Dynamically encircling exceptional points: in situ control of encircling loops and the role of the starting point

Transcription

1 Dynamcally encrclng exceponal pons: n su conrol of encrclng loops and he role of he sarng pon Xu-Ln Zhang, 1, Shubo Wang, 1,4 Bo Hou, 1,3 and C.. Chan 1 1 Deparmen of Physcs, he Hong Kong Unversy of Scence and echnology, Clear Waer Bay, Hong Kong, Chna Sae Key Laboraory of Inegraed Opoelecroncs, College of Elecronc Scence and Engneerng, Jln Unversy, Changchun, Chna 3 College of Physcs, Opoelecroncs and Energy & Collaborave Innovaon Cener of Suzhou Nano Scence and echnology, Soochow Unversy, Suzhou, Chna 4 Deparmen of Physcs, Cy Unversy of Hong Kong, Kowloon, Hong Kong, Chna (Daed: Aprl 16, 018) he mos nrgung properes of non-herman sysems are found near he exceponal pons (EPs) a whch he Hamlonan marx becomes defecve. Due o he complex opologcal srucure of he energy emann surfaces close o an EP and he breakdown of he adabac heorem due o non-hermcy, he sae evoluon n non-herman sysems s much more complex han ha n Herman sysems. For example, recen expermenal work [Doppler e al. Naure 537, 76 (016)] demonsraed ha dynamcally encrclng an EP can lead o chral behavors,.e., encrclng an EP n dfferen drecons resuls n dfferen oupu saes. Here, we propose a coupled ferromagnec wavegude sysem ha carres wo EPs and desgn an expermenal seup n whch he rajecory of sae evoluon can be conrolled n su usng a unable exernal feld, allowng us o dynamcally encrcle zero, one or even wo EPs expermenally. he unably allows us o conrol he rajecory of encrclng n he parameer space, ncludng he sze of he encrclng loop and he sarng/ pon. We dscovered ha wheher or no he dynamcs s chral acually deps on he sarng pon of he loop. In parcular, dynamcally encrclng an EP wh a sarng pon n he pary-me-broken phase resuls n non-chral behavors such ha he oupu sae s he same no maer whch drecon he encrclng akes. he proposed sysem s a useful plaform o explore he opology of energy surfaces and he dynamcs of sae evoluon n non-herman sysems and wll lkely fnd applcaons n mode swchng conrolled wh exernal parameers. I. INODUCION Exceponal pons (EPs) are degeneraces n non- Herman sysems [1-4]. Unlke degeneraces n Herman sysems such as dabolc pons (DPs) [5,6] whose egenvalues bu no egenvecors coalesce, a EPs boh he egenvalues and he egenvecors coalesce, leadng o varous couner-nuve phenomena and fascnang applcaons such as loss-nduced ransmsson enhancemen [7], lasng effecs [8-11], unusual beam dynamcs [1,13], enhanced sensng [14-16], robus wreless power ransfer [17], and ohers [18-3]. he mos nrgung feaure of he EP s perhaps s opologcal srucure n he sense ha adabacally encrclng an EP can resul n an exchange of he egensae [4,5], unlke he encrclng of a DP n Herman sysems where he egensae would only acqure a geomerc phase [5,6]. he so-called sae flp acheved by adabacally encrclng an EP s made possble by he degeneracy-nduced nersecon of complex emann shees [4,5]. hs phenomenon has been demonsraed expermenally n mcrowave caves [6], excon-polaron sysems [7] and acousc sysems [8], where sac measuremens of he specra and egenmodes successfully revealed he opologcal srucure of EPs. However, he oucome s compleely dfferen f an EP s encrcled n a dynamcal process. In dynamcal encrclng, he oupu sae has been predced o be deermned solely by he drecon of roaon n he parameer space regardless of he npu sae. Such chral behavor [9] s a manfesaon of he breakdown of he adabac heorem n non-herman sysems n he presence of gan and loss [30,31]. he chral naure of he dynamcs has also been heorecally 1 nvesgaed from he vewpon of sably loss delay [3] and he Sokes phenomenon of asympocs [33], and a full analycal model has been proposed for a beer undersandng [34]. I was no unl recenly ha he dynamcal encrclng of an EP was realzed expermenally n mcrowave [35] and opomechancal sysems [36]. he chral behavor s expeced o have promsng applcaons n asymmerc mode swchng [35,37] and on-chp nonrecprocal ransmsson [38]. Alhough he dynamcal encrclng of an EP has been demonsraed boh heorecally and expermenally, prevous sudes focused exclusvely on encrclng loops wh he sarng/ pon near he pary-me-symmerc (P-symmerc) phase [34-38], where he magnary pars of he egenvalues coalesce. Bu wha f he sarng pon of he dynamcal process les somewhere else n he parameer space? Would he chral behavor perss f he dynamcal encrclng sars from a pon near he P-broken phase where he real pars of he egenvalues coalesce? hese quesons reman open. Furhermore, he dynamcal evoluon of saes n non-herman sysems n whch nonadabac ransons (NAs) may occur due o he breakdown of he adabac heorem s of fundamenal neres. hs area s, however, largely unexplored especally expermenally due o he complexy n sysem desgn. he recen poneerng work [35] used a modulaed wavegude sysem o realze EP encrclng. he sysem offers an excellen plaform o sudy he dynamcs n non-herman sysems as he sae evoluon and NAs can be undersood nuvely from he feld profles n he wavegudes. However, he encrclng loop n he expermen s fxed once

2 FIG. 1. (a) Schemac dagram of a coupled yrum ron garne (YIG) wavegude sysem wh a mcrowave absorber aached o wavegude-. A bas magnec feld s appled along he negave x-axs. (b) Calculaed bas magnec feld a whch a dabolc pon (DP) emerges n he lossless sysem as a funcon of frequency wh g = 0.5 mm (black lne) and 1 mm (red lne). Oher srucural parameers are W = 8 mm, H = 4 mm, and α = 1. (c) Calculaed effecve mode ndex as a funcon of he bas feld and scale facor of he lossless sysem. A DP appears a B0 = 0.09 and α = 1 due o accdenal degeneracy. (d)-(e) Calculaed real par (d) and magnary par (e) of he effecve mode ndex as a funcon of he bas feld and scale facor of he lossy sysem. he wo fgures show self-nersecng emann surfaces wh wo exceponal pons (EPs) a B0 = 0.06, α = and B0 = 0.13, α = he whe dashed lne n (d)/(e) marks he broken/symmerc phase lne. he black and yellow lnes represen he rajecory of sae evoluon for case I wh Bm = 0.08 (encrclng one EP) and 0.17 (encrclng wo EPs), respecvely. (f)-(g) Same as panels (d)-(e) excep ha he rajecores are for case II. In he smulaons of (c)-(g), he frequency s 9 GHz and he sysem parameers are W = 8 mm, H = 4 mm, g = 1 mm, w = 1.5 mm, and h = mm. he relave permvy of he absorber s 3+3. he sample s fabrcaed, and changng he loop n fac requres fabrcang new samples. A new plaform on whch he encrclng loop could be conrolled n su usng, for example, exernal parameers s hghly desrable. In hs work, we propose a plaform o sudy he dynamcal process n non-herman sysems and he dynamcal encrclng of EPs. On hs plaform, he rajecory of sae evoluon n he parameer space can be conrolled n su usng an exernal parameer. Our sysem consss of a par of ferromagnec wavegudes appled wh ransverse bas magnec felds. he wavegude wdh and he exernal magnec feld are non-unform so ha when wave scaers hrough he sysem, s effecvely ravelng along a rajecory n a pre-desgned wo-varable parameer space, where a par of EPs wh oppose chraly resde. he opologcal srucure of he sysem can be desgned by choosng approprae sysem parameers, allowng us no only o dynamcally encrcle dfferen numbers of EPs (e.g., zero, one or even wo) whou changng or movng he sample, bu also o sudy he depence of he dynamcs on he sarng/ pon of he encrclng loop. We frs realzed expermenally he prevously dscovered chral ransmsson behavor [35] by dynamcally encrclng an EP wh he sarng pon n he symmerc phase. Moreover, our sysem has wo EPs whch allow us o dynamcally encrcle wo EPs o reveal he more complex opologcal srucure of energy surfaces. he man fndng of hs work s ha we nvesgaed he dynamcal encrclng of an EP wh he sarng pon n he broken phase and dscovered a nonchral behavor, ndcang ha wheher he dynamcs s chral or no deps on he sarng pon. A heorecal model was used o nvesgae he underlyng physcs and reveal he role of he sarng pon. II. IN SIU CONOL OF ENCICLING LOOPS WIH AN EXENAL FIELD We sar by nroducng a plaform for sudyng he dynamcal process n non-herman sysems. As shown n Fg. 1(a), he sysem consss of a par of yrum ron garne

3 FIG.. (a) Schemac dagram of a coupled YIG wavegude sysem wh a lengh L = 400 mm, where he bas feld generaed by he wo magnes and he wdh of YIG wavegude- vary connuously along he z-axs. (b) Sde vew of he coupled sysem. (c) Expermenally measured bas feld dsrbuons along he z- axs (crcles), fed usng for numercal smulaons (sold lne). (d) Varaon n he scale facor α along he z-axs. he mnmum α s a z = 100 mm and he maxmum s 1.15 a z = 300 mm. Injecons from z = 0 and z = 400 mm correspond o couner-clockwse and clockwse loops, respecvely. (YIG) wavegudes separaed by a small gap. We apply a ransverse bas magnec feld along he negave x-axs. A mcrowave absorber s aached o he sde of YIG wavegude- o nroduce asymmerc losses [39] no he sysem. he background s ar. he wdh of YIG wavegude- s conrolled by a scale facor α, correspondng o a deunng of he sysem. We frs calculaed he effecve mode ndex of he wavegude par sysem (W = 8 mm, H = 4 mm, g = 1 mm) as a funcon of he scale facor α and he bas feld usng COMSOL [40]. In he smulaon, he relave permvy of YIG s se o ~15.6, and he relave permeably ensor of YIG s modeled wh a dagonal erm 1 / and off-dagonal erms b m /, where ω m = μ 0γ M s deermned by m he gyromagnec rao γ and he magnezaon M, and ω 0 =γ B 0 s deermned by he bas magnec feld B 0 [41]. he effecve mode ndex s defned as n eff = β z/k 0, where β z and k 0 are he mode propagaon consan and vacuum wave number, respecvely. he resuls for he lossless sysem (.e., whou he absorber) a 9 GHz are shown n Fg. 1(c). We fnd ha wo egenmodes are suppored n he sysem. A DP emerges (B 0 = 0.09, α = 1) due o he accdenal degeneracy of he wo egenmodes [4]. When he mcrowave absorber s aached (w = 1.5 mm, h = mm, ε = 3+3), he effecve mode ndex becomes a complex number, and he DP spls no a par of EPs [4], exhbng a selfnersecng emann surface as shown n Fgs. 1(d) (real par) and 1(e) (magnary par). he whe dashed lne n Fg. 1(d) marks he broken phase lne on whch he real pars of he wo egenvalues coalesce (also refer o he sde vew). he wo pons of hs broken phase lne are EPs, beyond whch are wo symmerc phase lnes (see he wo whe dashed lnes n Fg. 1(e)) on whch he magnary pars of he wo egenvalues coalesce. he symmerc phase lne s a branch cu ha connecs he lower-loss emann shee (see he blue shee n Fg. 1(e)) wh he hgher-loss emann shee (see he red shee n Fg. 1(e)). As we have a emann surface conanng a par of EPs, formng an encrclng loop requres changng wo parameers (he bas feld and he scale facor α) connuously n space. o mplemen he encrclng, we desgn a sysem ha s ~400 mm long as shown n Fgs. (a) (op vew) and (b) (sde vew) wh he wo parameers varyng connuously along he wavegudng drecon (.e., 3 z-axs). he bas feld s expermenally generaed wh a vbrang sample magneomeer (VSM) whch has wo magnes wh a dameer of ~00 mm. he expermenally measured bas feld dsrbuon along he z-axs s ploed wh crcles n Fg. (c), and B m denoes he maxmum feld srengh a he cener of he wavegudes (.e., z = 00 mm). he feld s essenally unform along he x- and y-axs n our expermenal seup. he feld dsrbuon s well fed usng a snusodal funcon (sold lne n Fg. (c)) for furher numercal smulaons. he scale facor α s desgned o vary along he z-axs wh a mnmum of a z = 100 mm and a maxmum of 1.15 a z = 300 mm (see Fg. (d); also see Supplemenal Maeral for a dscusson on he wo corners). A wo-parameer space s defned n Fg. 3(a), where he locaons of he wo EPs are also marked. We noe ha he wave scaerng hrough he sysem s analogous o a loop n he wo-parameer space, wh he sarng/ pon a B 0 = 0 and α = 1. Injecons from he lef (z = 0) and he rgh sde (z = 400 mm) of he wavegude sysem (see he schemac dagram n Fg. (a)) correspond o couner-clockwse and clockwse loops, respecvely. Seleced examples of he loops are llusraed n Fg. 3(a), where he green, black, and yellow loops are generaed a bas feld srenghs B m = 0.01, 0.08, and 0.17, correspondng o a dynamcal encrclng of zero, one, and wo EPs, respecvely. he encrclng loop n he proposed sysem can be uned n su along he B 0-axs of he parameer space and he loop sze s deermned by an adabacally unable parameer, B m. Alhough he loop canno be uned along he α-axs, such unably can already enable us o conrol n su he number of EPs encrcled. hs was no possble n prevous expermenal work (see, for example, ef. [35]), where he encrclng loop s fxed once he samples are fabrcaed. he opologcal srucure of our sysem s also more complex han prevous ones [34-38] due o he presence of wo EPs, and he locaons of he EPs can be specfed by choosng approprae sysem parameers. o demonsrae hs pon, we show n Fg. 1(b) he calculaed bas felds requred o access he DP n he lossless sysem as a funcon of frequency wh wo dfferen gap dsances. he red crcle corresponds o he case n Fg. 1(c). When loss s nroduced, he DP spls no wo EPs and her locaons can be specfed by choosng approprae absorbers. Hgher-loss absorbers can resul n a broader broken phase regon whereas lower-loss absorbers can lead o a narrower regon

4 FIG. 3. (a) hree loops n parameer space, generaed as examples wh Bm = 0.01 (green loop no enclosng any EP), 0.08 (black loop enclosng one EP), and 0.17 (yellow loop enclosng wo EPs). he sarng/ pon les a B0 = 0 and α = 1, correspondng o he symmerc phase. he black dashed lne represens he broken phase lne where he real pars of he egenvalues coalesce. (b) Calculaed ransmsson nenses for he four cases (see ex for defnon) as a funcon of Bm. he shaded regon represens he feld srenghs where one EP s dynamcally encrcled, and a sae flp occurs for cases I and IV only. Ousde he shaded regon, zero (lef regon) or wo EPs (rgh regon) are encrcled. he number of non-adabac ransons (NAs) n he dynamcal process, denoed by NNA, s gven n dfferen regons. (c)-(h) Numercally smulaed Hy feld dsrbuons n he wavegude sysem wh dfferen npu modes and njecon drecons. he resuls wh Bm = 0.08 for cases I-IV are shown n (c)- (f), respecvely, correspondng o an encrclng of one EP. Panels (g) and (h) show resuls for cases I and II, respecvely, wh Bm = 0.01, correspondng o an encrclng of zero EP. In all of he smulaons, he frequency s 9 GHz and he sysem parameers are he same as hose gven n Fg. 1. [4]. Our sysem serves as a conrollable plaform o sudy he dynamcal process of sae evoluon on complex energy surfaces n non-herman sysems. III. SAING/END POIN IN HE SYMMEIC PHASE: CHIAL DYNAMICS We performed numercal smulaons o demonsrae he effecs arsng from he dynamcal encrclng of EPs. We frs consder encrclng loops (Fg. 3(a)) wh sarng/ pon n he symmerc phase where one egenmode s symmerc and he oher one ansymmerc. As he encrclng can proceed eher n he clockwse or he couner-clockwse drecon and we can choose o exce eher he symmerc or he ansymmerc mode a he sarng pon, here are four possble cases. Cases I and II correspond o counerclockwse loops and cases III and IV clockwse loops. he njecon s a symmerc mode for cases I and III and an ansymmerc mode for cases II and IV. We calculaed he modal ransmsson nenses nm ( ' ), whch are defned nm as he ransmsson from mode m o mode n n a counerclockwse (clockwse) loop, where he subscrp s denoes he symmerc mode and a he ansymmerc mode. he 4 modal ransmsson nenses can reveal he behavor of mode swchng. Fgure 3(b) plos he calculaed ransmsson nenses of he proposed sysem a 9 GHz as a funcon of B m for he four cases. In each plo, he lef, mddle (shaded), and rgh regons correspond o he bas feld srenghs a whch zero, one, and wo EPs are encrcled respecvely. We noe ha he sysem s sll recprocal (.e., ' ) n he presence of he ransverse bas feld snce he nm mn cross secon of he coupled wavegudes has a mrror symmery wh respec o he plane y = 0 (see [43]; also see Supplemenal Maeral for dealed descrpons of he mrror symmery). We frs sudy he dynamcs of encrclng one EP for couner-clockwse loops. Case I n Fg. 3(b) shows ha as n he shaded regon (correspondng o one EP ss beng encrcled), so ha he ansymmerc mode domnaes he oupu. hs means ha a symmerc mode a he sarng pon s up beng an ansymmerc mode once he sysem has raveled one couner-clockwse loop n he parameer space. hs phenomenon s represenave of sae flppng due o he self-nersecng emann energy surface n non- Herman sysems. We noe ha he oupu s also an ansymmerc mode n case II, ndcang ha here s no

5 FIG. 4. Calculaed ampludes of he egensaes along he wavegudng drecon for (a) case I wh Bm = 0.08, (b) case II wh Bm = 0.08, (c) case I wh Bm = 0.01, and (d) case II wh Bm = 0.01, where cg and cl represen he coeffcen of he egensae projeced ono he lower-loss and hgher-loss emann shees, respecvely. he black dashed lnes n (c) and (d) show he exsence of a branch cu va whch he sae can cross from one emann shee o he oher. he NA s characerzed by he crossng of wo curves. sae flp n hs case. We ake he loop generaed a B m = 0.08 ha encrcles one EP (see Fg. 3(a)) as an example o explan he dynamcs. he smulaed feld dsrbuons (H y componen) n he wavegude sysem for cases I and II are shown n Fgs. 3(c) and 3(d), respecvely. In Fg. 3(c), we see he mode swchng,.e., a symmerc mode exced a he lef becomes an ansymmerc mode a he ex on he rgh. Bu mode swchng s no observed n Fg. 3(d). o beer undersand he dynamcs, we expand he ransverse feld dsrbuons f z no a lnear combnaon of he egenfelds (.e., rgh egenvecors) rg z and rl z of he confguraon a a parcular value of z. ha s, we wre f z c r z c r z, where c G and c L are ampludes, G G L L and he subscrps G and L are assocaed wh he egenmode wh a lower loss (a relave gan mode) and he egenmode wh a hgher loss (a relave loss mode), respecvely. he rgh egenvecors r z are ypcally no rg z and L orhogonal snce he sysem s non-herman. We consruc her correspondng lef egenvecors va l r r r G L G L G L LG r, and hen deermne he LG ampludes by projecng he ransverse feld dsrbuons ono he lef egenvecors (see Appx A for deals). he calculaed ampludes for cases I and II wh B m = 0.08 are ploed n Fgs. 4(a) and 4(b), respecvely. We fnd ha n case I he encrclng process s sable and adabac snce he sae evoluon akes place on he lower-loss emann shee (also see he black lne n Fgs. 1(d) and 1(e)) so ha c G domnaes n he whole process. For case II, however, he sae frs propagaes on he hgher-loss emann shee on whch he sae s known o be unsable [9-38]. here s a delay me [3] afer whch a NA occurs (also see he black lne n Fgs. 1(f) and 1(g)), correspondng o he breakdown 5 of adabacy [30-3]. Afer he NA, he sae propagaes on he lower-loss emann shee and no furher NAs occur. As a resul, he sae reurns o self a he of he loop because of he one NA. he oupu for couner-clockwse loops s herefore always an ansymmerc mode, ndepen of he symmery of he npu mode, when one EP s encrcled. By he same argumen, he oupu for clockwse loops (.e., cases III and IV) s always a symmerc mode (see Fgs. 3(e) and 3(f); also see Supplemenal Maeral for rajecores on he emann surface). hs s he so-called chral behavor of he ransmsson when one EP s encrcled [9,30,3-38],.e., he oupu deps solely on he encrclng drecon regardless of he symmery of njecon. As we can vary he bas feld srengh o conrol he sze of he loop and our sysem has wo EPs, we can hen sudy he dynamcs when zero or wo EPs are encrcled. Fgure 3(b) ndcaes ha n he wo non-shaded regons, he oupu mode s he same as he njecon for all four cases as long as he loop s nowhere near he EP. he dynamcs urns ou o be raher complex. We ake he loops generaed a B m = 0.01 and B m = 0.17 as examples o nvesgae he dynamcs. Fgures 3(g) and 3(h) show respecvely he H y feld dsrbuons n he wavegude sysem for cases I and II a B m = Alhough n boh cases he sae reurns o self afer compleng he loop, hey exhb dfferen dynamcs. o llusrae hs pon, we plo n Fgs. 4(c) and 4(d) he correspondng ampludes of he egenmodes n he evoluon process. he evoluon process n case I s adabac so ha he sae reurns o self snce he loop does no enclose any EP. In case II, however, he dynamcs s hghly non-adabac and wo NAs occur hroughou he process. As a resul, he mode symmery says he same. he dfference n he number of NAs can be undersood nuvely by drawng he rajecores of he sae evoluon on he emann surface for cases I and II a B m = 0.17, correspondng o an encrclng of wo EPs. Consderng he opologcal srucure of our sysem, encrclng zero or wo EPs should no make any dfference o he behavor of mode swchng because he chraly of one EP cancels he chraly of he oher snce hey are derved from he same DP. he sae acqures a geomerc phase when wo EPs are encrcled, alhough hs s unrelaed o he symmery of he oupu mode. We frs consder case II (yellow lnes n Fgs. 1(f) and 1(g)). A he begnnng, he sae says on he hgherloss emann shee unl he frs NA occurs, afer whch he sae jumps o he lower-loss shee on whch becomes sable. Laer a z = ~00 mm, he sae re-eners he hgherloss emann shee va he branch cu (also see Fg. 4(d)) and becomes unsable agan unl he second NA occurs. A oal of wo NAs occur n hs hghly non-adabac process. he evoluon process n case I (yellow lnes n Fgs. 1(d) and 1(e)) s que dfferen snce a frs he sae propagaes on he lower-loss shee. I s no unl he sae crosses over he branch cu (also see Fg. 4(c)) ha eners he hgher-loss shee. Ineresngly, he expeced NA does no occur alhough n he res of he process he sae s no sable. hs s because he delay me exceeds he me spen on he hgher-loss shee, ndcang ha he expeced NA may occur f we ncrease he lengh of he sysem (see Supplemenal Maeral for a dealed dscusson). he resuls

6 FIG. 5. (a) A phoograph of he fabrcaed coupled YIG wavegudes. Wavegude-1 measures W H L = 8 mm 4 mm 400 mm, whle wavegude- measures α(z)w H L wh he profle of α(z) shown n Fg. (d). he gap dsance s g = ~0.5 mm. Mcrowave absorbers wh he dmensons of ~ mm 1 mm 00 mm are aached o he sde of wavegude- o nroduce loss. (b)-(c) Expermenally measured phase dfferences a varous bas felds Bm and frequences for case I (b) and case III (c). (d)-(e) Numercally smulaed phase dfferences as a funcon of he bas feld Bm and frequency for case I (d) and case III (e). he wo dashed lnes mark he calculaed locaons of EPs whch paron he map no hree regons depng on he number of EPs encrcled. he phase dfference was calculaed based on he obaned ransmsson nenses (e.g., for case I). of cases III and IV can be smlarly undersood (see Supplemenal Maeral for rajecores on he emann surface). he number of NAs, denoed by N NA, s summarzed n Fg. 3(b) for he four cases. We performed mcrowave expermens o demonsrae he above effecs. A phoograph of he fabrcaed samples s shown n Fg. 5(a) (see he fgure capon for dealed parameers). he YIG wavegudes were made from pure YIG wh a sauraon magnezaon of 4πM s 1884 G (produced by Nanjng B ao Elecronc echnology Co., Ld.). he YIG wavegude- was creaed from a larger sample usng a hand polshng machne and followed he shape desgned n Fg. (d). he mcrowave absorber s aached o only half of YIG wavegude-. hs has been shown o be an effecve way o mnmze he dsspaon of he sysem whle keepng he opology of he sysem nac (see ef. [35]; also see Supplemenal Maeral for a dscusson on he performance of such a sysem). We consder he phase dfference 1 as he creron o deermne he symmery of he oupu mode, where 1 ( ) s he phase measured a he oupu sde of wavegude-1 (wavegude-). By defnon, 0 corresponds o a symmerc mode whereas 180 an ansymmerc mode. In he expermen, he symmerc FIG. 6. (a)-(b) Expermenally measured phase dfferences a varous bas felds Bm and frequences for case II (a) and case IV (b). (c)-(d) Numercally smulaed phase dfferences as a funcon of he bas feld Bm and frequency for case II (c) and case IV (d). njecon was exced usng an ~0 mm long anenna, whle he ansymmerc njecon was exced usng wo ~8 mm long anennas whch were conneced o he source va a oneo-wo power spler and placed along oppose drecons so ha her currens were oscllang ou of phase. An anenna ~8 mm n lengh was placed a he ex of wavegude-1 and wavegude- o deec her correspondng phases 1 and. All of he anennas were conneced o an Aglen echnologes 870ES Nework Analyzer o record he ransmsson nensy and phase. he measured phase dfferences as a funcon of he exernal feld srengh (B m) and frequency are shown n Fgs. 5(b) and 5(c), respecvely, for cases I and III n whch a symmerc mode s njeced. We noe n Fg. 5(b) ha for each frequency above ~8 GHz here s a specfc range of B m (n red) whn whch he sysem exhbs a sae flp. hs specfc range shfs owards larger bas felds for lower frequences. Fgure 5(d) shows numercal smulaon resuls for case I, whch agrees well wh he measuremen. In he smulaon, he relave permvy of he absorber s se o 3+10 whch can bes mach he expermenal resuls. We also deermne for each frequency he locaon of he EPs n he parameer space and mark hem wh he wo whe dashed lnes n Fg. 5(d). he whole map s paroned wh hese EP rajecores no hree regons depng on he number of EPs encrcled. he varaon n he oupu mode symmery wh ncreasng bas feld ndeed reflecs a change n he number of EPs encrcled n he parameer space. In conras, he oupu n case III s always a symmerc mode regardless of he number of EPs encrcled (Fgs. 5(c) and 5(e)). hs hus demonsraes expermenally he breakdown of adabacy. he devaon beween expermenal and numercal resuls comes from he mperfecness of sample whch s made by hand polshng. In addon, he npu mode s exced usng anennas placed ousde he wavegude, and as such, s symmery can only be approxmaely correc. However, even by jus comparng he expermenal resuls 6

7 sysem can be appled o he swchng of modes conrolled wh exernal felds,.e., manpulang he symmery of he oupu sae by dynamcally encrclng dfferen numbers of EPs. Noe ha he mcrowave absorber n our desgn s aached on he YIG wavegude- wh a varyng wdh. We can also aach he absorber on he sragh YIG wavegude- 1 and he physcs s he same. FIG. 7. (a) Calculaed magnary par of he effecve mode ndex as a funcon of he bas feld and scale facor of he sysem a 11.5 GHz wh srucure parameers: W = 8 mm, H = 4 mm, g = 0.5 mm, w = mm, and h = 3 mm. he relave permvy of he absorber s se o he yellow and black lnes mark he sae evoluon rajecory for confguraons A and B (see ex for defnon), respecvely, and he whe dashed lne marks he branch cu. (b) Same as hose n (a) excep ha he rajecores are for confguraons C and D. hemselves (Fgs. 5(b) and 5(c)), here s obvously a marked dfference for he case of encrclng one EP. he phase dfferences for cases II and IV njeced wh an ansymmerc mode are shown n Fg. 6. All hese resuls are conssen wh he analyss n Fg. 3, convncngly demonsrang he behavor of mode swchng when dfferen numbers of EPs are encrcled,.e., a chral behavor s found when one EP s encrcled and no sae flp occurs when zero or wo EPs are encrcled. esuls of a conrol expermen are gven n Supplemenal Maeral. he chral naure of he dynamcs of encrclng one EP has been exploed for asymmerc mode swchng [35,37]. Snce he exernal feld n hs work can be uned connuously, our IV. SAING/END POIN IN HE BOKEN PHASE: NON-CHIAL DYNAMICS In he prevous secon, we have explored he dynamcal behavor when zero, one, or wo EPs are dynamcally encrcled. he sarng/ pon lay n he symmerc phase, whch s also he confguraon explored n all prevous works [34-38]. In hs secon, we show ha when he sarng/ pon moves o he broken phase, he dynamcal encrclng would resul n a non-chral ransmsson behavor, n sark conras o he chral behavor when he sysem sars from a pon n he symmerc phase. We frs descrbe he prncple behnd he sysem desgn. he sarng/ pon s sll fxed a B 0 = 0 and α = 1 for ease of expermenal realzaon. o fulfl hs requremen, he DP n he lossless sysem should be locaed close enough o he zero-bas feld. We fnd n Fg. 1(b) ha hgher frequences mee hs requremen so we se he frequency o 11.5 GHz and choose he followng sysem parameers: W = 8 mm, H = 4 mm, and g = 0.5 mm. he DP s hen locaed a B 0 = (black crcle n Fg. 1(b)), whch s also approxmaely he cener of he broken phase when mcrowave absorbers are aached [4]. We should choose a sronger absorber o ensure ha he lossy sysem says n he FIG. 8. (a) Loops n he parameer space generaed wh Bm = 0.03 (yellow loop no enclosng any EP) and 0. (black loop enclosng one EP). he sarng/ pon les a B0 = 0 and α = 1, correspondng o he broken phase. he black dashed lne represens he broken phase lne where he real pars of he egenvalues coalesce. (b) Calculaed ransmsson nenses as a funcon of Bm for couner-clockwse loops and clockwse loops wh a gan mode as he njecon. he shaded regon represens he area where one EP s dynamcally encrcled. he number of NAs n he dynamcal process s gven n dfferen regons. (c) Same as hose n (b) excep ha he njecon s a loss mode. (d)-(g) Numercally smulaed Hy feld dsrbuons n he wavegude sysem for confguraons A-D (see ex for defnon). he black dashed lnes and red dashed lnes mark he NA and branch cu, respecvely. Sysem parameers are he same as hose gven n Fg. 7. 7

8 FIG. 9. (a)-(d) Calculaed ampludes of he egensaes along he wavegudng drecon for confguraons A-D. broken phase regon a B 0 = 0 and α = 1. o verfy he desgn concep, we calculaed he effecve mode ndex of he sysem wh a sronger absorber (w = mm, h = 3 mm, ε = 4+15) and show he emann surface n Fg. 7(a) (magnary par). here s a large gap beween he wo emann shees a B 0 = 0, confrmng ha he sarng/ pon ndeed les n he broken phase, where one egenmode s nearly lossless (see he blue shee) and he oher one more lossy (see he red shee). hs s a resul of symmery breakng,.e., he power flow of he lossless/lossy mode manly propagaes n he lossless/lossy YIG wavegude. As expeced, here s only one EP whch les a B 0 = 0.11 and α = 1.0 (also see he parameer space n Fg. 8(a)). he ransmsson nenses of he proposed sysem (see Fgs. (a) and (b) for he schemac dagram) wh he parameers menoned above are calculaed as a funcon of B m o nvesgae he behavor when he EP s encrcled wh he sarng/ pon n he broken phase. he ransmsson nensy nm ( ' ) s defned n he same way as ha n Fg. nm 3(b), excep ha here we use subscrps G and L o denoe he nearly lossless (.e., a relave gan ) mode and he lossy mode, respecvely. he resuls wh a gan mode njecon and a loss mode njecon are ploed n Fgs. 8(b) and 8(c), respecvely, n whch he regon where one EP s encrcled s shaded. Under each njecon, he resuls of counerclockwse loops and clockwse loops look almos he same, ndcang a non-chral ransmsson behavor whch s dsnc from he chral behavor found when he sarng/ pon s n he symmerc phase (see Fg. 3(b)). More neresngly, we fnd ha he oupu s always a gan mode, regardless of he deals such as he npu, encrclng drecon, or even he number of EPs encrcled. o nvesgae he underlyng physcs, we sudy four confguraons n hs secon. Confguraons A and B are couner-clockwse loops generaed a B m = 0.03 and 0., correspondng o an encrclng of zero and one EP, respecvely, wh a gan mode as he njecon (see Fg. 8(b)). I s he same for confguraons C and D bu wh a loss mode as he npu (see Fg. 8(c)). Fgures 8(d)-8(g) show he H y feld profles n he 8 wavegude sysem for confguraons A-D, and he ampludes of her egenmodes exraced from he feld profles are ploed n Fgs. 9(a)-9(d), respecvely. We frs analyze he small encrclng loop ha excludes he EP. Confguraon A s he smples case n he sense ha he sae evoluon says all he me on he lower-loss emann shee (see he yellow lne n Fg. 7(a)). As a resul, he dynamcal process s sable and adabac (Fg. 9(a)), as verfed by he calculaed resuls showng a concenraon of power flow n YIG wavegude-1 n he whole process (Fg. 8(d)). Confguraon C s dfferen n ha a loss mode s njeced. he process s unsable a frs unl a NA o he lower-loss emann shee occurs, and he sae becomes sable for he res of he process (see he yellow lne n Fg. 7(b)). hs NA can be seen from he feld profles n Fg. 8(f). I s characerzed by a power ransfer from wavegude- o wavegude-1 (see he black dashed lne; also refer o Fg. 9(c)). Confguraons B and D n whch one EP s encrcled exhb raher complex dynamcs. Confguraon B sars wh a sable evoluon process. As he sae encrcles he EP, eners he hgher-loss emann shee va he branch cu. A NA hen occurs afer some delay me, causng he sae o jump o he lower-loss shee, afer whch he sable sae arrves a he pon as a gan mode. he rajecory of hs process s ploed wh a black lne n Fg. 7(a), accordng o he smulaed feld profles n Fg. 8(e) and ampludes of he egenmodes n Fg. 9(b). he branch cu s characerzed by a power ransfer from wavegude-1 o wavegude- (see he red dashed lne n Fg. 8(e)). Confguraon D has he mos complex dynamcs. he sae s unsable a frs so ha jumps o he lower-loss shee va a NA. he followng process s he same as ha of confguraon B,.e., he sae re-eners he hgher-loss shee va he branch cu, experences a second NA and reaches he pon as a gan mode (see he black lne n Fg. 7(b); also see Fgs. 8(g) and 9(d)). he number of NAs obaned from he above analyss s summarzed n Fgs. 8(b) and 8(c), whch shed lgh on he complex ransmsson behavor. When a gan mode s njeced, confguraon A exhbs he hghes ransmsson nensy snce he sae evoluon s always on he lower-loss shee. As he bas feld s ncreased o enlarge he encrclng loop, he EP can be encrcled. he sae s hen able o clmb up o he hgher-loss shee so ha he ransmsson drops consderably. he delay me of he unsable sae on he hgher-loss shee s deermned by he sysem parameers, especally he absorber properes. he ransmsson dp n Fg. 8(b) (~B m = 0.17 ) can hus be nerpreed as a process feaurng he larges energy aenuaon consderng boh he encrclng loop and he delay me. I s also evden ha he ransmsson nensy should be much lower when a loss mode s njeced, e.g., confguraons C and D. We performed expermens o verfy he above analyss. In he expermens, he gan mode and loss mode were exced by pung an ~8 mm anenna near he enrance of wavegude-1 and wavegude-, respecvely. he measured ransmsson specra a 11.5 GHz are shown n Fgs. 10(a) and 10(b), whch agree well wh he numercal resuls n Fgs. 8(b) and 8(c), confrmng he non-chral ransmsson behavor. We also measured he elecrc feld nensy o elucdae he NAs n he dynamcal process. In he

9 FIG. 10. (a)-(b) Expermenally measured ransmsson nenses a 11.5 GHz as a funcon of Bm wh a gan mode (a) or a loss mode (b) as he njecon. he sysem parameers are W = 8 mm, H = 4 mm, and g = 0.5 mm. A mcrowave absorber sronger han he one used n Fg. 5(a) wh he dmensons of ~3 mm mm 400 mm s aached o wavegude-. (c)-(f) Expermenally measured surface elecrc feld nenses along he wavegudng drecon a 11.5 GHz for dfferen values of Bm. esuls for couner-clockwse loops wh a gan mode as he njecon are shown n (c) and (d) for wavegude-1 (WG1) and wavegude- (WG), respecvely, whle resuls for couner-clockwse loops wh a loss njecon are shown n (e) and (f). expermenal measuremen, we pu an ~8 mm long anenna on op of each YIG wavegude o measure her correspondng elecrc feld nensy as a funcon of z. he measured resuls of couner-clockwse loops wh a gan njecon a dfferen B m values are shown n Fgs. 10(c) and 10(d), respecvely, for wavegude-1 and wavegude-. We fnd n Fg. 10(d) ha he feld nensy n wavegude- s very weak a z = 0. In he range B m > ~0.15, here s a consderable ncrease n he feld nensy a he cener of he sysem (z = ~0 mm). hs s a ypcal feaure of he branch cu (see he dashed ellpse) and confrms he dynamcal encrclng of one EP n expermen. he sae hen clmbs up o he hgher-loss emann shee va he branch cu so ha becomes unsable allowng a NA o occur, as shown by he drasc decrease n he feld nensy a z = ~30 mm (see he dashed ellpse). he number of NAs s herefore a good ndcaor of he number of EPs encrcled. he same measuremens bu wh a loss njecon are shown n Fgs. 10(e) and 10(f). he frs NA appears a z = ~5 mm for all values of B m (see he whe dashed ellpse n Fg. 10(f)), afer whch he sae jumps o he lower-loss emann shee assocaed wh a sudden ncrease n he feld nensy n wavegude-1 (see Fg. 10(e)). he followng dynamcs s he same as ha wh a gan njecon,.e., he sae re-eners he hgher-loss shee va he branch cu and experences a 9 second NA (see he wo dashed ellpses n Fg. 10(f)), for he loops enclosng one EP only. he expermenally measured ransmsson specra and number of NAs exraced from he feld profles srongly suppor he numercal smulaons and demonsrae he non-chral behavor when he sarng/ pon les n he broken phase. V. HEOEICAL DEMONSAION OF NON- CHIAL DYNAMICS In hs secon, we consder he me evoluon of a smple non-herman Hamlonan o show ha he dynamcs s non-chral when he sarng pon les n he broken phase. We consder a wo-sae sysem governed by H, where he generc me-depen Hamlonan has he form g H, (1) g and a, b s he sae vecor a me. I s easy o see ha g() and δ() represen respecvely he amoun of gan/loss and deunng, and he couplng srengh s denoed by κ whch for smplcy s se o be -1. We use hs smple Hamlonan o hghlgh he fac ha he

10 FIG. 11. (a) he g-δ parameer space where a par of EPs locaes a g = ±1 and δ = 0. he crcle wh a radus ρ depcs a rajecory o encrcle he EP wh he sarng pon n he broken phase. he red lne and green lne denoe he broken phase and symmerc phase, respecvely. (b) Calculaed as a funcon of ρ wh γ = 0.1 (counerclockwse loops) and -0.1 (clockwse loops). he regon wh ρ < corresponds o he dynamcal encrclng of one EP whereas ha wh ρ > corresponds o he encrclng of wo EPs. We performed four calculaons (.e., counerclockwse/clockwse loop wh a gan/loss npu) and he resuls are all he same as shown by he black crcles. he red lne shows he value of as a funcon of ρ, whch maches well wh he black crcles, ndcang ha he fnal sae s always a gan sae. (c)-(f) Calculaed ampludes of he egenvecors for (c) couner-clockwse loop wh a gan npu, (d) couner-clockwse loop wh a loss npu, (e) clockwse loop wh a gan npu, and (f) clockwse loop wh a loss npu. In he calculaons, we choose ρ = 1 and γ = ±0.1, correspondng o he dynamcal encrclng of an EP. phenomenon we have observed s raher generc, and no jus specfc o our parcular expermenal confguraon. A woparameer space wh g and δ s shown n Fg. 11(a), where we have a par of EPs a g = ±1 and δ = 0. he red lne and green lne correspond o he broken phase and symmerc phase, respecvely. We consder an encrclng loop parameerzed by g 1 cos, sn, () where ρ denoes he radus of he loop (see Fg. 11(a)), and γ s a measure of adabacy. A posve γ leads o a counerclockwse loop whereas a negave γ a clockwse loop. he sarng pon and pon are chosen a 0 / and, respecvely, so ha hey boh le n he broken / phase. here are wo egenmodes,.e., a gan mode and a loss mode, a he sarng/ pon. he correspondng egenvalues are G and L, whle he rgh egenvecors are G 1, 1 and L 1, 1. We frs calculaed he evoluon of he sae vecor by numercally solvng he me-depen equaon. he sae vecor a each me sep can be decomposed as a sum of he nsananeous rgh egenvecors,.e., CG G CL L, where G and L are nsananeous rgh egenvecors ha can be solved from he nsananeous Hamlonan, and her correspondng ampludes C and G C can be obaned by projecng he L sae vecor ono he lef egenvecors. hs process s exacly he same as ha for calculang he ampludes of he nsananeous egenmodes n he coupled wavegude sysem (see Fgs. 4 and 9). he ampludes of he nsananeous egenvecors wh ρ = 1 and γ = ±0.1, correspondng o he dynamcal encrclng of an EP, are shown n Fgs. 11(c)-11(f) for dfferen npu modes and encrclng drecons as ndcaed n he fgures. he blue lnes are assocaed wh he gan egensae whle he red dashed lnes he loss egensae. We can nfer from he resuls ha he oupu s always domnaed by he gan egensae, regardless of he npu sae and encrclng drecon. For any npu sae, he dynamcs for couner-clockwse and clockwse loops are nearly he same. here s one NA when a gan sae s njeced whle wo NAs wh a loss sae njecon. he resuls of hs smple model well reproduce he feaures of he coupled wavegude sysem (see Fgs. 9(b) and 9(d)). A more rgorous way o denfy he oupu sae s o calculae b / a as a funcon of ρ. We fnd no maer whch sae s njeced and whch drecon he encrclng akes, he resuls are he same as shown by he black crcles n Fg. 11(b) where we fx γ = ±0.1. We know he gan sae has he egenvecor G 1, 1 so ha he correspondng rao b/ a 1. hs expresson s ploed as a funcon of ρ n Fg. 11(b) by he b / a, ndcang red lne whch concdes wh ha he fnal sae s always a gan sae when he sarng pon les n he broken phase, no maer wheher one or wo EPs are encrcled (.e., ρ < for encrclng one EP and ρ > for wo EPs). In fac, hs preferred fnal sae and he correspondng non-chral dynamcs can be proved mahemacally by b / a. he above dervng an analycal form of model Hamlonan and rajecory n he parameer space have been analyzed recenly [34], where he auhors suded he dynamcs wh he sarng pon n he symmerc phase and derved a closed-form expresson of he sae evoluon. We adop he same mehod o sudy our case. he key o he dervaon s o frs recas Eq. (1) no a second order dfferenal equaon for a(), e.g., d a / d e e a 0, whch can furher be reduced o a degenerae hypergeomerc 10

11 dfferenal equaon. We frs consder γ > 0 and he soluon can be wren as a sum of confluen hypergeomerc funcons of he frs knd F and second knd U. By applyng nal condons, he sae vecor can be expressed n he form of a ransfer marx, a b M1 M M3 a0, b0, (3) / e 1 / e wh Г beng he gamma where funcon and he marces are 0 0 F U M1, (4a) F e F / U e U / U / / U / / U / M, (4b) F / / F / / F / 1 0 M 3, (4c) 1 / / where F (n) and U (n) represen confluen hypergeomerc F n /, n 1, e / and funcons [44] U n /, n 1, e / respecvely. he mahemacal echnques used o solve he dfferenal equaon can be found n ef. [34]. Our formulas are slghly dfferen from hose n ef. [34] (see Eqs. (6a)-(6c) here) snce here we have he nal condon 0 / (.e., sarng pon n he broken phase) whereas he sarng pon n ef. [34] les n he symmerc phase wh 0 0. We now ake a closer look a Eqs. (4a)-(4c). We focus on he fnal me sep / and we nroduce a marx M M M M wh marx elemens (see Appx B 1 3 for deals) m 4 F F F / / F / / / / F / U / F / U / 0 0 m1 F / F, (5b) / / 8 m F F F F / / / / / / 8 F 3 / 1 1 / F / 4 m F F F F / / / / / /, (5a), (5c). (5d) F / U / F / U / I s dffcul o furher smplfy he above formulas bu s nsrucve o consder some lmng cases. Here we choose a fne γ and le, correspondng o he dynamcal encrclng of wo EPs. A bg enough ρ can make he sysem parameers change slowly enough so ha wll no nroduce non-adabacy no he sysem, whch means he non-adabacy (f any) only comes from he non- Hermcy nduced by he gan and loss. In he lm F p, p, z and, we have z for 1 U p p z snce z / 1,,. hen we can use he 11 asympoc expansons of p F p, p, z z 1 p / p p and 1 1 1,, n he lm z (see Eqs. (4.1.3) U p p z z p 1 and (4.1.1) n ef. [44]), whch leads o 0 F / / / /, (6a) 1 / / 0 / / /, (6c) / F / / / / 4, (6b) 1 U U / / /. (6d) Inserng hese asympoc forms no Eqs. (5a)-(5d) can help smplfy he expressons of he marx elemens. We fnd m m m and only m1 0 (see Appx C for deals). he fnal sae / a b m a b a hen akes he form m b / 0 m11a 0 m1b 0 m1b 0. (7) hs analyc resul demonsraes ha no maer wha sae s b / a 0 njeced, he fnal sae always has when. Meanwhle, he rao of he egenvecor elemen b/ a for he gan sae (.e., a 1, b 1 ) and loss sae (.e., a 1, b 1 ) s, respecvely, 0 and n he lm. We can herefore conclude ha he fnal sae s always a gan mode for he generc model descrbed by Eq. (1). he case of γ < 0 can also be proved o have b / a 0 usng a smlar process. hs demonsraes he non-chral dynamcs when he sarng pon les n he broken phase. VI. DISCUSSION ON HE OLE OF HE SAING/END POIN As we have demonsraed chral and non-chral dynamcs n Secs. III and IV, we dscuss he role of he sarng/ pon n hs secon. he key o undersandng he dynamcs n he encrclng process s he NA, whch may occur f here s more han one egensae n he non-herman sysem and he predomnan egensae s no he one wh he lowes loss. he sae n he dynamcal process s sable only f s on he emann shee wh he lowes loss. Once he sae clmbs up o a hgher-loss emann shee va he branch cu (e.g., see confguraons B and D n Fgs. 7-10), becomes unsable bu a NA does no occur mmedaely. here s a ceran sysem parameer-depen delay before a NA occurs, and hs delay me plays a key role n he dynamcal process. We have demonsraed boh numercally and expermenally ha he delay me can always be accessed n he sysems suded n hs work when one EP s encrcled (see he sae rajecores n Fgs. 1 and 7). hs fac mples ha when he sae approaches he pon, would be on he lower-loss emann shee (.e., he blue shee n Fgs. 1 and 7), and he deals of he prevous dynamcal process such as he njeced mode and he number

12 of NAs would all be forgoen by he sysem. As a resul, he fnal sae s solely deermned by he encrclng drecon. We noe n Fg. 1(d) ha n he symmerc phase lne, he blue shee s dsconnuous so ha when he sarng/ pon les here, couner-clockwse loops resul n an ansymmerc oupu whereas clockwse loops a symmerc oupu, correspondng o a chral ransmsson behavor. When he sarng/ pon moves o he broken phase where he blue shee s connuous (see Fg. 7(a)), counerclockwse loops and clockwse loops gve he same oupu,.e., he gan mode, showng a non-chral ransmsson behavor. he chral and non-chral dynamcs can also be undersood usng he heorecal model proposed n Sec. V. I was shown n ef. [34] ha when he encrclng drecon s reversed, he fnal sae can be obaned by smply employng a ransformaon o he sae vecor * * a, b a, b. When he sarng/ pon les n he symmerc phase (.e., = 0), he egenvecors are 1 1, e and 1, e, where arcsn 1. I s easy o fnd 1 and by dong he above ransformaon. he 1 dynamcs s chral,.e., changng he encrclng drecon flps he fnal sae. he suaon s que dfferen f he sarng pon s n he broken phase where he egenvecors are G 1, 1 L 1, 1 and. Performng he above ransformaon leads o G and G L, L ndcang ha reversng he encrclng drecon does no affec he fnal sae, whch s exacly he non-chral dynamcs found n hs work. hs mahemacal nerpreaon shows ha he chral and non-chral dynamcs are relaed o he properes of he egenvecors n he symmerc and broken phase. he above analyss acually apples o loops ha enclose any number of EPs, provded ha he NA occurs each me when he sae s on he hgher-loss shee. In fac, we have observed he non-chral dynamcs when zero EP (see Fgs. 8(b) and 8(c)) and wo EPs (see Fg. 11(b) and Eq. (7)) are dynamcally encrcled wh he sarng pon n he broken phase. For he case wh he sarng pon n he symmerc phase, we noe n Fg. S3(a) (see Supplemenal Maeral) ha when he wavegude s longer (L = 1000nm), he dynamcs s always chral, ndepen of wheher zero, one or wo EPs are encrcled. However, he chral dynamcs s no observed n our expermenal sysem (L = 400nm) when zero and wo EPs are encrcled (see Fgs. 5 and 6) whch s due o he fac ha our sysem s no long enough for he requred NA o occur. We noe ha a very recen paper [45] (wh sarng pon n he symmerc phase) also poned ou ha he chral dynamcs can be observed when he loop does no encrcle any EP n he lm of very slow cycles, whch s conssen wh our analyss. A naural queson o ask s wha he fnal sae would be f he sarng/ pon les somewhere far away from boh he symmerc and broken phases. Alhough he above 1 analyss ndcaes ha he oupu s lkely o be he mode wh a lower loss, hs s sll an open queson snce he delay me s no always accessble. A sably loss delay was nroduced n ef. [3] o sudy he dynamcal encrclng of EPs and analycal form of he delay me for smple examples was derved. However, deermnng he delay me n realsc non-herman sysems remans a very complcaed ssue ha needs furher nvesgaon. VII. CONCLUSION In summary, we have shown boh numercally and expermenally ha a par of ferromagnec wavegudes appled wh non-unform bas magnec felds serves as a good plaform o sudy dynamcal processes n non- Herman sysems. Such a sysem has wo EPs and hence energy surfaces wh a more complex opology. he rajecory of he sae n he parameer space can be conrolled n su, as demonsraed expermenally. Usng he proposed sysem, we have demonsraed expermenally he chral dynamcs when one EP s encrcled. We can also dynamcally encrcle more han one EP expermenally o reveal he opologcal srucure of he sysem possessng mulple EPs. More mporanly, we revealed ha wheher he so-called chral behavor can be observed deps on he locaon of he sarng/ pon of he encrclng loop. When he sarng/ pon moves o he broken phase, he sysem exhbs non-chral dynamcs. We have proposed a heorecal model o nerpre he underlyng physcs. Our resuls clarfy he role of he sarng/ pon n he dynamcal process of encrclng EPs. he proposed sysem can be appled o mode swchng conrolled wh an exernal parameer whou changng or movng he sample. he plaform can also be used o sudy more complex dynamcs n non-herman sysems such as he encrclng of hghorder EPs. ACKNOWLEDGEMENS We hank Prof. Z.Q. Zhang and Dr..Y. Zhang for her valuable commens and suggesons. hs work was suppored by he Hong Kong esearch Grans Councl hrough gran no. AoE/P-0/1. X.-L.Z. was also suppored by he Naonal Naural Scence Foundaon of Chna (gran no ) and he Chna Posdocoral Scence Foundaon (gran no. 016M591480). B.H. was also suppored by he Naonal Naural Scence Foundaon of Chna (gran no ) and he vsng scholarshp program for young scenss n Collaborave Innovaon Cener of Suzhou Nano Scence and echnology and he Prory Academc Program Developmen (PAPD) of Jangsu Hgher Educaon Insuons. APPENDIX A: CONSUCING LEF EIGENVECOS here are wo egenmodes n he wavegude sysem propagang along he posve z-axs. her ransverse elecrc and magnec felds are denoed by E, E and H, H, where and he superscrp ndcaes ha hey are rgh egenvecors. he nner produc of he wo

13 rgh egenvecors n he wavegude confguraon s defned as an negraon over he enre wavegude cross secon S: 1 * * x, y x, y x, y x, y ds 4 S E H E H z. (A1) We have 0 snce he sysem s non-herman. he correspondng lef egenvecor can hen be consruced va where we have defned E E E L H H H / L, (A) / 1 e, *, S E x y H x y z ds. (A3) I s easy o verfy ha S S * * L L E x, y H x, y E x, y H x, y zds 0, * * L L E x, y H x, y E x, y H x, y zds 1 (A4) whch sasfes he orhogonal relaon beween lef egenvecors and rgh egenvecors. Consder he ransverse feld dsrbuons as a lnear combnaon of he egenfelds: x, y c x, y c x, y x, y c x, y c x, y E E E H H H. (A5) he amplude coeffcens can hen be solved by projecng he ransverse feld dsrbuon ono he lef egenvecors: 1 L * L c,,, * x y x y x y x, y ds 4 S E H E H z. (A6) In he smulaons, we frs performed full wave calculaons o oban all he feld componens n he sysem such as hose n Fgs. 8(d)-8(g). hen we performed egenmode analyss a each poson z o ge he rgh egenvecors of a unform wavegude of he same cross secon. Afer ha we consruced he lef egenvecors usng Eq. (A). Fnally, we projeced he ransverse feld dsrbuons a each poson z ono he correspondng lef egenvecors usng Eq. (A6) and we go he ampludes of he egenmodes whch were hen shown n, for example, Fgs. 9(a)-9(d) o help undersand he number of nonadabac ransons occurred n he process m11 U / F / F / U / F / U / U / F / m1 U / F / F / U / m1 F / U / U / F / 3 F / U / U / F / F / U / U / F / F / U / U / F / m F / U / U / F / F / U / U / F /. (B1) We use he properes of confluen hypergeomerc funcons o smplfy hese formulas. I s easy o fnd 0 0 F / F / and 1 1 F / F. On he oher hand, he prncpal value / of U p, p, z s n he nerval arg z 1. Apparenly, 0 U / and 1 U / are ou of hs range so ha we have o use a connecon formula (see Eq. (..0) n ef. [44]) U / F / U / /. (B) U / F / U / / Inserng Eq. (B) no Eq. (B1), we oban more smplfed expressons m11 F / F / F / F / / / F / U / F / U / 0 0 m1 F / F / /, (B3) m1 F / F / F / F / / / F 3 / F / / m F / F / F / F / / / F / U / F / U / whch are exacly Eqs. (5a)-(5d) of he man ex. APPENDIX C: DEEMINAION OF m11, m1, m1 and m Inserng Eqs. (6a)-(6d) no Eqs. (5a)-(5d), we have APPENDIX B: DEIVAION OF EQ. 5 Sarng from Eqs. (4a)-(4c), he elemens of he marx M a he fnal me sep / are 13