Symmetry breaking in binary chains with nonlinear sites

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  • PHYSICAL REVIEW E 88, 032901 (2013)

    Symmetry breaking in binary chains with nonlinear sites

    Dmitrii N. Maksimov and Almas F. SadreevInstitute of Physics, 660036 Krasnoyarsk, Russia

    (Received 4 June 2013; published 3 September 2013)We consider a system of two or four nonlinear sites coupled with binary chain waveguides. When a

    monochromatic wave is injected into the first (symmetric) propagation channel, the presence of cubic nonlinearitycan lead to symmetry breaking, giving rise to emission of antisymmetric wave into the second (antisymmetric)propagation channel of the waveguides. We found that in the case of nonlinear plaquette, there is a domain in theparameter space where neither symmetry-preserving nor symmetry-breaking stable stationary solutions exit. Asa result, injection of a monochromatic symmetric wave gives rise to emission of nonsymmetric satellite waveswith energies differing from the energy of the incident wave. Thus, the response exhibits nonmonochromaticbehavior.

    DOI: 10.1103/PhysRevE.88.032901 PACS number(s): 05.45.a, 03.75.b, 42.65.Ky, 42.65.Pc

    I. INTRODUCTION

    To the best our knowledge, symmetry breaking (SB) innonlinear systems was first predicted by Akhmediev, whoconsidered a composite structure of a single linear layerbetween two symmetrically positioned nonlinear layers [1].One could easily see that if the wavelength is larger than thethickness of the nonlinear layers, then Akhmedievs modelcould be reduced to a dimer governed by the nonlinearShrodinger equation. Independently, the SB was discoveredfor the discrete nonlinear Schrodinger equation with a finitenumber of coupled cites (nonlinear dimer, trimer, etc.) [28].For example, in the case of the nonlinear Schrodinger dimer,Eilbeck et al. found two different families of stationarysolutions [2]. The first family is symmetric (antisymmetric)(|1| = |2|), while the second is nonsymmetric (|1| = |2|).This consideration was later extended to a nonlinear dimerembedded into an infinite linear chain [9] with the samescenario for the SB. Multiple bifurcations to the symmetry-breaking solutions were demonstrated by Wang et al. [10] forthe nonlinear Schrodinger equation with a square four-wellpotential. Remarkably, the above system can also support astable state with a nodal point, i.e., quantum vortex [11]. Inthe framework of the nonlinear Schrodinger equation, one canachieve bifurcation to the states with broken symmetry, varyingthe chemical potential which is equivalent to the variationof the population of the nonlinear sites or, analogously, ofthe constant in the nonlinear term of the Hamiltonian. Inpractice, however, one would resort to the optical counterpartsof the quantum nonlinear systems where the variation of theamplitude of the injected wave affects the strength of Kerrnonlinearity (see Refs. [1222] for optical examples of the SB).

    In the present paper we consider a nonlinear dimer anda square four-site nonlinear plaquette coupled with twochannel waveguides in the form of binary tight-binding chains.The latter system is analogous to that recently consideredin Ref. [23], where the plaquette, however, was set upwithout mirror symmetry with respect to the center line ofthe waveguide. Due to nonlinearity of the plaquette, thesystem can spontaneously bifurcate between diode-antidiodeand bidirectional transmission regimes. In our case, we willshow that when a monochromatic wave is injected into thefirst (symmetric) propagation channel the presence of the

    cubic nonlinearity can lead to symmetry breaking, givingrise to emission of antisymmetric waves into the second(antisymmetric) propagation channel of the waveguides.

    This result raises an important question about the effect ofa probing wave on the SB. To allow for symmetry breaking,the architecture of the open system should support somesymmetries of its closed counterpart. Then the growth ofthe injected power can result in bifurcations into the statesviolating the symmetry of the probing wave by the amplitudeof the scattering function [1214,17,18,20,21,24] or by itsphase [25].

    The second important question is whether the solutionsin the linear waveguides could be stationary monochromaticplane waves, reflected (n,t) = R exp (ikn iE(k)t) andtransmitted (n,t) = T exp (ikn iE(k)t), where R and Tare the reflection and transmission amplitudes. If the answeris positive, then we can apply the Feshbach projectiontechnique [2630] and implement the formalism of the non-Hermitian effective Hamiltonian (now nonlinear) which actson the nonlinear sites only, thus truncating the Hilbert spaceto the scattering region [21,30]. When the radiation shiftsof the energy levels are neglected, that formalism reduces to thewell-known coupled mode theory (CMT) equations [3134].In the present paper we will show that there are domains in theparameter space where there are no stable stationary solutions.We will demonstrate numerically that a plane wave incidentonto a nonlinear object gives rise to emission of multiplesatellite waves with energies (frequencies) differing from theenergy (frequency) of the probing wave.

    II. STRUCTURE LAYOUT AND BASIC EQUATIONSWe consider the two tight-binding structures shown in

    Fig. 1, the nonlinear dimer [Fig. 1(a)] and nonlinear four-sitesquare plaquette [Fig. 1(b)] coupled with linear binary chains.Each chain, left and right, supports two continua of planewaves,

    ()1 (n,t) =

    12| sin k1|

    (11

    )exp (ik1n iE(k1)t),

    (1)

    ()2 (n,t) =

    12| sin k2|

    (1

    1)

    exp (ik2n iE(k2)t),

    032901-11539-3755/2013/88(3)/032901(8) 2013 American Physical Society

  • DMITRII N. MAKSIMOV AND ALMAS F. SADREEV PHYSICAL REVIEW E 88, 032901 (2013)

    (a)u2 u1 u0 u2

    v2 v0 v1 v2v1

    u1

    u2 u1

    v2 v1

    u0 u1 u2 u3

    v0 v1 v2 v3

    (b)

    FIG. 1. (Color online) (a) A two-channel waveguide in the formof a binary tight-binding chain which holds a nonlinear dimer (shownby filled circles). (b) The same as in (a) but with four nonlinearsites (four-site plaquette). A green dotted box is placed around thenonlinear scattering region.

    where indices p = 1,2 enumerate continua or channels withthe propagation bands given by

    E(kp) = 2 cos kp 1, kp . (2)In order to control the resonant transmission we adjust thehopping matrix element between the nonlinear sites andwaveguides < 1 [30] shown in Fig. 1 by dashed lines.The coupling between nonlinear sites (shown in Fig. 1 by adash-dot line) is controlled by constant .

    In the case of the nonlinear dimer the Schrodinger equationtakes the following form:

    iun = Jnun+1 Jn1un1 Knvn + n,0|un|2un, (3)ivn = Jnvn+1 Jn1vn1 Knun + n,0|vn|2vn,

    where Jn = 1 + ( 1)(n,1 + n,0), Kn = 1 + ( 1)n,0.In the case of a nonlinear square plaquette the Shcrodingerequation could be written down essentially in the same way.

    III. NONLINEAR DIMER

    First, let us follow Refs. [3541] and search for the solutionof the Schrodinger equation (3) in the form of a stationarywave,

    un(t) = uneiEt , vn(t) = vneiEt , (4)where the discrete space variable n and the time t areseparated. The absence of nonlinearity in the waveguidesdrastically simplifies analysis of Eq. (3). Assuming that asymmetric-antisymmetric wave is incident from the left, wecan write the solutions in the left L(n,t) and right R(n,t)waveguides as

    L(n,t) = A0 (+)p (n,t) + Rp,1 ()1 (n,t) + Rp,2 ()2 (n,t),(5)R(n,t) = Tp,1 (+)1 (n,t) + Tp,2 (+)2 (n,t),where parameter A0 is introduced to tune the intensity of theprobing wave. Notice that Eq. (5) implicitly defines reflectionand transmission amplitudes Rp,p and Tp,p with the first

    subscript p indexing the channels and, respectively, the sym-metry of the incident wave. One can now match the solutionsEq. (5) to the equations for the nonlinear sites to obtain a setof nonlinear equations for on-site and reflection-transmissionamplitudes. Computationally, however, it is more convenientto use the approach of the non-Hermitian Hamiltonian [28,30]in which the number of unknown variables equals the numberof nonlinear sites. Following Ref. [21] we write the equationfor the amplitudes on the nonlinear sites,

    (E Heff)| = A0|in, (6)where

    Heff = H0 C

    VC

    1E + i0 HC VC

    =(2(eik1 + eik2 ) + |u0|2 2(eik1 eik2 )

    2(eik1 eik2 ) 2(eik1 + eik2 ) + |v0|2).

    (7)Here vector | = (u0,v0) is the state vector of the dimer,H0 is the nonlinear Hamiltonian of the dimer decoupled fromthe waveguides, VC is the coupling operator [30] between thenonlinear sites and the left and right waveguidesC = L,R withthe Hamiltonian HC , and in| = (1, 1) is the source termfor symmetric-antisymmetric incident waves correspondingly.One can easily see that in the limit 0 the dimer isdecoupled from the waveguides and Eq. (6) limits to thestandard nonlinear Schrodiner equation of the closed nonlineardimer. After Eq. (6) is solved, one easily obtains transmission-reflection amplitudes from Eq. (3).

    In Fig. 2 we present the response of the nonlinear dimerto a symmetric probing wave both in form of transmissionprobabilities |T11(E)|2, |T12(E)|2 [Fig. 2(a)] and phase 2 1and intensity I1 I2 differences [Fig. 2(b)]. On-site phases andintensities are defined through u0 =

    I1e

    i1 ,v0 =

    I2ei2

    .

    There are two principally different families of solutions similarto those obtained in Ref. [21]. The symmetry-preservingfamily inherits the features of the linear case in which asymmetric probing wave excites only the symmetric modes = 12 (1,1) with eigenvalue Es = . In fact, we see atypical example of nonlinear transmission through a reso-nant state. The transmission probability |T11(E)|2 for thesymmetry-preserving family of solutions with I1 = I2,1 = 2are shown in Fig. 2(a) by blue dashed lines. The energybehavior of the transmission is very similar to the case ofa linear waveguide coupled with one nonlinear in-channelsite [35,36,42]. However, one can see that the peak of theresonant transmission is now unstable, which is a key tomode conversion, as will be shown below. The stability of thesolutions was examined by standard methods based on smallperturbations technique (see, for example, Refs. [43,44]).

    The SB occurs due to excitation of the second (antisym-metric) mode of the dimer a = 12 (1,1) with eigenvalueEa = . The phase and intensity differences for the SBsolutions are shown in Fig. 2(b). One can see that the symmetryis broken simultaneously by both intensity I1 = I2 and phase1 = 2 of the wave function. This scenario of SB differs fromthe case of the closed dimer [2,3,5] or the dimer openedto a single chain (off-channel architecture) [21], where the

    032901-2

  • SYMMETRY BREAKING IN BINARY CHAINS WITH . . . PHYSICAL REVIEW E 88, 032901 (2013)

    1 0.5 0 0.5 10

    0.2

    0.4

    0.6

    0.8

    1

    E

    trans

    miss

    ions

    (a)

    0.1 0.2 0.3 0.4 0.5

    3

    2

    1

    0

    1

    2

    3

    E

    I 1I

    2, 1

    2

    (b)

    FIG. 2. (Color online) Response of nonlinear dimer to symmetric probing wave. (a) Transmission probability |T11|2 for the symmetry-preserving solution (blue dash-dotted line), |T11|2 for SB solutions (solid red line), and |T12|2 for SB solutions (dashed black line). (b) Phasedifference 2 1 for the pairs of SB solutions (dashed red line); intensity difference I1 I2 (solid blue line). Thicker lines mark stablesolutions. The parameters are as follows: = 0.2,A0 = 1, = 0.025, = 0.

    symmetry was broken either by the intensities or by the phasesof the on-site amplitudes.

    In Fig. 3 we show the case when the antisymmetric waveis injected to the system. Although the energy behavior ofthe on-site intensities, on-site phases, and the transmissionprobabilities to the first and second channels |T21|2, |T22|2 issimilar to the case of the symmetric probing wave, the stabilitypattern of the symmetry-preserving solution is now typicalfor the nonlinear transmission through a single nonlinear siteRef. [42]. Finally, following Ref. [45], we present the plotsoutput vs. input, i.e., transmission probabilities |Tp,p |2 vs.A20, for both symmetric and antisymmetric probing waves.In Fig. 4 one can see that the input-output curves for thesymmetry-preserving family of solutions have typical bifurca-tion behavior while the symmetry-breaking family exists onlywithin a bounded domain of the input.

    As mentioned above, the SB phenomenon results in themode conversion, i.e., emission of scattered waves into adifferent channel. Let us consider the time evolution of thewave front given by

    |in(n) = f (n) (+)1 (n,0), (8)where

    f (n) ={

    1 if n 100 ;e(n+100)

    2/250 if n > 100, (9)

    is an auxiliary function that provides a smooth increase of theincident plane wave amplitude. As the wave front propagatesto the right, the input signal converges to (+)1 (n,t) so onecould expect that in the course of time the system wouldstabilize in one of the solutions shown in Fig. 2. In order

    1 0.5 0 0.5 10

    0.2

    0.4

    0.6

    0.8

    1

    E

    trans

    mis

    sion

    s

    (a)

    0.2 0.3 0.4 0.54

    3

    2

    1

    0

    1

    2

    3

    4

    I 1I

    2, 1

    2

    E

    (b)

    FIG. 3. (Color online) Response of a nonlinear dimer to an antisymmetric probing wave. (a) Transmission probability |T22|2 for thesymmetry-preserving solution (blue dash-dotted line), |T22|2 for SB solutions (solid red line), and |T21|2 SB solutions (dashed black line).(b) Phase difference 2 1 for the pairs of SB solutions (dashed red lines) and intensity difference I1 I2 (solid blue lines). Thicker linesmark stable solutions. The parameters are as follows: = 0.2, A0 = 1, = 0.03, = 0.

    032901-3

  • DMITRII N. MAKSIMOV AND ALMAS F. SADREEV PHYSICAL REVIEW E 88, 032901 (2013)

    0 5 10 150

    0.2

    0.4

    0.6

    0.8

    1

    1.2

    1.4

    input

    outp

    ut

    (a)

    0 2 4 6 8 100

    0.5

    1

    1.5

    2

    input

    outp

    ut

    (b)

    FIG. 4. (Color online) Input A20 vs. outputs |Tp,p |2 for nonlinear dimer. (a) Symmetric probing wave in| = (1,1), |T11|2 for symmetry-preserving solutions (blue dash-dotted line), |T11|2 SB solutions (solid red line), and |T12|2 SB solutions (dashed black line). (b) Antisymmetricprobing wave in| = (1, 1), |T22|2 for symmetry-preserving solutions (blue dash-dotted line), |T22|2 SB solutions (solid red line), |T21|2 SBsolutions (dashed black line). The parameters are as follows: = 0.2, E = 0.45, = 0.025, = 0.

    to perform numerical computations, we directly applied bothstandard fourth-order Runge-Kutta method and the Besse-Crank-Nicolson relaxation scheme [46] to Eq. (3) with bothtechniques yielding the same result (Crank-Nicolson approachbeing slightly advantageous in terms of computational time).The absorbing boundary conditions were enforced at far endsof the waveguides according to Ref. [47] to truncate theproblem to a finite domain.

    Let us, first, choose the energy in the domain where thesymmetry-preserving solution is stable, for example,E = 0.35[shown in Fig. 2(a) by a red circle]. To detect the SBwe use the populations As and Aa of the symmetric sand antisymmetric a modes correspondingly. Figure 5(a)shows that although stable SB solutions exist at E = 0.35the response to the probing signal [Eq. (8)] contains onlysymmetric contribution and after several oscillations convergesto the stationary transmission. On the other hand, if onechooses the energy of the incident wave from the domain where

    the symmetry-preserving solutions are unst...