††thanks: These authors contributed equally to this work††thanks: These
authors contributed equally to this work

# Observation of Momentum Space Josephson Effects

Annesh Mukhopadhyay  Department of Physics and Astronomy, Washington State
University, Pullman, WA 99164-2814, USA  Xi-Wang Luo  CAS Key Laboratory of
Quantum Information, University of Science and Technology of China, Hefei,
Anhui 230026, China  Synergetic Innovation Center of Quantum Information and
Quantum Physics, University of Science and Technology of China, Hefei, Anhui
230026, China  Hefei National Laboratory, University of Science and Technology
of China, Hefei 230088, China  Colby Schimelfenig  Department of Physics and
Astronomy, Washington State University, Pullman, WA 99164-2814, USA  M. K. H.
Ome  Department of Physics and Astronomy, Washington State University,
Pullman, WA 99164-2814, USA  Sean Mossman  Department of Physics and
Astronomy, Washington State University, Pullman, WA 99164-2814, USA
Department of Physics and Biophysics, University of San Diego, San Diego, CA
92110, USA  Chuanwei Zhang<EMAIL_ADDRESS>Department of Physics,
The University of Texas at Dallas, Richardson, TX 75080-3021, USA  Department
of Physics, Washington University in St. Louis, St. Louis, MO 63130, USA
Peter Engels<EMAIL_ADDRESS>Department of Physics and Astronomy, Washington
State University, Pullman, WA 99164-2814, USA

###### Abstract

The momentum space Josephson effect describes the supercurrent flow between
weakly coupled Bose-Einstein condensates (BECs) at two discrete momentum
states. Here, we experimentally observe this exotic phenomenon using a BEC
with Raman-induced spin-orbit coupling, where the tunneling between two local
band minima is implemented by the momentum kick of an additional optical
lattice. A sudden quench of the Raman detuning induces coherent spin-momentum
oscillations of the BEC, which is analogous to the a.c. Josephson effect. We
observe both plasma and regular Josephson oscillations in different parameter
regimes. The experimental results agree well with the theoretical model and
numerical simulation, and showcase the important role of nonlinear
interactions. We also show that the measurement of the Josephson plasma
frequency gives the Bogoliubov zero quasimomentum gap, which determines the
mass of the corresponding pseudo-Goldstone mode, a long-sought phenomenon in
particle physics. The observation of momentum space Josephson physics offers
an exciting platform for quantum simulation and sensing utilizing momentum
states as a synthetic degree.

### Introduction.

The Josephson effect describes supercurrents flowing between two reservoirs
with a weak tunneling link (e.g., flow through a thin insulating barrier) [1,
2]. Josephson effects have experimentally been observed in many platforms,
ranging from solid state superconductors [3] to superfluid Helium [4, 5, 6, 7,
8], exciton polaritons [9], and ultra-cold atomic gases [10, 11, 12, 13, 14,
15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26]. Important applications of
Josephson effects include superconducting quantum interference devices
(SQUIDs) [27, 28], superconducting qubits [29, 30, 31, 32], and precision
measurements [27].

In recent years, momentum states of ultracold bosons have emerged as a new
synthetic degree of freedom for quantum matter and simulation. In this
context, the Josephson effect in momentum space has been theoretically
predicted for BECs located at two momentum states with a weak coupling induced
by momentum kicks of laser beams [33]. Such momentum space tunneling has been
implemented in experiments using a Bragg transition for a single component BEC
[34, 35] or an optical lattice in a spin-orbit coupled BEC [36]. Despite
significant experimental progress in the observation of various forms of
quantum dynamics in momentum space lattices (e.g., macroscopic quantum self-
trapping or phase-driven nonlinear dynamics [34, 35]), the momentum space
Josephson oscillation has not been observed in experiments due to the
challenge of realizing a coherent ground state BEC occupying two momentum
states with a long lifetime.

In this Letter, we show experimental evidence for the momentum space Josephson
effect in a spin-orbit coupled BEC [37, 38, 39], whose double well band
dispersion possesses two band minima at different momentum states, in analogy
to real space Josephson junctions. The incorporation of a weak optical lattice
induces a coupling between BECs located at two band minima, leading to the
experimental observation of the long-lived ($>100$ ms) superfluid stripe
ground state [36]. Starting from the stripe ground state, a supercurrent
through the momentum space junction is induced by a sudden quench of the Raman
detuning between two band minima, similar to applying a voltage in a
superconducting Josephson junction. The detuning quench displaces the initial
stripe state from the ground state for the final detuning parameter, leading
to periodic spin-momentum oscillations observed in experiments that are
Josephson oscillations.

We observe two types of Josephson oscillations: i) Josephson plasma
oscillations, which are characterized by a small change in population and
small phase differences between the two BECs, excited through a weak change of
the system ground state; ii) regular Josephson oscillations with a large
population oscillation and a continuous increase (or decrease) of the phase
difference, excited through a large change of the ground state. Our
experimental results show good agreement with a theoretical model based on a
two-mode approximation and numerical simulation based on the nonlinear Gross-
Pitaevskii (GP) equation. The observed constant plateau of the plasma
oscillation frequency in the weak lattice region showcases the important role
of nonlinear interactions. Furthermore, we find that the observed Josephson
plasma oscillation frequency corresponds to the zero quasimomentum gap of the
Bogoliubov excitations in the superfluid stripe phase, which, in our system,
represents the mass of a pseudo-Goldstone mode [40] emerging from explicit
symmetry breaking (here, the weak optical lattice breaks spatial translational
symmetry). Pseudo-Goldstone modes, first proposed in particle physics [41],
have been a long-sought phenomenon in many different fields, and our work
provides one direct experimental evidence for observing such an exotic mode.

### Description of the system.

The experimental setup for the spin-orbit coupled BEC
[[SeeSupplementaryMaterialsat][formoretheoreticaldetailsaboutthemodel,
Josephson&GPdynamics, therelationwithBogoliubovspectrum,
aswellassomeexperimentaldetailsabouttheparametersandBraggspectrum.]sup_mat]
has been described in our previous work [36]. Briefly, a 87Rb BEC is confined
in a cigar-shaped crossed optical dipole trap (Fig. 1 (a)). An external
magnetic field applied along the $x$-axis lifts the degeneracy among the three
Zeeman states ($m_{F}$) in the $F=1$ hyperfine manifold. A pair of 789nm Raman
beams intersecting at approximately $None$ angles with the $x$-axis couples
the $\lvert\uparrow\rangle\equiv|1,-1\rangle$ and
$\lvert\downarrow\rangle\equiv|1,0\rangle$ Zeeman split states (Fig. 1 (b)),
which generates spin-orbit coupling (SOC) in the $x$-direction. The Raman
coupling provides an effective momentum offset of $2\hbar k_{\text{R}}$
between these two pseudospin states. Due to the quadratic Zeeman splitting,
the $|1,+1\rangle$ state is sufficiently decoupled and does not play a
significant role
[[SeeSupplementaryMaterialsat][formoretheoreticaldetailsaboutthemodel,
Josephson&GPdynamics, therelationwithBogoliubovspectrum,
aswellassomeexperimentaldetailsabouttheparametersandBraggspectrum.]sup_mat].
Additionally, two 1064nm laser beams co-propagating with the Raman beams
create a weak stationary optical lattice
$V_{\text{L}}(x)=2\hbar\Omega_{\text{L}}\sin^{2}(k_{\text{L}}x)$ along the
$x$-direction, which provides a $2\hbar k_{\text{L}}$ momentum kick while
keeping the spin unchanged. Additional experimental details involving the
atomic states and energy scales are provided in
[[SeeSupplementaryMaterialsat][formoretheoreticaldetailsaboutthemodel,
Josephson&GPdynamics, therelationwithBogoliubovspectrum,
aswellassomeexperimentaldetailsabouttheparametersandBraggspectrum.]sup_mat].

Figure 1: Illustration of the experimental setup and the momentum space
Josephson effect. (a) A crossed optical dipole trap (red) with two Raman laser
beams (green) co-linear with two optical lattice beams (red) intersecting at
the BEC position in the center. (b) Two-photon Raman transitions within the
$F=1$ hyperfine manifold of 87Rb. (c) Band structure of $H_{\text{SOC}}$ for
$\hbar\Omega_{\text{R}}=2.7\,E_{\text{R}}$ and
$\delta=2\pi\times\qty{500}{Hz}$. (d) Phase space diagram demonstrating
Josephson dynamics for $\hbar\Omega_{\text{L}}=0.5\,E_{\text{R}}$, and
$gn=0.25\,E_{\text{R}}$.

The dynamics of the system can be described by the one-dimensional GP equation

$i\partial_{t}\psi=\left[H_{0}+g|\psi(x,t)|^{2}\right]\psi,$ (1)

where $\psi=\left(\psi_{\uparrow},\psi_{\downarrow}\right)^{T}$ is the two-
component spinor wavefunction with the normalization to the total number of
atoms $N=\int\left|\psi\right|^{2}dx$, $g$ is the density interaction
strength, and $H_{0}=H_{\text{SOC}}+{V}_{\text{L}}(x)$ is the single particle
Hamiltonian with

$H_{\text{SOC}}=(i\partial_{x}+\sigma_{z})^{2}-\frac{\delta}{2}\sigma_{z}+\frac{\Omega_{\text{R}}}{2}\sigma_{x}.$
(2)

Here $\Omega_{\text{R}}$ is the Raman coupling strength, $\delta$ is the
detuning of the two-photon Raman transition, and $\hbar k_{\text{R}}$ and
$E_{\text{R}}=\frac{\hbar^{2}k_{\text{R}}^{2}}{2m}=h\times\qty{1.96}{kHz}$ are
the momentum and energy units.

Fig. 1 (c) shows the momentum-space double-well band dispersion of
$H_{\text{SOC}}$. In the experiment, the period of the optical lattice is set
such that $2\hbar k_{\text{L}}$ equals the separation between the two spin-
orbit band minima [36]. The optical lattice leads to the hopping between the
BECs at the two band minima, in analogy to the tunneling between two
superconductors separated by an insulating barrier in a Josephson junction.
While the optical lattice produces multiple off-resonance couplings in
addition, the momentum space Josephson junction can be more intuitively
understood using a two-mode approximation, i.e., considering only two BEC
modes at two band minima with
$\psi=\left(\phi_{l}\chi_{l}e^{-ik_{\text{L}}x}+\phi_{r}\chi_{r}e^{ik_{\text{L}}x}\right)e^{ik_{\text{b}}x}$,
where $\chi_{j}$ are the spinor wavefunction at two band minima, $\phi_{j}(t)$
are the mode population coefficients, and $k_{\text{b}}$ is the bias momentum
induced by the detuning $\delta$. This model also neglects modes in the
excited band. Denoting $\phi_{j}=\sqrt{n_{j}}e^{i\theta_{j}}$, the GP equation
can be projected as

$\displaystyle\partial_{\tau}{z}$ $\displaystyle=$
$\displaystyle-\sqrt{1-z^{2}}\sin{\theta}$
$\displaystyle\partial_{\tau}{\theta}$ $\displaystyle=$ $\displaystyle\Lambda
z+\frac{z}{\sqrt{1-z^{2}}}\cos{\theta}-\Delta E$ (3)

in terms of the phase difference $\theta=\theta_{l}-\theta_{r}$ and relative
population difference $z=\left(n_{r}-n_{l}\right)/n$, where $n=n_{l}+n_{r}$ is
taken as a constant by neglecting populations in other modes. $\tau=2Kt$,
where $K=\frac{\Omega_{\text{L}}}{2}\chi_{l}^{\ast}\chi_{r}$ describes the
hopping between the two modes ($K$ is chosen to be real without loss of
generality). $\Lambda=-Un/(2K)$, where $U=g|\chi_{l}^{\ast}\chi_{r}|^{2}$
represents the interaction strength of the BECs with two modes. $\Delta
E=(E_{l}-E_{r})/(2K)$, with $E_{j}=\int
dx\chi_{j}^{\ast}H_{0}\chi_{j}+(g+U)n$, is the energy difference between the
two modes. Eq. (3) describes a Bose Josephson junction governed by the
effective Hamiltonian
$H_{\text{eff}}=\frac{\Lambda}{2}z^{2}-\sqrt{1-z^{2}}\cos{\theta}-\Delta Ez$
[15].

A typical phase space diagram is shown in Fig. 1 (d) to illustrate the
momentum space Josephson dynamics. The red fixed point ($z_{0},\theta_{0}$)
corresponds to the equilibrium ground state that can be obtained by finding
the minima of $H_{\text{eff}}$. When the BEC is initially prepared away from
($z_{0},\theta_{0}$), ($z,\theta$) oscillate following periodic orbits in
phase space, corresponding to Josephson oscillation between BECs at two band
minima. There are two different types of oscillating behavior: i) when the BEC
is initially prepared not far from the fixed point, ${\theta}$ only changes
within a small range for the closed periodic orbits, representing Josephson
plasma oscillation shown as the orange dashed line; ii) when the BEC is far
from the fixed point, ${\theta}$ increases (or decreases) continuously through
$[0,2\pi]$, corresponding to regular Josephson oscillation shown as the orange
solid line [15].

### Observation of Josephson oscillation.

In our experiments, we observe momentum space Josephson dynamics after a
sudden quench of the Raman detuning from an initial value $\delta_{i}$ to a
final value $\delta_{f}$, in analogy to the voltage-driven ac Josephson
effect. After the quench, the initially prepared superstripe state at
$\delta_{i}$ is no longer the ground state at $\delta_{f}$. The BEC will then
evolve under the two-mode approximation along a periodic orbit around the
fixed point for $\delta_{f}$, demonstrating the Josephson oscillation. The
starting point of the experiment is the preparation of a superfluid stripe
state through Raman and optical lattice dressing of the BEC [36]. SOC is
generated by adiabatically ramping on the Raman beams such that the Raman
coupling strength $\hbar\Omega_{\text{R}}$ increases from $0$ to
$2.7\,E_{\text{R}}$ in 50ms. During this time, the Raman coupling is far
detuned (typically, $\delta=2\pi\times(\qty{5.5}{kHz}\pm\qty{100}{Hz}$)) from
the resonance. The optical lattice beams are then adiabatically applied,
increasing $\hbar\Omega_{\text{L}}$ from $0$ to the desired strength in 50ms.
Following that, $\delta$ is linearly decreased to a desired value $\delta_{i}$
in 50ms. This adiabatic process prepares the BEC in the superfluid stripe
ground state for $\delta_{i}$.

Figure 2: Spin-momentum oscillation in a double-well SOC BEC. (a)–(c)
Absorption images for $\hbar\Omega_{\text{L}}=0.4\,E_{\text{R}}$ after an
evolution time of $t=0.15$, $1.05$, and $\qty{1.65}{ms}$.

In the case of a Josephson plasma oscillation, $\left(z,\theta\right)$
oscillate along a small closed orbit around $\left(z_{0},\theta_{0}\right)$ in
phase space. We choose a fixed $\delta_{f}=2\pi\times\qty{500}{Hz}$ and
different $\delta_{i}=2\pi\times\left(500+Q\right)$ Hz for different
$\hbar\Omega_{\text{L}}\in\\{0.2,0.4,\cdots,1.4\\}\,E_{\text{R}}$. Proper
reasoning for choosing a finite value of the final Raman detuning
($\delta_{f}$) can be found in
[[SeeSupplementaryMaterialsat][formoretheoreticaldetailsaboutthemodel,
Josephson&GPdynamics, therelationwithBogoliubovspectrum,
aswellassomeexperimentaldetailsabouttheparametersandBraggspectrum.]sup_mat].
Suitable values of the quench frequency $Q$ are chosen such that spin
oscillations are still observable, but the initial
$\left(z_{i},\theta_{i}\right)$ do not deviate significantly from
$\left(z_{0},\theta_{0}\right)$, leading to plasma oscillation. After the
sudden quench $\delta_{i}\rightarrow\delta_{f}$, we let the BEC evolve for a
time $t$ in the presence of the Raman and optical lattice couplings.
Subsequently, the Raman and lattice beams are switched off; the BEC is
released from the crossed optical dipole trap, and a 17.5ms long time of
flight (TOF), along with a briefly applied Stern-Gerlach field, resolves the
BEC into different bare spin-momentum eigenstates. In the absorption images of
the BEC, the two spin states are separated vertically, and for each spin
state, the momentum components are resolved horizontally [36] (Fig. 2). We
measure the total spin polarization
$\langle\sigma_{z}\rangle=(N_{\uparrow}-N_{\downarrow})/(N_{\uparrow}+N_{\downarrow})$
at each time $t$, where $N_{\uparrow}$ and $N_{\downarrow}$ are the total
number of atoms in spin $\lvert\uparrow\rangle$ and $\lvert\downarrow\rangle$
respectively. Notice that
$\langle\sigma_{z}\rangle=\sum_{j}\frac{n_{j}}{n}\langle\chi_{j}|\sigma_{z}|\chi_{j}\rangle$,
which can be written as $\langle\sigma_{z}\rangle=a+bz$ under the two-mode
approximation, with
$2a=\langle\chi_{l}|\sigma_{z}|\chi_{l}\rangle+\langle\chi_{r}|\sigma_{z}|\chi_{r}\rangle$,
and
$2b=\langle\chi_{r}|\sigma_{z}|\chi_{r}\rangle-\langle\chi_{l}|\sigma_{z}|\chi_{l}\rangle$.
Therefore, the spin polarization oscillates with the same frequency as the
Josephson oscillation. For the parameters in Fig. 3, we have
$a=0.0493,b=0.732$
[[SeeSupplementaryMaterialsat][formoretheoreticaldetailsaboutthemodel,
Josephson&GPdynamics, therelationwithBogoliubovspectrum,
aswellassomeexperimentaldetailsabouttheparametersandBraggspectrum.]sup_mat].

Figure 3: Josephson plasma oscillation after Raman detuning quench. (a)–(c)
Oscillation of the spin polarization for three different lattice coupling
strengths (see main text for parameters). The solid black (light blue) curves
represent the sinusoidal fitting of the experimental data (GP simulation
results), while the symbols with error bars are the experimental data points.
(d) Comparison between the observed $\Delta$ (symbols with error bars) and
predicted $\Delta$ obtained from analyzing the Bogoliubov spectrum (solid
green), quench dynamics using the GP equation (thick solid green),
perturbation analysis of the two-mode Josephson model (densely dotted
magenta), and the quench dynamics of Josephson model (dash-dotted red). (e)
Comparison of the Bogoliubov spectrum analysis for $gn=0.25\,E_{\text{R}}$ and
final Raman detuning $\delta_{f}=2\pi\times\qty{500}{Hz}$ (green solid line),
$\qty{400}{Hz}$ (green dashed line), and $\qty{300}{Hz}$ (green densely dash-
dot-dotted line) with the experimental data points for
$\delta_{f}=2\pi\times\qty{500}{Hz}$. The solid orange line represents the
calculated variation of $\Delta$ with $\hbar\Omega_{\text{L}}$ for
$\delta_{f}=2\pi\times\qty{500}{Hz}$ and $gn=0$ (non-interacting case). The
distinct markers represent the oscillation frequency experimentally obtained
for the corresponding $\hbar\Omega_{\text{L}}$ from the time-dependent
$\langle\sigma_{z}\rangle$ plots in (a)–(c) and similar additional plots in
[[SeeSupplementaryMaterialsat][formoretheoreticaldetailsaboutthemodel,
Josephson&GPdynamics, therelationwithBogoliubovspectrum,
aswellassomeexperimentaldetailsabouttheparametersandBraggspectrum.]sup_mat].
The star marker at $\hbar\Omega_{\text{L}}=0$ in (d) and (e) are obtained from
Bragg spectroscopic measurements on a SOC BEC
[[SeeSupplementaryMaterialsat][formoretheoreticaldetailsaboutthemodel,
Josephson&GPdynamics, therelationwithBogoliubovspectrum,
aswellassomeexperimentaldetailsabouttheparametersandBraggspectrum.]sup_mat].

Fig. 3 (a)–(c) shows the oscillations of $\langle\sigma_{z}\rangle$ during the
post-quench time $t$, measured for three different lattice coupling strengths.
The corresponding quench, optical lattice coupling strength, and the spin-
polarization oscillation frequency $(Q,\,\hbar\Omega_{\text{L}},\,\Delta)$ for
the three cases are (a)
($2\pi\times\qty{1.3}{kHz},\,0.2\,E_{\text{R}},\,(0.287\pm
0.007)\,E_{\text{R}}$), (b)
($2\pi\times\qty{200}{Hz},\,1.0\,E_{\text{R}},\,(0.530\pm
0.016)\,E_{\text{R}}$), and (c)
($2\pi\times\qty{400}{Hz},\,1.4\,E_{\text{R}},\,(0.734\pm
0.039)\,E_{\text{R}}$), where the errors in oscillation frequencies represent
the standard errors in $\Delta$ obtained from the sinusoidal fitting of the
corresponding data set. The experimentally observed time evolution of
$\langle\sigma_{z}\rangle$ agrees reasonably well with the numerical results
from directly simulating the quench dynamics using the GP Eq. (1)
[[SeeSupplementaryMaterialsat][formoretheoreticaldetailsaboutthemodel,
Josephson&GPdynamics, therelationwithBogoliubovspectrum,
aswellassomeexperimentaldetailsabouttheparametersandBraggspectrum.]sup_mat].
In Fig. 3 (d), we show experimentally measured Josephson plasma oscillation
frequencies with respect to $\hbar\Omega_{\text{L}}$ and their comparison with
several theoretical models and numerical calculations for the interaction
strength $gn=0.25\,E_{\text{R}}$. The constant Josephson plasma frequency for
small values of $\hbar\Omega_{\text{L}}$ indicates a nonlinear interaction in
the system, as evidenced by the comparison to the non-interacting case (Fig. 3
(e)). Additionally, Fig. 3 (e) shows the dependence of the plasma oscillation
frequencies ($\Delta$) on the Raman detuning ($\delta$), which is an
experimental parameter
[[SeeSupplementaryMaterialsat][formoretheoreticaldetailsaboutthemodel,
Josephson&GPdynamics, therelationwithBogoliubovspectrum,
aswellassomeexperimentaldetailsabouttheparametersandBraggspectrum.]sup_mat].
The different theoretical models and numerical calculations are described
below:

Figure 4: Regular Josephson oscillation after Raman detuning quench. (a)
Oscillation of the spin polarization. The solid black (light blue) curve
represents a sinusoidal fit of the experimental data (GP simulation results).
The dark blue curve represents the GP simulation results for a Raman detuning
quench of $\delta_{i}=2\pi\times\qty{30}{Hz}$ to
$\delta_{f}=-2\pi\times\qty{570}{Hz}$. (b) $z-\theta$ phase space diagram for
$\hbar\Omega_{\text{L}}=0.2\,E_{\text{R}}$ and $gn=0.25\,E_{\text{R}}$ along
with the experimental data. The points shown are obtained by calculating
$\theta$ theoretically for the experimentally measured $z$ values according to
$\langle\sigma_{z}\rangle=a+bz$
[[SeeSupplementaryMaterialsat][formoretheoreticaldetailsaboutthemodel,
Josephson&GPdynamics, therelationwithBogoliubovspectrum,
aswellassomeexperimentaldetailsabouttheparametersandBraggspectrum.]sup_mat].
Contour lines correspond to orbits with different energy.

i) Numerical simulations of GP Eq. (1), which agree well with the experimental
data for all $\hbar\Omega_{\text{L}}$.

ii) Numerical simulations of the effective dynamics (3) from the two-mode
approximation. In such detuning quench-driven dynamics, we have
$\theta_{i}=0$, as seen from the fixed point position in the phase space
diagram. Starting from the initial $z_{i}$ for $\delta_{i}$, we numerically
integrate Eq. (3) and determine the oscillation frequency.

iii) Perturbation analysis of the two-mode dynamics. For the case of a small
quench, we can treat the dynamics around the fixed point
$(z_{0},\theta_{0}=0)$ for $\delta_{f}$ as a perturbation, i.e.,
$z=z_{0}+\delta z$, $\theta=\delta\theta$, yielding

$\displaystyle\partial_{\tau}{\delta z}$ $\displaystyle=$
$\displaystyle-\sqrt{1-z_{0}^{2}}~{}\delta\theta$
$\displaystyle\partial_{\tau}{\delta\theta}$ $\displaystyle=$
$\displaystyle\left[\Lambda+(1-z_{0}^{2})^{-3/2}\right]~{}\delta z,$ (4)

with initial conditions $\delta z(0)=z_{i}-z_{0}$ and
$\delta\theta(0)=\theta_{i}-\theta_{0}=0$. Therefore the oscillation frequency
$\Delta=2K\left[\sqrt{1-z_{0}^{2}}~{}\Lambda+\left(1-z_{0}^{2}\right)^{-1}\right]^{1/2}$.
Note that the initial imbalance $z_{0}$ also depends on the lattice strength,
and as $\hbar\Omega_{\text{L}}\rightarrow 0$, one has $z_{0}\rightarrow-1$ and
$K\rightarrow 0$, leading to a finite $\Delta$. A plot of $z_{0}$ with respect
to $\hbar\Omega_{\text{L}}$ is shown in
[[SeeSupplementaryMaterialsat][formoretheoreticaldetailsaboutthemodel,
Josephson&GPdynamics, therelationwithBogoliubovspectrum,
aswellassomeexperimentaldetailsabouttheparametersandBraggspectrum.]sup_mat].
The analytic expression agrees well with the numerical results in ii) but
deviates from the experimental results and GP simulation significantly in the
large $\hbar\Omega_{\text{L}}$ regime, where higher energy modes in the SOC
band are coupled by the optical lattice, leading to the failure of the two-
mode approximation.

iv) The zero quasimomentum gap of the Bogoliubov excitation spectrum. The
sudden quench of the detuning leads to collective Bogoliubov quasiparticle
excitations of the BEC located at quasimomentum $q=0$ due to the lack of
momentum transfer. When the quench is weak, only the lowest quasiparticle band
is excited. Therefore, the plasma oscillation frequency can be determined from
the lowest Bogoliubov band gap. Details of this analysis and its connection
with the pseudo-Goldstone mode are discussed in the next section.

While the phase varies only within a small range for a Josephson plasma
oscillation, it can continuously vary through $[0,2\pi]$ for a regular
Josephson oscillation. We access this regime by choosing a larger quench
$\delta_{i}=2\pi\times\qty{100}{Hz}$ and $\delta_{f}=-2\pi\times\qty{500}{Hz}$
with $\hbar\Omega_{\text{L}}=0.2\,E_{\text{R}}$ so that the initial $z_{i}$ is
far from the fixed point $z_{0}$. Fig. 4 (a) shows the spin-polarization
oscillation as a function of the post-quench time $t$, demonstrating a large
amplitude of oscillation with a change of sign in $\langle\sigma_{z}\rangle$
and an oscillation frequency of $0.275\pm 0.011\,E_{\text{R}}$. In Fig. 4 (b),
we show a contour plot for the variation of the phase $\theta$ with $z$. The
data points in Fig. 4 (b) are obtained by calculating $\theta$ theoretically
for the experimentally measured $z$ values according to
$\langle\sigma_{z}\rangle=a+bz$
[[SeeSupplementaryMaterialsat][formoretheoreticaldetailsaboutthemodel,
Josephson&GPdynamics, therelationwithBogoliubovspectrum,
aswellassomeexperimentaldetailsabouttheparametersandBraggspectrum.]sup_mat].
The error in $z$ represents the standard error calculated from the standard
deviation in $\langle\sigma_{z}\rangle$. Fig. 4 (a) shows the regular
Josephson oscillation in momentum space, with a suboptimal agreement between
the theory and experiment. The dark blue curve shows the numerical GP
simulation results for a Raman detuning ($\delta$) shift of $\qty{70}{Hz}$,
which is within the range of $\pm\qty{100}{Hz}$ uncertainty and demonstrates a
better agreement.

### Connection with Bogoliubov spectrum and pseudo-Goldstone mode.

As discussed in _iv_), a small sudden Raman detuning quench generates
collective excitations of the BEC from the ground band to the first excited
band. The modified band structure with the incorporation of the optical
lattice and mean field interaction is shown in Fig. 5 (a). Without the optical
lattice and in the absence of a Raman detuning $\delta$, the system has
translational symmetry, and the interaction leads to two gapless Goldstone
modes for the stripe phase [43]. The weak optical lattice breaks the
translational symmetry explicitly, causing one Goldstone mode to become gapped
at zero quasimomentum $q=0$. This mode is then referred to as a pseudo-
Goldstone mode, which is highly relevant in the context of particle and
condensed matter physics [40]. The dependence of this pseudo-Goldstone gap on
the mean-field interaction is shown in Fig. 5 (b). The comparison between the
experimentally measured Josephson plasma oscillation frequency and the
Bogoliubov excitation gap is shown in Figs. 3 (d) and (e). We see that the
Bogoliubov gap agrees well with the experimental measurement as well as the GP
simulation. Clearly, mean-field interactions play an important role in the
zero quasimomentum band gap
[[SeeSupplementaryMaterialsat][formoretheoreticaldetailsaboutthemodel,
Josephson&GPdynamics, therelationwithBogoliubovspectrum,
aswellassomeexperimentaldetailsabouttheparametersandBraggspectrum.]sup_mat].
As expected, the Bogoliubov gaps agree with those obtained from two-mode
Josephson dynamics in shallow lattice regimes. The gap deviates significantly
in deep lattice regimes due to the coupling with higher momentum modes.

Figure 5: (a) Bogoliubov spectrum of $H_{0}$ for
$\hbar\Omega_{\text{R}}=2.7\,E_{\text{R}}$,
$\hbar\Omega_{\text{L}}=1\,E_{\text{R}}$, $\delta=2\pi\times\qty{500}{Hz}$,
and $gn=0.25\,E_{\text{R}}$. (b) Dependence of zero quasimomentum ($q=0$) band
gap ($\Delta$) on $gn$ for $\hbar\Omega_{\text{L}}=0$ (solid) and
$\hbar\Omega_{\text{L}}=1\,E_{\text{R}}$ (dashed).

The connection between the Josephson oscillation frequency and the pseudo-
Goldstone gap can be understood as follows. Consider a small deviation of the
wave function away from the ground state

$\psi=\left[(\phi_{l}^{0}+\delta\phi_{l})\chi_{l}e^{-ik_{\text{L}}x}+(\phi_{r}^{0}+\delta\phi_{r})\chi_{r}e^{ik_{\text{L}}x}\right]e^{ik_{\text{b}}x},$
(5)

that is, Bogoliubov excitations with the two-mode approximation.
$\delta\phi_{l}$ and $\delta\phi_{r}$ can be obtained by solving their
corresponding Bogoliubov equation
[[SeeSupplementaryMaterialsat][formoretheoreticaldetailsaboutthemodel,
Josephson&GPdynamics, therelationwithBogoliubovspectrum,
aswellassomeexperimentaldetailsabouttheparametersandBraggspectrum.]sup_mat,
44]. Comparing with Eq. (4), we find $\delta
z=-2\sum_{j}\text{Re}[(-1)^{j}\delta\phi_{j}^{\ast}\phi_{j}^{0}]$ and
$\delta\theta=-\sum_{j}\text{Im}[(-1)^{j}\delta\phi_{j}/\phi_{j}^{0}]$. By
examining the two lowest modes at $q=0$, one finds that the gapped (gapless)
one gives rise to non-vanishing (vanishing) $\delta z$ and $\delta\theta$.
Therefore, the Josephson plasma oscillation frequency is just the zero
quasimomentum Bogoliubov roton gap.

The symmetry-breaking origin of the pseudo-Goldstone mode can also be
intuitively understood in the effective two-mode dynamics. Without the
coupling $K$ (induced by the optical lattice), Eq. 3 has a $U_{s}(1)\times
U_{a}(1)$ symmetry, where $U_{s}(1)$ corresponds to the simultaneous rotation
of two modes and $U_{a}(1)$ represents equal but opposite phase rotation of
two modes. The spontaneous symmetry breaking leads to two uncoupled gapless
modes (i.e., the Goldstone modes). The introduction of a coupling $K\neq 0$
breaks the symmetry $U_{a}(1)$ and reduces the system symmetry to $U_{s}(1)$.
This $U_{s}(1)$ symmetry is spontaneously broken, and the attendant Goldstone
boson is absorbed and removed from the spectra via the Higgs mechanism [45].
Only the second Goldstone boson corresponding to symmetry $U_{a}(1)$ appears.
The parameter $K$ is the soft breaking parameter of the symmetry $U_{a}(1)$,
and the corresponding excitations become pseudo-Goldstone Bosons.

### Conclusion & discussion.

Our work offers a new experimental platform for designing exotic quantum
matter and engineering quantum simulators utilizing momentum states as a
synthetic degree of freedom. For instance, applying Bragg scattering to the
superfluid stripe ground state, one can measure the Bogoliubov excitation
spectrum at finite quasimomentum, leading to the full characterization of the
long-sought pseudo-Goldstone mode. Because of the spin-momentum coupling, the
density interaction in the superfluid stripe phase could induce strong spin
squeezing, which may be realized in our platform and used for quantum sensing.
Applying small but periodic modulations of the Raman detuning can lead to the
observation of a Shapiro resonance in the momentum space Josephson junction.
These novel quantum phenomena enabled by the momentum space Josephson junction
could potentially be useful for quantum technologies.

###### Acknowledgements.

Acknowledgements: A.M., C.S., M.K.H.O., S.M., and P.E. acknowledge funding
from NSF through Grant No. PHY-1912540. P.E. also acknowledges support through
the Ralph G. Yount Distinguished Professorship at WSU. We acknowledge
experimental support from Ethan Crowell during the initial stage of this
project. C.Z. is supported by AFOSR (FA9550-20-1-0220) and NSF (PHY-2409943,
OMR-2228725, ECC-2411394). X.L. acknowledges support from the Innovation
Program for Quantum Science and Technology (Grant No. 2021ZD0301200), the
National Natural Science Foundation of China (No. 12275203), and the USTC
start-up funding.

##  I Supplementary Materials

### Two-mode model and Josephson dynamics.

To introduce the two-mode model, we begin by writing the wavefunction of the
spinor BEC in the following form

$\psi(x,t)=\sum_{j}\phi_{j}(t)\chi_{j}e^{i(2j-1)k_{\text{L}}x+ik_{\text{b}}x},$
(6)

where the integer $j=-J,...,J+1$ represent the reciprocal lattice vectors and
$J$ is the cutoff of the plane-wave modes. $\chi_{j}$ is the expansion spinor
and $\phi_{j}(t)$ are the coefficients. $k_{\text{b}}$ is the bias momentum
induced by detuning $\delta$. The dynamics are governed by

$i\partial_{t}\psi(x,t)=H_{0}\psi(x,t)+g|\psi(x,t)|^{2}\psi(x,t)$ (7)

with $g$ as the interaction strength.

We are interested in the low energy dynamics with sufficiently weak lattice
and interaction strength, and thus make a two-mode approximation, that is,
keeping only two modes at two band minima with $j=0,1$ and treating the
lattice and interactions as perturbations. Notice that $j=0$ ($j=1$)
corresponds to the left (right) minimum labeled by $l$ ($r$) in the main text.
Under the two-mode approximation, we will use $j=l$ and $j=r$ to represent
$j=0$ and $j=1$. The lattice couples the two modes while the interactions
induce effective attractive interactions for each mode. First, we solve for
the two band minima with $\Omega_{\text{L}}=0$ to obtain $k_{\text{b}}$,
$k_{L}$, $\chi_{l}$, and $\chi_{r}$. Then, we rewrite the dynamic equation by
substituting Eq. 6 into Eq. 7

$\displaystyle i\partial_{t}\phi_{l}(t)$ $\displaystyle=$
$\displaystyle(E_{l}-Un_{l})\phi_{l}(t)-K\phi_{r}(t)$ $\displaystyle
i\partial_{t}\phi_{r}(t)$ $\displaystyle=$
$\displaystyle(E_{r}-Un_{r})\phi_{r}(t)-K^{\ast}\phi_{l}(t)$ (8)

where $E_{j}=\int dx\chi_{j}^{\ast}H_{0}\chi_{j}+(g+U)n$,
$U=g|\chi_{l}^{\ast}\chi_{r}|^{2}$ and
$K=\frac{\Omega_{\text{L}}}{2}\chi_{l}^{\ast}\chi_{r}$ (we set $K$ to be real
without loss of generality), with $n=n_{l}+n_{r}$ a constant and
$n_{j}=|\phi_{j}|^{2}$. The above equation is the Bose Josephson junction
tunneling equation. We can write the wave function as
$\phi_{j}=\sqrt{n_{j}}e^{i\theta_{j}}$, and in terms of the phase difference
$\theta=\theta_{l}-\theta_{r}$ and population difference
$z=\frac{n_{r}-n_{l}}{n}$, Eq. (8) becomes

$\displaystyle\partial_{\tau}{z}$ $\displaystyle=$
$\displaystyle-\sqrt{1-z^{2}}\sin{\theta}$
$\displaystyle\partial_{\tau}{\theta}$ $\displaystyle=$ $\displaystyle\Lambda
z+\frac{z}{\sqrt{1-z^{2}}}\cos{\theta}-\Delta E$ (9)

with $\tau=2Kt$, $\Delta E=(E_{l}-E_{r})/(2K)$ and $\Lambda=-Un/(2K)$. The
effective Hamiltonian reads

$H_{\text{eff}}=\frac{\Lambda}{2}z^{2}-\sqrt{1-z^{2}}\cos{\theta}-\Delta Ez.$
(10)

The ground state ($z_{0},\theta_{0}$) can be obtained by finding the minima of
$H_{\text{eff}}$. The variation of $z_{0}$ with respect to
$\hbar\Omega_{\text{L}}$ is shown in Fig. 6. Without the coupling $K$, the
two-mode system Eq. 8 has a $U_{s}(1)\times U_{a}(1)$ symmetry. The
spontaneous symmetry breaking leads to two uncoupled gapless modes (i.e., the
Nambu-Goldstone modes). The introduction of a coupling $K\neq 0$ breaks the
symmetry $U_{a}(1)$ and reduces the system symmetry to $U_{s}(1)$ (i.e.,
simultaneous rotation of the two modes). This $U_{s}(1)$ symmetry is
spontaneously broken, and the attendant Nambu-Goldstone boson is absorbed and
removed from the spectra via the Higgs mechanism [45]. Only the second Nambu-
Goldstone boson corresponding to symmetry $U_{a}(1)$ (i.e., equal but opposite
phase rotation of the two modes) appears. The parameter $K$ is the soft
breaking parameter of the symmetry $U_{a}(1)$, and the corresponding
excitations become pseudo Nambu-Goldstone Bosons. The frequency associated
with these excitations is correspondingly small.

Figure 6: Initial imbalance ($z_{0}$) as a function of the optical lattice
coupling strength ($\hbar\Omega_{\text{L}}$). All other parameters are the
same as those in Fig. 3 in the main text.

### Treatment of quench dynamics using the GP equation.

Consider the spinor wave function
$\psi(x,t)=[\psi_{\uparrow}(x,t),\psi_{\downarrow}(x,t)]^{T}$. The dynamics
are characterized by the Gross-Pitaevskii (GP) equation

$i\frac{\partial\psi(x,t)}{\partial t}=(H_{0}+V_{\text{int}})\psi(x,t),$ (11)

with

$V_{\text{int}}=\left(\begin{array}[]{cc}g_{\uparrow\uparrow}|\psi_{\uparrow}|^{2}+g_{\uparrow\downarrow}|\psi_{\downarrow}|^{2}&0\\\
0&g_{\downarrow\downarrow}|\psi_{\downarrow}|^{2}+g_{\uparrow\downarrow}|\psi_{\uparrow}|^{2}\end{array}\right).$
(12)

Here, $g_{ss^{\prime}}$ denotes the interaction strength between atoms in $s$
and $s^{\prime}$ states. We can adopt the isotropic approximation for the
interaction and set
$g_{\uparrow\uparrow}=g_{\downarrow\downarrow}=g_{\uparrow\downarrow}=g$,
which is an excellent approximation for 87Rb atoms. Before the quench, the
system is in the ground state, which can be obtained by the imaginary time
evolution of the GP equation. With this ground state as the initial state, the
quench dynamics (i.e., the polarization oscillation) can be obtained by
solving the real-time GP equation.

### Bogoliubov spectrum.

To connect the Josephson oscillation frequency with the Bogoliubov spectrum,
we first calculate the spectrum of Bogoliubov excitations. We write the
deviations of the wavefunctions with respect to the ground state as

$\psi_{s}(x,t)=e^{-i\mu t}\left[\psi_{0s}(x)+\delta\psi_{s}(x,t)\right]$ (13)

where $\psi_{0s}(x)$ is the ground state, and the deviations take the form
$\delta\psi_{s}(x,t)=u_{s}(x)e^{-i\varepsilon
t}+v_{s}^{\ast}(x)e^{i\varepsilon t}$. The amplitudes $u_{s}(x)$ and
$v_{s}(x)$ satisfy the normalization condition
$\sum_{s}\int_{0}^{d}dx[|u_{s}(x)|^{2}-|v_{s}(x)|^{2}]=1$, with
$d=2\pi/k_{\text{L}}$ being the lattice period and $\mu$ the chemical
potential. Substituting Eq. (13) into Eq. (11), we obtain the Bogoliubov
equation as
$\mathcal{H}[u_{\uparrow},u_{\downarrow},v_{\uparrow},v_{\downarrow}]^{T}=\varepsilon[u_{\uparrow},u_{\downarrow},v_{\uparrow},v_{\downarrow}]^{T}$,
where the Bogoliubov Hamiltonian is given by

$\mathcal{H}=\left(\begin{array}[]{cccc}\mathcal{H}_{\uparrow}&\frac{\Omega_{\text{R}}}{2}+g_{\uparrow\downarrow}\psi_{0\uparrow}\psi_{0\downarrow}^{\ast}&g\psi_{0\uparrow}^{2}&g_{\uparrow\downarrow}\psi_{0\uparrow}\psi_{0\downarrow}\\\
\frac{\Omega_{\text{R}}}{2}+g_{\uparrow\downarrow}\psi_{0\uparrow}^{\ast}\psi_{0\downarrow}&\mathcal{H}_{\downarrow}&g_{\uparrow\downarrow}\psi_{0\uparrow}\psi_{0\downarrow}&g\psi_{0\downarrow}^{2}\\\
-g\psi_{0\uparrow}^{\ast
2}&-g_{\uparrow\downarrow}\psi_{0\uparrow}^{\ast}\psi_{0\downarrow}^{\ast}&-\mathcal{H}_{\uparrow}^{\ast}&-\frac{\Omega_{\text{R}}}{2}-g_{\uparrow\downarrow}\psi_{0\uparrow}^{\ast}\psi_{0\downarrow}\\\
-g_{\uparrow\downarrow}\psi_{0\uparrow}^{\ast}\psi_{0\downarrow}^{\ast}&-g\psi_{0\downarrow}^{\ast
2}&-\frac{\Omega_{\text{R}}}{2}-g_{\uparrow\downarrow}\psi_{0\uparrow}\psi_{0\downarrow}^{\ast}&-\mathcal{H}_{\downarrow}^{\ast}\end{array}\right),$
(14)

with

$\mathcal{H}_{\uparrow}=-\partial^{2}/\partial x^{2}+2i\partial/\partial
x-\delta/2+V_{\text{L}}(x)-\mu+2g|\psi_{0\uparrow}|^{2}+g_{\uparrow\downarrow}|\psi_{0\downarrow}|^{2},$
(15) $\mathcal{H}_{\downarrow}=-\partial^{2}/\partial
x^{2}-2i\partial/\partial
x+\delta/2+V_{\text{L}}(x)-\mu+2g|\psi_{0\downarrow}|^{2}+g_{\uparrow\downarrow}|\psi_{0\uparrow}|^{2}.$
(16)

The excitation spectra can be calculated numerically by expanding $u_{s}(x)$
and $v_{s}(x)$ in the Bloch basis. Each excitation spectrum is periodic in
momentum space with the Brillouin zone $[0,2k_{\text{L}}]$ determined by the
lattice period.

The ground state takes the following form

$\psi_{0}(x)=\sum_{j}\phi^{0}_{j}\chi_{j}e^{i(2j-1)k_{\text{L}}x+ik_{\text{b}}x},$
(17)

where the integer $j=-J,...,J+1$ represent the reciprocal lattice vectors and
$J$ is the cutoff of the plane-wave modes. $\chi_{j}$ is the expansion spinor
and $\phi^{0}_{j}$ are the coefficients. $k_{\text{b}}$ is the bias momentum
induced by the detuning $\delta$. In the quasi-momentum frame, the two band
minima become asymmetric with respect to the zero quasi-momentum if the
detuning is non-zero. From the form of the ansatz, one can see that $k_{b}$ is
located at the center of the two band minima. So when there is a non-zero
detuning, $k_{b}$ is also nonzero [36].

The perturbation amplitudes $u_{s}(x)$ and $v_{s}(x)$ are expanded in the
Bloch form in terms of the reciprocal lattice vectors:

$u_{s}(x)=\sum_{m=-M}^{M+1}U_{s,m}e^{i(k_{\text{b}}+q)x+i(2m-1)k_{\text{L}}x},$
(18)
$v_{s}(x)=\sum_{m=-M}^{M+1}V_{s,m}e^{i(k_{\text{b}}+q)x-i(2m-1)k_{\text{L}}x},$
(19)

where $q$ is the Bloch wavevector of the excitations, and $M$ is the cutoff of
the expansion. The whole spectrum can be obtained by substituting the
expansions into the Bogoliubov equation.

The ground state of the system is a striped phase due to the presence of the
optical lattice. The stripe is enforced by the optical lattice, and the
translational symmetry (equivalent to the $U_{a}(1)$ symmetry) is explicitly
broken by the lattice, which is different from the interaction-induced
supersolid stripe phase where the translational symmetry is spontaneously
broken. At zero-momentum, one of the two lowest Bogoliubov bands corresponds
to a gapless Nambu-Goldstone mode (related to the spontaneous breaking of
$U_{s}(1)$ symmetry), the other is a gapped pseudo-Nambu-Goldstone mode
(related to the explicit breaking of $U_{a}(1)$ symmetry).

As mentioned above, without the lattice, the translational symmetry (i.e.,
$U_{a}(1)$ symmetry) may be spontaneously broken by interaction. For example,
anti-ferromagnetic interaction ($g_{\uparrow\downarrow}-g_{\uparrow\uparrow}$)
may lead to the supersolid stripe phase [46, 47, 37, 48]; however, it is very
weak, and the supersolid stripe phase only exits for extremely low values of
$\Omega_{\text{R}}$ and $\delta$, making it difficult to observe in an
experiment. For the dynamics studied in this paper, the effect of such anti-
ferromagnetic interaction is negligible, and we can safely adopt the isotropic
approach by setting
$g_{\uparrow\uparrow}=g_{\downarrow\downarrow}=g_{\uparrow\downarrow}=g$.

We note that the calculation of the correct Bogoliubov spectrum requires
including high momentum modes with large $J,M$. We have set $J=9$ and $M=7$ in
calculating the Bogoliubov spectrum. To show that the Bogoliubov gap at $q=0$
is connected with the Josephson junction oscillation frequency, we consider
only the corresponding excitation. Initially ($t=0$), the wave function of the
excitation reads $\delta\psi_{s}(x,t=0)=u_{s}(x)+v^{*}_{s}(x)$, and we examine
the distributions at the two band minima. That is, we consider the ground
state
$\psi_{0s}\simeq(\psi_{0,ls}e^{-ik_{\text{L}}x}+\psi_{0,rs}e^{ik_{\text{L}}x})e^{ik_{\text{b}}x}$
and the excitation
$\delta\psi_{s}\simeq(\delta\psi_{ls}e^{-ik_{\text{L}}x}+\delta\psi_{rs}e^{ik_{\text{L}}x})e^{ik_{\text{b}}x}$.
The distribution amplitudes $\psi_{0,js}$ and $\delta\psi_{js}$ can be real
under a proper gauge choice and are shown in Fig. 7. We find that
$\delta\psi_{ls}\propto-\psi_{0,ls}$ and $\delta\psi_{rs}\propto\psi_{0,rs}$,
so we can write the wave function as

$\psi=\left[(\phi_{l}^{0}+\delta\phi_{l})\chi_{l}e^{-ik_{\text{L}}x}+(\phi_{r}^{0}+\delta\phi_{r})\chi_{r}e^{ik_{\text{L}}x}\right]e^{ik_{\text{b}}x},$

and initially ($t=0$) we have $\delta\phi_{l}(0)\propto-\phi^{0}_{l}(0)$ and
$\delta\phi_{r}(0)\propto\phi^{0}_{r}(0)$. We find that the gapped Bogoliubov
mode corresponds to $\delta n_{l}\simeq-\delta n_{r}$ with $\delta
n_{j}=\delta\phi_{j}\phi^{0}_{j}$, leading to nonzero initial $\delta z$
(notice that the initial phase difference $\delta\theta=0$ since both
$\phi_{j}^{0}(0)$ and $\delta\phi_{j}(0)$ are real).

Figure 7: Wave functions of the ground state and the gapped Bogoliubov
excitation. (a) Wave function of ground state $\psi_{0,js}$ (unnormalized with
arbitrary unit). (b) Wave function of gapped Bogoliubov excitation
$\delta\psi_{js}$ (unnormalized with arbitrary unit). Parameters are
$\hbar\Omega_{\text{L}}=0.4\,E_{\text{R}}$ and $gn=0.25\,E_{\text{R}}$.

### Explanation of oscillation frequency for $\hbar\Omega_{\text{L}}=0$.

The Raman detuning quench technique leads to the excitation of the gapped
Bogoliubov mode (corresponding to spin excitation), and the interference with
the ground state leads to spin oscillation with an identical frequency as the
Bogoliubov gap. In the weak lattice region, the system is well described by
the two-mode Josephson model. As one quenches the detuning, the system is
quenched away from the fixed point and enters plasma oscillation (i.e., the
spin oscillation). Therefore, in the weak lattice region, when
$\hbar\Omega_{\text{L}}$ is small, both the Bogoliubov gap and the Josephson
plasma frequency are determined by the observed spin oscillation. At
$\hbar\Omega_{\text{L}}=0$, there would be no Josephson oscillation at all, so
the numerical two-mode Josephson oscillating frequency is obtained by taking
the limit $\hbar\Omega_{\text{L}}\rightarrow 0$. Experimentally, this
gap/frequency is obtained using Bragg spectroscopy instead of quench dynamics,
where two Bragg laser beams, collinear with the Raman laser beams of varying
detuning, are pulsed onto a SOC BEC. The obtained frequency from this Bragg
measurement corresponds to the star marker at $\hbar\Omega_{\text{L}}=0$ in
Fig. 3 (d) and (e) in the main text. Also, it is worth noting that the real-
space Josephson dynamics usually rely on the tight trap and do not have an
energy-momentum spectrum. Here, the momentum space Josephson dynamics does not
require a real-space trap. Hence, the momentum is a good quantum number, and
the pseudo-Goldstone excitation can have an energy-momentum spectrum similar
to a pseudo-Goldstone boson.

### Additional experimental details.

In our experiments, we have approximately $2.2\times 10^{5}$ atoms of 87Rb
confined in a harmonic trap with trap frequencies
$\bm{\omega}=(\omega_{x},\omega_{y},\omega_{z})=2\pi(23,154,192)$ Hz. The BEC
is prepared in the $\lvert 1,-1\rangle$ Zeeman state. A 10G external magnetic
field leads to Zeeman splitting of the $F=1$ states. The quadratic Zeeman
shift for the $\lvert 1,+1\rangle$ state is 14.6kHz. Therefore, this state is
out of resonance when near-resonant Raman coupling is applied between the
$\lvert 1,-1\rangle$ and $\lvert 1,0\rangle$ states. The Raman and optical
lattice coupling strengths are denoted as $\hbar\Omega_{\text{R}}$ and
$\hbar\Omega_{\text{L}}$ respectively, with $\hbar k_{\text{R}}$ and $\hbar
k_{\text{L}}$ being the corresponding recoil momentum where
$k_{\text{L}}<k_{\text{R}}$.

At the end of the experiment, the BEC is released from the crossed optical
dipole trap and finally imaged after a 17.5 ms time of flight (TOF). The
absorption images resolve the spin-momentum states, as shown in Fig. 2 in the
main text. These images involve six atomic clouds, of which two are strongly
populated while the others are faint due to the weak optical lattice coupling
to these states. The two-photon coupling between the $\lvert 1,-1\rangle$ and
$\lvert 1,0\rangle$ states effected by the Raman lasers separates the two
strongly populated clouds by $2\hbar k_{\text{R}}$ in momentum space.
Additionally, the optical lattice, coupling the two band minima of the SOC
BEC, induces a momentum kick of $2\hbar k_{\text{L}}$ in each of the spin
states, which effectively leaves the two BECs in the $\lvert\uparrow\rangle$
and $\lvert\downarrow\rangle$ spin-states separated by
$2\hbar\,(k_{\text{R}}-k_{\text{L}})$ in the undressed picture. The lattice
coupling to the other four states in the SOC-dressed and undressed picture is
shown via schematic diagrams in [36].

### Reasoning for the choice of a finite final detuning.

In our system, the repulsive inter-mode interaction is equivalent to
attractive intra-mode interaction. Therefore, a negative curvature is expected
at small $K$ values if we work at zero detuning, as shown in Fig. 8. When
$\delta_{f}=0$, the ground state would be $\left(z_{0},\,\theta_{0}=0\right)$
at large $K$ values, where $K\propto\Omega_{\text{L}}$. Then, the plasma
frequency reads $\sqrt{2K\,(2K-Un)}$, which decreases to zero at $K=Un/2$
(i.e., $\Lambda=-1$). This indicates a phase transition in the weak lattice
regime shown in Fig. 8 by the dashed vertical black line at
$\hbar\Omega_{\text{L}}\simeq 0.17\,E_{R}$. For $K<Un/2$ (i.e., $\Lambda<-1$),
the ground state has $z_{0}^{2}=1-\Lambda^{-2}\neq 0$ ($z_{0}$ can be negative
or positive due to the spontaneous symmetry breaking). So, in the weak $K$
regime, the plasma frequency reads $\sqrt{U^{2}n^{2}-4K^{2}}$, which will
display a negative curvature and hence a nonlinear effect. In this experiment,
we work at a finite final detuning ($\delta_{f}=2\pi\times\qty{500}{Hz}$) due
to the following reasons.

The interaction in our system is not very strong, so its effect is more
significant in the weak lattice regime. However, the phase transition occurs
at a weak lattice depth of $\hbar\Omega_{\text{L}}\simeq 0.17\,E_{R}$.
Therefore, in a weak lattice regime, the plasma frequency is very small in a
large interval, making it hard to measure with high precision. Also, the
ground state preparation is more challenging around the phase-transition
point, requiring long ramp times of the parameters. Moreover, considering the
$\qty{100}{Hz}$ uncertainty in our detuning, the dynamics become more
sensitive to the detuning in the weak lattice regime.

Hence, $\delta_{f}=0$ is not a good choice for measuring the oscillation
dynamics. Also, choosing a finite final detuning does not prevent us from
observing the two types of Josephson oscillation. We chose
$\delta_{f}=2\pi\times\qty{500}{Hz}$, which is not too large but dominates
over the uncertainty.

Figure 8: Comparison of the Bogoliubov spectrum analysis for
$gn=0.25\,E_{\text{R}}$ and Raman detuning $\delta=2\pi\times 500$ Hz (green
solid line), $400$ Hz (green dashed line), and $300$ Hz (green densely dash-
dot-dotted line) with the experimental data points for $\delta=2\pi\times 500$
Hz. The solid orange line represents the variation of $\Delta$ with respect to
$\hbar\Omega_{\text{L}}$ for $\delta=2\pi\times 500$ Hz and $gn=0$ (i.e.,
single particle analysis without interactions). The blue solid line is for
$\delta=0$ and $gn=0.25\,E_{\text{R}}$, where a phase transition occurs at
$\hbar\Omega_{\text{L}}\simeq 0.17\,E_{\text{R}}$ which is represented by the
dashed vertical black line.

### Role of atomic interactions.

Figure 9: (a)–(d) Additional data for the oscillation of
$\langle\sigma_{z}\rangle$. The solid black curves represent the sinusoidal
fitting of the experimental data represented by the distinct markers, and the
light blue curves are the GP simulation results. See text for parameters.

The coherent coupling of two macroscopic quantum states gives rise to the
Josephson phenomenon. In this experiment, we are working with a residual
detuning ($E_{l}\neq E_{r}$), and the constant plasma frequency at small $K$
(proportional to $\Omega_{\text{L}}$) values capture the interaction effect
for $gn=0.25\,E_{\text{R}}$. For the non-interacting case ($gn=0$), shown by
the orange line in Fig. 3 (e) in the main text, the plasma frequency would be
$\sqrt{(E_{l}-E_{r})^{2}+4K^{2}}$ in the weak lattice regime. The observed
constant plateau cannot be produced from the non-interaction curve by
rescaling or shifting experimental parameters.

The real-space atom-atom interaction is repulsive, but in the momentum space,
it is effectively attractive. This can be seen by considering the two momentum
modes. The inter-mode interaction is repulsive because when both modes are
populated, the formed real-space density stripes have high interaction energy,
so the BEC prefers to occupy a single momentum mode due to interaction.
Effectively, this corresponds to attractive intra-mode interaction (i.e., the
effective interaction is negative). As mentioned in the previous section, the
interactions modify the dynamics and lead to a constant plateau of plasma
frequency in a weak lattice regime, which is observed experimentally. Such a
plateau is unique for the negative interaction, making it more difficult to
mix the two modes. Thus, the negative interaction will weaken the effects of
the coupling $K$, leading to the plateau structure.

Figure 10: (a) Schematic of the transitions corresponding to the observed
resonance peaks in Fig. 11 (a)–(c). (b) A summed image of three absorption
images corresponding to the three Bragg resonance peaks in Fig. 11 (a)–(c)
showing all the bare spin-momentum eigenstates $\lvert i\rangle$. Box 1, 2, 3,
5, and 6 encloses atoms in $\lvert\uparrow,\,-2\hbar
k_{\text{L}}\,\hat{x}\rangle$, $\lvert\uparrow,\,0\rangle$,
$\lvert\uparrow,\,2\hbar k_{\text{L}}\,\hat{x}\rangle$,
$\lvert\downarrow,\,-2\hbar(k_{\text{R}}-k_{\text{L}})\,\hat{x}\rangle$, and
$\lvert\downarrow,\,2\hbar k_{\text{R}}\,\hat{x}\rangle$ bare state,
respectively. The number of atoms in box 4 is negligible.

### Experimental parameters.

For our experimental parameters $\hbar\Omega_{\text{R}}=2.7\,E_{\text{R}}$,
$E_{\text{R}}=1960$ Hz, $\delta=2\pi\times\qty{500}{Hz}$, and
$k_{\text{L}}=\sqrt{1-(\Omega_{\text{R}}/4E_{\text{R}})^{2}}k_{\text{R}}$, we
find the right band minimum at $k_{m_{r}}=0.78k_{\text{R}}$ and the left
minimum at $k_{m_{l}}=-0.664k_{\text{R}}$, so that the ideal lattice has
$k_{\text{L}}^{\text{ideal}}=0.722k_{\text{R}}$. However, in the realized
experiment, we have used
$k_{\text{L}}=\sqrt{1-(\Omega_{\text{R}}/4E_{\text{R}})^{2}}k_{\text{R}}=0.7378k_{\text{R}}$.
Since the BEC is initially prepared with momentum $k_{m_{r}}$, the two modes
should be at momenta $k_{m_{r}}=0.78k_{\text{R}}$ and
$k_{m_{r}}-2k_{\text{L}}=-0.6956k_{\text{R}}$. Then we have
$\chi_{r}=[0.331,-0.9436]^{T}$ and $\chi_{l}=[0.9123,-0.4096]^{T}$. $\Delta
E=-0.1874E_{\text{R}}/(2K)$, $\Lambda=-0.4613gn/(2K)$,
$2K=0.6792\Omega_{\text{L}}$. For a given $gn$ and $\Omega_{\text{L}}$, we can
solve for $z_{0}$ and obtain the oscillation frequency. The corresponding
polarization is $\langle\sigma_{z}\rangle=a+bz$ with coefficients
$a=0.0493,b=0.732$.

We note that the dynamics are not sensitive to $k_{m_{r}}$ and $k_{\text{L}}$,
so the effect of slight deviations of $k_{\text{L}}$ from the ideal value is
negligible. On the other hand, the ideal $k_{\text{L}}$ depends on the Raman
detuning. Here, we set $k_{\text{L}}$ to be the ideal value corresponding to
zero detuning.

### Additional plasma oscillation results.

In Fig. 9, we present additional results corresponding to the Bogoliubov
spectrum using the Raman detuning quench technique. This demonstrates the
robust nature of our experiment, which provides access to study the dependence
of $\Delta$ on $\hbar\Omega_{\text{L}}$ for a range of
$\hbar\Omega_{\text{L}}$. The corresponding
$Q,\,\hbar\Omega_{\text{L}},\,\Delta$ (see main text for definition) are (a)
$(2\pi\times\qty{1.0}{kHz},0.4\,E_{\text{R}},(0.287\pm 0.008)\,E_{\text{R}})$,
(b) $(2\pi\times\qty{900}{Hz},0.6\,E_{\text{R}},(0.382\pm
0.004)\,E_{\text{R}})$, (c)
$(2\pi\times\qty{500}{Hz},0.8\,E_{\text{R}},(0.438\pm 0.012)\,E_{\text{R}})$,
and (d) $(2\pi\times\qty{500}{Hz},1.2\,E_{\text{R}},(0.628\pm
0.02)\,E_{\text{R}})$. Each experimental data point presented here and in the
main text is an average over three measurements, and the error bars represent
the standard deviation over those measurements.

Here, we are primarily interested in short-time Josephson oscillations. Within
this small time scale ($\sim$ 2 to 4 ms), decoherence induced by imperfections
such as the atom loss and magnetic field fluctuation still have small effects,
and coherent oscillation of the spin-polarization ($\langle\sigma_{z}\rangle$)
is observed. At long times, the magnetic fluctuations and atom-atom
scatterings would break the coherence between BECs at the two band minima,
leading to damped spin oscillation for long-time dynamics.

Figure 11: A plot of $\langle N^{\text{frac}}\rangle$ with respect to
$\Delta\nu_{\text{Bragg}}$ for a SOC BEC under Bragg spectroscopy. The
parameters are $\hbar\Omega_{\text{R}}=2.7\,E_{\text{R}}$ and
$\delta=2\pi\times\qty{500}{Hz}$. Panels (a)-(f) correspond to box no. 1, 3,
5, 2, 4, and 6 in Fig. 10 (b), and the red lines are Gaussian fits to the
peaks T3, T2, T1, and T4. Here, each gray circle is an average of over three
measurements. An error bar represents the standard deviation of the three
measurements. The observed resonance frequencies corresponding to the peaks
are $(-7.432\pm 0.062)$ kHz, $(4.193\pm 0.031)$ kHz, $(-0.495\pm 0.044)$ kHz,
and $(4.448\pm 0.141)$ kHz in (a)-(c), respectively. Panel (d) corresponds to
Box 2, where most atoms were before pulsing the optical lattice. After the
optical lattice pulse, atoms transfer to the other states enclosed by other
boxes, and hence, there is a dip in $\langle N^{\text{frac}}\rangle$. Panels
(e) and (f) show that the atom numbers are negligible in Box 4 and 6.

### Bragg spectroscopy.

Experimentally, $\hbar\Omega_{\text{L}}=0.2\,E_{\text{R}}$ is the lowest
lattice strength for which a clearly measurable quench response could be
obtained. To extend our measurements to zero lattice strength, Bragg
spectroscopy is employed [49]. For this, an optical lattice of small coupling
strength ($\hbar\Omega_{\text{L}}=0.13\,E_{\text{R}}$) and of a varying
detuning ($\Delta\nu_{\text{Bragg}}$) is pulsed onto the SOC BEC for $1$ ms.

The fractional population $N^{\text{frac}}_{i}$ of each bare spin-momentum
eigenstate $\lvert i\rangle$ is measured after 17.5ms of TOF. It is defined as
$N^{\text{frac}}_{i}=N_{i}/N_{\text{total}}$, where $N_{i}$ is the number of
atoms in the eigenstate $|i\rangle$ and $N_{\text{total}}$ is the total number
of atoms (Fig. 10 (b)). Three experimental data sets are collected, and the
expectation value of the fractional population $\langle
N^{\text{frac}}_{i}\rangle$ is computed for each $\Delta\nu_{\text{Bragg}}$.
In Fig. 11, a plot of $\langle N^{\text{frac}}_{i}\rangle$ with respect to
$\Delta\nu_{\text{Bragg}}$ is shown, where each observed peak can be explained
by referring to the transitions shown in Fig. 10 (a) using the single-particle
SOC band structure. The observed resonant frequencies -0.495kHz, 4.193kHz,
-7.432kHz, and 4.448kHz correspond to the transitions T1, T2, T3, and T4 in
Fig. 10 (a), where the $(\pm)$ signs refers to positive and negative
$\Delta\nu_{\text{Bragg}}$. At resonance, a positive detuning imparts a
$2\hbar k_{\text{L}}\,\hat{x}$ momentum kick, whereas a negative detuning
imparts a $-2\hbar k_{\text{L}}\,\hat{x}$ momentum kick. According to the
single-particle SOC band structure, the predicted resonant frequencies for the
transitions T1, T2, T3, and T4 are -0.368kHz, 3.61kHz, -7.645kHz, and 4.448kHz
respectively. The difference between the theoretical values and experimental
observations is attributed to the mean-field interactions. The measured energy
$h\times\lvert(-0.495\pm 0.044)\rvert$ kHz associated with the transition T1
is the limiting value of the Josephson plasma frequency and the zero
quasimomentum band gap for $\hbar\Omega_{\text{L}}=0$.

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