# All-optical linear polarization engineering in single and coupled exciton-
polariton condensates

I. Gnusov<EMAIL_ADDRESS>Skolkovo Institute of Science and
Technology, Moscow, Territory of innovation center “Skolkovo”, Bolshoy
Boulevard 30, bld. 1, 121205, Russia.  H. Sigurdsson<EMAIL_ADDRESS>Skolkovo Institute of Science and Technology, Moscow, Territory of innovation
center “Skolkovo”, Bolshoy Boulevard 30, bld. 1, 121205, Russia.  School of
Physics and Astronomy, University of Southampton, Southampton, SO17 1BJ, UK.
J. D. Töpfer  Skolkovo Institute of Science and Technology, Moscow, Territory
of innovation center “Skolkovo”, Bolshoy Boulevard 30, bld. 1, 121205, Russia.
S. Baryshev  Skolkovo Institute of Science and Technology, Moscow, Territory
of innovation center “Skolkovo”, Bolshoy Boulevard 30, bld. 1, 121205, Russia.
S. Alyatkin  Skolkovo Institute of Science and Technology, Moscow, Territory
of innovation center “Skolkovo”, Bolshoy Boulevard 30, bld. 1, 121205, Russia.
P. G. Lagoudakis  Skolkovo Institute of Science and Technology, Moscow,
Territory of innovation center “Skolkovo”, Bolshoy Boulevard 30, bld. 1,
121205, Russia.  School of Physics and Astronomy, University of Southampton,
Southampton, SO17 1BJ, UK.

###### Abstract

We demonstrate all-optical linear polarization control of exciton-polariton
condensates in anisotropic elliptical optical traps. The cavity inherent TE-TM
splitting lifts the ground state spin degeneracy with emerging fine structure
modes polarized linear parallel and perpendicular to the trap major axis with
the condensate populating the latter. Our findings show a new type of
polarization control with exciting perspectives in both spinoptronics and
studies on extended systems of interacting nonlinear optical elements with
anisotropic coupling strength and adjustable fine structure.

Introduction. — Exciton-polaritons (polaritons hereafter) arise in the strong
coupling regime between quantum well excitons and cavity photons in
semiconductor microcavities Kavokin  _et al._ (2007). Being composite bosons,
they can undergo a power-driven nonequilibrium phase transition into a highly
coherent many-body state referred as polariton condensation Kasprzak  _et al._
(2006). An essential characteristic of polaritons is their spin projection
($\pm\hbar$) onto the growth axis of the cavity which corresponds to the right
and left circular polarizations of their photonic part.

The strong nonlinear nature of polaritons through their spin-anisotropic
excitonic Coulomb interactions results in numerous intriguing spinor
condensate properties. This includes spin bistability Pickup  _et al._ (2018);
del Valle-Inclan Redondo  _et al._ (2019); Sigurdsson (2020) and
multistability Paraïso  _et al._ (2010), switches Amo  _et al._ (2010); Cerna
_et al._ (2013), optical spin Hall effect Leyder  _et al._ (2007), polarized
solitons Hivet  _et al._ (2012); Sich  _et al._ (2018) and vortices Lagoudakis
_et al._ (2009); Donati  _et al._ (2016), bifurcations Ohadi  _et al._ (2015),
and topological phases Bleu  _et al._ (2016); Sigurdsson  _et al._ (2019).
Aforementioned opens great prospects for the utilization of the polariton spin
degree of freedom in future spinoptronic technologies Shelykh  _et al._
(2009); Liew  _et al._ (2011). Different parts for future polariton based spin
circuitry have already been realized Amo  _et al._ (2010); Cerna  _et al._
(2013); Gao  _et al._ (2015); Dreismann  _et al._ (2016); Askitopoulos  _et
al._ (2018) with some recent exciting theoretical proposals Sedov  _et al._
(2019); Mandal  _et al._ (2020), but many challenges are still yet to be
solved. Indeed, optical applications such as data communication or sensing
benefit from precise control over a laser’s polarization and modulation
speeds, ideally using nonresonant excitation schemes like spin-VCSEL
technologies Ostermann and Michalzik (2013); Lindemann  _et al._ (2019); Drong
_et al._ (2021). In this spirit, a great deal of effort has been devoted to
generating sources of linearly polarized light such as colloidal nanorods Hu
_et al._ (2001), materials with anisotropic optical properties Wang  _et al._
(2015), quantum dots integrated into exotic structures Lundskog  _et al._
(2014), and with optical parametric oscillators in the strong coupling regime
Krizhanovskii  _et al._ (2006).

Under nonresonant excitation in inorganic semiconductors, spin transfer from
the pumping laser to the condensate is possible by creating a spin-imbalanced
gain media for the circularly polarized polaritons (i.e., optical orientation
of excitons) using an elliptically polarized beam del Valle-Inclan Redondo
_et al._ (2019); Gnusov  _et al._ (2020). This allows generating polariton
condensates of high degree of circular polarization aligned with the pump.
However, in such systems the linearly polarized polariton modes experience
isotropic gain, making it not possible to influence the linear polarization of
the condensate under nonresonant excitation Ohadi  _et al._ (2012); Baumberg
_et al._ (2008) except in the presence of cavity strain and birefringence
Martín  _et al._ (2005); Kłopotowski  _et al._ (2006); Kasprzak  _et al._
(2007); Balili  _et al._ (2007); Read  _et al._ (2009); Gnusov  _et al._
(2020) or anisotropic confinement Gerhardt  _et al._ (2019); Klaas  _et al._
(2019) inherent to the engineering of the cavity. The same also applies for
VCSEL cavities, where the linear polarization of the emission is engineered by
etching asymmetric masks Xiang  _et al._ (2018); Gayral  _et al._ (1998) or
electrodes Choquette and Leibenguth (1994), heating Pusch  _et al._ (2017), or
by applying mechanical stress Lindemann  _et al._ (2019). Alternatively, in
organic polaritonics, single-molecule Frenkel excitons can be excited by a
linearly polarized pump co-aligned with their dipole moment with condensation
into a mode with the same linear polarization as the pump Plumhof  _et al._
(2013). However, control over both circular and linear polarization degrees of
freedom in a polariton condensate through nonresonant all-optical means,
instead of engineering specific cavity systems, remains elusive.

Here, we demonstrate in-situ optical engineering of the linear polarization in
inorganic polariton condensates in a cavity with polarization-dependent
reﬂectivity, or TE-TM splitting Panzarini  _et al._ (1999); Leyder  _et al._
(2007). By spatially shaping the nonresonant excitation laser transverse
profile into the form of an ellipse, we are able to fully control the
direction of the condensate linear polarization. Our elliptically shaped
pumping profile induces an anisotropic in-plane trapping potential and gain
media for the condensate. Such an excitation profile along with the cavity TE-
TM splitting leads to condensation (lasing) into a mode of definite linear
polarization parallel to the minor axis of the trap ellipse. The optical
malleability of the trap geometry allows for non-invasive, yet deterministic,
control over the linear polarization of the condensate by just utilizing the
nonresonant excitation laser. Moreover, we investigate the effects of the
anisotropic coupling mechanism between two spatially separated condensates and
identify regions—as a function of coupling strength—of correlated high degree
of random linear polarization between the condensates, and otherwise complete
depolarization.

Figure 1: Spatial in-plane profiles of the (a,b) excitation laser and (c,d)
condensate PL. The excitation laser induces a trapping potential with
horizontal and vertical radii denoted $a,b$ respectively. (e) Momentum
distribution of the condensate PL. Panels (c,d,e) correspond to a condensate
pumped twice above its condensation threshold (i.e., $P=2P_{th}$).

Results. — Our experiments are conducted on an inorganic $2\lambda$
GaAs/AlAs0.98P0.02 microcavity with embedded InGaAs quantum wells Cilibrizzi
_et al._ (2014). The sample is excited nonresonantly by a linearly polarized
continuous wave (CW) laser ($\lambda=783.6$ nm). The optical excitation beam
is chopped using an acousto-optic modulator to form 10 $\mu$s square pulses at
1 kHz repetition rate to diminish heating of the sample held at a temperature
of 4 K. The exciton-cavity mode detuning is $-3$ meV. A reflective, liquid-
crystal spatial light modulator (SLM) transforms the transverse profile of the
pump laser beam to have an elliptically shaped confinement region [see Fig.
1(a) and dashed ellipse in Fig. 1(b)]. We investigate the sample PL in real
[Fig. 1(c,d)] and reciprocal [Fig. 1(e)] space, and record the time- and
space-averaged polarization of the PL by simultaneously detecting all
polarization components Gnusov  _et al._ (2020). Our results are independent
on the angle of linear polarization of the pump laser [see Sec. S1 in the
Supplemental Information (SI)].

The polariton condensate can be described by an order parameter written in the
canonical spin-up and spin-down basis $\Psi=(\psi_{+},\psi_{-})^{T}$
corresponding to left- and right-circularly polarized condensate emission,
respectively. It is then convenient to represent the condensate as a
pseudospin on the Poincaré sphere corresponding to the Stokes vector
(polarization) of the emitted light
$\mathbf{S}=(S_{1},S_{2},S_{3})^{T}=\langle\Psi^{\dagger}\boldsymbol{\hat{\sigma}}\Psi\rangle/\langle\Psi^{\dagger}\Psi\rangle$
where $\boldsymbol{\hat{\sigma}}$ is the Pauli matrix vector. The PL is
analyzed in terms of time-averaged Stokes components which are written as,

$S_{1}=\frac{I_{H}-I_{V}}{I_{H}+I_{V}},\
S_{2}=\frac{I_{D}-I_{A}}{I_{D}+I_{A}},\
S_{3}=\frac{I_{\sigma^{+}}-I_{\sigma^{-}}}{I_{\sigma^{+}}+I_{\sigma^{-}}},$
(1)

where $I_{H,V,D,A,\sigma^{+},\sigma^{-}}$ are the time-averaged intensities of
horizontal, vertical, diagonal, antidiagonal, right- and left circular
polarization projections of the emitted light.

Figure 2: (a) Power dependence of the condensate $S_{1,2,3}$ Stokes parameters
(black, red, blue markers) for the annular trap with resultant cyllindrically
symmetric condensate profile. (b) Same but now for a trap/condensate with a
major axis orientated at $90^{\circ}$, (c) $45^{\circ}$, and (d) $0^{\circ}$.
Black lines in the insets depict the orientation of the trap major axis.

We start by exciting with a symmetric ring-shaped pump profile [see inset in
Fig. 2(a)], creating a two-dimensional trap for the polaritons and obtaining
condensation with polaritons dominantly populating the trap ground state by
ramping the pump power above the polariton condensation threshold denoted
$P_{th}$. The optical trap is realized by the strong polariton repulsive
interactions with the background laser-induced cloud of incoherent excitons
which, in the mean field formalism, form a blueshifting potential onto the
polaritons Askitopoulos  _et al._ (2013), while at the same time providing
gain to the condensate. Such an optical trapping technique has the advantage
of reducing the overlap between the condensate and uncondensed excitons,
minimizing detrimental dephasing effects. By scanning the excitation position
with the ring-shaped pump profile we locate a spot on our sample with small
degree of polarization $\text{DOP}=\sqrt{S_{1}^{2}+S_{2}^{2}+S_{3}^{2}}$ [see
Fig. 2(a)]. The small $S_{1,2}$ implies that the trap ground state is spin-
degenerate such that from realization to realization random linear
polarization builds up which averages out over many shots. The small $S_{3}$
component confirms that our laser excitation is (to a good degree) linearly
polarized and doesn’t break the spin parity symmetry of the system. We
additionally investigate the condensate pumped with elliptical polarization in
Sec. S2 in the SI.

We then transform the excitation profile to the one shown in Figs. 1(a) and
1(b). Non-uniform distribution of the intensity in the excitation leads to the
formation of an elliptically shaped optical trap denoted by the dashed
ellipse, squeezing the condensate as shown in Fig. 1(d). We now observe a
massive increase of the condensate’s linear polarization components above
1.2$P_{th}$ at the same sample position. The direction of the linear
polarization of the emission is found to follow the trap minor axis. Namely,
for the vertically elongated condensate in Fig. 2(b) we observe an increase of
the $S_{1}$ Stokes component (horizontal polarization). The same effect is
present for the horizontally and diagonally elongated condensates in Figs.
2(c) and 2(d).

Figure 3: (a) Condensate Stokes parameters for different orientations of the
condensate major axis in real space (insets) at $P=1.94P_{th}$. Black lines
depict the major axis of the trap. Colored regions show error of the
measurement. (b) $S_{1}$ and (c) $S_{2}$ for varying pump powers and major
axis orientation.

By rotating the excitation profile with the SLM, we can engineer any desired
linear polarization in the condensate. In Fig. 3(a), we present the measured
polarization components of the condensate as a function of the condensate
major axis angle. We observe a continuous rotation of the condensate
polarization close to the equatorial plane of the Poincaré sphere following
the minor axis of the trap. We also tested a different geometrical
construction of the elliptical excitation profile with the same outcome (see
Sec. S4 in SI). We point out that $\text{DOP}<1$ appears from various
depolarizing effects such as noise due to scattering from the incoherent
reservoir to the condensate Read  _et al._ (2009), polariton-polariton
interactions in the condensate causing self-induced Larmor precessions Ryzhov
_et al._ (2020), and mode competition Redlich  _et al._ (2016). We also note
that the finite $S_{3}$ component comes from optical elements in the detection
path of our setup.

Figures 3(b) and 3(c) show pump power and trap orientation dependence of the
$S_{1,2}$ Stokes parameters. Interestingly, with increasing pump power we
observe counterclockwise rotation of the pseudospin in the equatorial plane of
the Poincaré sphere. The rotation is approximately $30^{\circ}$ between 1.2
and $2.2P_{th}$. This effect appears due to a small amount of circular
polarization in our pump which creates a spin-imbalanced trapping potential
and gain media which acts as a complex population-dependent out-of-plane
magnetic field $\boldsymbol{\Omega}_{\perp}$ that applies torque on the
condensate pseudospin. This is confirmed through simulations using the
generalised Gross-Pitaevskii equation (see Sec. S10 in SI). Further analysis
on this power dependent trend of the $S_{1,2}$ is beyond the scope of the
current study.

Our observations can be interpreted in terms of photonic TE-TM splitting
acting on the optically confined polaritons which, when the trap
$V(\mathbf{r})$ has broken cylindrical symmetry, leads to fine structure
splitting in the trap transverse modes. This determines a state of definite
polarization which the polaritons condense into. In the noninteracting
(linear) regime the polaritons obey the following Hamiltonian,

$\hat{H}=\frac{\hbar^{2}\boldsymbol{k}^{2}}{2m}-\boldsymbol{\hat{\sigma}}\cdot\boldsymbol{\Omega}+V(\mathbf{r})-\frac{i\hbar\Gamma}{2},$
(2)

where $m$ is the polariton mass, $\mathbf{k}=(k_{x},k_{y})$ is the in-plane
cavity momentum, $\Gamma^{-1}$ is the polariton lifetime, and

$\boldsymbol{\Omega}=\hbar^{2}\Delta\left(k_{x}^{2}-k_{y}^{2},\ 2k_{x}k_{y},\
0\right)^{\text{T}},$ (3)

is the effective magnetic field [see Fig. 4(a)] coming from the TE-TM
splitting of strength $\Delta$ Leyder  _et al._ (2007). In the considered case
of an elliptical confinement, which we assume to be harmonic for simplicity
$V(\mathbf{r})=m(\omega_{x}^{2}x^{2}+\omega_{y}^{2}y^{2})/2$, the TE-TM
splitting results in an effective magnetic field acting on the polariton
pseudospin which splits the trap spin-levels. For the lowest (fundamental)
harmonic state where most of the polaritons are collected this field can be
written as follows (see Sec. S5 in SI),

$\boldsymbol{\Omega}_{\text{trap}}=\frac{\hbar
m\Delta\delta\omega}{2}\begin{pmatrix}\cos{(2\theta_{\text{min}})}\\\
\sin{(2\theta_{\text{min}})}\\\ 0\end{pmatrix}.$ (4)

Here, $\theta_{\text{min}}$ is the angle of the trap minor axis from the
horizontal, and $\delta\omega=|\omega_{x}-\omega_{y}|\propto|a^{-1}-b^{-1}|$
is the absolute difference between the trap oscillator frequencies along the
major and the minor axis [Fig. 1(d)]. We point out that $\Delta<0$ in our
sample Maragkou  _et al._ (2011) (see Sec. S3 in SI).

The direction of the effective magnetic field is controlled by the angle of
our elliptical trap, $\theta_{\text{min}}$ which consequently rotates the
condensate pseudospin in the equatorial plane of the Poincaré sphere such that
it stabilizes antiparallel to the magnetic field
$-\mathbf{S}\parallel\boldsymbol{\Omega}_{\text{trap}}$. This leads to smooth
changes in the $S_{1,2}$ Stokes components of the emitted light as the trap
rotates like shown in Fig. 3.

Figure 4: (a) Distribution of the in-plane effective magnetic field
$\boldsymbol{\Omega}(\mathbf{k})$ (red arrows) in momentum space due to TE-TM
splitting given by Eq. (3). (b) Horizontal (black) and vertical (red)
polarization resolved normalized spectrum of the condensate emission at $k=0$
for a vertically elongated trap ($\theta_{\text{min}}=0$) favouring
condensation into the horizontal mode. Splitting between levels is $\approx
20$ $\mu$eV. (c) Linear polarization components ($S_{1}$ and $S_{2}$) and DOP
for different real-space ellipticities of the condensate. Data was taken at
$P\approx 1.8P_{th}$

The results of our experiment are accurately reproduced through a mean-field
theory using a generalized Gross-Pitaevskii model describing the polariton
condensate spinor order parameter $\Psi$ coupled with a background excitonic
reservoir (see Sec. S6 in SI).

Interestingly, in a recent experiment Gnusov  _et al._ (2020) we observed
condensation into the spin ground state of a circular trap, where the fine
structure splitting originated from the cavity birefringence
$\boldsymbol{\Omega}_{\text{bir}}(\mathbf{r})$. This meant that the condensate
pseudospin stabilized parallel to the magnetic field
$\mathbf{S}\parallel\boldsymbol{\Omega}_{\text{bir}}(\mathbf{r})$. In the
current experiment however, we instead observe condensation into the excited
spin state, i.e. antiparallel to the magnetic field
$-\mathbf{S}\parallel\boldsymbol{\Omega}_{\text{trap}}$. This can be directly
evidenced in Fig. 4(b) where we show the normalized polarization-resolved
spectrum of a vertically elongated trap which obtains a horizontally polarized
condensate. The horizontal component is higher in energy in Fig. 4(b), in
agreement with Eq. (4).

Performing linear stability analysis on a Gross-Pitaevskii mean field model
(see Sec. S7 in SI) we determine that repulsive polariton-polariton
interactions normally leads to condensation in the fine structure ground state
Read  _et al._ (2009). However, the additional presence of an uncondensed
background of excitons (referred as the reservoir) contributes to an effective
attractive mean-field interaction in the condensate Estrecho  _et al._ (2018)
which causes the ground state to become unstable, favouring condensation into
the excited state as we observe in the current experiment. Another effect is
the different penetration depths of the linearly polarized polariton modes
(due to their different effective masses) into the excess gain region about
the trap short axis. This leads to higher gain for the fine structure excited
state which facilitates its condensation. Several parameters of the polariton
system such as exciton-photon detuning, the quantum well material, and shape
of the pump profile allow tuning from one stability regime to another which
explains why some experiments show ground-state condensation Gnusov  _et al._
(2020) while other, like ours, show exited-state condensation Maragkou  _et
al._ (2010). We stress that regardless of whether system parameters favour
condensation into the spin ground- or excited state of the optical trap, the
main result of our study remains valid.

Figure 5: (a,b) Spatially resolved $S_{1}$ for two different realizations of
coupled condensates oriented horizontally and separated by 27.5 $\mu$m, and
(c) orientated vertically at 26.5 $\mu$m. Insets in (a) and (c) depict the
corresponding $k$-space PL. (d,e) Show 100 time-integrated realizations
(shots) of the $S_{1}$ for the left (blue) and right (red) condensate in the
horizontal-horizontal major axis configuration and (f,g) in the vertical-
vertical configuration. The experimental data is taken at $\approx$1.8
$P_{th}$. Distance dependence of the $S_{1}$ from Gross-Pitaevskii simulations
corresponding to (h) horizontal-horizontal and (i) vertical-vertical major
axes configuration. Each datapoint represents one shot (time averaged). Green
and purple backgrounds in (d)-(i) illustrate regions of similar behaviour
between experiment and theory.

In Fig. 4(c) we continuously change the trap ellipticity from a vertically
elongated trap ($a<b$) to a horizontally elongated one ($a>b$). The PL
ellipticity axis denotes the ratio of width of the PL along x- and y-axis [see
Fig. 1(d)]. The pseudospin of the condensate changes from horizontal to
vertical polarization going through a low DOP regime. We stress that the data
in Fig. 4(c) is obtained at a different position of the cavity sample compared
to Figs. 1-3 which leads to finite $S_{1}$ at zero PL ellipticity ($a=b$) even
though $\boldsymbol{\Omega}_{\text{trap}}=0$. This is because of local
birefringence in the cavity mirrors giving rise to an additional static in-
plane magnetic field $\boldsymbol{\Omega}_{\text{bir}}(\mathbf{r})$.
Therefore, one needs to account for a net field
$\boldsymbol{\Omega}_{\text{net}}=\boldsymbol{\Omega}_{\text{bir}}(\mathbf{r})+\boldsymbol{\Omega}_{\text{trap}}$
orientating the condensate pseudospin. The point of low DOP in Fig. 4(c)
corresponds then to near cancellation between the local birefringence and TE-
TM splitting $\boldsymbol{\Omega}_{\text{net}}\approx 0$.

Coupled condensates. — Networks of coupled polariton condensates can be seen
as an attractive platform to study the synchronization pheonomena in laser
arrays, and to investigate the behaviour of complex nonequilibrium many-body
systems and excitations in non-Hermitian lattices Ohadi  _et al._ (2017);
Mandal  _et al._ (2020); Töpfer  _et al._ (2021); Pieczarka  _et al._ (2021).
Inspired by these studies, we create two identical, spatially separated,
optical traps utilizing two SLMs resulting in the formation of two coupled
condensates [Fig. 5]. The trap anisotropy [see Fig1(b)] allows polaritons to
escape faster along its major axis[Fig1(e)]. This leads to stronger coupling
when the traps major axes are orientated longitudinally to the coupling
direction, and weaker when orientated transverse (estimated as 3 times weaker
from energy resolved spatial PL). This can be evidenced from the different
visibility in the momentum space interference fringes (implying
synchronization) [see insets in Figs. 5(a) and 5(c)].

Polarization resolving 100 quasi-CW 50 $\mu$s excitation shots, we observe
distinct regimes depending on the condensates separation distance and
orientation. For strongly coupled traps [Figs. 5(a) and 5(b)] at $26.5$ $\mu$m
distance we observe zero DOP in each CW shot [Fig. 5(d)] where the blue and
red curves correspond to the left and right condensate. At a $27.5$ $\mu$m
distance we now observe a strong $S_{1}$ component stochastically flipping
from shot to shot [Fig. 5(e)] with small $S_{2,3}$ (see S9 in SI)).
Interestingly, the $S_{1}$ components of the condensates are almost perfectly
correlated (Pearson correlation coefficient $\rho$ equals 0.99) which implies
that they are strongly coupled. The linear polarization flipping suggests
bistability in our system Sigurdsson (2020), triggered by the spatial coupling
mechanism. This interpretation is supported through Gross-Pitaevskii
simulations on time-delay coupled spinor condensates presented in Fig. 5(h)
(see Sec. S8 in SI). For weakly coupled traps [Fig. 5(c)] we observe
qualitatively different behaviour. Choosing again the same distances, we now
see regimes of strong positive $S_{1}$ component [Fig. 5(f)] and then semi-
depolarized behaviour [Fig. 5(g)]. Due to the weaker spatial coupling the
condensates are no longer strongly correlated in their $S_{1}$ components
($\rho=0.5$ and $0.26$ respectively). We note that the different mean $S_{1}$
values in Fig. 5(g) can be attributed to the position-dependent birefringence
$\boldsymbol{\Omega}_{\text{bir}}(\mathbf{r})$. We reproduce the experiment
from simulation [Fig. 5(i)] by only decreasing the coupling strength by a
factor of 3. We note that the ballistic (time-delayed) nature of the polariton
condensate coupling Töpfer  _et al._ (2020) distinguishes them from
evanescently coupled quantum fluids. Indeed, the distance between the
radiating condensates dictates their interference condition (in analogy to
coupled laser systems) which—in our system—leads to distance-periodic
appearance of the classified polarization regimes as seen in Fig. 5(h) and
5(i). Full dynamical trajectories from simulation, and a wider distance-power
scan, are shown in Secs. S8 and S9 in the SI.

Conclusion. — We have investigated the steady state polarization dynamics of a
polariton condensate in an elliptically shaped trapping potential created
through optical nonresonant linearly polarized injection. We have demonstrated
that the polarization of the condensate is determined by the lifted spin-
degeneracy of the trap levels due to the geometric ellipticity of the trap and
inherent cavity TE-TM splitting. The condensate always forms in a higher
energy spin state of the lowest trap level with a linear polarization that
follows the minor axis of the trap ellipse. By rotating the excitation
profile, we can rotate the condensate linear polarization around the
equatorial plane of the Poincaré sphere. We have extended our system to
coupled condensates, revealing rich physics of synchronization and
desynchronization by tuning the condensate coupling strength through the
optical trap anisotropy and/or spatial separation. Our results pave the way
towards all-optical spin circuitry in spinoptronic applications, and coherent
light sources with on-demand switchable linear polarization.

The data presented in this paper are openly available from the University of
Southampton repository.

Acknowledgements. — The authors acknowledge the support of the UK’s
Engineering and Physical Sciences Research Council (grant EP/M025330/1 on
Hybrid Polaritonics) and by RFBR according to the research project No.
20-02-00919.

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Supplementary Information

##  S1 Condensate polarization dependence on the linear polarization of the
excitation laser

The experimental data presented in the main manuscript are acquired using a
horizontally polarized pump laser which excites the optically trapped
polariton condensate. In this supplemental section, we evidence that the
linear polarization direction of the pump laser does not affect our presented
results. In Fig. S1 we show the measured condensate photoluminescence (PL)
Stokes components $S_{1,2,3}$ for varying power and linear polarization
direction of the pump laser, the latter being controlled by a half-waveplate
(HWP) in the excitation path. The four columns in Fig. S1 correspond to
different spatial orientations of the elliptically shaped pump profile (i.e.,
the optical trap). Figures. S1(a-c) are taken for $0^{\circ}$, (d-f)
$-45^{\circ}$, (g-i) $90^{\circ}$, and (j-l) $45^{\circ}$ degrees of the trap
ellipse major axis rotated counterclockwise from the horizontal direction (as
defined in the main manuscript). We observe that the condensate polarization
always dominantly follows the minor axis of the trap ellipse [see Fig.
S1(a),(e),(g), and (k)].

The small amount of $S_{3}$ component emerging for diagonally oriented traps
in Figs. S1(f) and S1(l) is due to optical elements in the detection path of
our setup. For example, different reflectivities of the mirrors for $s$\- and
$p$-polarized light and small birefringence in the cryostat window glass. We
have measured the effective retardance of the detection path in our setup to
be $\approx$0.06$\pi$ at the condensate emission wavelength ($\approx 856$
nm).

Figure S1: Condensate Stokes components for different pump powers and
directions of linear polarization of the excitation laser. The labels H,A,V,D
on the vertical axis denote horizontal, antidiagonal, vertical, and diagonal
polarization respectively. The condensate PL is depicted on the top row with
the black line denoting the trap major axis oriented at (a-c) 0, (d-f) -45,
(g-i) 90, and (j-l) 45 degrees with respect to the cavity plane $x$-axis
(horizontal direction).

##  S2 Condensate polarization dependence on the polarization ellipticity of
the excitation laser

In this section we quantitatively investigate the dependence of the condensate
polarization on the pump laser ellipticity. We install a quarter waveplate
(QWP) in the excitation path so that by rotating the QWP, we can control the
ellipticity and handedness of the excitation polarization. In Fig. S2 we show
the measured condensate PL Stokes components $S_{1,2,3}$ depending on the pump
polarization ellipticity and power. Overall, we obtain a similar behavior of
the condensate polarization that was reported for annular optical traps Gnusov
_et al._ (2020).

As expected, circular polarization transfers to the condensate from our
nonresonant excitation through the optical orientation of the background
excitons feeding the condensate. It can also be seen that the linear
polarization of the condensate is sensitive to the pump polarization
ellipticity. This is effect is theoretically modeled and discussed further in
Sec. S10. In agreement with the findings presented in the main manuscript,
when the pump is almost purely linearly polarized ($\text{QWP}\approx 0$) we
observe that the condensate aligns along the short axis of the optical trap
[see e.g. blue coloured region in Fig. S2(a)]. Our additional measurements in
this supplemental section underline the richness of polarization regimes
accessible in polariton condensates where, in this study, we have focused on
anisotropic trapping conditions around $\text{QWP}\approx 0$.

Figure S2: Condensate Stokes components for different pump powers and angles
of the QWP (pump polarization ellipticity). The ellipticty of the pump can be
written in normalized form as $S_{3}^{\text{pump}}=\sin{(2\cdot\text{QWP})}$.
The negative and positive values of the QWP angle correspond to left and right
handedness of the circular polarization. The condensate PL is depicted on the
top row with the black line denoting the trap major axis oriented at (a-c) 0,
(d-f) -45, (g-i) 90, and (j-l) 45 degrees with respect to the cavity plane
$x$-axis (horizontal direction).

##  S3 TE-TM splitting

We experimentally measure the TE-TM splitting of the sample by polarization
resolving the lower polariton branch in the linear regime (i.e., below
condensation threshold) along the $k_{y}$ momentum axis. We observe that
vertically polarized polaritons possess higher energy than horizontally
polarized polaritons [see Fig. S3(a)]. The energy difference between these
branches gives the TE-TM splitting which follows the expected parabolic
trajectory [see Fig. S3(b)].

Figure S3: (a) Fitted lower polariton branch in horizontal (blue) and vertical
(red) linear polarization detection. (b) Experimentally obtained TE-TM
splitting for different wavevectors.

##  S4 8-point excitation

In this supplemental section, we demonstrate the precise shape of our optical
excitation beam is not important as long as it introduces different
confinement strengths in the two orthogonal spatial directions. Here, we shape
the overall laser profile using 8 Gaussians distributed in the form of an
ellipse [see Fig. S4(b)] leading to the formation of an elliptical condensate
[see Fig. S4(c)]. In agreement with the results presented in the main text,
such an excitation profile also favors the formation of a condensate with
linear polarization aligned along the ellipse minor axis. By rotating the
excitation profile in the cavity plane Fig. S4(a), we observe the same
rotation of the linear polarization of the condensate as in the main text. The
deviations from the sinusoidal fits in Fig. S4(a) occur due to a some
differences in power and shape of the individual Gaussian spots.

Figure S4: (a) Condensate polarization for different spatial orientations of
the 8-Gaussian excitation profile major axis in real space at $P=2P_{th}$. (b)
Excitation laser intensity profile. (c) Condensate PL.

##  S5 The single-particle polariton Hamiltonian

In the non-interacting (linear) regime the polaritons obey the following
Hamiltonian (same as Eq. (2) in the main text),

$\hat{H}=\frac{\hbar^{2}k^{2}}{2m}-\boldsymbol{\sigma}\cdot\boldsymbol{\Omega}+V(\mathbf{r})-\frac{i\hbar\Gamma}{2},$
(S1)

where $m$ is the polariton mass, $\mathbf{k}=(k_{x},k_{y})$ is the in-plane
cavity momentum, $\Gamma^{-1}$ is the polariton lifetime,
$\boldsymbol{\sigma}$ is the Pauli matrix vector, and

$\boldsymbol{\Omega}=\hbar^{2}\Delta(k_{x}^{2}-k_{y}^{2},\ 2k_{x}k_{y},\
0)^{T},$ (S2)

is the effective magnetic field [see Fig. S5(e)] coming from the TE-TM
splitting of strength $\Delta$. We will consider that our laser generated
potential in experiment can be approximated by an elliptically shaped harmonic
oscillator (HO),

$V(\mathbf{r})=\frac{1}{2}m\omega_{x}^{2}x^{2}+\frac{1}{2}m\omega_{y}^{2}y^{2}.$
(S3)

Using the shorter momentum operator expression
$p_{x(y)}=-i\hbar\partial_{x(y)}=\hbar k_{x(y)}$ for brevity, our Hamiltonian
becomes:

$\hat{H}=\begin{pmatrix}\dfrac{p_{x}^{2}}{2m}+\dfrac{p_{y}^{2}}{2m}+V(\mathbf{r})-\dfrac{i\hbar\Gamma}{2}&-\Delta(p_{x}-ip_{y})^{2}\\\
-\Delta(p_{x}-ip_{y})^{2\dagger}&\dfrac{p_{x}^{2}}{2m}+\dfrac{p_{y}^{2}}{2m}+V(\mathbf{r})-\dfrac{i\hbar\Gamma}{2}\end{pmatrix}.$
(S4)

We will diagonalize this problem in the basis of the harmonic oscillator modes
written for spin-up and spin-down particles as,

$\displaystyle|\psi_{+}\rangle$
$\displaystyle=\sum_{n_{x},n_{y}}c^{(+)}_{n_{x},n_{y}}|n_{x},n_{y}\rangle$
(S5) $\displaystyle|\psi_{-}\rangle$
$\displaystyle=\sum_{n_{x},n_{y}}c^{(-)}_{n_{x},n_{y}}|n_{x},n_{y}\rangle,$
(S6)

where $|n_{x},n_{y}\rangle=|n_{x}\rangle\otimes|n_{y}\rangle$ are the harmonic
oscillator eigenmodes in the ladder operator $\hat{a}_{x(y)}$ formalism. These
are defined in the standard way through the position and momentum operators,

$\displaystyle x$
$\displaystyle=\sqrt{\frac{\hbar}{2m\omega_{x}}}(\hat{a}^{\dagger}_{x}+\hat{a}_{x}),\qquad
y=\sqrt{\frac{\hbar}{2m\omega_{y}}}(\hat{a}^{\dagger}_{y}+\hat{a}_{y}),$ (S7)
$\displaystyle p_{x}$
$\displaystyle=i\sqrt{\frac{m\hbar\omega_{x}}{2}}(\hat{a}_{x}^{\dagger}-\hat{a}_{x}),\qquad
p_{y}=i\sqrt{\frac{m\hbar\omega_{y}}{2}}(\hat{a}_{y}^{\dagger}-\hat{a}_{y}).$
(S8)

Our Hamiltonian can then be expressed,

$\hat{H}=\begin{pmatrix}\hbar\omega_{x}\left(\hat{a}_{x}^{\dagger}\hat{a}_{x}+\dfrac{1}{2}\right)+\hbar\omega_{y}\left(\hat{a}_{y}^{\dagger}\hat{a}_{y}+\dfrac{1}{2}\right)-\dfrac{i\hbar\Gamma}{2}&-\Delta(p_{x}-ip_{y})^{2}\\\
-\Delta(p_{x}-ip_{y})^{2\dagger}&\hbar\omega_{x}\left(\hat{a}_{x}^{\dagger}\hat{a}_{x}+\dfrac{1}{2}\right)+\hbar\omega_{y}\left(\hat{a}_{y}^{\dagger}\hat{a}_{y}+\dfrac{1}{2}\right)-\dfrac{i\hbar\Gamma}{2}\end{pmatrix},$
(S9)

where the following holds,

$\displaystyle p_{x(y)}^{2}=(p_{x(y)}^{2})^{\dagger}$
$\displaystyle=-\frac{m\hbar\omega_{x(y)}}{2}(\hat{a}_{x(y)}^{\dagger}\hat{a}_{x(y)}^{\dagger}-\hat{a}_{x(y)}^{\dagger}\hat{a}_{x(y)}-\hat{a}_{x(y)}\hat{a}_{x(y)}^{\dagger}+\hat{a}_{x(y)}\hat{a}_{x(y)}),$
(S10) $\displaystyle p_{x}p_{y}=(p_{x}p_{y})^{\dagger}$
$\displaystyle=-\frac{m\hbar\sqrt{\omega_{x}\omega_{y}}}{2}(\hat{a}_{x}^{\dagger}\hat{a}_{y}^{\dagger}-\hat{a}_{x}^{\dagger}\hat{a}_{y}-\hat{a}_{x}\hat{a}_{y}^{\dagger}+\hat{a}_{x}\hat{a}_{y}).$
(S11)

The diagonal harmonic oscillator terms can be written more neatly as,

$E^{(0)}_{n_{x},n_{y}}=\hbar\omega_{x}\left(n_{x}+\frac{1}{2}\right)+\hbar\omega_{y}\left(n_{y}+\frac{1}{2}\right).$
(S12)

The TE-TM terms will operate on our states as follows,

$\displaystyle p_{x}^{2}|n_{x},n_{y}\rangle=$
$\displaystyle-\frac{m\hbar\omega_{x}}{2}(\hat{a}_{x}^{\dagger}\hat{a}_{x}^{\dagger}-\hat{a}_{x}^{\dagger}\hat{a}_{x}-\hat{a}_{x}\hat{a}_{x}^{\dagger}+\hat{a}_{x}\hat{a}_{x})|n_{x},n_{y}\rangle$
$\displaystyle=$
$\displaystyle-\frac{m\hbar\omega_{x}}{2}(\sqrt{n_{x}+1}\sqrt{n_{x}+2}|n_{x}+2,n_{y}\rangle-
n_{x}|n_{x},n_{y}\rangle-(n_{x}+1)|n_{x},n_{y}\rangle+\sqrt{n_{x}}\sqrt{n_{x}-1}|n_{x}-2,n_{y}\rangle),$
(S13) $\displaystyle p_{x}p_{y}|n_{x},n_{y}\rangle=$
$\displaystyle-\frac{m\hbar\sqrt{\omega_{x}\omega_{y}}}{2}(\hat{a}_{x}^{\dagger}\hat{a}_{y}^{\dagger}-\hat{a}_{x}^{\dagger}\hat{a}_{y}-\hat{a}_{x}\hat{a}_{y}^{\dagger}+\hat{a}_{x}\hat{a}_{y})|n_{x},n_{y}\rangle$
$\displaystyle=$
$\displaystyle-\frac{m\hbar\sqrt{\omega_{x}\omega_{y}}}{2}\bigg{(}\sqrt{(n_{x}+1)(n_{y}+1)}|n_{x}+1,n_{y}+1\rangle-\sqrt{n_{x}+1}\sqrt{n_{y}}|n_{x}+1,n_{y}-1\rangle$
$\displaystyle-\sqrt{n_{x}}\sqrt{n_{y}+1}|n_{x}-1,n_{y}+1\rangle+\sqrt{n_{x}n_{y}}|n_{x}-1,n_{y}-1\rangle\bigg{)}.$
(S14)

We will give a special notation to TE-TM terms which do not mix levels,

$\epsilon_{n_{x},n_{y}}=-\frac{m\hbar\Delta}{2}[\omega_{x}(2n_{x}+1)-\omega_{y}(2n_{y}+1)].$
(S15)

We can write a truncated version of our Hamiltonian for just the spins in the
trap ground state $|0,0\rangle$ which reads (i.e., coupling to other HO levels
is neglected),

$\hat{H}\approx\begin{pmatrix}E^{(0)}_{0,0}-\dfrac{i\hbar\Gamma}{2}&\epsilon_{0,0}\\\
\epsilon_{0,0}&E^{(0)}_{0,0}-\dfrac{i\hbar\Gamma}{2}\end{pmatrix}.$ (S16)

The eigenvectors are the horizontally (H) and vertically (V) polarized states
of light with eigenvalues,

$E_{0,0}^{(H,V)}=\frac{\hbar}{2}\left[\omega_{x}+\omega_{y}\mp
m\Delta(\omega_{x}-\omega_{y})\right]-\frac{i\hbar\Gamma}{2}.$ (S17)

We remind that $\Delta<0$ in our cavity sample Maragkou  _et al._ (2011) (see
Fig. S3). This expression confirms experimental observations of the cavity
energy resolved emission in Fig. 4(b) in the main text. When the laser induced
trap has a major axis along the vertical direction (i.e.,
$\omega_{x}>\omega_{y}$) then we observe higher frequency in the horizontally
emitted light as opposed to the vertical light, in agreement with
$E_{0,0}^{(H)}>E_{0,0}^{(V)}$. When $\omega_{x}<\omega_{y}$ the vice versa
appears. In Fig. S5 we put $\Gamma=0$ for simplicity and compare the
calculated spectrum obtained from diagonalizing Eq. (S9) (black lines) for
$\text{max}{(n_{x(y)})}=15$ modes against our truncated lowest HO level
Hamiltonian Eq. (S17). The generalization of Eq. (S16) to arbitrary angles of
the potential orientation in the $x$-$y$ plane is straightforward and
presented in Eq. (4) in the main text.

Figure S5: Spectrum of the harmonic oscillator with TE-TM splitting and
$\Gamma=0$ for simplicity. Black lines correspond to diagonalization of Eq.
(S9) for $\text{max}{(n_{x(y)})}=15$ and $\omega_{x}=1.75\omega_{y}=0.55$ ps-1
which were estimated from the full-width-half-maximum of the trapped
condensate, and for a typical polariton mass of $m=0.3$ meV ps2 $\mu$m-2. The
red lines are outcome of Eq. (S17).

##  S6 Generalized Gross-Pitaevskii simulations

We will now model the dynamics of the polariton condensate spinor
$\Psi=(\psi_{+},\psi_{-})^{T}$ using the generalized (driven-dissipative)
Gross-Pitaevskii equation, coupled to a semiclassical rate equation describing
a reservoir of low-momentum excitons $\mathbf{X}=(X_{+},X_{-})^{T}$ which
scatter into the condensate Wouters and Carusotto (2007):

$\begin{split}i\hbar{\partial\psi_{\pm}\over\partial
t}&=\left[-{\hbar^{2}\nabla^{2}\over 2m}-\Delta(p_{x}\mp
ip_{y})^{2}+g\left(X_{\pm}+\frac{P}{W}\right)+\alpha|\psi_{\pm}|^{2}+i\hbar{RX_{\pm}-\Gamma\over
2}\right]\psi_{\pm},\\\ {\partial X_{\pm}\over\partial
t}&=P-(R|\psi_{\pm}|^{2}+\Gamma_{R})X_{\pm}+\Gamma_{s}(X_{\mp}-X_{\pm}).\end{split}$
(S18)

Here, $\nabla^{2}$ denotes the two-dimensional Laplacian operator, $\Delta$
the TE-TM splitting, $m$ is the polariton effective mass, $g$ and $\alpha$ are
the interaction constants describing the polariton repulsion off the exciton
density and polariton-polariton repulsion, $R$ governs the stimulated
scattering from the reservoir into the condensate, $\Gamma$ and $\Gamma_{R}$
are the polariton and active exciton decay rates, and $\Gamma_{s}$ is the rate
of spin relaxation. As we are working in continuous wave regime, and the pump
is linearly polarized at all times, we do not need to take into account the
polarization- and time-dependence of a high-momentum (inactive) reservoir
describing excitons that are too energetic to scatter into the condensate
Antón  _et al._ (2013). Instead, the contribution of photoexcited high-
momentum excitons to the condensate appears through the blueshift term $P/W$
where $W$ describes the conversion rate of high-momentum excitons into low-
momentum excitons $X_{\pm}$ that sustain the condensate.

We will use the pseudospin formalism (analogous to the Stokes parameters
describing the cavity photons) to describe the polarization of the condensate
(note that in Eq. (1) in the main manuscript we have used the normalized
definition),

$\mathbf{S}=\begin{pmatrix}S_{1}\\\ S_{2}\\\
S_{3}\end{pmatrix}=\Psi^{\dagger}\boldsymbol{\sigma}\Psi.$ (S19)

Here, $\boldsymbol{\sigma}$ is the Pauli matrix vector and the total density
of the condensate is expressed as $S_{0}=|\psi_{+}|^{2}+|\psi_{-}|^{2}$.

We will study the dynamics of Eq. (S18) for three different excitation
profiles shown in Figs. S7(a,d,f). The profiles shown in Fig. S7(d) and S7(f)
represent the experimental configurations shown in Fig. 1(a,b) in the main
manuscript and in Fig. S4(b). We also introduce, for completeness, a third
type of an elliptical excitation profile in the numerical analysis shown in
Fig. S7(a). The three excitation profiles can be written as follows,

$\displaystyle P_{I}(\mathbf{r})$
$\displaystyle=P_{0}\frac{L_{I}^{4}}{\left(\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}-r_{0}^{2}\right)^{2}+L_{I}^{4}},$
(S20) $\displaystyle P_{II}(\mathbf{r})$
$\displaystyle=P_{0}\frac{L_{II}^{4}}{\left(x^{2}+y^{2}-r_{0}^{2}\right)^{2}+L_{II}^{4}}[1-\eta\cos{(2\varphi+\phi_{\text{maj}})}\sin^{2}{(\pi
r/2r_{0})}],$ (S21) $\displaystyle P_{III}(\mathbf{r})$
$\displaystyle=P_{0}\sum_{n=1}^{8}\frac{L_{III}^{2}}{(x-x_{n})^{2}+(y-y_{n})^{2}+L_{III}^{2}}.$
(S22)

Here, $L_{I,II,III}$ denote the spread (thickness) of the potentials. For
pumps (S20) and (S21) the common radius $r_{0}$ defines the length of the
ellipse minor and major axis. Specifically, the minor and major axis are given
by the parameters $0<a,b<1$ for (S20) and $\eta,\phi_{\text{maj}}$ for (S21).
For the third pump profile (S22) we use coordinates $x_{n},y_{n}$ of the eight
tightly focused pump spots corresponding to the experiment. The laser power
density is given by $P_{0}$.

In Fig. S6 we show the obtained steady state wavefunction $\Psi$ obtained from
random initial conditions while driving the system above the pump threshold
$P_{0}>P_{\text{th}}$ using the first pump profile $P_{I}(\mathbf{r})$. The
threshold $P_{\text{th}}$ is defined as the transition point where the normal
state $|\Psi|=0$ becomes unstable and instead a condensate forms $|\Psi|>0$.
Indeed, choosing parameters corresponding to the experiment we obtain complete
match between experimental observations and simulations. Testing 100 different
random initial conditions we find that the simulated condensate always
converges to a steady state corresponding to the excited spin state of the
trap. The parameters of the simulation are: $m=0.3$ meV ps2 $\mu$m-2;
$\Gamma=\frac{1}{5.5}$ ps-1; $\hbar\alpha=3$ $\mu$eV $\mu$m2, $g=\alpha$;
$\Gamma_{R}=\Gamma/4$; $R=0.67\alpha$; $W=0.05$ ps-1;
$\Gamma_{s}=\Gamma_{R}/2$; $\hbar^{2}\Delta=-0.03$ meV $\mu$m2; $L_{I}=5$
$\mu$m; $L_{II}=6$ $\mu$m; $L_{III}=2$ $\mu$m; $r_{0}=5$ $\mu$m; $\eta=0.2$;
and $P_{0}=4.625$ $\mu$m-2 ps-1 which is around $\approx 20\%$ above threshold
for each configuration.

Figure S6: (Top row) Steady state solution from simulation of Eq. (S18). Here
we used the $P_{I}(\mathbf{r})$ pump profile with a major axis along the
vertical direction, $b/a=1.3$. The results show a condensate forming with
horizontal polarization implying occupation of the excited trap spin state.
(Bottom row) Same simulations but now the pump major axis is orientated
$\pi/4$ from the horizontal resulting in antidiagonally polarized condensate
which again corresponds to the excited spin state of the trap. Figure S7:
(a,d,f) Pump profiles (normalized) corresponding to Eqs. (S20)-(S22). In (a)
we have set $a/b=1.4$ and in (d) $\phi_{\text{maj}}=0$ and $\eta=0.2$. The
green lines are contours which represent the isoenergy lines of the optical
trap. The inset in (e) shows schematically the energy structure of the
linearly polarized modes in the trap s-orbital. In the panels on the right we
show the corresponding condensate $\bar{S}_{0}$ dynamics for the two ansatz
$\Psi^{H,V}$ in Eq. (S23) separately (circles and squares respectively) for
fixed pumping value $P_{0}$ above threshold. The red-blue color denotes the
area-integrated linear polarization $\bar{S}_{1}$. In panels (b,c) and (g,h)
we split the time axis to better show the early and late dynamics. In panel
(b) the early dynamics show that the $\Psi^{H}$ ansatz rises faster and the
condensate saturates into a horizontally polarized state (red circles) with
higher particle number as compared to the vertically polarized state (blue
squares), implying it has larger gain. At later times shown in panel (c), the
horizontal polarized solution always collapses into the vertically polarized
solution, indicating that it is only metastable and that a vertically
polarized condensate survives at long times. In panels (e) and (g) we observe
a qualitatively different early dynamics where now the horizontally polarized
condensate rises slower. Eventually, as we see in (e) and (h), the
horizontally polarized solution destabilizes and converges into a vertically
polarized condensate.

###  S6.1 Condensate metastability

We will here scrutinize the early and late condensate dynamics using two
simple initial conditions (ansatz). We will use a trap with a horizontal major
axis (i.e., $\omega_{x}<\omega_{y}$) since all other major axis orientations
are completely generalizable. The two initial conditions for Eq. (S18) are
written,

$\Psi(t=0)=\Psi^{H,V}=Ae^{-c(x^{2}+y^{2})}\begin{pmatrix}1\\\ \pm
1\end{pmatrix}.$ (S23)

The parameter $c$ is chosen to have $\Psi^{H,V}$ localized dominantly within
the trap and $A\ll 1$ is a small number to minimize nonlinear effects in the
initial dynamics. We then solve Eq. (S18) for each initial condition and plot
the spatially integrated particle number $S_{0}$ and $S_{1}$ (normalized)
Stokes parameter as a function of time,

$\bar{S}_{0}(t)=\frac{1}{\mathcal{A}}\int_{\mathcal{A}}S_{0}(\mathbf{r},t)d\mathbf{r}\qquad\bar{S}_{1}(t)=\frac{\int_{\mathcal{A}}S_{1}(\mathbf{r},t)d\mathbf{r}}{\int_{\mathcal{A}}S_{0}(\mathbf{r},t)d\mathbf{r}},$
(S24)

where $\mathcal{A}$ is the area enclosed by the pump profile ridge (i.e.,
$\text{max}\,{P(\mathbf{r})}$).

Simulations using the generalised Gross-Pitaevskii equation (S18) for fixed
power $P_{0}$ (above threshold) and the three pump profiles $P_{I,II,III}$ are
shown in Figs. S7(b,c), and S7(e), and S7(g,h), respectively. We plot the
area-integrated particle number $\bar{S}_{0}$ for the two different initial
conditions $\Psi^{H,V}$ (circles and squares, respectively). The color of the
markers indicates the area-integrated linear polarization $\bar{S}_{1}$ of the
condensate. In Figs. S7(b,c) and S7(g,h) we split the time axis to show better
the early and late dynamics. The inset in Fig. S7(e) shows schematically the
energy splitting between the polarizations. For pump profiles $P_{II}$ and
$P_{III}$ we see that in the early dynamics a vertically polarized condensate
(blue squares) rises faster and saturates at a higher particle number than a
horizontally polarized condensate (red circles). This can be understood from
the fact that the vertical and horizontal polarized modes of the condensate
have different effective masses and, thus, have different penetration depths
into the gain region of the pump. In particular, pumps $P_{II}$ and $P_{III}$
lead to an excess density of reservoir excitons about the short axis of the
ellipse. Since the penetration depth of the confined mode in the potential
well is larger in the direction of the linear polarization axis, this would
increase the overlap of the mode co-polarized with the short-axis of the
potential well with the gain region, i.e. the ’excited state’ in the fine
structure. In the late dynamics [Fig. S7(e) and S7(h)] the horizontal solution
destabilizes and converges into the vertically polarized solution. This is in
agreement with our experimental observations showing robust condensation into
the excited spin state of the optical traps.

Interestingly, for pump $P_{I}$ the early dynamics [Fig. S7(b)] are reversed
with respect to $P_{II,III}$. Now the horizontally polarized condensate rises
faster and saturates at a higher particle number. This pump profile does not
generate a strong excess of excitons about the ellipse short axis and, as a
consequence, the higher energy polaritons, co-polarized with the short axis,
escape (leak) faster from the trap. Nevertheless, in the late dynamics [Fig.
S7(c)] the horizontally polarized solution destabilizes and collapses into the
vertically polarized solution. This observation underlines that the stable
solution of the condensate does not necessarily correspond to the one with
maximum particle number $\bar{S}_{0}$. In the next section we address the
different parameters of our model that determine the stability of the excited
state and the ground state.

##  S7 Stability analysis on a two mode problem

Here, we will analyse the stability properties of the condensate by projecting
our order parameter on only the lowest trap state $|0,0\rangle$. Let us here
denote $\psi_{\pm}(\mathbf{r},t)\to\psi_{\pm}(t)$ as the condensate order
parameter describing polaritons only in the HO ground state and neglect
contribution from higher HO modes,

$\displaystyle\begin{split}i&\frac{d\psi_{\pm}}{dt}=\Big{[}\alpha|\psi_{\pm}|^{2}+gX_{\pm}+i\frac{RX_{\pm}-\Gamma}{2}\Big{]}\psi_{\pm}+(\epsilon+i\gamma)\psi_{\mp},\\\
&\frac{dX_{\pm}}{dt}=P-\left(\Gamma_{R}+R|\psi_{\pm}|^{2}\right)X_{\pm}+\Gamma_{s}(X_{\mp}-X_{\pm}).\end{split}$
(S25)

The parameters here have the same meaning as their counterparts in Eq. (S18),
but we stress that we have absorbed $\hbar$ into their definition for brevity
and some will obtain modified values after integrating out the spatial degrees
of freedom depending on the precise shape of the condensate and reservoir. We
have removed the $P/W$ term as it only induces an overall blueshift to both
spins which does not affect the stability properties of the system. The spin-
coupling parameter $\epsilon$ corresponds to $\epsilon_{0,0}$ from Eq. (S15).
We additionally include an imaginary coupling parameter $\gamma$ which
physically represents different linewidths of the horizontal and vertical
polarized modes (i.e., different decay rates).

The two steady state solutions of interest correspond to spin-balanced
reservoirs $X_{+}=X_{-}$ supporting either purely horizontally or vertically
polarized condensate written,

$\Psi_{\text{st}}^{H,V}=\sqrt{\frac{S_{0}^{H,V}}{2}}\begin{pmatrix}1\\\ \pm
1\end{pmatrix}e^{-i\omega^{H,V}t}$ (S26)

where,

$\omega^{H,V}=\frac{\alpha}{2}S_{0}^{H,V}+gX^{H,V}\pm\epsilon_{0,0},\qquad
S_{0}^{H,V}=\frac{P}{\Gamma\mp 2\gamma}-\frac{\Gamma_{R}}{R},\qquad
X^{H,V}=\frac{\Gamma\mp 2\gamma}{R}.$ (S27)

The power to reach the lower threshold solution is
$P_{\text{th}}=\Gamma_{R}(\Gamma-2|\gamma|)/R$. The stability analysis of the
$\Psi_{\text{st}}^{H,V}$ solutions is exactly the same as in Ref. Sigurdsson
(2020) where a $5\times 5$ Jacobian matrix $\mathbf{J}$ corresponding to
linearisation of Eq. (S25) around its steady state solutions was derived. This
allows determining the stability of the solutions in terms of their Jacobian
eigenvalues $\lambda_{n}$ (also known as Lyapunov exponents in nonlinear
dynamics or Bogoliubov elementary excitations in the context of Bose-Einstein
condensates). If a single eigenvalue of $\mathbf{J}$ has a positive real part
then the solution is said to be asymptotically unstable. It was found in Ref.
Sigurdsson (2020) that the relative strength between the mean field energy
coming from the condensates $\alpha|\psi_{\pm}|^{2}$ and the reservoir
blueshift $gX_{\pm}$ played a big role in whether the excited state or the
ground state was stable.

To understand this better, we will first consider the stability of the two
solutions $\Psi_{\text{st}}^{H,V}$ using a more general coupled Gross-
Pitaevskii equations (similar to coupled amplitude oscillators),

$i\frac{d\psi_{\pm}}{dt}=\alpha|\psi_{\pm}|^{2}\psi_{\pm}+\epsilon\psi_{\mp},\qquad\epsilon>0.$
(S28)

Clearly, $\Psi_{\text{st}}^{H,V}$ is a solution of the above equation for any
particle number $S_{0}=S_{0}^{H}=S_{0}^{V}$ with frequency
$\omega^{H,V}=\alpha S_{0}/2\pm\epsilon$. The Lyapunov exponents of these
solutions are written:

$\begin{split}\lambda_{1}^{H,V}&=0,\\\
\lambda_{2}^{H,V}&=\,\,\,\,\sqrt{2\epsilon(-2\epsilon\pm\alpha S_{0})},\\\
\lambda_{3}^{H,V}&=-\sqrt{2\epsilon(-2\epsilon\pm\alpha S_{0})}.\end{split}$
(S29)

We only have three eigenvalues because our two level system (S28) can be
described with the three-dimensional pseudospin state vector [see Eq. (S19)],
analogous to the Stokes vector of light. It is clear that only
$\lambda_{2}^{H,V}$ can have real values greater than zero (the signature of
instability) and there are only two cases when this happens:

  1. 1.

If $\alpha>0$ (repulsive particle interactions) then
$\text{Re}{(\lambda_{2}^{\color[rgb]{1,0,0}H})}>0$ when $\alpha
S_{0}>2\epsilon$.

  2. 2.

If $\alpha<0$ (attractive particle interactions) then
$\text{Re}{(\lambda_{2}^{\color[rgb]{0,0,1}V})}>0$ when $|\alpha
S_{0}|>2\epsilon$.

This simple result shows that the stability of the excited state and the
ground state changes when the mean field energy $\alpha S_{0}$ exceeds the
fine structure splitting $2\epsilon$. In our case, polariton interactions are
repulsive $\alpha>0$ and the condensate should always form in the ground state
when $S_{0}>2\epsilon/\alpha$ Shelykh  _et al._ (2006). Note that
$2\hbar\epsilon\approx 20$ $\mu$eV in our experiment which is small compared
to the typical polariton mean field energies $\alpha S_{0}$. It therefore
appears puzzling that we observe stable excited state condensation when the
above simple consideration dictates that only the ground state should be
stable.

In the following, we will address two different mechanisms that fight against
ground state condensation. First, if the ground state is lossier than the
excited state (i.e., $\gamma/\epsilon<0$) then polaritons will preferentially
condense into the excited state until $S_{0}$ exceeds a critical value
Sigurdsson (2020). However, as we can see from Fig. S7(b,c), even if the
excited state is lossier than the ground state the condensate can still
preferentially populate and stabilize in the excited state. Second, a stable
excited spin state condensation can appear due to an effective attractive
nonlinearity coming from the reservoir [i.e., the term $gX_{\pm}$ in Eq.
(S25)]. Indeed, it is well established that the presence of the condensate
“eats away” the reservoir density analogous to the hole burning effect in
lasers Estrecho  _et al._ (2018). In the adiabatic regime where the reservoir
is assumed to adjust to the condensate density dynamics very fast it can be
approximated as follows,

$X_{\pm}=\frac{P}{\Gamma_{R}}\left[1-\frac{R|\psi_{\pm}|^{2}}{\Gamma_{R}}+\mathcal{O}(|\psi_{\pm}|^{4})\right].$
(S30)

The nonlinearity of the condensate can therefore described by an effective
interaction parameter,

$\alpha_{\text{eff}}=\alpha-\frac{gPR}{\Gamma_{R}^{2}}.$ (S31)

To test our hypothesis, we numerically solve the eigenvalues of the Jacobian
for Eq. (S25) and plot their maximum real part for the
$\Psi_{\text{st}}^{H,V}$ solutions in Fig. S8 (red and blue curves,
respectively) as a function of varying $\alpha$ and several values of
$\Gamma_{R}$ [Fig. S8(a)] and $\gamma$ [Fig. S8(b)]. Indeed, we see that there
exist three distinct regimes which we schematically illustrate with the blue-
white-red color gradient:

  1. i.

Ground state unstable and excited state stable.

  2. ii.

Both states stable.

  3. iii.

Ground state stable and excited state unstable.

Figure S8: Maximum real eigenvalues (Lyapunov exponents) from the Jacobian of
Eq. (S25) around the two steady state solution $\Psi_{\text{st}}^{H,V}$ (red
and blue curves, respectively) for $\epsilon=0.01$ ps-1. Regimes where
$\text{Re}{(\lambda_{\text{max}})}>0$ imply instability of the solution. The
color gradient at the top is added for illustration purposes. Parameters:
$\Gamma=1/5$ ps-1, $\Gamma_{s}=\Gamma/4$, $\epsilon=\Gamma/20$,
$R=0.015\Gamma$, $g=5R/6$, and $P=2P_{\text{th}}$. In (a) we fix $\gamma=0$
and in (b) we fix $\Gamma_{R}=0.5\Gamma$.

As expected from Eq. (S31), the stability range of the excited state increases
as $\Gamma_{R}$ decreases [see Fig. S8(a)] due to the nonlinearity
$\alpha_{\text{eff}}$ becoming more negative. Moreover, the stability range of
the excited state also increases when $\gamma/\epsilon$ becomes more negative
corresponding to the ground state becoming lossy [see Fig. S8(b)]. We point
out that it is not possible to determine separately the contribution of
$\alpha S_{0}^{H,V}$ and $gX^{H,V}$ in experiment since we can only measure
the net blueshift in condensate energy. Nevertheless, our experimental results
indicate that the current pump configuration favours the far-left regime in
Fig. S8 where the ground state is unstable. Recently, we reported results
corresponding to the far-right regime where robust ground state condensation
was instead observed Gnusov  _et al._ (2020) using the same cavity sample but
somewhat different experimental configuration. How exactly one can tune from
one regime to the other is difficult to tell, but the clearest path would
either involve changing the detuning between the photon and exciton mode. This
is possible because $\alpha\propto|\chi|^{4}$ and $g\propto|\chi|^{2}$ where
$|\chi|^{2}$ is the exciton Hopfield fraction of the polariton quasiparticle.
Another method would be to design an excitation profile $P(\mathbf{r})$ which
changes the mean field rate $R$ of particles scattering into the condensate.

Finally, to see if our hypothesis agrees with the full spatial calculations of
Eq. (S18) we repeat the simulation from Fig. S6 in a new Fig. S9 with the
strength of polariton-polariton interactions doubled, i.e. $\alpha\to 2\alpha$
while keeping all other parameters unchanged. We now find, in agreement with
our predictions, that the steady state solution (tested over 100 random
initial conditions) converges to the spin ground state instead of the excited
state. This can be evidenced from the opposite polarization appearing in the
Stokes components in Fig. S9 as compared to Fig. S6.

Figure S9: Same simulations as in Fig. S6 but with doubled interactions
strength $\alpha\to 2\alpha$ (keeping all other parameters unchanged) such
that the excited spin state now becomes unstable and only the ground state
becomes populated.

##  S8 Modeling of coupled optically trapped spinor condensates

In this section we modify Eq. (S25) to describe coupling between two spatially
separated condensates as shown in Fig. 5 in the main manuscript. The inter-
condensate-coupling is sometimes referred to as ballistic coupling because
energetic polaritons escape from each trap and undergo finite-time free-space
propagation before they reach their neighboring condensate. Such coupling is
qualitatively different from evanescent coupling (e.g., tunneling between
Bose-Einstein condensates) since the propagation time of polaritons between
the condensates is comparable to their intrinsic frequencies. This implies
that the coupling between ballistic condensates is time delayed Töpfer  _et
al._ (2020) which we can introduce explicitly to Eq. (S25),

$\displaystyle\begin{split}i&\frac{d\psi_{\pm}^{(1,2)}}{dt}=\Big{[}\omega_{0}+\alpha|\psi_{\pm}^{(1,2)}|^{2}+gX_{\pm}^{(1,2)}+i\frac{RX_{\pm}^{(1,2)}-\Gamma}{2}\Big{]}\psi_{\pm}^{(1,2)}+(\epsilon+i\gamma)\psi_{\mp}^{(1,2)}+J\psi_{\pm}^{(2,1)}(t-\tau)+\mathcal{J}\psi_{\mp}^{(2,1)}(t-\tau),\\\
&\frac{dX_{\pm}^{(1,2)}}{dt}=P-\left(\Gamma_{R}+R|\psi_{\pm}^{(1,2)}|^{2}\right)X_{\pm}^{(1,2)}+\Gamma_{s}(X_{\mp}^{(1,2)}-X_{\pm}^{(1,2)}).\end{split}$
(S32)

Here, the indices $(1,2)$ refer to the two different condensates. We have also
introduced the condensate intrinsic energy $\omega_{0}$ since a suitable
rotating reference frame cannot be chosen for time delayed coupling between
oscillators. As was previously demonstrated Töpfer  _et al._ (2020), the
strength of the coupling $J$ depends on the separation distance $d$ between
the condensates,

$J(d)=J_{0}|H_{0}^{(1)}(k_{c}d)|,$ (S33)

where $H_{0}^{(1)}$ is the zeroth order Hankel function, $J_{0}\in\mathbb{C}$
quantifies the non-Hermitian coupling strength dictated by the overlap of the
condensates over the optical trap region, and $k_{c}$ is the complex
wavevector of the polaritons propagating outside the optical trap,

$k_{c}=k_{c}^{(0)}+i\frac{\Gamma m}{2\hbar k_{c}^{(0)}}.$ (S34)

From experiment, we have estimated $k_{c}^{(0)}\approx 1.35$ $\mu$m-1 by
spatially filtering the polariton PL outside the pump spots. The imaginary
term in Eq. (S34) describes the additional attenuation of polaritons due to
their finite lifetime. We also account for coupling between the spins of the
two condensates due to the TE-TM splitting which is captured with the
parameter $\mathcal{J}$. The time delay parameter is approximated from the
polariton phase velocity which gives,

$\tau=\frac{2dm}{\hbar k_{c}^{(0)}}.$ (S35)

Figure S10: (a-d) Time averaged Stokes parameters of $\psi_{\pm}^{(1)}$ and
(e-h) $\psi_{\pm}^{(2)}$ from random initial conditions and varying pump power
$P$ and distance $d$. We normalise the pump power in units of the threshold
power for a single condensate $P_{\text{th}}=(\Gamma-2|\gamma|)\Gamma_{R}/R$.
The parameters of the simulation are: $\epsilon=-0.01$ ps-1;
$\gamma=\epsilon/2$; $\Gamma=0.25$ ps-1; $\Gamma_{R}=\Gamma/4$;
$\Gamma_{s}=\Gamma_{R}/2$; $\alpha=0.15\epsilon$; $R=0.05\epsilon$;
$\omega_{0}=5.5\Gamma$; $g=\alpha$; $m=0.3$ meV ps2 $\mu$m-2;
$J_{0}=0.67e^{1.8i}$ ps-1; and $\mathcal{J}=0.2J$. Note that the large value
of $J_{0}$ (as compared to $\epsilon$) is due to the smallness of the Hankel
function in Eq. (S33). At, e.g., $d=27$ $\mu$m we have $|J|=2.64|\epsilon|$.

We show in Fig. S10 results of numerically integrating Eq. (S32) from random
initial conditions. Each pixel in the data is one realization of the
condensate for the given power $P$ and distance $d$. The angled brackets of
the Stokes parameters $\langle S_{n}^{(1,2)}\rangle$ represent time-average.
We applied a constant step size Bogacki-Shampine algorithm Flunkert (2011) (a
3rd order Runge-Kutta). The timestep was chosen $\Delta t=0.05$ ps and the
integration was over $T=5000$ ps for each condensate realization. The results
reveal periodic polarization regimes similar to the phase-flip transitions
recently reported in Töpfer  _et al._ (2020). At high powers we observe the
condensates stabilizing into the ground state (horizontal) polarization
[yellow colors in Figs. S10(a,e)] whereas at low power we retrieve stable
excited state (vertical) polarization condensation [blue colors in Figs.
S10(a,e)]. Between these bright yellow and blue regions we observe an
intermediate region (seen as a mixture of blue and yellow datapoints) where
the ground and excited state condensates are both stable and the random
initial condition determines the winner. Such linear polarization bistability
was already reported in Sigurdsson (2020) for a single condensate. We also
observe regions of complete depolarization (sea-green color) which correspond
to condensate destabilization. Comparing Figs. S10(a-d) with S10(e-h) we
observe, in the stable regime, that the condensates are strongly correlated in
polarization (i.e., $\Psi^{(1)}$ and $\Psi^{(2)}$ always co-polarize). Our
theoretical modeling gives good agreement with experimental observations
presented in Fig. 5 in the main manuscript. In Fig. S11 we additionally show
example dynamical trajectories from the unstable and stable regions marked by
the red square and circle in Fig. S10(a), respectively.

Figure S11: Evolution of the Stokes parameters for a single realization of the
condensate at the (a-d) the red square and (e-h) the red circle in Fig. S10,
corresponding to the unstable and stable regimes, respectively.

##  S9 Additional experimental data for coupled condensates

Here we present additional experimental data on the coupled elliptical
condensates. In this experiment, we measure the Stokes components for 100
quasi-CW 50 $\mu$s shots. Each experimental point in Fig. S12 represents the
polarization component averaged within one excitation shot. We note that the
$S_{1}$, $S_{2}$ and $S_{3}$ in Figs. S12(c)-(n) are not measured
simultaneously but consequently under the same pumping conditions and at the
same position on the sample.

As expected, when isolated, the individual condensates stay dominantly
polarized linearly parallel to the minor axis of the pump trap, as we describe
in the main text [see Figs. S12(a) and S12(b) for a horizontal trap and
vertical trap, respectively]. However, some fluctuations can be sometimes
observed, for example, in the horizontally elongated trap in Fig. S12(a). This
happens due to some noise in our system as well due to mode competition. It is
worth noting that such fluctuations decrease the values of the Stokes
components presented in Figs. 2-4 in the main text since there we
integrate/average over hundreds of shots.

In Figs. S12(c)-(h) we present the Stokes components for coupled horizontally
elongated ellipses separated by 27.2 $\mu$m in Figs. S12(c)-(e) and 24 $\mu$m
in Figs. S12(f)-(h). Blue and red colors correspond to the ”right” and ”left”
condensate, respectively. For 27.2 $\mu$m, the condensate flips randomly from
horizontal to vertical polarization from shot-to-shot, whereas the $S_{2}$ has
smaller values (less than 0.5) but also flips from shot to shot. Overall the
$S_{3}$ component stays close to zero. For a different separation distance 24
$\mu$m shown in Figs. S12(f)-(h) we observe that all Stokes components are
close to zero in each shot. This means that the condensate pseudospin
fluctuates rapidly in time within one excitation pulse with a zero mean
polarization just like in simulation in Figs. S11(a)-(d). Notice that the
Stokes components still remain correlated indicating the condensate are
coupled together.

We also plot all polarization components for two coupled vertically elongated
condensates [Figs. S12(i)-(n)]. The weaker coupling of such mutual trap
configuration is evidenced through less correlations between the left and
right condensates (i.e., the red and blue curves fluctuate more
independently). For a distance of 26.5$\mu$m both condensates have strong
horizontal polarization — i.e. big $S_{1}$ component and small $S_{2}$ and
$S_{3}$ components. This corresponds to Figs. S11(e)-(h) in simulations. At a
distance of 25 $\mu$m the condensates are in a semi-depolarized regime with
oscillating $S_{1}$ and $S_{2}$ from shot to shot, and small $S_{3}$.

Figure S12: 100 realizations (shots) of the condensates Stokes components
time-integrated in each 50 $\mu$s excitation shot. (a) and (b) shows $S_{1}$
of single horizontally and vertically elongated condensate (trap)
respectively. $S_{1}$, $S_{2}$, and $S_{3}$ for two coupled condensates with
their trap’s major axes orientated longitudinally to the coupling direction
and separated by 27.2 $\mu$m (c)-(e) and 24 $\mu$m (f)-(h), respectively.
$S_{1}$, $S_{2}$, and $S_{3}$ for two coupled condensates with their trap’s
major axes orientated perpendicularly to the coupling direction, separated by
26.5 $\mu$m (i)-(k) and 25 $\mu$m (l)-(n), respectively. Blue and red colors
correspond to right and left condensate respectively. Background green and
purple colors depict different coupling regimes.

##  S10 Power dependent pseudospin rotation

In this section we explain the results of Fig. 3(b) and 3(c) in the main
manuscript where we can observe noticeable change in the linear polarization
of the condensate as we increase the power. It manifests as counterclockwise
rotation of the pseudospin in the equatorial plane of the Poincaré sphere.

The explanation for the power dependent torque effect is due to slight
polarization ellipticity in the excitation laser. This creates an imbalance
between the spin-up and spin-down exciton populations in the system and a
consequent out-of-plane effective magnetic field $\boldsymbol{\Omega}_{\perp}$
which rotates the pseudospin. This is confirmed through Gross-Pitaevskii
simulations using Eq. (S25) where we introduce a slight pumping imbalance by
redefining a spin-dependent pumping rate $P\to P_{\pm}$ where $P_{+}\neq
P_{-}$. We present our simulation in Fig. S13 where we show the time-
integrated Stokes (pseudospin) components of the condensate at increasing mean
pump power [$P=(P_{+}+P_{-})/2$] where each datapoint is averaged over 100
random initial conditions. We have set $P_{+}/P_{-}=1.0355$ and other
parameters of the model (specified in the caption) are taken similar to the
ones used in Fig. S8 and S10. The shaded area is one standard deviation in the
pseudospin dynamics (calculated over 5 ns) indicating nonstationary and
stationary behaviour at low and high powers, respectively. We point out that
Eq. (S25) must now include the additional pump induced blueshift $gP_{\pm}/W$
like in Eq. (S18) since it contributes to $\boldsymbol{\Omega}_{\perp}$. The
results show precisely the counterclockwise rotation of the pseudospin in the
$(S_{1},S_{2})$ plane like in Fig. 3(b) and 3(c) in the main manuscript. We
have also confirmed that if $P_{+}<P_{-}$ then the pseudospin rotates
clockwise in the $(S_{1},S_{2})$ plane.

Moreover, this change in the $S_{1}$ and $S_{2}$ can also be evidenced from
the experimental data presented in Fig. S2. There, a small change in the
polarization ellipticity of our excitation beam dramatically affects the
$S_{1}$ and $S_{2}$ distributions.

Figure S13: Generalized Gross-Pitaevskii simulations of the condensate under
spin-imbalanced pumping. (a)-(d) Time-integrated Stokes (pseudospin)
components of the condensate for increasing pump power. Each datapoint is
averaged over 100 random initial conditions. The shaded area is one standard
deviation in the pseudospin dynamics (calculated over 5 ns) indicating
nonstationary and stationary behaviour at low and high powers, respectively.
Parameters are: $P_{+}/P_{-}=1.0355$, $P=(P_{+}+P_{-})/2$,
$P_{\text{th}}=\Gamma\Gamma_{R}/R$, $\Gamma=1/5$ ps-1,
$\Gamma_{R}=0.35\Gamma$, $\Gamma_{s}=\Gamma_{R}$, $\epsilon=\Gamma/20$,
$R=0.015\Gamma$, $W=\Gamma_{R}$, and $g=R$.

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