Dark polariton-solitons in semiconductor microcavities
aa r X i v : . [ phy s i c s . op ti c s ] O c t Dark polariton-solitons in semiconductor microcavities
A.V. Yulin , , O.A. Egorov , , F. Lederer , and D.V. Skryabin Centre for Photonics and Photonic Materials, Department of Physics,University of Bath, Bath BA2 7AY, United Kingdom Department of Engineering Mathematics, University of Bristol, Bristol, BS8 1TR, United Kingdom Institute of Condensed Matter Theory and Solid State Optics,Friedrich Schiller University Jena, Max-Wien-Platz 1, 07743 Jena, Germany (Dated: October 31, 2018)We report the existence, symmetry breaking and other instabilities of dark polariton-solitons insemiconductor microcavities operating in the strong coupling regime. These half-light half-mattersolitons are potential candidates for applications in all-optical signal processing. Their excitationtime and required pump powers are a few orders of magnitude less than those of their weakly coupledlight-only counterparts.
Polaritons are mixed states of photons and material ex-citations and are well-known to exist in many condensedmatter, atomic and optical systems [1, 2, 3, 4, 5]. We aredealing below with a semiconductor microcavity, wherepolaritons exist due to mixing of quantum well excitonsand resonant microcavity photons [3, 4, 5]. In the strongcoupling regime photons, emitted as a result of electrontransitions, excite the medium and are re-emitted in acascaded manner, which gives rise to so-called Rabi oscil-lations [1, 3, 4]. This phenomenon results in the two peakstructure of the microcavity absorption spectrum. Themeasured spectral width of the peaks corresponds to thepicosecond polariton life time [3]. This is in contrast withthe more usual weak-coupling regime (typical for opera-tion of vertical cavity surface emitting lasers (VCSELs)[3]), where the slow (nanosecond) carrier dynamics doesnot catch up with the fast (picosecond) photon decay.Thereby most of the photons leave the cavity as soon asthey are emitted. In this regime the response to a pulse,resonating with a cavity mode, results in a single spectralpeak. Thus any potential application of microcavity po-laritons in optical information processing leads to a 2-3orders of magnitude response time reduction relative tothe VCSEL-like operating regimes.One of the topics of the recent research into the weaklycoupled semiconductor microcavities has been the lo-calised structures of light or cavity solitons [6, 7, 8, 9,10, 11, 12], which have demonstrated rich physics andhave been proposed for information processing applica-tions [7, 8]. In the weak coupling regime formation ofpolaritons is irrelevant, since the dispersion of linear ex-citations is purely photonic. Slowness of the light-onlycavity solitons is an outstanding problem, which can berectified in the strong-coupling regime, where potentiallymuch faster, but not yet reported, light-matter solitonsare expected.In the last few years extensive studies of the polari-tons in strongly coupled microcavities have been stronglymotivated by the smallness of the polariton mass lead-ing to observation of the polariton Bose-Einstein con-densation at few Kelvin temperatures [13, 14]. Polari-tons have also been recently observed even at the roomtemperatures, see, e.g., [15], which has further boosted their potential for practical applications. Another veryimportant feature of polaritons in semiconductor micro-cavities is their strong repulsive interaction (two-bodyscattering) resulting in a substantial defocusing nonlin-earity [2, 3]. Amongst nonlinear effects predicted orobserved with microcavity polaritons are optical bista-bility [16, 17, 18, 19, 20] and parametric conversion[18, 21, 22, 23, 24]. Observation of these effects withpolaritons requires pump intensities of ∼ / cm orbelow (see, e.g., Fig. 1 in [17]), which is less thanthe typical pump of 10kW / cm required for semiconduc-tor microcavities operating in the weak-coupling regime[6, 7, 8, 9, 10, 11] (see, e.g., Fig. 5 in [9]).Solitonic effects with polaritons in bulk media have at-tracted a significant (mostly theoretical) attention since70s till now, see, e.g., Ref. [2, 25]. In the latest wave ofresearch on exciton-polaritons in strongly coupled micro-cavities the solitonic effects have not been much of a focusyet, with an important exception of a recent experimen-tal paper [26]. In this work the authors claim observationof dark and bright localized structures or cavity solitonsin a strongly coupled semiconductor microcavity. Someother papers have reported localisation of microcavitypolaritons due to linear defects [14, 27], as a result ofswitching between two polarizations [28], or neglectingsuch important requisites of passive cavities as losses, ex-ternal pump and hence bistability [29]. For studies of spa-tially dependent polariton dynamics, see, e.g., [30]. Ourwork is aimed at filling an existing gap in the theoreticalknowledge about microcavity polariton-solitons. This isnecessary not only for backing so far limited experimen-tal observations [26], but also and mainly for guiding thefuture work in this direction.The widely accepted dimensionless mean-field modelfor excitons strongly coupled to the circularly polarizedcavity photons is [2, 3, 21] ∂ t E − i ( ∂ x + ∂ y ) E + ( γ c − i ∆) E = E p + i Ψ ,∂ t Ψ + ( γ − i ∆ + i | Ψ | )Ψ = iE. (1)Here E and Ψ are the averages of the photon and ex-citon creation or annihilation operators. Normalizationis such that (Ω R /g ) | E | and (Ω R /g ) | Ψ | are the photon -5 -2.5 0 2.5 5 k x [ m m -1 ] -3-1.501.534.5 R e D– [ m e V ] LPUP
FIG. 1: Polariton dispersion calculated from Eq. (2), Re ∆ ± ( k x ), and renormalized back into physical units. Pa-rameters are n = 3 .
5, operating wavelength λ = 0 . µ m, γ c, = 0 . ~ Ω R = 2 . and exciton numbers per unit area. Here, Ω R is the Rabifrequency and g is the exciton-exciton interaction con-stant. ∆ = ( ω − ω r ) / Ω R describes detuning of the pumpfrequency ω from the identical resonance frequencies ofexcitons and cavity, ω r . Time t is measured in units of1 / Ω R . γ c and γ are the cavity and exciton dampingconstants normalized to Ω R . Transverse coordinates x , y are normalized to the value x = p c/ kn Ω R where c is the vacuum light velocity, n is the refractive indexand k = nω/c is the wavenumber. The normalized am-plitude of the external pump E p is related to the phys-ical incident intensity I inc as | E p | = gγ c I inc / ~ ω Ω R [31]. As a guideline for realistic estimates one can use pa-rameters for a microcavity with a single InGaAs/GaAsquantum well: ~ Ω R ≃ . meV , ~ g ≃ − eV µm , see[17, 21, 31]. Assuming the relaxation times of the pho-tonic and excitonic fields to be 2 . ps gives γ c, ≃ . | E p | = 1 physically correspondsto the external pump intensity ∼ kW/cm . Opticalbistability appears for | E p | ∼ .
1, it gives the input inten-sity ∼ W/cm . Experimentally the polariton bista-bility has been observed for values close or even less than100 W/cm [16, 17, 18, 19, 20].First we briefly summarize important aspects of thelinear dispersion and bistability properties of the aboveequations. Assuming that E, Ψ ∼ e ik x x + ik y y and neglect-ing pump and nonlinearity we find the dispersion law ofcavity polaritons∆ ± = k − i ( γ c + γ )2 ± r k + i ( γ − γ c )) , (2)where k = k x + k y . Re ∆ + corresponds to the fre-quency of the upper polariton (U-polariton) and Re ∆ − to the lower polariton (L-polariton) branch, see Fig. 1.In the strong coupling regime the gap between U- andL-polaritons is greater than the linewidth of the branchdue to Im ∆ ± = 0.If E p = 0, then solitons can exist only on a finite am- plitude background ( E ( ±∞ ) = 0), simply because thezero homogeneous solution is absent. Therefore we pro-ceed with a brief consideration of spatially homogeneoussolutions (HSs) and their stability. Then we report theexistence of various cavity polariton solitons (CPSs) andstudy their stability and instability scenarios. HS havingbistable dependence from E p is an important prerequisitefor the soliton existence. E ( E p ) is multivalued providedthat f (∆) >
0, where f (∆) ≡ ∆(∆ + γ c − − √ γ (∆ + γ c + γ c γ ) . (3)The cumbersome expressions for the roots of f (∆) = 0simplify for γ c = γ = 0 and give two bistability inter-vals ∆ > − < ∆ <
0. These two intervals overlapwith the ∆ intervals allowed by the dispersion relation,see Eq. (2) and Fig. 1. The bistability in the interval − < ∆ < >
0, see, e.g. [12]. Below wefocus our attention on the solitons linked to L-polaritons,therefore our studies are unique to the strong couplingregime. Stability analysis of the HS L-polaritons (∆ < E p ,while the upper state is generally stable, see Figs. 2(a)and 3(a). Here, modulational instability (MI) we under-stand as the growth of linear perturbations in the form e ik x x + ik y y + κt ( Reκ is the growth rate). As ∆ is chang-ing from the bottom of the L-polariton branch towardsthe linear exciton resonance, ∆ = 0, the point of MI ismoving towards the left edge of the bistability loop andfinally goes beyond the latter, cf. Figs. 2(a) and 3(a).Restricting ourselves to the structures independent onthe polar angle ( θ = arg ( x + iy )) we find that the time-independent CPSs obey − i (cid:18) d Edr + 1 r dEdr (cid:19) + ( γ c − i ∆) E = E p + i Ψ (4)where r = p x + y , Ψ = iE/ [( γ − i ∆) + iz ] and z ≡ | Ψ | is found solving the real cubic equation ( γ +( z − ∆) ) z = | E | . z turns out to be a single valuedfunction of | E | throughout the range of parameters cor-responding to the bistability of L-polaritons. Thus thepotential problem of ambiguity in choosing a root for z is avoided.We start our analysis of cavity polariton solitons fromthe case, when the MI point of low state L-polaritonsis within the bistability interval. In many previouslystudied models bifurcation points of the homogeneoussolutions have been the sites where localized structuresbranch off [12]. Applying the Newton iterative methodto Eq. (4) we have found a family of small amplitude E p |Y| HSBrightDark MI (a) (b) -50 -25 0 25 50 x
343 4 -100 -50 0 50 100 x |Y| (c) (d) FIG. 2: (a) Amplitude of the homogeneous state (HS)(black line), max | Ψ( x, y ) | for bright solitons (blue line) andmin | Ψ( x, y ) | for dark solitons (red line) shown as functionsof E p : ∆ = − . γ ,c = 0 .
1. (b) is the zoom of the rectan-gular area from (a) showing bifurcations of the dark solitons.(c,d) Exciton density distribution | Ψ( x, y = 0) | across thebright (c) and dark (d) solitons for the points marked by 1,2, 3 and 4. Full and dashed lines in (a)-(d) mark stable andunstable solutions, respectively. bright CPSs emerging from the MI point, see the dashedred line in Fig. 2(a). Going towards smaller values of E p , the CPSs become more intense, see Fig. 2(c). The E p value, at which the lower and upper homogeneousstates can be connected by a standing 1D front, is calledMaxwell point and this is the point where the branch ofthe bright CPSs terminates ( E p = 0 . ǫ + ( r ) e iJθ + κt + ǫ ∗− ( r ) e − iJθ + κ ∗ t , where J = 0 , , , . . . [32].The resulting Jacobian operator is analysed using finitedifferences in r . The linear stability analysis shows thatthe bright CPSs are unstable with respect to the per-turbation with the azimuthal index J = 0 and that thedevelopment of the instability splits the CPS into 2Dmoving fronts. When E p is close to the Maxwell pointthis instability is relatively weak and bright CPSs canbe easily stabilized by the spatial inhomogeneities of thepump or cavity detuning. This problem deserves moredetailed investigation and it will be analyzed elsewhere.Because of the defocusing nature of the polaritonicnonlinearity dark CPSs, see, e.g., [12], are expected tobe naturally selected by our system and the instability ofbright CPSs is not surprising. Dark cavity solitons, havebeen previously studied both theoretically and experi-mentally for semiconductor microcavities in the weak-coupling regime, see, e.g. [9, 33, 34]. Unlike fiber solitons,the dark cavity solitons have no conceptual disadvantage E p |Y| (a) (b) -50 0 50 x -50 -25 0 25 50 x |Y| (c) (d) B2B2
43 56
Hopf y
50 100 x y
50 100 x
50 100 x (e)(f) t=0 t=1000 t=7000 t=0 t=5000 t=9000 FIG. 3: a) Amplitude of the homogeneous state (HS) (blackline) and min | Ψ( x, y ) | for dark solitons (red and blue lines)shown as functions of E p : ∆ = − . γ ,c = 0 . B B | Ψ( x, y = 0) | across B1 (c) and B2 (d) CPSs for the points marked by 1,2, 3 and 4 in panels (a) and (b). Full and dashed lines in(a)-(d) mark stable and unstable solutions, respectively. (e,f)show development of the symmetry breaking instabilities ofthe CPSs marked as 5 and 6 in (b). over the bright ones as information carriers. The branchof dark CPSs have been found to detach from the leftfolding point of the bistability loop and tend towards theMaxwell point, see Fig. 2(a) and the zoomed area in (b).At the onset of their existence the dark solitons are seenonly as a very deviation from the homogeneous back-ground. As E p tends towards the Maxwell point fromthe right, they become much dipper. Near the Maxwellpoint the dark solitons become very broad and can beroughly considered as superpositions of infinitely sepa-rated 1D fronts (1 /r term in Eq. (4) can be disregardedfor large distances and the equation becomes effectively1D). It is important to note that the relaxation of thefronts towards the upper state happens without oscilla-tions, however the relaxation towards the lower state isoscillatory, see Fig. 2(c). Thus pinning of the two frontsand hence stabilization of CPSs is possible only for thedark structures (see the thick line in Fig. 3(c)). Thestable branches of dark CPSs are shown by full lines inFig. 2(b). The unstable ones correspond to the instabil-ities with J = 0.In the case when the lower branch HS is unstablewithin the whole range of the bistability (∆ = − . J = 0) CPSs become stable after the turning point.Close to this turning point the B1 CPSs have a deeplike shape, while later they transform into dark rings ofgrowing radius Fig. 3(c). Note, that close to the turn-ing points additional destabilization of dark CPS hap-pens due to linear eigenmodes with complex κ and J = 0(Hopf instability, see Fig. 3(a)) resulting in the formationof oscillating dark CPSs.The branch B2 consists of ring shaped structures, seeFig. 3(d). The linear stability analysis shows that the B2 CPSs can be stable (see the interval marked by 4in Fig. 3(b)). However, more often, they are unstablewith respect to perturbations breaking the radial sym-metry, i.e. with J = 0. An example of this instabilitydevelopment is shown in Figs. 3(e,f). The dark ring CPSshown in Fig. 3(e) is unstable against linear eigenmodewith J = 3. The broader CPSs undergo azimuthal insta-bilities with larger azimuthal numbers J . For example, J = 8 for the concentric ring CPS shown in Fig. 3(f).In summary: Following a series of recent experimentson observation of microcavity polaritons, we have stud-ied the formation of spatially localised polariton-solitonstructures in the strong coupling regime. In particular,our results can be used for the interpretation of the ex-perimental measurements reported in [26], where the po-lariton Rabi splitting has been observed simultaneouslywith the formation of various bright and dark localisedstructures. Microcavity polariton solitons reported hereexhibit a picosecond excitation time and can be observedat pump powers few orders of magnitude lower than thoserequired in the weak coupling regime of the semiconduc-tor microcavities [6, 7, 8, 9, 10, 11]. Thus the light-matterpolariton solitons have potentially significant advantagesin all-optical signal processing applications over the light-only cavity solitons [7, 11, 12]. [1] Confined Electrons and Photons: New Physics and Appli-cations , edited by E. Burstein and C. Weisbuch, (Plenum,New York, 1995).[2] S.A. Moskalenko and D.W. Snoke,
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