# Dynamical mean-field driven spinor condensate physics beyond the single-mode
approximation

Jianwen Jie  Homer L. Dodge Department of Physics and Astronomy, The
University of Oklahoma, 440 W. Brooks Street, Norman, Oklahoma 73019, USA
Center for Quantum Research and Technology, The University of Oklahoma, 440 W.
Brooks Street, Norman, Oklahoma 73019, USA  Shenzhen Institute for Quantum
Science and Engineering (SIQSE), Southern University of Science and
Technology, Shenzhen, P. R. China.  Shan Zhong  Homer L. Dodge Department of
Physics and Astronomy, The University of Oklahoma, 440 W. Brooks Street,
Norman, Oklahoma 73019, USA  Center for Quantum Research and Technology, The
University of Oklahoma, 440 W. Brooks Street, Norman, Oklahoma 73019, USA
Qimin Zhang  Homer L. Dodge Department of Physics and Astronomy, The
University of Oklahoma, 440 W. Brooks Street, Norman, Oklahoma 73019, USA
Center for Quantum Research and Technology, The University of Oklahoma, 440 W.
Brooks Street, Norman, Oklahoma 73019, USA  Isaiah Morgenstern  Homer L. Dodge
Department of Physics and Astronomy, The University of Oklahoma, 440 W. Brooks
Street, Norman, Oklahoma 73019, USA  Center for Quantum Research and
Technology, The University of Oklahoma, 440 W. Brooks Street, Norman, Oklahoma
73019, USA  Hio Giap Ooi  Homer L. Dodge Department of Physics and Astronomy,
The University of Oklahoma, 440 W. Brooks Street, Norman, Oklahoma 73019, USA
Center for Quantum Research and Technology, The University of Oklahoma, 440 W.
Brooks Street, Norman, Oklahoma 73019, USA  Q. Guan  Homer L. Dodge Department
of Physics and Astronomy, The University of Oklahoma, 440 W. Brooks Street,
Norman, Oklahoma 73019, USA  Center for Quantum Research and Technology, The
University of Oklahoma, 440 W. Brooks Street, Norman, Oklahoma 73019, USA
Department of Physics and Astronomy, Washington State University, Pullman,
Washington 99164-2814, USA  Anita Bhagat  Homer L. Dodge Department of Physics
and Astronomy, The University of Oklahoma, 440 W. Brooks Street, Norman,
Oklahoma 73019, USA  Center for Quantum Research and Technology, The
University of Oklahoma, 440 W. Brooks Street, Norman, Oklahoma 73019, USA
Delaram Nematollahi  Homer L. Dodge Department of Physics and Astronomy, The
University of Oklahoma, 440 W. Brooks Street, Norman, Oklahoma 73019, USA
Center for Quantum Research and Technology, The University of Oklahoma, 440 W.
Brooks Street, Norman, Oklahoma 73019, USA  A. Schwettmann  Homer L. Dodge
Department of Physics and Astronomy, The University of Oklahoma, 440 W. Brooks
Street, Norman, Oklahoma 73019, USA  Center for Quantum Research and
Technology, The University of Oklahoma, 440 W. Brooks Street, Norman, Oklahoma
73019, USA  D. Blume  Homer L. Dodge Department of Physics and Astronomy, The
University of Oklahoma, 440 W. Brooks Street, Norman, Oklahoma 73019, USA
Center for Quantum Research and Technology, The University of Oklahoma, 440 W.
Brooks Street, Norman, Oklahoma 73019, USA

###### Abstract

23Na spin-1 Bose-Einstein condensates are used to experimentally demonstrate
that mean-field physics beyond the single-mode approximation can be relevant
during the non-equilibrium dynamics. The experimentally observed spin
oscillation dynamics and associated dynamical spatial structure formation
confirm theoretical predictions that are derived by solving a set of coupled
mean-field Gross-Pitaevskii equations [J. Jie et al., Phys. Rev. A 102, 023324
(2020)]. The experiments rely on microwave dressing of the $f=1$ hyperfine
states, where $f$ denotes the total angular momentum of the 23Na atom. The
fact that beyond single-mode approximation physics at the mean-field level,
i.e., spatial mean-field dynamics that distinguishes the spatial density
profiles associated with different Zeeman levels, can—in certain parameter
regimes—have a pronounced effect on the dynamics when the spin healing length
is comparable to or larger than the size of the Bose-Einstein condensate has
implications for using Bose-Einstein condensates as models for quantum phase
transitions and spin squeezing studies as well as for non-linear SU(1,1)
interferometers.

##  I Introduction

Spinor Bose-Einstein condensates (BECs) provide an exciting platform for
exploring—among other phenomena—the dynamics of a quantum pendulum pendulum1 ,
thermal and quantum phase transitions spinorreview1 ; spinorreview2 ;
phasetransition0 ; phasetransition1 ; phasetransition2 ; phasetransition3 ;
phasetransition4 ; phasetransition5 ; phasetransition6 ; phasetransition7 ,
SU(1,1) interferometers interferometer1 ; interferometer2 ; interferometer3 ;
interferometer4 ; interferometer5 ; interferometer6 ; interferometer7 ;
interferometer8 , and the interplay of symmetry and interactions evrard2021a .
Compared to a single-component BEC, the spin degrees of freedom of spinor BECs
lead to rich mean-field and beyond mean-field phases that are characterized by
non-trivial order parameters ho1998 ; spinorreview1 ; spinorreview2 . In some
instances, the spatial orbitals of the different spinor components are, to a
good approximation, the same: While the number of atoms occupying each spinor
component may be different, the shape of the spatial orbital is approximately
independent of the spinor component spinorreview1 ; spinorreview2 ; sma1 ;
sma2 ; sma3 ; sma4 . This single-mode regime is said to be realized when the
spin healing length $\xi_{s}$ is comparable to or larger than the size $R$ of
the BEC dalfovo . If $\xi_{s}\gtrsim R$, then the BEC is too small to support
a ground or low-energy state that exhibits long wavelength inhomogeneities of
the order of the size of the BEC, besides those that exist due to the
finiteness of the BEC. In this case, the densities of the spinor components
all have a maximum at the center of the BEC and decrease monotonically till
they are zero at the edge of the cloud.

This work presents experimental data for a spin-1 BEC, which—in conjunction
with simulations based on a set of coupled mean-field Gross-Pitaevskii
equations—confirm the existence of an alternative mechanism for the creation
of long wavelength density deformations (i.e., density deformations with
characteristic length scale of the size of the BEC). This dynamical mean-field
driven mechanism, which is beyond the single-mode approximation (SMA), has
been recently predicted theoretically jie2020 . It is distinct from the
quantum fluctuation driven processes discussed in Refs. scherer2010 ;
evrard2021  and also distinct from the moving lattice driven process discussed
in Ref. shaw2022 . While our work employs a 23Na BEC, the effect should also
be observable in other spin-1 BECs as well as higher-spin BECs with $s$-wave
contact interactions. The presence of non-local potentials such as dipolar
interactions or spin-orbit coupling, which couple different partial waves,
would likely modify the observations and interpretation thereof.

The phenomenon described in this paper hinges critically on the microwave
tunability of the $f=1$ hyperfine energy levels via coupling to the $f=2$
states zhao2014 ; gerbier2006 ; here, $f$ denotes the total angular momentum
of the atom. Specifically, a combination of external microwave and magnetic
fields is used to adjust the single-particle detuning $q$ between the $m=0$
and $m=\pm 1$ states of the $f=1$ hyperfine manifold. It is well established
that spin-spin interactions, characterized by the spin interaction energy
$c_{s}$, are associated with projection quantum number preserving collisions
between two $m=0$ atoms and a pair of $m=\pm 1$ atoms [see Fig. 1(a)]. These
collisions play an important role in quench-induced oscillations of the
fractional populations $\rho_{m}$ (so-called spin oscillations)
phasetransition3 ; zhao2014 ; chang2005 ; black2007 ; schmaljohann2004 ;
chang2004 ; kron2005 ; kron2006 ; zibold2016 ; gerbier2019 , which are—in the
SMA framework—governed by the ratios $q/c_{s}$ and $c_{s}t/\hbar$, where $t$
denotes the time. In the beyond-SMA scenario considered in this paper, the
single-particle detuning $q$ is adjusted such that projection quantum number
preserving population transfer is facilitated by “activating” a long
wavelength excitation. The resonance occurs at $q/c_{s}$ values that are
larger than the critical value at which the spin oscillation period—predicted
within the SMA—diverges zhang2005 ; spinorreview1 ; spinorreview2 .

Specifically, when $q$ is tuned such that
$E_{\text{gr}}^{(0)}+E_{\text{exc}}^{(0)}$ is equal to
$E_{\text{gr}}^{(+1)}+E_{\text{gr}}^{(-1)}$, the pathway
$|m=0,n_{\rho}=0\rangle+|m=0,n_{\rho}=1\rangle\leftrightarrow|m=+1,n_{\rho}=0\rangle+|m=-1,n_{\rho}=0\rangle$
becomes resonantly enhanced [see Fig. 1(b)] jie2020 . Here,
$E_{\text{gr/exc}}^{(m)}$ denotes the energy of the ground/excited state
(labeled by $n_{\rho}$) that is supported by the effective mean-field
potential associated with the $m$th channel. The effective potentials have a
spatial extent that is set by the density interaction energy $c_{n}$, thereby
supporting an excited state with energy $E_{\text{exc}}^{(m)}$ that sits by an
energy that is comparable to the Thomas-Fermi energy above the ground state
with energy $E_{\text{gr}}^{(m)}$. When the resonance condition is fulfilled,
the quench-induced spin oscillation dynamics is no longer fully captured by
the SMA but instead displays, as illustrated in this work, oscillations that
are characterized by an amplitude and oscillation period that change with
time; we use the term “drifting” to refer to this beyond-the-SMA dynamics.
Since the drifting is captured by the coupled Gross-Pitaevskii equations, the
dynamically induced beyond SMA physics discussed here is mean-field in nature;
quantum fluctuations are not at play.

Figure 1: Schematic illustration of population changing processes in spin-1
BECs for positive $q$. (a) “Standard” spin interaction energy driven process.
The horizontal lines show the single-particle energy levels of the $m=0$ and
$m=\pm 1$ states with single-particle detuning $q$. The relative shift of the
energy levels is due to the effective quadratic Zeeman shift; the energy
contributions due to the linear Zeeman shift are not shown. Spin-changing two-
body collisions, characterized by the spin interaction energy $c_{s}$, lead to
population transfer between the spin-components. (b) Mean-field driven beyond
SMA process. The effective mean-field potentials $V_{\text{eff}}^{(m)}$ (solid
lines; not to scale) felt by the $m=\pm 1$ components (left) and $m=0$
component (right) support ground and radially excited states (the
corresponding densities are represented schematically by dashed and dotted
lines, respectively). If the excitation energy is equal to $2q$, then the
resonant energy condition facilitates (A) population transfer from the $m=\pm
1$ ground state to the $m=0$ ground and excited states (and vice versa) and
(B) population transfer from the $m=0$ excited state to the $m=\pm 1$ ground
and excited states (and vice versa). Process (A) (thick curved black arrows),
which involves one excited state, is “activated” dynamically before process
(B) (thin curved black arrows), which involves two excited states.

The remainder of this paper is organized as follows. Section II outlines the
experimental procedure. Section III summarizes the employed mean-field
formulation and highlights mean-field predictions relevant to the experiment.
Section IV presents and interprets experimental data that evidence the
dynamical emergence of beyond-single-spatial-mode behavior in the mean-field
regime where the spin healing length is comparable to or larger than the size
of the BEC. Last, Sec. V summarizes.

##  II Experimental procedure

Our experiment starts with a nearly pure 23Na BEC in the $f=1$, $m=-1$
hyperfine state in a crossed-beam optical dipole trap. The trapping potential
near the minimum is approximately harmonic and approximately axially
symmetric. The stronger confinement direction aligns with the direction of
gravity. The center-of-mass sloshing motion, induced either by letting the BEC
fall for a short time before recapturing it or by applying a magnetic field
gradient along the $z$-direction, is used to calibrate the angular frequency
$\omega_{z}$. To measure $\omega_{x}$ and $\omega_{y}$, we simultaneously
excite sloshing motions along the $x$\- and $y$-directions. From the combined
motion we deduce that $\omega_{x}$ and $\omega_{y}$ are approximately equal.
In our theory calculations, we set $\omega_{x}=\omega_{y}=\omega_{\rho}$.
Calibration measurements yield trap frequencies with an uncertainty of 3 Hz.
While the trap frequencies are stable for each experimental run, variations on
the order of up to about $10$ % arise over the course of a measurement
campaign that lasts around 100 hours due to fluctuations in the laser power
and room temperature (which leads to changes in the alignment during the
course of the day/night). While the majority of our Gross-Pitaevskii
simulations (see below) utilize $\omega_{z}=2\pi\times 246$ Hz and
$\omega_{\rho}=2\pi\times 140$ Hz, the dependence of the spin oscillations on
$\omega_{\rho}$ is illustrated for select cases.

The $f=1$ hyperfine levels are split by a constant magnetic field of $0.430$
G. The magnetic field corresponds to a quadratic Zeeman shift of the $f=1$,
$m=\pm 1$ levels (in units of $h$) by $51.4$ Hz. For the analysis of the data,
the linear Zeeman shift is irrelevant since it can be removed by going to a
rotated basis spinorreview2 ; sma4 . To prepare the initial state, we apply a
radio-frequency pulse, which transfers atoms from the $m=-1$ state to the
$m=0$ and $m=+1$ states (see Fig. 2). The pulse length is chosen such that the
fractional populations of the $m=+1$, $0$, and $-1$ hyperfine states are, to
within a few percent, equal to $1/4$, $1/2$, and $1/4$, respectively zhao2014
; black2007 .

Figure 2: State preparation via radio-frequency pulse. Fractional populations
$\rho_{m}$ of the $f=1$ hyperfine states are shown as a function of the pulse
length. The symbols show the average of three experimental runs. Starting with
all atoms in the $m=-1$ state (blue triangles), the radio-frequency pulse with
frequency $300$ kHz transfers atoms to the $m=0$ (red circles) and $m=+1$
(black squares) states. The lines are the result of a non-interacting three-
state model, which treats the coupling strength of the radio-frequency pulse
and the magnetic field strength as fitting parameters. The fit yields a
coupling strength, in units of $h$, of $34.2$ kHz and $B=0.430$ G. We estimate
the fluctuations from one initial state preparation to another to be $0.3$ kHz
and $0.001$ G for the coupling strength and magnetic field strength,
respectively.

At the end of the radio-frequency pulse ($t=0$), we quench the system by
rapidly turning on a microwave field, which dresses (i.e., shifts) the $m=\pm
1$ hyperfine states relative to the $m=0$ state. We parametrize the effective
energy shift due to the magnetic field induced quadratic Zeeman shift and the
microwave field induced ac-Stark shift by $q$ zhao2014 ; gerbier2006 . Our
versatile microwave source morgenstern2020 , which has the capability to
modulate the power and frequency on fast time scales, provides access to a
wide range of $q$ values, including positive and negative values zhao2014 .
Throughout this work, we restrict ourselves to positive $q$; we stress,
however, that resonances also exist for negative $q$ jie2020 . Our
experimental determination of the value of $q$ is associated with an
uncertainty of $1$ to $2$ Hz. The value of $q$ is kept constant for
$0<t<t_{\text{hold}}$. The in-trap dynamics of the $f=1$ spinor BEC, i.e., the
quench induced population transfer from the $m=0$ state to the $m=+1$ and $-1$
states, is then monitored as a function of $t_{\text{hold}}$.

At $t=t_{\text{hold}}$, the confining potential is turned off. After $1.5$ ms
of free expansion, a $9$ ms long Stern-Gerlach pulse is applied. After a total
of $10.5$ ms time-of-flight expansion, destructive absorption imaging of the
$m=1$, $0$, and $-1$ components is performed in the plane spanned by the unit
vectors $\hat{e}_{xy}=(\hat{x}+\hat{y})/\sqrt{2}$ and $\hat{z}$. Using
standard techniques, we extract the number of atoms in each of the three spin
components ($m=0$ and $\pm 1$) as well as the two-dimensional density.

##  III Theoretical framework and results

To describe the spin oscillations that ensue in response to the quench at
$t=0$ from $q/h=51.4$ Hz for $t<0$ to its final value, we employ two different
theory frameworks: the mean-field SMA spinorreview1 ; spinorreview2 ; sma1 ;
sma2 ; sma3 ; sma4  and a set of coupled mean-field Gross-Pitaevskii equations
spinorreview1 ; sma2 ; sma3 ; sma4 ; pu2000 . The former assumes that the
spatial orbitals of the three spinor components have an identical shape that
is independent of time. The frozen spatial orbital assumption implies a
decoupling of the spatial and spin degrees of freedom. The spin degrees of
freedom are treated at the mean-field level zhang2005 , i.e., the $m=+1$, $0$,
and $-1$ components are characterized by
$\sqrt{\rho_{m}(t)}\exp[\mathrm{i}\theta_{m}(t)]$, where $\rho_{m}(t)$ and
$\theta_{m}(t)$ denote the population and phase of the $m$th component.
Normalization implies $\rho_{+1}(t)+\rho_{0}(t)+\rho_{-1}(t)=1$. The
differential equations that govern the spin dynamics [$\rho_{0}(t)$ and the
relative phase $\theta(t)$, where $\theta(t)$ is defined as
$2\theta_{0}(t)-\theta_{+1}(t)-\theta_{-1}(t)$] depend on two dimensionless
parameters, namely the ratios $q/c_{s}$ and $c_{s}t/h$. The spin interaction
energy $c_{s}$ is determined by the spin interaction strength $g_{s}$,
$g_{s}=4\pi\hbar^{2}(a_{2}-a_{0})/(3M)$ ($M$ denotes the atom mass), and the
mean spatial density $\overline{n}$ before the application of the radio-
frequency pulse, $c_{s}=g_{s}\overline{n}$. Here, $a_{0}$ and $a_{2}$ denote
the two-body scattering lengths in the two-particle angular momentum channels
$0$ and $2$, $a_{0}=48.91$ $a_{B}$ and $a_{2}=54.54$ $a_{B}$ knoop2011
($a_{B}$ denotes the Bohr radius). The shape of the spatial orbital, and
correspondingly the mean spatial density $\overline{n}$, is determined by
solving a single-component time-independent Gross-Pitaevskii equation, which
depends on the aspect ratio $\lambda$ ($\lambda=\omega_{z}/\omega_{\rho}$) and
the dimensionless interaction strength
$(N-1)g_{n}/(\hbar\omega_{z}a_{\text{ho},z}^{3})$, where
$g_{n}=4\pi\hbar^{2}(2a_{2}+a_{0})/(3M)$ and
$a_{\text{ho},z}^{2}=\hbar/(M\omega_{z})$. For typical atom numbers and trap
frequencies considered in this paper (i.e., $N=2.3\times 10^{4}$,
$\omega_{\rho}=140$ Hz, and $\lambda=1.75$), the associated density
interaction energy $c_{n}$, $c_{n}=g_{n}\overline{n}$, is (in units of $h$)
equal to $589$ Hz. The fact that $c_{n}$ is $28.1$ times larger than $c_{s}$
is typically used to justify the applicability of the SMA. Within the SMA
framework, the spin oscillations are fully periodic (time-independent
oscillation period and time-independent minimum/maximum amplitude) zhang2005 .

To go beyond the SMA, we solve a set of three coupled time-dependent mean-
field Gross-Pitaevskii equations, which depend on five dimensionless
parameters: $(N-1)g_{n}/(\hbar\omega_{z}a_{\text{ho},z}^{3})$, $g_{n}/g_{s}$,
$\lambda$, $q/c_{s}$, and $tc_{s}/\hbar$ jie2020 . This framework allows for
the coupling of the spin and spatial degrees of freedom, which—in the regime
where the SMA breaks down—can lead to modifications of the spin oscillations.
In particular, previous theory work jie2020  predicted that the interplay
between these degrees of freedom induces, for certain parameter combinations,
a resonance-like effect that leads to drifting, i.e., spin oscillations whose
oscillation amplitude, frequency, and mean value are not—as predicted by the
SMA—constant in time. Figures 3-5 show examples of this behavior.

The physical picture behind the drifting is illustrated in Fig. 1. Within the
coupled Gross-Pitaevskii equation framework, the $m$th spinor component feels
an effective time-dependent mean-field potential that is created by its own
spinor wave function as well as the spinor wave functions of the other
components. Neglecting some small terms so that the effective potentials
depend only on the densities of the spinor components and treating the time as
an adiabatic parameter jie2020 , the effective potential
$V_{\text{eff}}^{(m)}(\vec{r},t)$ of the $m$th component supports a ground
state with energy $E_{\text{gr}}^{(m)}$ and excited states with energies
$E_{\text{exc},j}^{(m)}$ at each time. Specializing to positive $q$, a
resonance condition is realized when
$E_{\text{gr}}^{(+1)}+E_{\text{gr}}^{(-1)}$ is equal to
$E_{\text{gr}}^{(0)}+E_{\text{exc},j}^{(0)}$, i.e., when a pair of $m=\pm 1$
atoms is energetically degenerate with two $m=0$ atoms, one in the ground and
one in the excited state of the effective potential felt by the $m=0$
component. The energetic degeneracy enhances projection quantum number
preserving population transfer between the $m=0$ and $m=\pm 1$ modes (and vice
versa). Since the excited state associated with the energy
$E_{\text{exc},1}^{(0)}$ has a “wavelength” or “density modulation” that is of
the order of the size of the BEC, the considered $m=0\leftrightarrow m=\pm 1$
population transfer mechanism leads to dynamically induced density
deformations of the spinor components; for $j\geq 2$, the associated density
deformation is characterized by a smaller length scale. The competition
between the dimensionless energy scales $q/c_{s}$ and
$[E_{\text{gr}}^{(0)}-E_{\text{exc},j}^{(0)}]/c_{s}$ triggers the drifting of
the spin oscillations. For fixed
$(N-1)g_{n}/(\hbar\omega_{z}a_{\text{ho},z}^{3})$, $g_{n}/g_{s}$, and
$\lambda$, the single-particle detuning $q$ provides a knob for tuning the
spinor BEC into and out of resonance.

Figure 3 shows the fractional population $\rho_{0}$, obtained by evolving the
initial state using the time-dependent three-component Gross-Pitaevskii
equation for various $q/h$ for a 23Na BEC consisting of (a) $N=2.3\times
10^{4}$, (b) $N=1.7\times 10^{4}$, and (c) $N=3.1\times 10^{4}$ particles. For
$N=2.3\times 10^{4}$ [Fig. 3(a)], the divergence of the oscillation period at
$q^{*}/h\approx 20$ Hz—which is associated with a separatrix in classical
phase space—is clearly visible and well described by the mean-field spin
model. If the SMA was valid, the spacing of the “green” and “blue” stripes
would be decreasing monotonically as one moves away from the divergence. While
this holds for $q<q^{*}$, “irregularities” are observed for $q/h\approx 40$
Hz; these irregularities are indicative of the drifting that is caused by a
resonance (see the discussion in the context of Fig. 1 above). Figures
3(b)-3(c), which show the dynamics of the spin oscillations for two other $N$
(using a smaller $q$-region), demonstrate that the irregularities depend quite
sensitively on the particle number. The sensitivity of the drifting on the
particle number is an important consideration when interpreting the
experimental data (see Sec. IV).

Figure 3: Fractional population $\rho_{0,\text{GPE}}(t)$ as functions of time
$t$ and Zeeman energy $q$ for $\omega_{z}=2\pi\times 246$ Hz,
$\omega_{\rho}=2\pi\times 140$ Hz, and (a) $N=2.3\times 10^{4}$, (b)
$N=1.7\times 10^{4}$, and (c) $N=3.1\times 10^{4}$; the color coding is given
by the color bar on the right. In (a), the oscillation period diverges at
$q/h\approx 20$ Hz, in agreement with the mean-field SMA result of $q/h\approx
c_{s}/h=20.9$ Hz. Note that the range of $q$ values considered in (a) is
larger than in (b) and (c). This paper focuses on the regime where the spin
dynamics deviate from regular oscillatory behaviors; the red color for
$q/h\approx 40$ Hz signals drifting. It can be seen that the $q$-values for
which the drifting occurs (“fuzzy” red region) move to smaller values with
increasing $N$.

To quantify the deviations between the fractional populations obtained by the
mean-field Gross-Pitaevskii framework [$\rho_{0,\text{GPE}}(t)$] and the SMA-
based mean-field spin model [$\rho_{0,\text{SMA}}(t)$; time-independent
oscillation period and maximum/minimum amplitude], Fig. 4 shows the absolute
value of the normalized difference between the maximum
$\rho_{0,\text{GPE}}^{\text{max}}$ of $\rho_{0,\text{GPE}}(t)$ and the maximum
$\rho_{0,\text{SMA}}^{\text{max}}$ of $\rho_{0,\text{SMA}}(t)$, calculated
using fractional population data for $0<t\leq 80$ ms, as functions of $N$ and
$q$. The quantity
$|\rho_{0,\text{GPE}}^{\text{max}}-\rho_{0,\text{SMA}}^{\text{max}}|/\rho_{0,\text{GPE}}^{\text{max}}$
is obtained for the same trap frequencies as those considered in Fig. 3. A
larger value of
$|\rho_{0,\text{GPE}}^{\text{max}}-\rho_{0,\text{SMA}}^{\text{max}}|/\rho_{0,\text{GPE}}^{\text{max}}$
signals larger drifting. Figure 4 shows that the drifting depends sensitively
on both $N$, which unavoidably fluctuates in experiment, and $q$, which can be
tuned via microwave dressing.

Figure 4: Absolute value of the normalized difference
$[\rho_{0,\text{GPE}}^{\text{max}}-\rho_{0,\text{SMA}}^{\text{max}}]/\rho_{0,\text{GPE}}^{\text{max}}$,
obtained by considering $0<t\leq 80$ ms, as functions of $N$ and $q/h$ for
$\omega_{z}=2\pi\times 246$ Hz and $\omega_{\rho}=2\pi\times 140$ Hz.

Motivated by the fact the experimental trap frequencies can change by up to
about 10 % over the course of a day (see Sec. II), Fig. 5 illustrates the
dependence of the spin oscillations on the angular trapping frequency
$\omega_{\rho}$ along the $\rho$-direction. For both $N$ considered, the
deviations between the fractional populations for different trap frequencies
but otherwise identical parameters initially increases with time ($t\lesssim
30$ ms). For some of the parameter combinations [see, e.g., the red line for
$2\pi\times\omega_{\rho}=130$ Hz in Fig. 5(a) and the black line for
$2\pi\times\omega_{\rho}=120$ Hz in Fig. 5(b)], the upward drift slows after a
finite number of oscillations, followed by a downward drift; this behavior is
indicative of a competition of two energy scales, namely the spin and the
density interaction energies. Looking ahead to the interpretation of the
experimental data, a key message of Fig. 5 is that the spin oscillation
dynamics depends more strongly on the trap frequencies in the vicinity of the
resonance than away from the resonance, i.e., the amount of drifting
depends—when it occurs—sensitively on $\omega_{\rho}$.

Figure 5: Fractional population $\rho_{0,\text{GPE}}(t)$ as a function of time
$t$ for $q/h=40$ Hz, $\omega_{z}=2\pi\times 246$ Hz, and (a) $N=1.7\times
10^{4}$ and (b) $N=2.3\times 10^{4}$. The different lines show results for
different $\omega_{\rho}$, ranging from $2\pi\times 120$ to $2\pi\times 160$
Hz (see the legend on the right of the figure).

##  IV Experimental results

Figure 6: Fractional population $\rho_{0,\text{Exp}}$ as functions of the hold
time $t=t_{\text{hold}}$ and the Zeeman energy $q/h$ determined in two
different experimental campaigns. The trap frequencies are (a)
$(\omega_{x},\omega_{y},\omega_{z})=2\pi\times(147,132,246)$ Hz and (b)
$(\omega_{x},\omega_{y},\omega_{z})=2\pi\times(140,122,255)$ Hz. The red
regions are interpreted as signatures of the drifting, in qualitative
agreement with what is predicted by mean-field Gross-Pitaevskii simulations
(see Fig. 3).

This section presents experimental data that confirm the dynamically induced
beyond SMA spin-oscillation dynamics. Analogous to Fig. 3, Figs. 6(a) and 6(b)
show the fractional population $\rho_{0}(t)$ of the $f=1$, $m=0$ hyperfine
state as functions of the hold time $t$ and $q$ for two separate data
campaigns, corresponding to somewhat different mean atom numbers and trap
frequencies. The two data campaigns used, several months apart, the same
apparatus [the data from Fig. 6(a) are also shown in the Supplemental Material
supplement ]. The data sets shown in Figs. 6(a) and 6(b) are characterized by
mean atom numbers of $2.3\times 10^{4}$ and $2.7\times 10^{4}$, respectively,
with BECs ranging from about $N=1.5\times 10^{4}$ to $N=3.2\times 10^{4}$. The
atom number distributions follow Gaussians with standard deviations of
$\sigma_{N}=1.7\times 10^{3}$ and $1.8\times 10^{3}$ in Figs. 6(a) and 6(b),
respectively. A BEC with $N=2.4\times 10^{4}$, $\omega_{z}=2\pi\times 246$ Hz,
and $\omega_{\rho}=2\pi\times 140$ Hz, e.g., corresponds to Thomas-Fermi radii
along the $\rho$\- and $z$-directions of $R_{\text{TF},\rho}\approx 7.06$
$\mu$m and $R_{\text{TF},z}\approx 0.57R_{\text{TF},\rho}$, respectively. For
comparison, the spin healing length $\xi_{s}$,
$\xi_{s}=\hbar/\sqrt{2M|c_{s}|}$, is about $3.20$ $\mu$m, i.e., the spin
healing length is comparable to the Thomas-Fermi radii in the $\rho$\- and
$z$-directions. In Fig. 6, the hold time and $q$-value are varied in steps of
2 ms and $1$ Hz, respectively. Each experimental data point is the average of
10 measurements. Drifting can be seen in both data sets for $q/h$ values
around $31-32$ Hz, in qualitative agreement with the theoretical mean-field
Gross-Pitaevskii simulations.

To map out the resonance in more detail, the gray histograms in Fig. 7 show
the distribution of the fractional population $\rho_{0,\text{Exp}}(t)$ at
$t=t_{\text{hold}}=60$ ms for nine $q/h$-values; for each $q/h$, the
experiment is repeated 90 times. The data are for the same conditions (i.e.,
same mean particle number and trap frequencies) as Fig. 6(a). The blue solid
lines show normalized Gaussian distributions that are obtained from the mean
value and standard deviation of the experimental data. It can be seen that
$\rho_{0,\text{Exp}}$ (i.e., the fractional population at which the blue lines
take their maximum) changes—as expected—smoothly with $q$: it increases
monotonically for $q/h=27-29$ Hz and subsequently decreases monotonically for
$q/h=29-35$ Hz. The shape of the gray histograms, in contrast, varies
intricately with $q/h$. The distributions for $q/h=27$ Hz and $q/h\geq 33$ Hz
are approximately Gaussian and comparatively narrow; for the other
$q/h$-values, the distributions are less well described by a Gaussian
distribution (fits, not shown, result in larger $\chi^{2}$) and comparatively
broad. The non-Gaussian behavior near $q/h=31$ Hz is attributed to the
resonance, which triggers the drifting that is—as evidenced by our mean-field
Gross-Pitaevskii simulations shown in Fig. 4—associated with a comparatively
strong sensitivity to the particle number. As a consequence, repeated
experiments with a Gaussian atom number distribution lead to a non-Gaussian
distribution of the fractional population in the $m=0$ component.

Figure 7: Distribution of fractional population $\rho_{0,\text{Exp}}$ at
$t=t_{\text{hold}}=60$ ms for
$(\omega_{x},\omega_{y},\omega_{z})=2\pi\times(147,132,246)$ Hz for (a)
$q/h=27$ Hz to (i) $q/h=35$ Hz. The blue lines show Gaussian distributions,
using the mean value and standard deviation of the experimentally measured
$\rho_{0,\text{Exp}}$.

If the system was described accurately by the SMA, the maximum of the
densities $n_{m}(\vec{r},t)$ would always be located at $(\rho,z)=(0,0)$,
where $\rho^{2}$ is equal to $x^{2}+y^{2}$. However, Gross-Pitaevskii
simulations for an axially-symmetric trap and axially-symmetric $m$-dependent
mean-field orbitals show that the beyond SMA spin oscillation dynamics are
associated with density deformations (peak densities that are located at
$\rho\neq 0$), which develop dynamically with increasing time
$t=t_{\text{hold}}$ jie2020 . These deformations oscillate back and forth
between the $m=0$ and $m=\pm 1$ components. While the density deformations are
most pronounced on resonance, they also occur for $q/h$-values below and above
the resonance.

Figures 8(a)-8(d) show integrated two-dimensional Gross-Pitaevskii component
densities $\overline{n}_{m}(e_{xy},z,t)$ as functions of $z$ and
$e_{xy}=(x+y)/\sqrt{2}$ (this is the same representation as employed in the
experimental imaging system) for $q/h=31$ Hz, $\omega_{\rho}=2\pi\times 140$
Hz, $\omega_{z}=2\pi\times 246$ Hz, and $N=4\times 10^{4}$ at two times,
namely $t=50$ ms [Figs. 8(a)-8(b)] and $t=58$ ms [Figs. 8(c)-8(d)]. For this
particle number, the system is close to resonance, as can be seen by
extrapolating the simulation results shown in Fig. 4 to larger $N$. The two
images at the top left show the $m=\pm 1$ densities while the two images at
the bottom left show the $m=0$ density. Since the $(e_{xy},z)$-representation
is “inconsistent” with the axial symmetry of the system, the density
deformations are being partially averaged over. They lead to an elongation
along the $e_{xy}$-direction of the $m=\pm 1$ density at $t=50$ ms [Fig. 8(a)]
and a double-peak structure of the $m=0$ density at $t=58$ ms [Fig. 8(d)].

The experimental images shown in Figs. 8(e)-8(g) are for
$t=t_{\text{hold}}=50$ ms while those shown in Figs. 8(h)-8(j) are for
$t=t_{\text{hold}}=58$ ms; they correspond to the same $q/h$-value as the
theory calculations but smaller atom number. It can be seen that the
experimental data are in qualitative agreement with the Gross-Pitaevskii
simulation results, thereby confirming the beyond SMA dynamics. Gross-
Pitaevskii simulations for the same atom numbers as measured experimentally
show significantly smaller density deformations. This is attributed to
multiple effects. The experimental set-up breaks the axial symmetry, which is
assumed to hold strictly in the theory calculations, weakly. Moreover, the
experimental data may be impacted by small trap frequency variations. Our
simulations show that the resonance position and shape depend—due to the
intricate interplay between the kinetic and potential energy
contributions—sensitively on the exact trap parameters and atom number (see
Figs. 3-5), rendering fully quantitative side-by-side comparisons of theory
and experiment challenging.

Figure 8: Theoretical and experimental spatial integrated two-dimensional
densities $\overline{n}_{m}$ [defined through
$\overline{n}_{m}(e_{xy},z,t)=\int n_{m}(\vec{r},t)d\eta$, where
$\eta=(x+y)/\sqrt{2}$] for $q/h=31$ Hz. (a)-(d) Theoretical densities,
obtained by solving the coupled Gross-Pitaevskii equations for $N=4\times
10^{4}$, $\omega_{\rho}=2\pi\times 140$ Hz, and $\omega_{z}=2\pi\times 246$
Hz; (a) $m=\pm 1$ and $t=t_{\text{hold}}=50$ ms, (b) $m=0$ and
$t=t_{\text{hold}}=50$ ms, (c) $m=\pm 1$ and $t=t_{\text{hold}}=58$ ms, and
(d) $m=0$ and $t=t_{\text{hold}}=58$ ms. The normalization is $\sum_{m}\int
n_{m}(\vec{r},t)d\vec{r}=N$. The time-of-flight sequence is not included in
the simulations. (e)-(j) Experimental two-dimensional images for $10.5$ ms
time-of-flight expansion; (e) $m=-1$ and $t=t_{\text{hold}}=50$ ms, (f) $m=0$
and $t=t_{\text{hold}}=50$ ms, (g) $m=+1$ and $t=t_{\text{hold}}=50$ ms, (h)
$m=-1$ and $t=t_{\text{hold}}=58$ ms, (i) $m=0$ and $t=t_{\text{hold}}=58$ ms,
(j) $m=+1$ and $t=t_{\text{hold}}=58$ ms. The particle numbers $N$ are
$2.6\times 10^{4}$ and $2.2\times 10^{4}$ for (e)-(g) and (h)-(j),
respectively. To aid with the visualization, the three images were centered
individually. The side bar on the right defines the color code for the
experimental images shown in (e)-(j).

##  V Conclusions

This paper presented theory predictions and experimental data for a sodium
spinor condensate that confirm the existence of a dynamically-induced mean-
field driven resonance mechanism that is not captured by the SMA. The physical
picture behind the resonance mechanism is quite simple: When the density and
spin interaction strengths are such that the effective mean-field potentials
support an excited state that leads to an energetic degeneracy, population
transfer between the $m=0$ and $m=\pm 1$ modes is enhanced. For a fixed
single-particle detuning $q$, the mean-field parameters can be adjusted by,
e.g., changing the particle number or trap frequencies. This population
transfer mechanism is distinctly different from the “usual” collision induced
population transfer in which the spin-changing two-body collision term
“triggers” the transfer of population. This process is, in the case where the
density interaction energy is much larger than the spin interaction energy,
captured by the SMA. The resonance mechanism studied in this paper, in
contrast, is not captured by the SMA since it leads to the dynamical
occupation of excited spatial modes. While the experimental observations
reported here are for a spin-1 23Na BEC, the same mechanism exists—according
to the theory—in spin-1 87Rb BECs, which are characterized by a much larger
density-to-spin-interaction-energy ratio.

The results presented in this paper are of relevance to a broad range of
dynamical studies involving spinor BECs. Spinor BECs have, e.g., been used to
study quench-induced dynamical quantum phase transitions, which are supported
by the quantum spin Hamiltonian that is derived by treating the spatial
degrees of freedom within the SMA. The quantum spin Hamiltonian also forms the
starting point of spin squeezing studies and interferometer protocols. The
work presented in this paper shows that attention needs to be paid to the
mean-field parameters to ensure that the SMA provides a faithful description.
The dynamically induced transfer of population to excited modes, which can be
controlled by adjusting the single-particle detuning via microwave dressing,
provides a novel route for studying the coupling between the spin and spatial
degrees of freedom.

## Acknowledgement

We gratefully acknowledge support by the National Science Foundation through
grant numbers PHY-1806259, PHY-2110158, and PHY-1846965 (CAREER), and from the
Department of Defense through AFOSR Grant No. FA9550-20-1-0071. JJ is
supported by the National Natural Science Foundation of China under Grant No.
12104210 and the China Postdoctoral Science Foundation under Grant No.
2022M711496. This work used the OU Supercomputing Center for Education and
Research (OSCER) at the University of Oklahoma (OU).

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