Omega$ for a fixed
chemical potential [labeled by the horizontal line in (a)]. The two types are
completely degenerate in the ($N,\Omega$) plane. The coordinate-space and
momentum-space density distributions of the first-typed gap solitons [labeled
by the dot in (b)] are demonstrated in (c) and (d) respectively. The shadow
areas in (c) represent $-V\cos(2x)/2>0$. (e) The stable time evolution of the
first-typed gap soliton [labeled by the dot in (b)] with $10\%$ random noise.

These polar-like and ferromagnetic-like types can exist for a broad range of
the Raman coupling $\Omega$ as long as the linear energy gap keeps opening.
The evolution of linear spectrum as a function of $\Omega$ is demonstrated in
Fig. 4(a). It shows that the energy gap between the second and third bands is
widely opened as the change of $\Omega$. We fix the chemical potential and
numerically search for gap solitons along the horizontal line shown in Figs.
4(a). The results are described in Fig. 4(b). Two types are still degenerate
in the $(N,\Omega)$ plane. The solitonic atom number $N$ increases as the
increase of $\Omega$. When $\Omega$ is small, the two-photon detuning $\delta$
completely dominates leading to the zero occupation in the $|0\rangle$
component. Since the Raman coupling works via the $|0\rangle$ component,
without which, it has no effect on solitons. Therefore, the number $N$ keeps
as a constant for small $\Omega$. However, as it increases, the Raman coupling
manages to flip spins to allow for the small occupation in the $|0\rangle$
component. This causes to the increase of solitonic atom number with the
increase of the Raman coupling as shown in Fig. 4(b). The typical coordinate-
space and momentum-space density profiles of the first-typed spinor gap
soliton at a large Raman coupling $\Omega=4$ are shown in Fig. 4(c) and 4(d).
This soliton looks like the previous result of $\Omega=1$ in Fig. 3. However,
the difference between them is that there is a small occupation in the
$|0\rangle$ component [see the olive line in Fig. 4(c)]. The Raman coupling
transfers a small amount of atoms to the $|0\rangle$ component from the
$|-1\rangle$ and $|1\rangle$ components, and during the transfer momentum is
conserved. Consequently, the $|0\rangle$ component becomes a superposition of
two very small packets locating at $-\gamma$ and $\gamma$ in the momentum
space.

The stability of the whole branch in Fig. 4 is also checked by the nonlinear
evolution. All solitons are stable. We present a typically stable evolution of
the first-typed solution [represented by the dot in Fig. 4(b)] in Fig. 4(e).
The soliton can exist stably for a very long time.

##  V Spin-orbit-coupled spinor gap solitons with a large effective quadratic
Zeeman shift

In above, due to a very large negative $\delta$, the occupation of the
$|0\rangle$ component is almost negligible, $|\psi_{0}|^{2}=0$. The first type
of spinor gap solitons has $|\psi_{-1}|^{2}=|\psi_{1}|^{2}$ obeying the spin-
dependent parity symmetry $\mathcal{O}$ . The second type spontaneously
chooses to occupy the one of $|\psi_{-1}|^{2}$ and $|\psi_{1}|^{2}$,
therefore, is ferromagnetic-like. This type breaks the spin-dependent parity
symmetry. In the following, we consider $\delta=0$, which leads to a large
effective quadratic Zeeman shift $\epsilon=15.2$ in the GP equations. There is
no constraint on the occupation of the $|0\rangle$ component. The gap solitons
are searched inside the energy gap shown in Fig. 2(f).

We find that there are two different families for the fundamental gap
solitons. They are demonstrated in Fig. 5. The two families have different
atom numbers in the ($N,\mu$) plane; the first family has a slightly larger
number [see the dotted line in Fig. 5(a)]. Both families exist in whole energy
gap, excepting for a regime very close to the third linear band. The typical
density profiles of the first family are shown in Figs. 5(b) and 5(d) in the
coordinate and momentum space respectively. The first family features a
dominating occupation of the $|0\rangle$ component, and the $|-1\rangle$ and
$|1\rangle$ components have the same density with a very small population,
$|\psi_{0}|^{2}\gg|\psi_{-1}|^{2}=|\psi_{1}|^{2}$. The momentum of the
$|0\rangle$ component lays at $k=0$. This can be seen from the momentum-
density peak in Fig. 5(d) [noting that the small amplitude splitting still
exists due to the tail of solitons]. The density profiles of the second family
are described in Figs. 5(c) and 5(e). This family features
$|\psi_{-1}|^{2}=|\psi_{1}|^{2}\gg|\psi_{0}|^{2}$. The momentum of the
$|-1\rangle$ component lays at $k=-\gamma$, and the $|1\rangle$ component is
peaked around $k=\gamma$. It is very interesting to note that both families
respect the spin-dependent parity symmetry $\mathcal{O}$. The symmetry wants
to wave functions to satisfy $\mathcal{O}\psi(x)=\pm\psi(x)$ with the
eigenvalues being $\pm 1$, leading to

$\psi_{-1}(x)=\pm\psi_{1}(-x),\ \psi_{0}(x)=\pm\psi_{0}(-x).$ (14)

This result further gives rise to $\langle F_{y}\rangle=\langle
F_{z}\rangle=0$. The first family belongs to the eigenstate with $+1$
eigenvalue, i.e., even spin-dependent parity, this can be inferred from that
there is no node in the density of the $|0\rangle$ component as shown by the
olive line in Fig. 5(b). While the second family is odd spin-dependent parity
with the eigenvalue $-1$, as there is a node at $x=0$ in the density of the
$|0\rangle$ component [see the olive line in the inset in Fig. 5(c)].

Figure 5:  Two different families of spin-1 spin-orbit-coupled spinor gap
solitons with a large effective quadratic Zeeman shift $\epsilon=15.2$ [i.e.,
$\delta=0$]. Other parameters are $V=3$ and $\Omega=1$. (a) The existence of
two families in the ($N,\mu$) plane. The first (second) family is represented
by the dotted (solid) line. The wave functions of the labeled points shall be
shown in (b) and (c). The shadow areas correspond to linear bands. (b) A
density profile of the first family in the coordinate space,
$|\psi_{-1}|^{2}=|\psi_{1}|^{2}\ll|\psi_{0}|^{2}$. (c) A density profile of
the second family in the coordinate space,
$|\psi_{-1}|^{2}=|\psi_{1}|^{2}\gg|\psi_{0}|^{2}$. In (b) and (c), the red,
olive, and blue-dashed lines correspond to $|\psi_{-1}|^{2}$, $|\psi_{0}|^{2}$
and $|\psi_{1}|^{2}$, respectively. The shadow areas indicate the regimes for
the lattices $-V\cos(2x)/2>0$. In (c), the inset zooms in the density
distributions around $x=0$. (d) and (e) The momentum-space-density
distributions correspond to (b)-(c).  Figure 6:  (a) The ground state and
first excited state of $H^{\prime}_{\text{sin}}$ in Eq. (16). The wave
functions of labeled points shall be shown in the following plots. (b) The
spectrum of $H_{\text{soc}}$ in the momentum space. Three local minima in the
lowest band are labeled by circles. (c) and (d) The density distributions of a
ground state in the coordinate and momentum space respectively. The olive line
is $|0\rangle$ component, and the red (blue-dashed) line is the $|-1\rangle$
($|1\rangle$) component. (e) and (f) The density distributions of a first
excited state in the coordinate and momentum space. In (e), the inset shows
the zooming-in of the density distribution of the $|0\rangle$ component around
$x=0$.

The spin-dependent parity symmetry of the two families indicates their origin.
We note that densities of these two fundamental families are mainly confined
inside a unit cell of the optical lattice. To give a simple physical picture
for these two families, we develop an approximate model to replace the optical
lattice by a Harmonic trap. With the approximation, the single-particle
Hamiltonian becomes,

$H^{\prime}_{\text{sin}}=H_{\text{soc}}+\frac{1}{2}m\omega^{2}x^{2},$ (15)

where, $\omega$ is the Harmonic trap frequency. We use the Harmonic oscillator
basis to diagonalize $H^{\prime}_{\text{sin}}$ by introducing
$x=\sqrt{\hbar/(2m\omega)}(a^{\dagger}+a)$ and $p_{x}=i\sqrt{\hbar\omega
m/2}(a^{\dagger}-a)$, here $a$ and $a^{\dagger}$ are the annihilation and
creation operators. Consequently, the single-particle Hamiltonian becomes,

$\displaystyle\frac{H^{\prime}_{\text{sin}}}{\hbar\omega}=a^{\dagger}a+i\frac{\gamma^{\prime}}{\sqrt{2}}(a^{\dagger}-a)F_{z}+\frac{\gamma^{\prime
2}}{2}F_{z}^{2}+\sqrt{2}\Omega^{\prime}F_{x}.$ (16)

We have renormalized the energy by using the energy unit as $\hbar\omega$. The
spin-orbit coupling coefficient is
$\gamma^{\prime}=2k_{\text{Ram}}\sqrt{\hbar/(m\omega)}$ and
$\Omega^{\prime}=\bar{\Omega}/(\hbar\omega)$. Using the Harmonic trap
$\omega/(2\pi)=1$kHz, we have $\gamma^{\prime}=5.4$. Diagonalizing the
Hamiltonian in Eq. (16) gives rise to the energy and associated eigenstates.
The energies of ground state and the first excited state are shown in Fig.
6(a). When $\Omega^{\prime}=0$, these two states are degenerate. They decrease
as the increase of $\Omega^{\prime}$. The wave functions of the ground state
and the first excited state are demonstrated in Figs. 6(c),(d) and 6(e),(f)
respectively. It is well-known that the eigenstates of a Harmonic trapped
system have the well-defined parity symmetry and neighboring two states have
opposite signs of parity. In the presence of the spin-orbit coupling, the
symmetry becomes the spin-dependent parity $\mathcal{O}$. The spin-dependent
parity of the ground state is even and it is odd for the first excited state.
This can be inferred from the density distributions of the $|0\rangle$
component in Figs. 6(c) and 6(e); for the ground state, there is no node in
the density of the $|0\rangle$ component, while for the first excited state
there is a node at $x=0$ in the $|0\rangle$ component [see the olive lines in
Fig. 6(c) and in the inset in Fig. 6(e)]. The ground state has only a
population in the $|0\rangle$ component. While the first excited state has an
equal occupation of the $|-1\rangle$ and $|1\rangle$ components and has a
little occupation in the $|0\rangle$ component.

Figure 7: Two families of fundamental spin-orbit-coupled spinor gap solitons
with a large effective quadratic effect $\epsilon=15.2$ can exist in a broad
range of the Raman coupling $\Omega$. The optical lattice depth is $V=3$. (a)
Linear Bloch spectrum as a function of $\Omega$. The shadow areas represent
energy bands. We shall search for gap solitons along the horizontal line. (b)
The dependence of the solitonic atom number on $\Omega$ for a fixed chemical
potential [labeled by the horizontal line in (a)]. The dotted (solid) line is
for the first (second) family. The coordinate-space and momentum-space density
distributions of two families [labeled by the dots in (b)] are demonstrated in
(c), (e) and (d),(f) respectively. The shadow areas in (c) and (d) represent
$-V\cos(2x)/2>0$.

Without the Harmonic trap, the Hamiltonian $H_{\text{soc}}$ can be directly
diagonalized in the momentum space. The spectrum of $H_{\text{soc}}$ has three
energy bands, as shown in Fig. 6(b). The lowest band features a three-
parabolic-like structure with three local minima which are labeled by circles
in the figure. The right and left minima are degenerate and their energy is a
bit higher than the middle one. The ground state of $H^{\prime}_{\text{sin}}$
comes from the occupation of the middle minimum. Since the middle parabola is
mainly generated by the $|0\rangle$ component, the ground state possesses the
dominating population in the $|0\rangle$ component and its momentum is peaked
at $k=0$, which is demonstrated in Fig. 6(d). The first excited state of
$H^{\prime}_{\text{sin}}$ originates from an equal superposition of the right
and left minima. In order to take the odd spin-dependent parity symmetry, the
superposition should be out of phase. As the right (left) parabola is
generated by the $|-1\rangle$ ($|1\rangle$) component, the first excited state
becomes
$\propto\exp(-ik_{\min}x)(\psi_{-1},0,0)^{T}-\exp(ik_{\min}x)(0,0,\psi_{1})^{T}$,
where $k_{\min}=\gamma^{\prime}$ is the location of the left minimum.
Therefore, in the momentum space, the first excited state features two
momentum peaks locating at $\pm\gamma^{\prime}$, which are demonstrated in
Fig. 6(f).

The first (second) family of spin-orbit-coupled spinor gap solitons is
strongly reminiscent of the ground state (first excited state) of the Harmonic
trapped system. For a fixed atom number, the chemical potential of the first
family should be smaller than the second family. This causes that the first
family lays above the second one in the $(N,\mu)$ plane in Fig. 5(a). All
features of two families, such as the spin-dependent parity symmetry, spin
composition and momentum distribution, are the same as those of the ground
state and first excited state. Therefore, the features are single particle
effect and are induced solely by the spin-orbit coupling. While the spinor
interactions guarantee the existence of gap solitons.

By examining the stability of these two families, we find that the whole
branch of the second family in Fig. 5(a) is dynamically stable. While the
first family is also stable, but except for a very small regime close to the
third band where they are weakly unstable.

The two families with the opposite spin-dependent parity can exist and are
stable in a wide range of the Raman coupling $\Omega$. The linear Bloch
spectrum as a function of $\Omega$ is shown in Fig. 7(a). It demonstrates that
the size of the energy gap between the third and fourth bands decreases as the
increasing of $\Omega$. Therefore, the existence of gap solitons in this gap
is not preferred in the regime of large $\Omega$. We find their existence in
the regime of $0\leq\Omega\leq 2.5$ for a fixed chemical potential as $\mu=0$.
The dependence of the solitonic atom number $N$ on $\Omega$ is demonstrated in
Fig. 7(b). The first family is always higher than the second one. As $\Omega$
increases, the difference of the atom number between two families becomes
large. This can be understood from the approximate model of the Harmonic trap.
The energy of the ground state decreases more dramatically than the first
excited state as the increase of $\Omega$, which is described in Fig. 6(a).
For the gap solitons in Fig. 7(b), the chemical potential of two families is
fixed to be equal. The atom number must be adjusted to generate the same
effect as the energy in the Harmonic trapped case. Typical coordinate-space
and momentum-space densities at $\Omega=2$ are shown in Figs. 7(c),(e) for the
first family and in Figs. 7(d),(f) for the second one. All features of
densities are the same as these shown in Figs. 5(b)-(e).

##  VI conclusion

Stimulated by the experimental realization of the spin-1 spin-orbit-coupled
BEC in Ref. [50], we study the spin-orbit-coupled spinor gap solitons in this
system. They can exist in the wide energy gaps of corresponding linear system.
The spinor interactions possess the spin rotation symmetry. While the spin-
orbit coupling breaks this symmetry by respecting a particular spin-dependent
parity symmetry which is a specific spin rotation accompanying with an orbital
parity. In the case of a large effective quadratic Zeeman shift $\epsilon$, we
uncover two different families of gap solitons having opposite spin-dependent
parity. By developing a spin-orbit-coupled model with a Harmonic trap, these
two families of gap solitons can be considered as the ground state and first
excited state of the Harmonic trapped system. The spin-orbit coupling brings
intriguing features, including spin compositions and momentum distributions,
to the two families. For the zero effective quadratic Zeeman shift, the two-
photon detuning $\delta$ dominates, which prevents the population in the
$|0\rangle$ component. In this case, we reveal that spin-orbit-coupled spinor
gap solitons have two types; one type obeys the spin-dependent parity symmetry
and is polar-like, and the other type spontaneously breaks the symmetry and is
ferromagnetic-like. All gap solitons in our study are fundamental modes whose
main density peaks confines inside a unit cell of the optical lattice. Using
the fundamental modes as building blocks, higher-order modes with rich spatial
structures are expected to be constructed.

In our parameter regimes, there always exists a kind of gap solitons. They
occupy the $|-1\rangle$ and $|1\rangle$ components and have a negligible
population in the $|0\rangle$ component. Using the idea of synthetic
dimension, the $|-1\rangle$ and $|1\rangle$ components are the two edges of
three-leg ladders. Therefore, this kind of solitons belongs to edge states.
However, different from usual edge states which are extended waves along the
edges, due to solitary property, these states are also spatially localized
along the edges. Physically, the spin-orbit coupling induces a nonzero
magnetic flux to the three-leg ladders. The magnetic flux guarantees the
existence of the edge states which can have a dispersion along the edges.
While, the nonlinearities can be used to balance the dispersion to generate
solitons along the edges. It is of interest that for the zero effective
quadratic Zeeman shift, such “edge state solitons” can spontaneously choose to
occupy only the one of two edges. Our study fully relies on the experimental
system and all parameters are within the experimental accessibility. These
make it possible to observe “edge state solitons” experimentally in the
future.

##  VII Acknowledges

This work was supported by National Natural Science Foundation of China with
Grants No.11974235 and 11774219.

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