# Quantum phases in spin-orbit-coupled Floquet spinor Bose gases

Yani Zhang  Department of Physics, Shanghai University, Shanghai 200444, China
Yuanyuan Chen<EMAIL_ADDRESS>Department of Physics, Shanghai University,
Shanghai 200444, China  Hao Lyu<EMAIL_ADDRESS>Quantum Systems Unit,
Okinawa Institute of Science and Technology Graduate University, Onna, Okinawa
904-0495, Japan  Yongping Zhang<EMAIL_ADDRESS>Department of
Physics, Shanghai University, Shanghai 200444, China

###### Abstract

We propose a spin-orbit-coupled Floquet spinor Bose-Einstein condensate (BEC)
which can be implemented by Floquet engineering of a quadratic Zeeman field.
The Floquet spinor BEC has a Bessel-function-modulated Rabi frequency and a
Floquet-induced spin-exchange interaction. The quantum phase diagram of the
spin-orbit-coupled Floquet spinor BEC is investigated by considering
antiferromagnetic or ferromagnetic spin-spin interactions. In comparison with
the usual spin-orbit-coupled spin-1 BEC, we find that a stripe phase for
antiferromagnetic interactions can exist in a large quadratic Zeeman field
regime, and a different stripe phase with an experimentally favorable contrast
for ferromagnetic interactions is uncovered.

##  I introduction

Ultracold neutral atoms provide a fertile playground for engineering
artificial gauge fields [1, 2, 3, 4]. Synthetic spin-orbit coupling, utilizing
atomic hyperfine levels as pseudo-spins, can be realized by coupling these
states via Raman lasers [5, 6, 7]. Spin-orbit-coupled Bose-Einstein
condensates (BECs) open a new route to explore exotic superfluid states and
simulate topological matter [8, 9, 10, 11, 12, 13]. One interesting feature is
that the spin-orbit coupling modifies the dispersion relation of a BEC. The
spin-orbit-coupled dispersion may have multiple energy minima. Condensations
into these energy minima present exotic quantum phases, such as the plane-wave
(PW) phase and stripe phase [14, 15, 16, 17, 18, 19]. The PW phase occupies
one of the minima and possesses a nonzero quasimomentum, which breaks the
time-reversal symmetry [15]. The phase transition and excitations of PW states
have been experimentally observed [8, 20]. The stripe phase, condensing at
least two minima, represents a combination of spatial density modulation and
superfluidity and is identified as having supersolid properties [21]. The
realization of the stripe phase requires miscibility of the two spin
components and a low Rabi frequency of the Raman lasers [19, 22]. This is
quite a challenge in 87Rb atoms experiments since atomic interactions are
insensitive to the hyperfine states [23, 24, 25]. Recently, the spin-orbit-
coupling-induced stripe phase was observed in atoms loaded into superlattices
[26], in which the sub-lattice sites are treated as pseudo-spins.

A spinor BEC has more degrees of freedom and intriguing interactions which
lead to a rich ground-state phase diagram [27]. A spin-orbit-coupled spin-1
BEC has been experimentally realized [28]. Quantum phases in spin-orbit-
coupled spin-1 BECs depend on antiferromagnetic and ferromagnetic spin-spin
interactions and show salient features [29, 30, 31, 32, 33]. Three different
kinds of stripe phases have been revealed to exist, and phase transitions
between different phases are so abundant that tricritical points emerge [31,
32]. One of outstanding features is that the quadratic Zeeman field plays an
important role in the existence of stripe phases. Especially, in a
ferromagnetic spinor BEC, stripes appear in the limited regime of low Rabi
frequency and quadratic Zeeman field [28].

On the other hand, Floquet engineering is a powerful tool in quantum physics
for controlling system parameters and manipulating quantum states [34, 35,
36]. In a periodically driven system, an effective static Hamiltonian can be
tailored which depends on the driving parameters. The driving could lead to
dramatic changes in the system properties. Ultracold atoms provide an ideal
platform for Floquet engineering due to the tunability and purity of the
system, which has been used to explore artificial gauge fields, topological
insulators, and soliton dynamics [37, 38, 39, 40, 41, 42, 43, 44, 45]. In
spin-orbit-coupled ultracold atoms, a coherently periodic modulation of Raman
laser intensities can produce a tunable spin-orbit coupling strength [46, 47,
48], which provides a practical way for dynamical control. A periodic
modulation of Raman laser frequencies is employed to manipulate the emergence
of the Dirac point in Floquet bands and thus to change band topology [49]. A
shaking Raman lattice that generates high-dimensional spin-orbit coupling is
implemented to tune Floquet topological bands [50]. Recently, a Floquet spinor
BEC was proposed using a periodically driven quadratic Zeeman field [51].
Compared with the usual spinor BEC, the Floquet spinor BEC has an additional
spin-exchange interaction which is induced by the high-frequency driving. Such
an induced spin-exchange interaction can have a profound effect in
ferromagnetic condensates and can generate unconventional quantum phases [51].

In this paper, we study a Floquet spin-1 BEC with spin-orbit coupling. In
spin-1 spin-orbit coupling experiments, three external Raman lasers are used
to couple three hyperfine states, and the quadratic Zeeman effect is
proportional to the two-photon detunings between Raman lasers and hyperfine
states [28]. We propose to drive the quadratic Zeeman effect periodically
around a constant value via periodic modulation of Raman laser frequencies.
Under high-frequency driving, the spin-orbit-coupled spinor BEC is effectively
described by a static Floquet spinor BEC, in which the Rabi coupling is
modulated by a Bessel function and a unique spin-exchange interaction emerges.
Quantum ground phases are investigated in such a spin-orbit-coupled Floquet
spinor BEC with antiferromagnetic or ferromagnetic spin-spin interactions. Our
main results are as follows.

(i) Due to the Bessel-function-modulated Rabi frequency, each quantum phase
can exist in a broad region of the Rabi frequency. Previous studies showed
that the stripe phases in antiferromagnetic and ferromagnetic spinor BECs
exist in a small regime of the Rabi frequency, say,
$\Omega_{c1}<\Omega<\Omega_{c2}$, where $\Omega$ is the Rabi frequency and
$\Omega_{c1,c2}$ are two critical values, with $\Omega_{c2}-\Omega_{c1}$ being
a small quantity [28, 31, 32]. In the Floquet spinor BEC, the Rabi frequency
is modulated as $\Omega J_{0}$ with $J_{0}$ being the zero-order Bessel
function of the first kind. We find that the corresponding phases appear in
$\Omega_{c1}/J_{0}<\Omega<\Omega_{c2}/J_{0}$. Since $J_{0}$ is tunable and
less than 1, the $\Omega$ region for the existence of the stripe phase is
enlarged significantly. This extension of the Rabi frequency for the stripe
phases is beneficial for their experimental observations.

(ii) For antiferromagnetic interactions, the appearance of the Floquet-induced
spin-exchange interaction extends the second stripe phase to broaden the
quadratic Zeeman field domain, which exists in an extremely narrow region of
the quadratic Zeeman field in a typical spin-orbit-coupled spinor BEC.

(iii) For ferromagnetic interactions, a different stripe phase is induced by
the combined effects of the Floquet-induced spin-exchange interaction and the
Rabi coupling. These stripes have a very high density contrast. Their
Bogoliubov excitations are identified as having two gapless Nambu-Goldstone
modes.

This paper is organized as follows. In Sec. II, we present the theoretical
model for a spin-orbit-coupled Floquet spinor BEC. It features the Floquet-
induced spin-exchange interaction and the Bessel-function-modulated Rabi
frequency. In Sec. III, the phase diagram of a noninteracting spin-orbit-
coupled Floquet spinor BEC is analyzed. In Sec. IV, phase diagrams for
antiferromagnetic and ferromagnetic spin-spin interactions are studied
separately. Finally, the conclusion follows in Sec. V.

##  II MODEL

We consider an experimentally realizable spin-orbit-coupled spin-1 BEC. The
spin-orbit coupling is implemented by coupling three hyperfine states with
total angular momentum $F=1$ ($m_{F}=0,\pm 1$) via three Raman lasers
propagating along the $x$ axis [28]. Adjusting two-photon detunings between
Raman lasers and hyperfine states so that they are equal can mimic an
effective quadratic Zeeman field. We propose to periodically drive it by a
periodic oscillation of the Raman laser frequencies. The mean-field energy
functional of the oscillating system is

$\displaystyle E[\Phi]$ $\displaystyle=\int
d\bm{r}\Phi^{\dagger}\left[H_{\text{SOC}}+\xi(t)F_{z}^{2}\right]\Phi$
$\displaystyle\phantom{={}}+\int
d\bm{r}\Phi^{\dagger}\left[\frac{c_{0}}{2}\Phi^{\dagger}\Phi+\frac{c_{2}}{2}\Phi^{\dagger}\bm{F}\Phi\cdot\bm{F}\right]\Phi,$
(1)

with $\Phi=(\Phi_{1},\Phi_{2},\Phi_{3})$ being the spin-1 spinor describing
three-component wave functions. $\bm{F}=(F_{x},F_{y},F_{z})$ are the spin-1
Pauli matrices. $H_{\text{SOC}}$ is the single-particle spin-orbit-coupled
Hamiltonian,

$\displaystyle H_{\text{SOC}}=\left(-i\frac{\partial}{\partial
x}+2F_{z}\right)^{2}+\varepsilon F^{2}_{z}+\frac{\Omega}{\sqrt{2}}F_{x},$ (2)

where $\Omega$ is the Rabi frequency depending on the laser intensities and
$\varepsilon$ is a constant quadratic Zeeman shift which is induced by the
detunings of the Raman lasers [28]. The spin-1 spin-orbit coupling is
represented by the second term in Eq. (2) with a fixed coupling strength due
to the experimentally chosen gauge. In our dimensionless equations, the units
of momentum, length, and energy are $\hbar k_{\text{Ram}}$,
$1/k_{\text{Ram}}$, and $E_{\text{R}}=\hbar^{2}k^{2}_{\text{Ram}}/2m$,
respectively. Here, $m$ is the atom mass, and
$k_{\text{Ram}}=2\pi/\lambda_{\text{Ram}}$ is the wave number of the Raman
lasers, with $\lambda_{\text{Ram}}$ being the wavelength. Considering the
typically experimental parameter $\lambda_{\text{Ram}}=790$ nm, we have
$E_{\text{R}}=2\pi\hbar\times 3.67$ kHz as the units of energy for rubidium
atoms [28]. The quadratic Zeeman field is periodically driven,

$\xi(t)=\alpha\cos(\omega t),$ (3)

with $\omega$ and $\alpha$ being the frequency and amplitude of the driving,
respectively. $c_{0}$ and $c_{2}$ in Eq. (1) denote density-density and spin-
spin interactions, respectively, which depend on the $s$-wave scattering
lengths in the total spin channels. In this work, we assume a repulsive
density-density interaction ($c_{0}>0$), while the spin-spin interaction
$c_{2}$ can be either positive (antiferromagnetic) or negative
(ferromagnetic).

For high-frequency driving, we can derive an effective static Hamiltonian by
averaging the time-dependent Hamiltonian over one modulation period [35]. We
transform the system into an oscillating frame by using the transformation,

$\displaystyle U(t)=\exp\left(-i\frac{\alpha}{\omega}\sin(\omega
t)F_{z}^{2}\right).$ (4)

After applying the transformation $\Phi=U(t)\Psi$, the resultant time
oscillating terms are dropped due to the average in a period. Finally, we end
up with the following time-independent energy functional:

$\displaystyle E[\Psi]$ $\displaystyle=\int
d\bm{r}\Psi^{\dagger}\left[H^{\prime}_{\text{SOC}}+\frac{c_{0}}{2}\Psi^{\dagger}\Psi+\frac{c_{2}}{2}\Psi^{\dagger}\bm{F}\Psi\cdot\bm{F}\right]\Psi$
$\displaystyle\phantom{={}}+c_{f}\int
d\bm{r}\left(\Psi_{1}^{\dagger}\Psi_{3}^{\dagger}\Psi^{2}_{2}+\Psi_{1}\Psi_{3}\Psi_{2}^{\dagger
2}\right).$ (5)

The energy functional describes a spin-orbit-coupled Floquet spinor BEC with
the spinor $\Psi=(\Psi_{1},\Psi_{2},\Psi_{3})$. The modulated single-particle
Hamiltonian is

$\displaystyle H^{\prime}_{\text{SOC}}$
$\displaystyle=\left(-i\frac{\partial}{\partial
x}+2F_{z}\right)^{2}+\varepsilon
F^{2}_{z}+\frac{\Omega}{\sqrt{2}}J_{0}\left(\frac{\alpha}{\omega}\right)F_{x}.$
(6)

Note that the only difference between the Floquet spin-orbit coupled
Hamiltonian $H^{\prime}_{\text{SOC}}$ and the original one $H_{\text{SOC}}$ is
that the Rabi frequency is modulated by the zero-order Bessel function of the
first kind $J_{0}(\alpha/\omega)$. The density-density and spin-spin
interactions in Eq. (5) are the same as those in the usual spinor BEC.
Nevertheless, there is a new spin-exchange interaction with the coefficient
$c_{f}$ which is a pure effect of Floquet modulation [51],

$c_{f}=c_{2}\left[1-J_{0}\left(2\alpha/\omega\right)\right].$ (7)

The spin-orbit-coupled Floquet spinor BEC is reduced back to the usual spin-
orbit-coupled spinor BEC if the driving disappears, i.e., $\alpha/\omega=0$.

##  III Phase diagram of the noninteracting spin-orbit-coupled Floquet spinor
BEC

We study quantum phases in a spin-orbit-coupled Floquet spinor BEC. First, we
analyze the single-particle phase diagram, which was addressed in Refs. [29,
30, 31]. An analysis of the single-particle phase diagram can provide insight
into ground states in an interacting system. The dispersion of
$H^{\prime}_{\text{SOC}}$ can be calculated by direct diagonalization.
Depending on spin-orbit coupling parameters, the lowest band in the dispersion
may have one, two or three local minima. Ground states choose one of minima to
occupy. Therefore, a general ground-state wave function should be

$\displaystyle\Psi=\sqrt{\bar{n}}e^{ikx}\left(\begin{array}[]{c}\cos\theta\cos\varphi\\\
-\sin\theta\\\ \cos\theta\sin\varphi\end{array}\right),$ (11)

where $\bar{n}=N/V$, with $N$ being the total atom number and $V$ the volume
of the system; $k$ is the quasimomentum; and $\theta$ and $\varphi$ are two
parameters. By substituting Eq. (11) into Eq. (5) (with $c_{0}=c_{2}=0$), we
obtain the energy per particle,

$\displaystyle
E_{k}=k^{2}-\left(\frac{A_{k}^{\prime}}{54}\right)^{\frac{3}{2}}-A_{k}\left(\frac{2}{27A_{k}^{\prime}}\right)^{\frac{3}{2}}+\frac{2}{3}A_{0},$
(12)

with

$\displaystyle A_{k}$
$\displaystyle=48k^{2}+(\varepsilon+4)^{2}+\frac{3}{2}J^{2}_{0}\left(\frac{\alpha}{\omega}\right)\Omega^{2},$
$\displaystyle A^{\prime}_{k}$
$\displaystyle=(\varepsilon+4)A_{k}^{\prime\prime}+\sqrt{(\varepsilon+4)^{2}A_{k}^{\prime\prime
2}-4A_{k}^{3}},$ $\displaystyle A_{k}^{\prime\prime}$
$\displaystyle=-288k^{2}+2(\varepsilon+4)^{2}+\frac{9}{2}J^{2}_{0}\left(\frac{\alpha}{\omega}\right)\Omega^{2}.$

Then the quasimomentum can be determined by solving $\partial E_{k}/\partial
k=0$. The occupation of $k=0$ is the zero-momentum (ZM) state, and the
occupation of a nonzero quasimomentum is the PW state.

Figure 1: Quantum ground-state phase diagram of a noninteracting spin-orbit-
coupled Floquet spinor BEC in the space of the Rabi frequency $\Omega$ and the
quadratic Zeeman field $\varepsilon$. The driving is $\alpha/\omega=2$
[$J_{0}(\alpha/\omega)=0.224$]. The background corresponds to values of the
tensor magnetization $\langle{F}^{2}_{z}\rangle$. The black and white solid
lines with dots represent first-order and second-order phase transitions,
respectively. Below these lines is the plane-wave phase, and beyond is the
zero-momentum phase. The red star denotes a tricritical point. Insets show the
lowest bands of the single-particle dispersion. The black dashed lines
separate different regions where the lowest band of the dispersion has one,
two or three local energy minima.

Figure 1 shows the ground-state phase diagram in the $(\Omega,\varepsilon)$
plane, in which the tensor magnetization
$\langle{F}^{2}_{z}\rangle=\cos^{2}\theta$ is chosen as the order parameter.
The solid lines with dots are the transition lines between PW and ZM phases,
above which is the ZM phase and below which is the PW phase. We also show the
lowest band of $H^{\prime}_{\text{SOC}}$ in Fig. 1. The dashed line in the ZM
regime is a separation, above which the lowest band has only one minimum at
$k=0$ and below which it has three local minima but the lowest one at $k=0$.
In the PW regime, the lowest band may have two or three local minima. The
separation between these two cases is demonstrated by the black dashed lines.
Two dashed lines merge together with the phase transition line at the so-
called tricritical point, which is labeled by the red star in Fig. 1. The
location of the tricritical point can be analytically calculated from
$\partial^{2}E_{k}/\partial k^{2}=0$ and the equal energy between the PW and
ZM states [30, 31]. The calculated value for the tricritical point is
$(\Omega^{\ast},\varepsilon^{\ast})=(30.14,-1.66)$. When
$\Omega<\Omega^{\ast}$ the PW-ZM transition is first-order and when
$\Omega>\Omega^{\ast}$ the phase transition is second-order.

##  IV Phase diagram of the interacting spin-orbit-coupled Floquet spinor BEC

For a spin-orbit-coupled spin-1 BEC, previous works revealed ground states,
including PW, ZM, and stripe phases and rich phase transitions between them
[28, 30, 31, 32]. The single-particle dispersion of spin-orbit coupling
provides diverse arrangements of energy minima, and interactions determine how
they condense into these minima. Since the dispersion of
$H^{\prime}_{\text{SOC}}$ has three energy-minima at most, we construct
ground-state wave functions as a superposition of the spinors at these minima,
which can be assumed to be

$\displaystyle\Psi$
$\displaystyle=\sqrt{\bar{n}}C_{+}e^{ikx}\left(\begin{array}[]{c}\cos\theta_{1}\cos\varphi\\\
-\sin\theta_{1}\\\
\cos\theta_{1}\sin\varphi\end{array}\right)+\sqrt{\bar{n}}C_{0}\left(\begin{array}[]{l}\sin\theta_{2}/\sqrt{2}\\\
-\cos\theta_{2}\\\ \sin\theta_{2}/\sqrt{2}\end{array}\right)$ (19)
$\displaystyle\phantom{={}}+\sqrt{\bar{n}}C_{-}e^{-ikx}\left(\begin{array}[]{c}\cos\theta_{1}\sin\varphi\\\
-\sin\theta_{1}\\\ \cos\theta_{1}\cos\varphi\end{array}\right).$ (23)

The superposition coefficients satisfy the normalization condition,
$|C_{+}|^{2}+|C_{0}|^{2}+|C_{-}|^{2}=1$. The spinors are the eigenstates of
$H^{\prime}_{\text{SOC}}$, with the exact parameters $\theta_{1,2}$ and
$\varphi$ to be specified. The second state in Eq. (23) is the spinor at
$k=0$, and the first and third ones are spinors modulated by the plane waves
at $\pm k$. The symmetry of $H^{\prime}_{\text{SOC}}$ requires that the first
and third states have the same $\theta_{1}$ and $\varphi$. We substitute the
above variational wave functions (23) into the energy functional in Eq. (5).
The minimization of the resultant energy functional gives the values of
parameters $k$, $C_{0,\pm}$, $\theta_{1,2}$, and $\varphi$. From these
parameters, we can classify ground states: the ZM phase has $C_{\pm}=0$; the
PW phase has a nonzero $k$ and $C_{0}=0$, with one of $C_{\pm}$ being nonzero;
and the stripe phase requires $k\neq 0$ and at least two of $C_{\pm,0}$ are
nonzero. The stripe phases can be further classified according to relative
values of $C_{\pm,0}$ [31, 32]. Considering that the classification of ground
states depends strongly on $C_{\pm,0}$, we use the tensor magnetization
$\left\langle F^{2}_{z}\right\rangle$ as the order parameter to identify
different phases,

$\left\langle
F^{2}_{z}\right\rangle=\left(\left|C_{+}\right|^{2}+\left|C_{-}\right|^{2}\right)\cos^{2}\theta_{1}+\left|C_{0}\right|^{2}\sin^{2}\theta_{2}.$
(24)

We find that antiferromagnetic and ferromagnetic spin-spin interactions have
very different ground-state phase diagrams, which are studied separately.

###  IV.1 Antiferromagnetic interactions

Figure 2: Quantum ground-state phase diagram of a spin-orbit coupled Floquet
spinor BEC with an antiferromagnetic spin-spin interaction ($\bar{n}c_{0}=1$
and $\bar{n}c_{2}=0.1$). The background corresponds to values of the tensor
magnetization $\langle{F}^{2}_{z}\rangle$ defined in Eq. (24). The black and
white solid lines with dots represent the first-order and second-order phase
transitions, respectively. The different tricritical points are denoted by the
red and purple stars. The driving is $\alpha/\omega=2$
[$J_{0}(\alpha/\omega)=0.224$ and $J_{0}(2\alpha/\omega)=-0.397$].

The antiferromagnetic interaction is $c_{2}>0$, which is typical for the 23Na
BEC. Figure 2 demonstrates the phase diagram for antiferromagnetic
interactions with driving $\alpha/\omega=2$ in the space of the quadratic
Zeeman field $\varepsilon$ and the Rabi frequency $\Omega$. When $\varepsilon$
is negative, the single-particle dispersion has two lowest minima located at
$\pm k_{m}$ (see the inset in Fig. 1); the antiferromagnetic interaction
allows atoms to simultaneously occupy these two minima to form a stripe for
low $\Omega$. This stripe phase labeled as S1 in Fig. 2, has
$|C_{+}|=|C_{-}|=1/\sqrt{2}$ and $C_{0}=0$. Using the wave functions in Eq.
(23) with $C_{0}=0$ and considering the single-particle spinors at $\pm k_{m}$
with $\varphi=\pi/2$, we get the energy of the antiferromagnetic interaction
$\langle E\rangle_{c_{2}}$ and Floquet-induced spin-exchange interaction
$\langle E\rangle_{c_{f}}$,

$\displaystyle\langle E\rangle_{c_{2}}+\langle E\rangle_{c_{f}}$
$\displaystyle=\frac{c_{2}\bar{n}^{2}}{2}\cos^{4}\theta_{1}+c_{2}\bar{n}^{2}|C_{-}|^{2}|C_{+}|^{2}$
$\displaystyle\phantom{={}}\times\left[(1+\frac{c_{f}}{c_{2}})\sin^{2}(2\theta_{1})-2\cos^{4}\theta_{1}\right].$
(25)

For a low $\Omega$, we have $\theta_{1}\approx 0$, and the minimization of
$\langle E\rangle_{c_{2}}+\langle E\rangle_{c_{f}}$ leads to
$|C_{+}|=|C_{-}|=1/\sqrt{2}$, corresponding to the S1 phase, the tensor
magnetization of which is $\left\langle F^{2}_{z}\right\rangle\approx 1$, as
shown in Fig. 2. $\theta_{1}$ prefers to be nonzero for a large $\Omega$.
Meanwhile, the first term $c_{2}\bar{n}^{2}/2\cos^{4}\theta_{1}$ in $\langle
E\rangle_{c_{2}}+\langle E\rangle_{c_{f}}$ allows $\theta_{1}$ to approach to
$\pi/2$, at which it is minimized, so that $\theta_{1}$ can grow from zero to
$\pi/2$ as $\Omega$ increases. Consequently, for a high $\Omega$, we may have
$(1+c_{f}/c_{2})\sin^{2}(2\theta_{1})-2\cos^{4}\theta_{1}>0$. Then the
minimization of $\langle E\rangle_{c_{2}}+\langle E\rangle_{c_{f}}$ requires
one of $C_{\pm}$ to be zero. Even though the single-particle dispersion has
two minima, the antiferromagnetic interaction chooses one of them to occupy,
generating the PW phase shown in Fig. 2. The phase transition between the S1
and PW phases is first order. Physically, $\langle E\rangle_{c_{2}}+\langle
E\rangle_{c_{f}}$ is proportional to
$c_{2}\bar{n}^{2}|C_{+}|^{2}|C_{-}|^{2}[(1+c_{f}/c_{2})\langle
F_{x}\rangle_{+}\langle F_{x}\rangle_{-}+\langle F_{z}\rangle_{+}\langle
F_{z}\rangle_{-}]$, where $\langle F_{x}\rangle_{\pm}$ and $\langle
F_{z}\rangle_{\pm}$ are the $x$ and $z$ polarizations of the spinors at $\pm
k_{m}$. The antiferromagnetic interaction generates $\langle
F_{z}\rangle_{+}\langle F_{z}\rangle_{-}<0$, and the Rabi coupling favors
$\langle F_{x}\rangle_{+}\langle F_{x}\rangle_{-}>0$. The competition between
these two effects gives rise to the S1-PW transition, and we have
$\left\langle F^{2}_{z}\right\rangle<1$ in the PW phase (see Fig. 2). The
emergence of the ZM phase in Fig. 2 is due to the fact that the lowest minimum
of the single-particle dispersion lays at $k=0$. There is a second stripe
phase labeled S2 which is unique only for antiferromagnetic interactions. The
S2 phase is featured with $|C_{-}|=|C_{+}|\neq 0,|C_{0}|\neq 0$, and
$\Theta\equiv\arg(C_{-})+\arg(C_{+})-2\arg(C_{0})=\pi$.

At first glance, the phase diagram shown in Fig. 2 is similar to that of the
usual spin-orbit-coupled BEC demonstrated in Refs. [31, 32] (i.e., Fig. 1(a)
in [31] and Fig. 1 in [32]). There are two tricritical points represented by
stars in Fig. 2. The first- (second-) order phase transitions between
different phases are shown by black (white) solid lines with dots. However,
there are two different features in our system. (i) All phases exist in a
broadened region of the Rabi frequency. This is a straightforward consequence
of the Bessel-function modulation $\Omega J_{0}$. (ii) The existence of the S2
phase is also extended in the $\varepsilon$ domain. In the usual spin-orbit-
coupled antiferromagnetic BEC the S2 phase exists in an extremely narrow
region of $\varepsilon$ (see Fig. 1(a) in [31] and Fig. 1 in [32]). Our
Floquet system has a large extension, which benefits from the Floquet-induced
interaction.

Figure 3: Phase diagram for an antiferromagnetic interaction ($\bar{n}c_{0}=1$
and $\bar{n}c_{2}=0.1$) as a function of the driving $\alpha/\omega$. The Rabi
frequency is $\Omega=2$. The background corresponds to values of the tensor
magnetization $\langle{F}^{2}_{z}\rangle$. The black and white solid lines
with dots represent the first-order and second-order phase transitions,
respectively.

To reveal the extension of the stripe regions more clearly, we study the phase
diagram as a function of the driving $\alpha/\omega$. The results are shown in
Fig. 3 for $\Omega=2$. It is clear from Fig. 3 that without the driving
($\alpha/\omega=0$) the S2 phase exists in an extremely narrow region of
$\varepsilon$. This leads to a challenge for its experimental implementation.
The upper boundary of the S2 phase corresponds to the degeneracy of three
minima of the single-particle dispersion, i.e.,
$E_{-k_{m}}=E_{k=0}=E_{k_{m}}$, and beyond the boundary $E_{k=0}$ becomes the
lowest one such that ground state is the ZM phase. Below the boundary, we have
$E_{-k_{m}}=E_{k_{m}}<E_{k=0}$. For low $\Omega$, the spinors at $\pm k_{m}$
have $\theta_{1}=0$ and $\varphi=\pi/2$ and the spinor at $k=0$ has
$\theta_{2}=0$. The general wave function becomes

$\Psi=\sqrt{\bar{n}}\begin{pmatrix}C_{-}e^{-ikx}\\\ -C_{0}\\\
C_{+}e^{ikx}\end{pmatrix},$ (26)

which is a good approximation for low $\Omega$. By using Eq. (26), we find
that the antiferromagnetic energy can be minimized as $\langle
E\rangle_{c_{2}}=0$ in both the S1 phase ($|C_{\pm}|=1/\sqrt{2},C_{0}=0$) and
the S2 phase ($|C_{-}|=|C_{+}|<1/\sqrt{2}$, $C_{0}\neq 0$, $\Theta=\pi$) .
However, the S2 phase is not a minimization of the quadratic Zeeman energy
$\langle E\rangle_{\varepsilon}=\varepsilon\bar{n}(|C_{-}|^{2}+|C_{+}|^{2})$
for $\varepsilon<0$, so the ground state is the S1 phase. A dominant Rabi
frequency $\Omega$ causes a small deviation $\theta_{1}$ from zero, i.e.,
$\theta_{1}=\delta\theta$, where $\delta\theta>0$ is a very small quantity.
This term leads to $\langle
E\rangle_{c_{2}}=8c_{2}\bar{n}^{2}|C_{+}|^{4}(\delta\theta)^{2}$ for both the
S1 and S2 phases. Considering the S1 phase with $|C_{+}|^{2}=1/2$ and the S2
phase with $|C_{+}|^{2}<1/2$, this extra antiferromagnetic energy prefers the
S2 phase as the ground state if the quadratic Zeeman energy is weak. If the
quadratic Zeeman energy exceeds this extra energy, the S1 phase is back as the
ground state. Since the extra energy is a small quantity of second order, the
S2 ground state exists in a very small $\varepsilon$ domain.

In the presence of the driving, the region of the S2 phase is dramatically
extended around $\varepsilon=0$ (see Fig. 3). The upward shift of the region
is due to the Bessel-function-modulated Rabi frequency. As the driving
$\alpha/\omega$ increasing from zeros, $\Omega J_{0}(\alpha/\omega)$ decreases
towards zero. As shown in Fig. 2, for a small $\Omega$, the S2 phase is
located around $\varepsilon=0$. The dramatic expansion of the existence area
is the consequence of the Floquet-induced spin-exchange interaction. The S2
phase can greatly minimize the spin-exchange-interaction energy, which can be
easily seen from the approximate wave function for low $\Omega$ in Eq. (26).
With the wave function, the spin-exchange-interaction energy becomes $\langle
E\rangle_{c_{f}}=2c_{f}\bar{n}^{2}|C_{-}||C_{+}||C_{0}|^{2}\cos(\Theta)$. The
S2 phase, having $0<|C_{-}|=|C_{+}|<1/\sqrt{2}$ and $\Theta=\pi$, minimizes
the spin-exchange energy. Other phases, such as the ZM phase ($C_{0}=1$), the
PW phase ($C_{0}=0$, $|C_{+}|+|C_{-}|=1$), and the S1 phase
($C_{0}=0,|C_{\pm}|=1/\sqrt{2}$), lead to $\langle E\rangle_{c_{f}}=0$, so
that the Floquet-induced spin-exchange energy cannot be minimized. Meanwhile,
the S2 phase also minimizes the antiferromagnetic interaction energy, $\langle
E\rangle_{c_{2}}=c_{2}\bar{n}^{2}/2(|C_{-}|^{2}-|C_{+}|^{2})^{2}+c_{2}\bar{n}^{2}[|C_{-}|^{2}|C_{0}|^{2}+|C_{+}|^{2}|C_{0}|^{2}+2|C_{-}||C_{+}||C_{0}|^{2}\cos(\Theta)]=0$.
The only obstacle to the existence of the S2 phase is the quadratic Zeeman
energy $\langle
E\rangle_{\varepsilon}=\varepsilon\bar{n}(|C_{-}|^{2}+|C_{+}|^{2})$. If
$\varepsilon>0$, the quadratic Zeeman energy prefers the ZM phase, and when
$\varepsilon<0$, it prefers the S1 phase. Therefore, the competition between
the Floquet-induced spin-exchange interaction and the quadratic Zeeman field
leads to the existence region for the S2 phase, which is dramatically extended
in comparison with the usual case with $\alpha/\omega=0$. The S2-ZM (white
lines with dots) and S2-S1 (black lines with dots) transition lines oscillate
as a function of $\alpha/\omega$. It is noted that the maxima of the
transition lines correspond to the zeros of $J_{0}(\alpha/\omega)$; therefore,
the oscillations come from $\Omega J_{0}(\alpha/\omega)$. It is also
interesting that without the driving the S2 phase always exists in the
negative-$\varepsilon$ area, but with the driving, it can exist even in
positive-$\varepsilon$ areas.

###  IV.2 Ferromagnetic interactions

The ferromagnetic interaction is $c_{2}<0$. We consider $c_{2}/c_{0}=-0.5$,
which is typical of 7Li atoms [32]. Figure 4 demonstrates the phase diagram
for ferromagnetic interactions with driving $\alpha/\omega=1.6$. In the
low-$\Omega$ region, there is a third stripe phase, which is labeled S3 in
Fig. 4. It has $|C_{-}|=|C_{+}|\neq 0,|C_{0}|\neq 0$, and $\Theta=0$. Using
the approximate wave function in Eq. (26), we know that the S3 phase minimizes
only the second term in the ferromagnetic interaction energy $\langle
E\rangle_{c_{2}}=c_{2}\bar{n}^{2}/2(|C_{-}|^{2}-|C_{+}|^{2})^{2}+c_{2}\bar{n}^{2}[|C_{-}|^{2}|C_{0}|^{2}+|C_{+}|^{2}|C_{0}|^{2}+2|C_{-}||C_{+}||C_{0}|^{2}\cos(\Theta)]$
($c_{2}<0$) and it cannot minimize the first term
$c_{2}\bar{n}^{2}/2(|C_{-}|^{2}-|C_{+}|^{2})^{2}$ which is minimized by the PW
phase. With the effect of the quadratic Zeeman field, the S3, PW and ZM phases
are distributed in the way shown in Fig. 4. These three phases are similar to
those in previous studies [31, 32] (i.e., Fig. 1(b) in [31] and Fig. 2 in
[32]), but with the outstanding feature that every phase exists in a broadened
region of $\Omega$ due to the Bessel-function modulation.

Figure 4: Quantum ground-state phase diagram of a spin-orbit-coupled Floquet
spinor BEC with a ferromagnetic spin-spin interaction $\bar{n}c_{0}=1$ and
$\bar{n}c_{2}=-0.5$. The background corresponds to values of the tensor
magnetization $\langle{F}^{2}_{z}\rangle$. The black and white solid lines
with dots represent the first-order and second-order phase transitions,
respectively. The different tricritical points are denoted by red and yellow
stars. The driving is $\alpha/\omega=1.6$ [$J_{0}(\alpha/\omega)=0.455$ and
$J_{0}(2\alpha/\omega)=-0.320$].

Different from the case of $\alpha/\omega=0$ in Refs. [31, 32, 28], we find in
the Floquet spinor BEC that there a new stripe phase, which is labeled as S4,
exists. The S4 phase is located inside the region where the single-particle
dispersion has two energy minima at $\pm k_{m}$, and they are equally occupied
by the S4 phase with $|C_{\pm}|=1/\sqrt{2}$ and $C_{0}=0$. This condition is
exactly the same as the S1 phase with antiferromagnetic interactions.
Nevertheless, the S1 phase exists in the low-$\Omega$ region (see Fig. 2),
while the S4 phase is in the high-$\Omega$ region (see Fig. 4). Such a
difference in the existence region related to $\Omega$ leads to different
features in the S4 phase. With $C_{0}=0$, the minimization of the
ferromagnetic energy and the Floquet-induced energy demonstrated in Eq. (25)
leads to $|C_{-}|=0$ or $|C_{+}|=0$ for low $\Omega$ ($\theta_{1}\approx 0$).
In this case, the ground state is the PW phase with $\left\langle
F^{2}_{z}\right\rangle\approx 1$, as shown in Fig. 4. For high $\Omega$, one
may have $\theta_{1}\neq 0$ and
$(1+c_{f}/c_{2})\sin^{2}(2\theta_{1})-2\cos^{4}\theta_{1}>0$. The minimization
of $\langle E\rangle_{c_{2}}+\langle E\rangle_{c_{f}}$ requires
$|C_{\pm}|=1/\sqrt{2}$, so that the ground state is the S4 phase. Due to the
existence of the S4 phase, there are two tricritical points, labeled by stars
in Fig. 4.

Figure 5: Phase diagram in a ferromagnetic interaction ($\bar{n}c_{0}=1$ and
$\bar{n}c_{2}=-0.5$) as a function of the driving $\alpha/\omega$. The Rabi
frequency is $\Omega=8$. The background corresponds to values of the tensor
magnetization $\langle{F}^{2}_{z}\rangle$. The black and white dotted lines
represent the first-order and second-order phase transitions. The different
tricritical points are denoted by red and yellow stars. Figure 6: Profiles of
the S4 phase in a ferromagnetic interaction ($\bar{n}c_{0}=1$ and
$\bar{n}c_{2}=-0.5$). The driving is $\alpha/\omega=1.6$
($J_{0}(\alpha/\omega)=0.455$ and $J_{0}(2\alpha/\omega)=-0.320$). The
quadratic Zeeman field is $\varepsilon=-2$. (a) Spatial density distributions,
$n_{i}=|\Psi_{i}|^{2}$, where $n$ is the total density $n=n_{1}+n_{2}+n_{3}$.
The Rabi frequency is $\Omega=8$. (b) The contrast
$(n_{\mathrm{max}}-n_{\mathrm{min}})/(n_{\mathrm{max}}+n_{\mathrm{min}})$ as a
function of $\Omega$.

We want to emphasize that without driving ($c_{f}=0$), the S4 phase cannot
exist [31, 32, 28]. In the absence of driving, Eq. (25) becomes $\langle
E\rangle_{c_{2}}=c_{2}\bar{n}^{2}/2\cos^{4}\theta_{1}+c_{2}\bar{n}^{2}|C_{-}|^{2}|C_{+}|^{2}\left[\sin^{2}(2\theta_{1})-2\cos^{4}\theta_{1}\right]$.
For $c_{2}<0$, the first term $c_{2}\bar{n}^{2}/2\cos^{4}\theta_{1}$ prefers
$\theta_{1}=0$. According to the second term, the realization of the S4 phase
needs a nonzero $\theta_{1}$ satisfying
$\sin^{2}(2\theta_{1})-2\cos^{4}\theta_{1}>0$, which can be achieved by
increasing the Rabi frequency. In addition, the negative Rabi coupling energy
is also beneficial for lowing the total energy. However, for the single-
particle dispersion with large $\Omega$ the energy minimum at $k=0$ will be
lower than the energy minima at $k=\pm k_{m}$, and the ground state prefers
the ZM phase. Thus, there is no way for the S4 phase to exist. The Floquet-
induced interaction has the nature of spin-exchange. It has two effects: the
spin-exchange interaction causes direct competition with the first term since
it prefers $\theta_{1}=\pi/4$, so the three components having equal
populations in each spinor; according to Eq. (25), the S4 phase requires
$(1+c_{f}/c_{2})\sin^{2}(2\theta_{1})-2\cos^{4}\theta_{1}>0$, and the positive
$c_{f}/c_{2}$ as a prefactor also increases the possibility of $\theta_{1}$
satisfying the requirement. Therefore, combined effects of the Rabi coupling
and the Floquet-induced interaction makes the existence of the S4 phase
possible.

In order to understand how the S4 phase emerges in the presence of driving, we
analyze the phase diagram as a function of the driving $\alpha/\omega$, which
is demonstrated in Fig. 5. The Rabi frequency is fixed as $\Omega=8$. For
$\alpha/\omega=0$, the ground state is the ZM phase, as shown in Fig. 5, which
is consistent with the results in Refs. [31, 32, 28]. As $\alpha/\omega$
increasing, the S3, S4 and PW phases appear and have an interesting
distribution shown in Fig. 5. Transition lines (white and black solid lines
with dots) have an oscillating behavior with the maxima matching the zeros of
$J_{0}(\alpha/\omega)$. The S3 and S4 phases locate between two transition
lines. Furthermore, the S4 phase exists in limited regions. The change in
$\alpha/\omega$ is equivalent to scanning $\Omega$. A high $\alpha/\omega$
leads to $\Omega J_{0}(\alpha/\omega)$ being confined around zero. According
to Fig. 4, the ground state around $\Omega=0$ is the S3 phase. Therefore, for
high $\alpha/\omega$ there is no S4 phase anymore (see Fig. 5).

In Fig. 6(a), we show density distributions $n_{i}=|\Psi_{i}|^{2}$ of a
typical S4 state. The outstanding feature is that the second component $n_{2}$
is comparable with the other components $n_{1}=n_{3}$. This is completely
different from the S1 phase with antiferromagnetic interactions, where
$n_{2}\ll n_{1}=n_{3}$. This is due to the low-$\Omega$ region for the S1
phase. For low $\Omega$, the spinors at $\pm k_{m}$ can be physically
approximated as $e^{ik_{m}x}(\delta^{2},\delta,1)^{T}$ and
$e^{-ik_{m}x}(1,\delta,\delta^{2})^{T}$, respectively, where $\delta$ is a
small quantity. The S1 phase is an equal superposition of the two spinors, and
we have $n_{1}=n_{3}=1+2\delta^{2}\cos(2k_{m}x)$ and
$n_{2}=4\delta^{2}\cos^{2}(k_{m}x)$. Therefore, the S1 phase has $n_{2}\ll
n_{1}=n_{3}$ and a very low contrast for $n_{1}$ and $n_{2}$ which is
proportional to a small quantity of second order. The contrast is defined as
$(n_{\mathrm{max}}-n_{\mathrm{min}})/(n_{\mathrm{max}}+n_{\mathrm{min}})$,
with $n_{\mathrm{max}}$ ($n_{\mathrm{min}}$) being the density maximum
(minimum). The low contrast of the S1 phase is unfavorable for experimental
observations. However, the S4 phase with ferromagnetic interactions exists in
the high-$\Omega$ region, and with further help from the Floquet-induced spin-
exchange, $\delta$ is not a small quantity anymore. Therefore, the contrast of
$n_{1}$ and $n_{3}$ is obviously high for the S4 phase. The advantage of the
second component is that its contrast is always maximized (it is equal to 1).
The dominant occupation in the second component makes it perfect for direct
experimental observations. In Fig. 6(b), we show the contrast in the full
$\Omega$ region. The contrast of $n_{1}$ and $n_{3}$ increases with the
increase of $\Omega$, and it is always 1 as expected for the second component
$n_{2}$.

Figure 7: Bogoliubov excitation spectrum $\zeta(q_{x})$ of a typical S4 state.
The parameters are $\bar{n}c_{0}=1$, $\bar{n}c_{2}=-0.5$, $\Omega=7$, and
$\varepsilon=-1.1$. The two lowest bands are gapless, corresponding to two
Nambu-Goldstone modes.

A closely related topic for ground states is their elementary excitations. The
excitation spectrum of each phase in typical spin-orbit-coupled spin-1 BECs
has been investigated [30, 31, 33]. The S4 phase exists only in Floquet spinor
BECs, and we study its Bogoliubov excitation. The stripe wave function ansatz
in Eq. (23) includes only low-order plane waves. It is known that such an
ansatz cannot precisely capture Bogoliubov excitation and high-order plane
waves should be involved [23, 21, 52, 53, 54]. Therefore, we use the ansatz
with high-order modes [33],

$\Psi=\sqrt{\bar{n}}\sum_{j=-L}^{L}e^{ijKx}\begin{pmatrix}\varphi_{1}^{(j)}\\\
\varphi_{2}^{(j)}\\\ \varphi_{3}^{(j)}\end{pmatrix},$ (27)

with the normalization condition
$\sum_{\sigma,j}|\varphi_{\sigma}^{(j)}|^{2}=1$. Here, $L$ is the cutoff of
the plane waves, and $K$is related to the period of the stripes. Spinors
$(\varphi_{1}^{(j)},\varphi_{2}^{(j)},\varphi_{3}^{(j)})^{T}$ and $K$ are
determined by minimizing the energy function in Eq. (5) using Eq. (27). In the
S4 phase parameter region, we first get the stripe wave function using
minimization procedures, and then we use the ground state to solve
Bogoliubov–de Gennes equation to get the elementary excitation energy $\zeta$
[33]. A typical excitation spectrum $\zeta(q_{x})$, i.e., the relation between
excitation energy $\zeta$ and excitation quasimomentum $q_{x}$, is
demonstrated in Fig. 7, in which only the three lowest bands are shown. The
size of the Brillouin zone is $2K$, which means that the period of stripes is
$\pi/K$. The lowest two bands are gapless, corresponding to two Nambu-
Goldstone modes. The physical origin of these two gapless modes is the fact
that stripes spontaneously break the continuously translational symmetry and
gauge symmetry [21].

Finally, we discuss possible experimental observations of our results. In the
spin-1 spin-orbit-coupled experiment in [28], the Rabi frequency $\Omega$ and
the constant quadratic Zeeman shift $\varepsilon$ are completely tunable.
$\Omega$ can be experimentally tuned up to $15$ with units of $E_{\text{R}}$,
and $\varepsilon$ can reach $\pm 5E_{\text{R}}$. $\Omega$ and $\varepsilon$ in
our study are completely within experimental accessibility. In a spin-orbit-
coupled degenerate Fermi gas experiment in [49], the periodic driving of the
frequency of a Raman laser was implemented to reach the high-frequency driving
regime. We expect that similar to this experiment, our driving can be realized
precisely. Under such a high-frequency driving regime, the possible
micromotions induced by high-order drivings are safely suppressed, as shown in
spin-orbit-coupled experiments [47, 49]. So far, experiment have realized
spin-1 spin-orbit coupling in a 87Rb BEC [28]. The 87Rb BEC is ferromagnetic
but with a very small ratio $c_{2}/c_{0}$. Such a small ratio makes the stripe
phase S3 exists in a negligible parameter regime [32]. In the presence of
driving, there is no the new S4 phase for the 87Rb BEC.

##  V Conclusion

Spin-orbit-coupled spin-1 BECs have been realized in experiments. Based on the
experimental platform, we proposed a spin-orbit-coupled Floquet spinor BEC by
periodically driving the quadratic Zeeman field with a high-frequency. In the
Floquet spinor BEC, the Rabi frequency is modulated by a Bessel function and a
Floquet-induced spin-exchange interaction emerges. We studied quantum ground-
state phase diagram of a spin-orbit-coupled Floquet spinor BEC while
considering antiferromagnetic and ferromagnetic spin-spin interactions
separately. A general result is that due to the Bessel-function modulation,
every phase in the diagram can exist in a broadened Rabi frequency region. For
antiferromagnetic interactions, we found that the existence of a stripe phase
can be dramatically extended in the $\varepsilon$ domain due to the Floquet-
induced spin-exchange interaction. For ferromagnetic interactions, a different
stripe phase was revealed, and its features, including high contrast and
Bogoliubov excitations, were identified. In all previous studies of spin-$1/2$
and spin-1 spin-orbit-coupled BECs, stripes have a very low contrast since
they exist in in the low-$\Omega$ regime and the contrast is proportional to
the Rabi frequency $\Omega$ [23]. This stripe phase in the Floquet spinor BEC
exists in the high-$\Omega$ region and its high contrast favors experimental
observations.

## Acknowledgments

We thank Prof. P. Engels for stimulating discussions. This work is supported
by the National Natural Science Foundation of China under Grants No. 11974235
and No. 11774219. H.L acknowledges support from Okinawa Institute of Science
and Technology Graduate University.

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