From the Topological Spin-Hall Effect to the Non-Hermitian Skin Effect in An

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Intensity 25 31 , 30 σ .Hwvr achiev- However, ]. ± .Sc neﬀect an Such ]. aeueulde- unequal have ) Sites σ ± 33 ) – 29 41 .Deto Due ]. n re- and ] 43 – 46 pil- ars ], s 2 where in some cases the nonlinearity alone induces topo- (a) J (b) logical behavior [47–50]. The inherent non-Hermitian nature of the polaritons has lead to the realization of excep- J tional points [51, 52], non-Hermitian topological phases [52, 53], and other related theoretical predictions [54– y 56]. The spin-orbit coupling due to the TE-TM split-

y Finite x ting inside the microcavity has played a vital role in ob- Periodic taining nontrivial spin related eﬀects, such as the optical Finite x spin Hall eﬀect [57–59], meron polarization textures [65], Periodic nontrivial bands with spin orbit interaction Hamiltonians (c) (d) [60–67]. 2 In this work, we propose a system based on an exciton- J 1 polariton elliptical micropillar chain, where both the TSHE and NHSE can be realized in a single set up (see 0

Fig. 1). By proper orientation of the otherwise identical Energy/ 1 elliptical micropillars along with the naturally present polarization splitting due to the shape anisotropy give 2 rise to a band structure, where the positive and negative momentum states have opposite circular polariza- 0 0 tions (σ±). As a result σ± polaritons propagate in the opposite directions in the chain giving rise to the TSHE. If the effective decay rates of the σ± polaritons are made FIG. 2: (a) The unit cell of a square lattice with real couplings unequal by subjecting the chain to a circularly polarized along the x direction and complex couplings along the y di- spatially uniform incoherent pump, the non-Hermitian rection such that each plaquette has total accumulated phase physics takes over. The behavior of the system changes φ = 2π/3. (b)The same lattice as in (a), but having only two sites along the y direction. (c-d) The band structure of the drastically and the modes of the system get localized at system presented in (a-b), respectively. The bulk bands are one end of the chain, giving rise to the NHSE [23], where shown in grey, whereas red and green being the edge states σ± polaritons propagate in the same direction, while the located at the opposite edges. Parameters: ∆/J = 1. propagation in the other direction is strongly suppressed. The key advantage of our scheme is that here both the topological Hermitian and non-Hermitian phases can be Hofstadter butterfly if the eigenenergies of the system studied on the same sample without the need of spatial are plotted against φ [17, 68], where for φ = 2π/q, each patterning of the pump. Besides the robust polariton Bloch band splits into q sub-bands. In Fig. 2(a) the unit propagation here is in a one dimensional lattice, which is cell of the system, which is periodic along the x direc- more compact and ideal for potential information trans- tion, is shown for φ = 2π/3. The band structure of the port applications. system is shown in Fig. 2(c), where the bulk bands are Topological spin-Hall effect. — In order to show the shown in grey, while red and green represent the edge TSHE, we first consider a square lattice, which is pe- states located at the two edges. As expected there are riodic along the x direction and has Ny sites along the y three bulk bands, which are connected by the topologi- direction (see Fig. 2(a)). The coupling between the near- cally protected edge states. est sites along the x direction is taken as J, while the Next, we reduce the width of the lattice and keep only coupling between the nearest sites along the y direction two sites along the y direction (see Fig. 2(b)). The band is taken as complex ∆einxφ, where n is the site num- x structure of such a lattice is shown in Fig. 2(d). Remark- ber along the x direction. The total phase in each pla- ably the band structure in this case is exactly the same as quette becomes φ ( φ) while hopping counter-clockwise the one Fig. 2(c), however all the bulk modes disappear (clockwise). Any nonzero− phase φ introduces an effective leaving behind only the edge modes. Achieving com- magnetic field, which breaks the time reversal (TR) sym- plex hopping between sites for polaritons is extremely metry in the system. The Hamiltonian of the system can challenging, which makes the realization of the system be represented as shown in Fig. 2(a) difficult in practice. However, consid- ering the red (green) sites as σ+ (σ−) spin states of the Hˆ = Jaˆ† aˆ + ∆einxφaˆ† aˆ nx,ny nx+1,ny nx,ny nx,ny+1 polaritons, the system shown in Fig. 2(b) can be mapped h− i nXx,ny to a chain of elliptical micropillars (see Fig. 3(a)), where + h.c. (1) ∆ is the polarization splitting inside the micropillars due to shape anisotropy [69] and φ represents the orienta- Here ny is the site index along the y direction, tion angle. With this interpretation the system becomes † aˆnx,ny aˆnx,ny is the particle creation (annihilation) op- TR symmetric and red (green) represents the σ+ (σ−) erator at site (nx,ny), and h.c. represents the Hermitian spin states in the band structure in Fig. 2(d). The states conjugate. The Hamiltonian in Eq. (1) gives rise to the having positive and negative momentum have opposite 3

y (a) polarizatons. In Fig. 3(c) the same band struture is ob- x 0 10 tained by Fourier transforming the eigenstates of a ﬁnite 6.0 (b) (c) +0.9 chain, which corresponds to the photoluminescence spec- 5.6 trum in the experiments. The colorbar corresponds to 5.2 the degree of circular polarization, which is deﬁned as S = ψ 2 ψ 2 / ψ 2 + ψ 2 . 4.8 z σ+ σ− σ+ σ− z | | − | | | | | |

S In order to see the eﬀect of the TSHE, we consider 4.4 Γ = 0, F = 0 keeping P = 0. The polaritons are 6 σ± 6 σ±

Energy, meV Energy, 4.0 injected using a tightly localized linearly polarized Gaus- 3.6 sian shaped coherent continuous pump positioned at the -0.9 middle of the chain. The full width at half maximum of 3.2 0 0 the pump is around 2.1 µm, which excites a wide range of momentum corresponding to a particular energy. In Fig. 4(a-b) the dynamics of the σ± polaritons are shown, FIG. 3: (a) A chain of elliptical micropillars, where three where they propagate in the opposite directions showing pillars oriented by π/3 with respect to each other form an the TSHE. This is expected from the band structure as unit cell. (b-c) Band structure of the system under consid- σ± states have opposite group velocities. The intensity eration using Bloch theorem (infinite chain) and by Fourier of the polaritons decreases with the propagation distance transforming the eigenstates of a finite chain. due to their finite lifetime. Fig. 4(c) shows the degree of circular polarization of the propagating polaritons. The full dynamics of the polariton under a linearly polarized circular polarizations. As a result σ± spins propagate in coherent pump and pulse is shown in movie1 [70]. Such the opposite directions giving rise to the TSHE. non-trivial propagation of polariton spins exists for an en- The complex hopping between the σ± states inside an ergy range around 1.5 meV (3.3 - 4.8 meV; above this the elliptical micropillar can be shown using a basis transfor- propagation direction for σ± polaritons changes). This mation, where an orientation angle θ induces 2θ phase in is around 15 times larger than the non-trivial bandgap of the hopping term [70]. The dynamics of the polaritons in the topological polaritons observed in [42]. a chain of elliptical micropillars can be described by the Non-Hermitian skin effect.— Until now the dynam- following driven-dissipative Gross-Pitaevskii equation ics of the polaritons were determined by the Hermitian physics, with the non-Hermitian decay term determining 2 2 ∂ψσ ~ −2Γt i~ ± = ∇ + V (x, y)+ i P Γ ψ the rate of loss of intensity with a factor e . Here ∂t − 2m σ± − σ± we choose different decay rates corresponding to the σ ± i(kpx−ωpt) modes. The Hamiltonian in Eq. (1) gets updated to + VT (x,y,θ)ψσ∓ + Fσ± (x, y)e . (2)

2 −5 Here represents the Laplacian, m =7.5 10 me is ˆ † † ∇ × H = J aˆnx,σ aˆnx+1,σ± +ân +1,σ aˆnx,σ± the polariton mass with me being the free electron mass. − ± x ± nXx,σ± h i V represents the potential corresponding to the chain of ±inxφ † † elliptical mircropillars shown in Fig. 3(a) with potential + ∆e aˆ aˆn ,σ iΓσ aˆ aˆn ,σ . nx,σ± x ∓ − ± nx,σ± x ± depth 10 meV. In order to realize 2π/3 phase in each nXx,σ± h i plaquette (see Fig. 2(b)), any two neighbouring elliptical (3) micropillars need to be oriented by an angle π/3 with respect to each other. To be consistent with experiment we Here Γσ± are the effective decay rates of the σ± modes. choose the micropillars having semi-major axis 1.3 µm We choose Γσ+ < Γσ− and diagonalize the Hamiltonian and semi-minor axis 0.85 µm [69], which gives a three having 90 sites along the x axis (see Fig. 5(a)). The micropillar unit cell with length a = 3 µm. Pσ± is the spatial profiles of all the modes corresponding to both gain corresponding to an incoherent pump and Γ = 0.032 σ± spins are plotted in Fig. 5(b-c). The modes having meV represents the polariton decay corresponding to a real energy < 0 (> 0) are localized at the left (right) lifetime around 10 ps. VT represents the linear polariza- end of the lattice. This is known as the NHSE, where tion splitting around 1 meV inside the pillars [69], which whole band collapse to one end of the finite lattice. Un- has the same spatial profile as V but multiplied by a like the TSHE, where the σ± spins propagate in opposite ±2iθ factor e corresponding to the orientation of each mi- directions, here both the σ± spins are localized at the cropillars. Fσ± is a coherent pump that serves to inject same end. σ± polaritons propagate in opposite direc- ~ polaritons having energy ωp and wave vector kp. tions. When σ+ polaritons hit the end of the chain, they In Fig. 3(b) we remove the pump and decay terms couple to counter-propagating σ− modes. However, since

(Pσ± =Γ= Fσ± = 0) and calculate the band structure Γσ+ < Γσ− , the σ− modes decay (much faster compared of the system using the Bloch theorem. The band struc- to σ+ modes) before propagating a signiﬁcant distance. ture is similar to the one in Fig. 2(d) with the positive Due to this a large population of the polaritons builds up and negative momentum states having opposite circular at the end of the chain giving rise to the non-Hermitian 4

(a) (b) (c) -40 -20 0 20 40 S Intensity z Min Max -0.9 +0.9

2 2 FIG. 4: Topological spin-Hall eﬀect: (a-c) |ψσ+ | , |ψσ− | , and Sz, respectively, at t = 60 ps under a linearly polarized coherent pump positioned at the middle (x = 0 µm) of the chain. Parameters: ~ωp = 3.6 meV and akp = 0.

6.0 (a) 2 (b) (c) (d) (e) Max

J 5.3 J 1 0 4.6 1 3.9

J Real Energy/

Real Energy, meV Real Energy, Min 2 3.2 1 45 90 1 45 90 -40-20 0 20 40 -40-20 0 20 40 Site number Site number (f) (g) (h) (i) (j) (k) -40 -20 0 20 40

FIG. 5: (a) Scheme of the non-Hermitian skin eﬀect where the decay rates of σ± modes are diﬀerent. (b-c) Mode localization using the tight binding model in Eq. (3). (d-e) Plot of Iσ± (x) for each eigen mode calculated from Eq. (2), which shows the mode localization. Polariton propagation at t = 200 ps under a coherent pump when the pump is positioned at the left end 2 2 (f-g), middle (h-i) and the right end of the chain (j-k). (f,h,j) correspond to |ψσ+ | and (g,i,k) correspond to |ψσ− | . Parameters:

(b-c) ∆/J =1, Γσ+ /J =0, Γσ− /J = 0.3. (d-e) Pσ+ = 1.4Γ, Pσ− = 0. All other parameters are the same as those in Fig. 4.

skin effect. The group velocity of the σ+ modes decides one-way propagation of the polaritons is independent of the localization end of the bands and if one considers the the position of injection. The full dynamics of the po- case, where Γσ+ > Γσ− , the localization is in the opposite laritons is shown in movie3 [70]. end (see movie2 [70]). Conclusion.— We have presented a scheme based on an In order to verify our claim, we introduce a spa- elliptical micropillar chain, where the naturally present tially uniform σ+ incoherent pump, which corresponds polarization splitting inside the mircropillars along with to Pσ+ > 0 and Pσ− = 0 in Eq. (2)[88, 89]. The main their orientation give rise to the topological Spin Hall purpose of the incoherent pump is to decrease the ef- effect, where σ± polaritons propagate in the opposite fective decay rate of the σ+ polaritons, while keeping direction. By making the decay rates of σ± polaritons the system below the condensation threshold. We define, different, the topological phase changes from topological E 2 E Iσ (x)= ψ dy, for each eigenstate ψ and plot it spin Hall effect to non-Hermitian skin effect, where the ± | σ± | σ± as a functionR of the real energies and the x axis in order polaritons accumulate at one end of the chain indepen- to see the loclization of the eigenstates corresponding to dent of the excitation position. The energy window in a finite chain (see Fig. 5(d-e)). Next we inject the polari- which these effects take place is around 1.5 meV, which tons using the same coherent pump as in the TSHE case. is about one order magnitude larger than that of the In Fig. 5(f-g) the coherent pump is positioned at the left edge states of the two dimensional topological polariton end of the chain, which shows that the polaritons do not system. Although, we have used coherent excitation for propagate from left to right even after 200 ps. Next the injecting polaritons into the system, it is possible to set coherent pump is positioned at the middle of the chain the strength of the circularly polarized spatially uniform in Fig. 5(h-i) and unlike the TSHE both σ± polaritons incoherent pump above the condenstation threshold [70]. propagate leftward. Finally in Fig. 5(j-k) the coherent For such a case the condensation spontaneously forms pump is positioned at the right end of the chain, where at the end of the chain. Because the incoherent pump σ± polaritons propagate from right end to left end and is distributed over the whole sample, it can in principle gets accumulated at the left end. This shows that the help to overcome the problem of having condensation in 5 some of the modern samples, where the sample burns out Sebastian Klembt for stimulating discussions. The work before reaching the condensation threshold. was supported by the Ministry of Education, Singapore Acknowledgment.— We thank Christian Schneider and (Grant No. MOE2019-T2-1-004).

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Matuszewski, Neuromorphic Computing in arXiv:2103.05480v1 [cond-mat.mes-hall] 9 Mar 2021 ai h ytmcnb ersne as represented be can system the basis pn,respectively. spins, aiainsltig tsosthat shows It splitting. larization stetasomto matrix, transformation the is Here where Here ob h nriso h oe aiglna oaiainaogtelon the along polarization represente linear be having can This modes respectively. the directions, of axis) energies (semi-minor the be to ‡ † ∗ h yaiso h oaiosi ie ytefloignnlna drive non-linear following the in by given non-linear is the regime polaritons this the of In dynamics threshold. the condensation the above is nue 2 induces i orsodn author: Corresponding author: Corresponding orsodn author: Corresponding rmtetplgclsi-aleett h o-emta s non-Hermitian the to effect spin-Hall topological the From ~ ftelniuia xso h litclmcoilrmksa angle an makes micropillar elliptical the of axis longitudinal the If oso h ope opn ewe the between hopping complex the show To nodrt hwteectnplrtncnesto,w e h in the set we condensation, exciton-polariton the show to order In ∂ψ ∂t α σ 1 ± and θ = hs ntehpigterm. hopping the in phase − α 2 (1 .RAIAINO H OPE OPN BETWEEN HOPPING COMPLEX THE OF REALIZATION I. iiino hsc n ple hsc,Sho fPhysical of School Physics, Applied and Physics of Division r h o-ierplrtnplrtnitrcincecet betw coefficients interaction polariton-polariton non-linear the are − α iλ [email protected] [email protected] [email protected] NL ) ~ stegi auaincue ytedpeino h xioi reser excitonic the of depletion the by caused saturation gain the is 2 2 m ∇ 2 ayn ehooia nvriy igpr 331 Sing 637371, Singapore University, Technological Nanyang i ~ + ε ∂t ∂ I XIO-OAIO CONDENSATION EXCITON-POLARITON II. V ( = ( σ ,y x, .Mandal, S. ψ ψ ± L σ σ − + i + ) pnsae r ope ihacmlxhpig hr norientation an where hopping, complex a with coupled are states spin + ~ ∂t T upeetlmtra for material Supplemental ∂ = i ) M / P ψ ψ ersnsteost nry n ( = ∆ and energy, onsite the represents 2 σ T † L ∗ M ± H irpla chain micropillar σ .Banerjee, R. ± − LT = = H Γ pnsae niea litclmcoilr econsider we micropillar, elliptical an inside states spin M √ LT L 0 1 + 2 ψ ψ T = 0 α σ σ − 1 − + e ie

L 0 − ψ ψ ψ − iθ † = σ T L T iθ ± 0 n ...Liew T.C.H. and

2 = ∆ as d ie . e + e iθ iθ H θ − ε oeetpumps coherent α eato em eoerlvn.Consequently, relevant. become terms teraction LT 2 ihthe with 2 iθ

ψ isptv rs-iavkiequation Gross-Pitaevskii dissipative n ∆ ψ iuia sm-ao xs n transverse and axis) (semi-major gitudinal ψ σ n ahmtclSciences, Mathematical and ∓ T L e ε

2 2 iθ , x − xs hni h iclrpolarization circular the in then axis, iα ‡ ψ ψ i ﬀc na elliptical an in eﬀect kin NL σ σ apore e h aesisadopposite and spins same the een − + P

σ ψ . + L σ σ ± ± 2 = − or echoose We voir.

MODES 2 T . Γand 2Γ ) ψ / σ ersnstepo- the represents 2 ± + V P T σ ( − ,y θ y, x, ,which 0, = α L 1 10 = angle ) and ψ (S2) (S4) (S3) (S1) (S5) σ ∓ − T . θ 3 2

2 −3 meVµm [S1], α2 = 0.1α1 [S2], and αNL = 0.3α1 [S3]. A phenomenological parameter λ = 5 10 is introduced to take into account− the relaxation process [S4]. This corresponds to higher energy states decaying× faster compared to the lower energy states. We keep all other parameters the same as those in the main text and solve the time dynamics in Eq. (S5) using a very small random noise as an initial condition. In Figs. S1(a-b) the intensities of the σ± modes are shown, respectively. Although Pσ+ is uniformly distributed over the whole lattice, the condensates form at the left end of the lattice due to the non-Hermitian skin eﬀect. In Fig. S1(c) the dispersion corresponding to the condensate is shown, which is slightly blueshifted due to the polariton-polariton interaction. Similar topological edge mode condensation, but based on Hermitian Su-Schrieﬀer-Heeger polariton lattice has been demonstrated recently [S5].

(a)

(b)

-40 -20 0 20 40 Intensity Min Max (c) 6.0

5.5 Max

5.0

4.5

4.0 Real Energy, meV Real Energy,

3.5 Min 3.0 0 ak x

FIG. S1: (a-b) Intensities of the condensate corresponding to the σ± modes. (c) Formation of the condensate in the reciprocal space. The dotted curve is the dispersion corresponding to the linear case shown in Fig. 3 in the main text.

III. COMPARISON OF THE POLARITON PROPAGATION UNDER A PULSED EXCITATION

In this section, we compare the dynamics of the polaritons under a pulsed excitation in the topological spin-Hall case and non-Hermitian skin eﬀect case. The dynamics of the polaritons is given by Eq. 2 in the main text, however the coherent continuous pump term is changed to a pulse

2 2 ∂ψσ ~ 2 2 i~ ± = ∇ + V (x, y)+ i P Γ ψ + V (x,y,θ)ψ + F (x, y)e[−(t−t0) /2τ ]ei(kpx−ωpt). (S6) ∂t − 2m σ± − σ± T σ∓ σ±

Here t0 = 0 ps is the time when the pulse is launched and τ = 10 ps is the width of the pulse in time. In Fig. S2 the dynamics of the polaritons is shown. In Figs. S2(a-c) the spin-Hall nature (Pσ± = 0) can be seen, where σ± polaritons propagate in the opposite directions. When the σ+ (σ−) polaritons hit the left (right) end of the chain, they couple with the σ− (σ+) polaritons and propagate towards the right (left) end. The whole dynamics is shown in movie1 lower panel.

Next in Figs. S2(d-f) the polariton dynamics in the non-Hermitian skin eﬀect regime Pσ+ =1.4Γ, Pσ− =0 is shown. In this case, both the σ± polaritons propagate in the same directions. 3

(a) t=1 ps

(b) t=30 ps

(d) t=1 ps

(e) t=30 ps

(f) t=60 ps

-40 -20 0 20 40 Intensity Min Max

FIG. S2: Polariton propagation under a pulsed excitation showing the topological spin-Hall eﬀect in (a-c) and non-Hermitian skin eﬀect in (d-f).

IV. SUPPLEMENTARY MOVIES

In this section we provide the description of the supplementary movies. Movie1.— In movie1, we show the topological spin-Hall nature using a linearly polarized continuous coherent pump (upper panel) and a pulse (lower panel). Both the pump and pulse are positioned at the middle of the chain. Under the pulsed excitation the coupling between the opposite circularly polarized modes at the edges of the sample can be seen, where the overall motion is anticlockwise. Such a non-trivial directional dependent motion of the spins, may find its application in connecting different spin-dependent polariton devices [S6, S7, S8]. Movie2.— Movie2 shows the localization of the bands in the tight binding model according to Eq. (3) in the main text. Even a small inequality in the decay rates give rise to the non-Hermitian skin effect. The localization of the modes starts to become more prominent as Γ Γ increases. When Γ > Γ , the modes get localized at the σ+ − σ− σ+ σ− opposite end to the one for case Γσ+ < Γσ− . Movie3. — In movie3, we show the polariton propagation for the non-Hermitian skin effect case Γσ+ < Γσ− . In the top panel a linearly polarized continuous coherent pump (the same pump used in the topological spin-Hall case) is placed at the left end of the chain. The polaritons spread a little from the excitation position according to the spatial profiles of skin modes, however they do not propagate from the left end to the right end. In the middle panel the same pump is positioned at the middle of the chain. At initial times σ− polaritons tries to propagate towards both the left and right ends, however the modes propagating right decay much faster. As a result large polariton population builds up at the left end at larger time. In the bottom panel the same pump is positioned at the right end of the chain. Unlike the top panel, polaritons can propagate from the right end to left end and get accumulated at the left end. This movie shows that polaritons can propagate one-way (right to left) independent of the excitation position, while the propagation in the opposite direction is reduced significantly. Such a one-way motion can be used to suppress 4 feedback in polariton networks [S9, S10, S11, S12] and connecting different components of a polariton information processing device [S13, S14, S15, S16, S17].

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