LETTER. Photonic Floquet topological insulators

Transcription

1 oi:1.138/nture1266 Photoni Floquet topologil insultors Mikel C. Rehtsmn 1 *, Juli M. Zeuner 2 *, Yontn Plotnik 1 *, Ykov Lumer 1, Dniel Poolsky 1, Felix Dreisow 2, Stefn Nolte 2, Morehi Segev 1 & Alexner Szmeit 2 Topologil insultors re new phse of mtter 1, with the striking property tht onution of eletrons ours only on their surfes 1 3. In two imensions, eletrons on the surfe of topologil insultor re not sttere espite efets n isorer, proviing roustness kin to tht of superonutors. Topologil insultors re preite to hve wie-rnging pplitions in fult-tolernt quntum omputing n spintronis. Sustntil effort hs een irete towrs relizing topologil insultors for eletromgneti wves 4 13.Oneimensionl systems with topologil ege sttes hve een emonstrte, ut these sttes re zero-imensionl n therefore exhiit no trnsport properties 11,12,14. Topologil protetion of mirowves hs een oserve using mehnism similr to the quntum Hll effet 15, y pling gyromgneti photoni rystl in n externl mgneti fiel 5. But euse mgneti effets re very wek t optil frequenies, relizing photoni topologil insultors with stter-free ege sttes requires funmentlly ifferent mehnism one tht is free of mgneti fiels. A numer of proposls for photoni topologil trnsport hve een put forwr reently 6 1. One suggeste temporl moultion of photoni rystl, thus reking time-reversl symmetry n inuing one-wy ege sttes 1.Thisis in the spirit of the propose Floquet topologil insultors 16 19,in whih temporl vritions in soli-stte systems inue topologil ege sttes. Here we propose n experimentlly emonstrte photoni topologil insultor free of externl fiels n with stter-free ege trnsport photoni lttie exhiiting topologilly protete trnsport of visile light on the lttie eges. Our system is ompose of n rry of evnesently ouple helil wveguies 2 rrnge in grphene-like honeyom lttie Prxil iffrtion of light is esrie y Shröinger eqution where the propgtion oorinte (z) ts s time 27. Thus the heliity of the wveguies reks z-reversl symmetry s propose for Floquet topologil insultors. This struture results in one-wy ege sttes tht re topologilly protete from sttering. Prxil propgtion of light in photoni ltties is esrie y the Shröinger-type eqution: il z yðx,y,zþ~{ yðx,y,zþ{ k Dnðx,y,zÞ yðx,y,zþ ð1þ 2k n where y(x,y,z) is the eletri fiel envelope funtion efine y E(x,y,z) 5 y(x,y,z)exp(ik z 2 ivt)x; E is the eletri fiel, x is unit vetor n t is time; the Lplin, = 2, is restrite to the trnsverse (x y) plne; k 5 2pn /l is the wvenumer in the mient meium; v 5 2p/l is the optil frequeny; n n l re respetively the veloity n wvelength of light. Our mient meium is fuse sili with refrtive inex n , n Dn(x,y,z) is the effetive potentil, tht is, the evition from the mient refrtive inex. The rry is frite using the femtoseon lser writing metho; eh elliptil wveguie hs ross-setion with mjor n minor xis imeters of 11 mmn4mm, respetively. The photoni lttie is n rry of evnesently-ouple wveguies rrnge in honeyom struture with nerest-neighour sping of 15 mm. The totl propgtion length (in the z iretion) is 1 m, whih orrespons to the wvefuntion y of single wveguie moe ompleting,2 yles in phse while propgting from z 5 toz51 m. The inrese in refrtive inex ssoite with the wveguies is Dn The quntum mehnil nlogue of eqution (1) esries the propgtion of quntum prtile tht evolves in time for exmple, n eletron in soli. The wveguies t like potentil wells, similrly to nulei of toms in solis. Thus, the propgtion of light in the rry of helil wveguies s it propgtes in the z iretion is equivlent to the temporl evolution of n eletron s it moves through two-imensionl lttie with toms tht re rotting in time. A mirosope imge of the input fet of the photoni lttie is shown in Fig. 1, n igrm of the helil wveguies rrnge in honeyom lttie is shown in Fig. 1. The perio (or pith) of the helil wveguies is suffiiently smll tht guie moe is itilly rwn long with wveguie s it urves. We therefore trnsform the oorintes into referene frme where the wveguies re invrint in the z iretion (i.e., stright), nmely: x9 5 x 1 Ros(Vz), y9 5 y 1 Rsin(Vz)nz95z,whereRis the helix rius n V 5 2p/F 5 2p/1 m is the frequeny of rottion (F 5 1 m eing the perio). In the trnsforme oorintes, the light evolution is esrie y: il z y ~{ 1 ð+ ziaðz ÞÞ 2 y { k R 2 V 2 y { k Dnðx,y Þ y ð2þ 2k 2 n where y9 5 y(x9,y9,z9), n A(z9) 5 k RV[sin(Vz9),2os(Vz9), ] is equivlent to vetor potentil ssoite with sptilly homogeneous eletri fiel of irulr polriztion. The itiity of the guie moes n the presene of the vetor potentil yiel ouple moe (tight-ining) eqution, vi the Peierls sustitution 1 : il z y n (z )~ X ia e ð z Þ: r mn y m (z ) ð3þ hmi where the summtion is tken over neighouring wveguies; y n (z9)is the mplitue in the nth wveguie, is the oupling onstnt etween wveguies n r mn is the isplement etween wveguies m n n. Beuse the right-hn sie of eqution (3) is z-epenent, there re no stti eigenmoes. Rther, the solutions re esrie using Floquet moes, of the form y n (z9) 5 exp(iz9)q n (z9), where the funtion Q n (z9) is F-perioi 18. This yiels the spetrum of (the Floquet eigenvlues or qusi-energies ) s funtion of the Bloh wvevetor, (k x, k y ), s well s their ssoite Floquet eigenmoes. Floquet eigenmoes in the z iretion re equivlent to Bloh moes in the x y plne. Therefore, our input em (initil wvefuntion) exites superposition of Floquet moes whose popultion oes not hnge over the ourse of propgtion 17,18. The n struture for the se of non-helil wveguies (R 5 ) is shown in Fig. 1. The onil intersetions etween the first n seon ns re the Dir points 28, feture of grphene tht mkes it semi-metlli. When the wveguies re me helil (R. ), ngp in the Floquet spetrum opens, s shown in Fig. 1, n the photoni lttie eomes nlogous to n insultor inee, to Floquet topologil insultor. As we show elow, this struture possesses topologilly protete ege sttes. 1 Deprtment of Physis n the Soli Stte Institute, Tehnion Isrel Institute of Tehnology, Hif 32, Isrel. 2 Institute of Applie Physis, Ae Center of Photonis, Frierih-Shiller-Universität Jen, Mx-Wien-Pltz 1, 7743 Jen, Germny. *These uthors ontriute eqully to this work. 196 NATURE VOL APRIL 213

2 RESEARCH 15 μm y x Bngp k x k y k x k y Figure 1 Geometry n n struture of honeyom photoni Floquet topologil insultor lttie., Mirosope imge of the input fet of the photoni lttie, showing honeyom geometry with zig-zg ege termintions on the top n ottom, n rmhir termintions on the left n right sies. Sle r t top right, 15 mm. The yellow ellipse inites the position n shpe of the input em to this lttie., Sketh of the helil wveguies. Their rottion xis is in the z iretion, with rius R n perio We lulte the ege n struture y using unit ell tht is perioi in the x iretion ut finite in the y iretion, ening with two zig-zg eges (infinite in the x iretion). The zig-zg ege is one of three typil ege termintions of the honeyom lttie; the other two re the rmhir ege n the ere ege. Note tht the presene of hirl ege sttes n e erive using the ulk ege orresponene priniple y lulting the Chern numer 4,5,17,29. In our smple (see Fig. 1), the top n ottom eges re zig-zg eges n the right n left eges re rmhir eges. The n struture of the zig-zg ege is presente in Fig. 2 for the se where the wveguies re not helil (R 5 ). There re two sets of sttes, one per ege. Their ispersion urves re flt n ompletely oinie (tht is, they re egenerte with one nother), resiing etween k x 5p2p/3 n k x 5 4p/3, oupying one-thir of k x spe, where 5 15 ffiffi 3 mm is the lttie onstnt. The Floquet n struture when the lttie is helil with R 5 8 mmis shown in Fig. 2. Here, the ege sttes re no longer egenerte, ut now hve opposite slopes. Speifilly, the trnsverse group veloity Z., Bn struture ( versus (k x, k y )) for the se of non-helil wveguies omprising honeyom lttie (R 5 ). Note the n rossings t the Dir point., Bulk n struture for the photoni topologil insultor: helil wveguies with R 5 8 mm rrnge in honeyom lttie. Note the ngp opening up t the Dir points (lelle with the re, oule-ene rrow), whih orrespons to the ngp in Floquet topologil insultor. (i.e., the group veloity in the (x y) plne) on the top ege is now irete to the right, n on the ottom ege to the left, orresponing to lokwise irultions. However, there re no ege sttes whtsoever irulting in the nti-lokwise iretion. Hene, the ege sttes presente in Fig. 2 re the topologilly protete ege sttes of Floquet topologil insultor. The lk of ounter-propgting ege stte on given ege iretly implies tht ny ege-efet (or isorer) nnot kstter, s there is no kwrs-propgting stte ville into whih to stter, ontrry to the se of R 5, where there re multiple sttes into whih sttering is possile. This is the essene of why topologil protetion ours. The trnsverse group veloity (for revity, we heneforth rop trnsverse ) of these ege sttes hs non-trivil epenene on the helix rius, R. For smll R, the group veloity of the ege sttes inreses, ut eventully it rehes mximum n ereses gin. Before the group veloity rosses zero, the Chern numer is lulte to e 21 (initing the presene of lokwise ege stte, s seen in Fig. 2). However, fter the group veloity rosses β/ 3 Ege sttes Bulk sttes 3 π/2 π 3π/2 k x Top ege Bottom ege 1 2 2π 3 π/2 π 3π/2 2π k x βz Ege group veloity (μm m 1 ) Helix rius, R (μm) Figure 2 Dispersion urves of the ege sttes, highlighting the unique ispersion properties of the topologilly protete ege sttes for helil wveguies in the honeyom lttie., Bn struture of the ege sttes on the top n ottom of the rry when the wveguies re stright (R 5 ). The ispersion of oth top n ottom ege sttes (re n green urves) is flt, thereforetheyhvezerogroupveloity. The ns of the ulk honeyom lttie re rwn in lk., Dispersion urves of the ege sttes in the Floquet topologil insultor for helil wveguies with R 5 8 mm: the n gp is open n the ege sttes quire non-zero group veloity. These ege sttes resie stritly within the ulk n gp of the ulk lttie (rwn in lk)., Group veloity (slope of green n re urves) versus helix rius, R, of the helil wveguies omprising the honeyom lttie. The mximum ours t R mm. 11 APRIL 213 VOL 496 NATURE 197

3 RESEARCH LETTER zero t whih point the n gp loses the Chern numer is 2 (initing the presene of two nti-lokwise ege sttes, s onfirme y lultions). The R epenene of the group veloity is shown in Fig. 2, where we plot the group veloity of the topologilly protete ege stte t k x 5 p/ versus R. The mximum lulte group veloity is t R mm. To emonstrte these ege sttes experimentlly, we lunh em with n ellipti profile of wvelength 633 nm suh tht it is inient on the top row of wveguies in n rry with helix rius R 5 8 mm. The position of the input em is inite y the ellipse in Fig. 1. The light istriution emerging from the output fet is presente in Fig. 3, with the shpe n position of the input em inite y yellow ellipse. In Fig. 3, the em emerges t the upper-right orner of the lttie, hving move long the upper ege. When we move the position of the input em horizontlly to the right, the output em moves own long the vertil right ege, s shown in Fig. 3. The em emerging from the lttie remins onfine to the ege, not spreing into the ulk n without ny ksttering. Moving the position of the input em further rightwr mkes the output em move frther own long the sie ege, s shown in Fig. 3 n. Clerly, the input em hs move long the top ege, enountere the orner, n then ontinue moving ownwr long the right ege. We show this ehviour in em-propgtion-metho (BPM) simultions 3, solving eqution (1) (see Supplementry Vieo 1). The entrl oservtion of these experimentl results is tht the orner (whih is in essene strong efet) oes not kstter light. Inee, no optil intensity is evient long the top ege t the output fet, fter hving ksttere from the orner. Furthermore, no sttering into the ulk of the rry is oserve (owing to the presene of ulk ngp). These oservtions provie strong eviene of topologil protetion of the ege stte. Figure 3 Light emerging from the output fet of the wveguie rry s the input em is move rightwrs, long the top ege of the wveguie rry. The yellow ellipse t the top of eh pnel shows the position of the input em (whih is t the top of the rry, see Fig. 1), whih is move progressively to the right in. The em propgtes long the top ege of the rry (whih is in the zig-zg onfigurtion), hits the orner, n lerly moves own the vertil ege (whih is in the rmhir onfigurtion). Note tht the wvepket shows no eviene of ksttering or ulk sttering ue to its impt with the orner of the lttie. This sttering of the ege stte is prevente y topologil protetion. Further eviene follows from the ft tht light stys onfine to the sie ege of the rry s it propgtes ownwrs. This ege is in the rmhir geometry, whih, for stright wveguies (R 5 ) oes not llow ege onfinement t ll (tht is, no ege sttes). However, when R., ege stte ispersion lultions revel tht onfine ege stte emerges. This is essentil for the topologil protetion euse it prevents trnsport into the ulk of the lttie. We now experimentlly exmine the ehviour of the topologil ege sttes s the helix rius, R, is vrie. We fin tht the group veloity rehes mximum n then returns to zero s R is inrese, in orne with Fig. 2. To investigte this, we frite series of honeyom ltties of helil wveguies with inresing vlues of R, ut in tringulr shpe (Fig. 4). We first exmine light propgtion in the lttie with non-helil wveguies (tht is, R 5 ; Fig. 4). Lunhing em into the wveguie t the upper-left orner of the tringle (irle) exites two types of eigensttes: (1) ulk sttes extening to the orner, n (2) ege sttes tht meet t the orner. As the light propgtes in the rry, the exite ulk sttes le to some egree of spreing into the ulk (the exittion of these ulk moes n e eliminte y engineering the em to only overlp with eigensttes onfine to the ege). In ontrst, the ege sttes o not spre into the ulk, n, euse the ege sttes re ll egenerte (Fig. 2), they o not use spreing long the eges either (tht is, zero group veloity). Figure 4 shows the intensity t the output fet highlighting this effet: while some light hs iffrte into the ulk, the mjority remins t the orner wveguie. This is lso shown in simultions (where the nimtion evolves y sweeping through the z oorinte from z 5 m to z 5 1 m); see Supplementry Vieo 2. When the helil wveguies hve lokwise rottion, the ege sttes re no longer egenerte. In ft, the lttie now hs set of ege sttes tht propgte only lokwise on the irumferene of the tringle. Light t the orner no longer remins there, n moves long the ege. Figure 4 j shows the output fet of the lttie for inresing rius R. For R 5 8 mm, the wve pket wrps roun the orner of the tringle n moves long the opposite ege (Fig. 4f) (the orresponing simultion is shown in Supplementry Vieo 3; the loss of intensity over the ourse of propgtion is ue to ening/rition losses). Importntly, the light is not ksttere even when it hits the ute orner, owing to the lk of ounter-propgting ege stte. This is key exmple of topologil protetion ginst sttering. For R 5 12 mm, the wvepket moves long the ege, ut with slower group veloity. This is onsistent with the preition tht the group veloity of the ege stte rehes mximum t R mm n therefter ereses with inresing rius. The experiments suggest tht the mximl group veloity is hieve etween 6 mm n 1mm, while the theoretil result (1.3 mm) is well within experimentl error, given tht this is preition from ouple-moe theory. Ext simultions onfirm the experimentl result. By R 5 16 mm, ening losses re lrge, leing to lekge of optil power into sttering moes (ounting for the lrge kgroun signl). The ening losses for R 5 4 mm, 8 mm, 12 mm n 16 mm were foun to e, respetively,.3 B m 21,.5 B m 21, 1.7 B m 21 n 3Bm 21. Rell tht eh lttie hs propgtion length z 5 1 m. The lrge kgroun signl prevents us from experimenting with lrger R, where we woul expet two nti-lokwise-propgting ege sttes, s isusse erlier. As shown in Fig. 4j, the group veloity of the wvepket pprohes zero n therefore the optil power remins t the orner wveguie. These oservtions lerly emonstrte the presene of one-wy ege sttes on the ounry of the photoni lttie tht ehve oring to theory. Note tht for ifferent initil ems the elliptil em of Fig. 3, n the single-wveguie exittion of Fig. 4 the topologil ege stte ehves extly s the moel preits, proviing experimentl proof of the existene of the topologil ege stte. To emonstrte the z epenene of the wvepket s it propgtes long the ege, we turn to omintion of experimentl results n 198 NATURE VOL APRIL 213

4 RESEARCH e R = μm 2 μm 4 μm 6 μm f g h i j 8 μm 1 μm 12 μm 14 μm 16 μm Figure 4 Experiments highlighting light irultion long the eges of tringulr-shpe lttie of helil wveguies rrnge in honeyom geometry., Mirosope imge of the honeyom lttie in the tringulr onfigurtion. j, Light emerging from the output fet of the photoni lttie (fter 1 m of propgtion) for inresing helix rius, R (given t ottom right of eh pnel), t wvelength 633 nm. The light is initilly lunhe into the wveguie t the upper-left orner (on the input fet of the rry, inite y yellow irle). At R 5 (), the initil em exites onfine efet moe t simultions of eqution (1) 3. We exmine lttie with efet on the ege in the form of missing wveguie (Fig. 5). Beuse of topologil protetion, the wvepket shoul simply propgte roun the missing wveguie (the efet) without ksttering. An experimentl imge of the output fet is shown in Fig. 5 (for R 5 8 mm). The exite wveguie is t the top right, n the ege stte propgtes lokwise, voiing the efet, n eventully hitting the next orner. In Fig. 5 h we show simultions for the optil intensity t z 5, 2, 4, 6, 8, 1 m, respetively. The wvepket lerly propgtes roun the Missing wveguie e z = m f g h z = 6 m z = 2 m z = 8 m z = 4 m z = 1 m R = 8 μm z = 1 m Figure 5 Experiments n simultions showing topologil protetion in the presene of efet. The lttie is tringulr-shpe, n the wveguies re helil with R 5 8 mm., Mirosope imge of photoni lttie with missing wveguie (ting s efet, rrowe) on the rightmost zig-zg ege. A light em of l nm is lunhe into the single wveguie t the upperright orner (on the fr sie of the rry, surroune y yellow irle)., Experimentl imge of light emerging from the output fet fter z 5 1 m of propgtion, showing no ksttering espite the presene of the efet ( signture of topologil protetion). h, Numeril simultions of light propgtion through the lttie t vrious propgtion istnes (respetively z 5 m, 2 m, 4 m, 6 m, 8 m n 1 m). After miniml ulk sttering, the light propgtes long the ege, enounters the efet, propgtes roun it, n ontinues pst it without sttering, in greement with. the orner. As the rius is inrese ( j), light is moving long the ege y virtue of topologil ege moe. It rehes its mximum isplement ner R 5 8 mm (f). The light wrps roun the orner of the tringle n is not ksttere t ll: this is ler exmple of topologil protetion ginst sttering. As R is inrese further, the light exhiits eresing group veloity s funtion of R, n finlly stops ner R 5 16 mm. The lrge egree of kgroun noise in i n j is ue to high ening losses of the wveguies s result of oupling to free-spe sttering moes. efet, ontinuing forwr without ksttering. Note tht the simulte wvepket hs progresse slightly frther thn tht in the experiment. This is result of smll unertinty in the oupling onstnt,. Tken together, these t show the progression of topologilly protete moes s they trvel long the ege. Photoni Floquet topologil insultors hve the potentil to provie n entirely new pltform for proing n unerstning topologil protetion. For exmple, our photoni ltties hve the sme geometry s photoni rystl fires, n thus these systems re likely to exhiit roust topologilly protete sttes. Mny interesting open questions re prompte, onerning (for exmple) the ehviour of entngle photons in topologilly protete system, the effet of intertions on the non-sttering ehviour, or the possiility of simulting photoni Mjorn fermions for pplitions in roust quntum omputing. 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