Wave modes trapped in rotating nonlinear potentials
We study modes trapped in a rotating ring with the local strength of the nonlinearity modulated as , where is the azimuthal angle. This modulation pattern may be of three different types: self-focusing (SF), self-defocusing (SDF), and alternating SF-SDF. The model, based on the nonlinear Schrödinger (NLS) equation with periodic boundary conditions, applies to the light propagation in a twisted pipe waveguide, and to a Bose-Einstein condensate (BEC) loaded into a toroidal trap, under the action of the rotating nonlinear pseudopotential induced by means of the Feshbach resonance in an inhomogeneous external field. In the SF, SDF, and alternating regimes, four, three, and five different types of stable trapped modes are identified, respectively: even, odd, second-harmonic (2H), symmetry-breaking, and 2H-breaking ones. The shapes and stability of these modes, together with transitions between them, are investigated in the first rotational Brillouin zone. Ground-state modes are identified in each regime. Boundaries between symmetric and asymmetric modes are also obtained in an analytical form, by of a two-mode approximation.
pacs:42.65.Tg; 03.75.Lm; 47.20.Ky; 05.45.Yv
Optical and matter waves exhibit a plenty of dynamical scenarios under the action of effective nonlinear potentials (which may sometimes be combined with usual linear potentials) YVKRevMod (). The dynamics of such systems is governed by the nonlinear Schrödinger equation (NLSE) in optical media, or Gross-Pitaevskii equation (GPE) in the context of Bose-Einstein condensates (BECs). In either case, the nonlinear pseudopotential Harrison () may be induced by a regular LCQian (); 2D () or singular singular () spatial modulation of the local nonlinearity. These systems have been studied theoretically in a variety of one- LCQian (); singular () and two-dimensional (1D and 2D) 2D () settings, and recently reviewed in Ref. YVKRevMod (). To the same general class belong models which predict that, in any dimension , stable fundamental and vortex solitons can be supported by a purely self-defocusing (SDF) nonlinearity growing towards the periphery () at any rate faster than Barcelona ().
In optics, such nonlinear potentials may be designed using the mismatch between the nonlinearity of the host material and solid Fluan () or liquid TTLarsen () stuff filling voids of photonic-crystal-fiber waveguides. Another possibility to create the effective nonlinear potential in optics is offered by inhomogeneous distributions of dopants which induce the resonantly enhanced nonlinearity Kip (). In particular, it is possible to use the Rhodamine B dopant added to the SU-8 polymer (a commonly used transparent negative photoresist) Liangbing (), or Pr ions doping the YSiO host medium Yongyao (); Fleischhauer ().
In BEC, the pseudopotential can be created with the help of the Feshbach resonance controlled by nonuniform magnetic SInouye () or optical optical () fields. In particular, the necessary pattern of the spatial modulation of the scattering length, which determines the local strength of the cubic nonlinearity in the respective GPE, can be induced by appropriately designed magnetic lattices magnetic ().
Another well-known tool for the creation of various dynamical states is provided by rotating potentials, which may trap optical and matter waves. Effects of the rotation have drawn a great deal of attention in the studies of BEC. A well-known results is that rotational stirring of the condensate with repulsive interactions leads to the formation of vortex lattices, see review vort-latt (). Under special conditions (the compensation of the trapping by the centrifugal force), giant vortices can be produced too, which were studied in detail experimentally Cornell () and theoretically giant (). In a binary immiscible BEC, vortex streets were predicted to form, instead of the vortex lattices street (). On the other hand, it was predicted that the rotation of self-attractive condensates gives rise to several species of localized modes, such as solitary vortices, mixed-vorticity states (“crescents”), and quasi-solitons rotating-trap ().
It is natural too to consider the dynamics of matter-wave modes trapped in rotating lattices, which can be created in BEC by broad laser beams transmitted through a revolving mask sieve (). Quantum states and vortex lattices have been studied in this setting in-sieve (), as well as the depinning of trapped solitons and solitary vortices when the rotation rate exceeds a critical value HS (). The nucleation of vortices in the rotating lattice was demonstrated experimentally nucleation ().
In optics, a setting similar to the rotating lattice can be realized in twisted photonic-crystal fibers twistedPC (); Russell (). In plain optical fibers, the twist affects the polarization dynamics RYChiao () and couples it to the transmission of temporal solitons rocking (). In helical photonic-crystal fibers, modified Bragg reflection and enhancement of the mode conversion and transport have been studied Sujia (), and the transformation of the linear moment of photons into the orbital angular momentum has been demonstrated recently Russell ().
The simplest version of the rotating lattice is represented by the revolving quasi-1D double-well potential (DWP). It gives rise to azimuthal Bloch bands Ueda (), and allows one to support solitons and solitary vortices even in the case of the SDF nonlinearity guiding (). The generation of a vortex lattice in the rotating DWP was studied too WenLuo (). Further, it is well known that the interplay of the DWP with the SF or SDF nonlinearity provides for the simplest setting for the study of the spontaneous symmetry breaking of even and odd states, respectively, in one dimension SSB (). In this connection, a natural problem, which was recently considered in Ref. we (), is a modification of the symmetry-breaking scenarios in a rotating ring carrying the DWP potential, along with the nonlinearity.
While the dynamics of nonlinear waves trapped in rotating linear potentials has been studied in detail, previous works did not tackle modes pulled by rotating nonlinear (pseudo-) potentials. This setting, which may be implemented in optics and BEC alike, is the subject of the present work. To analyze the basic features of the respective phenomenology, we here concentrate on the 1D nonlinear potential on a rotating ring, as shown in Fig. 1. In optics, this system is realized as a pipe (hollow) waveguide with an azimuthal modulation of the local nonlinearity, ( is the angular coordinate), twisted with pitch , where plays the role of the effective rotation speed. In BEC, a similar setting corresponds to a toroidal trap, which is available in the experiment torus (), combined with the rotating nonlinear potential, that can be superimposed on top of the trap Salasnich (). This combination realizes a rotating ring Stringari () carrying the nonlinear potential.
It is necessary to define the sign of the nonlinear potential. In this work, we consider three distinct cases, namely, the SF (attractive), SDF (repulsive), and SF-SDF (alternating) nonlinearities, all subject to the harmonic spatial modulation, which is defined as follows:
|The solution domain is set as . Note that, in all the cases the local nonlinearity coefficient (1) has its maxima at points and , and minima at . We aim to find basic types of modes trapped in rotating nonlinearity profiles (1), and establish their basic properties, such as symmetry/asymmetry, stability, and the identification of the respective ground states, varying two control parameters, viz., the rotation speed, , and the total power (norm), , which is defined below in Eq. (20).|
|The paper is structured as follows. In Section II, we formulate the system and methods of the stability analysis of the modes. In Section III, we present numerical results for the basic modes and their stability in each type of the nonlinear potential (1). In Section IV, we present analytical results, obtained by means of a two-mode approximation, which explain a boundary between the symmetric and asymmetric modes in each case. The paper is concluded by Section V.|
, where is the propagation distance in the case of the optical waveguide, and the radius of the ring is scaled to be . It is more convenient to rewrite Eq. (2) in the rotating reference frame, with :
while the solution domain is defined as above, i.e., . For the BEC trapped in the rotating potential, the respective GPE differs by replacing with time . Equation (3) conserves the total power (norm) of the field and its Hamiltonian (energy),
with the asterisk standing for the complex conjugate. Stationary modes with real propagation constant (in terms of BEC, is the chemical potential) are sought for as
with complex function obeying equation
The periodic boundary conditions, and , make Eq. (3) invariant with respect to the boost transformation, which allows one to change the rotation speed from to with arbitrary integer :
hence the speed may be restricted to interval . Furthermore, Eq. (3) admits an additional invariance, relating solutions with opposite signs of the speed: . If combined with boost , the latter transformation demonstrates that the solutions with
where , are tantamount to each other. Thus, the rotation speed may be eventually restricted to the fundamental interval,
which plays the role of the first rotational Brillouin zone, cf. Ref. Ueda ().
The dynamical stability of the stationary solutions has been investigated via numerical computation of eigenvalues for small perturbations, and verified by direct simulations of the perturbed evolution. The perturbed solutions are introduced as usual,
where is an infinitesimal amplitude of the disturbance, and are the corresponding eigenmodes, and the eigenfrequency. The substitution of ansatz (11) into Eq. (3) and linearization leads to the linear eigenvalue problem,
where is the single-particle Hamiltonian. The underlying solution is stable if Eq. (12) generates solely real eigenvalues. Lastly, stationary equation (7) was solved using numerical code “PCSOM” borrowed from Ref. YJK ().
Iii Numerical Results
iii.1 The classification of trapped modes
For all types of the nonlinear potentials defined in Eq. (1), the numerical solution of stationary equation (7) with different inputs (initial guesses) makes it possible to identify five basic species of stationary states, which are listed in Table 1.
|Inputs||Types of modes|
Following the symmetry of the nonlinearity-modulation patterns in Eq. (1), the modes are identified as symmetric (alias even) and antisymmetric (alias odd), with respect to the central point, . The mode of the “second-harmonic” (2H, also even) type refers to the dominant term in its Fourier decomposition. The names of the last two modes in Table 1 come from the types of their symmetry breaking. In particular, the named of 2H-breaking mode is come from its spontaneous symmetry breaking via the 2H mode, which will be demonstrated in the following. Note that all the inputs displayed in the table are real functions, while the numerically found solutions of Eq. (7) with are actually complex. Naturally, real parts of the solutions generated by the real inputs indicated in the table have the same parity (even/odd) as the inputs, while the imaginary parts feature the opposite parity (odd/even) (the input of the 2H-breaking type does not feature a certain parity). It is shown below too that maxima and minima of the local power (density) of the even mode defined in the table coincide with the maxima and minima of the local nonlinearity, , while for the odd and 2H modes the relation is the opposite, with the peak powers sitting in potential wells [see Eq. (1)], hence the odd and 2H modes tend to have lower values of the energy, and may play the role of the ground state, as confirmed below.
iii.2 The self-focusing nonlinearity
The SF nonlinear potential, defined as per Eq. (1a), gives rise to four types of dynamically stable trapped modes, viz., even (symmetric), odd (antisymmetric), 2H and 2H-breaking ones (i.e., only the symmetry-breaking species is missing in this case). Typical examples of these stable modes are displayed in Figs. 2 and 3. In addition to the above-mentioned fact that the maxima and minima of the local power coincide with those of the nonlinear potential for the even mode, and, on the contrary, coincide with minima and maxima of the potential for the odd and 2H modes, suggesting that either of the latter modes may be a ground state, the figures demonstrate that the 2H-breaking mode has one maximum and one minimum of the local power, both sitting in nonlinear-potential wells (minima).
Results of the numerical analysis for the stability of the modes in the model with the SF nonlinear potential are summarized in Fig. 4, in the form of diagrams drawn in the plane of (). They show that the stability regions of the even, odd, 2H and 2H-breaking modes strongly overlap between themselves. In particular, the asymmetry parameter of the 2H-breaking modes, which is defined as
The multistability, which is obvious in Fig. 4, makes it necessary to compare energies of the coexisting dynamically stable modes, defined by Eq. (5), in order to identify the ground state that realizes the energy minimum. First, in Fig. 6(a) we show the results along horizontal cuts of all the three panels of Fig. 4, made at a constant value of the total power, , with the rotation speed varying in the interval of . The tristability of the even, odd and 2H modes takes place along this segment. Further, Fig. 6(b) displays the results along vertical cuts of panel (c) of Fig. 4 made at , while the power is varying as [it is seen in panels (b) and (c) of Fig. 4 that the 2H-breaking mode also coexists with the stable odd one, but the energy of the odd mode is definitely larger, therefore it is not displayed in Fig. 6(b)]. In panel (b) of Fig. 6, the two branches merge into one, with , at , as the nonlinear potential vanishes in this limit.
From Fig. 6, we conclude that , while Fig. 6 shows that . Calculations of the energy, performed along other horizontal and vertical cuts, demonstrate that the following relation between the energies of the different modes, suggested by these inequalities, is always correct:
Thus, the 2H-breaking mode, if it exists [recall that, in the plane of shown in Fig. 4(b), it exists above the curve separating the bottom (red) and middle (yellow) areas], represents the ground state in the system with the self-focusing nonlinearity. If the latter mode does not exist, then the 2H state plays the same role.
In Fig. 6, we can see that in the limit of , the energies of the even and odd modes are very close, which seems in contradiction with the fact that these two modes are essentially different, having the opposite parities. The reason is that the total power that we chose here () is not large enough to show the energy difference between the two modes. To clarify the point, in Fig. 6 we display the energy curves for these two modes selected along the vertical segment with and . It shows that the energy difference indeed increases with the growth of .
iii.3 The self-defocusing nonlinearity
In the case of the SDF nonlinear potential, represented by Eq. (1b), the numerical solution of Eq. (7) reveals stable modes of three types, viz., even, symmetry-breaking, and 2H (recall that symmetry-breaking modes were not found in the system with the SF nonlinearity), while odd and 2H-breaking states do not exist in this case. Because the profiles of the even and 2H modes are quite similar to their counterparts presented above in Fig. 2, we here display, in Fig. 7, only a typical stable symmetry-breaking mode. The shape of this mode seems symmetric, centered at ; however, it is classified as an asymmetric mode, as the power profile of a true symmetric state would be double-humped, cf. Figs. 2(b,d) and 3(b), while the present one has a single maximum, similar to the intensity distribution in the 2H-breaking state in Fig. 3(d). Accordingly, the ASP (effective asymmetry measure) for the symmetry-breaking mode is introduced as follows, instead of the above definition (13):
to stress the lack of the asymmetry between the central and peripheral parts parts of the mode.
The stability and energy diagrams for the even, symmetry-breaking, and 2H modes are displayed in Fig. 8 [the 2H mode exists and is stable in the entire plane of , therefore it is not specially marked in panel 8(a)]. In particular, it is observed that the symmetry-breaking mode exists near the right edge of the rotational Brillouin zone, i.e., it does not exist in the stationary system (with ). The ASP of the even and symmetry-breaking modes, defined per Eq. (15), is displayed in Fig. 5 as a function of of the total power. The absence of the linkage between the branches representing these two modes implies that they are not related by any bifurcation.
Figure 8 displays the comparison of energies of these three kinds of the modes (even, symmetry-breaking, and 2H ones) along the vertical cut made at , with the power varying in interval . It demonstrates that the curve for the 2H mode goes close to but slightly lower than its counterpart for the symmetry-breaking mode. The analysis of more general data, produced by the numerical calculations for the system with the SDF nonlinear potential, demonstrates that energies of all the three dynamically stable modes existing in this case are ordered as follows:
cf. Eq. (14). Thus, the 2H mode plays the role of the ground state in the case of the SDF nonlinearity (recall this mode exists at all values of and ).
iii.4 The alternating self-focusing – self-defocusing nonlinear potential
In the case of the alternating SF-SDF nonlinearity, defined as per Eq. (1), the numerical solutions reveal the existence of all the five types of stable trapped modes indicated in Table 1. Profiles of these modes are quite similar to those of their counterparts displayed above in Figs. 2, 3, and 7, therefore we do not show them again here. The respective stability diagrams in the plane are presented in Fig. 9. In particular, the absence of a bistability area in panel (c) of Fig. 9 suggests that the transition between the 2H and 2H-breaking modes is supercritical, like in the system with the SF nonlinearity, cf. Fig. LABEL:fig_4_c.
The energies of the symmetry-breaking and odd modes are not compared in Eq. (17), as their stability regions do not overlap, see Fig. 9(a,b). Thus, the 2H-breaking mode, when it exists, plays the role of the ground state in the present case; otherwise, the ground state is represented by the 2H mode, see panels (b) and (c) in Fig. 9.
Iv The analytical approach
iv.1 The two-mode approximation
The present setting may be naturally approximated by a finite-mode truncation of the expansion of stationary field over the set of spatial harmonics. The simplest approximation reduces to the substitution of truncation
into Eq. (7). In this expression, may be fixed to be real, while amplitude is allowed to be complex. This approach is consistent if, in the linear approximation, each term in combination (18) is an exact solution of Eq. (7) for a common value of , hence the zeroth-order approximation exists, and the analysis of weakly nonlinear states can be developed around it. It is easy to see that such a case corresponds, in the zeroth approximation, to [which is exactly the right edge of zone (10)] and we (). Then, weakly nonlinear modes can be constructed in an approximate analytical form, assuming that , , and
are all small quantities. To this end, ansatz (18) and a particular expression (1) for are substituted into Eq. (7), and equations for amplitudes and are derived as balance conditions for the zeroth and first harmonics.
iv.2 The self-focusing nonlinearity
As shown above by the numerical analysis, the SF nonlinearity gives rise, inter alia, to the 2H-breaking mode, which is generated from input in Table 1. To capture the part of the solution corresponding to in the input, we set
Then, Eq. (21) yields
It demonstrates that this mode, as predicted by the analytical approximation, satisfied the Vakhitov-Kolokolov (VK) criterion, , which is a necessary stability condition for patterns supported by the SF nonlinearity VK ().
Furthermore, Eq. (23) predicts the emergence of the 2H-breaking modes at , i.e.,
This analytical result is compared with the corresponding numerical findings in Fig. 10(a), which shows a reasonably good agreement.
iv.3 The self-defocusing nonlinearity
The substitution of the same ansatz (18) into Eq. (7) in the case of the SDF nonlinear potential, taken as per Eq. (1b), yields the following algebraic equations, instead of Eq. (21) derived above for the SF nonlinearity:
The above numerical results for the SDF nonlinearity demonstrate the existence of the symmetry-breaking mode in this case, which is generated by input in Table 1. To capture this mode by means of ansatz (18), it is natural to set
cf. Eq. (24). This equation demonstrates that the asymmetric mode satisfies the anti-VK criterion, , which, as argued in Ref. anti (), may play the role of a necessary stability condition for modes supported by the SDF nonlinearity.
The analytical approximation predicts the existence boundary for the symmetry-breaking states, , in the same form (25) as it was obtained above for the SF nonlinearity. Consequently, the existence region for these states is predicted in the form coinciding with that given by Eq. (26). This result is compared with its numerical counterpart in Fig. 10(b).
iv.4 The alternating self-focusing – self-defocusing nonlinearity
In the case of the alternating SF-SDF nonlinear potential (1c), the substitution of ansatz (18) into Eq. (7) yields the algebraic equations in the form which is somewhat simpler than Eqs. (21) and (27) derived above for the “pure” SF and SDF nonlinearities:
The corresponding relation between and is
which satisfies the above-mentioned anti-VK criterion, cf. Eq. (30), i.e., it is possible to assume that the stability of the symmetry-breaking mode is supported by the SDF part of the alternating nonlinear potential. Further, Eq. (32) predicts the emergence of the symmetry-breaking mode () at . In terms of the total power related to by Eq. (20), this implies that this mode exists at
The numerical results presented in the previous section demonstrate that the alternating SF-SDF nonlinear potential support the stable 2H-breaking mode too. To describe it in the framework of the two-mode approximation, we now assume to be imaginary, as in Eq. (22). In this case, Eq. (32) yields
cf. Eq. (31), the respective relation between and being
cf. Eq. (33). The latter relation satisfies the VK criterion, which implies that the stability of the 2H-breaking mode is supported by the SF part of the alternating nonlinear potential. The emergence of the 2H-breaking mode corresponds to , i.e., again , as in the case of the symmetry-breaking mode, under the same alternating nonlinear potential. Finally, this means that the existence of the stable 2H-breaking mode is predicted in the same region (34) as for its symmetry-breaking counterpart. The latter prediction is compared to the numerical findings in Fig. 10(d).
This work aimed to study the existence and stability of modes trapped in the rotating nonlinear-lattice potentials of the SF, SDF (self-focusing and defocusing) and alternating SF-SDF types. The consideration was carried out for the first rotational Brillouin zone in the rotating reference frame, i.e., for , where is the rotation speed. The stability analysis was performed through the computation of eigenvalues for small perturbations, and verified by direct simulations. The model can be realized in the spatial domain, in terms of a twisted-pipe optical waveguide, with the built-in azimuthal modulation of the local Kerr coefficient, or, in the temporal domain, as the Gross-Pitaevskii equation for BEC loaded into a toroidal trap, under the action of a rotating optical or magnetic structure which affects the local value of the scattering length.
In the SF system, four types of different modes have been identified: even, odd, which are dominated by combinations of the fundamental and zeroth angular harmonic, and the 2H (second-harmonic) and 2H-breaking states. On the other hand, the SDF nonlinear potential supports three species of the trapped states: 2H, even, and symmetry-breaking ones, the latter existing only at the rotation speed close to the right edge of the Brillouin zone, , and in a limited interval of values of the total power, . As for the alternating SF-SDF nonlinear potential, it supports all the five species of the trapped modes. Transitions between the 2H and 2H-breaking modes are of the supercritical types. The energy comparison reveals that, in the SF and SF-SDF systems alike, the 2H-breaking mode, if it exists, represents the ground state; otherwise, this role is played by the 2H mode. The ground state of the SDF system is always represented by the 2H solution.
This work may be naturally developed in other directions. On the one hand, it is relevant to consider rotating nonlinear lattices with smaller azimuthal periods, , for integer , unlike the case of investigated here. On the other hand, it may be interesting to consider a two-dimensional version of the present one-dimensional model (with an entire rotating plane, rather than the thin ring, which will include effects of the Coriliolis’ force).
Acknowledgements.We appreciate a valuable discussion with Dimitri J. Frantzeskakis, and help in the use of numerical methods provided by Nir Dror and Shenhe Fu. This work was supported by CNNSF(grant No. 11104083,11204089,11205063), by the German-Israel Foundation through grant No. I-1024-2.7/2009, and by the Tel Aviv University in the framework of the “matching” scheme.
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