Symmetry breaking in a two-component system with repulsive interactions and linear coupling
aa r X i v : . [ n li n . PS ] A ug Symmetry breaking in a two-component sys-tem with repulsive interactions and linear cou-pling
Hidetsugu Sakaguchi and Boris A.Malomed , Department of Applied Science for Electronics and Materials,Interdisciplinary Graduate School of Engineering Sciences,Kyushu University, Kasuga, Fukuoka 816-8580, Japan Department of Physical Electronics, School of Electrical Engineering,Faculty of Engineering, and Center for Light-Matter Interaction,Tel Aviv University, Tel Aviv 69978, Israel Instituto de Alta Investigaci´on, Universidad de Tarapac´a, Casilla 7D, Arica, Chile
Spontaneous symmetry breaking (SSB) is a broad class of ef-fects occurring in systems combining spatial or inter-componentsymmetry and intrinsic nonlinearity [1]. While in linear systemsthe ground state (GS) exactly reproduces the underlying sym-metry [2], and the GS is always a single (non-degenerate) state,self-attraction in binary systems drives a phase transition whichdestabilizes the symmetric GS, replacing it by a pair of asym-metric ones that are mirror images of each other (so as to main-tain the overall symmetry). The transition takes place when thenonlinearity strength attains a certain critical value. Startingfrom early works [3,4], many realizations of the SSB phenomenol-ogy have been supplied by nonlinear optics and studies of Bose-Einstein condensates (BECs). The similarity of these areas isunderlain by the fact that the nonlinear Schr¨odinger equations[5] and Gross-Pitaevskii equations (GPEs) [6,7]), which are basicmodels for optics and BEC, respectively, are essentially identical,with the difference that the evolution variable is the propagation istance, z , in optical waveguides, and time, t , in BEC. In bothcases SSB effects have been predicted in two-components systems,with linear coupling between the components and self-attractionin each component. In optics, a natural realization of the nonlin-ear couplers and SSB phenomenology in them is offered, in termsof the temporal-domain propagation, by dual-core optical fibers.This setting was studied in detail theoretically [8]-[21] (see alsoreview [22]), and nonlinear switching in the couplers was demon-strated experimentally [23,24]. The same model may be realizedin the spatial domain, considering a planar dual-core nonlinearwaveguide [25,15,26]. Another class of nonlinear-optical symmet-ric systems and SSB dynamics in them, which were explored boththeoretically and experimentally, is provided by dual-cavity lasers[27]-[30].A similar BEC system may be realized by loading the condensateinto a pair of parallel tunnel-coupled elongated traps, filled byself-attractive BEC and linearly coupled by tunneling of atoms[31]-[33]. SSB effects, chiefly for matter-wave solitons, have beenpredicted in such dual traps.A more general two-component system includes, in addition tothe self-interaction of each component and linear mixing betweenthem, nonlinear cross-component interaction [34,35]. In optics,this model applies to the propagation of electromagnetic waveswith orthogonal circular polarizations in the fiber with linear-polarization birefringence (that may be induced by ellipticity ofthe fiber’s cross section), which induces the linear mixing [36]. InBEC, the realization is possible in a single elongated trap filledby a mixture of two different atomic states in the condensate,while the linear (Rabi) mixing is imposed by a radiofrequencyfield which resonantly couples the states [37]-[41].A majority of the above-mentioned works addressed effectivelyone-dimensional (1D) settings. In optics, 2D two-component spa-tiotemporal propagation may be realized in planar dual-core waveg-uides [42], and similar realizations were considered for BEC loaded n dual-core “pancake-shaped” traps [43,44,45]. The more generalsystem, which includes the inter-component nonlinearity, can benaturally implemented as the spatial-domain propagation in bulkoptical waveguides, with the linear mixing of orthogonal circu-lar polarizations induced by intrinsic anisotropy (linear birefrin-gence) in the transverse plane. In BEC, an implementation isoffered by a pancake trap filled by a binary condensate, with theradiofrequency-induced coupling between two atomic states.As said above, the SSB of GSs in previous works was drivenby attractive nonlinearity. Self-repulsion does not break the GSsymmetry, but it may drive spontaneous breaking of antisymme-try of the first excited state, an example being an antisymmetricbound state of matter-wave gap solitons in tunnel-coupled trapsequipped with periodic potentials [46,43,32]. The objective of thepresent work is to demonstrate that, nevertheless, the symmetryof the GS in 1D and 2D linearly-coupled systems with repulsivenonlinearity may be spontaneously broken, replacing it by a pairof asymmetric GSs, provided that the inter-component repulsiondominates over the intra-component nonlinearity, and the GS ismade localized by a confining potential. This mechanism is simi-lar to other effects in which the dominant repulsion between twointeracting fields (in the absence of the linear coupling betweenthem) makes their uniformly mixed state unstable, well-knownexamples being phase separation in binary mixtures [49] and themodulational instability in bimodal systems in which the cross-repulsion is stronger than self-repulsion [50].In terms of BEC, the system is represented by coupled GPEsfor wave functions φ , of the two components, written in the2D form, which includes the confining harmonic-oscillator (HO)potential with strength Ω , acting in the ( x, y ) plane, and scaled tomic mass m : i ∂φ ∂t = − m ∇ φ + ( g | φ | + γ | φ | ) φ + Ω (cid:16) x + y (cid:17) φ − ǫφ (1) i ∂φ ∂t = − m ∇ φ + ( g | φ | + γ | φ | ) φ + Ω (cid:16) x + y (cid:17) φ − ǫφ . (2)Here, ǫ is the coefficient of the linear coupling, while γ > g > γ > g , except forSection IV (the case of g <
0, i.e., self-attractive nonlinearity,may be considered too, although it is less interesting than thecase of the competition between the cross- and self-repulsion).By means of obvious rescaling of t , coordinates, and wave func-tions we fix g = Ω = ǫ = 1 , (3)while m and γ are kept as free parameters (a different normaliza-tion, with ǫ = 1, is adopted below in Section IV, where a differentpotential is considered).Equations (1) and (2) conserve the total norm, N = Z Z (cid:16) | φ | + | φ | (cid:17) dxdy, (4)Hamiltonian, H = Z Z " m (cid:16) |∇ φ | + |∇ φ | (cid:17) + 12 (cid:16) | φ | + | φ | (cid:17) + γ | φ | | φ | + 12 (cid:16) x + y (cid:17) (cid:16) | φ | + | φ | − ( φ φ ∗ + φ ∗ φ ) (cid:17) dxdy, (5)where normalization (3) is taken into regard, and ∗ stands for thecomplex conjugate, as well as the total angular momentum, M = i X j =1 , Z Z φ ∗ j y ∂∂x − x ∂∂y ! φ j dxdy. (6) n terms of optics, Eqs. (1) and (2) with t replaced by z and m replaced by the Fresnel number provide a model of the spatial-domain copropagation of waves with orthogonal circular polariza-tions in a bulk waveguide made of a self-defocusing material, pro-vided that relation g = γ/ γ > g . The relevant solutions areobtained in an analytical form by means of the Thomas-Fermiapproximation (TFA), and confirmed by full numerical solutions.Section III addresses vortex states, with topological charges S =1 and 2, in the 2D system, in which SSB is again considered bymeans of TFA and a full numerical solution. In Section IV weaddress the symmetry breaking of two-component gap solitons inthe 1D system, which is described by Eqs. (1) and (2) with the HOpotential replaced by the spatially periodic one, see Eqs. (38) and(39) below. In that case, the situation is drastically different, asSSB takes place in antisymmetric two-component solitons, underthe condition opposite to the one mentioned above, i.e., γ < g ,which means that the cross-repulsion is weaker than the intrinsicself-repulsion The paper is concluded by Section V. Eigenstates of the system based on Eqs. (1), (2), and (3), withchemical potential µ >
0, are looked for as φ , ( x, y ) = exp ( − iµt ) ϕ , ( x, y ) , (7) here functions ϕ , ( x, y ) satisfy stationary equations µϕ = − m ∇ ϕ + | ϕ | ϕ + γ | ϕ | ϕ + 12 (cid:16) x + y (cid:17) ϕ − ϕ , (8) µϕ = − m ∇ ϕ + | ϕ | ϕ + γ | ϕ | ϕ + 12 (cid:16) x + y (cid:17) ϕ − ϕ . (9)The degree of asymmetry of solutions produced by Eqs. (8) and(9) is quantified by parameter R = 2 N Z Z | ϕ ( x, y ) | dxdy − , (10)where N is the total norm defined as per Eq. (4). Obviously, R = 0 in the case when norms of both components are equal.In the 1D case, ∇ is replaced by ∂ /∂x , y is dropped in the HOpotential, and functions ϕ , ( x ) are always real. In this case, theasymmetry is defined by the 1D version of Eq. (10), definitionsof N and H , given by Eqs. (4) and (5) are also replaced by their1D versions, while the definition of the angular momentum [seeEq. (6)] is irrelevant. First, we apply the TFA to Eqs. (8) and (9), dropping, as usual,the derivatives in these equations [6,7] (which is, formally, tanta-mount to taking m → ∞ ): µϕ = gϕ + γϕ ϕ + 12 (cid:16) x + y (cid:17) ϕ − ϕ , (11) µϕ = gϕ + γϕ ϕ + 12 (cid:16) x + y (cid:17) ϕ − ϕ . (12) quations (11) and (12) admit two nonzero solutions: an obvious symmetric one, ϕ ( x, y ) = ϕ ( x, y ) = 2 ( µ + 1) − (cid:16) x + y (cid:17) γ + 1) , (13)which exists at 12 (cid:16) x + y (cid:17) < µ + 1 , (14)and a symmetry-broken (alias asymmetric) solution, which existsunder the condition adopted above, γ > ϕ + ϕ = µ − (cid:16) x + y (cid:17) , ϕ ϕ = 1 γ − . (15)Formally, solution (15) exists at γ < ϕ ϕ <
0, it will make the last (linear-coupling) term in Hamiltonian(5) positive, which definitely implies instability of the respectivestates [22].The asymmetric solution given by Eq. (15) exists in a regionwhere it complies with obvious condition ϕ + ϕ > ϕ ϕ , i.e., x + y < µ − γ − ! . (16)Note that condition (16) may hold if, at least, it is valid at x = y = 0, thus a condition necessary for the existence of theasymmetric solution is µ > µ (cr) ≡ γ − . (17)Thus, TFA produces the state with the spontaneously broken ymmetry which features a two-layer structure : asymmetric , given by Eq . (15) , in the inner ( central ) layer, ≤ x + y < (cid:16) µ − γ − − (cid:17) ;symmetric , given by Eq . (13) , in the outer ( surrounding ) layer, (cid:16) µ − γ − − (cid:17) ≤ x + y < µ + 1) ≡ r ; (18)and zero outside of both layers, i.e., at x + y > r . Of course,the exact solution has a small nonzero “tail” in the latter area,which is, as usual, ignored by TFA. Another difference of a nu-merically exact solution is that the symmetry is slightly brokenin the TFA-symmetric layer.If condition (17) does not hold, the inner layer does not exist,and the entire TFA solution keeps the symmetric form, as givenby Eqs. (13) and (14). Its total norm, defined as per Eq. (4) orits 1D version, is (cid:18) N (symm)TFA (cid:19) = 2 π ( µ + 1) γ + 1 , (19) (cid:18) N (symm)TFA (cid:19) = 2 / µ + 1) / γ + 1 , (20)With the increase of the cross-repulsion strength, γ , at a fixednorm, the SSB sets in when µ , expressed in terms of the norm ofthe symmetric GS by means of Eq. (20) or (19), attains criticalvalue (17). After a simple algebra, this condition leads to equa-tions which predict, in the framework of TFA, the critical value f γ , above which SSB takes place for given N ,3 N / = r γ (cr)1D + 1 (cid:18) γ (cr)1D − (cid:19) / , (21) N π = γ (cr)2D + 1 (cid:18) γ (cr)2D − (cid:19) . (22)In particular, for N large enough, when TFA is a natural ap-proximation, Eqs. (21) and (22) yield critical values of γ close to g ≡ viz ., γ (cr)1D ≈ / N ) / , γ (cr)2D ≈ q π/N . Numerical solutions for the 1D version of Eqs. (1), (2) and (8),(9) are presented here for the total norm N = 10, as definedby the 1D form of Eq. (4), since this value makes it possible toproduce generic results. Figure 1(a) displays profiles of compo-nents ϕ , ( x ) ≡ | φ , ( x ) | of the GS, obtained by means of theimaginary-time evolution method [51,52] applied to Eqs. (1), (2),for γ = 2 . m = 1. Further, Fig. 1(b) produces TFA profilesfor the same parameters, constructed as per Eqs. (13), (15), and(18), and provides explicit comparison of this approximate ana-lytical solution for ϕ ( x ) with its numerically found counterpart[the comparison for component ϕ ( x ) is provided by the juxta-position of panels (a) and (b) in Fig. 1].Figure 1(c) summarizes the numerical and analytical results byplotting the asymmetry degree R , defined as per Eq. (10), vs. thecross-repulsion strength, γ , at a fixed value of the norm, N = 10,and fixed mass, m = 1, as obtained from the numerical solution,and from TFA, i.e., as produced by the integration of expres-sions (13), (15), and (18). It is seen that SSB takes place at γ > γ (cr)numer ≈ .
83, while its TFA counterpart, obtained from Eq.(21), is γ (cr)TFA ≈ .
73 for N = 10. The analytically predicted curve R ( γ ) is reasonably close to its numerical counterpart, both show- ng the SSB transition which may be identified as a supercriticalbifurcation [53].Overall, TFA provides reasonable accuracy for m = 1, even if,formally speaking, this approximation applies for small values ofcoefficient 1 / (2 m ) in Eqs. (8) and (9). To focus on the role ofthis parameter, Fig. 2 displays R vs. 1 / (2 m ) at γ = 2 .
5. TheSSB disappears at 1 / (2 m ) > .
8. Another essential characteristicof the GS solutions is the critical value of γ , above which SSBsets in; recall that, in the framework of TFA, it is predicted byEq. (21). A set of curves of γ (cr)1D ( N ) for several fixed values ofthe inverse-mass parameter, viz ., 1 / (2 m ) = 2, 0 .
5, and 0 .
05, areplotted in Fig. 2(b), along with the corresponding TFA limit,corresponding to 1 / (2 m ) = 0.The SSB effect corresponding to the supercritical bifurcation im-plies that, when symmetric and symmetry-broken stationary so-lutions coexist as stationary ones, the symmetric solution shouldbe unstable. This expectation is corroborated in Fig. 3(a), whichshows the time evolution of the maximum values of | φ ( x ) | and | φ ( x ) | , produced by direct simulations of coupled equations (1)and (2), for γ = 2 . m = 1, and N = 10. The input is a sym-metric solution, with ϕ ( x ) = ϕ ( x ), and a small perturbationadded to it. It is unstable, clearly tending to spontaneously trans-form into a broken-symmetry state. On the other hand, Fig. 3(b)demonstrates stability of the asymmetric GS for the same valuesof the parameters. Solutions of the system of coupled 2D equations (1) and (2) withembedded angular momentum are looked for, in the polar coor- (a) R (c) | (cid:131) (cid:211) | (b) | (cid:131) (cid:211) | (cid:131)` Fig. 1. (a) Profiles of components | φ ( x ) | ans | φ ( x ) | (solid and dashed lines) inthe stable 1D ground state (GS) with broken symmetry between the components.The solution was obtained by means of the imaginary-time-evolution method, ap-plied to the 1D version of Eqs. (1) and (2) with γ = 2 . m = 1, for the totalnorm N = 10 [see Eq. (4)]. (b) Comparison of | φ ( x ) | (the solid line) taken fromthe same numerical solution, and its TFA counterpart, produced by Eqs. (13), (15),and (18) (the dashed line running close to the solid one); the TFA-produced compo-nent | φ ( x ) | is shown too, by the double-peaked dashed curve. (c) The asymmetrymeasure, defined as per Eq. (10), vs. γ , for fixed m = 10 and N = 10. The chainof rhombuses and the dashed line show, severally, the numerical results and theirTFA-produced counterparts. R (d) (cid:131) ` (a) (b) Fig. 2. (a) The numerically found asymmetry measure R for the 1D GS solutions[see Eq. (10)], as a function of the inverse-mass parameter, 1 / (2 m ). (b) The criticalvalues of γ , above which the GS symmetry is broken in 1D, as a function of N forfixed values of the inverse-mass coefficient, 1 / (2 m ) = 2 (pluses), 0 . .
05 (squares). The dotted curve is the analytical result predicted by TFA, asper Eq. (21). dinates ( r, θ ), as φ , = exp ( − iµt + iSθ ) ϕ , ( r ) , (23)where ϕ , ( r ) are real radial wave functions, and S = 1 , , , ... is the integer vorticity, S = 0 corresponding to the 2D GS. Forstationary states represented by ansatz (23), the angular mo-mentum, defined as per Eq. (6), is related to the total norm, M = SN . A p (a) A p (b) Fig. 3. The evolution of the largest values of | φ ( x ) | and | φ ( x ) | (solid and dashedlines), produced by direct simulations of the 1D version of Eqs. (1) and (2) for γ = 2 . m = 1, and total norm N = 10, starting from (a) the unstable symmetricalstate and (b) the stable GS with broken symmetry. In the framework of TFA (which was previously applied to delo-calized vortex states [54]), the substitution of ansatz (23) leadsto the following radial equations, instead of 1D equations (11)and (12): µϕ = gϕ + γϕ ϕ + 12 r + S r ϕ − ϕ , (24) µϕ = gϕ + γϕ ϕ + 12 r + S r ϕ − ϕ . (25)Straightforward consideration of Eqs. (24) and (25) demonstratesthat solutions are different from zero in annulus r < r < r , (26) r , core = µ + 1 ± r ( µ + 1) − S , (27)provided that µ + 1 > S . Note that the central “empty” core, inwhich the TFA solution is zero, is absent in the GS solution (with S = 0), as Eq. (27) yields r ( S = 0) = 0. In the same case, Eq.(27) yields r ( S = 0) = 2 ( µ + 1), which coincides with r given above by Eq. (18) for the GS.In annulus (26), the symmetric solution of Eqs. (24) and (25) is ϕ , ( r ) ≡ ϕ ( r ) = µ + 1 γ + 1 −
12 ( γ + 1) r + S r . (28)Note that setting S = 0 makes this solution identical to its GS ounterpart given by Eq. (13). The total norm of symmetric ex-pression (28) is N (symm)TFA ( S ) = 2 π µ + 1 γ + 1 r ( µ + 1) − S − πS γ + 1 ln µ + 1 + q ( µ + 1) − S S , (29)which, for S = 0 (the GS), is tantamount to Eq. (19).Points at which the broken-symmetry solution with S ≥ ϕ − ϕ →
0, and cancelling the common in-finitesimal factor ( ϕ − ϕ ): µ − − γ ) ϕ ( r ) + 12 r + S r . (30)The substitution of the symmetric TFA solution (28) in Eq. (30)leads to the conclusion that the asymmetric solution may existbetween the following branching points: r , min = µ − γ − ± vuuut µ − γ − ! − S (31)Thus, TFA predicts the vortex solution with a three-layer struc-ture. First, it vanishes in the empty core and in the peripheralzone, ϕ , ( r < r core ) = ϕ , ( r > r outer ) = 0 . (32)The symmetric solution, as given by Eq. (28), is supported in twoedge layers, r < r < r , (33) r < r < r , (34)with the edges determined by Eqs. (27) and (31). Finally, thebroken symmetry is featured by the TFA solution in the inner ayer, r < r < r . (35)In the framework of TFA, the condition necessary for the exis-tence of the asymmetric solution amount to the existence of realvalues (31), i.e., µ > µ (cr) ( S ) ≡ γ − g + S, (36)cf. Eq. (17) for the GS. Further, combining Eq. (36) with expres-sion (29) for the norm of the symmetric vortex state, one canderive an equation for the critical strength of the cross-repulsion,above which the symmetry of the vortex states is broken, cf. Eq.(22) for the 2D GS ( S = 0). In particular, for S = 1 the equationis N = 2 πγ (cr) S =1 (cid:18) γ (cr) S =1 − (cid:19) vuuuut γ (cr) S =1 − γ (cr) S =1 + 1 − πγ (cr) S =1 + 1 ln γ (cr) S =1 + s(cid:18) γ (cr) S =1 + 1 (cid:19) (cid:18) γ (cr) S =1 − (cid:19) γ (cr) S =1 − . (37) Proceeding to numerical results for the 2D system, in Fig. 4(a)we, first, display numerically found radial profiles, ϕ , ( r ) ≡ | φ , ( r ) | ,of the two components of the 2D GS (with S = 0), for γ = 2 . m = 1, and the total 2D norm N = 100. The numerical pro-files are compared to their TFA-predicted counterparts, obtainedfrom Eqs. (24) and (25), in Fig. 4(b). Further, the numerical andapproximate analytical radial profiles of a stable vortex state,with S = 1 and the same values of other parameters as in panels(a,b), are displayed in Figs. 4(c,d). In addition to the 2D stateswith S = 0 and S = 1, higher-order ones, with S ≥ | (cid:131) (cid:211) | | (cid:131) (cid:211) | (a) (b) | (cid:131) (cid:211) | (c) (d) | (cid:131) (cid:211) | Fig. 4. (a) 1D cross sections (radial profiles) of components | φ | ans | φ | (solidand dashed lines, respectively) of the stable 2D ground state ( S = 0) with brokeninter-component symmetry. The solution was obtained by means of the imaginary–time-evolution method applied to the axisymmetric reduction of Eqs. (1) and (2) inthe 2D form, with γ = 2 . m = 1, for the total 2D norm N = 100. (b) Com-parison of | φ | (the solid line) taken from the same numerical solution as in (a),and its TFA counterpart, produced by Eqs. (24) and (25) (the dashed line whichalmost completely overlaps with the solid one). The other dashed line shows theTFA-produced component | φ | . (c,d): The same as in (a,b), but for a stable vortexmode with S = 1 and the same values of γ , m , and N . The results are summarized in Figs. 5(a), (b), and (c) by meansof curves R ( γ ) for the dependence of asymmetry measure (10)on the cross-component repulsion strength, γ , for the fixed mass, m = 1, total norm, N = 100, and three values of the vorticity, S = 0 [(a), the GS], S = 1 (b), and S = 2 (c). The TFA-predicted analytical results are included too. In particular, for S = 1 the numerically found SSB point is γ (cr) S =1 ≈ .
55, whileits TFA counterpart, found from Eq. (37), is (cid:18) γ (cr) S =1 (cid:19) TFA ≈ . S = 0, 1, and 2 the SSB transitions, displayed inFigs. 5(a-c), may be categorized as the supercritical bifurcation[53].In comparison with the similar results for the 1D GS, displayedin Fig. 1(c), the relative error of TFA [in particular, in predicting γ (cr) ] is much larger in the 2D setting, being ≃
5% in 1D and ≃
15% in 2D. On the other hand, it is worthy to note that therelative error is smaller by a factor ≃ S = 1 and 2, in comparison to the 2D GS.Numerical and approximate analytical dependences of the valueof γ at the SSB point, γ (cr) , on the total norm, N , for the 2D R (cid:131)` (a) (c) R (cid:131)` R (cid:13) ga(cid:13) (b) Fig. 5. The asymmetry measure (10) vs. the cross-repulsion strength, γ , for familiesof 2D states with S = 0 [the ground state, (a)], S = 1 (b), and S = 2 (c), with fixedeffective mass, m = 1, and 2D norm, N = 100. Chains of rhombuses and dashedcurves represent, severally, numerical results produced by the imaginary-time-prop-agation method, and analytical results provided by TFA, viz ., Eqs. (15) and (18)for S = 0, and Eqs. (24), (25) for S = 1 and 2. GS ( S = 0) and vortex mode with S = 1 are plotted in Fig.6, fixing the effective mass to be m = 1. The analytical curvesare produced by TFA, i.e., Eq. (22) for S = 0, and Eq. (37) for S = 1. In comparison to similar results for the 1D GS, shownin Fig. 2(b), the accuracy of TFA in 2D is only slightly poorer(note, however, a great difference in the scale of N between the1D and 2D cases). (cid:131) ` (cid:131) ` (a) (b) Fig. 6. The critical value of the cross-repulsion strength at the symmetry-breakingpoint in the 2D system, as a function of N at m = 1. (a) The ground state, S = 0;(b) the vortex, with S = 1. The dashed lines denote the TFA prediction, produced,respectively, by Eqs. (22) and (37). Finally, systematic direct simulations of the perturbed evolutionof the 2D states clearly demonstrate that families of the asym-metric GS solutions with S = 0 and unitary-vortex ones with S = 1 are completely stable, while asymmetric double vortices,with S = 2, are not. A typical example of the instability devel-opment in displayed in Fig. 7, which shows spontaneous splitting f the double vortex into a pair of unitary ones, that keep theasymmetric structure, with respect to components φ and φ . Itis relevant to mention that double vortices may be unstable in theframework of the single GPE with the HO trapping potential andself-repulsive nonlinearity, while unitary vortices are completelystable in the same setting [55]. | (cid:131) (cid:211) | x 0500100015002000-10 -5 0 5 10 t x (a) (b) Fig. 7. (a) A numerically found 2D stationary state with double vorticity, S = 2and broken symmetry between the components. (b) Unstable evolution of this state,leading to its spontaneous splitting in two unitary vortices. The parameters are m = 1 and γ = 1 .
8, the total norm of the stationary state being N = 100. Both thestationary state and the perturbed evolution are displayed by means of 1D crosssections. Another possibility to create localized states in the presence ofa fully repulsive nonlinearity, is offered, instead of the HO trap-ping potential, by a periodic one – namely, an optical lattice inBEC [47,48], or a photonic-crystal structure in optics [56,57]. It iswell known that the interplay of the self-repulsion with a periodicpotential gives rise to stable gap solitons [59,58]. The analysis oftwo-component gap solitons in linearly-coupled dual-core systemswith a periodic potential and intrinsic self-repulsive nonlinearityin each layer (in the absence of the inter-component nonlinearity)was developed in Refs. [46,43,32]. As mentioned above, symmetricstates in such systems are not subject to SSB, while antisymmet-ric ones develop instability which replaces them by states withbroken antisymmetry. Here, we aim to demonstrate that SSB ccurs in the coupled system if it includes relatively weak repul-sion between the components (instead of the relatively stronginter-component repulsion, which was necessary for the SSB ef-fect considered above in the states trapped in the HO potential).In the 1D setting, the system of linearly coupled GPEs, includingthe periodic potential with amplitude U , is written, in the scaledform, as i ∂φ ∂t = − ∂ φ ∂x − U cos (2 πx ) φ + ( | φ | + γ | φ | ) φ − ǫφ , (38) i ∂φ ∂t = − ∂ φ ∂x − U cos (2 πx ) φ + ( | φ | + γ | φ | ) φ − ǫφ . (39)Here, the period of the potential is fixed to be 1, and, to clearlydemonstrate the SSB effects, it is convenient to fix the linear-coupling constant as ǫ = 0 .
05, instead of ǫ = 1 in Eqs. (1), (2).To develop an analytical approach to the study of the gap soli-tons, we resort to the averaging method [60], which looks for so-lutions in the form of a rapidly oscillating carrier wave, with theperiod equal to the double period of the lattice potential, modu-lated by slowly varying (envelope) wave functions, Φ , ( x, t ): φ , ( x, t ) = Φ , ( x, t ) cos ( πx ) . (40)It is known that this approach makes it possible to predict gapsolitons existing close to edges of the spectral bandgap. The sub-stitution of this ansatz in Eqs. (38) and (39) and application ofthe averaging procedure leads, in the lowest approximation, tothe following system of equations for the slowly varying wavefunctions: i ∂ Φ ∂t = − m eff ∂ Φ ∂x + 34 ( | Φ | + γ | Φ | )Φ − ǫ Φ , (41) i ∂ Φ ∂t = − m eff ∂ Φ ∂x + 34 ( | Φ | + γ | Φ | )Φ − ǫ Φ , (42) here the effective mass is m eff = | U | / ( | U | − π ) . (43)At U < π , the effective mass is negative, hence its interplaywith the repulsive sign of the nonlinearity in Eqs. (41) and (42)gives rise to bright solitons, similar to how usual solitons areproduced by the balance of the positive mass and attractive non-linearity.To use the results reported in Ref. [34], it is convenient to ad-ditionally rescale the variables in Eqs. (41) and (42), defining t ′ = − ǫt , x ′ = √− m eff ǫx , andΦ ′ = q ǫ/ (3 γ )Φ , Φ ′ = − q ǫ/ (3 γ )Φ , (44)thus replacing Eqs. (41) and (42) by i ∂ Φ ′ ∂t ′ = − ∂ Φ ′ ∂x ′ − γ | Φ ′ | + | Φ ′ | ! Φ ′ − Φ ′ . (45) i ∂ Φ ′ ∂t ′ = − ∂ Φ ′ ∂x ′ − γ | Φ ′ | + | Φ ′ | ! Φ ′ − Φ ′ . (46)Accordingly, the total norm of rescaled fields Φ ′ , is related tothe norm of the original ones, Φ , : N ′ = (3 γ/ q m eff /ǫN. (47)As shown in Ref. [34], Eqs. (45) and (46) with γ < ′ = Φ ′ , if γ belongs to interval γ < ( γ ′ ) (cr) , where the respectivecritical value is related to N ′ by equation N ′ = 256 (cid:20) / ( γ ′ ) (cr) + 1 (cid:21) − "r / ( γ ′ ) (cr) − − r / ( γ ′ ) (cr) + 1 / ( γ ′ ) (cr) − r / ( γ ′ ) (cr) + 1 . (48)Actually, Eq. (44) implies that the solitons subject to constraintΦ ′ = Φ ′ represent antisymmetric solitons, with Φ ( x ) = − Φ ( x ),in terms of the original equations (41) and (42). umerical solutions for asymmetric solitons can be readily gen-erated by the imaginary-time integration method applied to Eqs.(41) and (42). Figure 8(a) shows an example of a soliton withbroken antisymmetry, obtained at parameters γ = 0 . U = 5, ǫ = 0 .
05, and N = 2 .
5. Note that the signs of the two compo-nents are opposite, in accordance with what is said above, andthe breaking of the antisymmetry is exhibited by the differenceof their amplitudes. Further, Fig. 8(b) summarizes the results,by displaying the asymmetry measure R for the soliton family,defined as per Eq. (10) with | ϕ | replaced by | Φ | , and N re-placed by the total norm defined in terms of envelope wave func-tions Φ , , versus the relative strength γ of the inter-componentrepulsion. The R ( γ ) dependence is plotted for fixed N = 2 . U = 5, hence the respective effectivemass, given by Eq. (43), is m eff ≈ − .
34. As shown in Ref. [34],the R ( γ ) dependence may be accurately represented by meansof a variational approximation. It is seen in Fig. 8(b) that theantisymmetry of the two-component gap solitons is broken at γ < γ (cr) ≈ .
28. The respective analytical result, determined byEq. (48) is ( γ ′ ) (cr) ≃ . (a) (b) (cid:131)` -0.3-0.2-0.100.10.20.30.40.5-100 -50 0 50 100 x R (cid:13) (cid:131) ‡ Fig. 8. (a) Profiles of Φ and Φ (solid and dashed lines, respectively), obtained asa numerical solution to Eqs. (41) and (42), representing the two-component solitonwith broken antisymmetry, for U = 5, ǫ = 0 .
05, and total norm N = 2 .
5. (b) Theasymmetry measure R ( γ ), defined for the family of the two-component solitons, asproduced by the numerical solution for U = 5, ǫ = 0 .
05, and total norm N = 2 . The prediction of the breaking of antisymmetry in two-componentgap solitons was confirmed by direct simulations of underlyingequations (38) and (39), see an example shown in Figs. 9(a)-(c) or γ = 0 . < γ (cr) . The initial conditions were taken as per ansatz(40): φ , ( x, t = 0) = Φ , ( x, t = 0) cos( πx ) , (49)where Φ , ( x, t ) is the above-mentioned numerically exact broken-antisymmetry soliton solution of Eqs. (41) and (42), obtained bymeans of the imaginary-time simulations. Figures 9(a) and (b)show, severally, the resulting evolution of maximum values of | φ , ( x ) | , and of the asymmetry measure, R , defined accordingto Eq. (10). Periodic small-amplitude variations of the fields andasymmetry are caused by a deviation of ansatz (40) from a nu-merically exact form of the two-component gap soliton with bro-ken inter-component antisymmetry. A nearly exact shape of bothcomponents of the gap soliton is produced in Fig. 9(c), which dis-plays snapshots of | φ , ( x ) | at t = 500. These numerical resultsalso demonstrate that the gap solitons with broken antisymmetryare stable. Similar to the structure of the asymmetric localizedstates produced by Eqs. (8) and (9) in the presence of the HOtrapping potential, see Fig. 1(a), the two-component gap solitonsfeature broken antisymmetry in the central zone, and persistentapproximate antisymmetry in decaying tails.Thus, an essential difference from the results reported above forthe localized states, trapped in the 1D or 2D HO potential, is thatthe SSB transition, revealed by Fig. 8(b), is categorized as an inverted bifurcation [53] (alias an extreme subcritical bifurcation ,cf. Ref. [61]), in comparison with the supercritical bifurcationsobserved in Figs. 1(c) and 5. The inversion is explained by the factthat effective mass (43), which drives the bifurcation of the gapsolitons, is negative. Another obvious difference is that the two-component antisymmetric gap solitons undergo SSB at γ <
1, onthe contrary to the above condition, γ >
1, which is necessary toimpose SSB onto the HO-trapped states. On the other hand, itis known that, in the interval of 1 < γ <
3, the system of Eqs.(41) and (42) gives rise to a bifurcation which breaks symmetryof two-component solitons with Φ = Φ (the symmetric state,rather than the antisymmetric one considered here) [34]. The onsideration of the latter effect in terms of Eqs. (38) and (39) isbeyond the scope of the present paper. (a) (b) R t (c) Ap | (cid:131) (cid:211) (cid:129) b Fig. 9. (a) The evolution of maximum values of | φ ( x ) | and | φ ( x ) | (green and bluelines), as produced by direct simulations of Eqs. (38) and (39), initiated by theinput in the form of ansatz (49), with Φ , taken as a numerically exact broken–an-tisymmetry soliton solution of Eqs. (41) and (42) with γ = 0 .
1. (b) The evolution ofthe asymmetry measure for the solution from panel (a), defined as per Eq. (10). (c)Snapshots of | φ ( x ) | and | φ ( x ) | (green and blue lines, respectively) of the solutionfrom panels (b,c) at t = 500. The snapshots closely approximate an exact shape ofthe two-component gap soliton with broken antisymmetry between the components. In numerous works, SSB (spontaneous symmetry breaking) oftwo-component localized modes in linearly-coupled systems wasfound in the case of attractive self- and/or cross-component non-linear interactions in the system. On the contrary to that, insystems with repulsive interactions spontaneous breaking wasonly shown for antisymmetric two-component states, but not forsymmetric ones. Here, we have demonstrated that SSB in both1D and 2D symmetric states, trapped in the confining potential[taken as the HO (harmonic oscillator)], is possible if the cross-repulsion is stronger than intrinsic repulsion in each component.This setting may be realized in BEC and nonlinear optics. Inthe former case, it represents a binary condensate with naturalrepulsive contact interactions and radiofrequency-induced linearmixing between two atomic states, which compose the binaryBEC. In terms of self-defocusing optical waveguides, the systemis based on the copropagation of two orthogonal polarizations of ight with the linear mixing induced by linear-polarization bire-fringence of the material.For both one- and two-dimensional GSs (ground states), as wellas for 2D vortex states, the transition from symmetric states toasymmetric ones has been demonstrated analytically by means ofTFA (Thomas-Fermi approximation) and confirmed by system-atically collected numerical solutions of the underlying systemof linearly coupled GPEs (Gross-Pitaevskii equations). A char-acteristic feature of the asymmetric states is that they combinestrongly broken asymmetry in the inner area, while a surroundinglayer keeps the original symmetry, in the approximate form. TheSSB transition for all these states is identified as a supercriticalbifurcation. It produces stable 1D and 2D asymmetric GSs, aswell as stable asymmetric vortices with topological charge S = 1,while the vortices with S = 2 are unstable against splitting in apair of unitary vortices.The phenomenology of the spontaneous antisymmetry breakingwas also briefly considered for 1D antisymmetric two-componentgap solitons, maintained by the spatially periodic potential. Inthis case, the antisymmetry breaks in the inner region of the gapsoliton under the condition opposite to that necessary for theoccurrence of the SSB effect in the HO-trapped modes, viz ., thecross-component repulsion must be weaker than the self-repulsionin each component. The character of the antisymmetry-breakingtransition for the gap solitons is opposite too, namely, it amountsto an inverted bifurcation.The above analysis did not address motion of the trapped modes.Application of a kick to the 1D trapped state, as well as of a radialpush to 2D ones, should excite a dipole mode of oscillations of theperturbed states around the center. In this connection, the effectof the Rabi coupling on the oscillations may be an interestingfeature, as suggested by recently studied effects of the same termon the motion of spinor solitons in a random potential [62]. Gapsolitons feature mobility too, with a negative dynamical mass Acknowledgments
The work of H.S. is supported by the Japan Society for Promotionof Science through KAKENHI Grant No. 18K03462. The workof B.A.M. is supported, in part, by the Israel Science Foundationthrough grant No. 1286/17, and by grant No. 2015616 from thejoint program of Binational Science Foundation (US-Israel) andNational Science Foundation (US). This author appreciates hos-pitality of the Interdisciplinary Graduate School of EngineeringSciences at the Kyushu University (Fukuoka, Japan).
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