Instability-induced formation and non-equilibrium dynamics of phase defects inpolariton condensates.

T. C. H. Liew,1 O. A. Egorov,2 M. Matuszewski,3 O. Kyriienko,1, 4 X. Ma,2 and E. A. Ostrovskaya5

1Division of Physics and Applied Physics, Nanyang Technological University 637371, Singapore2Institute of Condensed Matter Theory and Solid State Optics, Abbe Center of Photonics,

Friedrich-Schiller-Universitat Jena, Max-Wien-Platz 1, 07743 Jena, Germany3Instytut Fizyki Polskiej Akademii Nauk, Aleja Lotnikow 32/46, PL-02-668 Warsaw, Poland

4QUANTOP, Danish Quantum Optics Center, Niels Bohr Institute,University of Copenhagen, Blegdamsvej 17, DK-2100 Copenhagen, Denmark

5Nonlinear Physics Centre, Research School of Physics and Engineering,The Australian National University, Canberra ACT 2601, Australia

(Dated: January 23, 2015)

We study, theoretically and numerically, the onset and development of modulational instabilityin an incoherently pumped spatially homogeneous polariton condensate. Within the framework ofmean-field theory, we identify regimes of modulational instability in two cases: 1) Strong feedbackbetween the condensate and reservoir, which may occur in scalar condensates, and 2) Parametricscattering in the presence of polarization splitting in spinor condensates. In both cases we investigatethe instability induced textures in space and time including non-equilibrium dynamics of phasedislocations and vortices. In particular we discuss the mechanism of vortex destabilization andformation of spiraling waves. We also identify the presence of topological defects, which take theform of half-vortex pairs in the spinor case, giving an “eyelet” structure in intensity and dipole typestructure in the spin polarization. In the modulationally stable parameter domains, we observeformation of the phase defects in the process of condensate formation from an initially spatiallyincoherent low-density state. In analogy to the Kibble-Zurek type scaling for nonequilibrium phasetransitions, we find that the defect density scales with the pumping rate.

PACS numbers: 71.36.+c,42.65.Sf,78.67.-n,03.75.Mn

I. INTRODUCTION

The formation of complex spatio-temporal patternsand textures is a particularly intriguing phenomenon oc-curring in a diverse range of physical systems.1,2 Thepatterns commonly arise in nonlinear dissipative sys-tems driven far from equilibrium. Amongst examplesof such systems, exciton-polaritons in semiconductor mi-crocavities have emerged as a hybrid light-matter sys-tem with strongly nonlinear properties. Effects such asBose-Einstein condensation,3–5 superfluidity6–8 and theformation of solitons9–13 have been well-documented inthe literature.14,15

A particularly important nonlinear effect in the con-text of pattern formation in microcavities is the paramet-ric scattering of resonantly excited polaritons in planarsemiconductor microcavities.16–19 The pair scattering ofpairs of polaritons to different states in reciprocal spaceallows a homogeneous polariton field to spontaneouslybreak translational symmetry. This effect enables theformation of ordered hexagonal/triangular lattices, pre-dicted theoretically20–22 and observed experimentally23

under different excitation conditions, as well as latticesof breathing solitons.24

Under non-resonant/incoherent excitation, vortex lat-tices were predicted to occur in harmonic traps25,26 andlater observed in experiments involving multiple excita-tion spots.27 The formation of multi-lobed28 and vortex-antivortex patterns29 under ring shaped excitation, as

well as sunflower ripples30 excited by a narrow pump spothave also been reported. In these examples, the transla-tional symmetry of the system is already broken by thechosen shape of the pump-spot and/or the presence of agradient in the potential of polaritons (as in, for example,the case of harmonic traps4).

In this work we consider the possibility of spontaneousbreaking of translational symmetry and pattern forma-tion in planar microcavities excited by a spatially homo-geneous incoherent pump. The pumping creates a reser-voir of “hot” exciton-like polaritons, which form a po-lariton condensate through a stimulated scattering pro-cess. The translational symmetry breaking is triggeredby linear instability of the homogeneous condensate tospatial modulations, and the nonlinear evolution of theunstable state leads to formation of spatial patterns. Weconsider two different mechanisms of such modulationalinstability (MI) in this system. The first one arises whenthe polariton condensate has a strong feedback effect onthe reservoir, in the form of reservoir depletion due tostimulated scattering of reservoir excitons into polari-tons. In this case, while polariton-polariton interactionsare repulsive, the essentially saturable nature of exciton-polariton interactions may lead to effectively attractivenonlinearity for sufficiently low pump powers.31 This ef-fective focusing nonlinearity in the system naturally leadsto MI of a spatially homogeneous state, which was estab-lished in several previous studies for both quasi-1D and2D geometry.15,26,31–35.

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Modulational instability is also known in spin-1 Bose-Einstein condensates of ultracold atoms due to paramet-ric coupling36–39 and nonlinear interactions between spincomponents,40 as well as in 1D exciton-polariton conden-sates with a spin (polarization) degree of freedom.41 Al-though polariton systems are non-conservative and non-equilibrium, the two-component spin degree of freedomof polaritons does allow a second mechanism of MI,which works also in the defocusing regime where a strongcondensate-reservoir feedback is unnecessary. A circu-larly polarized excitation splits the energy of the σ+ andσ− states due to anisotropic interactions42,43 occurringbetween the spin polarized reservoir and condensed po-laritons. This splitting sets the foundation for a paramet-ric scattering process as polaritons in an initially homoge-neous state with wavevector k = 0 on the upper spin-splitbranch can now scatter to degenerate non-zero wavevec-tor states on the lower branch, reminiscent of experi-ments under resonant excitation in triple microcavities44

or experiments in one-dimensional polariton systems.45

While such a scattering process is not strictly allowed inisotropic cavities as it would violate spin conservation,the presence of sample anisotropy, which typically causesan additional linear polarization splitting and hybridiza-tion of the σ+ and σ− branches, relaxes this limitation.

By considering the stability of the steady states of thesystem to weak perturbations, we find the zones of MIin the two different regimes. In the scalar case, whereMI is derived from the condensate-reservoir feedback, wefind that the homogeneous state breaks its translationalsymmetry and forms a turbulent state of phase disloca-tions, i.e., vortices. Unlike the previously studied cases,vortices do not form as the result of thermal fluctua-tions46 or scattering on disorder47. Rather, the spatialfragmentation of the initially homogeneous condensatedue to the development of MI creates multiple interfer-ence between polariton flows generated by the randomlydistributed sources, which leads to the development ofmultiple phase dislocations, similar to the scenario pre-viously considered for multiple pump spots48 and highlyinhomogeneous trapped polariton condensates26,49.

In the case of modulationally stable background, weshow that multiple phase singularities can appear as aresult of mean-field evolution of an initial white noisestate, which mimics a pre-condensate state lacking spa-tial and phase coherence. Remarkably, formation of mul-tiple vortices in this scenario seems to be analogous to,but not the same as, the Kibble-Zurek mechanism, whichacts during the quench through a phase transition to theBose-Einstein condensation (BEC)50,51. Indeed, the laterdescribes the formation of boundary defects between dif-ferent domains of condensate which develop an indepen-dent phase rather than inheriting it from the neighbour-ing spatial domains52,53.

We stress that the process of defect formation dur-ing nonequilibrium condensation of exciton-polaritonsdoes not follow the scenario of the Kibble-Zurek mecha-nism51,54,55. The main difference is that in the latter, it

is assumed that the system is initially in thermal equilib-rium, and is driven out of equilibrium only in the vicin-ity of the phase transition51,55. The process is dividedinto three phases, corresponding to adiabatic-impulse-adiabatic evolution. In nonequilibrium condensation, thesystem is far from equilibrium at the outset, and the tran-sition to the quasi-equilibrium (condensed) state occursonly after crossing the critical point.

Nevertheless, the Kibble-Zurek mechanism and the de-fect formation in nonequilibrium systems have much incommon. In both cases, defects are created due to sym-metry breaking in separate parts of the system whichcannot communicate in a finite time. In both cases, thereis a competition of two timescales existing in the system,which results in the same algebraic forms of power-lawscalings for the number of defects and their characteristiccreation time54. In the polariton condensation case, thequench time is replaced by the timescale of the forma-tion of the condensate, which is controlled by the exter-nal pumping rate. We refer the reader to Sec. IV D andRef. 54 for the detailed description of the process.

Regardless of the mechanism of the vortex formation,either as the result of the MI development or as a resultof transition to BEC, we show that the presence of theincoherent reservoir affects substantially both stability ofvortices and their collective dynamics even for the caseof a stable homogeneous background. As a consequence,the vortices can lose their radial symmetry and developeither into spatially localized rotating phase dislocationsor into non-localized spiraling waves.

A similar situation occurs in the spinor case, althoughmultiple branches of modulationally unstable and stablesolutions are present. Defects in the spin polarizationof the condensate may appear even in the modulation-aly stable regime. Such structures move randomly inthe microcavity plane and are composed of half-vortex56

half-antivortex pairs,57 exhibiting an associated dipoletype spin texture. We predict that the density of vor-tices grows with increasing pump power similarly to theKibble-Zurek scaling behaviour.

The paper is organised as follows. In Sec. II, we de-scribe the mathematical model of a semiconductor mi-crocavity operating in the strong coupling regime un-der incoherent homogeneous optical pump of a circularpolarization. Then, in Sec. III, we study the stabilityand collective dynamics of phase dislocations in a single-component polariton condensate. Here the dynamics ismostly affected by the modulational instability originat-ing from the strong feedback between the condensate andreservoir. In Sec. IV, we report a numerical analysis ofthe condensate dynamics in the presence of polarizationsplitting in spinor condensates. In Sec. IV D, we studythe defect formation and the scaling laws for their densityin analogy to the Kibble-Zurek mechanism.

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II. THEORETICAL MODEL

Let us begin by considering the incoherent excitation of aspinor polariton system, i.e., the system where the polar-ization degree of freedom is significant. The scalar case,which is valid when only one spin component is pop-ulated, is then easily obtained by removing one of thespin components. It is worth recalling that experimentswith a circularly polarized optical pump have resultedin the excitation of a circularly polarized polariton con-densate at the pump position, in both 2D58 and 1D59

samples. A theoretical model can be based on the gener-alized Gross-Pitaevskii approach,35 where a condensateof exciton-polaritons can be described by the wavefunc-tions, ψ+ and ψ−, of the σ+ and σ− circularly polarizedstates, respectively:

i~dψ+

dt=

(−~2∇2

2m+ g1nR + α1|ψ+|2 + α2|ψ−|2

)ψ+

+ i~ (rnR − Γ)ψ+ + ∆XYψ−, (1)

i~dψ−dt

=

(−~2∇2

2m+ g2nR + α2|ψ+|2 + α1|ψ−|2

)ψ−

− i~Γψ− + ∆XYψ+. (2)

Here we assume that the circularly polarized pumpingcreates a circularly polarized reservoir, nR, with dynam-ics described by the rate equation:

dnRdt

= −(ΓR + r|ψ+|2

)nR + P, (3)

where P represents the pumping rate. Non-linearinteractions between polaritons are characterized by α1

and α2, representing the interaction strengths betweenparallel and antiparallel spins42,43, respectively. Similarparameters, g1 and g2, characterize the blueshift causedby the circularly polarized reservoir. Γ is the polaritondecay rate. ∆XY represents a linear polarization split-ting, which has been reported in several experimentalstudies60–62 and can take values of 0-0.2 meV.60,63 Evenlarger values can be expected by the application ofmagnetic fields (in Voight configuration).

We note that in the limit ΓR � Γ, it is possi-ble to proceed by adiabatic elimination of the reservoirdynamics.26,64 However we do not make such an approx-imation here since the reservoir dynamics is importantfor our further analysis.

III. NON-EQUILIBRIUM DYNAMICS IN THESCALAR CASE

A. Modulational instability of the homogeneoussteady state.

If we consider a circularly polarized pump in a micro-cavity with negligible polarization splitting (∆XY = 0)

FIG. 1: (Color online) Steady-state solutions of the model (1-3) and their stability for a circularly polarized pump (∆XY =0). (a) The homogeneous solution (HS) (|ψ0|2) as functionsof pump P . The dotted line depicts the modulational in-stability (MI). (b) Growth rates [=mω (ps−1)] of the smallperturbations (around |ψ0|2) as a function of momentum kand HS density |ψ0|2. The vertical axis is the same as in (a)and the horizontal dashed (blue) line connecting (a) and (b)indicates that the boundary of modulational instability oc-curs for the same maximum homogeneous density. Snapshotsof the condensate phase in real space (c) and the density intwo-dimensional reciprocal space (d) within the MI domain ofa HS for pump P = 9 ps−1µm−2. kcutoff can be identified asthe magnitude of the in-plane wavevector at which all unsta-ble modes disappear. Parameters: α1 = 6 × 10−3 meVµm2,g1 = 4α1, Γ = 0.165 ps−1, r = 0.01 ps−1µm2, ΓR = 0.5 ps−1.The polariton effective mass was taken as ×10−4 of the freeelectron mass.

then the model can be reduced to a scalar one (ψ = ψ+)without the second polarization component (ψ− = 0).First, we study a scalar steady-state homogeneous solu-tion (HS) of the system of Eqs. (1)-(3) and discuss itsstability. For the sake of generality we allow also theHSs with nonzero transversal momenta k0 6= 0, whichhave the form of travelling waves

ψhs(x, t) = ψ0e−i µ(k0,|ψ0|)t+ik0x, (4)

where the condensate energy is given by ~µ (k0, |ψ0|) =(~2/

2m)k2

0 +α1|ψ0|2 +g1nR0. The HS becomes nontriv-ial provided that the external pump compensates for alllosses and overcomes the threshold value:35 Pth = ΓΓR/r.The coherent exciton-polariton density and incoherentreservoir density are given by |ψ0|2 = (P − Pth)/Γ andnR0 = Γ/r, respectively.

The linear stability analysis of the homogeneous steadystate of our scalar system and its modifications has been

4

previously performed by many authors.15,31–35 For ourchoice of the system parameters, the linear stability anal-ysis shows that the HS becomes modulationally (dynam-ically) unstable within a pump interval just above thethreshold value of the pump Pth [Fig. 1(a)] (details ofthe analysis are given in Appendix A). This MI is asso-ciated with the parametrical generation of field compo-nents with nonzero momenta k. Figure 1(b) presents thelinear growth rate =mω(k) > 0 of the unstable pertur-bations as a function of their momenta k.

It has been shown recently31 that this MI is associ-ated with the effective attractive nonlinearity inducedby the saturation of the incoherent reservoir. Indeed,based on the intuition gained from paradigm nonlinearmodels, such as the Schrodinger equation with a Kerr-nonlinearity, one expects that the existence of the MIrequires a focusing nonlinearity.65–68 However, owing torepulsive interactions between excitons, the nonlinear be-haviour of an exciton-polariton condensate is akin tothat of optical waves in a defocusing media. This seem-ing contradiction clearly elucidates the influence of theopen-dissipative nature of the system on the nonlinearbehaviour,31 which requires inclusion of an incoherentreservoir of “hot” excitons. To illustrate this influencewe consider a nonlinear energy shift induced by both thecoherent exciton-polaritons and the incoherent reservoir

~µnl(|ψ|2

)= α1|ψ|2 + g1

P

ΓR + r|ψ|2. (5)

We note that the polariton density |ψ|2 corresponds to asteady-state solution which, in general, is not necessar-ily given by the homogeneous value |ψ0|2. The reservoirintensity nR follows this steady-state solution |ψ|2. Thefirst term of Eq. (5) describes the blue shift originatingfrom the repulsive exciton-exciton interaction, whereasthe second term describes the saturation of the reser-voir. We define the effective nonlinearity coefficient asgeff = ~∂µnl/∂|ψ|2. Then the nonlinear response is ef-fectively focusing provided that the coefficient is nega-tive geff < 0. Otherwise, if geff > 0 the nonlinear re-sponse is defocusing or repulsive [see thick solid lines inthe Fig. 2(a)]. In the vicinity of the steady-state HS with

intensity |ψ0|2 this coefficient takes the form (cf. Eq. (36)in Ref.[31]):

geff = α1

(1− g1Γ2

α1rP

). (6)

The effective nonlinear coefficient [Eq. (6)] changes signfor a pump value PMI = g1Γ2/α1r. It means that in thevicinity of the HS the nonlinear response changes charac-ter from effectively focusing to effectively defocusing. Asa result the modulationally unstable HS becomes stableexactly at this point P = PMI . The condition PMI > Pthgives a general criterion for the appearance of MI derivedin Ref. [31]:

g1Γ

α1ΓR> 1. (7)

FIG. 2: (Color online) (a) Effective nonlinear energy shift~µnl versus the condensate density, |ψ|2, for different pump P .Dashed lines (with the negative slope) represent an effectivelyfocusing nonlinear response (or nonlinear red shift). The greytriangle represents the intensities |ψ|2 exceeding values givenby the steady-state homogeneous solutions (|ψ0|2). (b) Vortexwidth (FWHM) vs. the pump P . The dashed line representsthe width approximated by the value ≈ 3.5ξ. dmax depicts themaximal diameter allowed for stable vortices, see Eq. (9). (c)The radial dependence of the condensate and reservoir densitywithin the vortex profile for P = 20 ps−1µm−2. (d) The realand imaginary parts of the ψ within the vortex profile givenin (c). Other parameters are the same as in Fig. 1.

Within the MI interval (Pth < P < PMI) indicated inFig. 1(a), the HS experiences spontaneous translationalsymmetry breaking, resulting in the formation of non-uniform turbulent states of the condensate.

The result of a direct numerical calculation is shownin Fig. 1(c). In these calculations and those presentedthroughout the manuscript we use a square grid with atleast 128 × 128 points covering the plot area. We makeuse of an adaptive step Adams-Bashforth-Moulton pro-cedure, which was previously found to be consistent withfixed step methods with an integration step of 0.01ps.To double check the results of our numerical simula-tions we repeated them with higher precision using upto 1024× 1024 grid points. Unless stated otherwise, pe-riodic boundary conditions are applied.

The strongly non-equilibrium state in Fig. 1(c) in-cludes one- and two-dimensional phase dislocations whichmove chaotically and overlap. Therefore this dynamicscan be characterized as a “strong” turbulence regimewith overlapping defects (see Ref. [48] and referencestherein). Two-dimensional Fourier transformation showsthat the most part of the spatial momenta are boundedwithin the ring with the radius given by the cutoff mo-menta of the unstable modes kcutoff [Fig. 1(d)]. Notethat direct numerical simulations of the model (1,3) for

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different initial conditions did not reveal the formation ofstationary periodical patterns, known for the coherentlypumped polaritonic systems20–23. This is due to desta-bilization caused by fluctuations in the exciton reservoir.The modulational instability discussed later in the spinorcase (section IV) does not depend on having a dynamicreservoir and can largely be reproduced assuming a staticreservoir25. This allows an explicit testing of the effect ofthe dynamic reservoir in the spinor case, where we findthat periodic patterns are possible with a static reser-voir but are prevented as soon as the reservoir density isallowed to evolve spatiotemporally. We assume that itis the freedom for density fluctuations to appear in thereservoir that lead to the disruption of regular patternsin the condensate also in the scalar case.

B. Single vortices in the dynamically stable regime.

The reservoir contribution becomes negligible in thelimit of very strong pump P � Pth. We expect that inthis case the nonlinear dynamics is very similar to thatknown for the conservative systems, including the forma-tion of the stable phase dislocations and vortices. Indeedthe scalar version of the equations possesses vortex solu-tions within the stability interval of the HS for P > PMI

[see Figs. 2(b) and (c)]. Similar to the conservative case,one can define a characteristic length or an effective heal-ing length in the vicinity of the HS |ψ0|2:

ξ(P ) =~√

2mgeff |ψ0(P )|2. (8)

The healing length is a typical length scale over which ψcan change significantly. It also gives the typical size ofthe vortices. More precisely the full width at half maxi-mum (FWHM) of the vortices is approximately given byd ≈ 3.5ξ at least for a strong enough pumping [P � Pthin Fig. 2(b)]. It is worth to mention that the spatial oscil-lations of the real and imaginary parts of the amplitudeprofiles of the vortices go substantially beyond the heal-ing length [Fig. 2(d)], especially for a moderate pumpingin the vicinity of the MI. This indicates permanent en-ergy and polariton exchange within the vortex profile.The presence of these intrinsic fluxes essentially changesthe interaction dynamics between vortices69 providing apurely dissipative mechanism for their mutual repulsion.

The radial symmetry of the vortex phase is broken alsofor an inhomogeneous pump, for instance, a Gaussianpump13,26. As a result, the vortices have spiraling phasesindicating again the permanent exchange of particles be-tween different points within the resonator plane.

The stability analysis, discussed in the context of theHSs, can also be applied to the vortices, at least in thelimit of very broad states. Indeed, following the heal-ing length ξ(P ), the vortex width increases for smallervalues of the pump and diverges in the vicinity of theMI [P = PMI in Fig. 2(b)]. The formal condition for

the effective focusing nonlinearity (geff(|ψ|2) < 0) is al-ways satisfied within the vortex profile close enough to itscore. Therefore the steady-state condensate on a circle

with a fixed radius %c ≡√x2 + y2 and density |ψ(%c)|2

can become unstable against spatially modulated per-turbations. The periodical boundary conditions on thecircle restrict the number of available momenta to thevalues kc ' n/%c, where n is an integer. In the limit oflarge radius %c →∞ the values of the unstable momentacan be approximated by those values calculated for theHSs [see Fig. 1(b)]. Therefore the polariton condensatebecomes unstable provided that at least one of the avail-able momenta (kc ' n/%c) is smaller than the cut-offvalue kcutoff ≈ 0.19µm−1 for MI. This allows estimationof the vortex diameter dmax where the instability justsets in, i.e., kcutoff = 2/dmax. Therefore the vortices arestable provided that their diameters do not exceed themaximal value given by

dmax ' 2k−1cutoff . (9)

It is shown in Fig. 2(b) that the vortex diameter exceedsthe maximal value dmax within the pump interval PMI <P . 18 ps−1µm−2 in the vicinity of the MI domain.As we will see below (in the next subsection III C) theybecome unstable and lose their radial symmetry.

C. Collective dynamics of vortices

Beyond the MI instability interval for P > PMI thenon-trivial HS solution is stable. However, at the onset ofcondensation when it passes into the mean-field regime,the polariton state is spatially incoherent, with randomphase. This leads to the generation of phase defects asthe system goes through the condensate transition, inanalogy to the Kibble-Zurek theory. To go beyond themean-field approximation and describe first and secondorder spatial coherences when crossing the condensationthreshold, one can make use of stochastic classical fieldapproaches.70,71 These describe the pre-condensate stateas an ensemble of fluctuating white noise states (governedby a stochastic Gross-Pitaevskii equation). The projec-tion onto the classical condensate state upon condensa-tion selects a particular realization of the noise. Afterthe condensate has formed we assume that fluctuationsare weak in comparison to the condensate mean-field andcan be neglected. This approach reproduces the typicalestablishment and coherent evolution of topological de-fects in polariton condensate experiments57,72. We notethat the account of fluctuations throughout the evolutionwould be important for describing accurately spectral po-lariton properties or the polariton photoluminescence be-low threshold70. Fluctuations can in principle shift thephase boundaries between stable and unstable regions73,however, these shifts are expected to be limited and havelittle further effect on the dynamics.

For a pumping substantially above the MI thresholdthe dynamics is dominated by the defocusing nonlinear-

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FIG. 3: Nonlinear dynamics of the condensate beyond theMI threshold for (a,b,c) P = 25 ps−1µm−2 and (d,e,f)P = 17.5 ps−1µm−2. (a,d) Long-time evolution dynamics ofthe dislocations number. (b,e) Snapshots of the condensatedensity (µm−2) profiles. (c,f) Snapshots of the condensatephase profiles. See also the Supplemental Materials74. Pa-rameters are the same as in Fig. 1.

ity of a pure condensate (α). The influence of dissipa-tive dynamics of the reservoir becomes less important.Using the scalar version of the governing equations (1),(3) we calculated the condensate dynamics starting witha spatially incoherent state given by a small-amplitudewhite noise. Our initial condition mimics a particularrealisation of the stochastic pre-condensate state. Sim-ilar to the conservative limit we observed the forma-tion of a spatially coherent condensate accompanied bythe spontaneous formation of the vortices which movechaotically and interact with each other. Two vorticeswith equal (opposite) topological charges repel (attract)each other. Two attracting vortices mutually annihi-late if the distance between them becomes smaller thanthe healing length. Therefore, as discussed in Ref. [48],the whole number of vortices drops gradually with timeand approaches zero, at least for a very strong pumpP > 40 ps−1µm−2.

In contrast, for a weaker pump, the number of phasesingularities converges eventually to some constant valueindicating the formation of a coherent state with a fi-nite number of dislocations, i.e., a superfluid turbu-lence [Fig. 3(a)]. A snapshot of the intensity and phaseprofiles shows a state of well distinguishable vortices[Figs. 3(b),(c)]. The vortices move chaotically, inter-acting with their neighbours, and, in general, sustain a

dynamical equilibrium. It is remarkable that the aver-age separations between nearest vortices remains moreor less constant for this particular interval of pump val-ues. This means that there exists some equilibrium dis-tance between vortices. Apparently the influence of thedissipative effects and the condensate flows (mentionedin the previous subsection III B) are substantial.69 Theout-going condensate flows from the vortex centers hinderattraction between vortices and their annihilation. Foreven weaker pump the dissipative effects become strongerand, as a consequence, the average distance between vor-tices under dynamical equilibrium becomes even smaller[Figs. 3(d),(e) and (f)]. We note that this dynamicalequilibrium forms over a long time scale exceeding hun-dreds of nanoseconds. Therefore the Kibble-Zurek-likescaling law does not describe the number of vortices inthis regime (see Sec. IV D below).

Even though the HSs are stable, the nonlinear dynam-ics of vortices is strongly affected by the reservoir sat-uration dynamics. The numerical modelling shows thatthe vortices themselves become unstable and develop intoradially asymmetric rotating structures, as can be seenon the snapshot profiles in Figs. 3(e),(f) (a movie show-ing the time dynamics is available in the SupplementalMaterial74). This is in agreement with the destabiliza-tion scenario for the vortices discussed in the previoussubsection. Indeed the vortex size exceeds the maximaldiameter dmax given by Eq. (9) and therefore becomesunstable.

D. Formation of spiraling waves

In the vicinity of the MI the vortices are strongly unsta-ble and the initial noise first develops into non-uniformdynamical states similar to those which appear for themodulationally unstable background [Figs. 4(a,b)]. Inthis “strong” turbulence regime48 the characteristic dis-tance between vortices is substantially smaller than theirtypical core size, so the vortices are not structured andthe chaotic behavior is seen on the level of a singlevortex. However, after some sufficiently long time ofabout several thousand polariton lifetimes, the systemswitches spontaneously into a more regular regime, char-acterized by the formation of a single spiraling topo-logical dislocation [Fig. 4] (a movie showing the timedynamics is available in the Supplemental Material74).This spiraling topological state drives away other phasedislocations in the system and covers eventually thewhole computational window provided that PMI < P .16 ps−1µm−2. Similar spiraling waves are known forother non-equilibrium dissipative systems.75,76 In thesimplest case they are solutions of the complex Ginzburg-Landau equation.

It is worth to articulate the differences between the spi-raling topological states and the vortices. First, there isno rotational symmetry in the profiles [Figs. 4]. Second,these topological states are not stationary and experience

7

FIG. 4: (Color online) Snapshots of the condensate intensi-ties (a,c,e) and phase profiles (b,d,f) for different time pointsand P = 13.5 ps−1µm−2. (a,b) Initial turbulent state of thecondensate at t = 1000 ps. (c,d) Onset of a spiraling wave att = 5000 ps. (e,f) The spiraling waves at t = 11000ps. Therotation period of the spiraling wave is 144ps. See also theSupplemental Materials74. Other parameters are the same asin Fig. 1.

uniform rotation of the density with a typical rotation pe-riod of about 144ps. Third, far from the center the pro-file converges to the homogeneous traveling wave solutiongiven by Eq. (4) with the amplitude ψ0 and a nontrivialmomentum k0. Apparently this solution is characterizedby a permanent radial flux of exciton-polaritons from thevortex center to the periphery and therefore resembles apoint-like source of ring waves. We note that these spi-raling waves can exist only in nonequilibrium dissipativesystems.

An important question remains whether the spiralingstate with a non-zero orbital angular momentum canemerge from the initially non-rotating turbulent state.Indeed, according to the conservation law of the total or-bital angular momentum, the phase dislocations appearin pairs which is also valid for the turbulent state con-sidered here. Apparently the local intensity fluctuationsbreak the symmetry between the two dislocations withina pair in such a way that only one of them develops intothe spiraling wave. The periodical boundary conditions(in x and y directions) were used in our numerical mod-elling. However, we have confirmed that the spiralingwaves appear with the same probability independentlyon the computational window size and particular reali-sations of the initial seeding noise. Moreover, since somephase dislocations are always present [see Figs. 4(e) and(f)], the total orbital angular momentum of the conden-sate within the computational window remains zero. Weperformed additional numerical simulations of the con-densate dynamics under a localized pump with a “flat-

top” shape in the form of a super-Gaussian intensity dis-tribution. It turned out that the spiraling waves appearalso for the localized pump where the condensate densityvanishes at the boundaries of the computational window.These calculations serve as a solid proof of the existenceof the spiraling waves independently of a particular choiceof the numerical boundary conditions.

In general the out-going radiation from the center ofthe topological solution repels the local inhomogeneitiesof the profile and other topological solutions. This givesan additional purely dissipative mechanism which en-forces a long range spatial coherence in non-equilibriumsystems operating in the regime of “strong” turbulence.

IV. NON-EQUILIBRIUM DYNAMICS IN THESPINOR CASE

A. Stability of Homogeneous States

In the presence of non-zero polarization splitting(∆XY 6= 0), the population of σ− component appearseven for a fully polarized σ+ pumping. To emphasizethe difference to the nonlinear dynamics discussed above(see Sec. III) we consider system parameters which donot satisfy the MI criteria (7) and, therefore, guaranteethe stability of homogeneous solutions in the scalar limit.

Homogeneous stationary solutions can be found bysubstituting trial solutions in the form ψ±(t) = ψ±e

−iµt

into Eqs. (1) and (2). This gives four stationary equa-tions for real and imaginary parts of the amplitudes.These are supplemented by the requirement that the timederivative in Eq. (3) vanishes for a steady state, that is,

|ψ+|2 = 1r

(PnR− ΓR

). Noting that the phase reference

of the system can be freely chosen, we can for simplicityset =m {ψ−} = 0, which allows one to find the relation<e {ψ−} = ∆XY

Γ =m {ψ+} from one of the four station-ary equations. The remaining three equations can besolved for the remaining unknown quantities: <e {ψ+},=m {ψ+} and µ.

The dependence of the stationary HSs on the pumpingpower is shown in Figs. 5(a) and (b). Linear stabilityof the steady states can be determined by the standardextension of the Bogoliubov-de Gennes analysis15,31–35,41

by considering perturbations in the polariton and reser-voir fields of the form δψ± = u±e

i(kx−ωt)+v∗±e−i(kx−iω∗t)

and δnR = w(ei(kx−ωt) + e−i(kx−ω

∗t)), respectively. The

details of the derivation can be found in Appendix A.The stationary states labelled U1 and U2 in Fig. 5(a,b)

are unstable to fluctuations with k = 0. This “single-mode instability” indicates that even in a confined system(e.g., micropillar) the homogeneous steady state would belinearly unstable in the Lyapunov sense and any spatiallyhomogeneous perturbation would grow. The stationarystates labelled MI1 and MI2 are unstable to spatial mod-ulations with non-zero wavevectors k. These states wouldbe stable in a confined system, with no spatial degrees

8

FIG. 5: (Color online) Stationary solutions to Eqs. 1-3. (a)and (b) show the dependence of the σ+ and σ− polarizedpolariton populations on the pumping strength, respectively.The different branches are labelled stable (S), MI, and single-mode unstable (U). (c) Parametrically unstable zones of theMI1 and MI2 branches. The shading illustrates the gain ofthe instability (given by the imaginary part of ω) for the MI2

solution. The grey rectangle denotes a region below conden-sation threshold where ψ± = 0. The vertical line representsthe pump intensity considered in Figs. 7 and Fig. 9. Param-eters: α1 = 1.55 × 10−4 meVµm2, α2 = −0.1α1, g1 = α1,g2 = α2, Γ = 0.033 ps−1, r = 0.01 ps−1µm2, ΓR = 10 ps−1,∆XY = 0.1 meV, P = 92 ps−1µm−2. The polariton effectivemass was taken as 5× 10−5 of the free electron mass.

of freedom, however, in a 2D planar system the statesMI1 and MI2 undergo parametric scattering. The statelabelled S is fully stable, as the imaginary part of E = ~ωremains negative for all wavevectors (see Appendix A).

The “S”-shape of the curves MI1, MI2 and U2 is char-acteristic of multistability, which is a common featureof resonantly excited microcavities77–80 but less studiedunder the non-resonant or incoherent excitation81 thatwe consider here. While multistability is strictly onlypresent in the confined system, since the MI1 and MI2

states are unstable in the presence of spatial degreesof freedom, they can still give rise to different (non-stationary) configurations under the same excitation con-ditions.

Fig. 5(c) shows the regions of MI in the system whenthe MI1 or MI2 branches are excited. The MI1 branchbegins once the threshold for polariton condensation ispassed (indicated by the grey rectangle), such that onlyweak pump intensities are needed to see the effects of MI.

B. Spin Textures due to Modulational Instability

Due to the presence of MI, we can expect the frag-mentation of the homogeneous density of the condensateand spontaneous formation of spin textures, even in thepresence of homogeneous pumping. Solving Eqs. (1)-(3)numerically when the system is excited just above thresh-old on the MI1 branch, we obtain the spin texture shownin Fig. 6. In analogy to the scalar case, the texture com-prises a simultaneous modulation of the intensity andphase in the system, which oscillate with multiple fre-

quencies. In addition to the scalar case, there is alsoan appearance of a non-uniform polarization, despite thefact that the pumping of the system is homogeneous inboth intensity and polarization.

FIG. 6: (Color online) Spinor MI regime. (a) Density pat-tern of a polariton condensate under homogeneous incoherentpumping. (b) Distribution of the circular polarization degree,ρz =

(|ψ+|2 − |ψ−|2

)/(|ψ+|2 + |ψ−|2

). (c) phase of the ψ+

polariton field (the ψ− component (not shown) has a similardependence). (d) Time evolution of the polariton intensityalong a slice in real space. Parameters were the same as inFig. 5 (a small pump power was chosen so as to excite onlythe MI1 branch).

C. Spin Defects in the Dynamically Stable Regime

When the system is excited with a larger pump power,the system tends to follow the stable branch (S) in Fig. 5.In this case one can expect a spatially HS due to the sta-bility of the (S) branch, however, defects present in theinitial state after transition to condensation (in simula-tions taken as a low intensity white noise as in sectionsIII C and III D) are trapped in the system and stabilizewith the structure shown in Fig. 7.

The defects have a non-trivial density structure withlocalized maxima inside of an otherwise circular shapeddrop in density. These “eyelets” are not fixed in theirlocations, but move randomly in the plane with a typi-cal speed on the order of 0.05 µm/ps. The eyelets alsopossess a characteristic spin polarization with a dipoletype shape, as shown in Fig. 7(b). The structure of theeyelets can also be understood when looking at the phasedistribution, which is shown in Figs. 7(c) and (d) for the

9

FIG. 7: (Color online) Spin defect formation without MI. a)Density pattern of a polariton condensate under homogeneousincoherent pumping. b) Distribution of the circular polar-ization degree, ρz =

(|ψ+|2 − |ψ−|2

)/(|ψ+|2 + |ψ−|2

). The

black curves show the path traced out by the eyelets over1000 ps. c) and d) show the phase of the ψ+ and ψ− polari-ton fields, respectively. Black spots indicate the positions of(half) vortices. Parameters were the same as in Fig. 5 (withP = 84 ps−1µm−2).

σ+ and σ− polarized components, respectively. Here itis clear that each eyelet is formed from a pair of halfvortices56 – one appearing in each spin component.

In addition to the slow drift, the eyelets preserve theirshape while undergoing a faster periodic rotation with aperiod close to 2π/∆XY (a movie of the motion is avail-able in the Supplemental Material). Apparently this pe-riodical motion of two bound vortices with opposite spinsis induced by the polarization splitting in spinor conden-sates. It has been shown recently82 that similar spinoreffects can evoke a uniform motion of bound polaritonsolitons in coherently driven microresonators.

D. Scaling Laws for the Defect Density

The number of vortices generated in the system outsideof the MI region, during the mean-field evolution of aninitially noisy, low-density state, is found to grow withthe pump intensity, as shown in Fig. 8. The number ofvortices is counted at a time after spatial coherence isestablished in the system, where the average polaritonand reservoir density achieves a steady-state. Note thatat very long-times there may be further recombination ofvortex-antivortex pairs.

By taking advantage of the universality of dynamics of

FIG. 8: (Color online) (a) Same as in Fig. 7 under higherpower excitation. (b) Variation of the number of vortices withpump power obtained numerically. The numerical results areaveraged over ten different initial configurations at each valueof pump intensity. The scaling with pump power is comparedto the dependencies Nd ∼ Pth (1− Pth/P ) (green curve) andNd ∼ (P/Pth − 1) (magenta curve), see Eqs. (15) and (16).

the system in the vicinity of the phase transition, we findapproximate scaling laws governing the number of defectscreated during the transition, see solid line in Fig. 8(b).The scaling laws in the case of exciton-polaritons havesimilar forms as the ones obtained from the argument ofKibble and Zurek.50,51 However, the dynamics of defectformation is different due to the fact that the initial stateof the system is strongly out of equilibrium.54 While inthe Kibble-Zurek mechanism the phase transition is as-sumed to begin in the initial state that is close to thermalequilibrium, here the white-noise initial state is domi-nated by fluctuations. In both cases, spontaneous sym-metry breaking occurs differently in separate regions ofspace which cannot communicate on how the symmetryis broken due to the finite timescale of the process. Onthe borders between these separate regions defects canappear in the form of domain walls, vortices, or more so-phisticated structures, depending on the dimensionalityof the system and the form of the order parameter.50,51,83

In the case of dynamics described by (1)-(3), start-ing from the initial low density white-noise state, thegrowth of polariton density initially occurs without es-tablishing any coherence (a movie showing the dynamicscorresponding to the establishment of the state in Fig. 8acan be found in the supplemental material). This corre-sponds to the first stage of the process, where there ispractically no nonlinearity and no k-dependence of thegrowth rate. The defects are created in the second stage,when fluctuations are suppressed due to nonlinear inter-actions. In this process, regions of ordered phase (orpatches of a condensate) appear out of the initial dis-ordered strongly fluctuating phase. Due to spatial andpolarization symmetry breaking, defects are created onthe borders between these condensed regions.

We assume that in the emerging ordered regions de-scribed above the relative phase between the two po-larization components is approximately equal to π anddensities of |ψ+|2 and |ψ−|2 are similar. This assump-tion is in agreement with the numerical data presented

10

in Figs. 7(c) and 7(d), where the phase is equal to πaway from the eyelets, and is related to the fact thatsuch configuration minimizes the energy in Eqs. (1-2).We substitute ψ± = ± 1√

2ψ to obtain

i~dψ

dt=

(−~2∇2

2m+ g1nR +

α1 + α2

2|ψ|2

)ψ

+i~2

(rnR − 2Γ)ψ −∆XYψ,

dnRdt

= −(

ΓR +r

2|ψ|2

)nR + P. (10)

Next, we assume that the reservoir quickly adjusts to thechange of |ψ|2,

nR =P

ΓR + r|ψ|2/2. (11)

Let us now assume that there exists a patch of approxi-mately constant polariton field ψ0 (or a condensate seed)at t = t1. We can describe small fluctuations around ψ0

by

ψ =(ψ0 + u(t)eikx + v∗(t)e−ikx

)e−iµt+λ(t−t1) , (12)

where µ and λ are the chemical potential and the growthrate of the patch, and u, v represent the fluctuations. Ex-panding (11) in Taylor series around ψ0, we can rewrite(10) as

i~ψ =

[− ~2∇2

2m+ i

~γ0

2

(1− |ψ|

2

nsat

)+ geff |ψ|2 + µa

]ψ ,

(13)with the accuracy of the order of O(|ψ|4). Heregeff = (α1 + α2)(1 − g1rP/2Γ2

A)/2, γ0 = (rP/ΓA)(1 +r|ψ0|2/2ΓA)− 2Γ, and nsat = 2γ0Γ2

A/(r2P ) where ΓA =

ΓR + r|ψ0|2/2 is the effective exciton decay rate. Theabove equation has the same form as the one derived inRef. [54], apart from the negligible dependence of geff onP . The Bogoliubov-de Gennes modes are u, v∗ ∼ e−iω±t,and the mode frequencies

ω±γ0

= −iα2± i√(α

2

)2

+ (αβ)2 − (εk + αβ)2, (14)

where εk = ~k2/2mγ0, the saturation parameter α =|ψ0|2/nsat, and β = 2geffΓ2

A/~r2P . This spectrum hasthe property that modes with high momenta are stronglydamped (the imaginary part of the frequency is nega-tive), in contrast to the initial linear dynamics where nok-dependence of the imaginary part of the spectrum waspresent. We can define a characteristic momentum cutoffκ for the modes that are strongly damped, which scales

with the parameters as κ ∼ γ1/20 . Any fluctuations with

momenta higher than κ will be suppressed, while fluctu-ations with lower momentum can form regions of orderedcondensate phase. To estimate the scaling of the numberof defects, two different limiting cases can be considered.In the first case, we assume that the vortices are formed

when the condensate density is already near to its equi-librium value |ψ0|2 = (P − Pth) /Γ. Here we obtain

Nd ∼ κ2 ∼ γ0 ∼ Pth

(1− Pth

P

). (15)

Note that the above scaling does not have a power-lawform, which is due to the fact that the transition is effec-tively nonlinear.

In the opposite limit,54 we assume that the vortices areformed when the condensate density is still very small|ψ0|2 = 0. In this case we obtain

Nd ∼ κ2 ∼ γ0 ∼P

Pth− 1. (16)

The two estimates [Eqs. (15) and (16)] are compared tothe numerical results in Fig. 8(b). The numerically ob-tained scaling appears to be intermediate between thetwo extreme cases.

V. CONCLUSIONS

In this paper we presented a comprehensive theoreticalstudy of non-equilibrium dynamics of polariton conden-sates in incoherently pumped semiconductor microcav-ities. We have anticipated two different destabilizationmechanisms that govern nonlinear dynamics of this sys-tem. The first arises when the polariton condensate has astrong feedback effect on the reservoir of incoherent “hot”polaritons. The second one is associated with the para-metric scattering in the presence of polarization splittingin a spinor condensate. Both mechanisms result in theformation of phase defects, i.e. vortices, triggered by themodulational instability of the homogeneous condensate.

In the scalar case, we have shown that the presenceof the incoherent reservoir can affect substantially boththe vortex stability and their mutual collective dynamics.In particular this can lead to the formation of rotatingdislocations or delocalized spiraling waves.

In the spinor two-component condensate we have iden-tified the presence of topological defects, which take theform of half-vortex pairs, giving an “eyelet” structure inintensity and dipole type structure in the spin polariza-tion.

In the case when the phase defects are formed in thedynamically (modulationally) stable regime, as a resultof the condensate formation from an initial spatially andphase-incoherent state, we find that the defect densityscales with the pumping rate in analogy to the Kibble-Zurek type scaling for non-equilibrium phase transitions.

ACKNOWLEDGEMENTS

T.C.H.L. acknowledges support from the Lee KuanYew Fellowship. O.A.E. acknowledges financial supportby the Deutsche Forschungsgemeinschaft (DFG project

11

EG344/2-1) and the Thuringian Ministry for Education,Science and Culture (TMESC project B514-11027).M.M. acknowledges support from the National ScienceCenter grant DEC-2011/01/D/ST3/00482. E.A.O. ac-knowledges funding by the Australian Research Council(ARC).

Appendix A: Linear Stability Analysis

The stability of the solutions can be checked using thestandard approach of applying perturbations, in the formψ± 7→ e−iµt (ψ± + δψ±), nR 7→ nR + δnR. Substitutioninto Eqs. (1)-(3) and collecting terms linear in the smallamplitudes δψ± and δnR we have:

i~dδψ+

dt= c1δψ+ +

(∆XY + α2ψ

∗−ψ+

)δψ− + α1ψ

2+δψ

∗+

+ α2ψ−ψ+δψ∗− + (g1 + i~r)ψ+δnR, (A1)

i~dδψ−dt

= c2δψ− +(∆XY + α2ψ

∗+ψ−

)δψ+ + α1ψ

2−δψ

∗−

+ α2ψ+ψ−δψ∗+ + g2ψ−δnR, (A2)

dδnRdt

= −(ΓR + r|ψ+|2

)δnR − 2rnR<e

{ψ∗+δψ+

},

(A3)where we have defined:

c1 = g1nR + i~ (rnR − Γ) + 2α1|ψ+|2 + α2|ψ−|2, (A4)

c2 = g2nR − i~Γ + 2α1|ψ−|2 + α2|ψ+|2. (A5)

We consider perturbations in the polariton and reser-voir fields of the form δψ± = u±e

i(kx−ωt)+v∗±e−i(kx−iω∗t)

and δnR = w(ei(kx−ωt) + e−i(kx−ω

∗t)), respectively.

Here u± and v± are complex amplitudes, while w is a realamplitude. ω is a complex eigenvalue to be determined.Stability of the original solutions occurs if =m {ω} < 0such that the perturbation decays.

Substitution of δψ± and δnR into Eqs. (A1)-(A3) andcollection of terms oscillating as e−iωt and e−iω

∗t yieldsa set of five coupled equations, which represent an eigen-value problem for ω. In matrix form:

c′1 − ~ω α1ψ

2+ ∆XY + α2ψ

∗−ψ+ α2ψ−ψ+ (g1 + i~r)ψ+

−α1ψ∗2+ −c′∗1 − ~ω −α2ψ∗−ψ

∗+ −∆XY − α2ψ−ψ

∗+ −(g1 − i~r)ψ+

∆XY + α2ψ∗+ψ− α2ψ−ψ+ c′2 − ~ω α1ψ

2− g2ψ−

−α2ψ∗−ψ∗+ −∆XY +−α2ψ+ψ

∗− −α1ψ

∗2− −c′∗2 − ~ω −g2ψ

∗−

−i~rnRψ∗+ −i~rnRψ+ 0 0 −i~(ΓR + r|ψ+|2)− ~ω

u+

v+

u−v−w

= 0, (A6)

where c′1 = c1 + ~2k2

2m − ~µ and c′2 = c2 + ~2k2

2m − ~µ.The imaginary and real parts of the perturbation spec-

tra, E = ~ω, are shown in Fig. 9(a,b). The energies ofthe stationary states ~µ are illustrated by thin horizontal

lines in Fig. 9(a). Where the imaginary parts are posi-tive, thick curves denote modulational instability at thegiven wavevectors k. The stationary state labelled U2 isunstable to fluctuations with k = 0.

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