Spontaneous symmetry breaking in Schrödinger lattices with two nonlinear sites
We introduce discrete systems in the form of straight (infinite) and ring-shaped chains, with two symmetrically placed nonlinear sites. The systems can be implemented in nonlinear optics (as waveguiding arrays) and BEC (by means of an optical lattice). A full set of exact analytical solutions for symmetric, asymmetric, and antisymmetric localized modes is found, and their stability is investigated in a numerical form. The symmetry-breaking bifurcation (SBB), through which the asymmetric modes emerge from the symmetric ones, is found to be of the subcritical type. It is transformed into a supercritical bifurcation if the nonlinearity is localized in relatively broad domains around two central sites, and also in the ring of a small size, i.e., in effectively nonlocal settings. The family of antisymmetric modes does not undergo bifurcations, and features both stable and unstable portions. The evolution of unstable localized modes is investigated by means of direct simulations. In particular, unstable asymmetric states, which exist in the case of the subcritical bifurcation, give rise to breathers oscillating between the nonlinear sites, thus restoring an effective dynamical symmetry between them.
pacs:05.45.Yv; 03.75.Lm; 42.65.Tg; 42.82.Et
Spontaneous symmetry breaking is a fundamental effect caused by the interplay of nonlinearity with linear potentials featuring basic symmetries, such as double-well structures. In particular, while it is commonly known that the ground state in one-dimensional quantum mechanics follows the symmetry of the underlying double-well potential LL (), the self-attractive nonlinearity added to the respective Schrödinger equation [which transforms it into the Gross-Pitaevskii equation for a Bose-Einstein condensate (BEC) of interacting atoms loaded into the double-well potential BEC (), or into the nonlinear Schrödinger equation in optical counterparts of the system NLS ()] breaks the symmetry of the ground state, replacing it by a new asymmetric state minimizing the system’s energy, provided that the strength of the self-attraction exceeds a certain critical value (see, e.g., Ref. misc () for the general consideration, and Ref. misc2 () for the analysis of the symmetry-breaking self-trapping in BEC). The spontaneous symmetry breaking in double-well potentials was realized experimentally in BEC Markus () and in nonlinear optics, where the symmetry breaking was observed in a setup based on a photorefractive material photo ().
The symmetry-breaking bifurcation (SBB), which destabilizes the symmetric ground state and gives rise to an asymmetric one in the nonlinear system, was originally discovered in a discrete model of self-trapping Chris (). In nonlinear optics, a similar SBB was predicted in Ref. Snyder () for continuous-wave (spatially uniform) states in the model of dual-core fibers. For solitons in dual-core systems, this bifurcation was studied in detail in Ref. dual-core (). Subsequently, the SBB was studied for gap solitons in dual-core fiber Bragg gratings with the same Kerr (cubic) nonlinearity as in the ordinary fibers Mak (). In another physical setting which also features the cubic nonlinearity, the SBB was predicted for matter-wave solitons in the self-attractive BEC loaded into a dual-trough potential trap Arik ()-Luca ().
The self-focusing cubic nonlinearity gives rise to the soliton bifurcations of the subcritical (alias backward) type, in which the branches of asymmetric modes emerge as unstable ones, going backward and getting stabilized after switching their direction forward at turning points bif (). On the other hand, the combination of the self-focusing nonlinearity with a periodic potential acting in the free direction (perpendicular to the direction of the action of the double-well potential) changes the character of the bifurcation from subcritical to supercritical, with the asymmetric branches emerging as stable ones and immediately going in the forward direction Arik (); Warsaw2 (). The SBB in the model of the dual-core fiber Bragg grating is of the forward type too Mak (). The sub- and supercritical (alias backward/forward) SBBs may be regarded as examples of phase transitions of the first and second kinds, respectively bif ().
A physically interesting alternative to the linear double-well potential is the setting with an effective pseudopotential Harrison () induced by the double-peak spatial modulation of the local nonlinearity coefficient, which may be implemented in optics and BEC alike Pu (); Barcelona (). The ultimate form of such a setting is the one with the nonlinearity concentrated at two points, in the form of a symmetric pair of delta-functions or narrow Gaussians Thawatchai (); Nir (), as well as a two-dimensional counterpart of the system, in the form of two parallel troughs in which the nonlinearity is applied Hung () (the underlying model, with the self-attractive nonlinearity represented by a single delta-function, was introduced much earlier in Ref. Mark ()). The SBB of solitons in the symmetric double-well nonlinear pseudopotentials was studied recently, featuring the subcritical type of the symmetry breaking Thawatchai (); Hung (); Nir ().
The spontaneous symmetry breaking was also analyzed for solitons in dual-core discrete systems, with the uniform coupling between two parallel chains Herring (), or with the coupling established at a single site Ljupco (). In the former case, the SBB is subcritical, while in the latter case it is supercritical.
The objective of the present work is to consider the SBB in one-dimensional discrete lattices with the nonlinearity tightly concentrated at two symmetric sites, or in narrow regions around them. This is a straightforward discrete counterpart of the double-well nonlinear pseudopotential Thawatchai (); Nir (), which offers a simple testbed for the study of symmetry-breaking effects in discrete media. Physically, the linear chain with two nonlinear sites can be readily implemented in optics, by embedding two nonlinear cores into an arrayed linear waveguide, and in BEC, by means of the Feshbach-resonance technique applied locally to the condensate trapped in a deep optical lattice BEC (). An essential advantage of the system is that, as we demonstrate below, it admits a fully analytical solution (for the infinite chain), with an arbitrary separation between the two symmetric nonlinear sites, while the stability of the exact solutions may be efficiently predicted following general principles of the elementary bifurcation theory bif () and the Vakhitov-Kolokolov (VK) criterion VK ().
The paper is organized as follows. The model is introduced in Section 2, which is followed by producing exact solutions for symmetric, asymmetric, and antisymmetric localized modes in Section 3. Numerical results, obtained for finite lattices, are reported in Section 4 (in particular, it is demonstrated that the SBB is subcritical in the system with two nonlinear sites). In Section 5, we consider an essentially different version of the system, in the form of a ring-shaped chain, with the two nonlinear sites placed at diametrically opposite points, in which case an analytical solution for the SBB is available too. The paper is concluded by Section 6.
Ii The model
According to what is said above, the model is based on the linear discrete Schrödinger equation with two nonlinear sites embedded into it:
where is the integer distance between the two nonlinear sites, and is the Kronecker’s symbol. The evolutional variable, , is time in the application to BEC, or propagation distance in the case of an array of optical waveguides. In the former case, the nonlinearity at two sites can be induced by focusing laser beams, that may enhance the nonlinearity through the Feshbach resonance optical-FR (), at two particular droplets of the condensate trapped in a deep optical lattice. In the photonic realization of the setting, strong nonlinearity in two particular cores in the waveguiding array can be readily imposed by doping them with resonant atoms (see, e.g., Ref. doping ()), which does not affect the linear coupling of these sites to adjacent ones, as implied in Eq. (1). The same mechanisms can be used for inducing the local nonlinearity in circular configurations considered below in Section 5.
Stationary solutions to Eq. (1) are sought for as , where real stationary field obeys equation
Along with this model, we will also consider its version with a smoothed form of the discrete delta-function, viz.,
where or , and is the smoothing width.
A preliminary remark is that Eq. (2) with corresponds to the single nonlinear site with the double strength, placed at . In that case, an obvious exact solution is
where is connected to by the dispersion relation for evanescent waves in the linear lattice:
Note that this expression for satisfies the VK stability criterion, , hence solutions (4) may be stable MIW (). On the other hand, the continuous counterpart of Eq. (2) with the single nonlinear site is
where is the delta-function (this continuous equation was first introduced in Ref. Mark ()). It has an obvious localized solution,
whose norm is degenerate (it does not depend on ): . Being formally neutrally stable in terms of the VK criterion, all solutions (8) are actually unstable Nir (). The degeneracy and instability of solutions (8) resemble the classical properties of the Townes’ solitons Townes ().
Iii Exact solutions for the infinite lattice
iii.1 General analysis
Symmetric, antisymmetric, and asymmetric solutions to Eq. (2) can be found in an exact form, following the pattern of the exact solutions for the continuous counterpart of Eq. (2), which was considered in Ref. Thawatchai ():
Coordinate determines the location of the center of the intermediate part of the solution. Note that does not need to be an integer number. Symmetric and asymmetric modes correspond, respectively, to and .
Ansatz (10) automatically satisfies the linear discrete Schrödinger equation. There remains to check Eq. (2) at the nonlinear sites (of course, imposing the condition of the continuity of the solution at these sites). With regard to Eq. (5), the continuity condition yields relations between the amplitudes:
and the equation at the nonlinear sites amounts to the following relations:
iii.2 Symmetric modes
Two roots of quartic equation (13) are . The negative one is unphysical, while the positive root corresponds, according to Eq. (14), to , which represents the symmetric mode. The amplitudes of the symmetric solution, as given by Eqs. (11) and (12), are
|It is relevant to note that, in the case of , Eqs. (15) yield , which naturally coincides with as given by Eq. (4).|
Taking into regard Eq. (14), it is easy to check that roots (16) correspond to two values which are located mutually symmetrically around the center: , i.e., . Further, roots (16) are physical if they are real and positive, which means . Thus, the symmetry-breaking bifurcation (SBB), i.e., the appearance of the asymmetric modes from the symmetric one [see Fig. 1(a)], happens, with the increase of distance between the nonlinear sites (at fixed , i.e., fixed ), at the critical point, , or, in terms of the frequency of the localized mode, at
In other words, for fixed , the SBB occurs with the increase of at point (17), the asymmetric modes existing at .
The asymmetric mode keeps the double-peak shape if the minimum point in expression (10), , is a really existing minimum, rather than a virtual one, i.e., it stays in interval . Further, this means that must fall into the following interval: . It is easy to check, making use of Eq. (16), that the latter condition places and into narrow intervals of their values, namely, , and, accordingly,
cf. Eq. (17). At , i.e., [in other words, at , cf. Eq. (17)], leaves the region of , hence the asymmetric mode becomes single-peaked (“strongly asymmetric”). The transition from the double-peak asymmetric mode to the single-peak shape is illustrated by Fig. 3(b).
|where signs pertain to the two conjugate asymmetric modes. Note that, at the SBB point, , expressions (19) coincide with their counterparts (15) obtained above for the symmetric mode: at this point, the amplitudes are and .|
|As a natural measure of their asymmetry, we will use|
Obviously, at the SBB point, , and expression (20) makes sense past the bifurcation point, i.e., at . In particular, at the above-mentioned point of the transition from the double- to single-peak profile, , the asymmetry is still relatively small, .
iii.4 Antisymmetric modes
Antisymmetric solutions, and their counterparts with broken antisymmetry (if any) are looked for by dint of the following ansatz, cf. Eq. (10):
Looking for a bifurcation occurring to the antisymmetric mode, one arrives at the respective roots for [with defined as in Eq. (14)]:
which differs from Eq. (16) by the opposite sign in front of the first term in the square brackets. This difference makes both roots (22) negative (unphysical), hence the antisymmetric solutions do not undergo the bifurcation, similar to the situation known in the continuous model (Thawatchai ()).
The mode with the unbroken antisymmetry is given by the exact solution in the form of ansatz (21) with amplitudes
|Finally, it is natural to call the antisymmetric modes, corresponding to even and odd , on-site and inter-site ones, respectively, as they have the zero point, , either coinciding or not with the (central) site of the lattice.|
where may be defined as the top and bottom points of a vertical diameter cutting the circle. In fact, it is more convenient to replace Eq. (24) by a system of two identical equations for two semi-circles, left and right ones. In each equation, takes values ( and are assumed). In BEC, the circular chain can be realized as a combination of a toroidal trap and periodic potential created in it torus (), and in optics it corresponds to an array of waveguides created in a hollow cylindrical shell, or written in the form of the ring in a bulk sample Jena ().
In principle, one may expect two types of the symmetry breaking in this setting: between the top and bottom, which is a counterpart of what was considered above in the framework of the rectilinear chain, or between the left and right semi-circles.
The linear solutions for the left () and right () semi-circles can be looked for as
with related to by Eq. (5), and some constants and ( and are not necessarily integer numbers). Then, the continuity conditions should be imposed at points , where the two semi-circles are linked into the entire circle:
After simple manipulations, one may eliminate the amplitudes from two equations (26), which leads to the following consistency condition: It is obvious that the consistency condition can be met in the case of , which is trivial (zero length of the ring), or
However, the top-bottom symmetry breaking is possible. Substituting ansatz (25) with the left-right symmetry into Eq. (24) at points , the result of a straightforward analysis is a system of two equations, corresponding to and :
An obvious corollary of Eqs. (28) is relation
The SBB point can be found by setting , with infinitesimal , and demanding that the coefficient in front of in the respective expansion of the left-hand side of Eq. (30) vanishes. The result is that this happens at point which is equivalent to
At this point, the amplitude is
The asymmetric state exists, for given , i.e., given , for the length of the semi-circle, , which is larger than the one corresponding to Eq. (31). Alternatively, at given , the asymmetric state exists for exceeding the value defined by Eq. (31).
In Fig. 10 we summarize the results related to the ring configuration. In panel (a), families of the symmetric and asymmetric solutions are displayed for three different lengths of the ring. We have checked the linear stability of the corresponding solutions through the numerical computation of the eigenvalues for small perturbations. The respective stable and unstable regions of the existence curves are shown by the solid and dashed lines, respectively. As in the case of the linear chain, at critical frequency calculated from Eq. (31), the branches of the asymmetric solutions bifurcate from the branch of symmetric solutions. The amplitude of the symmetric solutions at the bifurcation point is displayed in Fig. 10(b) for different lengths of the ring.
Also for different lengths of the ring, we have analyzed the type of the corresponding SBB, calculating the asymmetry parameter as per Eq. (20). The dependences of on frequency and norm are shown in Figs. 10(c) and (d). As one can see, the length of the ring plays a crucial role in the determination of the bifurcation type. In the small ring (e.g., of length 6), the bifurcation is supercritical, while the increase the length to changes it into a subcritical one.
In Figs. 11 and 12, the evolution of stable and unstable asymmetric and symmetric solutions is shown. The direct simulations corroborate the linear-stability analysis. As one can see, the solutions taken on stable parts of the existence curves are stable indeed. The unstable asymmetric solution [see Fig. 11(b)] starts to oscillate between two asymmetric configurations, while the unstable symmetric solution in Fig. 12(b) rapidly transforms into an oscillating single-peak mode. It is relevant to stress that, similar to the situation in the straight chain (cf. Fig. 5), the instability of symmetric modes transforms them into asymmetric ones, while the instability of asymmetric modes leads to the emergence of the effective dynamical symmetry.
Finally, it is relevant to note that, in the case of the circle of a finite length, it is also possible to consider values , i.e., imaginary wavenumbers, , see Eq. (5). This means that the exponentially decaying discrete waves are replaced by oscillatory ones. After a simple analysis, the respective version of Eq. (28) can be derived in the following form:
A straightforward analysis of Eq. (33) demonstrates that, on the contrary to Eq. (28), it does not give rise to the symmetry breaking (the crucial difference is the opposite sign in front of in the square brackets).
The objective of this work is to elaborate the simplest setting for the study of the spontaneous symmetry breaking in dynamical chains. For this purpose, we have introduced two one-dimensional discrete systems, in the form of the straight and ring-shaped linear chains, with two symmetrically inserted nonlinear sites. This pair of sites introduces the symmetry that may be spontaneously broken. The chains, both rectilinear and circular ones, can be realized in BEC (with the help of optical lattices) and in optics, in the form of waveguiding arrays. A full set of analytical solutions for symmetric, asymmetric, and antisymmetric localized states has been obtained in the explicit form, for both geometries. The stability of the stationary modes was investigated through the numerical computation of the eigenvalues for small perturbations. The SBB (symmetry-breaking bifurcation), which is responsible for the asymmetric modes emerging from the symmetric ones, is of the subcritical type in the straight lattice, and in the circular one of a sufficiently large size. The bifurcation becomes supercritical if the system is made effectively nonlocal, i.e., in the ring of a smaller size, or if the nonlinearity is spread over relatively broad areas around the two central sites in the straight chain. The antisymmetric modes are not subject to bifurcations, although they too may be both stable and unstable. The development of the instability (when it occurs) was tested with the help of direct simulations. It was found that the unstable stationary asymmetric states, which are a part of the subcritical bifurcation, spontaneously transform into breathers oscillating between the two nonlinear sites, that may be considered as the restoration of an effective dynamical symmetry.
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