Bounded dynamics of finite PT-symmetric magnetoinductive arrays Mario I. Molina Departamento de F´ısica, MSI-Nucleus on Advanced Optics, and Center for Optics and Photonics (CEFOP), Facultad de Ciencias, Universidad de Chile, Santiago, Chile. Tel: 56-2-9787275, Fax: 56-2-2712973, email: [email protected] WeexaminetheconditionsfortheexistenceofboundeddynamicalphasesforfinitePT-symmetric arrays of split-ring resonators. The dimer (N = 2), trimer (N = 3) and pentamer (N = 5) cases are solved in closed form, while for N >5 results were computed numerically for several gain/loss spatial distributions. Itis found that theparameter stability windowdecreases monotonically with thesize of thearray. 3 1 The study of PT-symmetric systems has attracted a dow decreases with the size of the system. 0 lot of attention during the past few years. In these sys- Let us consider a simple model of a magnetic meta- 2 tems, the effects of loss and gain can balance each other material consisting of a finite one-dimensional array of n and, as a result, give rise to a bounded dynamics. These split-ring resonators including gain/loss terms: a J studies are based on the seminal work of Bender and 22 ctoonwioarnkserasr[1e,c2a]pwabhloe sohfowdiesdpltahyaintgnoanp-huerremlyitiraenalHeaigmenil-- ddτ22(qn+λ(qn+1+qn−1))+γnddτqn+qn =0 (1) valuespectrumprovidedthesystemissymmetricwithre- where λ is the magnetic interaction coefficient or cou- ] specttothe combinedoperationsofparity(P)andtime- S pling, and γ positive (negative) denotes a ring with n P reversal(T) symmetry. For one-dimensionalsystems the loss (gain). In order to satisfy the requirements for PT- . PTrequirementleadstotheconditionthattheimaginary symmetry,thespatialdistributionsofthegain/lossmust n part of the potential term in the Hamiltonian be an odd i beodd, γ−n = γn. Inthisworkwewillfocusinbinary- l function, while the real part be even. The system thus − n like systems with two gain/loss terms and thus examine [ dinegscfrribomedcaanPTexpseyrmiemnceetraicsppohnatsaene(oaullsesyigmenmveatlruyesbrreeaakl)- aarrrraayyss owfitthheafnormodd...n−umγb1,e−r γo2f,−rinγg1,s.0,γF1o,rγ2a,rγr1a,y.s..w, iftohr 1 to a broken phase (at least two complex eigenvalues), v even number of rings the distribution of gain/loss is of 1 as the gain/loss parameter is varied. To date, numer- the form ... γ1, γ2, γ1,γ1,γ2,γ1,.... This distinc- 9 ousPT-symmetricsystemshavebeenexploredinseveral tion is only m−eanin−gful−for small arrays and disappears 2 fields, from optics[3–7], electronic circuits[8], solid-state for system of infinite size. Hereafter, and without loss of 5 andatomicphysics[9,10],tomagneticmetamaterials[11], generality, we will focus on arrays with an odd number . 1 among others. The PT symmetry-breakingphenomenon of sites (except for the dimer case). Results for the case 0 has been observed in several experiments[7, 12, 13]. withanevennumberofsitesaresimilar. Sincethevalues 3 1 Magnetic metamaterials,on the other hand, consistof ofγ1 andγ2 arearbitrary,thegain/lossdistributionthus : artificial structures whose magnetic response can be tai- introducedallowsformanyinterestingcasestobe exam- v i lored to a certain extent. A common realization of such ined. Inparticularwewillfocusonthreecases. Thefirst X systemconsistofanarrayofmetallicsplit-ringresonators one is γ1 =γ, γ2 = γ, giving rise to the distribution − r (SRRs) coupled inductively[14–16]. This type of system ...γ, γ,γ, γ,0,γ, γ,γ, γ.... The second one is a − − − − canfeaturenegativemagneticresponseinsomefrequency γ1 = γ and γ2 = 0, which gives rise to the distri- window, making them attractive for use as a constituent bution ...0, γ,0, γ,0,γ,0,γ,0,.... Another interest- − − in negative refraction index materials[17]. A common ing case is the one with γ1 = γ = γ2 that gives problem with SRRs is the heavy ohmmic and radiative ... γ, γ, γ, γ,0,γ,γ,γ,γ.... Forthislastcase,we’ll − − − − losses. One possible solution is to endow the SRRs with see that in spite of the concentration of loss and gain on external gain, such as tunnel (Esaki) diodes[18, 19] to oppositesides,thedynamicsdoespossessastabilitywin- compensate for such losses. dow for finite arrays lengths. In the case of some one-dimensional coupled discrete Dimer case (N =2): In this case, the only possible case systems, such as a harmonic oscillator array,it has been is γ, γ. The dynamical equations read: observedthatinthelimitofaninfinitesizearray,thesys- − tem belongs to the brokenPT phase,i.e., there arecom- plexeigenvaluesmakingthedynamicsunbounded[11,20]. d2 d In this work we examine the case of short SRRs arrays, dτ2(q1+λq2)+γdτq1+q1 = 0 (2) and determine the parameter window inside which the d2 d system exhibits a bounded dynamics, and how this win- dτ2(q2+λq1)−γdτq2+q2 = 0 (3) 2 Π 1.6 1.2 oss1.2 E (cid:144)GainL0.8 W21.1 HAS 0 cal 1. P Criti0.4 0.9 -Π 0 0 0.1 0.2 0.3 0 0.4 0.8 1.2 0 0.1 0.2 0.3 lambda Gain(cid:144)loss Γ Figure 1: Dimer array. Left: Stability region (shaded) in Figure 2: Dimer array. Mode phase as a function of the gain/loss-coupling space. Right: Mode frequency squared as gain/loss parameter. Solid: Ω++ mode. Dotted: Ω+− mode. a function of the gain/loss parameter for λ=0.3. Short dash: Ω−− mode. Long dash: Ω−+ mode. (λ=0.33) We look for stationary modes q1,2(τ) = q1,2exp(iΩτ). with eigenvalues This leads to the equations, Ω2(q1+λq2)+iγΩq1+q1 = 0 (4) Ω = 1 (10) − ± −Ω2(q2+λq1)−iγΩq2+q2 = 0. (5) 2 γ2 4γ2+γ4+8λ2 1/2 Ω = − ± − (11) Theconditionofthevanishingofthedeterminantofthis ±" p2−4λ2 # linear system leads to a quadratic equation for Ω2, with solutions: The condition that all Ω be real leads to the condi- tions λ < 1/√2 and γ < γ = 2 √4 8λ2. Figure 1/2 c − − 2 γ2 γ4 4γ2+4λ2 3 shows the window of real eigenvalues in gain/loss- Ω= − ± − . (6) p ±" 2p(1−λ2) # ccoausep,libnugtstphaecea.lloIwteisdqcuoaulpitliantgiveinlytesrivmalilaisrstmoatlhleer.diTmheer We denote the four solutions as Ω++,Ω+−,Ω−+ and figure also shows the square of the real frequency as a −− function of gain/loss,for a given coupling value. Ω . The stable phase (unbroken PT symmetry) cor- responds to the cases where Ω is a real quantity. From Pentamer case (N = 5): Here the gain/loss distribu- straightforward examination of Eq.(6), one concludes tion can have three possible forms: γ, γ,0,γ, γ, or − − thatthestabilitywindowinγ-λspaceisgivenbythearea 0, γ,0,γ,0, or γ, γ,0,γ,+γ. We will focus on the − − − first case since is more amenable to an exact form so- under the curve γ = 2(1 √1 λ2), for 0 < λ < 1. c − − lution (Numerical results show that the other two cases Outside γ = γc(λ), thqe system is unstable. Figure 1 displaysimilarbehavior). Thestationarystateequations showsthestabilityregionandalsothesquarefrequencies asafunctionofthegain/lossparameter. Due tosymme- try considerations, only the positive γ-λ sector needs to be explored. 1.6 FromEq.(4)(orEq.(5))andEq.(6),itiseasytoobtain q2/q1 = 1 and thus, q1 and q2 differ by a phase only. oss1.2 1.2 | | L We have four branches for the phase, corresponding to (cid:144)n each of the four solutions. Figure 2 shows the phase of Gai0.8 W2 all solutions as a function of the gain/loss parameter. Critical0.4 1. Trimer case (N = 3): Here the gain/loss distribution 0 has the form γ,0,γ. The stationary state equations 0 0.4 0.8 1.2 0 0.2 0.4 have the form− lambda Gain(cid:144)loss Ω2(q1+λq2)+iγΩq1+q1 = 0 (7) Figure 3: Trimer array. Left: Stability region (shaded) in − gain/loss-coupling space. Right: Mode frequency squared as Ω2(q2+λ(q2+q3))+q2 = 0 (8) a function of thegain/loss parameter, for λ=0.3. − Ω2(q3+λq2) iγΩq3+q3 = 0. (9) − − 3 1.6 1.5 1.5 1.4 N=7 N=9 ss1.2 o L 1 1 (cid:144)n 1.2 Gai0.8 W2 Γ cal 0.5 0.5 Criti0.4 1. 0 0 00 0.4 0.8 1.2 0.80 0.2 0.4 0 0.4 0.9 0 0.4 0.9 lambda Gain(cid:144)loss 1.5 1.5 N=11 N=17 Figure 4: Pentamer array. Left: Stability region (shaded) in gain/loss-coupling space. Right: Mode frequency squared as 1 1 a function of the gain/loss parameter, for λ=0.3. Γ 0.5 0.5 have the form 0 0 Ω2(q1+λq2)+iγΩq1+q1 = 0 (12) 0 0.4 0.9 0 0.4 0.9 − Λ Λ Ω2(q2+λ(q1+q3)) iγΩq2+q2 = 0 (13) − − Ω2(q3+λ(q2+q4))+q3 = 0 (14) Figure 5: Stability regions (area under curve) in gain/loss- − −Ω2(q4+λ(q3+q5))+iγΩq4+q4 = 0 (15) c.o..u,p−liγn,g+γsp,−acγe, 0fo,r+γse,v−erγa,l+γa,r.ra.y.. lengths, for the case Ω2(q5+λq4) iγΩq5+q5 = 0. (16) − − with eigenvalues Case a. We start with the case γ1 = Ω = 1 (17) ± γ2 γ that gives rise to the distribution 2 γ2 4γ2+γ4+4λ2 1/2 −...,−γ≡,+γ,−γ,O,+γ,−γ,+γ,.... Some results Ω = − ± − (18) are shown in Fig.5. It is clear that, as the size of ±" p2−2λ2 # the array increases, the stability region shrinks, and 1/2 disappears altogether around N =20. This is consistent 2 γ2 4γ2+γ4+12λ2 Ω = − ± − (19) with a previous result[11] stating that in the infinity ±" p2−6λ2 # size limit, the dynamics is always unstable (that is, the system belongs to the broken PT phase). The condition that all Ω be real leads to the conditions Case b. Now we take γ2 0 and γ1 γ, that is, the λ < 1/√3 and γ < γ = 2(1 1 λ2). Figure 4 → ≡ c distribution ...,0, γ,0, γ,O,+γ,0,+γ,0,+γ,0,.... − − − − shows the stability windowqin gain/loss-coupling space, Notice how the gainandloss portions arenow separated p aswellasthesquarefrequencyasafunctionofgain/loss, and on each side they are rather diluted. The stability for a fixed coupling value. As we can see, the stabil- phase diagrams for this case (not shown) are qualita- ity window is substantially smaller than the one for the tively similar to the previous case, although, for a given dimer and trimer cases. array size, the stability windows are smaller. Case c. Now we take γ2 =γ1 γ. The gain/loss distri- ≡ Short chains (N > 5): Let us consider now the case bution is of finite arrays,where the stationary-state equations are ..., γ, γ, γ, γ,O,+γ,+γ,+γ,+γ,...Now the gain − − − − given by and loss on each side are densely populated and the re- sulting area of the stability windows (not shown) while Ω2[qn+λ(qn+1+qn−1)]+(1+iΩγn)qn =0.. (20) qualitatively similar to the previous cases, drop even − faster. The γ distribution we consider has the general form: n Figure 6 shows a summary of the results obtained for { } ... γ1, γ2, γ1,0,γ1,γ2,γ1,.... We will compute the thesizeofthestabilitywindowasafunctionofthearray − − − relevant eigenvalues numerically from the vanishing of length,forthethreecasesconsidered. 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