Revisiting the PT-symmetric Trimer: Bifurcations, Ghost States and Associated Dynamics
K. Li, P. G. Kevrekidis, D. J. Frantzeskakis, C. E. Ruter, D. Kip
aa r X i v : . [ qu a n t - ph ] J un Revisiting the PT -symmetric Trimer: Bifurcations, Ghost States and AssociatedDynamics K. Li and P. G. Kevrekidis
Department of of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003-9305, USA
D. J. Frantzeskakis
Department of Physics, University of Athens, Panepistimiopolis, Zografos, Athens 157 84, Greece
C. E. R¨uter and D. Kip
Faculty of Electrical Engineering, Helmut Schmidt University, 22043 Hamburg, Germany
In this paper, we revisit one of the prototypical PT -symmetric oligomers, namely the trimer. Wefind all the relevant branches of “regular” solutions and analyze the bifurcations and instabilitiesthereof. Our work generalizes the formulation that was proposed recently in the case of dimers forthe so-called “ghost states” of trimers, which we also identify and connect to symmetry-breakingbifurcations from the regular states. We also examine the dynamics of unstable trimers, as wellas those of the ghost states in the parametric regime where the latter are found to exist. Finally,we present the current state of the art for optical experiments in PT -symmetric trimers, as well asexperimental results in a gain-loss-gain three channel waveguide structure. PACS numbers:
I. INTRODUCTION
The study of PT -symmetry in both linear and nonlinear systems has received a continuously growing amount ofattention over the past 15 years. This effort originally stemmed from the realm of quantum mechanics [1, 2], where PT -symmetric Hamiltonians were proposed as presenting a viable alternative – also due to the fundamental natureof the corresponding parity ( P ) and time-reversal ( T ) symmetries – to the postulate of Hermiticity. Nevertheless,and while experimental realizations in the quantum mechanical framework are still less clear, a critical observationthat significantly advanced the field was made in the context of optics both theoretically [3] and experimentally [4, 5].Indeed, in this field, since losses are abundant and controllable gain is possible, an experimental synthesis of PT -symmetric Hamiltonians was demonstrated. This opened a new chapter in the relevant investigations by enablingthe interplay of linear such PT -symmetric Hamiltonians with the effects of nonlinearity. The latter is a featureubiquitously present in such optical systems and a theme of particular interest and complexity in its own right. This,in turn, spearheaded not only additional experimental investigations in optics [6] and in electrical circuit analogues ofsuch systems [7], but also paved the way for numerous significant theoretical contributions on the subject. As a smallsample among the many relevant topics, we highlight the facilitation of unidirectional dynamics [8], the analysis ofthe universality of the dynamics [9], the exploration of symmetry breaking effects [5, 10], the study of switching ofbeams [11], of solitons [12], the formation of symmetric and asymmetric bright solitary waves [13, 14], of breathers [15]and their stability [16], of dark solitons [17, 18], of vortices [17], as well as the emergence of ghost states [19–21] and thegeneralization of such ideas into vortex type configurations [22], PT -symmetric plaquettes [23] and higher-dimensionalmedia [24].One of the themes of particular interest within these studies concerns the so-called PT -symmetric “oligomers”.While most of the relevant attention was focused on dimers [8, 11, 19–21] (arguably due to the corresponding experi-mental explorations of [4, 5, 7]), configurations with more sites, such as trimers [10, 25] and quadrimers [10, 23, 25, 26],have also attracted recent interest. In the present work, our aim is to revisit one of these configurations, namely the PT -symmetric trimer. Both the optical [5] and the electrical [7] implementation of the corresponding dimer stronglysuggest that the experimental realization of such a trimer system may be feasible. Hence, it is particularly interest-ing and relevant to fully explore the PT -symmetric trimer, and provide analytical results complementing the earlierfindings of Ref. [10], as well as to report on the current state-of-the-art regarding a possible experimental realizationthereof in optics.Our presentation will be structured as follows. In section II, we will present the model and theoretical setup of thetrimer. We will devise analytical algebraic conditions that are relevant towards identifying the full set of standingwave solutions for this configuration. Importantly, in addition to the more standard stationary solutions, we willalso identify the so-called “ghost states” of the model [19–21]. These are states that, remarkably, albeit solutionsof the steady state equations, due to their complex propagation constant, are not genuine solutions of the originaldynamical equations. Nevertheless, as has been argued in the case of the dimer [21], these are waveforms of potentialrelevance in understanding the system’s dynamics. In section III, we will present the corresponding numerical results.In particular, we will seek both regular standing wave states and ghost states, and will build a full state diagramas a function of the gain/loss parameter γ of our PT -symmetric trimer. In addition to the existence propertiesof the obtained solutions, we will consider their stability (and potential instabilities/bifurcations) and, finally, wewill examine the system’s dynamics, how the instabilities are manifested, both in the case of the “regular” standingwave solutions and in that of the ghost states identified herein. In section IV we discuss possible realizations of PT symmetric optical systems (with a particular view towards trimers) and describe actual experimental limitations thathave to be overcome. Finally, in section V, we will summarize our findings and present our conclusions, as well assome directions for future study. II. MODEL AND THEORETICAL SETUP
The prototypical dynamical equations for the PT -symmetric trimer model read [10]: i ˙ u = − ku − | u | u − iγu i ˙ u = − k ( u + u ) − | u | u i ˙ u = − ku − | u | u + iγu . (1)Here, u j ( t ) ( j ∈ { , , } ) are complex amplitudes, dots denote differentiation with respect to the variable t (whichis the propagation distance in the context of optics), while k and γ represent, respectively, the inter-site couplingand the strength of the PT -symmetric gain/loss parameter. In the above equations it is assumed that the first sitesustains a loss at rate γ , while the third site sustains an equal gain. The middle site suffers neither gain, nor loss.Following the spirit of Refs. [10, 25], we start our analysis by seeking stationary solutions of Eqs. (1) in the form u = a exp( iEt ), u = b exp( iEt ) and u = c exp( iEt ), where E represents the nonlinear eigenvalue parameter. Thisway, we obtain from Eqs. (1) the following algebraic equations: Ea = kb + | a | a + iγa,Eb = k ( a + c ) + | b | b,Ec = kb + | c | c − iγc. (2)Let us now use a polar representation of the three “sites”, namely, a = A exp( iφ a ), b = B exp( iφ b ), and c = C exp( iφ c ).Then, from Eqs. (2), one can immediately infer that A = C , i.e., the amplitudes of the two “side-sites” of the trimerare equal. In addition, the amplitude of the central site is given, as a function of A , by: B = E ± p E − A ( E − A )2 . (3)In turn, the algebraic polynomial equation for the squared amplitude of A ≡ x is given by x [ γ + ( E − x ) ] − k E [ γ + ( E − x ) ] − k x + 2 k E = 0 (4)Once A is determined from Eq. (4) and subsequently B from Eq. (3), then the two relative phases between the threesites of the trimer have to satisfy: sin( φ b − φ a ) = − sin( φ b − φ c ) = − γAkB (5)cos( φ a − φ b ) = cos( φ b − φ c ) = EA − A kB (6)The above formulation provides [via Eqs. (3)-(4) and (5)-(6)] the full set of stationary solutions of the trimer system,for given values of the coupling strength k , nonlinear eigenvalue parameter E , and gain/loss strength γ . In whatfollows in our numerical section below, we will fix two of these parameters ( E and k ) and vary γ to explore thedeviations from the Hamiltonian limit of γ = 0. As an important aside, let us note here that the global freedom ofselecting a phase (due to the U(1) invariance of the model) can be used to choose φ b = 0. Then, it is evident that φ c = − φ a , which combined with the amplitude condition A = C implies that u = ¯ u , where the overbar denotescomplex conjugation. Clearly, this condition is in line with the demands of PT -symmetry for our system.As mentioned above, in addition to the regular stationary solutions for which E is real, one can seek additionalsolutions with E being complex, i.e., E = ˆ E exp( iφ e ) The resulting waveforms are quite special in that they aresolutions of the stationary equations of motion (2), yet they are not solutions of the original dynamical evolutionequations (1), because of the imaginary part of E . Such “ghost state” solutions have recently been identified in thecase of the PT -symmetric dimer [19–21] and have even been argued to play a significant role in its correspondingdynamics therein. In the present case of the trimer, to the best of our knowledge, they have not been previouslyexplored. Such ghost trimer states will satisfy the following algebraic conditions:sin φ a = A (cid:0) B + 2 C (cid:1) γB ( A + B + C ) k (7)cos φ a = A ( B − C )( B + C ) (cid:0) − A + B + C (cid:1) B ( − A + B − C ) k (8)sin φ c = − (cid:0) A + B (cid:1) CγB ( A + B + C ) k (9)cos φ c = (cid:0) − A + B (cid:1) C (cid:0) A + B − C (cid:1) B ( − A + B − C ) k (10)sin φ e = ( A − C )( A + C ) γ ( A + B + C ) ˆ E (11)cos φ e = A − B + C ( A − B + C ) ˆ E , (12)From these equations, the amplitudes A , B , C can be algebraically identified by applying the identity sin φ +cos φ = 1for each of the above angles. The relevant six algebraic equations lead to the identification of the six unknowns, namelythe three amplitudes, as well as the phases φ a , φ c and φ e (for simplicity we have set φ b = 0 hereafter, without lossof generality). It should be noted here that should such ghost state solutions be present with φ e = 0, these willspontaneously break the PT symmetry, given that they will have A = C .Notice that for each branch of solutions that we identify in what follows, we will also examine its linear stability.This will be done through a linearization ansatz of the form u i = e iEt [ v i + ǫ ( p i e λt + ¯ q i e ¯ λt )]. Here the v i ’s for i = 1 , , λ are the corresponding eigenvalues and( p i , q i ) for i = 1 , , ǫ ); the overbar will be used to denote complex conjugation. When the eigenvalues λ of the resulting 6 × III. NUMERICAL RESULTS
Since Eq. (4) is a polynomial of degree 5, we expect at most 5 distinct real roots (and at least 1 such). Indeed forsuitable choices of the free parameters (
E, k ), we identify five branches of stationary solutions. Figure 1 illustratesa situation with five branches under E = 0 . k = 0 .
1. Two of them, denoted by blue circles and red diamonds,collide and terminate at γ = 0 .
1. The blue circles are essentially stable while the red diamonds are unstable. Anotherpair of branches, namely the magenta squares and green pluses collide and terminate at γ = 0 .
02, with the magentasquares being stable and the green pluses being unstable (i.e., both of the above collisions are examples of saddle-center bifurcations). The black crosses branch, which is essentially unstable, persists beyond γ = 0 .
1. Notice that theamplitudes of the different nodes for this branch shown in the top left panel of the figure are not constant: the upperline (standing for B ) is slightly increasing and the lower line (standing for A = C ) is slightly decreasing.In the following, we focus on a typical example of the branches (both stationary and ghost ones) for a selection ofthe free parameters of order unity, more specifically for E = k = 1; cf. Fig. 2. We identify three distinct examples ofstationary states denoted by the blue circle, red diamond and black cross branches. The blue circle and red diamondbranches stem from the corresponding “+0–” and “–+–” branches, respectively, namely the second and third excitedstate of the Hamiltonian trimer problem of γ = 0; cf. with Ref. [27]. The blue circles branch is mostly unstable,except for a small interval of γ ∈ [1 , . γ ∈ [1 . , . γ = 1 and split as imaginary thereafter. One of these pairs exits as real for γ > . γ = 1 . γ = 1 and persists for all values of γ thereafter. This is quite interesting in its own right as an observation since, ashighlighted in Ref. [10], the linear critical point for the PT phase transition is γ = √ k . Thus, this branch presentsthe simplest oligomer example (ones such are absent in the case of the dimer) whereby nonlinearity enables a solutionfamily to persist past the point of the linear limit PT phase transition. Additionally, it should be noted that thebranch is stable for all values of γ < .
13, but destablizes for all larger values of γ .In Fig. 2, however, in addition to the standard stationary solutions, the ghost state solutions are also shown. Theseare designated by the magenta squares and green pluses in the figure. These ghost solutions are also obtained forˆ E = k = 1, and importantly (and contrary to what is the case for the stationary states), they bear distinct amplitudesin all three sites. The two (magenta and the green) branches shown in the figure are mirror images of each other, i.e., A, B, C in the magenta branch are the same as
C, B, A in the green branch, respectively, and their phase differenceand eigenvalues are opposite to each other. Notice that as indicated above the difference in the magnitudes of A and C supports the fact that these branches defy the expectations of the PT symmetry. Indeed, both of the branches arisethrough a symmetry-breaking bifurcation from the blue branch when it becomes unstable at γ = 1 . γ = 1 . both of them to be unstable. Case examplesof the linearization results for both the regular states and the ghost ones are shown for three different values of γ inFig. 3. For γ = 0 .
5, the red diamond branch is (marginally) stable, while the blue circle branch bears the instabilitythat we discussed above for γ <
1. For γ = 1 .
5, only the black branch is present among the stationary ones andthe magenta and green ghost state branches manifest their respective asymmetries with spectra that are asymmetric around the imaginary axis. This is a characteristic feature of the ghost states; see also [21, 24]. Although amongthe two branches, the magenta is more stable and the green highly unstable, even the magenta branch is predictedto be weakly unstable with a small real positive eigenvalue. We will examine the dynamical implications of theseinstabilities in what follows. The last panel similarly shows the case of γ = 1 . γ = 0 . γ = 1 . γ = 1 .
1. Notice that the cases of (b) and (c), thecorresponding branches cease to exist at γ = 1 . γ = 1 . γ = 1 .
1. Importantly, we note that in the unstableevolution of cases (b) and (c), two of the sites end up growing indefinitely while the lossy site ends up decaying. Onthe contrary, in the case (a), only the site with gain is led to growth, while the other two are led to eventual decay. In γ | u | γ ∆ φ γ | λ r | γ | λ i | FIG. 1: (Color online) The solution profile of Eq. (1) with E = 0 . k = 0 . φ b = 0. The four panels denote the solutionamplitude (top left), phase differences between adjacent nodes (top right), real and imaginary parts (second row) of eigenvalues. γ | u | γ ∆ φ γ λ r γ λ i FIG. 2: (Color online) In a way similar to that of the previous figure (i.e., with top left denoting amplitudes, top right relativephases, bottom left real and bottom right imaginary part of the linearization eigenvalues), the 4 panels show the existence andstability of solutions for a trimer with parameters E = k = 1. There are three regular standing wave branches: the blue, thered and the black; the blue and red are the ones disappearing hand-in-hand at γ = 1 . γ = 1 . γ = 1 . −0.5 0 0.5−505 γ =0.5 λ r λ i (a) −2 0 2−202 γ =1.5 λ r λ i (b) −2 0 2−202 γ =1.7 λ r λ i (c) FIG. 3: (Color online) The spectral planes ( λ r , λ i ) of the eigenvalues λ = λ r + iλ i of the solutions shown in Fig. 2. The firstpanel shows the case of γ = 0 . γ = 1 . γ = 1 . panel (d), we show the black crosses branch, the third among the standing wave solutions identified herein for γ = 1 . γ = 1 . γ = 1 .
5. With E = E r + iE i being complex, the ghost state solutions under theform u = exp( iEt ) a , u = exp( iEt ) b and u = exp( iEt ) c should evolve exponentially, as indicated by dashed linesin panels (e)-(h). In particular, the magenta squares branch with negative E i is expected to lead to growth (for allthree nodes of the trimer), while the green plus branch with positive E i is anticipated to decay (again for all nodes).The slopes of these growth/decay features are given by − E i = − E sin φ e . However, in line with their anticipated −2 γ =0.5t | u | (a) blue −2 γ =1.1t | u | (b) blue circles −2 γ =1.1t | u | (c) red diamonds −2 γ =1.5t | u | (d) black crosses −2 γ =1.1t | u | (e) magenta squares −2 γ =1.5t | u | (f) magenta squares −2 γ =1.1t | u | (g) green pluses −2 γ =1.5t | u | (h) green pluses FIG. 4: (Color online) The dynamical evolution of the amplitudes of the three sites for the solutions shown in Fig. 2. Noticethat all solutions are plotted in semilog. The first row shows the evolution of the three stationary branches. In (b) and (c),since these branches are absent for γ = 1 .
1, their profile for γ = 1 .
043 is initialized. The second row shows dynamics of the twoghost state solution branches. The dashed lines are the predicted dynamics of the ghost states on the basis of their growth (formagenta squares) or decay (for green pluses) rates. linear “instability”, neither of these follows exactly the dynamics anticipated above. Both of them evolve for a shortperiod according to the expected growth or decay, and then the gain sites start to grow and the loss sites start todecay, regardless of the trend predicted by the form of the ghost state (discussed above). Moreover, it is relevant tonote as regards the corresponding dynamics that the cases of the blue circle and red diamond branches of γ = 1 . γ = 1 . PT -symmetry in optical systems especiallyas regards trimers, but also more generally. IV. EXPERIMENTAL REALIZATION OF PT OPTICAL SYSTEMS
General requirements for realization of PT optical systems are the availability of adequate methods for formation ofcoupled waveguide systems or arrays with the additional opportunity to spatially tailor loss and gain in the substrate.In other words, suitable fabrication conditions should allow for spatial manipulation of both real and imaginary partof the dielectric constant.Laser crystals and glasses are amplifying media that may provide the necessary optical gain by using differentphysical mechanisms. Examples are doped bulk crystals and fibers that make use of stimulated emission to amplifya weak signal, electron-hole recombination in semiconductor optical amplifiers (SOAs), parametric amplification innonlinear crystals, or stimulated Raman scattering (SRS). Furthermore, many laser gain materials allow for theformation of guiding index structures. However, although many amplifying materials exist, the most challengingaspect is the necessity to achieve optical gain while at the same time the real part of the refractive index shouldremain constant (or change only in a negligible way in order to maintain PT symmetry), which is difficult to achievebecause of the Kramers-Kronig relation. Besides thermo-optic effects in case of high optical powers, it turns outthat other mechanisms like self-and cross-phase modulation are limiting the suitability of most laser gain media forapplication in PT optics.Another well-known amplification mechanism is optical beam coupling in photorefractive media like photovoltaiclithium niobate (LiNbO ) crystals, which exists already at quite low optical light powers. Due to advanced waveguideformation techniques, LiNbO is a favorite material for use in integrated optics [28]. Besides diffraction of weaksignal beams on recorded index gratings which leads to gain, the point symmetry 3 m of LiNbO enables interactionof orthogonally polarized waves too [29]. A polarization grating recorded by a pump and signal beam having mutualorthogonal polarization allows for optical signal gain; the small-signal gain can reach several tens per cm for strong Fedoping [29]. At the same time, the spatially varying polarization pattern of pump and signal beam does not inducesignificant phase changes for the interacting beams. Using this mechanism, the first experimental demonstration of PT symmetry in optics has been achieved in Fe:LiNbO using Ti in-diffusion to form coupled waveguide structures [5].However, there exist still some limitations that have to be overcome in order to realize more advanced PT symmetricoptical settings.While spatial tailoring of optical gain may be achieved by limiting an optical pump beam to certain waveguidechannels, this can be hardly done for loss (a technologically quite challenging solution consists in the formationof metallic stripes of precisely defined width on certain channels, see [4]). Due to such difficulties, a more realisticexperimental approach consists in allowing for a constant loss in all coupled channels, while this loss is overcompensatedby adjustable gain in some selected channels only. FIG. 5: (Color online) (a) Evolution of powers in a three-channel coupler. Left and right channels experience equal gain; thelossy central channel is excited from the input facet. Gain develops due to holographic recording of a polarization gratingaccording to γ ( t rec ) = γ (1 − exp( − t rec /τ )) with the photorefractive (Maxwell) time constant τ . (b,c) Power distributions (leftpanel) on the end-facets for different recording times (b: t rec = 0; c: t rec = 34 min) and corresponding interferograms (rightpanels) showing the phase relation of central and outer channels. An experimental example of a three-channel coupled waveguide structure having distributed gain and loss is shownin Fig. 5. While the precisely PT -symmetric pattern of loss-(neither gain nor loss)-gain is not directly realizablein the above described experiments, here we focus on a somewhat different configuration featuring an alternationof gain-loss-gain which is, arguably, the nearest experimentally realizable optical variant. To achieve that, threeparallel single-mode waveguide channels for a wavelength of λ = 532 nm are formed on a Fe-doped x-cut LiNbO substrate by in-diffusion of a stripe-like Ti film. The propagation length is 20 mm and the linear coupling coefficientis k ≈ . − . In the sample overall but constant loss is due to absorption of the used green light by in-diffused Feions. Similar to the arrangement in [5], optical gain for the extraordinarily polarized signal is achieved by pumpingthe sample from the top using a plane wave of ordinary polarization. An amplitude mask on top of the substrateshields the central channel ( ♯ ♯
1) and right ( ♯
3) channels only. The signallight is coupled from the end-facet into the central channel. As can be seen, when the pump beam is switched onat time t rec = 0, power in the two outer channels start to increase. Simultaneously the total power (black symbolsin Fig. 5) increases due to buildup of the polarization grating. Beside some asymmetries in the temporal evolutiondiscussed below, for longer recording symmetry improves again, and the corresponding interferograms on the rhs show the development of relative phase of central and outer channels, starting from the in-phase condition at t rec = 0(Fig. 5b) towards a final phase difference of ± π/ t rec ≈
34 min (Fig. 5c). This relative phase development is in linewith the earlier theoretical expectations on the basis of Eqns. such as (5)-(6). This behavior is also in good agreementwith simulations of this gain-loss-gain system based on Runge-Kutta integration of the corresponding coupled-waveequations in Fig. 6. Of course, these runs also manifest the partial differences of this experimental realization from thegenuinely PT -symmetric case in that ultimately all three waveguides feature growing optical power in the simulationsof Fig. 6, a trait which is absent e.g. in Fig. 4 (where at least one waveguide is not growing indefinitely in power).Obviously, some experimental problems still exist. The temporal evolution in the left and right channels is farfrom being perfectly symmetric, especially for intermediate recording times. In most experiments, for long recordingtimes (i.e. high gain) output powers of the three channels start to fluctuate slightly. Possible explanations for this FIG. 6: (Color online) (a) Simulation of normalized intensity of the three-channel coupler as a function of gain (normalizedto k ) using integration of the coupled-wave equations. The vertical line corresponds to γ/k = 2 √
2, which is the “break even”point of this gain-loss-gain system. The four panels (b-e) on the right show the light power evolution during propagation for γ/k = 0, (b), γ/k = 0 . γ/k = 2 . γ/k = 3 . √ I toimprove visibility of the amplified signal light in the channels. behavior are large induced space-charge fields that may lead to spark plugs across the sample surface, or weak phaseinstabilities of the setup. A general problem is the small achievable gain, which is rarely sufficient to reach typical PT symmetry-breaking thresholds in most of the fabricated samples. Gain is also limited because of low powers ofsignal and pump light: Higher signal power would lead to decoupling of the excited channel due to nonlinear indexchanges, while higher pump power would record a distorting phase gradient at the boundaries of the used amplitudemask (i.e. between illuminated/non-illuminated channels).For future experiments using Fe:LiNbO waveguide samples, a main objective will thus be to increase optical gainby optimizing material properties. However, when doping LiNbO substrates with Fe using in-diffusion, the physicalmechanisms that may lead to high gain when coupling orthogonally polarized waves are not yet fully understood: Theinfluence of certain diffusion atmospheres, interference from simultaneous Ti in-diffusing used for waveguide formation,or the effect of Li out-diffusion at high temperatures and consequences on possible lattice sites of in-diffused Fe ionshave not been investigated in detail. In particular, the high gain found in some Fe-doped bulk LiNbO crystals hasnot been observed in waveguide samples so far.An alternative experimental PT -symmetric model system that also uses LiNbO with its well-developed waveguidefabrication technology as a substrate is Er doping to achieve gain in the optical communication window at 1 . µ m.Although no detailed data on cross-phase modulation when pumped e.g. with 980 nm wavelength is available yet, datafrom Er-doped fiber amplifiers (EDFAs) indicates that induced phase changes might be sufficiently small [30]. Workon such systems is currently in progress, which may perhaps allow avoiding the described unwanted nonlinear effectsthat disturb the symmetry condition of the (real) refractive index in Fe-doped photorefractive LiNbO for higherinput powers. V. CONCLUSIONS & FUTURE CHALLENGES
In the present work, we have revisited the theme of one of the prototypical PT -symmetric oligomers, namely thetrimer. We have illustrated the different number of branches (at least one and at most five) of standing wave solutionsthat exist for this system. We have thereafter focused on a case example of parameters of order unity and have shownthat two of these standing wave branches terminate in a pairwise disappearance, while the third one exists for valuesof the gain/loss parameter, in fact, extending past the point of the PT -symmetry breaking phase transition of thelinear limit occuring at γ = √ k .Additionally, we have also presented the formulation of the so-called ghost states in this system and have explicitlycomputed them, showing how they emerge through a symmetry breaking bifurcation (asymmetrizing the amplitudesof the two side sites A and C ) from one of the standing wave branches. As expected on the basis of such aneffective pitchfork bifurcation, the two resulting ghost state branches are mirror-images of each other (and so are theircorresponding spectra). The dynamics of both the unstable stationary states and those of the ghost states revealedtwo possible dynamical scenaria. In one of these, the “neutral” site (without gain or less) sided with the gain one,while the other corresponded to the case where it sided with the lossy site.Additionally, we have explored the possibility of creating PT symmetric systems in nonlinear optics, revealing thatit is arguably simpler to create e.g. a gain-loss-gain three-channel system, rather than the genuinely PT symmetricsituation where a waveguide with gain and one with loss straddle a middle one without either gain or loss. Onthe other hand, for this gain-loss-gain setting, we presented both physical experiments and corroborating numericalsimulations revealing the partition of the fraction of optical power (initially placed at the central channel) and how ittransfers more to the outer gain channels as the gain is increased beyond γ/k = 1.There are many interesting questions that arise from the present study and are worthy of further exploration. Itwould be interesting to generalize our considerations herein to the case of quadrimers and to appreciate how thecomplexity of the relevant configurations expands, especially since in the latter case, there is generally the potentialof two gain/loss parameters [26]; nevertheless per the above discussion on experimental possibilities in optics, the caseof two waveguides with equal gain and two with equal loss would appear as the most realistic one presently. At thesame time, further experimental implementations of oligomer systems, either at the electrical circuit level, or at theoptical waveguide level discussed in the last section would be particularly desirable and highly interesting towards anincreased understanding of the systems’ dynamics. Efforts in these directions are currently in progress and will bereported in future publications. Acknowledgments
The work of P.G.K. is partially supported by the US National Science Foundation under grants NSF-DMS-0806762,NSF-CMMI-1000337, by the US AFOSR under grant FA9550-12-1-0332, as well as the Binational Science Foundationunder grant 2010239 and the Alexander von Humboldt Foundation. D.K. thanks the German Research Foundation(DFG, Grant KI 482/14-1) for financial support of this research. D.J.F. is partially supported by the Special ResearchAccount of the University of Athens. [1] C. M. Bender and S. Boettcher, Phys. Rev. Lett. , 5243 (1998); C. M. Bender, S. Boettcher and P. N. Meisinger, J.Math. Phys. , 2201 (1999).[2] C. M. Bender, Rep. Prog. Phys. , 947 (2007).[3] A. Ruschhaupt, F. Delgado, and J. G. Muga, J. Phys. A: Math. Gen. , L171 (2005).[4] A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N.Christodoulides, Phys. Rev. Lett. , 093902 (2009).[5] C. E. R¨uter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nature Phys. , 192 (2010).[6] A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, Nature , 167 (2012).[7] J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, Phys. Rev. A , 040101 (2011).[8] H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, Phys. Rev. A , 043803 (2010).[9] M. C. Zheng, D. N. Christodoulides, R. Fleischmann, and T. Kottos, Phys. Rev. A , 010103(R) (2010).[10] K. Li and P. G. Kevrekidis, Phys. Rev. E , 066608 (2011).[11] A. A. Sukhorukov, Z. Xu, and Yu. S. Kivshar, Phys. Rev. A , 043818 (2010).[12] F. Kh. Abdullaev, V.V. Konotop, M. ¨Ogren, and M. P. Sørensen, Opt. Lett. , 4566 (2011).[13] R. Driben and B. A. Malomed, Opt. Lett. , 4323 (2011).[14] R. Driben and B. A. Malomed, Europhys. Lett. , 51001 (2011).[15] I. V. Barashenkov, S. V. Suchkov, A. A. Sukhorukov, S. V. Dmitriev, and Yu. S. Kivshar, Phys. Rev. A , 053809 (2012).[16] N. V. Alexeeva, I. V. Barashenkov, A. A. Sukhorukov, and Yu. S. Kivshar, Phys. Rev. A , 063837 (2012).[17] V. Achilleos, P. G. Kevrekidis, D. J. Frantzeskakis, and R. Carretero-Gonz´alez Phys. Rev. A , 013808 (2012).[18] D. A. Zezyulin and V. V. Konotop, Phys. Rev. A , 043840 (2012); Yu. V. Bludov, V. V. Konotop, and B. A. Malomed,Phys. Rev. A , 013816 (2013).[19] H. Cartarius and G. Wunner, Phys. Rev. A , 013612 (2012); H. Cartarius, D. Haag, D. Dast, and G. Wunner, J. Phys.A , 444008 (2012).[20] E.-M. Graefe, J. Phys. A , 444015 (2012).[21] A. S. Rodrigues, K. Li, V. Achilleos, P. G. Kevrekidis, D. J. Frantzeskakis, C. M. Bender, Rom. Rep. Phys. , 5 (2013).[22] D. Leykam, V. V. Konotop, and A. S. Desyatnikov, Opt. Lett. , 371 (2013).[23] K. Li, P. G. Kevrekidis, B. A. Malomed, and U. G¨unther, J. Phys. A: Math. Theor. , 444021 (2012).[24] V. Achilleos, P. G. Kevrekidis, D. J. Frantzeskakis, R. Carretero-Gonz´alez, arXiv:1208.2445.[25] M. Duanmu, K. Li, R. L. Horne, P. G. Kevrekidis and N. Whitaker, Phil. Trans. Roy. Soc. A , 20120171 (2013).[26] D. A. Zezyulin and V. V. Konotop Phys. Rev. Lett. , 213906 (2012). [27] T. Kapitula, P. G. Kevrekidis and Z. Chen, SIAM J. Appl. Dyn. Sys. , 598 (2006).[28] D. Kip, Appl. Phys. B: Laser and Optics , 131 (1998).[29] A. Novikov, S. Odoulov, O. Oleinik, and B. Sturman, Ferroelectrics , 295 (1987).[30] S. Jarabo, J. Opt. Soc. Am. B , 1846 (1997). .5 1k 0 0.50123 k B | λ ii