# Spontaneous spin bifurcations and ferromagnetic phase transitions in a
spinor exciton-polariton condensate

H. Ohadi<EMAIL_ADDRESS>Department of Physics, Cavendish Laboratory,
University of Cambridge, Cambridge CB3 0HE, United Kingdom  A. Dreismann
Department of Physics, Cavendish Laboratory, University of Cambridge,
Cambridge CB3 0HE, United Kingdom  Y. G. Rubo  Instituto de Energías
Renovables, Universidad Nacional Autónoma de México, Temixco, Morelos, 62580,
Mexico  F. Pinsker  Department of Applied Mathematics and Theoretical Physics,
University of Cambridge, Cambridge CB3 0WA, United Kingdom  Clarendon
Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, United Kingdom
Y. del Valle-Inclan Redondo  Department of Physics, Cavendish Laboratory,
University of Cambridge, Cambridge CB3 0HE, United Kingdom  S. I. Tsintzos
Foundation for Research and Technology–Hellas, Institute of Electronic
Structure and Laser, 71110 Heraklion, Crete, Greece  Z. Hatzopoulos
Foundation for Research and Technology–Hellas, Institute of Electronic
Structure and Laser, 71110 Heraklion, Crete, Greece  Department of Physics,
University of Crete, 71003 Heraklion, Crete, Greece  P. G. Savvidis
Department of Physics, Cavendish Laboratory, University of Cambridge,
Cambridge CB3 0HE, United Kingdom  Foundation for Research and
Technology–Hellas, Institute of Electronic Structure and Laser, 71110
Heraklion, Crete, Greece  Department of Materials Science and Technology,
University of Crete, 71003 Heraklion, Crete, Greece  J. J. Baumberg
<EMAIL_ADDRESS>Department of Physics, Cavendish Laboratory, University of
Cambridge, Cambridge CB3 0HE, United Kingdom

###### Abstract

We observe a spontaneous parity breaking bifurcation to a ferromagnetic state
in a spatially-trapped exciton-polariton condensate. At a critical bifurcation
density under nonresonant excitation, the whole condensate spontaneously
magnetizes and randomly adopts one of two elliptically-polarized (up to 95%
circularly-polarized) states with opposite handedness of polarization. The
magnetized condensate remains stable for many seconds at
$5\text{\,}\mathrm{K}$, but at higher temperatures it can flip from one
magnetic orientation to another. We optically address these states and
demonstrate the inversion of the magnetic state by resonantly injecting
100-fold weaker pulses of opposite spin. Theoretically, these phenomena can be
well described as spontaneous symmetry breaking of the spin degree of freedom
induced by different loss rates of the linear polarizations.

Condensation of exciton-polaritons (polaritons) spontaneously breaks the
global phase symmetry Deng  _et al._ (2002); Kasprzak  _et al._ (2006); Balili
_et al._ (2007); Baumberg  _et al._ (2008); Ohadi  _et al._ (2012). Owing to
their easy optical interrogation, high-speed (ps) interactions, and
macroscopic coherence (over hundreds of microns) Nelsen  _et al._ (2013),
polariton condensates are excellent candidates to probe and exploit for
sensing Sturm  _et al._ (2014); Dreismann  _et al._ (2014), spinoptronics Amo
_et al._ (2010); Ballarini  _et al._ (2013); Cerna  _et al._ (2013), new
optoelectronic devices Nguyen  _et al._ (2013); Bhattacharya  _et al._ (2013);
Schneider  _et al._ (2013), and quantum simulators Buluta and Nori (2009). The
driven-dissipative multicomponent polariton system can undergo additional
bifurcations and condense into states which are not eigenstates of the single-
particle Hamiltonian, but many-body states with reduced symmetry Aleiner  _et
al._ (2012); Zhang  _et al._ (2015). Thus, we should expect that two-component
exciton-polariton condensates can also show spontaneous symmetry breaking
bifurcations in their polarization state. Spin studies of microcavity
polaritons have been of great interest in recent years Lagoudakis  _et al._
(2002); Martín  _et al._ (2002); Kavokin  _et al._ (2003, 2004); Renucci  _et
al._ (2005); Krizhanovskii  _et al._ (2006); Gippius  _et al._ (2007); Leyder
_et al._ (2007); Paraïso  _et al._ (2010); Gao  _et al._ (2012); Kammann  _et
al._ (2012); Takemura  _et al._ (2014a). However, spontaneous symmetry-
breaking bifurcation of spin has not been observed before.

Here, we demonstrate spontaneous magnetization in an exciton-polariton
condensate, as a direct result of bifurcations in the spin degree of freedom.
Utilizing an optically trapped geometry, condensates spontaneously emerge in
either of two discrete spin-polarized states that are stable for many seconds,
$>10^{10}$ longer than their formation time. These states emit highly
circularly-polarized coherent light (up to 95%) and have opposite circular
polarizations. The condensate stochastically condenses in a left- or right-
circularly polarized state, with an occurrence likelihood that can be
controlled by the ellipticity of the nonresonant pump. The two spin-polarized
states can be initialized and switched from one state to another with weak
resonant optical pulses. Our system has potential applications in sensing,
optical spin memories and spin switches, and it can be implemented for
studying long-range spin interactions in polariton condensate lattices.

This article is structured as follows: in Section I we review trapped
polariton condensates and the current understanding of polarization in
untrapped polariton condensates. In Section II we present the key theme of
this work, which is the spontaneous buildup of stochastic circular
polarization. In Section III we propose a theoretical framework for the
phenomena discussed in this work. We show that stochastic circular
polarization is a signature of spontaneous parity breaking. In Section IV we
time-resolve the coherent driving of the spin, with resonant excitation. We
furthermore investigate the stability of the spin-polarized states against
thermal noise and conclude in Section V.

##  I Introduction

Figure 1: (a) Scanning electron microscopy image of a membrane microcavity,
cleaved across the middle. The membrane allows resonant excitation from the
back side of the cavity. (b) Above threshold hot polaritons condense into a
single coherent state located at the center of the trap (pump spots shown by
dashed circles), which is found at $k_{\parallel}=0$ (c). The spin state of
the condensate is determined by studying the emission polarization.

Exciton-polaritons are spinor particles formed by the strong coupling of
excitons in a semiconductor quantum well with photons in the microcavity in
which they are embedded Kavokin  _et al._ (2007). We create optically-trapped
polariton condensates by nonresonant excitation of a semiconductor microcavity
membrane [see FIG. 1(a) and Appendix A]. The excitation beam is shaped into a
4-spot pattern [shown by dashed circles in FIG. 1(b)]. The short-wavelength
continuous wave (CW) linearly-polarized pump injects an electron-hole plasma
at each pump spot, which rapidly relaxes to form excitons, in the process
losing all phase information. These reservoir excitons then scatter into
polariton states via multiple phonon-polariton and stimulated polariton-
polariton collisions Savvidis  _et al._ (2000), and they feed the zero-
momentum ground state at the center of the trap. Because of their large
effective mass, excitons typically diffuse only very small distances and stay
within $1\text{\,}\mathrm{\SIUnitSymbolMicro m}$ of the pump spots.
Microcavity polaritons, however, are $10000\text{\,}\mathrm{t}$imes lighter
giving longer diffusion lengths. Driven by their repulsive excitonic
interactions, polaritons can thus travel large distances away from the pump
spots within their lifetime Wertz  _et al._ (2010). Once the density inside
the trap exceeds the condensation threshold, a macroscopically coherent
condensate is formed [FIG. 1(c)]. Because the condensate overlaps only weakly
with the pump spots, it shows a narrower linewidth and less decoherence than
unconfined condensates Askitopoulos  _et al._ (2013). The optical trapping
method used here is similar to optical lattices in cold atomic systems Bloch
(2005), but with the major difference that the optical potential also provides
gain Wertz  _et al._ (2010); Wouters and Carusotto (2007); Tosi  _et al._
(2012).

Polaritons in quantum-well microcavities have two $J_{z}=\pm 1$ (spin-up or
spin-down) projections of their total angular momentum along the growth axis
of the structure, which correspond to right- and left-circularly polarized
photons emitted by the cavity, respectively. When the excitation is linearly-
polarized an equal population of spin-up and spin-down excitons forms in the
reservoir. An initially spin-balanced reservoir, in the absence of pinning to
any crystallographic axis, is expected to give a condensate with a stochastic
linear polarization Shelykh  _et al._ (2006). In most experiments with
polariton lasers the condensates have been found to be linearly polarized
along one of the crystallographic axes Deng  _et al._ (2002); Kasprzak  _et
al._ (2006). Nevertheless, in some cases a circularly polarized polariton
lasing has also been observed Martín  _et al._ (2002); Shelykh  _et al._
(2004); Krizhanovskii  _et al._ (2006); Roumpos  _et al._ (2009); Ohadi  _et
al._ (2012). Formation of a circularly polarized condensate is usually
associated with the effects of TE-TM splitting, or bosonic amplification of
the seed polarization of condensates. In all these cases, a circularly
polarized condensate is observed when the symmetry between the spin-up and
spin-down polaritons has been _explicitly_ broken in some fashion, by either
the pumping geometry (seeding with a circularly polarized pump) or by the
imposed rotation of the Stokes parameters due to polarization splitting. As a
result, the observed circular polarization was never stochastic (i.e. it is
fixed each time the condensate is excited, but it is different on each
realization).

##  II Spontaneous buildup of circular polarization

Figure 2:  (a) Polarization-resolved spatial image of the only two
realizations observed in a single trapped condensate in a 4-spot trapping
geometry. (b) At $T=$5\text{\,}\mathrm{K}$$ the spin-polarized states remain
stable for many seconds. (c) Simultaneously measured components of the
pseudospin for 1000 realizations. The total degree of polarization is $0.93\pm
0.03$ and the two average pseudospin states are
$s_{\downarrow}=[-0.22,0.19,-0.94]$ (blue vector) and
$s_{\uparrow}=[-0.22,-0.14,0.96]$ (orange vector). The measurement error for
each component is $<5\%$, and the variance is $\sim 1\%$.

Our experiments reveal a completely different behavior to the previous reports
on the polarization of polariton condensates. We observe a strong degree of
circular polarization $60\%<|s_{z}|<95\%$ [FIG. 2(a)] when the excitation is
linearly polarized (to better than 1 in $10^{5}$), which is stable for many
seconds [FIG. 2(b)]. The condensate stochastically adopts either of two
opposite circular-polarization states in each realization of the experiment.
We call these two states the spin-up ($s_{\uparrow}$) and spin-down
($s_{\downarrow}$) states. By mapping the polarization of the
photoluminescence (PL) we measure the polarization vector
$s_{x,y,z}=(I_{H,D,\circlearrowright}-I_{{V,A,\circlearrowleft}})/(I_{H,D,\circlearrowright}+I_{{V,A,\circlearrowleft}})$,
where $I$ is the measured intensity for horizontal ($H$), vertical ($V$),
diagonal ($D$), anti-diagonal ($A$), right-circular ($\circlearrowright$) and
left-circular ($\circlearrowleft$) polarizations. We measure all components of
the polarization vector (pseudospin) simultaneously and plot the mean of the
spin-up and spin down states on the Poincaré sphere separately for 1000
realizations [FIG. 2(c)]. In each realization the wavefunction of the
condensate spontaneously collapses into one of the two discrete spin-polarized
states which have opposing circular and diagonal components, marked by blue
and orange vectors in FIG. 2(c). The linear axis along which the pseudospin
flips (marked here as diagonal) does not depend on the geometry of the trap,
and it changes direction with the position of the condensate on the sample.

To demonstrate that the buildup of circular polarization is truly spontaneous,
we illuminate the sample with a long-duration pulse of $9\text{\,}\mathrm{ms}$
using an acousto-optic modulator (AOM). A condensate builds up and picks a
random state (e.g. $s_{\downarrow}$) and stays in that particular state as
long as the pump pulse lasts [FIG. 3(a)]. We then repeat the same measurement
but this time we modulate the pump intensity so that a condensate is created
and destroyed every $2\text{\,}\mathrm{\SIUnitSymbolMicro s}$. FIG. 3(b) shows
how the condensate, although being stable for many seconds, picks a random
polarization at every successive realization due to random initial conditions
at the onset of condensation. In this case the parity symmetry is broken
spontaneously. The buildup of circular polarization is independent of the
precise position on the sample, of the sample orientation, of the pump spot
orientation, and of the polariton detuning.

Figure 3:  (a) Emitted circular $\sigma_{+}$ and $\sigma_{-}$ intensities for
a condensate realization when $P>P_{c}$, created by a long pulse (marked by
P). The condensate randomly picks the $s_{\downarrow}$ state here. Note the
opposite axis directions for $\sigma_{+}$ and $\sigma_{-}$ components. (b)
Same as (a), with pump intensity modulated by square wave (shown on top by P).
A cut from the middle of a $9\text{\,}\mathrm{ms}$ exposure is shown, where
each square pulse corresponds to a single realization. The condensate randomly
picks $s_{\uparrow}$ or $s_{\downarrow}$ states (marked by grey rectangles).
(c) Occurrence contrast $\xi$ of spin-up and spin-down realizations measured
in (b) vs the ellipticity of the pump. (d) Measured circular polarization of
condensate vs pump ellipticity. Dashed lines mark the region where the
condensate initializes stochastically in spin-up or spin-down states in each
realization.

We can also _explicitly_ break the symmetry in our experiments by changing the
ellipticity of the pump laser and measure the contrast of the occurrence
frequency of spin-up ($f_{\uparrow}$) to spin-down ($f_{\downarrow}$)
condensate realizations,
$\xi=(f_{\uparrow}-f_{\downarrow})/(f_{\uparrow}+f_{\downarrow})$. The
probability of a realization resulting in state $f_{\uparrow\downarrow}$ is
equal to $(1\pm\xi)/2$ respectively. FIG. 3(c) shows $\xi$ as a function of
the pump circular polarization ($s_{z,\text{pump}}$) averaged over 1000
realizations for each point. As the ellipticity of the pump is increased from
linear ($s_{z,\text{pump}}=0$) to right circular ($s_{z,\text{pump}}>0$) the
probability of creating a condensate in the spin-up state increases;
conversely, for the left-circularly polarized pump ($s_{z,\text{pump}}<0$),
the probability of creating a spin-down condensate increases. With a linearly-
polarized pump we have an equal probability of creating a spin-up or spin-down
condensate. Although the pump laser is nonresonant with the final polariton
states, the initially created carrier spin is not entirely randomized during
their multiple carrier-carrier and exciton-phonon scatterings Roumpos  _et
al._ (2009); Ohadi  _et al._ (2012); Kammann  _et al._ (2012); Li  _et al._ .
As a result of this incomplete spin relaxation of the excited carriers,
changing the ellipticity of the pump breaks the symmetry of the condensate
toward the same circular polarization as that of the pump. For pump circular
polarizations far greater than 0.1 the condensate is formed deterministically
in the same polarization state as that of the pump [FIG. 3(d)].

An interesting question here is why we observe the spontaneous buildup of
circular polarisation, as opposed to the linear polarisation that is widely
reported in the literature Deng  _et al._ (2002); Kasprzak  _et al._ (2006).
The key difference between the experiments presented in this work and other
studies of polariton condensation lies in the excitation geometry. For our
trapped condensates, the pump and condensate are spatially separated, which
critically reduces the contaminating interactions between the condensate and
reservoir. The large interaction between untrapped condensates and the
unpolarized exciton reservoir results in spin-flip scattering of polaritons
with reservoir excitons. If there is a depolarized reservoir on top of the
condensate, the spin-flip scattering processes minimize any imbalance between
circular components of the condensate. This minimization leads to quenching of
the buildup of circular polarization, forcing the polaritons to condense only
with linear polarization. Moreover it has been shown previously that, because
of a smaller overlap with the reservoir, trapped condensates have a smaller
linewidth than untrapped condensates Askitopoulos  _et al._ (2013). Our
careful studies with temperature and our theoretical calculations (see Section
IV.4), show how spin noise in the system results in spin flipping of the
condensate. The spin flip rate scales exponentially with noise. Larger
linewidth untrapped condensates have higher spin flip rates, which wash out
circular polarization effects observed here.

Figure 4: (a) Nonresonant linearly polarized weak probe overlapped with the
condensate. (b) Average circular polarization degree vs power of the probe
beam. The blue line is a guide to the eye.

To show the crucial role of the reservoir excitons we place a weak nonresonant
linearly-polarized probe beam on top of the condensate as shown in FIG. 4(a).
The probe beam which has just a small fraction of the 4-spot pump power
($<0.025\;P_{th}$), induces a reservoir of excitons that overlap with the
condensate but it does not stop the condensation or reduce the condensate
density below the critical circular polarization density. We then measure the
absolute average circular polarization degree in each realization, as a
function of the weak probe power. As shown in FIG. 4(b), the circular
polarization degree decreases monotonously as the probe power increases,
demonstrating the quenching of circular polarization due to the influence of
the overlapping reservoir. This measurement evidences the importance of the
separation of the pump-induced reservoir and the trapped condensate in the
observation of ferromagnetic condensates.

Four scenarios might explain the buildup of the stochastic circular
polarization depending on populations of same- and opposite- (or cross-) spin
polaritons:  (i) a higher interaction energy for cross-spin than for same-spin
polaritons in a condensate which is in thermal equilibrium,  (ii) a higher
gain from the cross-spin reservoir,  (iii) density-dependent losses enhanced
by cross-spin condensed polaritons, and  (iv) linear polarization energy
splitting accompanied by a dissipation rate splitting which destabilizes the
linearly-polarized condensate (‘spin bifurcation’) . As we will show now, our
experimental data strongly suggest that scenario (iv) is the correct
explanation. In the first scenario (‘energy-minimization’), the condensate
free energy is minimized when it acquires circular polarization. This
situation could happen in a condensate in thermal equilibrium if the
interaction energy of cross-spin polaritons is stronger than that of same-spin
polaritons, meaning the coexistence of cross-polarized polaritons is not
favored Rubo  _et al._ (2006). However, the interaction with opposite spin
polaritons is well known to be weaker than, and opposite to, that of same-spin
polaritons Vladimirova  _et al._ (2010); Takemura  _et al._ (2014a, b).
Moreover, being externally driven and having short lifetimes, polariton
condensates are generally far from thermal equilibrium Kasprzak  _et al._
(2008). We note that “energetical” mechanisms cannot explain why states with
elliptical polarization (i.e. not fully circular) are formed. In the second
scenario (‘cross-gain’), in order to acquire a circular degree of
polarization, the condensate must experience larger gain from the cross-spin
reservoir than from the same-spin reservoir. However, the scattering rate from
the cross-spin reservoir into the condensate is measured to be significantly
smaller than the same-spin Lagoudakis  _et al._ (2002). The third scenario
(‘cross-loss’) requires the condensate loss rate to increase when opposite
spins are present. If polaritons are directly excited by light, observations
have suggested biexciton formation can produce such enhanced losses Paraïso
_et al._ (2010). However, this is only significant when the relative spectral
detuning of cavity and exciton is small ($<$2\text{\,}\mathrm{meV}$$), which
is far from the case here. We observe the spontaneous buildup of circular
polarization throughout the entire $+2$ to $-10\text{\,}\mathrm{meV}$ detuning
range. In the remaining scenario (‘spin bifurcation’), which we present in
Section III for the first time, the buildup of circular polarization is caused
by small differences of the energy and dissipation rates of two orthogonal
linearly-polarized polariton modes, which is present even at $k=0$.

Strain-induced splitting of the linear components of polariton condensates has
been demonstrated previously Balili  _et al._ (2010); Kłopotowski  _et al._
(2006). In our sample this splitting varies depending on the position on the
sample. We observe a linear polarization energy splitting of up to
$$100\text{\,}\mathrm{[}$\sim]{}$ depending on the position of excitation on
the membranes. Note that, all the effects reported here are also seen on
unetched samples, so strain from patterning is not crucial. However, we
observe a higher splitting at the edges of the membrane than in the middle, as
there is more stress in the structure at the edges. Due to the curvature of
the cavity stopband, any energy splitting is accompanied by a difference of
the linewidth (dissipation rate), as explained in Appendix B. This energy
splitting between the two linear components combined with a difference in
dissipation rates causes the polarization of the condensate to change from
linear polarization to circular at a critical density (see Section III).

We emphasize that the stochastic circular polarization here cannot be
explained in the framework of the optical spin Hall effect Kavokin  _et al._
(2005); Kammann  _et al._ (2012). In our trapping geometry the condensate is
formed at the ground state with ($\bar{k}=0,\;\delta
k=$0.4\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}^{-1}$$) [FIG. 1(c)],
where the transverse-electric and transverse-magnetic (TE-TM) splitting
vanishes Shelykh  _et al._ (2004). The trap diameter here is 6 times smaller
than the observed spin-ring patterns measured for a nonequilibrium condensate
formed at much higher in-plane wave-vectors in the same sample Kammann  _et
al._ (2012). Moreover, the geometry or the orientation of the trap does not
affect the polarization state of the condensate. We also see the stochastic
circular polarization with a ring-shaped trap and also in high-order spatial
mode condensates Cristofolini  _et al._ (2013); Askitopoulos  _et al._ (2014).
Finally, it should be noted that any theoretical picture that assumes the
buildup of circular polarization arises only because of the geometrical
arrangement of the pump, would necessarily fail to explain the most essential
part of this work, which is the spontaneous symmetry breaking (stochastic
behavior).

##  III Spin Bifurcation Theory (Broken Parity)

Figure 5:  (a) Numerical calculations using Eq. 3 for the case when the
condensate occupation is below the critical occupation $S_{c}$ (dotted lines)
and when above $S_{c}$ (solid lines). Blue lines show the occupation of the
spin-down component and orange lines show the spin-up component. (b) Numerical
calculations of the Stokes parameters at different powers using Eq. 3. (c)
Measured power dependence of the condensate Stokes parameters. (d) 2D
simulations, with asymmetric initial conditions [$s_{z}(t=0)<0.01$] show
density of the $\psi_{\pm}$ components of the wave function. Pump spots are
marked by dashed lines (see Appendix G for parameters).

Our theory is a development of the theory of polariton weak lasing in two
coupled condensation centers Aleiner  _et al._ (2012) now for the case of the
spin degree of freedom. Here, we have right and left circular polarizations
instead of two separated condensates, and we also allow for the gain-
saturation nonlinearity in the system.

The order parameter for an exciton-polariton condensate is a two-component
complex vector $\Psi=[\psi_{+1},\psi_{-1}]^{\mathrm{T}}$, where $\psi_{+1}$
and $\psi_{-1}$ are the spin-up and spin-down wave functions. The components
of the order parameter define the measurable condensate pseudospin
$\mathbf{S}=(1/2)(\Psi^{\dagger}\cdot\bm{\sigma}\cdot\Psi)$, and the
normalized spin vector $\mathbf{\hat{s}}=\mathbf{S}/S$, where $\sigma_{x,y,z}$
are the Pauli matrices. The components of this vector contain information
about the intensities and relative phases of the emitted light. The order
parameter evolves according to the driven dissipative equation

$\displaystyle\begin{split}\mathrm{i}\frac{d\Psi}{dt}=&-\textstyle{\frac{\mathrm{i}}{2}}g(S)\Psi-\frac{\mathrm{i}}{2}(\gamma-\mathrm{i}\varepsilon)\sigma_{x}\Psi\\\
&+\textstyle{\frac{1}{2}}\left[(\alpha_{1}+\alpha_{2})S+(\alpha_{1}-\alpha_{2})S_{z}\sigma_{z}\right]\Psi,\end{split}$
(1)

or in components:

$\displaystyle\begin{split}\dot{\psi}_{+1}=&-\textstyle{\frac{1}{2}}g(S)\psi_{+1}-\textstyle{\frac{1}{2}}(\gamma-\mathrm{i}\varepsilon)\psi_{-1}\\\
&-\textstyle{\frac{\mathrm{i}}{2}}(\alpha_{1}|\psi_{+1}|^{2}+\alpha_{2}|\psi_{-1}|^{2})\psi_{+1},\end{split}$
(2a)
$\displaystyle\begin{split}\dot{\psi}_{-1}=&-\textstyle{\frac{1}{2}}g(S)\psi_{-1}-\textstyle{\frac{1}{2}}(\gamma-\mathrm{i}\varepsilon)\psi_{+1}\\\
&-\textstyle{\frac{\mathrm{i}}{2}}(\alpha_{1}|\psi_{-1}|^{2}+\alpha_{2}|\psi_{+1}|^{2})\psi_{-1}.\end{split}$
(2b)

Here, $g(S)=\Gamma-W+\eta S$ is the pumping-dissipation balance, $\Gamma$ is
the (average) dissipation rate, $W$ is the incoherent in-scattering (or
‘harvest’ rate), and $\eta$ captures the gain-saturation term with
$S=(|\psi_{+1}|^{2}+|\psi_{-1}|^{2})/2$ Keeling and Berloff (2008). Here, this
gain saturation depends on the total occupation of the condensate (treated
more generally in Appendix C). It is assumed now that $X$ (horizontal) and $Y$
(vertical) linearly-polarized single-polariton states have different energies
and dissipation rates. The energy of the $X$-polarized state is shifted by
$-\varepsilon/2$, and the energy of the $Y$-polarized state by
$+\varepsilon/2$. The dissipation rate from the $X$-polarized state is
$\Gamma+\gamma$, while the dissipation rate from the $Y$-polarized state is
$\Gamma-\gamma$ (see also Appendix B). Finally, $\alpha_{1}$ is the repulsive
interaction constant for polaritons with the same spin, and $\alpha_{2}$ is
the interaction constant for polaritons with opposite spins.

From Eq. 1 we obtain for the components of the pseudospin vector
($\alpha=\alpha_{1}-\alpha_{2}$):

$\displaystyle\begin{split}\dot{S_{x}}=-g(S)S_{x}-\gamma S-\alpha
S_{z}S_{y},\end{split}$ (3a)
$\displaystyle\begin{split}\dot{S_{y}}=-g(S)S_{y}+\varepsilon S_{z}+\alpha
S_{z}S_{x},\end{split}$ (3b)
$\displaystyle\begin{split}\dot{S_{z}}=-g(S)S_{z}-\varepsilon
S_{y},\end{split}$ (3c)

and the related equation for the total spin $\dot{S}=-g(S)S-\gamma S_{x}$.
There are two sets of solutions, which we call here the paramagnetic and
ferromagnetic solutions.

#### Paramagnetic solutions.

These give simple condensation into either $X$ or $Y$ linearly-polarized
states. The $Y$ state possesses the longest lifetime, and the condensation
threshold is reached for this state first at $W_{1}=\Gamma-\gamma$. There is
no parity breaking for this condensate: $S_{y}=S_{z}=0$, $S_{x}=-S$ with
$S=(W-W_{1})/\eta$, so that the occupations of $+1$ and $-1$ components are
equal. However, this condensate solution becomes unstable for $W>W_{2}$. The
values of the critical occupation $S_{c}$ and the critical pumping rate are

$S_{c}=\frac{\gamma^{2}+\varepsilon^{2}}{\alpha\varepsilon},\qquad
W_{2}=W_{1}+\eta S_{c}.$ (4)

Note that this instability is present also for equal dissipation rates, i.e.
when $\gamma=0$. In this case, the system (Eq. 3a-c) describes the self-
induced Larmor precession of the pseudospin vector. Incorporating energy
relaxation (e.g., using small negative $\gamma$) then leads to the formation
of the $X$-polarized condensate—an intuitively expected result.

#### Ferromagnetic solutions.

The key ingredient of our theory is the presence of the $\gamma>0$ parameter
describing the variation of dissipation rates. This parameter allows the
formation of the ‘weak lasing’ regime Aleiner  _et al._ (2012), which is
characterized by two important features: (i) the $X$-polarized condensate is
also unstable, and (ii) when the $Y$-polarized condensate loses stability at
the critical occupation $S_{c}$, it continuously transforms into one of the
two ferromagnetic states. While Eqs. (2) are parity symmetric, i.e., they are
not affected by the interchange of left and right circular polarization, the
new solutions are characterized by broken parity symmetry and by spontaneous
formation of either left or right elliptical polarization. These solutions are

$S_{x}=-\frac{g(S)}{\gamma}S,\qquad S_{y}=-\frac{g(S)}{\varepsilon}S_{z},$
(5a)
$S_{z}=\pm\frac{\varepsilon}{\gamma}\sqrt{\frac{\gamma^{2}-g(S)^{2}}{\varepsilon^{2}+g(S)^{2}}}\,S,\qquad
S=\frac{\gamma[\varepsilon^{2}+g(S)^{2}]}{\alpha\varepsilon g(S)},$ (5b)

where the positive root of the second equation in Eq. (5b) should be taken. We
note that while the sign of $S_{x}$ is always negative, $S_{y}$ and $S_{z}$
have opposite signs for the two solutions. This means that the left-circular
component is accompanied by a diagonal component, and the right-circular by
anti-diagonal. Moreover, if these components change for some reason, they
mirror each other as long as the total condensate occupation stays fixed. We
label these two solutions as the $s_{\downarrow}$ and $s_{\uparrow}$ spin
states.

The spin-independent model for the gain saturation used in this Section is
sufficient to describe the experimentally observed features. However, the spin
relaxation in the reservoir can be slow Li  _et al._ , and in this case the
model should be modified to allow the saturation terms to depend on the
individual occupations of the left- and the right-circular polarization
components rather than the total occupation only. The parity breaking is still
present after this modification; however, the stability of solutions becomes
more complex. The ferromagnetic solutions can now become unstable and
transform into periodic cycles Rayanov  _et al._ (2015), the dynamics of the
pseudospin can become irregular, and this can also result in the formation of
the stable $X$-polarized condensate at high pumping powers. See Appendix C for
more details.

Numerical calculations for the occupation of the two circular components of
the wavefunction when $S<S_{c}$ (dotted lines) and when $S>S_{c}$ are shown in
FIG. 5(a). Here, the condensate is initialized with a small asymmetry in spin-
up and spin down occupations (<1%). Below the critical occupation $S_{c}$, the
condensate is linearly polarized, but when the occupation is increased above
the threshold $S_{c}$, the condensate adopts one of two elliptically polarized
configurations depending on the initial conditions. In the experiment, the
stochastic behavior is due to random spin fluctuations at the onset of the
condensation. In theory, we can reproduce it by randomly setting the initial
conditions. Numerical calculations of the condensate polarization versus
excitation power are shown in FIG. 5(b). Directly at the condensation
threshold the condensate is linearly polarized, but once it reaches the
critical occupation ($P_{c}=1.3P_{\mathrm{th}}$, marked by a dashed grey
line), the linear component is quenched and circular polarization builds up.
This behavior reproduces the experimental data, as shown in FIG. 5(c). We
observe an initial buildup and subsequent quenching of linear polarization
with the continuing increase of circular polarization at $P_{c}=1.25P_{th}$
(marked by a dashed grey line, with total intensity marked by a dotted red
line). Once circular polarization is achieved, the orientation of the
condensate circular polarization becomes stochastic under linearly polarized
pumping.

We can extend Eq. (1) and account for 2D real-space degrees of freedom by
using complex Ginzburg-Landau-type equations Wouters and Carusotto (2007);
Keeling and Berloff (2008); Dreismann  _et al._ (2014), which in addition to
the pump and decay also incorporate a repulsive potential due to the excitons
in the pump spots and an energy relaxation Wouters  _et al._ (2010) for
polaritons in the trap:

$\mathrm{i}\frac{d\Psi}{dt}=-\frac{\mathrm{i}}{2}\left[g(S)+\gamma\sigma_{x}\right]\Psi\\\
+(1-\mathrm{i}\Lambda)\Big{\\{}\frac{1}{2}\left[(\alpha_{1}+\alpha_{2})S+(\alpha_{1}-\alpha_{2})S_{z}\sigma_{z}\right]\Psi\\\
-\frac{1}{2}\varepsilon\sigma_{x}\Psi-\frac{\nabla^{2}}{2m^{*}}\Psi+V_{p}\Psi\Big{\\}},$
(6)

where $m^{*}$ is the effective mass of the polaritons. The harvest rate is
given by $W=rP/\Gamma_{R}$, where $P$ is the spin-independent spatial profile
of the excitation, $\Gamma_{R}$ is the decay rate of the exciton reservoir,
and $r$ is the incoming rate of polaritons into the condensate. The gain
saturation is given by $\eta=r^{2}P/\Gamma_{R}^{2}$. The repulsive potential
due to the interaction of polaritons with the exciton reservoir is given by
$V_{p}=\textstyle{\frac{1}{2}}g_{r}N+\textstyle{\frac{1}{2}}g_{P}P$, where
$g_{r}$, and $g_{P}$ are the interaction constants of polaritons with the
exciton reservoir and the pump spot respectively, and $N=g(S)/r$ is the
density of the exciton reservoir. Here, $\Lambda\ll 1$ is a phenomenological
constant that gives the energy relaxation. The density profile of the two
circular components of the wave function in the steady state [FIG. 5(d)], for
the case of a trapped condensate in the middle of the 4 pump spots, exhibits a
circular polarization degree of $|s_{z}|=0.69$. Note that with $\gamma=0$ and
only polarization splitting (including TE-TM splitting), our 2D simulations do
_not_ show bistable condensation.

It is important to note the differences between the ferromagnetic states we
discuss here and the magnetization transition in equilibrium cold atom systems
Sadler  _et al._ (2006); Stamper-Kurn and Ueda (2013). First, the parity
breaking bifurcation described above does not reduce the energy of the system
(unlike for atoms). In fact the energy of elliptically polarized states is
higher than that of linearly polarized states. Second, the in-plane components
of the spin do not vanish completely. Third, we have a magnetized condensate
with only two possible orientations, whereas in atomic systems ferromagnetic
domains with continuous variable orientation are observed.

If the Hamiltonian and the initial state of a system are symmetric under the
exchange of spin-up and spin-down components, but the final state is not, the
parity symmetry is spontaneously broken. This is indeed the case here: We
excite an equal population of spin-up and spin-down polaritons, which
spontaneously condense, but form highly circularly polarized macroscopic
states in the absence of any external magnetic field.

##  IV Resonant excitation

The GaAs substrate commonly used in the fabrication of GaAs microcavities is
opaque at the emission wavelength ($\sim 800\text{\,}\mathrm{nm}$) of the
cavity polaritons. As a result, the back side of the cavity is resonantly
inaccessible. Resonant excitation of the cavity from the front side has
complications with backscatter from the laser, especially in high finesse
cavities and at normal incidence, where the condensate emission mode is
located. To circumvent this problem we chemically etch the substrate to form
membranes of $8.8\text{\,}\mathrm{\SIUnitSymbolMicro m}$ thickness and
$300\text{\,}\mathrm{\SIUnitSymbolMicro m}$ diameter [see FIG. 1(a)]. For
resonant excitation, we use a narrow-linewidth ($<$2\text{\,}\mathrm{GHz}$$)
CW laser, which is amplitude modulated with a second AOM. We call this
resonant laser the ‘gate’. We use two photomultipliers and a fast oscilloscope
to time resolve the left- ($\sigma_{-}$) and right-circular ($\sigma_{+}$)
polarization intensity of the condensate emission. The resonant excitation
laser, the nonresonant pump laser, the cameras and the oscilloscope are all
synchronized, which allows us to vary the delay time and amplitudes of each
laser pulse on demand.

###  IV.1 Resonant initialization of spin states

Figure 6:  Measured circular polarization of condensate vs pump ellipticity,
for three different gate polarizations. Dashed lines mark the region where the
condensate initializes stochastically in spin-up or spin-down states in each
realization.

We can control the polarization of the trapped condensate with the gate laser.
In this case, we additionally _resonantly_ excite the condensate (which is
generated by the four nonresonant pump spots) with a second laser from the
backside of the microcavity membrane. This gate can be linearly, left-, or
right-circularly polarized. FIG. 6 shows the condensate circular degree of
polarization versus that of the pump. In the green curve which shows the
behavior when the gate is linearly polarized, we reproduce the same result as
that in FIG 3(d). However, when a right (or left) circularly polarized gate is
applied, the curve shifts to the left (or right). In other words, with a
linearly polarized pump, the condensate initializes in a right (or left)
circularly polarized state. The imbalance caused by the resonant gate cancels
out with an opposite circularly polarized pump with a circular polarization
degree of $|s_{z,\text{pump}}|\sim 0.1$.

###  IV.2 Elliptically-polarized pump: coherent driving

Figure 7:  (a) Measured intensity of circular polarization components
($\sigma_{\pm}$) for adding gate pulse $G$ (intensity shown on above inset).
The condensate initializes in the $s_{\downarrow}$ state, due to the
asymmetric pumping condition $s_{z,\text{pump}}=-0.1$. A gate pulse with an
opposite circular polarization ($\sigma_{+}$) is applied. (b) Numerical
calculations for (a), showing that during the gate pulse $G=A^{2}$ the spin-up
and spin-down densities oscillate with a $\pi$ phase shift. (c) Time evolution
of the pseudospin, showing how the pseudospin orbits around the spin-down
stationary state (marked by the blue arrow) when the gate pulse is applied.
(d) Same as in (a) but with a fivefold increase in the intensity of the gate.
The time-averaged condensate emission is almost circularly unpolarized when
the pulse is applied ($\langle s_{z}\rangle=0$). (e) Numerical calculations
for (d), showing almost equal amplitude oscillation of the spin-up and spin-
down densities. (f) Trajectory of pseudospin, showing that with a strong gate
pulse large orbits commence around the two spin-up and spin-down stationary
states. (g) Average circular polarization when the gate is on, at different
gate intensities. When an opposite polarization gate is applied, $\langle
s_{z}\rangle$ converges to zero, whilst it remains unchanged for the same-spin
gate. The orange line is a guide to the eye. (h) Numerical calculations of
$\langle s_{z}\rangle$ vs normalized pump intensity ($n$ is the condensate
occupation).

In a first experiment, we initialize a condensate in the spin-down state
($s_{\downarrow}$) by making the pump laser slightly left-circularly polarized
(‘asymmetric pumping’). Once the condensate is created, we excite the sample
resonantly from the back side with an oppositely circularly polarized
($\sigma_{+}$) gate. FIG. 7(a) shows the intensity of $\sigma_{+}$ and
$\sigma_{-}$ components of the condensate emission during this gating. We see
a reduction in intensity for the component which is opposite to the gate laser
polarization and an increase for the same spin polarization component. To
account for this in the theory, we add a new term to Eq. 2, corresponding to
the resonant laser:

$\displaystyle\begin{split}\dot{\psi}_{+1}=&-\textstyle{\frac{1}{2}}g_{+1}(S)\psi_{+1}-\textstyle{\frac{1}{2}}(\gamma-\mathrm{i}\varepsilon)\psi_{-1}\\\
&-\textstyle{\frac{\mathrm{i}}{2}}(\alpha_{1}|\psi_{+1}|^{2}+\alpha_{2}|\psi_{-1}|^{2})\psi_{+1}\\\
&-\mathrm{i}A\,\Pi(t,t_{0},\delta t)e^{-\mathrm{i}\omega_{g}t},\end{split}$
(7a)
$\displaystyle\begin{split}\dot{\psi}_{-1}=&-\textstyle{\frac{1}{2}}g_{-1}(S)\psi_{-1}-\textstyle{\frac{1}{2}}(\gamma-\mathrm{i}\varepsilon)\psi_{+1}\\\
&-\textstyle{\frac{\mathrm{i}}{2}}(\alpha_{1}|\psi_{-1}|^{2}+\alpha_{2}|\psi_{+1}|^{2})\psi_{-1},\end{split}$
(7b)

where $A\,\Pi(t,t_{0},\delta t)=AH(t-t_{0})H(t_{0}+\delta t-t)$ uses the
Heaviside step function $H$ to give a square pulse with amplitude $A$ which
starts at time $t_{0}$ and lasts for $\delta t$, and $\omega_{g}$ is the
excitation frequency. To account for the elliptically-polarized pumping we
modify $g(S)$ to $g_{\pm 1}(S)=\Gamma-W_{\pm 1}+\eta S$. Numerical
calculations [FIG. 7(b)] show the case $W_{-1}=1.09\;W_{+1}$, $A=5\times
10^{-4}n$, where $n$ is the occupation of the condensate, and
$\omega_{g}=-0.3\varepsilon$. The dotted line marks the time average of the
oscillations. The condensate pseudospin vector precesses around the stationary
state pseudospin vector at a frequency of
$\omega_{L}/2\pi\sim$10\text{\,}\mathrm{GHz}$$ in a limit cycle [see FIG.
7(c)]. In the case where the condensate is highly spin-polarized and the gate
is at resonance ($\omega_{g}=0$), the oscillation frequency is equal to the
self-induced Larmor precession frequency $\omega_{L}=\gamma\varepsilon/g$
(Appendix F). This sets the fastest possible spin dynamics in the system, and
does not depend on the gate intensity. Here, the condensate pseudospin
oscillates faster than our detection time resolution and consequently we only
see the average effect in FIG. 7(a). The coherent driving reported here
strongly depends on the detuning of the gate laser frequency relative to that
of the condensate. The resonance width at which we can drive the condensate is
determined to be 10-$20\text{\,}\mathrm{\SIUnitSymbolMicro eV}$, as explained
in Appendix E.

Figure 8:  (a) Intensity of the spin-up and -down states as a function of
time. Resonant pulse $G$, with opposite circular polarization ($\sigma_{+}$)
switches the circular polarization of the condensate. (b) Despite the slow
rise time of our AOM ($$\sim 100\text{\,}\mathrm{ns}$)$, the condensate
switches to the opposite spin state in less than $10\text{\,}\mathrm{ns}$ (the
limit of our detection), as soon as the intensity crosses the spin-switching
threshold. (c) Numerical calculations for (a), showing that the spin switches
with a gate pulse with intensity 6 orders of magnitude smaller than that of
the condensate. The evolution of the pseudospin is shown in (d). The two
attractors, marked by the blue and orange arrows, are the two stationary
$s_{\uparrow\downarrow}$ spin-polarized states.

In the case of FIG. 7(d) the intensity of the gate laser is increased by five
times with respect to that in (a). Under these conditions, the time average of
the circular component of the condensate emission becomes almost unpolarized.
If the amplitude of the resonant excitation is large enough, the condensate
pseudospin can cross over to the second attractor (the spin-up state) and form
a large trajectory which encompasses both spin-up and spin down stationary
states with a characteristic period doubling Alexeeva  _et al._ (2014) [FIG.
7(e,f)]. The dotted lines in (e) mark the time average of the oscillations.
FIG. 7(g) shows the power dependence of the average circular polarization as a
function of the intensity of the gate. As the oppositely polarized gate
intensity is increased, we see the condensate average circular polarization
converges to zero. Here, we have the gate resonant with the condensate.
However, the coherent driving strongly depends on the detuning of the gate
laser frequency to that of the condensate. The resonance width where we can
drive the condensate is determined to be
10-$20\text{\,}\mathrm{\SIUnitSymbolMicro eV}$, as explained in Appendix E.
That theory and experiment agree regarding the observed average polarization
over the whole range of gate powers, and also the observation of a resonance
(see Appendix E) strongly suggests that our description in terms of coherent
driving is valid.

In summary, the fixed points of broken parity symmetry
($s_{\downarrow},s_{\uparrow}$) become unstable due to small perturbations and
convert to limiting circles around stationary states. The precession is linear
for small gate amplitudes but it becomes nonlinear for large amplitudes. This
is manifested by oscillation of the condensate parameters, occupation and
polarisation with time. In this linear regime, the gate pulse appears to act
as if it induces an effective magnetic field, around which the condensate
polarisation precesses. It is worth noting that this system can exhibit
chaotic behavior with increased amplitude of the pulse. Increasing the
amplitude first results in period doubling, and eventually leads to chaos
(Feigenbaum scenario) Alexeeva  _et al._ (2014). The ability to coherently
control the spin of the condensate suggests its utility for computational
operations.

###  IV.3 Linearly-polarized pump: spin switching

We now explain the case when the nonresonant pump is linearly polarized. In
this case there is no external ‘force’ to drive the condensate to a specific
spin state. As a result, once the circular component of the condensate
pseudospin crosses zero, it falls into the opposite spin state’s attractor.
FIG. 8(a) shows a realization which lasts for
$4\text{\,}\mathrm{\SIUnitSymbolMicro s}$ with the condensate first
initialized in the spin down state ($s_{\downarrow}$). We resonantly excite
the condensate with an oppositely polarized gate ($\sigma_{+}$). This causes
the switching of the polarization of the condensate to the same polarization
as that of the gate. The gate, which is
$1.5\text{\,}\mathrm{\SIUnitSymbolMicro s}$ in duration, is then turned off
but the condensate remains in the switched polarization because the pump is
linearly polarized and the symmetry is not explicitly broken. This shows that
we are capable of manipulating the polarization of the condensate on demand,
while also reinforcing our observation that the condensate picks a
polarization spontaneously when it is formed. It should be noted that the
state of the condensate does not change when the circular polarization of the
gate is the _same_ as that of the condensate.

If the symmetry breaking was somehow set by the parameters of the experiment,
as in the previous example where the pump was slightly elliptical, the
condensate would switch back to its original state after the gate beam was
turned off. Instead, a very small gate field in the cavity can initiate
coherent switching. The upper panel in FIG. 8(a) shows the transmitted gate
intensity when the pump is blocked. Indeed, the gate power before the cavity
is $600\text{\,}\mathrm{\SIUnitSymbolMicro W}$, which is 50 times weaker than
the pump power. However, only a small fraction of this resonant beam couples
to the cavity in our setup. Therefore, a suitable comparison of the intensity
of the gate to that of the condensate is to compare the transmitted intensity
of the gate to the intensity of the condensate in the same direction. We find
that the gate intensity is more than 60 times weaker than the condensate
intensity, when the gate laser frequency is tuned to the bottom of the
polariton dispersion (accounting for the blueshift of the condensate).

By estimating the occupation number of the condensate (see Appendix D) we find
that only 13 polaritons are enough to reverse the spin state of the
condensate. The condensate switching time is less than
$10\text{\,}\mathrm{n}\mathrm{s}$ (our detection limit), and is 10 times
faster than the switching time of the AOM [FIG. 8(b)]. This important fact
shows that the condensate does not follow the optical gate adiabatically.
Simulations of the spin-switching with minimum resonant gate intensity are
shown in FIG. 8(c). The polarization of the gate is opposite to that of the
condensate and we have assumed symmetric pumping rates for the spin-up and
spin down condensates i.e. $W_{+1}=W_{-1}$. The condensate polarization
reverses within $\sim 200\text{\,}\mathrm{ps}$ once the gate is applied and
remains in that state after the pulse is turned off during continued spin
evolution [FIG. 8(d)]. Switching requires a minimum gate intensity, to twist
the condensate pseudospin onto the equator in the Poincaré sphere. This sets a
threshold for the gate power. By measuring the gate flux, and using the
switching time of $10\text{\,}\mathrm{ns}$, an _upper_ bound of the minimum
energy for switching the state of the condensate is found to be
$$1\text{\times}{10}^{-15}\text{\,}\mathrm{[}$\sim]{J}$, comparable to state-
of-the-art optoelectronic switches with similar speeds Nozaki  _et al._
(2010). We emphasise, however, that the theoretical limit for the minimum
switching energy is 50 times smaller because the spin dynamics of the system
($\sim$200\text{\,}\mathrm{ps}$$) is 50 times faster than our detection limit.
This polaritonic system is then an extremely low-power switch.

Multistability has indeed been demonstrated before in resonantly pumped
condensates Paraïso  _et al._ (2010); Cerna  _et al._ (2013). However, there
are several major differences between our system and that of the Deveaud
group. In Deveaud’s experiments the physical process causing multistability is
the nonlinear nonradiative losses in the polariton gas due to the formation of
biexcitons. These losses are only significant when the polariton gas energy is
close to the biexciton energy ($<$2\text{\,}\mathrm{meV}$$) Wouters  _et al._
(2013). In contrast here, the bistability that we observe is present
$>$10\text{\,}\mathrm{meV}$$ below the biexciton energy. There is no phase
transition in Paraïso et al. work Paraïso  _et al._ (2010) on multistability
and symmetry is _not_ spontaneously broken. In contrast, we observe spin
symmetry breaking while the power thresholds of the left- and right-circularly
polarized states are the same, allowing us to observe spontaneous
magnetization. In the Deveaud multistable system, in order to switch the
polarization one has to inject an opposite-spin polariton density equal to the
density difference of spin-up and spin-down polaritons. In contrast, in our
experiments, the condensate switches with a gate intensity 60 times weaker
than the condensate. Theoretically, the gate intensity can be much weaker and
the reason that we do not see switching at even weaker powers experimentally
is due to spin fluctuations in the condensate, as discussed below. Finally, we
note that other microstructured systems such as microdisk lasers can also show
bistability. The whispering gallery modes in microdisk lasers can have
clockwise or counterclockwise propagating lasing modes Hill  _et al._ (2004).
When the coupling between the two modes is small, a cross-gain saturation
causes one of the modes to dominate. The system can therefore operate in
“flip-flop” mode and the two modes can be excited and switched with weak
optical pulses Liu  _et al._ (2010).

###  IV.4 Thermal noise: spin flipping

Figure 9:  (a) Measured spontaneous spin flip rate at different temperatures
of the sample. Above the temperature $T_{c}\simeq$15\text{\,}\mathrm{K}$$, the
flip rate increases exponentially. The solid line shows the theoretical fit.
(b) Calculated dependence of flip rate with normalized noise amplitude. The
flip rate increases exponentially as the noise amplitude increases (light
region), and eventually becomes linear (dark region). Comparison with (a)
shows that the measured flip rates remain below this linear regime.

Above the spin bifurcation occupation threshold ($S_{c}$), any small
perturbation induces a ‘restoring force’, which drives the condensate toward
the attractor once the perturbation is stopped. This restoring force keeps the
condensate around the attractor for small spin fluctuations. However, for
sufficiently large perturbations, the condensate can flip to the other
attractor with the opposite circular polarization. In the experiment, for
sample temperatures near $5\text{\,}\mathrm{K}$ the observed spin-polarized
states remain stable for many seconds (longer than the stability of our
experiment can be maintained). However, increasing the temperature above
$T_{c}\simeq$15\text{\,}\mathrm{K}$\pm 1$ induces spin flips in the condensate
during a measurement time window of $t_{m}=$1.5\text{\,}\mathrm{ms}$$. The
measured rate of the spin flipping beyond $T_{c}$ is a nonlinear function of
the sample temperature as shown in FIG. 9(a). It is important to note that we
do not have a real threshold for the occurrence of spin flipping here. In
fact, the spin-flip rate increases exponentially with temperature right from
$T=0$, until it becomes significant in the finite-time measurement window. We
can account for this phenomenon by adding a thermally induced noise to our
theory. This thermal noise $f(t)$, which is similar to the Johnson noise, is
included in Eq. (2) to give Langevin-type equations:

$\displaystyle\begin{split}\dot{\psi}_{+1}=&-\textstyle{\frac{1}{2}}g(S)\psi_{+1}-\textstyle{\frac{1}{2}}(\gamma-\mathrm{i}\varepsilon)\psi_{-1}\\\
&-\textstyle{\frac{\mathrm{i}}{2}}(\alpha_{1}|\psi_{+1}|^{2}+\alpha_{2}|\psi_{-1}|^{2})\psi_{+1}+f_{+1}(t),\end{split}$
(8a)
$\displaystyle\begin{split}\dot{\psi}_{-1}=&-\textstyle{\frac{1}{2}}g(S)\psi_{-1}-\frac{1}{2}(\gamma-\mathrm{i}\varepsilon)\psi_{+1}\\\
&-\textstyle{\frac{\mathrm{i}}{2}}(\alpha_{1}|\psi_{-1}|^{2}+\alpha_{2}|\psi_{+1}|^{2})\psi_{-1}+f_{-1}(t),\end{split}$
(8b)

where $f_{\sigma}(t)$ with $\sigma=\pm 1$ is a realization of Gaussian random
processes with zero mean $\langle f_{\sigma}(t)\rangle=0$ and $\delta$-like
two-point correlation function

$\langle f_{\sigma}(t)f_{\sigma^{\prime}}(t^{\prime})\rangle=0,\quad\langle
f_{\sigma}(t)f^{*}_{\sigma^{\prime}}(t^{\prime})\rangle=2D\delta_{\sigma,\sigma^{\prime}}\delta(t-t^{\prime}).$
(9)

At finite temperature $T$ the intensity of the noise can be written
approximately as $D=\textstyle{\frac{1}{2}}(W+aT)$, with the $W$-contribution
being a shot noise from the reservoir Aleiner  _et al._ (2012) and the thermal
part $aT$ defined by spin-flip polariton-phonon scattering.

The flip rate vs noise amplitude using stochastic simulations [FIG. 9(b)]
reveals an Arrhenius-like increase at a critical threshold followed by a
crossover to a linear regime. This model gives an excellent account of the
dynamics [FIG 9(a)]. For temperatures beyond $18\text{\,}\mathrm{K}$ we reach
the time resolution of our detection. As a result, we cannot completely span
the crossover to the linear spin-flip regime in our experiment. A fit of the
simulation results to the experimental data in FIG. 9(a) gives
$a=$0.17\text{\,}{\mathrm{ps}}^{-1}\text{\,}{\mathrm{K}}^{-1}$$, which sets
the dependence of spin flip rate on $D(T)$. We can then explore how this noise
perturbs the spin system given by equations (8).

Inclusion of noise produces a spin flip rate that can overcome the effective
spin potential barrier (see Appendix F). For the case in which the circular
degree of polarization is high, the spin-flip process can be considered as a
one-dimensional Kramers transition. The spin-flip rate $R_{K}$ can then be
estimated as

$R_{K}=\frac{\sqrt{\varepsilon
g(S)}}{2\pi}\exp\left\\{-\frac{g(S)n}{4D}\ln\left(\frac{\varepsilon}{g(S)}\right)\right\\}.$
(10)

We note that for $\varepsilon=$30\text{\,}\mathrm{\SIUnitSymbolMicro eV}$$ and
the condensate occupation $n=800$, the zero-temperature shot-noise spin-flip
rate (set by $W$) is negligible for our observation timescales
($10\text{\,}\mathrm{s}$). The critical temperature $T_{c}(t_{m})$ given by
$R_{K}(T_{c})=1/t_{m}$ depends dramatically on the measurement window time
$t_{m}$. While the phonon-polariton interaction $a$ in our system gives
$T_{c}($1.5\text{\,}\mathrm{ms}$)=$15\text{\,}\mathrm{K}$$, at lower
temperatures the condensate spin lifetime rapidly exceeds the stability time
of our experimental apparatus (many seconds). Modifying the phonon-polariton
interaction $a$ thus has an enormous effect on the spin stability. Finally, we
note that in a similar fashion to thermal noise, an overlapping reservoir can
also induce spin noise in the condensate. The spin noise causes condensate
spin flips, which result in the reduction of the time-averaged circular
polarization. Theoretically, this can be studied by introducing a noise term
similar to Eq. 9, but instead of depending on temperature the noise intensity
depends on the overlapping reservoir density Read  _et al._ (2009).

##  V Concluding remarks

In summary, we showed how spin can emerge spontaneously in nonresonantly-
pumped polariton condensates. We found that for trapped condensates, in the
case where the pump is linearly polarized, parity symmetry is spontaneously
broken by spin fluctuations at the onset of condensation. Fluctuations are
amplified by nonlinearities in the condensate formation due to the energy and
lifetime splitting of the linear polarization components, producing a spin-up
or spin-down condensate. The symmetry can be explicitly broken by applying a
slightly elliptically polarized pump, which increases the likelihood of
forming condensates with the same spin as the pump. In the case where the pump
is linearly polarized, we switched the condensate state using a 60-fold weaker
resonant gate pulse with an opposite circular polarization. This situation
changes when the pump is elliptically polarized, where instead of switching,
the condensate pseudospin precesses around stationary states in limit cycles.
Finally, we showed how thermal excitations can induce spin flips with a rate
that increases exponentially with sample temperature.

We demonstrated here one way to explicitly break symmetry utilizing elliptical
polarization pumping geometries. One could also break the symmetry by
introducing a magnetic field to split the energy of the spin-down and spin-up
condensates. Alternatively, the pumping symmetry could be broken with spin
current injection. These experiments thus exhibit rich physics with potential
applications in sensing.

The observation of spontaneous discretized spin-polarized states also has
interesting consequences in the physics of condensate lattices. The
possibility of shared reservoirs, and a Josephson type tunneling Lagoudakis
_et al._ (2010) between adjacent sites, could provide new phenomena previously
unobserved in driven bosonic systems. Magnetic phase transitions, geometric
frustration and spontaneous pattern formation of spin in lattices, domain
formation, topological spin insulators, and topological defects are a few
examples of magnetic systems that could be studied, all within a highly-
controlled bosonic many-body system.

While the condensed polariton lattice resembles nanomagnet arrays Jamet  _et
al._ (2001); Hai  _et al._ (2010), it has the inherent advantages of tunable
nonlinearity, longer spin relaxation time, ps response, rapid optical
addressing and manipulation, and adaptable scalability. The spin-polarized
state at zero magnetic field is retained for many seconds, 10 orders of
magnitude longer than the condensation time, making it a suitable candidate
for optical spin-based memories.

Spin switching with only a fraction of the condensate density, which is a
direct result of the nonlinearity in our system, can be used for low power
optical and electrical sensing and spin switches. Finally, the possibility of
coherent driving allows the realization of superpositions of spin-up and spin-
down states, which are the key requirements for quantum information
processing.

## Acknowledgments

We acknowledge discussions with B. L. Altshuler and N. G. Berloff. This work
was supported by EPSRC Grant No. EP/G060649/1, EP/L027151/1, EU Grant No.
CLERMONT4 235114, EU Grant No. INDEX 289968, ERC Grant No. LINASS 320503, the
Leverhulme Trust Grant No. VP1-2013-011, Spanish MEC (Grant No.
MAT2008-01555), the Greek GSRT ARISTEIA Apollo program and Fundación La Caixa,
and Mexican CONACYT Grant No. 251808. F. P. acknowledges financial support
from EPSRC at the University of Cambridge and a Schrödinger grant at the
University of Oxford. The data corresponding to all the figures in this paper
can be found at https://www.repository.cam.ac.uk/handle/1810/248036

.

##  Appendix A Experimental methods

The cavity’s top (bottom) distributed Bragg reflector (DBR) is made of 32 (35)
pairs of Al0.15Ga0.85As/AlAs layers of
$57.2\text{\,}\mathrm{nm}$/$65.4\text{\,}\mathrm{nm}$. Four sets of three
$10\text{\,}\mathrm{n}\mathrm{m}$ GaAs quantum wells (QW) separated by
$10\text{\,}\mathrm{n}\mathrm{m}$ thick layers of Al0.3Ga0.7As are placed at
the maxima of the cavity light field. The 5$\lambda$/2
($583\text{\,}\mathrm{nm}$) cavity is made of Al0.3Ga0.7As. The microcavity
sample is chemically etched from the substrate side to form
$300\text{\,}\mathrm{\SIUnitSymbolMicro m}$ diameter membranes allowing
optical access from the back of the sample for resonant excitation [FIG.
1(a)]. The sample shows condensation under nonresonant excitation Tsotsis  _et
al._ (2012). The excitation laser is a single-mode CW Ti:Sapphire, which is
amplitude modulated using an AOM with a rise time of
$100\text{\,}\mathrm{ns}$. To pattern the pump intensity a spatial light
modulator was used Cristofolini  _et al._ (2013).

##  Appendix B Strain-induced linear polarization splitting

Figure 10: (a) Energy-resolved emission of the ground state polariton for
horizontal (H) and vertical (V) polarizations, split in energy by $\sim
30\text{\,}\mathrm{\SIUnitSymbolMicro eV}$ (b) Transfer matrix calculations
(red) and experimental reflectivity of sample. The lower polariton mode
(marked by an arrow), is offset by $10\text{\,}\mathrm{meV}$ below the center
of the stopband. (c) Transfer matrix calculations showing the linewidth
difference versus the energy splitting. The linewidth difference is $10\%$ of
the energy splitting.

The measured energy of the ground state $X$\- and $Y$-polarized
photoluminescence far below threshold is shown in FIG. 10(a). The energy
splitting of the linearly polarized modes varies across the sample surface,
reaching its maximum at the edges and minimum at the center of the membranes.
This observation suggests that the splitting is correlated to the level of
strain across the microcavity structure, since the latter is expected to
possess a similar spatial dependence: namely to be strongest at the boundary
between etched and non-etched regions and relax towards the central parts of
the membranes. Note that we observe all the same phenomena even in unetched
samples, as strain is universally present from the III-V heterostructure
growth.

Strain-induced splitting of the initially degenerate polariton states at $k=0$
into orthogonal linearly polarized modes has been demonstrated in previous
works Balili  _et al._ (2010); Kłopotowski  _et al._ (2006). Both the
excitonic and the photonic parts of the polariton can be affected by strain to
produce such an anisotropy. In the former case, the initial splitting of the
bright exciton states due to exchange interactions are enhanced by strain-
induced mixing of the heavy and light hole valence bands, thereby reducing the
symmetry of the QW Aleiner and Ivchenko (1992); Ivchenko  _et al._ (1996);
Balili  _et al._ (2010). In the latter case, strain induces a small
birefringence in the cavity and/or DBRs, hence lifting the degeneracy between
the $[110]$ and $[1\bar{1}0]$ axes Kłopotowski  _et al._ (2006).

Because of the finite curvature of the cavity stopband, splitting the lower
polariton into two orthogonal modes necessarily induces a difference of their
linewidths as well. For the microcavity structures studied in this work the
polariton modes are located on the low-energy side of the stopband [FIG.
10(b)]. In this case the mode possessing higher energy will exhibit a narrower
linewidth, since it is located closer to the stopband center, where the DBR
reflectivity is at its maximum. Correspondingly, the lower energy mode will
exhibit a larger linewidth, since it is located closer to the edge of the
stopband, where the DBR reflectivity starts to drop. For an energy splitting
of the order of $30\text{\,}\mathrm{\SIUnitSymbolMicro eV}$, transfer matrix
calculations predict a linewidth difference of approximately
$3\text{\,}\mathrm{\SIUnitSymbolMicro eV}$ [FIG. 10(c)].

Figure 11: (a) Absolute degree of polarization vs $\gamma$ for a fixed energy
splitting of $\varepsilon=$0.045\text{\,}{\mathrm{ps}}^{-1}$$. (b) Absolute
degree of polarization vs energy splitting $\varepsilon$, for the case when
$\gamma=0.1\varepsilon$.

This small difference in linewidth is not resolvable with our instruments.
However, it should be noted that any nonzero linewidth difference, as long as
it is negative sign relative to the energy splitting, eventually leads to the
bifurcation of the circular polarization at a critical threshold [FIG. 11(a)].
Moreover, if the ratio of the linewidth difference to energy splitting is kept
constant (as we have $\gamma=0.1\varepsilon$ according to our transfer matrix
simulations), the condensate becomes linearly polarized at high energy
splittings [FIG. 11(b)]. This is indeed what we observe at the edges of the
membrane, where the energy splitting is as high as
$100\text{\,}\mathrm{\SIUnitSymbolMicro eV}$.

##  Appendix C Variable cross spin saturation

For the case when the same-spin and cross-spin gain-saturation nonlinearities
are different we have:

$\displaystyle\begin{split}\dot{\psi}_{+1}=&\textstyle{\frac{1}{2}}\left[w-\textstyle{\frac{1}{2}}(\mu|\psi_{+1}|^{2}+(2\eta-\mu)|\psi_{-1}|^{2})\right]\psi_{+1}\\\
&-\textstyle{\frac{1}{2}}(\gamma-\mathrm{i}\varepsilon)\psi_{-1}-\textstyle{\frac{\mathrm{i}}{2}}\left[\alpha_{1}|\psi_{+1}|^{2}+\alpha_{2}|\psi_{-1}|^{2}\right]\psi_{+1},\end{split}$
(11a)
$\displaystyle\begin{split}\dot{\psi}_{-1}=&\textstyle{\frac{1}{2}}\left[w-\textstyle{\frac{1}{2}}(\mu|\psi_{-1}|^{2}+(2\eta-\mu)|\psi_{+1}|^{2})\right]\psi_{-1}\\\
&-\textstyle{\frac{1}{2}}(\gamma-\mathrm{i}\varepsilon)\psi_{+1}-\textstyle{\frac{\mathrm{i}}{2}}\left[\alpha_{1}|\psi_{-1}|^{2}+\alpha_{2}|\psi_{+1}|^{2}\right]\psi_{-1},\end{split}$
(11b)

where $w=W-\Gamma$ and we have now two saturation parameters, $\eta$ and
$\mu$. The saturation is controlled by individual occupations of circular
components when $\mu=2\eta$, and the saturation is controlled by the total
occupation when $\mu=\eta$. In general, $\eta\leqslant\mu\leqslant 2\eta$.
Equations above in the matrix form read

$\displaystyle\begin{split}\frac{d\Psi}{dt}=&\frac{1}{2}\left[w-\eta
S-(\mu-\eta)S_{z}\sigma_{z}\right]\Psi-\frac{1}{2}(\gamma-\mathrm{i}\varepsilon)\sigma_{x}\Psi\\\
&-\frac{\mathrm{i}}{2}\left[(\alpha_{1}+\alpha_{2})S+(\alpha_{1}-\alpha_{2})S_{z}\sigma_{z}\right]\Psi,\end{split}$
(12)

and the equations for pseudospin components are

$\displaystyle\dot{S}_{x}=(w-\eta S)S_{x}-\gamma S-\alpha S_{z}S_{y},$ (13a)
$\displaystyle\dot{S}_{y}=(w-\eta S)S_{y}+\varepsilon S_{z}+\alpha
S_{z}S_{x},$ (13b) $\displaystyle\dot{S}_{z}=(w-\mu S)S_{z}-\varepsilon
S_{y}.$ (13c)

###  C.1 Linearly polarized condensates (parity conserved)

These particular solutions are given by $S_{y}=S_{z}=0$, $S_{x}=\pm S$, and
$S=(w\mp\gamma)/\eta$. $X$-polarized condensates (upper sign) can exist for
$W>\Gamma+\gamma$. $Y$-polarized states (lower sign) appear for
$W>\Gamma-\gamma$. Considering their stability with respect to small
fluctuations: $S_{y}=y$, $S_{z}=z$, $S_{x}=\pm S_{0}+x$, and $S=S_{0}\mp x$,
with $S_{0}=(w\mp\gamma)/\eta$, we have linearized equations

$\displaystyle\dot{x}=-\eta S_{0}x,$ (14a) $\displaystyle\dot{y}=\pm\gamma
y+(\varepsilon\pm\alpha S_{0})z,$ (14b)
$\displaystyle\dot{z}=(\pm\gamma-(\mu-\eta)S_{0})z-\varepsilon y.$ (14c)

Taking $x,y,z\propto e^{\lambda t}$ we see that fluctuations in $x$ always
decay, while (14b,c) produce the equation for the Lyapunov exponent $\lambda$,

$(\lambda\mp\gamma)^{2}+(\mu-\eta)S_{0}(\lambda\mp\gamma)+\varepsilon(\varepsilon\pm\alpha
S)=0.$ (15)

The stability of the $Y$-polarized condensate is lost when one root of this
equation crosses zero. This corresponds to the critical occupation $S_{c}$ and
critical pumping $W_{2}$,

$S_{c}=\frac{\gamma^{2}+\varepsilon^{2}}{[\alpha\varepsilon-(\mu-\eta)\gamma]},\qquad
W_{2}=\Gamma-\gamma+\eta S_{c}.$ (16)

The stability of $X$ polarized state depends on the interrelation between
$\gamma$ and $(\mu-\eta)S_{0}$. For $\mu=\eta$ this state is always unstable.
However, if $\mu>\eta$ the stable $X$ polarized state can be formed for large
enough condensate occupations (far above the threshold).

###  C.2 Elliptically polarized condensates (parity broken)

These solutions with $S_{z}\neq 0$ are given by

$\displaystyle S_{x}=-\frac{1}{\alpha\varepsilon}\left[\varepsilon^{2}+(w-\eta
S)(w-\mu S)\right],\qquad$ (17a) $\displaystyle S_{y}=\frac{(w-\eta
S)}{\varepsilon}S_{z},\qquad$ (17b) $\displaystyle
S_{z}=\pm\varepsilon\sqrt{\frac{S^{2}-S_{x}^{2}}{\varepsilon^{2}+(w-\mu
S)^{2}}}.$ (17c)

Substitution into (13a) gives

$\displaystyle\begin{split}&(\mu-\eta)\varepsilon\left[\varepsilon^{2}+(w-\eta
S)(w-\mu S)\right]\\\ &+\alpha^{2}\varepsilon(w-\mu
S)S+\alpha\gamma\left[\varepsilon^{2}+(w-\mu S)^{2}\right]=0.\end{split}$ (18)

The positive root of this equation for $S$ should be taken. Also, it is
necessary to satisfy the condition $|S_{x}|\leqslant S$, which gives
$w\geqslant(W_{2}-\gamma)=w_{c}$. This means that the weak lasing solutions
appear continuously from the $Y$-polarized solution at the critical pumping
$W_{2}$.

The stability of weak lasing states can also be lost. Numerical analysis shows
the following typical scenario of evolution of the condensate polarization
state with increasing $w$ for $\mu>\eta$. First, the $Y$ linearly polarized
state is formed. Then it transforms into a weak lasing, elliptically polarized
state. The stability of the latter is also lost with increasing $w$, resulting
in some irregular, quasi-chaotic dynamics and/or in the oscillatory motion of
the pseudospin vector. Finally, for large $w$ the stable $X$ polarized state
is formed.

##  Appendix D Estimate for particle number

The condensate particle number is experimentally measured by:

$n=\frac{\Phi\tau}{|C|^{2}},$ (19)

where $\Phi$ is the photon flux, $\tau=$10\text{\,}\mathrm{ps}$$ is the
polariton lifetime, and $|C|^{2}=0.4$ is the photon Hopfield coefficient. The
photon flux is measured by:

$\Phi=\frac{\alpha R}{\eta},$ (20)

where $\alpha=3.5\;\mathrm{e^{-}/count}$ is the photoelectron sensitivity of
the CCD, $\eta=0.0021$ is the total detection efficiency including the camera
quantum efficiency and the total optical transmission efficiencies, and
$R=$1.9\text{\times}{10}^{10}\text{\,}{\mathrm{s}}^{-1}$$ is the spatially-
integrated count rate of the CCD. Inserting these values in Eq. (19) gives the
particle number $n\simeq 800$, at $P=1.7P_{\text{th}}$.

##  Appendix E Resonance width

Figure 12:  (a) The reduction of the spin-down intensity versus the detuning
of the gate laser frequency is shown. A sharp resonance is visible when the
laser is in resonance with the condensate. (b) The numerical calculations show
a similar resonance effect, with the center of the peak slightly redshifted
due to the linear polarization energy splitting $\varepsilon$.

We study the reduction of the opposite component of the condensate circular
polarization to the gate laser [marked by $\Delta s_{-}$ in FIG. 7(d)] as the
‘detuning’ of the gate varies, while the gate power remains constant. In order
to change the detuning of the gate laser frequency with respect to the
condensate, instead of changing the frequency of the laser, we tune the
condensate frequency by changing the pump intensity. The trapped condensate
‘blueshifts’ as the pump intensity is increased, mainly due to the repulsive
interaction between the polaritons. This contrasts with the case of untrapped
condensates, where interactions of polaritons with the excitons in the
reservoir is the source of blueshifts Wouters  _et al._ (2008); Wertz  _et
al._ (2010); Askitopoulos  _et al._ (2013). In the 4 spot trapped geometry, we
have a blueshift of
$6\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{eV}\mathrm{/}\mathrm{m}\mathrm{W}$
and as a result we can tune the condensate energy accurately with respect to
that of the gate. FIG. 12(a) shows $\Delta s_{-}$ with respect to the detuning
of the gate laser. We observe a sharp resonance, with a full width at half
maximum (FWHM) of $11\pm 6\text{\,}\mathrm{\SIUnitSymbolMicro eV}$. FIG. 12(b)
shows the theoretical curve versus the gate detuning ($\hbar\omega_{g}$),
which resembles a Lorentzian profile with a linewidth of
$17\text{\,}\mathrm{\SIUnitSymbolMicro eV}$. Theoretically, the linewidth of
the resonance is defined by the noise in Eqs. (8). The FWHM of the Lorentzian
resonance peak is $D/(1-s_{z}^{2})n$. We note that the resonance frequency
$\omega_{c}$ is slightly redshifted, due to the effect of the linear
polarization splitting and the decay rate splitting,

$\omega_{c}=\frac{1}{2}\left[(\alpha_{1}+\alpha_{2})S-\frac{(\gamma
S_{y}S_{z}+\varepsilon SS_{x})}{(S_{x}^{2}+S_{y}^{2})}\right].$ (21)

##  Appendix F The Kramers flip rate

From Eq. (8) we can obtain the spin vector equations:

$\displaystyle\dot{S}_{x}=-g(S)S_{x}-\gamma S-\alpha S_{z}S_{y}+F_{x}(t),$
(22a) $\displaystyle\dot{S}_{y}=-g(S)S_{y}+\varepsilon S_{z}+\alpha
S_{z}S_{x}+F_{y}(t),$ (22b) $\displaystyle\dot{S}_{z}=-g(S)S_{z}-\varepsilon
S_{y}+F_{z}(t),$ (22c)

where the correlators of real-function noise $F_{i}(t)$ are

$\left<F_{i}(t)F_{j}(t^{\prime})\right>=2DS\delta_{ij}\delta(t-t^{\prime}),\qquad
i,j=x,y,z.$ (23)

Here we consider limiting case when two parity breaking states are formed near
the north and south poles of the Poincaré sphere, i.e., $|S_{z}|\gg|S_{x,y}|$.
The spin components of fixed states in this limit are
$S_{x0}\simeq-\varepsilon/\alpha$, $S_{y0}\simeq-\gamma/\alpha$,
$S_{z}\simeq\pm S_{0}$, where $S_{0}$ is the root of
$S=\gamma\varepsilon/\alpha g(S)$. In what follows we also denote
$g_{0}=g(S_{0})$ and by assumption $g_{0}\ll\varepsilon,\gamma$. Being excited
away from the fixed state, the spin exhibits fast self-induced Larmor
precession and slow relaxation. The precession frequency is $\omega=\alpha
S_{z}$, so that $\omega=\gamma\varepsilon/g_{0}$ near the stationary states.

The spin should be driven by noise into the equatorial plane ($S_{z}=0$) in
order to flip. This Kramers problem can be simplified if we perform averaging
over fast precession of the spin vector. This treatment is valid as long as
$|S_{z}|\gg|S_{x,y}|$. Consider the motion in the north hemisphere. We assume
that the number of polaritons does not change during the flip, i.e., $S$ is
fixed to $S_{0}$, and we consider the case of large occupations, $\ln S_{0}\gg
1$. Omitting the $g$-terms from Eqs. (22a,b) we find that for a given value of
$S_{z}>0$ the averages over one cycle of the other two components are
$\left<S_{x}\right>=-\varepsilon/\alpha$ and $\left<S_{y}\right>=-\gamma
S_{0}/\alpha S_{z}$. Then, from (22c) we obtain the equation for slow
evolution of $S_{z}$

$\frac{dS_{z}}{dt}=-g_{0}S_{z}+g_{0}\frac{S_{0}^{2}}{S_{z}}+F_{z}(t)=-\frac{dU(S_{z})}{dS_{z}}+F_{z}(t),$
(24a) $U(S_{z})=g_{0}\left(\textstyle{\frac{1}{2}}S_{z}^{2}-S_{0}^{2}\ln
S_{z}\right).$ (24b)

This expression for the effective potential $U(S_{z})$ is not valid for small
$S_{z}$, where it diverges logarithmically. The top of the barrier should be
cut off when $S_{z}$ becomes comparable to $|\left<S_{y}\right>|$, that is for
$S_{z}\simeq S_{0}\sqrt{g_{0}/\varepsilon}$. The value of potential on the top
of the barrier is then $U_{b}\simeq-
g_{0}S_{0}^{2}\ln(S_{0}\sqrt{g_{0}/\varepsilon})$. The bottom of the well is
positioned at $U_{0}=g_{0}S_{0}^{2}(\textstyle{\frac{1}{2}}-\ln S_{0})$. As a
result, the spin should overcome the barrier

$\displaystyle\Delta U=U_{b}-U_{0}$
$\displaystyle=\frac{1}{2}g_{0}S_{0}^{2}\left[\ln\left(\frac{\varepsilon}{g_{0}}\right)-1\right]$
$\displaystyle\simeq\frac{1}{2}g_{0}S_{0}^{2}\ln\left(\frac{\varepsilon}{g_{0}}\right).$
(25)

Using the known result for the Kramers first passage time in the one-
dimensional problem Zwanzig (2001), we obtain the spin-flip rate

$\displaystyle R_{K}=\frac{R_{0}}{2\pi}\exp\left\\{-\frac{\Delta
U}{DS_{0}}\right\\}$
$\displaystyle=\frac{R_{0}}{2\pi}\exp\left\\{-\frac{g_{0}S_{0}}{2D}\ln\left(\frac{\varepsilon}{g_{0}}\right)\right\\},$
(26) $\displaystyle S_{0}$ $\displaystyle=\frac{\gamma\varepsilon}{\alpha
g_{0}}.$

This expression assumes $g_{0}S_{0}/D\gg 1$ and $\varepsilon\gg g_{0}$. The
pre-exponent $R_{0}$ cannot be written exactly by this method, since we do not
know the shape of the effective potential near the top of the barrier. It can
be estimated as $R_{0}\simeq\sqrt{\varepsilon g_{0}}$.

##  Appendix G Simulations and numerical parameters

The parameters used for all the simulations are as follows:
0D simulations: $\eta=$0.01\text{\,}\mathrm{p}\mathrm{s}^{-1}$$;
$\Gamma=$0.1\text{\,}\mathrm{p}\mathrm{s}^{-1}$$;
$\hbar\varepsilon=$30\text{\,}\mathrm{\SIUnitSymbolMicro eV}$$;
$\gamma=0.1\varepsilon$;
$\hbar\alpha_{1}=$10\text{\,}\mathrm{\SIUnitSymbolMicro eV}$$;
$\alpha_{2}=-0.5\alpha_{1}$
2D simulations:
$\hbar\alpha_{1}=$3\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{eV}\mathrm{\SIUnitSymbolMicro}\mathrm{m}^{2}$$;
$\alpha_{2}=-0.5\alpha_{1}$; $\hbar
g_{r}=$46\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{eV}\mathrm{\SIUnitSymbolMicro}\mathrm{m}^{2}$$;
$g_{P}=g_{R}/4$; $\Lambda=0.1$; $m^{*}=5.1\times 10^{-5}m_{e}$;
$\gamma_{R}=$10\text{\,}\mathrm{p}\mathrm{s}^{-1}\mathrm{\SIUnitSymbolMicro}\mathrm{m}^{2}$$;
$\hbar\varepsilon=$7\text{\,}\mathrm{\SIUnitSymbolMicro eV}$$;
$\gamma=0.2\varepsilon$.

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