Non-resonant optical control of a spinor polariton condensate

A. Askitopoulos,1, ∗ K. Kalinin,2 T.C.H. Liew,3 P. Cilibrizzi,1 Z.

Hatzopoulos,4, 5 P.G. Savvidis,4, 6 N.G. Berloff,2, 7 and P.G. Lagoudakis1, †

1Department of Physics & Astronomy, University of Southampton, Southampton, SO17 1BJ, United Kingdom2Skolkovo Institute of Science and Technology Novaya St., 100, Skolkovo 143025 , Russian Federation

3 School of Physical and Mathematical Sciences,Nanyang Technological University, 637371, Singapore

4Microelectronics Research Group, IESL-FORTH, P.O. Box 1527, 71110 Heraklion, Crete, Greece5Department of Physics, University of Crete, 71003 Heraklion, Crete, Greece

6Department of Materials Science and Technology, University of Crete, Crete, Greece7Department of Applied Mathematics and Theoretical Physics,

University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK(Dated: April 19, 2016)

We investigate the spin dynamics of polariton condensates spatially separated from and effectivelyconfined by the pumping exciton reservoir. We obtain a strong correlation between the ellipticity ofthe non-resonant optical pump and the degree of circular polarisation (DCP) of the condensate atthe onset of condensation. With increasing excitation density we observe a reversal of the DCP. Thespin dynamics of the trapped condensate are described within the framework of the spinor complexGinzburg-Landau equations in the Josephson regime, where the dynamics of the system are reducedto a current-driven Josephson junction. We show that the observed spin reversal is due to theinterplay between an internal Josephson coupling effect and the detuning of the two projections ofthe spinor condensate via transition from a synchronised to a desynchronised regime. These resultssuggest that spinor polariton condensates can be controlled by tuning the non-resonant excitationdensity offering applications in electrically pumped polariton spin switches.

PACS numbers:

I. INTRODUCTION

Implementation of all optical, integrable spin-polarization switching devices is one of the essential in-gredients for the realisation of novel solid state optoelec-tronic spin-logic architectures [1]. Semiconductor micro-cavities operating in the strong coupling regime, haverecently emerged as ideal systems for achieving this goaldue to their inherent spin multi-stabilities [2, 3] and fastspin dynamics [4, 5] that arise from their strong op-tical non-linearities [6]. Exciton-polaritons, the eigen-states of these systems, are bosonic light-matter quasi-particles that inherit the spin properties of their excitoncomponent while their decay gives rise to photons witha polarization defined by the polariton spin. This ro-bust spin to photon polarization conversion is favourablefor fast non-destructive readout of the spin state of thesystem. Moreover, the advancement of fabrication tech-niques has yielded microcavity structures of high finessefeaturing polariton lifetimes of the order of tens of pi-coseconds [7], enabling the emergence of optically excitedpolariton lasing and condensation [8, 9] even at roomtemperature [10]. In state of the art microcavity struc-tures, polariton condensates propagate ballistically [11]and this has enabled the first demonstrations of polari-ton condensate optical switches [12–14].

Initial realizations of resonant polariton spin-switches [15], followed by experimental demonstrationsof extensive polariton spin transport in planar [16] as

well as 1D structures [17], has emphasized the potentialof theoretical propositions for fully integrated polaritonbased spin circuits[18, 19]. The short lifetime of polari-tons considered to be an impediment for thermodynamicequilibrium and bosonic condensation can now be seenas an advantage in creating ultra-fast spin switchingdevices. In this regard, as non-resonant excitationschemes more closely resemble polariton formationunder electrical injection, they hold greater promise forthe implementation of electrically controlled polaritonspin logic devices. This remarkable possibility has beenfurther highlighted by the recent development of novelelectrically pumped polariton lasers [20, 21].

In polarization resolved experiments under resonantexcitation [15], parametric amplification [4] as well asin the OPO configuration [5, 22], correlations betweenthe polarization of the excitation beam and the result-ing polariton spin have been extensively studied. InRefs. [3, 23], the emergence of a circularly polarized con-densate was observed at near-resonant linearly polarizedoptical excitation. For non-resonant excitation, priorstudies have demonstrated that upon condensation, thepolarization of the condensate is pinned to the crystal-lographic axis of the microcavity structure independentof the exciting polarization [24]. This effect has beenoriginally attributed to a linear polarization splitting of≈ 0.1meV (at k‖ = 0), disputing that the emission polar-ization is inherited from the excitation [24]. The sponta-neous emergence of a strong linear polarization upon con-

2

densation threshold has in-fact been treated as a suitableorder parameter for determining the emergence of coher-ence in the system [25, 26]. However, in more recentexperiments, a higher total degree of polarization was re-ported under circularly polarised non-resonant pulsed ex-citation rather than for linearly polarised excitation [27].Moreover in polariton spin textures generated under cir-cularly polarised excitation a high DCP was reported inthe vicinity of the pump spot [16, 17]. These experi-ments confirmed that there are correlations between thepolarisation of the optical excitation with the polarisa-tion of the condensate, which in GaAs/AlGaAs MCs arenot screened from optical disorder.

In this letter, we demonstrate that the spin imbalancein a polariton condensate can be vigorously modulatedby the polarization properties of the non-resonant ex-citation beam when the polariton condensate and exci-ton reservoir are spatially separated. In the non-linearregime we observe a strong linear to circular polarisa-tion conversion controlling the condensate polarizationby tuning the angle of the linear polarization of the exci-tation beam. Moreover, the condensate density is shownto strongly effect the resulting spin imbalance leading toa spin reversal at high densities. We explain this reversalwithin the framework of a coherent internal Josephsoncoupling mechanism between the overlapping spin statesof the condensate and present results of analytical ap-proximations and numerical simulations that closely re-produce the experiment. These experiments confirm thatthe polarization of non-resonant excitation indeed sur-vives the relaxation mechanisms from electron and holesto polaritons and its effects are exacerbated when thestrong interactions from the reservoir are spatially de-coupled. Furthermore, internal Josephson coupling ef-fects between spin up and spin down coherent states aredemonstrated to be an important factor governing thepolarization build up in the condensate.

II. EXPERIMENTAL RESULTS

To study the polarization properties of a polariton con-densate we utilize the optical trap configuration fromour previous work [28]. Using a non-resonant continu-ous wave (CW) linearly polarized pump spatially shapedin the form of an annular ring and focused on the samplesurface through a 0.4 numerical aperture (NA) objective,we create the Ψ00 coherent state in the optically inducedtrap, as shown schematically in Fig.1a [29]. The S+ andS− components of the emission below and above thresh-old are resolved with the use of a λ/4 wave-plate and alinear polariser and are then used to reconstruct a realspace image of the Sz stokes component [30]. We useda high Q (16000) 5λ/2 microcavity with a cavity life-time of ∼ 7ps composed of GaAs/AlGaAs DBRs, whilethe cavity is embedded with four triplets of 10nm GaAs

FIG. 1: (a) Schematic representation of reservoir and con-densate in the microcavity. Real space map of the degree ofcircular polarisation (b) below (P = 0.95Pth), and (c) above(P = 1.08Pth) condensation threshold.(d) Normalised inten-sity of the polariton emission above threshold (red line, rightaxis), below threshold (blue line, right axis) and DCP (blackline, left axis) across the white dotted line profile of (c).

quantum wells with Al0.3Ga0.7As barriers.Below threshold we do not detect any significant cir-

cular polarization from polaritons inside the trap, shownon Fig.1b, however, just above the threshold we detect astrong circularly polarised emission of about 0.6 DCP asshown in Fig.1c. This is in stark contrast with the lin-ear polarization build-up previously reported for polari-ton condensates [24]. It is worth noting that the circularcomponent originates from the emission of the conden-sate while the DCP of the surrounding excitation region,where polaritons interact with the incoherent reservoir isnegligible. Interestingly, this strong Sz stoke componentpersists far from the peak of the condensate and exhibitsa DCP> 0.5 even where the condensate intensity is 10%of its peak intensity as shown in Fig.1d.

Although the emergence of a strong spin imbalancefrom a linearly polarized optical pump appears counterintuitive, it can nevertheless be interpreted if one takes

3

into account that the polarization of a monochromatictightly focused optical field does not have a uniform po-larization. For high numerical aperture lenses, scalardiffraction geometry is insufficient to accurately describethe electromagnetic field in the focal plane and a vec-torial analysis of light is required in order to preciselydescribe the electromagnetic field in the focal plane [31].Indeed it is a well studied effect that the electric fieldin the focal plane of a high numerical aperture aplanaticsystem can induce a small degree of ellipticity even fora 100% linearly polarized source [32, 33]. In our setup,the microscope objective (NA= 0.4), yields a measuredDCP at the center of a focused linearly polarized beamof 0.1 [34]. Moreover, this ellipticity is calculated to begreater towards the periphery of the focal plane, whereour ring-like excitation pattern is found, than in the cen-tral region [31].

The ellipticity of the pump does not play an importantrole below threshold where the emission is stronger in theexcitation region. Indeed, it has been predicted that spinnoise fluctuations are extremely sensitive to the statis-tics and occupation number in polariton systems [35].Therefore, in the incoherent state a small spin imbalance(pump with small ellipticity) in the injection of carriersis not expected to have a noticeable effect in the polari-sation of the emission. This has been demonstrated evenfor fully circular optical excitation schemes [27]. How-ever, upon crossing the condensation threshold, a dras-tically different picture emerges. Spin fluctuations aregreatly suppressed [35] and density dependent bosonicamplification favours the polariton spin state with higheroccupation. In the absence of any spin relaxation mech-anism between S↑ and S↓ polaritons, the bosonic ampli-fication of the dominant spin population gives rise to astrongly circularly polarised polariton condensate. Fur-thermore, the spatial separation of the condensate fromthe exciton reservoir amplifies this behaviour as the spindecoherence channels associated with polaritons interact-ing with particles within the reservoir are considerablysuppressed [28]. In excitation schemes where the reser-voir is spatially overlapped with the condensate (e.g. tophat or Gaussian spot excitation) the scattering processeswithin the reservoir suppress externally imposed spin im-balances. However even in such cases, careful analysisof the spatiotemporal polarization dynamics have shownspin phenomena that are driven by the spin imbalance inthe exciton reservoir, such as polariton spin whirls andintricate spin textures [34, 36].

In order to further characterise the origin of the cir-cular polarization in the condensate it is meaningful toestablish whether it is isotropic and independent of theangle of the linear polarization of the optical pump. Toachieve this, we initialize the condensate above thresh-old (P = 1.1Pth) as before and using a half wave-platewe rotate the linear polarisation of the excitation. Fig-ure 2(a) displays the spatial profile of the DCP under

horizontal (0) linear excitation showing a high DCP inthe emission. Rotating the angle initially has a marginaleffect on the circular stokes component of the conden-sate. However, for polarization rotation above 90, theDCP is reduced eventually leading to a complete reversalof the polarisation at 130 as shown in Fig. 2(b). After180 rotation of the linear polarisation angle the conden-sate has regained its original polarisation. Figure 2(c)shows the spatial DCP near the tipping point. In thisregime it has been shown that individual realizations ofthe trapped condensate are stochastically spin up or spindown [37]. Furthermore, perturbation of the system bya polarised femtosecond pulse can also induce temporaryas well as permanent spin flips [37, 38], depending on thepolarization of the pulse and of the trapped condensate.The DCP of the system has almost a 2θ relation with theangle of the linear polarisation of the excitation, shownin Fig. 2(d), where the value of each point is the averageof the DCP in the central region of the condensate.

FIG. 2: Variation of the condensate’s DCP with the linear po-larisation angle of the excitation beam. Circular polarisationmaps of the confined polariton condensate just above thresh-old for linear polarisation angle: (a) θ = 0, (b) θ = 150 and(c) θ = 270 . (d) The average DCP is plotted versus externalpolarisation angle featuring a 2θ dependence (red dots markthe data at the above angles. Inset: schematic representationof the induced ellipticity to the excitation beam vs the linearpolarisation angle.

The demonstrated 2θ dependence on the angle ofthe excitation polarization is an indication of opticalanisotropy present in the sample. Light focused on thesurface of the sample propagates through the top DBRmirror, where there is no absorption for the excitationwavelength (λlaser = 752nm), before exciting the exci-tons in the QWs. In our sample the DBR consists of pairsof AlAs/Al0.3Ga0.7As layers for which there have been a

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number of works, for similar multi-layered structures re-porting linear birefringence [39–41]. The birefringence inthe DBR will therefore exacerbate the ellipticity of theexcitation beam and result in the 2θ dependence. There-fore the observed symmetry breaking at threshold is ex-plicit, brought about by the imbalance in the populationof S↑,↓ particles. However, when the ellipticity inducedby the objective is counterbalanced by the birefringenceof the sample, any appearance of circular polarizationbecomes implicit to the system (spontaneous symmetrybreaking) [37].

In the previously discussed results, the excitationpower of the non-resonant, non-local pump and thereforethe condensate density was kept constant (P=1.1xPth).In polariton condensates, increasing the density has beenshown to have a strong impact on its polarization throughthe increase of spin dependent polariton-polariton inter-actions that tend to depolarize the emission [27, 42]. Weinvestigate the condensate density dependence in a po-larization resolved power dependent experiment. Fur-ther increase of the density above threshold initially re-sults in reducing the DCP of the emission. However at2.2 times the threshold power we observe a reversal ofthe DCP and thus of the polariton pseudo-spin Fig. 3(a).It is worth noting here that we do not observe any de-viation from the ground state of the trap within thepower range that we studied. This is demonstrated inthe inset of Fig. 3(a) that shows the real space 1D pro-file of the 2 spin components of the condensate withrespect to power over threshold. Figure 3(b) depictsthe power dependence of a spatial cross-section of theemission for increasing power, while the black line de-notes the photoluminescence from the trap barriers be-low threshold that outlines the trap profile. The points ofFig. 3(a) are taken from the point of maximum intensityof the condensate (I↑ + I↓). Due to the intensity non-linearity, inherent in polariton condensation, the signalto noise ratio (SNR) that we record from the condensate(8.75dB 6 SNRc 6 14dB) is much greater than thatat the excitation region (SNRres ≈ 0.1dB). In Figure3(b) we plot the DCP in the region of confidence whereSNR≥ 1dB. The spin reversal dependence of a polaritoncondensate with excitation density under non-resonantoptical pumping cannot be described within the frame-work of the linear optical spin Hall effect [43].

III. JOSEPHSON COUPLING MODEL

Our theoretical model consists of a system of spinorGinzburg-Landau equations (GLE) [44] written in thebasis of left- and right-circular polarized polariton wave-

FIG. 3: (a) DCP as a function of excitation power vs thresh-old power. (b) Power dependence vs a spatial cross-section ofthe DCP across the trap. Note the change in scale at 2.15.The figure shows the region where SNR≥ 1dB. The black lineoutlines the emission below threshold defining the trap. Theinset in (a) shows the 2 normalised spin components of the1D real space profile of the condensate with respect to powerabove threshold.

functions (spin-up and spin-down), denoted by ψ±:

i~∂ψ±∂t

=− ~2

2m∇2 + U0|ψ±|2 + (U0 − 2U1) |ψ∓|2+

+ ~gRn± +i~2

(RRn± − γC)ψ± + Ωψ∓,

(1)

where in the Hamiltonian part of the equations we takeinto account an interaction with the total polariton den-sity HU0 = U0/2(|ψ+|2 + |ψ−|2)2 as well as an attrac-tive interaction between opposite spin species, HU1 =−2U1|ψ+|2|ψ−|2, and the symmetry-breaking term HΩ =Ω(ψ+ψ

∗− + H.c.), which arises due to asymmetry at the

quantum-well interfaces, mechanical stresses, or due tothe anisotropy-induced splitting of linear polarisations inthe microcavity. Josephson coupling has been extensivelystudied and experimentally observed in polariton con-densates, both in the intrinsic case [23, 45] as well as theextrinsic case [46–50]. In this model there is no explicit

5

dependence on the shape of the pump, nevertheless thesymmetry breaking term HΩ, appears exactly becausethe interactions with the reservoir are filtered out due tothe geometry of the experiment [51]. The rate equationfor the reservoir represents the evolution of the excitondensities, n± is:

∂n±(r, t)

∂t= −

(γR +RR|ψ±|2

)n±(r, t) + P±(r, t), (2)

where we have neglected the hot excitons diffusion andadvection. In equations 1 and 2, γR and γC are thereservoir and condensate decay rates respectively, RR isthe scattering rate to/from reservoir, P± is the pumpingrate and gR is the condensate blueshift from interactionswith reservoirs.

Two-mode model. We non-dimensionalize (1) and (2),neglect the spatial variation of all parameters and obtainthe following ccGLEs and rate equations:

2i∂ψ±∂t

=|ψ±|2 + (1− Uα)|ψ∓|2 + gn±

+i

2(Rn± − 1)

ψ± + Jψ∓, (3)

2∂n±∂t

= −(γ + b|ψ±|2)n± + p±, (4)

where the following dimensionless parameters were used[51]:

Uα = 2U1

U0; g =

2mgR~

; R =2mRR

~; J =

Ω

~γC;

γ =γRγC

; b =~RRU0

; p± =~

2mγ2C

P±.

(5)

These equations can be conveniently re-parametrisedusing:

ψ± =√ρ±e

i(φ±Θ2 ), ρ =

ρ+ + ρ−2

, z =ρ+ − ρ−

2,

(6)where ρ± are the densities of polaritons with spin-up andspin-down, φ is the global phase which doesn’t have aninfluence on our solution due to non-resonant incoherentpump, Θ is the phase difference between the polaritonsof different species, ρ is the averaged density. Separatingreal and imaginary parts gives the coupled equations forΘ, ρ, and z:

Θ = −Uαz −g

2(n+ − n−) +

Jz cos Θ√ρ2 − z2

,

z =R

4

(n+(ρ+ z)− n−(ρ− z)

)− J

√ρ2 − z2 sin Θ− z

2,

ρ =R

4

(n+(ρ+ z) + n−(ρ− z)

)− ρ

2, (7)

where, in the view of the fast reservoir relaxation in com-parison with the condensate relaxation (γ ≈ 10 [52–54])we replaced Eq. (4) with the equilibrium values

n± =p±

γ + b(ρ± z). (8)

We are interested in the resulting degree of the circularpolarization, ξ = z/ρ, assuming a small discrepancy inthe pumping η = p+/p− > 1.

0.8

0.6

0.4

0.2

0.0

-0.2

Deg

ree

of C

ircul

ar P

olar

izat

ion

2.42.22.01.81.61.41.21.0

P/Pth

Synchronized Desynchronized

FIG. 4: The degree of circular polarization, ξ, as a functionof the pumping strength P/Pth obtained by numerical inte-gration of Eqs. (7). The blue dots correspond to the desyn-chronized state where the polarization degree and condensatedensity are oscillating in time (the values of these functionsare averaged over the period of their oscillation), the red dotscorresponds to fixed points. Even for a small discrepancy inthe pumping intensity the degree of the circular polarizationcan vary from elliptical polarization to linear polarization topolarization reversal. The parameters used are γ = 10, g =0.7, b = 15, R = 1.0, Uα = 1.1, J = 0.045, η = 1.1.

Josephson Regime. The behaviour of the system(Eqs. 7) can be understood by considering the so-calledJosephson regime: J Uαρ, z ρ, and where we con-sider a small pumping discrepancy η = 1+ε, where ε 1.Eq. (7) reduces to an equation for a driven damped pen-dulum (or a current-based Josephson junction):

Θ +

(1

2− γ

2Rp−(1 + ε)

)Θ = −Uα

ρstε

2(2 + ε)+

+ Jρst

(Uα −

gb

(1 + ε)R2p−

)sin Θ,

(9)

where the density ρst corresponds to the stationary case(a Bloch surface):

ρst =(ε+ 2)Rp− − 2γ

2b. (10)

The behaviour of the driven damped pendulum hasbeen studied in detail [55]. Depending on the sign of thedriving amplitude A = Jρst[Uα − gb/(1 + ε)R2p−] the”pendulum” acts to drive Θ to either 0 (if A > 0) or to π

6

(if A < 0). The first term on the right-hand side of Eq.(10) represents a constant driving torque. The thresholdcondition for condensation is γ/Rp− < 1, so the secondterm on the left-hand side of Eq. (10) represents dampingwith a rate α = (1 − γ/Rp−(1 + ε))/2 that grows fromthe threshold approaching 1/2. We transform Eq. (10)into a set of the first order equations:

Θ = χ, χ = −αχ− Uαρstε

2(2 + ε)+A sin Θ. (11)

From the analysis of this system it is clear that depend-ing on the values of the coefficients the trajectory of thesystem is attracted to either a fixed point or a limit cycle.A stable fixed point exists if

Uαε

2(2 + ε)≤ J |Uα −

gb

(1 + ε)R2p−|. (12)

We can refer to the corresponding fixed point solutionas a synchronized solution since the relative phase of twocomponents becomes fixed in time. If this condition is notsatisfied, there is no fixed point and the relative phasebetween the components evolves periodically. We willrefer to this regime as a desynchronized solution. The de-synchronization occurs when Eq. (12) is violated whichfor small epsilon occurs when

ε > 4J

∣∣∣∣1− gb

UαR2p−

∣∣∣∣, (13)

and in terms of the pumping takes place around

p− ≈gb

UαR2. (14)

Spin Reversal. The presence of the internal Josephsoncoupling term, J, is essential for obtaining the spin re-versal. If J = 0 then the equation on the relative phaseΘ in Eqs. (7) becomes decoupled from the rest of thesystem. The regime for negligible J is relevant whenthe reservoir shields the condensate from the mechanicalstresses and asymmetries of the underlying crystal (forinstance for a single spot excitation). As the pumpingstrength grows the condensate enters the desynchronisedregime which leads to small z getting close to the lin-early polarized state. If J is non-zero, as one would ex-pect for trapped condensates, the fixed point Θ = 0 inthe Josephson regime requires z to take negative valuesz ≈ −g(n+ − n−)/2Uα. Therefore, the transition fromthe synchronized regime at low pumping to desynchro-nised at higher pumping makes it possible for the systemto reach the linear polarized state. The transition to asteady state is not possible without polarization spin flip.To verify this analysis we integrate Eqs. (7) and showthe dependence of the DCP, ξ, on the pumping strength,p = p−, over the threshold in Fig. 4).

IV. FULL SYSTEM

Finally, we check the dynamics of the DCP for atrapped condensate without neglecting the spatial varia-tion. We solve

2i∂ψ±∂t

=−∇2 + |ψ±|2 + (1− Uα)|ψ∓|2 + gn± +

+i

2(Rn± − 1)

ψ± + J(r)ψ∓, (15)

n±(r) =p±(r)

γ + b|ψ±|2, (16)

where the pumping profile, p−, and the internal Joseph-son coupling, J, are given by

p−(r) = P exp[−α(r − r0)2],

J(r) = min(Jmax, 1/p−).(17)

The resulting condensate profile and DCP for a sin-gle power density, as well as the pump profile used arepresented in Fig. 5(a). In Fig. 5(b) we show the depen-dence of the DCP, ξ, on the pumping strength above thethreshold P/Pth for a condensate pumped in a ring. Thesolution was obtained by numerically solving system (15)for different P and using the spatial averaging of densitiesacross the condensate. The initial pump spin discrepancywas assumed to be 5% (η = 1.05). The full 2D simula-tions support the conclusions of Section III. The systemstarts at a synchronized state for small pumping with ahigh DCP. The desynchronized regime brings the systemcloser to the linearly polarized state. The synchronizedstate at high pumping gives the polarization reversal ac-cording to the analysis of fixed points of Section III.

V. CONCLUSIONS

We demonstrate the density dependence of the spinstate of a polariton condensate under non-resonant op-tical excitation. We utilise the spatial separation of theexciton reservoir from the polariton condensate in an an-nular optical trap scheme, and show the emergence ofa strongly polarised spinor condensate. The latter weattribute to the inherent spin imbalance induced by op-tical excitation in combination with the suppression ofspin relaxation through scattering with the exciton reser-voir. Most interestingly we observe a density dependentnon-linear spin reversal at moderate optical excitationdensities. The non-linearity of polariton spin dynamicsis described and theoretically reproduced by introduc-ing an internal Josephson coupling term in the spinorGLE, where the dynamics of the system are reduced toa current-driven Josephson junction. We show that theobserved spin reversal is due to the interplay between

7

0.8

0.6

0.4

0.2

0.0-10 0 10

X [µm]

1.0

0.8

0.6

0.4

0.2

0.0

Norm

alised Intensity

Pump Condensate DCP

0.6

0.4

0.2

0.0

2.42.22.01.81.61.41.21.0P/Pth

Synchronized DesynchronizedD

egre

e of

Circ

ular

Pol

ariz

atio

n

FIG. 5: (a) Simulated pump (dotted black line, right axis) andcondensate (solid red line, right axis) real space profile plottedtogether with the condensate DCP profile (solid blue line, leftaxis) for Pth=1.1 that has been multiplied by an appropriatehyperbolic tangent function to avoid the irrelevant DCP be-haviour for the points where the condensate density is toosmall (less than 10−2). (b) DCP, ξ, as a function of P/Pth fora trapped condensate pumped in a ring. The red points cor-respond to the fixed points, while the blue points to the aver-aged values of the desynchronized solutions. The blue circle in(b) denotes the point from where the profiles presented in (a)are extracted. Parameters are γ = 10, g = 0.7, b = 15, R =1.0, Uα = 1.1, Jmax = 0.07, η = 1.05, α = 0.06, r0 = 5.

an internal Josephson coupling effect and the detuningof the two projections of the spinor condensate via tran-sition from a synchronised to a desynchronised regime.These results facilitate the design and implementation ofpolariton based non-linear spinoptronic devices such aselectrically pumped polariton spin switches.

ACKNOWLEDGEMENTS

P.G.L acknowledges support by the Engineer-ing and Physical Sciences Research Council of UKthrough the Hybrid Polaritonics Programme Grant(EP/M025330/1). P.G.S. acknowledges funding fromthe EU Social Fund and Greek National Resources(EPEAEK II, HRAKLEITOS II), N.G.B acknowledgesthe financial support by Ministry of Education and

Science of Russian Federation 1425320 (Project DOI:RFMEFI58114X0006). The authors acknowledge fruit-ful discussions with Prof. Alexey Kavokin and Dr HamidOhadi. The data from this paper can be obtained fromthe University of Southampton e-Print research reposi-tory.

∗ correspondence address:alexis.askitopoulos@soton.ac.uk† correspondence address: pavlos.lagoudakis@soton.ac.uk

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