Topological edge states of nonequilibrium polaritons in hollow honeycomb arrays
Xuekai Ma, Yaroslav V. Kartashov, Albert Ferrando, Stefan Schumacher
LLetter Optics Letters 1
Topological edge states of nonequilibrium polaritons inhollow honeycomb arrays X UEKAI M A , Y AROSLAV
V. K
ARTASHOV , A
LBERT F ERRANDO , AND S TEFAN S CHUMACHER Department of Physics and Center for Optoelectronics and Photonics Paderborn (CeOPP), Universität Paderborn, Warburger Strasse 100, 33098 Paderborn,Germany ICFO-Institut de Ciències Fotòniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain Institute of Spectroscopy, Russian Academy of Sciences, Troitsk, Moscow, 108840, Russia Russian Quantum Center, Skolkovo 143025, Russia Interdisciplinary Modeling Group, Departament d’Òptica, Universitat de València, Doctor Moliner 50, E-46100 Burjassot (València), Spain College of Optical Sciences, University of Arizona, Tucson, AZ 85721, USA * Corresponding author: [email protected] September 18, 2020
We address topological currents in polariton conden-sates excited by uniform resonant pumps in finite hon-eycomb arrays of microcavity pillars with a hole in thecenter. Such currents arise under combined action ofthe spin-orbit coupling and the Zeeman splitting thatbreak the time-reversal symmetry and open a topologi-cal gap in the spectrum of the structure. The most repre-sentative feature of this structure is the presence of twointerfaces, inner and outer ones, where the directions oftopological currents are opposite. Due to the finite sizeof the structure polariton-polariton interactions lead tothe coupling of the edge states at the inner and outerinterfaces, which depends on the size of the hollow re-gion. Moreover, switching between currents can be re-alized by tuning the pump frequency. We illustrate thatcurrents in this finite structure can be stable and studybistability effects arising due to the resonant characterof the pump. © 2020 Optical Society of America http://dx.doi.org/10.1364/ao.XX.XXXXXX
The attention to new topological phases of matter has growndramatically during the last decade, fueled mainly by potentialapplications of topologically protected edge states, which existin such materials and demonstrate unusual propagation proper-ties, resistant to disorder and inhomogeneities, and the abilityto traverse even sharp material bends and corners. Discoveredfirst in electronic systems [1, 2], topological insulators were pre-dicted and successfully demonstrated in diverse areas of science,including photonic and optoelectronic systems [3, 4]. Multi-ple approaches to realization of photonic topological phasesare known, some of which make use of gyromagnetic photoniccrystals [5, 6], coupled microresonator arrays [7, 8], modulatedFloquet [9, 10] and many other structures. While in hybrid po-laritonic systems technologically fabricated micropillar arrays[11, 12] or lattices induced by acoustic waves [13] can supporttopological currents, the predictions recently culminated in ob- servation of polariton topological insulators in two- [14] andone-dimensional [15] geometries.Polariton topological insulators are intrinsically nonlinear sys-tems, where polariton-polariton interactions play an importantrole in the condensation process. This perfectly fits into moderntrends in topological photonics that now turns toward investiga-tion of nonlinear effects, see a recent review [16]. Nonlinearitybrings a number of new effects in topological systems, amongwhich are the formation of topologically protected solitons [17–21], predicted for polaritons in [22–24], nonlinearity-inducedinversion of topological currents [25], topological phases in-duced by vortex lattices [26], coupling between corner modesin higher-order polariton insulators [27], as well as bistability[28, 29], all of which considerably extend the tools for the control
Fig. 1.
Eigenstates of a hollow honeycomb array. (a) Eigenfre-quencies ω n of linear modes versus mode index n . The grayarea represents the band gap of the bulk array, where only theedge states appear. (b) Hollow honeycomb array. (c-e) | Ψ − | distributions corresponding to points of different color in (a).White arrows indicate the propagation direction of polaritons. a r X i v : . [ phy s i c s . op ti c s ] S e p etter Optics Letters 2 Fig. 2.
Edge states in the nonlinear regime. (a) Dependence ofthe maximum density ( ρ max in µ m − ) of Ψ − on the frequencyof the pump ( ω in THz) for different pump amplitudes: E ± = Ψ − at different pump amplitudes and fre-quencies. The color markers shown in (b-e) correspond to dotsin (a). Here, γ c = − .of topologically protected currents. One of the most convenientplatforms for the investigation of nonlinear effects in polari-ton condensates is offered by the arrays of microcavity pillars[30, 31], which can be used for the creation of insulators of vari-ous symmetries, from conventional honeycomb [14, 32], to Lieb[24, 33], and kagome [23, 34] ones.In this Letter we investigate the impact of nonlinearity on theproperties of topological edge currents in polariton condensatesin hollow honeycomb arrays, which simultaneously feature twointerfaces - inner and outer ones. We show that a resonant uni-form pump allows selective excitation of the topological currentseither at one or at both interfaces, and that nonlinearity may cou-ple two states at the opposite interfaces due to their proximity,thereby offering the control over shape and determining stabilityof the currents.The dynamics of nonequilibrium polariton condensates inthe array of microresonator pillars under the resonant pump canbe described by the equation [31]: i ¯ h ∂ Ψ ± ∂ t = (cid:34) − ¯ h m (cid:79) ⊥ − i ¯ h γ c ± Ω + g c | Ψ ± | + V ( r ) (cid:35) Ψ ± + β (cid:0) i ∂ x ± ∂ y (cid:1) Ψ ∓ + E ± ( t ) . (1) Here, the indices ± denote the right-/left-circular polarizationcomponents, m = − × m e ( m e is the free electron mass) is theeffective polariton mass, γ c is the polariton loss, Ω = Ψ − component dominates and the Ψ + com-ponent is substantially weaker), g c = µ eV · µ m is the polaritoninteraction strength that affects the shapes of both polarizationcomponents, but not the coupling of them, β = · µ m represents the strength of TE-TM splitting (leading to spin-orbitcoupling) intrinsically present in microcavities, E ± represent thecoherent pump, and V ( r ) = ∑ m,n V ( x − x m , y − y n ) is the po-tential energy landscape created by the array of microresonatorpillars, where V = V e − [( x − x m ) +( y − y n ) ] / d describes contri-bution from individual pillars with the diameter 2 d = µ m, Fig. 3.
Switching between the two edge states. (a) Time evo-lution of the peak density ( ρ max in µ m − ) of Ψ − at E ± = ω = − t < ω = − t ≥ Ψ − at different timemoments, corresponding to the black points in (a) from left toright, respectively. Here, γ c = − .depth V = − µ m. The pillars arearranged into a finite honeycomb array with a hole in the center,as illustrated in Fig. 1(b). The hollow array has two edges, theouter and the inner ones, each with hexagonal shape, so that thewidth of the ribbon between two edges remains constant. In thesimulations, the periodic boundary condition is applied.To find the edge states supported by this structure, we firstanalyse the linear, loss-free ( γ c = E ± =
0) Eq. (1) by applying the ansatz Ψ ± ( r , t ) = u ± ( r ) e − i ω t . Theeigenmodes can be obtained by solving the eigenvalue problem:¯ h ω u ± = − ¯ h m (cid:79) ⊥ u ± ± Ω u ± + Vu ± + β (cid:0) i ∂ x ± ∂ y (cid:1) u ∓ . (2) A part of the dependence of the eigenfrequency ω n of the modeson the mode index ( n ) in this structure relevant for edge states ispresented in Fig. 1(a), where we sorted modes by increasing ω n .The examples of linear edge states at the outer and inner edgesare presented in Figs. 1(c) and 1(d), respectively. Remarkably,the polaritons excited at the outer edge propagate clockwise,while the polaritons excited at the inner edge propagate counter-clockwise. This is the case for all edge states found within thefrequency range corresponding to forbidden topological gapof the infinite array [shaded region in Fig. 1(a)]. The states atthe outer and inner edges alternate (but in an irregular fashion)in the ω n ( n ) dependence, see example in Fig. 1(e). The statesoutside shaded region correspond to bulk modes.The excitation of the edge states can be realized using reso-nant pumps. Here, we consider the plane-wave pump E ± = E ± e − i ω t , whose frequency ω can drive the solution to the desiredstate. In this case, the stationary edge states in the nonlinear etter Optics Letters 3 Fig. 4.
Nonlinear edge states at larger loss rate γ c = ps − . (a) Dependence of the peak density ( ρ max in µ m − ) of Ψ − on thepump frequency ( ω in THz) in (b) honeycomb array with larger hole at E ± = Ψ − on ω in (h) honeycomb array with smaller hole at E ± = (cid:34) − ¯ h m (cid:79) ⊥ − i ¯ h γ c ± Ω + g c | u ± | + V ( r ) (cid:35) u ± + β (cid:0) i ∂ x ± ∂ y (cid:1) u ∓ + E ± − ¯ h ω u ± = (3) We first assume that the system approaches the conserva-tive regime with a very long polariton lifetime (1000 ps, i.e., γ c = − ), and the pump is linearly polarized with E + = E − . Figure 2(a) shows the dependence of the peak densityof Ψ − on the frequency and the amplitude of the pump. Tocheck the stability of the edge states, we perturbed the obtainedsolutions by adding complex (amplitude and phase) broad-band noise and let them evolve over long times. The solid(dashed) lines represent the stable (unstable) solutions. Twodistinguished resonances can be seen in the frequency domain ω = − ∼ − γ c is thus required to clearly distinguish tworesonances. When the pump amplitude is small [see the blueline in Fig. 2(a)], the two peaks are almost independent with theleft resonance corresponding to strongly excited outer edge andpractically unexcited inner edge [Fig. 2(b)], while in the rightresonance the situation is inverted, and polaritons concentrateexclusively on the inner edge as shown in Fig. 2(c). Increasingthe pump amplitude strengthens the coupling of the two edgestates [see the red lines in Fig. 2(a)], since corresponding reso-nances broaden and start to partially overlap. As a result, in Fig.2(d) one can observe a mixed state containing comparable contri-butions from both inner and outer edges of the structure. Withfurther increase in frequency [Fig. 2(e)] the nonlinear edge statelocalizes at the inner edge. Thus, even at such large lifetimes,nonlinearity substantially affects the structure and location ofthe edge states in the hollow honeycomb structure.Due to the nonequilibrium nature of polaritons, one can re-alize switching between two edge states by suddenly varyingthe frequency of the pump, as shown in Fig. 3. When t < ω = − t = ω = − ∼
10 ns, it is just 10 times longer than the polaritonlifetime. This ratio takes place also for much smaller polaritonlifetimes, where transitions to a new steady state after changingthe parameters of the structure occurs much faster, an exampleof which is presented in Fig. 5. This mechanism of switching isvery robust in comparison with previously suggested schemesbased on array modulations [35].For larger polariton loss rates the resonance peaks in thedependence on frequency ω substantially broaden as shownin Fig. 4(a), where we show polariton densities at the differ-ent edges separately to distinguish the two peaks. The thick(thin) lines represent the maximum density at the outer (inner)edge. In this case, the coupling of the two edge states becomesmuch stronger [Figs. 4(c,d)], and solutions become more stable.Increased coupling between the two edge states leads to theircoexistence, especially when the outer edge is predominantlyoccupied [Fig. 4(c)]. To decouple the two edge states and toincrease the separation between corresponding resonances tomake them more distinguishable, one can reduce the size of thecentral hole, increasing the width of the ribbon between innerand outer edges. This is done in Fig. 4(h) where the outer edgeis unchanged but the inner one is substantially reduced. As aconsequence, the peak density of the state on the inner edgeincreases, nonlinearity results in more pronounced shift of thetip of corresponding resonance curve to the right, so that theseparation between two resonances increases, as shown in Fig.4(g). In Figs. 4(i,j) the contrast between states residing at twodifferent edges becomes higher. For both structures, due to non-linear tilt of right resonance in Figs. 4(a,g), in the tip of thisresonance the state on the inner edge clearly dominates [Figs.4(e,k)]. At the right side of the right peaks in Figs. 4(a,g) (seethe cyan markers) the solutions in Figs. 4(f,l) show that morepolaritons lie between the two edges, although the edges aremainly occupied. Corresponding frequency values are close tothe border of the topological gap, hence corresponding statesdeeply penetrate into the structure.The influence of the polarization of the pump on the edge etter Optics Letters 4 Fig. 5.
Defect-induced coupling of two edge states. (a) Timeevolution of the peak density ( ρ max in µ m − ) of Ψ − at theouter edge (green line), inner edge (blue line), and in the bulk(red line). The potential at t <
500 ps is shown in Fig. 4(h) andthe potential with a missing pillar, indicated by the blue arrow,is shown in (b). (c-e) Density profiles at different time scales in(a), corresponding to the black points from left to right, respec-tively. The white arrows indicate the polariton current at theouter edge. The density profile at t <
500 ps is shown in Fig.4(i). Here, γ c = − .states was studied too. Switching the pump from linearly polar-ized ( E + = E − ) to circularly polarized ( E + = E − = E + = E − = t <
500 ps, before the pillar is removed, areshown in Fig. 5(c). The disappearance of the selected pillar at t >
500 ps does not lead to scattering into bulk [see the red linein Fig. 5(a) showing amplitude in the bulk] or counter-clockwisepolariton propagation [see the white arrows in Figs. 5(c-e)]. Re-moval of the pillar practically does not affect the amplitude atthe inner edge, even though the amplitude on the outer edgeslightly reduces after some time [Fig. 5(a)].In summary, we have demonstrated that hollow honeycombarrays support topological polariton currents at the outer andinner edges that are coupled by nonlinearity and that can beswitched on and off in a controllable fashion by varying thefrequency of the pump in this dissipative system.
Funding.
Deutsche Forschungsgemeinschaft (DFG) (No.231447078, 270619725); Paderborn Center for Parallel Comput-ing, PC ; Russian Science Foundation (Project 17-12-01413- Π );Spanish MINECO through Project No. TEC2017-86102-C2-1-R. Disclosures.
The authors declare no conflicts of interest.
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