Incoherent excitation and switching of spin states in exciton-polariton condensates

0.87 MB·English

by  G. Li

Incoherent excitation and switching of spin states in exciton-polariton condensates
G. Li,1 T. C. H. Liew,2 O. A. Egorov,3 and E. A. Ostrovskaya1
1Nonlinear Physics Centre, Research School of Physics and Engineering,
The Australian National University, Canberra ACT 0200, Australia
2Division of Physics and Applied Physics, Nanyang Technological University 637371, Singapore
3Institute of Condensed Matter Theory and Solid State Optics, Abbe Center of Photonics,
5 Friedrich-Schiller-Universit¨at Jena, Max-Wien-Platz 1, 07743 Jena, Germany
1
We investigate, theoretically and numerically, the spin dynamics of a two-component exciton-
0
polariton condensate created and sustained by non-resonant spin-polarized optical pumping of a
2
semiconductormicrocavity. Usingtheopen-dissipativemean-fieldmodel,weshowthattheexistence

n of well defined phase-locked steady states of the condensate may lead to efficient switching and
a controlofspin(polarization)stateswithanon-resonantexcitation. Spatiallyinhomogeneouspulsed
J excitations can cause symmetry breaking in the pseudo-spin structure of the condensate and lead

2 toformationofnon-trivialspintextures. Ourmodelisuniversallyapplicabletotwoweaklycoupled
2 polaritoncondensates,andthereforecanalsodescribethebehaviourofcondensatepopulationsand
phases in ’double-well’ typepotentials.

]
s
a PACSnumbers: 42.65.Pc,71.36.+c,42.65.Tg,42.65.Sf
g
-
t I. INTRODUCTION reservoircanbecontrollablyreplenishedbyapumplaser
n
a of the selected polarization, thus opening up the way
u Created in semiconductor microcavities, the two- to create and manipulate spinor condensates of exciton-
q component exciton-polariton condensates1,2 provide a polaritons in the regime of incoherent far off-resonant
t. rich playground for exploring the effects of internal optical excitation9.
a
m (pseudo-spin) degrees of freedom on the dynamics of Inthiswork,weconsidertheexciton-polaritonconden-
non-equilibriumsuperfluids. Aremarkablerangeofspin- sateintheregimeofcwfaroff-resonantincoherentexcita-
-
d related phenomena has been explored in systems with a tion, in the absence of external magnetic field. We anal-
n coherentresonantopticalpump,whichoffersthepossibil- yse the non-stationary dynamics of condensate pseudo-
o ity to directly inject the spin state of exciton-polaritons spinandtherelaxationdynamicstowardspossiblesteady
c and explore the ultrafast spin-switching dynamics3–6. states for both spatially homogeneous and inhomoge-
[ On the contrary, the non-resonant incoherent excitation neous cw pump. We show that the spin-polarized in-

1 responsible for the spontaneous formation of coherent coherent reservoir allows efficient switching between dif-
v exciton-polaritoncondensatesisonlybeginningtobeex- ferentspinstatesofthepolaritoncondensate,bothinthe
5 perimentally explored in the context of spin-dependent spatially homogeneous and some inhomogeneous pump-
5 effects. The reasonfor this disparity is a long-held belief ing regimes. Furthermore, we analyse steady-state spin
3
that the incoherent reservoir (uncondensed hot exciton- texturesanddemonstratethatexcitationofsteadystates
5
0 polaritons), inevitably created by the incoherent exci- by a spatially inhomogeneous pulsed optical pump can
. tations far above the energy of the condensed exciton- lead to the creation of stable non-trivial spin textures.
1
polaritonquasiparticles,lacksspin-selectivity,i.e.,owing
0
5 tothe fastspin-relaxationprocesseswithinthe reservoir,
1 the pump polarization cannot affect the spin of the con- II. THE MODEL
: densed polaritons. Thus, the theoretical modelling of an
v
incoherently excited exciton-polariton system with spin Wemodelthespontaneouslycreatedpolaritonconden-
i
X degrees of freedom usually treats the reservoir as either satebytheopen-dissipativeGross-Pitaevskiiequations10
r fast3 or slow7 response scalar density state, without any derived from the Hamiltonian for a quantum fluid with
a
spin discrimination. pseudo-spindegreesoffreedomobtained,indifferentcon-
Recent experimental studies of the optical spin-Hall texts, in Refs.3,4,7,11. The dynamical equations for the
effect8 employedmicrocavitieswith relativelyshortexci- condensate wavefunctions corresponding to the left ψ−
ton lifetimes and succeeded in creating a spin-polarized and right ψ hand circular polarization states are writ-
+
reservoir of high-energy ’excitonic’ polaritons. Such a ten as follows (σ = )8:
±
∂ψ ~2 i~
i~ σ = 2+ua ψσ 2+ub ψ−σ 2+gRnσ+ [Rnσ(r) γc] ψσ+Jψ−σ. (1)
∂t (cid:18)−2m∇ | | | | 2 − (cid:19)
2
Here m is the effective massof the lowerpolarition, γ is III. HOMOGENEOUS STEADY STATES
c
the loss rate of polaritons which is the inverse of the po-
laritionlifetime,andRcharacterisesthestimulatedscat- Westartwiththeanalysisofthespin-mixingdynamics
tering rate from the reservoir into the condensate. The foraspatiallyhomogeneouscondensate. Inexperiments,
interaction constants ua and ub characterise the scatter- the quasi-homogeneous distribution of condensate den-
ing between the polaritons of the same and different po- sity can be achieved, e.g., by a flat-top ’super-Gaussian’
larization(spin)states,respectively. Itiswellestablished excitation. In this case, it is convenient to separate the
that ub < ua 12. Also,gR characterisesinteractionsbe- amplitudeandphasedynamicsoftheorderparameterby
| | | |
tweenthe reservoirandcondensate,where forsimplicity, usingthetransformation: ψ (t)=Φ (t)exp(iφ (t))and
σ σ σ
we assume that interactions between oppositely spin po- introduce the density of the spinor components ρ (t)
σ
lTahriezeldinreeasrersvpoinir-manixdincgontdeermnsepdroppoolartriiotonnasl atoreJne(gJloigseibplhe-. Φtu2σtianngdtthheesereslpaatitviaelplyhahsoemdoiffgeerneenocuesθst=atφe−s−inφto+.ESqu.b(s3t≡)i-,
son coupling) accounts for the sample-specific linear po- we obtain the dynamical equations for the condensate
larization splitting observed in experiments3,4 and usu- density ρ , the relative phase θ, and the reservoir den-
σ
allyarisingduetostressatthesemiconductorinterfaces. sity n :
σ
The equations for polariton wave functions are cou-
pled to the equations for the reservoirs created by the ρ˙σ =(Rnσ γc)ρσ+σ2J√ρσρ−σsinθ
−
incoherent cw pump with intensity P (r,t). It is usually Jcosθ
assumedthat all memory of the polarσizationof the inco- θ˙ =(cid:18)U − √ρ+ρ−(cid:19)(ρ+−ρ−)+gRδnR (4)
herent pump is lost3,11. However, as mentioned above,
n˙ =P (γ +Rρ )n ,
it was shown that the creation of a spin-polarized exci- σ σ − R σ σ
ton reservoir is possible with a spin-imprinting optical
pump for sufficiently short-lived excitons. Because the where U = ua−ub and δnR = n+−n−. The dynamics
describedbyEq. (4)isratheruniversal,andwouldapply
precise spin-relaxation mechanisms for the reservoir are
tothedescriptionofanytwo-modepolaritonsystemwith
unknown, we will consider the simplest model of spin-
linear(Josephson-type)couplingsuchascondensatesina
dependent reservoir,which was introduced in8:
double-wellpotential14 ora two-levelsystem15, provided
that their spatial variation can be ignored.
∂n The amplitude and phase of each component in the
σ =P (r,t) γ +Rψ 2 n , (2)
σ R σ σ condensate determine the polarization of the coherent
∂t − | |
(cid:0) (cid:1) photoluminescence emitted from the cavity. It is there-
fore helpful to use the pseudo-spin representation (see,
where γR is the loss rate of the reservoir polaritons and e.g.,11) which allows us to visualise the polarization of
P (r,t) is the pump. The dynamics of the reservoir is
σ thecondensateasapointonthePoincar´e(Bloch)sphere.
assumed to be fast on the scale of the polaritonlifetime,
Thecoordinatesofthepointaregivenbythecomponents
i.e., comparable to the dynamics of the condensate, so
of the Stokes vector defined in the standard manner:
that the model of slow-response, static reservoir7 is no
longer applicable. We note that the coherent resonant Sx =√ρσρ−σcosθ, Sy =√ρσρ−σsinθ,
injection of spin4 can be readily introduced into these 1 1 (5)
equations. Sz = (ρ+ ρ−), S0 = (ρ++ρ−)
2 − 2
ByintroducingthecharacteristictimeT =γ−1,length
L = ~/(mγ ), and energy E = ~γ scales1c3, we can with the total condensate density nc =ρ++ρ− defining
c u c thelengthoftheStokesvector: S2 =S2+S2+S2 =n2/4.
rewritpe Eq. (1) and (2) in the dimensionless form: 0 x y z c
Notethatboththepseudo-spinandthetotalcondensate
density, S = n /2, are functions of time, therefore it
0 c
makes sense to consider the evolution of this vector in
1
i∂tψσ =(cid:26)−2∇2+ua|ψσ|2+ub|ψ−σ|2+gRnσ the normalized pseudo-spin space, si(t) = Si(t)/S0(t),
i.e., on the Poincar´e sphere of unit radius. Although we
i (3) will use the normalized Poincar´e sphere representation
+ [Rnσ γc] ψσ +Jψ−σ
2 − (cid:27) throughout the text, one should keep it in mind that its
∂ n =P (r) (γ +Rψ 2)n (not normalized) radius can change dynamically.
t σ σ R σ σ
− | | The stationary states of the system (4) define fixed
points on the Poincar´esphere, which correspond to syn-
where u u /(E L2), u u /(E L2), g chronized, phase-locked solutions of the Eq. (3) with a
a a u b b u R
g /(E L2),→and J J/E . →In these scaling unit→s, well-defined polarization state of the polaritons. Desyn-
R u u
→
γ = 1, however, we will retain it in the subsequent for- chronisedstateswithatime-dependentphasecorrespond
c
mulas for clarity. In the following discussions, except to Josephson oscillations, and, in the case of steady
some specified cases, all parameters take the numerical undamped oscillations to limit cycles on the Poincar´e
values indicated in Ref.13. sphere.
3
In principle, steady states can be obtained in a closed
analytical form, however the expressions are cumber-
some,andamoreusefulinsightintothesystem’sdynam-
ics is obtained in some limiting cases. As noted in7,16,
the following conditions are of particular interest: (a)
the excitons decay into the polariton channel dominates
the relaxationdynamics of the reservoir,Rρ± γR; (b)
≫
the polariton interaction energy is greater than the tun-
nelling energy, Un J, which approximately defines
c
≫
the boundary of the synchronization region; (c) popu-
lations of the two spin components are similar (semi-
linearly polarized pumping), n S . Following the
c z
≫
procedure outlined in7, one can derive the approximate
self-consistent equation for the relative phase, which de-
scribes a Josephson junction driven by a constant bias
current (see17, Ch. 8.5):
1 γ δP
θ¨+ c θ˙ =(U +g ) +UJn0sinθ, (6)
2 2 R 2 c
here δP = P+ − P−, n0c = (P+ + P−)/γc. Analy-
sis of the phase space of this dynamical system shows
that, for the equal pumping of the two spin compo- FIG.1: (Coloronline). (a)-(c)FixedpointsolutionsofEq.(4)
nents, i.e. linearly polarized pump, δP = 0, only phase- underbalanced(linearlypolarized)pumping. Branch1and3
locked states exist. The system becomes desynchro- are elliptically polarized solutions; branch 2 and 4 are bond-
nised for an elliptically polarized pump when Ic = ing (s1) and anti-bonding (s2) solutions respectively. Only
| | the solid branch is sable (see text). Green dots: correspond-
[δP /(P++P−)][γc(U +gR)/(2UJ)] > 1, whereby the
| | ingpointstotheinhomogeneoussimulation. (d)(seeSec.IV)
dynamics is dominated by limit cycles. For I < 1 the
| c| InhomogeneoussimulationresultsofEq.(3). Evolutionofthe
system admits fixed points determined by the condition
integrated Stokes parameters in phase space under balanced
sinθ =I andcorrespondingtothedistinctsteadystates
c Gaussian-shape pumps. Parameters (see Sec.IVfor thedefi-
of the homogeneous system for each value of the pump nitions): P¯ =5,PiLm =0.5,ax =ay =5,andJ=0.5. Reddot:
intensity. initial state of evolution; green dot: final state of evolution.
Although Eq. (6) is obtained under a very restric- Inset: energy evolution.
tive assumption of a nearly linear polarization of the
polaritonic state, it can describe some experimental re-
sults such as in Ref.14. Notably, when the pump power first of Eq. (4), it follows that the synchronized state
is large, conditions (a) and (b) can be automatically can be reached when Rn =γ and the internal Joseph-
σ c
fulfilled once condition (c) is satisfied. Physically the son current IJ = 2J√ρ+ρ−sinθ becomes zero. Conse-
condition (c) means that the system effectively evolves quently, the phases are locked to θ = 0 or θ = π (mod
on a fixed-radius Poincar´e sphere, where its evolution 2π). The two fixed points on the Poincar´e sphere corre-
is then given by the competition between the nonlin- sponding to these phase locked linearly polarized states
ear polarition-polarition interaction and the Josephson states are s = (1,0,0) (corresponding to θ = 0), and
1
tunnelling between two components. Meanwhile, if the s = ( 1,0,0) (corresponding to θ = π). The density
2
−
pump power is large, then the polarition-polariton in- of the steady state grows linearly with the pump rate:
teraction dominates and Eq. (6) is applicable (detailed ρ = (P /γ )(P /P 1), see Fig. 1 (a-c), where the
σ th c σ th
discussion can be found in Ref.7). normalized pump powe−r is defined as P¯ = P /P and
σ σ th
Ingeneral,however,Eq.(6)does notalwaysapply,es- for the linearly polarized pump particularly P¯+ = P¯− =
peciallywhentheJosephsoncouplingdominates. There- P¯.
fore,inwhatfollowswewillconsidersteadystatesofEq. Following the standard Lyapunov stability analysis17,
(4)givenby thepumping conditionsofdifferentpolarza- we deduce that, for our choice of the sign of J, the π
tion and strength. out-of-phase state s (the so called anti-bonding state)
2
is stable and the in-phase state s is unstable. Direct
1
numerical integration of Eq. (4) with different initial
A. Linearly polarized pump conditions also confirm that s is stable Fig. 2(a-b). We
2
note that this selection of the anti-bonding steady state
Mostcommonly, the opticalpump producing a polari- in a weakly (linearly) coupled two-state system has also
ton condensate in non-spin-resolved experiments is lin- been recently confirmed in18,19. This effect persists even
earlypolarized. Inourmodel, this meansthatthe pump for the inhomogeneous (Gaussian) pumping19, as shown
is balanced: P+ = P−. Under this condition, from the in Fig. 1(d) and discussed in Sec. IV below.
4
FIG. 2: (Color online). Trajectories of the pseudo-spin vec-
torobtainedfromnumericalsolutionsofEq. (4)withJ =0.5
andvariousinteractionstrengthsbetweenthecondensateand
reservoir gR. Red dots: initial states of evolution; Green
dots: finalstatesofevolution. Initialconditionsarearbitrary. FIG. 3: (Color online). (a)-(c) Selected fixed point solutions
Parameters are: (a)gR = 1.5×10−2,θ(0) = 0.5; (b)gR = of Eq. (4), with an elliptically polarized pump for P¯+ = 2.4.
1.5×10−2,θ(0)=2;(c)ArtificiallyenlargedgR =1.5×10−1; Curves with darker color are stable while those with lighter
(d) Artificially reduced gR =1.5×10−3. color are unstable. Red (green) dot: the steady state be-
fore (after) a pulse is added. (d) Inhomogeneous simula-
tion results of Eq. (3) (see Sec. IV). Evolution of integrated
The other steady states arise when ρ > Stokes parameters in phase space. Pumping parameters:
σ
i(sPtthh/eγocp)p(Posσi/tePtfhor−th1e),owthheerrepotlhaerizsaigtnionofcothmepionneeqnuta,lsietye Pt¯ers=: P4¯,sPeiL=m 0=.120,.4P,iLmax==1,aaypx==5,aJpy==05..5;(sPeuelsSeecp.aIrVamfoer-
thedefinitions.) Reddot: initialstateofevolution;greendot:
Fig. 1 (a-c) dashed branch 1 and 3. They are sustained
finalstate of evolution. Inset: energy evolution.
byanon-zerointernalJosephsoncurrentwhichwillgrow
with the pumping power. Lyapunov analysis reveals
that they are unstable.
Introduced by the interaction between the condensate
and the reservoir, the blue shift term which is propor-
tionaltog playsanimportantroleinthespindynamics. tion against small pumping imbalance, where all unsta-
R
Itcanmodifythefixedpointpropertieswhenitsenergyis ble solution branches with at least one vanishing ρσ are
compatible with the Josephson tunnelling energy. Start- not shown for clarity. Stable fixed points exist in semi-
ing with some arbitrarily chosen small amplitude initial circularpolarizedpumpingregion(darkblue),whichwill
conditions, Fig. 2(c-d) illustrates how different values of bediscussedinthenextsubsection,andthesemi-linearly
g canaffecttheevolutionofStokesparametersinphase polarized pumping region(dark red), where at the point
R
space. Indeed, both the positions of fixed points and P− = P+ the fixed point coincides with the single anti-
their stability can change, e.g., when gR = 1.5 10−3, bonding stable state ρ+ =ρ− shown in Fig. 1.
×
boths ands becomestable,seeFig.2(d); highervalue
2 1
of g will lead to the system in an elliptically polarized One should note that the above discussion is based
R
state, see Fig. 2(c). Here, the trajectories are obtained on the Lyapunov stability analysis, which describes the
by numerical integration of Eqs. (4). linear stability to a long-wavelength (spatially homoge-
neous) perturbation with the wave vector k = 0. The
steady state may become unstable to a spatially mod-
B. Elliptically polarized pump ulated perturbation with k = 0, see, e.g., discussions
6
aboutthemodulational instabilityin2D7 and1D20 cases.
When the imbalanced polarization-selective pumping
is introduced, the dependence of the density of the or- Asaverificationoftheexistenceofthestablebranches,
thogonalspincomponentsonthe pumppowerdisplaysa weperformedapulsedexcitationofasteadystateinFig.
hysteresis-likeprofile. However,itdoesnotcorrespondto 3 (a-c) where the red and green dots represent stable
the typical bistability behaviour,since mostof the (light steady states before and after the pulse. They fit the in-
blue)curve corresponds to an oscillatory unstable fixed homogeneous simulation result, Fig. 3 (d), qualitatively
point. Fig. 3 (a-c) shows a typical fixed point distribu- well. (see Sec. IV for details.)
5
IV. INHOMOGENEOUS SPIN-SWITCHING
DYNAMICS
In the spatially inhomogeneous situation, it is well
knownthatthe in-plane magneticfields, either being ex-
ternallyappliedorcausedbytheTM-TEsplitting24,can
leadto nontrivialspatialstructures for both density and
spindistributions7,8,25. Thelinearspincouplingcanalso
be regardedas an effective magnetic field pointing along
the xaxis,whichcanleadto the appearanceofspinpat-
terns. However, here we analyse steady states of the
polariton spinor systems sustained by a cw pump with
a Gaussian shape and show that the spatially averaged
evolution of the spin components qualitatively follows
the prediction of the homogeneous model discussed in
Sec. III. The results obtained within the homogeneous
approximationcan therefore be used as a guide for spin-
switching manipulation with a realistic, spatially inho-
mogeneous pump.
To determine the evolution of pseudo-spin under spa-
tially inhomogeneous excitation, we solve directly the
open-dissipative GP equations (3) with a radially sym-
FIG. 4: (Color online). (a)-(c) Fixed point solutions of metric Gaussianpump by using the split-step method26.
Eq. (4), with P¯− = 0. Only the solid branch is stable (see The total intensity of the pump is defined as P(x,y) =
tneexotu)s.siGmrueleantiodno.ts(:d)c(osrereesSpeocn.dIVin)gInphooinmtosgteonetohuessinimhoumlaotigoen- P0e−(x2/a2x+y2/a2y), where ax,y is the width along the x
and the y direction. The pump for each circularly po-
results of Eq. (3). Evolution of integrated Stokesparameters
larized component is given by splitting the energy from
in phase spaceundercircularly polarized pump. Parameters:
P¯ = 3,PiLm = 0,ax = ay = 5, and J=0.5. Red dot: initial P(x,y), which is P(x,y) = P+(x,y)+P−(x,y), since in
state of evolution; green dot: final state of evolution. Inset: manyexperimentsthepumpisgivenbyasinglelaser8. It
energy evolution. (see Sec. IV for thedefinitions.) isconvenienttousethespinbiasdefinedasPiLm =P0−/P0
to denote those two pumps, then P−(x,y) =PiLmP(x,y)
and P (x,y) = (1 PL )P(x,y). Also, throughout the
+ − im
text, the notation of the normalized (inhomogeneous)
pump is givenby, P¯ =P /P (normalizationagainstits
0 th
peak value), where the pump threshold P is the same
C. Circularly polarized pump th
as that in Sec.III.
By adjusting the pumping power and the imbalance
value, with an arbitrary small amplitude initial condi-
The steady state shown in Fig. 3 (a-c) at P− = 0 tion, the system can be driven to a synchronized steady
corresponds to the situation when only one spin compo-
state without any external potential. The condensate
nent is pumped, which is physically akin to the strongly
mean-field energy functional, E = E+ + E−, serving
imbalanceddissipativedouble-wellJosephsonjunction18.
as the measure of kinetic and interaction energy of the
(The oppositecasewhereP+ =0 issimilar.) Inthis lim- condensate27, can be used to detect whether the steady
iting case, the fixed point solutions of Eq. (4) are shown
state is reached:
inFig.4,wherethesteadystateforP¯ =2.4isthesame
+
asthatrepresentedbythedarkbluecurveinFig. 3(a-c)
E = dr[ ψ 2/2+u ψ 4+g n ψ 2+
at P− =0. σ Z |∇ σ| a| σ| R σ| σ| (7)
The stable branch whose θ is small but never van- +ub|ψσ|2|ψ−σ|2+JRe(ψσ∗ψσ)],
ishes corresponds to the so called self-trapped state with where the integrationis over the areawhere the conden-
a strongly imbalanced population. It is a consequence sate density is non-negligible.
of nonlinear interactions21,22 and, in the context of a To map the evolution of the inhomogeneous conden-
double-well polariton system, has been studied both sate to that of the Stokes parameters, we trace the evo-
experimentally23 and theoretically18. lution of the spatially integrated quantities as sint(t) =
i
S (x,y)dr/Sint, where Sint is given by the spatially
i 0 0
Again, Fig. 4 (d) shows us the inhomogeneous simula- Rintegrated length of the Stokes vector. The integration
tionresultverifyingthepositionoffixedpointspredicted area is the same as that used to determine the steady
by the homogeneous model. state energy (7). In the following, unless it is indicated
6
FIG. 6: (Color online). Steady state integrated quantities
against the pump imbalance PiLm. Parameters: P¯ = 4,ax =
ay = 5,J = 0.5. Between the gap 0.42 < PiLm < 0.58, one
needstoincreasetheP¯valuetoreachsteadystates(seetext).
thepolarizationstateisalmosthomogeneousacrossmost
of the integration area.
Thedegreeofcontroloftheaveragedpolarizationstate
of the condensate attainable by an incoherent, far off-
resonantspin-polarizedexcitationaccordingtoourmodel
is presented in Fig. 6 and could be tested in an exper-
FIG. 5: (Color online). (a)-(c) Synchronized steady state,
withP¯ =4,PiLm =0.4,ax =ay =5,andJ =0.5: (a)Density iment, which would validate our assumptions. Due to
profiles (along y = 0) and spatial distribution of sz for the ourchoiceofthe pump powerdistributionconfiguration,
steady state. (b) Evolution of integrated Stokes parameters in Fig. 6, between 0.42 < PL < 0.58 one needs to in-
im
and energy. (c) Evolution of integrated Stokes parameters in crease P¯ in order to reach steady states. With a larger
phasespace. Startingformthereddot;endingwiththegreen pump power, however, the degree of control of polariza-
dot. (d) desynchronised state, with P¯ = 8,PiLm = 0.4,ax = tion decreases because the system will easily fall into a
ay = 5, and J = 0.5: evolution of integrated Stokes param- desynchronisedstateevenwithsmallvalueofimbalance.
eters. Starting form the red dot; ending with a closed or-
Forexample,foranellipticallypolarisedpump, within
bit.Inset: energy evolution. The integrations were over areas
where nc(x, y)>10−3. the regionmarkedwith ’D’ (which stands for desynchro-
nised)inFig. 3,noneofthefixedpointbranchesaresta-
ble, which does not rule out the existence of the closed
orbits(limitedcycles)onthePoincarespherecorrespond-
explicitly; we shall refer to the normalized siint(t) sim- ing to desynchronised states. As shown in Ref.16, for a
ply as integrated Stokes parameters. This normalization
desynchronised state the density ρ (t) maintains peri-
σ
procedure, although eliminating information on the spa-
odic oscillations and the evolution of its Stokes vector
tial distribution of the pseudo-spin, allows us to recover
is similar to Fig. 5 (d) where the angle of the trajec-
the spatially averaged polarization state. This averaged
tory plane can vary depending on the system parame-
spin dynamics could be observable experimentally even
ters. This regime cannot be accessed by Eq. (6) because
without performing spatially resolvedpolarimetry of the
the latter canonly describe limit cycles inthe vicinity of
cavity photoluminescence.
the s =0 plane. Indeed, it can be revealed by inhomo-
z
Polarization dynamics for the linearly and circularly geneous simulations of the full model equations that, in
polarized pump are shown in Fig. 1(d) and Fig. 4(d), such a state, the spatial distribution of the spin pattern
respectively. As we can see, the spatially averaged dy- keeps breathing. The corresponding Stokes parameters
namics in the case of inhomogeneous pumping fits the circle around the Poincar´esphere28, reducing the degree
homogeneous predictions well. of (time-averaged) polarization of its luminescence. A
For a typical steady state under elliptically polarized similar pinning and depinning effect has been observed
pumping,thevalueofthecrosssectiondensitiesandspa- experimentally3. Here this effect is associatedwithmov-
tialdistribution ofthe pseudo-spinis showninFig. 5(a). inginandoutofthephase-lockedsynchronizedregimeof
Its build up process is similar to that of the linearly- thecondensatedynamicscausedbyintrinsicinteractions
polarized pumping case, see Fig. 5(b-c). The linear cou- between the two spin components, so that it can arise
plingJ causesswitchingofthedominantdensitybetween even without taking into account the structural disorder
the two components as polaritons spread out from the in the sample.
pump source, which leads to the appearance of radially The coexistence of stable and unstable fixed points
symmetricdomainsofpolarizationdensitys . Thelarger described in Sec. III indicates that an inhomogeneous
z
J leads to the denser spin pattern and loss of correspon- pump might be used to switch the system between
dence between the averagedspin dynamics in inhomoge- different density and polarization states. Indeed, we
neous and homogeneous cases. If J is sufficiently small, find that the efficient switching between different spin
7
E x 104 E x 104 E x 104
8 15 3
(a) (b) (c)
6
10 2
4
5 1
2
0 0 0
0 50 100 150 t 0 50 100 t 150 0 50 100 t 150
nc8×103 sx1nc8×103 sy1nc8×103 sz1
(d) (e) (f)
6 6 6
s s s
x y z
4 0 4 0 4 0
2 2 2
0 10 10 1
0 50 100 t 15(cid:1)0 0 50 100 t 15(cid:1)0 0 50 100 t15(cid:1)0
FIG. 7: (Color online). Spin switching dynamicsfor thecase
of Gaussian pump. (a-c) Energy of theinitial and final state
aftertheswitchingpulse. Dashline: pulseswitchedon. Dash-
dotline: pulseswitchedoff;(d-f)Integratedtotaldensityand
integratedStokesparametersduringtheswitchingprocessfor
switchingbetweenthesteadystatesshownin(c). Parameters:
the same as Fig. 3 (d), except for (a) P¯se =0.4,PL =0.65;
im
and (b) P¯se =0.9,PL =0.65.
im
and density states can be achieved by applying a radi-
ally symmetric incoherent pulsed excitation of the form:
P (r,t)=(1/4)Pse 1+tanh[τ (t t )] 1+tanh[τ (t
t)s]e}e−(x2/a2px+y2/a2p0y){, whereP0spe is−th0e p}{eak value, appx1,p−y
determines the pulse width, t and t (t > t ) are the
0 1 1 0
pulse switch on and switch off time respectively, and τ
p
is a coefficient controlling adiabaticity of the excitation FIG. 8: (Color online). Spin wave excitation for the case
(mlaarlgizeeτdpncootrarteisopnofnodrsthtoeapunlosne-iasdgiiavbeantbicypPu¯lssee=). PThsee/nPor-. aoftaignhtelylonfogcauteseddpruamdipallwyitshymthmeetirmicbaplualnsceewPitiLhmP=iLm0=.4 0a.n6d5
Note that the combined intensity after the pul0se wtahs and different widths (see text). (a) Density profile of the ψ+
componentshowingadensitydipassociatedwithaphasefold
added is given by the interference between pump and
(b). (c)Cross-sectionofdensityandphaseacrossthefold. (d)
pulse modes. Unless specified, they are regarded as be-
Density profile of the ψ− component corresponding to (a);
ing in phase. A radially symmetric non-adiabatic pulse,
(e) Density profile of the ψ+ component showing stationary
apx =apy,causesanabruptchangeinthecondensateen- configuration of vortex pairs connected by a phase fold (f);
ergy and a strong outward density flows combined with (g) Cross-section of density and phase across the phase fold.
the internal Josephson currents leading to spin mixing (h)Densityprofileoftheψ− componentcorrespondingto(e).
dynamics[Fig. 7(d-f)]. As aresult, the systemcanenter All density plots are logarithmic.
a different steady state [Fig. 7(a,c)], or a phase desyn-
chronised state [Fig. 7(b)]. If the original steady state
isstable,theperturbedsystemrestoresitsinitialdensity ments,while the value ofpump powermightbe givenby
and polarization state after the pulse is switched off. a spatially averagedone depending on the specific shape
of the pumping laser.
Regardless of the fact that with strong spatial vari-
ation the homogeneous results would not be generally
applicable, one can still compare the spatially-averaged
inhomogeneousresultsofFig. 7(c-f)withthe fixedpoint V. EXCITATION OF NON-TRIVIAL SPIN
TEXTURES
solutions of Eq. (4), as long as the former begins and
ends with steady states. Fig. 3 shows two of the rele-
vant fixed point solution branches that are qualitatively While the density and spin wave excitation in the
comparabletothe simulationdata. Theinitial(red)and regime of a radially-symmetric cw pump and pulse leads
final (green) steady states correspond to the flipping of to the efficientswitching betweendifferent spinandden-
s shown in Fig. 7(d) before and after the pulse was sity states, the spatial dynamics can be captured by the
x
added. These results are ready to be tested in experi- homogeneousorspatialaveragingapproximation,aslong
8
as the radial symmetry of the spatial density distribu- of the system under the assumption that the incoher-
tions for both spin components is preserved. However, ent pump creates a fast-responding and rapidly decay-
the situation changes dramatically when the radial sym- ing reservoir, which significantly affects the condensate
metry of the system is broken. This can be achieved by dynamics. Our analysis includes effectively spatially ho-
combining a radially symmetric normal incidence pump mogeneous pumping conditions, as well as spatially in-
with elongated excitation, e.g., at a steep angle to the homogeneous pumping with a Gaussian profile. The po-
sample surface, or vice versa. Fig. 8 demonstrates the larization dynamics in the latter case can be mapped to
consequence of pulsed perturbation of a steady state es- the homogeneous dynamics through a spatially averag-
tablished with an elongated (a = 16, a = 5) pump. ing procedure. We have demonstrated that the phase-
x y
The pulseis astrong,off-resonant,tightly focusedGaus- locking conditions for existence of stationary states nat-
sian beam with a = a = 4 (a-d) or a = a = 3 urally leads to the self-trapped states with a nontrivial
px py px py
(e-h)andpeakpowercomparabletothatofthecwpump. phase relationship between the condensate components,
It was set to having π/2 phase difference with the pump including π out-of-phase(’anti-bonding’)states. The ex-
and thus no interference pattern appeared. Remarkably, istence of both phase-lockedand spin-beating states can
the symmetry of the density flows caused by the pulse enable efficient control and switching of the polarization
is now broken, leading to different flow speeds along the states in the system. Experimentally, these findings can
two symmetry axes. As a result, larger area pulse pro- be tested with spatially averaged polarization measure-
duces a phase fold associated with a pronounced dip in ments under incoherent spin-selective excitation condi-
the condensate density Fig. 8 (a-c). An even tighter tions.
pulse will result in the two phase folds terminating on In addition, we have analysed formation of non-trivial
vortices, leading to formation of a stable and stationary spin-textures under pulsed, spatially inhomogeneous ex-
configuration of two vortex-antivortexpairs. citation, and demonstrated numerically reliable creation
Although nontrivial phase structures appear only in of stable vortex-antivortexpairs.
oneofthespincomponents[seeFigs. 8(d),(h)],theyare Ouranalysis,withsuitablemodificationofparameters,
maintained and stabilised by internal Josephson current is widely applicable to a general two-state polariton sys-
between the spin components and do not exist in the tem with weak linear coupling, including weakly linked
absenceoflinearcoupling(J =0). Wenotethatnophase spatially separated condensates or multi-mode conden-
singularitiesappearinthelinearly-polarizedbasis,which sates in shallow potential traps.
sets thesestructuresapartfromthe previouslydescribed
half-charge vortices29–31
VII. ACKNOWLEDGEMENTS
VI. CONCLUSIONS This work was supported by the Australian Research
Council (ARC). O.A.E. acknowledges financial support
Inconclusion,wehaveinvestigatedindetailtheforma- by the Deutsche Forschungsgemeinschaft (DFG project
tionandpolarizationstructure ofthe steady states ofan EG344/2-1)andthe ThuringianMinistry for Education,
exciton-polariton condensate created by far off-resonant ScienceandCulture(TMESCprojectB514-11027). G.L.
spin-polarizedpumpinasemiconductormicrocavitywith acknowledgesthesupportoftheChinaScholarshipCoun-
linear polarization splitting. We model the dynamics cil (CSC).
1 H. Deng, H. Haug, and Y. Yamamoto, Rev. Mod. Phys. P. Tsotsis, Z. Hatzopoulos, P. G. Savvidis, A. V. Ka-
82, 1489 (2010). vokin,andP.G.Lagoudakis,Phys.Rev.Lett.109,036404
2 I.CarusottoandC.Ciuti,Rev.Mod.Phys.85,299(2013). (2012).
3 J. Levrat, R. Butt´e, T. Christian, M. Glauser, E. Feltin, 9 C. Ant´on, S. Morina, T. Gao, P. S. Eldridge, T. C. H.
J.-F. Carlin, N. Grandjean, D. Read, A. V. Kavokin, and Liew,M.D.Mart´ın, Z.Hatzopoulos, P.G.Savvidis,I.A.,
Y. G. Rubo,Phys. Rev.Lett. 104, 166402 (2010). and Shelykh,L. Vin˜a, arXiv:1410.8417.
4 T.K.Para¨ıso, M.Wouters,Y.L´eger, F.Mourier-Genoud, 10 M.WoutersandI.Carusotto,Phys.Rev.Lett.99,140402
and B. Deveaud-Pl´edran,NatureMat. 9, 655 (2010). (2007).
5 R.Cerna,Y.L´eger,T.K. Para¨ıso, M. Wouters,F.Morier- 11 D.Read,T.C.H.Liew,YuriG.Rubo,andA.V.Kavokin,
Genoud, M.T. Portella-Oberli, and B. Deveaud, Nature Phys. Rev.B 80, 195309 (2009).
Comm., 4, 2008 (2013). 12 M. Vladimirova, S. Cronenberger, D. Scalbert, K. V. Ka-
6 S.S.Gavrilov,A.V.Sekretenko,S.I.Novikov,C.Schnei- vokin,A. Miard, A. Lematre, J. Bloch, D. Solnyshkov,G.
der, S. H¨ofling, M. Kamp, A. Forchel, and V. D. Ku- Malpuech, and A. V. Kavokin, Phys. Rev. B 82, 075301
lakovskii, Appl.Phys.Lett. 102, 011104 (2013). (2010).
7 M. O. Borgh, J. Keeling, and N. G. Berloff, Phys. Rev. B 13 In the numerical calculations, the following physical
81, 235302 (2010). values of the parameters are chosen as experimentally
8 E. Kammann, T. C. H. Liew, H. Ohadi, P. Cilibrizzi, accessible7,32,33: mLP = 10−4 me (where me is the free
9
electron mass), ua = 6×10−3 meVµm2, ub = −0.1ua, Phys. Rev.Lett. 95, 010402 (2005).
gR =2ua,γc =0.33ps−1, γR =1.5γc,R=0.01µm2ps−1, 23 M. Abbarchi, A. Amo, V. G. Sala, D. D. Solnyshkov, H.
and J ≈ 0.1∼0.2meV. So, the dimensionless quantities Flayac, L. Ferrier, I. Sagnes, E. Galopin, A. Lemaˆıtre, G.
are: ua =7.7×10−3, ub =−7.7×10−4, gR =1.5×10−2, Malpuech, and J. Bloch, Nat. Phys.9, 275 (2013).
γR =1.5, R=8.4×10−3, and J ≈0.1∼1. This choice of 24 K.V.Kavokin,I.A.Shelykh,A.V.Kavokin,G.Malpuech,
parameters corresponds to an assumption of a very short and P. Bigenwald, Phys.Rev.Lett. 92, 017401 (2004.)
lifetime of reservoir excitons, however the results do not 25 A.Kavokin,G.Malpuech,andM.Glazov,Phys.Rev.Lett.
depend on a particular valueof γR. 95, 136601 (2005).
14 K. G. Lagoudakis, B. Pietka, M. Wouters, R. Andr´e, and 26 J.Javanainen andJ.Ruostekoski,J.Phys.A:Math.Gen.
B.Deveaud-Pl´edran,Phys.Rev.Lett.105,120403(2010). 39, L179 (2006).
15 P.R. Eastham, Phys.Rev.B 78, 035319 (2008). 27 W. Bao and Y. Cai, Kinetic and Related Models, 6, 1
16 M. Wouters,Phys. Rev.B 77, 121302(R) (2008). (2013).
17 S. Strogatz, Nonlinear Dynamics and Chaos (Westview, 28 S. S. Gavrilov, A. S. Brichkin, S. I. Novikov, S. Hfling, C.
2001). Schneider, M. Kamp, A. Forchel, and V. D. Kulakovskii,
18 J.Y.LienY.N.Chen,N.Ishida,H.B.Chen,C.C.Hwang, Phys. Rev.B 90, 235309 (2014).
and F. Nori, arXiv:1407.1271v1. 29 Y. G. Rubo,Phys. Rev.Lett. 99, 106401 (2007).
19 G. Christmann, G. Tosi, N. G. Berloff, P. Tsotsis, P. S. 30 K. G. Lagoudakis, T. Ostatnicky´, A. V. Kavokin, Y. G.
Eldridge, Z.Hatzopoulos, P.G.Savvidis,andJ.J.Baum- Rubo, R. Andr´e, and B. Deveaud-Pl´edran, Science 326,
berg, NewJ. Phys.16, 103039 (2014). 974 (2009).
20 P.-E´.Larr´e,N.Pavloff,andA.M.Kamchatnov,Phys.Rev. 31 F. Manni, K. G. Lagoudakis, T. C. H Liew, R. Andr´e, V.
B 88, 224503 (2013). Savona, and B. Deveaud,Nat. Comm. 3, 1309 (2012).
21 A. J. Leggett, Rev.Mod. Phys.73, 307 (2001). 32 G. Roumpos, M. D. Fraser, A. L¨offler, S. H¨ofling, A.
22 A. Smerzi, S. Fantoni, S. Giovanazzi, and S. R. Shenoy, Forchel, and Y.Yamamoto, Nat.Phys. 7, 129 (2011).
Phys.Rev.Lett.79,4950(1997);E.A.Ostrovskaya,Yu.S. 33 J. Kasprzak, R. Andr´e, L. S. Dang, I. A. Shelykh, A. V.
Kivshar, M. Lisak, B. Hall, F. Cattani, and D. Anderson, Kavokin, Y. G. Rubo, K. V. Kavokin, and G. Malpuech,
Phys.Rev.A61,031601(R)(2000);M.Albiez,R.Gati,J. Phys. Rev.B 75, 045326 (2007).
Folling,S.Hunsmann,M.Cristiani,andM.K.Oberthaler,