# Expansion dynamics of Bose-Einstein condensates in a synthetic magnetic
field

Juan Wang  Department of Physics, School of Physics and Electronic Science,
East China Normal University, Shanghai 200241, China  Hongguang Liang
Department of Physics, School of Physics and Electronic Science, East China
Normal University, Shanghai 200241, China  Yan Li<EMAIL_ADDRESS>Department of Physics, School of Physics and Electronic Science, East China
Normal University, Shanghai 200241, China  Chuan-Hsun Li  Department of
Physics and Astronomy, Purdue University, West Lafayette, Indiana 47907, USA
Chunlei Qu<EMAIL_ADDRESS>Department of Physics, Stevens Institute of
Technology, Hoboken, NJ 07030, USA  Center for Quantum Science and
Engineering, Stevens Institute of Technology, Hoboken, NJ 07030, USA

###### Abstract

We investigate the expansion dynamics of spin-orbit-coupled Bose-Einstein
condensates subjected to a synthetic magnetic field, after their release from
an external harmonic trap. Our findings reveal that the condensate experiences
a spin-dependent rotation and separation due to the rigid-like rotational
velocity field, which leads to a spin density deflection. The deflection angle
reaches a peak at a time that is inversely related to the frequency of the
harmonic trap. When the detuning gradient is below a critical value for vortex
nucleation, our analytical results derived from a spinor hydrodynamic theory
align closely with numerical results using the coupled Gross-Pitaevskii
equations. Beyond this critical value, we also numerically simulated the
expansion dynamics of the condensates containing vortices with negative
circulation. Our findings highlight the pivotal role of the rigid-like
rotational velocity field on the dynamics of the condensate and may stimulate
further experimental investigations into the rich superfluid dynamics induced
by synthetic magnetic fields.

##  I Introduction

The investigation of expansion dynamics of quantum gases released from
external traps has significantly advanced our understanding of their quantum
states and the roles of interactions [1]. In 1995, the detection of the
anisotropic velocity distribution in an atomic cloud after expansion provided
pivotal evidence for the existence of Bose-Einstein condensates (BECs) [2, 3].
Since then, the measurement of expansion dynamics has become a powerful tool
for probing a plethora of physical phenomena in both bosonic and fermionic
quantum gases [4, 5, 6, 7, 8, 9]. For example, the observation of quantized
vortices in BECs released from rotating traps has revealed the superfluid
characteristic of Bose gases [10, 11].

In recent years, the experimental and theoretical exploration of synthetic
gauge fields in ultracold atomic gases have made rapid progress and opened new
frontiers in modern quantum physics [12, 13, 14, 15, 16, 17, 18, 19, 20, 21,
22, 23, 24, 25, 26, 27]. The implementation of Raman coupling via couter-
propagating laser beams to couple two hyperfine states of alkaline atoms has
enabled the realization of an equal superposition of Rashba and Dresslhaus
spin-orbit (SO) coupling [16]. SO coupling significantly affects the
superfluidity of Bose condensates, including the quenching of the superfluid
density along the SO coupling direction near the critical transition point
between the zero-momentum phase and plane-wave phase [28, 26]. Additionally,
introducing a position-dependent detuning to the SO-coupled Hamiltonian
enables the generation of a synthetic magnetic field for the neutral atoms
[14]. This has led to the observation of quantized vortices without a rotating
trap [19]. Even before the appearance of vortices, the synthetic magnetic
field induces a rigid-like rotational velocity field, violating the
irrotational superfluid velocity field constraint for traditional condensates
[29]. Synthetic magnetic fields have also facilitated novel investigations
into the non-equilibrium dynamics of SO-coupled BECs. Recently, it was found
that a synthetic magnetic field couples various low-energy collective modes of
the condensate, resulting in intriguing beat phenomena [30, 31].

In this work, we study the expansion dynamics of the condensate in the
presence of a synthetic magnetic field. Figure 1 illustrates the key
characteristics of the expansion dynamics across various condensates. For a
non-rotating, single-component BEC, an inversion of the aspect ratio is
observed during the expansion (Fig. 1(a)). This is attributed to the mean-
field interaction energy and the phenomenon can be accurately described by the
hydrodynamic theory [32, 33, 34]. In contrast, as shown in Fig. 1(b), a
rotating, vortex-free condensate does not exhibit the inversion of aspect
ratio due to the irrotational superfluid flow and the conservation of angular
momentum [35, 36]. Figure 1(c) depicts the expansion of a SO-coupled BEC which
is characterized by a significantly slower expansion along the SO coupling
direction $x$, due to a large effective mass [37]. Moreover, the spin-momentum
locking dictates that the two spin components (shown by yellow and blue
curves) separate along the SO coupling direction during the expansion,
creating a spin density along $x$ [23].

The expansion dynamics of a condensate in a synthetic magnetic field exhibit a
different behavior. As shown in Fig. 1(d), the two spin components exhibit
both rotation and separation in the expansion, leading to a deflection of the
spin density that is intimately related to the rigid-like rotational velocity
field. This deflection dynamics can be analytically determined by solving the
hydrodynamic equations for a vortex-free BEC in a weak synthetic magnetic
field. Moreover, when the synthetic magnetic field is sufficiently strong, by
solving the coupled Gross-Pitaevskii (GP) equations, we find that the
expansion dynamics exhibit an additional rotation of the vortices.

Figure 1: Schematic of the expansion dynamics of different types of
condensates. From left to right, the three columns illustrate the initial
state, short-time expansion, and long-time expansion, respectively. (a) An
ordinary BEC shows an inversion of the aspect ratio. (b) A rotating BEC never
becomes circular (indicated by the dotted circle) during the expansion. (c) A
SO-coupled BEC expands significantly slower along the SO direction, with the
two spin components (yellow and blue lines) separating as they expand. (d) A
BEC in a synthetic rotational field exhibits both rotation and separation
during the expansion. For (c) and (d), the SO coupling is along $x$. The red
arrows in (b) and (d) indicate the direction of the rotating trap or
rotational velocity field.

##  II Hydrodynamic theory

We consider the following single-particle Hamiltonian for a BEC in a synthetic
magnetic field [12, 14]:

$\displaystyle\hat{H}_{sp}$ $\displaystyle=\frac{1}{2m}(\hat{p}_{x}-\hbar
k_{r}\sigma_{z})^{2}+\frac{\hat{p}_{y}^{2}}{2m}+V_{trap}-\frac{\Omega}{2}\sigma_{x}-\delta(y)\sigma_{z},$
(1)

where $m$ represents the atomic mass, $\hat{p}_{\mu}=-i\hbar\nabla_{\mu}$
($\mu=x,y$) is the canonical momentum along $\mu$-direction, $\hbar k_{r}$ is
the recoil momentum, and $\Omega$ denotes the strength of the Raman coupling.
Without loss of generality, we have assumed a two-dimensional (2D) condensate
confined in a harmonic trapping potential
$V_{trap}(x,y)=m(\omega_{x}^{2}x^{2}+\omega_{y}^{2}y^{2})/2$. The nonlinear
interaction between atoms can be taken into account through the mean-field
approximation $V_{int}=(1/2)\int
d\mathbf{r}{\textstyle\sum_{\alpha\beta}}g_{\alpha\beta}n_{\alpha}n_{\beta}$,
where $g_{\alpha\beta}=4\pi\hbar^{2}a_{\alpha\beta}/m$ are the coupling
constants in different spin channels related to the corresponding s-wave
scattering lengths $a_{\alpha\beta}$. In this work, we assume isotropic
coupling constants $g_{\alpha\beta}\equiv g$. The last term in Eq. (1)
corresponds to a spatially varying detuning $\delta(y)=\eta k_{r}y$ that
induces a synthetic magnetic field, with $\eta$ being the detuning gradient.
Vortices appear when the detuning gradient exceeds a critical value
$\eta>\eta_{c}$, whereas for $\eta<\eta_{c}$, the formation of quantized
vortices is energetically inhibited, resulting in vortex-free condensates.

For simplicity, we focus on the dynamics within the zero-momentum phase,
characterized by $\Omega>\Omega_{c}\equiv 4E_{r}$, where
$E_{r}=\hbar^{2}k_{r}^{2}/2m$ is the recoil energy. The order parameter for
each component is parameterized as
$\psi_{\alpha}=\sqrt{n_{\alpha}(\mathbf{r})}e^{i\phi_{\alpha}(\mathbf{r})}$
[38], with $n_{\alpha}$ and $\phi_{\alpha}$ representing the density and phase
of the $\alpha$-th component. For ground states and low-frequency excitations,
the relative phase of the two components is locked due to the large excitation
gap, resulting in $\phi_{1}=\phi_{2}\equiv\phi$ [28]. Furthermore, when the
emergent spin density $s_{z}=n_{1}-n_{2}$ is much smaller than the total
density $n=n_{1}+n_{2}$, one can obtain a simple relation

$s_{z}=(\frac{2\hbar^{2}k_{r}}{m\Omega}\nabla_{x}\phi+\frac{2\eta
k_{r}}{\Omega}y)n$ (2)

which is intimately related the spin-momentum locking. Neglecting the
influence of the quantum pressure in the kinetic energy, the hydrodynamic
equations for BECs in the synthetic magnetic field can be derived as follows
[30, 31]

$\displaystyle\frac{\partial n}{\partial t}+\nabla\cdot(n\mathbf{v})=0,$ (3)
$\displaystyle\hbar\frac{\partial\phi}{\partial
t}+\frac{1}{2}m^{\ast}v_{x}^{2}+\frac{1}{2}mv_{y}^{2}+V_{trap}+ng-\frac{\Omega}{2}=0$
(4)

where the effective mass for the zero-momentum phase is given by
$m^{\ast}=m(1-\Omega_{c}/\Omega)^{-1}$. The physical velocity field

$\mathbf{v}=(\frac{\hbar}{m^{\ast}}\nabla_{x}\phi-\frac{\eta}{\hbar}\frac{\Omega_{c}}{\Omega}y,\frac{\hbar}{m}\nabla_{y}\phi)$
(5)

includes the canonical velocity field and a novel term associated with the
spin density. In the presence of a synthetic magnetic field, the spin
contribution becomes nonzero, leading to a rigid-like rotational velocity
field [29, 30, 31].

Figure 2: Evolution of total density $n$ and spin density $s_{z}$ of a 2D BEC
in a synthetic magnetic field at various expansion times $t$ (indicated in
each panel). The results are obtained from GP simulations with Raman coupling
$\Omega=6E_{r}$, detuning gradient $\eta=0.005E_{r}$, and anisotropic trapping
frequencies $(\omega_{x},\omega_{y})=2\pi\times(50\sqrt{3},50)$Hz. The Thomas-
Fermi radii at $t=0$ are $R_{x0}\approx 12.3\mu m$ and $R_{y0}\approx 21.5\mu
m$, respectively. As shown in (d), the deflection angle of the spin density is
$\theta(t)$, which is defined as a clockwise rotation angle. At $t=0$,
$\theta(t)=0$. The dark blue and light yellow of the colorbar correspond to
the density minimum and maximum, respectively.

At equilibrium, the total density of a vortex-free condensate is well
described by the Thomas-Fermi distribution, as shown in Fig. 2(a). For
$\eta=0$, the condensate is unpolarized in the zero-momentum phase, implying a
zero spin density. However, for finite $\eta$, the detuning gradient
introduces a spin-dependent potential along the $y$-direction:
$\frac{1}{2}m\omega_{y}^{2}y^{2}-\delta(y)\sigma_{z}=\frac{1}{2}m\omega_{y}^{2}(y\pm
y_{0})^{2}$ (up to a constant) with $y_{0}=\eta k_{r}/(m\omega_{y}^{2})$. As
shown in Fig. 2(b), this results in a separation of the two spin components
along $y$-direction. For small detuning gradient, the total density can be
written as the parabolic form
$n(\mathbf{r},0)=n_{0}(1-x^{2}/R_{x0}^{2}-y^{2}/R_{y0}^{2})$ [32, 33, 34] and
the phase takes the form $\phi_{0}=\alpha_{0}xy$. Here, $n_{0}=2N/(\pi
R_{x0}R_{y0})$ is the peak density, $N$ is the number of atoms, and $R_{\mu
0}$ is the Thomas-Fermi radius along $\mu$-direction. According to Refs. [29,
30], the velocity field and spin density can be explicitly expressed as
$\mathbf{v}=(y(\hbar\alpha_{0}/m^{\ast}-\eta\Omega_{c}/\hbar\Omega),\hbar\alpha_{0}x/m)$
and $s_{z}=2\beta_{0}yn$, respectively, with $\alpha_{0}=2\eta
k_{r}^{2}\omega_{x}^{2}/(\Omega(\omega_{x}^{2}m/m^{\ast}+\omega_{y}^{2}))$ and
$\beta_{0}=k_{r}(\eta+\alpha_{0}\hbar^{2}/m)/\Omega$.

##  III Expansion dynamics in the absence of vortices

Upon turning off the harmonic trap, the condensate starts to expand. Figure 2
shows the typical evolution of the total density and spin density of a BEC in
a synthetic rotational field after various expansion times obtained by
numerically solving the GP equations. We note that the total density of the
condensates can still be reasonably depicted by the Thomas-Fermi distribution
after an expansion time up to $t\sim 4ms$. During the expansion, the two spin
components rotate and separate, resulting in a rapid deflection of the spin
density which is characterized by the angle $\theta(t)$ (as illustrated in
Fig. 2(d)). After reaching a maximum value at the expansion time $t=3.182ms$,
the deflection angle starts to decrease slowly. Unlike the spin density, the
total density only begins to deviate significantly from the Thomas-Fermi
distribution after a much longer expansion time.

The short-time expansion dynamics and the deflection of the spin density can
be accurately predicted from the hydrodynamic theory. Since the total density
maintains the Thomas-Fermi distribution for a short expansion time, it can be
assumed to be

$n(\mathbf{r},t)=n_{0}(t)\left(1-\frac{x^{2}}{R_{x}^{2}(t)}-\frac{y^{2}}{R_{y}^{2}(t)}\right),$
(6)

where $n_{0}(t)=2N/(\pi R_{x}(t)R_{y}(t))$ and $R_{\mu}(t)$ are the peak
density and Thomas-Fermi radii during the expansion.

The phase of the expanding condensates takes the following ansatz

$\phi(t)=\frac{1}{2}\alpha_{x}(t)x^{2}+\frac{1}{2}\alpha_{y}(t)y^{2}+\alpha(t)xy.$
(7)

Substituting this into Eq. (5), we find the velocity field becomes
$v_{x}(t)=\hbar/m^{\ast}(\alpha_{x}(t)x+\alpha(t)y)-\eta\Omega_{c}/(\hbar\Omega)y$,
$v_{y}(t)=\hbar(\alpha_{y}(t)y+\alpha(t)x)/m$. Substituting these expressions
into the hydrodynamic Eq. (3), we obtain

$\displaystyle\alpha_{x}(t)$ $\displaystyle=$
$\displaystyle\frac{m^{\ast}\dot{b}_{x}(t)}{\hbar b_{x}(t)}$ (8)
$\displaystyle\alpha_{y}(t)$ $\displaystyle=$
$\displaystyle\frac{m\dot{b}_{y}(t)}{\hbar b_{y}(t)}$ (9)
$\displaystyle\alpha(t)$ $\displaystyle=$ $\displaystyle\frac{2\eta
k_{r}^{2}\omega_{x}^{2}b_{y}^{2}(t)}{\Omega(\frac{m}{m^{\ast}}\omega_{x}^{2}b_{y}^{2}(t)+\omega_{y}^{2}b_{x}^{2}(t))}$
(10)

where we have introduced $b_{\mu}(t)=R_{\mu}(t)/R_{\mu 0}$ as the expansion
parameter along $\mu$-direction. After turning off the trapping potential
($V_{trap}=0$), using the hydrodynamic Eq. (4) and neglecting higher-order
terms of $\eta$, we find that the expansion parameter $b_{\mu}(t)$ satisfies
the following equation of motion

$\ddot{b}_{\mu}(t)-\frac{\tilde{\omega}_{\mu}^{2}}{b_{\mu}(t)b_{x}(t)b_{y}(t)}=0,$
(11)

with $\tilde{\omega}_{x}=\omega_{x}\sqrt{m/m^{\ast}}$,
$\tilde{\omega}_{y}=\omega_{y}$. The initial conditions can be determined from
the properties of the system at equilibrium. It is easy to find
$\alpha_{x}(0)=\alpha_{y}(0)=0$, $\alpha(0)=\alpha_{0}$, $b_{\mu}(0)=1$ and
$\dot{b}_{\mu}(0)=0$.

Figure 3: Time evolution of the expansion parameters
$b_{\mu}=R_{\mu}(t)/R_{\mu 0}$ (a) and the deformation parameter $\Delta$ (b)
after turning off the harmonic trap. The solid and dashed lines are the
hydrodynamic predictions and the symbols are the results obtained from the GP
simulation. The system parameters are the same as that for Fig. 2.

A particularly interesting scenario occurs when the trapping frequencies
satisfy $\sqrt{m/m^{\ast}}\omega_{x}=\omega_{y}\equiv\omega_{0}$, i.e., when
the two dipole oscillation frequencies are degenerate. Under this condition,
the expansion parameters are equal and can be analytically obtained as
$b_{x}(t)=b_{y}(t)=\sqrt{1+\omega_{0}^{2}t^{2}}$. In the following, we will
focus on this parameter regime.

To better understand the time evolution of the condensate density during the
expansion, we introduce the deformation parameter which is defined as:

$\Delta(t)=\frac{\left\langle y^{2}-x^{2}\right\rangle}{\left\langle
y^{2}+x^{2}\right\rangle}=\frac{R_{y}^{2}(t)-R_{x}^{2}(t)}{R_{y}^{2}(t)+R_{x}^{2}(t)}.$
(12)

For the specific trap frequencies
$(\omega_{x},\omega_{y})=2\pi\times(50\sqrt{3},50)$Hz, the aspect ratio
$R_{x}(t)/R_{y}(t)=1/\sqrt{3}$ and the deformation parameter $\Delta=1/2$. As
shown in Fig. 3, the time evolution of the expansion parameters $b_{\mu}(t)$
and the deformation parameter $\Delta(t)$ from the hydrodynamic theory
predictions agree very well with those obtained from the GP simulations. Note
that in the GP simulation, we numerically compute the Thomas-Fermi radii
during the expansion using the related $R_{\mu}(t)=\sqrt{12\left\langle
x_{\mu}^{2}\right\rangle}$ and subsequently calculate the expansion parameters
and deformation parameter.

Substituting the expression for the total phase (Eq. (7)) into the spin
density expression $s_{z}(t)$ (Eq. 2), we find the following explicit form

$\displaystyle\frac{s_{z}(t)}{n(t)}=\frac{2\hbar^{2}k_{r}}{m\Omega}\alpha_{x}(t)x+2(\frac{k_{r}\eta}{\Omega}+\frac{\hbar^{2}k_{r}}{m\Omega}\alpha(t))y.$
(13)

Since $\alpha(t)\propto\eta$, in the absence of a position-dependent detuning
($\eta=0$), the spin density is zero in the ground state and it becomes
$x$-dependent in the expansion, resulting in a spin separation along $x$ [37].
When $\eta\neq 0$, the two spin components are separated along $y$ even at
equilibrium because $\alpha(t=0)\neq 0$. After the trapping potential is
turned off, the spin density starts to develop an $x$-dependent term. The
presence of both $x$\- and $y$-dependent terms in Eq. (13) implies a rotation
of the spin density characterized by the deflection angle. This angle is
related to the ratio of the $x$\- and $y$-dependent terms in Eq. (13). Thus,
it can be analytically obtained as

$\theta(t)=\frac{\pi}{2}-\arctan(\frac{\eta}{2\hbar}(2-\frac{\Omega_{c}}{\Omega})\frac{1+\omega_{0}^{2}t^{2}}{\omega_{0}^{2}t}).$
(14)

Figure 4: (a) The deflection angle of the spin density as a function of the
expansion time for different detuning gradients $\eta=0$ (red), $0.005E_{r}$
(black), and $0.008E_{r}$ (blue). The maximum deflection angle as a function
of $\eta$ and Raman coupling strength $\Omega$ is shown in (b) and (c),
respectively. In all the panels, the solid lines are the results obtained from
the hydrodynamic theory, while the symbols are obtained by numerically solving
the GP equations. The black and blue dashed horizontal lines in panel (a) mark
the maximum deflection angle $\theta_{max}$ which occurs $t_{c}=1/\omega_{0}$
(indicated by the green dashed vertical line). In (a), the step-wise behavior
of the circles at longer expansion times is related to the numerical
resolution in the GP simulation. The dashed gray line in (c) corresponds to
the asymptotic limit $\Omega\to\infty$.  Figure 5: The center-of-mass motion
of the two spin states after switching off the external trapping potential,
where red (blue) circles and solid lines represent the center-of-mass position
of the spin up (down) state calculated by GP and hydrodynamic equations,
respectively. (a) and (b) show the center-of-mass position versus the
expansion time along $x$ and $y$, respectively, with $\eta=0.005E_{r}$,
$\Omega=6E_{r}$. The trajectory of
$(\bar{x}_{n_{\alpha}},\bar{y}_{n_{\alpha}})$ obtained from (a) and (b) is
shown in (c). The inset panels in (c) show the 2D densities $n_{\alpha}(x,y)$
of the two spin components at an expansion time $t=0$ and $t=3.182ms$, with
the corresponding center-of-mass position indicated by the brown and green
squares, respectively. The black star is the center-of-mass position of total
density, which remains unchanged during the expansion.

The deflection angle predicted from the hydrodynamic theory (Eq. (14)) and
calculated by numerically solving the coupled GP equations are shown in Fig.
4. Without a synthetic magnetic field, the two spin states are overlapped and
they are located at the minimum of harmonic trap at $t=0$. Once released from
the trap, the two components only separate along $x$ and thus the defined
deflection angle for the expanding spin density becomes $\pi/2$ (red line in
Fig. 4(a)). However, in the presence of a synthetic magnetic field, the two
components do not overlap and are separated along $y$ even at $t=0$. Once
released from the trap, the deflection angle increases from zero to the peak
value at $t_{c}=1/\omega_{0}$, and then it slowly decreases. The expression
for the maximum deflection angle $\theta_{max}$ can be easily obtained from
the hydrodynamic theory

$\theta_{max}=\frac{\pi}{2}-\arctan\frac{\eta}{\hbar\omega_{0}}(2-\frac{\Omega_{c}}{\Omega}).$
(15)

Figure 4(b) illustrates the quantitative dependence of $\theta_{max}$ on the
detuning gradient $\eta$ for Raman coupling strength $\Omega=6E_{r}$. The
maximum deflection angle decreases as the detuning gradient increases. Fig.
4(c) shows $\theta_{max}$ versus $\Omega$ for a given detuning gradient
$\eta=0.005E_{r}$. We find that as $\Omega\to\infty$,
$\theta_{max}\to\pi/2-\arctan(2\eta/\hbar\omega_{0})$ (indicated by the dashed
grey line in Fig. 4(c)). Conversely, as $\Omega\to\Omega_{c}$,
$\theta_{max}\to\pi/2-\arctan(\eta/\hbar\omega_{0})$.

Thus far, we have focused on the evolution of total density and spin density
in the expansion dynamics. We may gain further insights by studying the
center-of-mass motion of each component. The density of the individual
components can be easily obtained as $n_{\alpha}=(n\pm s_{z})/2$. From this,
we can derive the time evolution of the center-of-mass position of the two
spin states. Using the definitions
$\bar{x}_{n_{\alpha}}=\int(xn_{\alpha})d\mathbf{r}$,
$\bar{y}_{n_{\alpha}}=\int(yn_{\alpha})d\mathbf{r}$, we find

$\displaystyle\bar{x}_{n_{1,2}}(t)$ $\displaystyle=$
$\displaystyle\pm\frac{N\hbar
k_{r}}{6(\Omega-\Omega_{c})}R_{x0}^{2}\omega_{0}^{2}t,$ (16)
$\displaystyle\bar{y}_{n_{1,2}}(t)$ $\displaystyle=$
$\displaystyle\pm\frac{Nk_{r}\eta}{12\Omega}(1+\frac{m^{\ast}}{m})R_{y0}^{2}(1+\omega_{0}^{2}t^{2}).$
(17)

From these results, we see again that the two components are overlapped along
$x$ at $t=0$ and then they are separated with the separation increasing
linearly in time (Fig. 5(a)). On the other hand, the two components are
separated along $y$ even at $t=0$ and then their separation increases
quadratically in time (Fig. 5(b)). Consequently, the trajectory of the two
expanding components in the $x-y$ plane follows a parabolic curve (Fig. 5(c)).
The deflection angle $\theta(t)$ for the spin density is related to the slope
of the line connecting the origin $(0,0)$ and the position at time $t$, i.e.,
$(\bar{x}_{n_{\alpha}}(t),\bar{y}_{n_{\alpha}}(t))$. Apparently, the
deflection angle becomes maximum when the line is tangential to the trajectory
which happens at $t_{c}=1/\omega_{0}$.

Figure 6: Expansion dynamics in the presence of vortices when $\eta>\eta_{c}$.
(a) and (c) show the expanding total density at $t=0$ and $t=5.019ms$,
respectively, with the corresponding velocity field (arrows) near the center
of vortices shown in (b) and (d). The color of the arrows reflects the
magnitude of the velocity field, where red arrows represent large velocities
and blue ones represent small velocities. The color map of the background in
(b) and (d) illustrates the phase distributions of order parameters, where the
phase changes by $2\pi$ after completing a closed loop around a vortex.
$\gamma(t)$ in (c) and (e) is the rotation angle of the two vortices, defined
positive counterclockwise. (e) and (f) respectively show the total density and
spin density after a long expansion time $t=16.85$ms. The GP simulation
results for the rotation angle of the vortices and the deflection angle of the
spin density are shown in (g) with $\eta=0.0013E_{r}$ and $\Omega=6E_{r}$.

##  IV Expansion dynamics in the presence of vortices

In this section, we consider the expansion dynamics in the presence of
vortices for $\eta>\eta_{c}$. Because of the emergence of vortices, the
density of the condensate is not smooth and one cannot apply the hydrodynamic
theory. Therefore, we focus on the numerical simulation of the GP equations to
demonstrate the expansion dynamics.

The number of the vortices appearing in the condensate depends on the detuning
gradient. With $\eta=0.0013E_{r}$ and $\Omega=6E_{r}$, through imaginary time
evolution of the coupled GP equations, we find that the ground state contains
two vortices located on the $x$-axis and near the origin (see Fig. 6(a)). The
corresponding velocity field and the phase around the vortex cores are
illustrated in Fig. 6(b), showing a negative circulation
$\mathcal{C}=\oint\mathbf{v}\cdot d\mathbf{l}$ as it is opposite to the
direction of the rigid-like rotational velocity field of the background
condensate [19]. After turning off the external trapping potential, in
addition to the separation and rotation of the two components, we find that
the two vortices begin to rotate counterclockwise which is the same as the
direction of the rotational velocity field (Fig. 6(c)). However, the
corresponding velocity field near the vortex cores remains clockwise (Fig.
6(d)). After a long-time expansion of $t=16.85$ms, the total density and spin
density are shown in Fig. 6(e) and (f), respectively. The deflection angle of
the spin density $\theta(t)$ reaches the maximum value at $t=t_{c}$, similar
to the case when $\eta<\eta_{c}$. However, as shown in Fig. 6(g), the rotation
angle $\gamma(t)$ of the two vortices increases linearly with time when the
expansion time $t>t_{c}$. The rotation angle of the two vortices is
predominately related to the rotation of the total density due to the
synthetic rotational field while the deflection of the spin density is mainly
related to the center-of-mass motion of the two spin components.

##  V Conclusion

In summary, we have investigated the expansion dynamics of SO-coupled BECs in
the presence of a position-dependent detuning. We demonstrate how the
generated synthetic magnetic field and rigid-like rotational velocity field
lead to the interesting phenomena of separation and rotation of the two
components. By developing a hydrodynamic theory, we have analytically
determined the maximum deflection angle that occurs at a characteristic time
$t_{c}=1/\omega_{0}$. Our hydrodynamic theory predictions align very well with
numerical GP simulation results. We also have extended the study to the
expansion dynamics after the formation of vortices when the detuning gradient
exceeds a critical value. In addition to the rotation of the spin density, the
vortices undergo additional rotation dynamics, with the rotation angle
increasing approximately linearly for expansion time $t>t_{c}$.

Our finding offers valuable insights into the effects of the synthetic
magnetic field on the expansion dynamics. It may catalyze further experimental
and theoretical explorations on the non-equilibrium dynamics of SO-coupled
BECs in the presence of synthetic magnetic fields and rigid-like rotational
velocity fields. The fact that the rotation of the spin density depends on the
detuning gradient suggests that this system may be a promising candidate for
the development of a magnetic field gradiometer.

###### Acknowledgements.

J.W., H.L., and Y.L. are supported by National Natural Science Foundation of
China (Nos. 11774093, 12074120), Natural Science Foundation of Shanghai (No.
23ZR118700), Innovation Program of Shanghai Municipal Education Commission
(No. 202101070008E00099) and Program of Chongqing Natural Science Foundation
(CSTB2022NSCQ-MSX0585). C.H.L. acknowledges support from Quantum Science
Center (QSC), a Department of Energy (DOE) quantum information science (QIS)
research center, and related earlier experimental work supported in part by
NSF Grant No. PHY-2012185.

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