Rabi oscillation of azimuthons in weakly nonlinear waveguides

Rabi oscillation, an inter-band oscillation, depicts the periodic flopping between two states that belong to different energy levels in the presence of an oscillatory driving field. In photonics, Rabi oscillation can be mimicked by applying a weak longitudinal periodic modulation to the refractive index change of the system. However, the Rabi oscillation of nonlinear states has yet to be discussed. We report Rabi oscillations of azimuthons---spatially modulated vortex solitons---in weakly nonlinear waveguides with different symmetries, both numerically and theoretically. The period of Rabi oscillation can be determined by applying the coupled mode theory, which largely depends on the modulation strength. Whether the Rabi oscillation between two states can be obtained or not is determined by the spatial symmetry of the azimuthons and the modulating potential. In this paper we succeeded in obtaining the Rabi oscillation of azimuthons in the weakly nonlinear waveguides with different symmetries. Our results not only enrich the Rabi oscillation phenomena, but also provide a new avenue in the study of pattern formation and spatial field manipulation in nonlinear optical systems.


I. INTRODUCTION
The Rabi oscillation originated in quantum mechanics [1], but by now is much investigated in a variety of optical and photonic systems that include fibers [2,3], multimode waveguides [4][5][6], coupled waveguides [7], waveguide arrays [8][9][10], and two-dimensional modal structures [11,12]. Recently, Rabi oscillations of topological edge states [13,14] and modes in fractional Schrödinger equation [15] were also reported. Rabi oscillations are inter-band oscillations that require an ac field to be applied as an external periodic potential. In optics, the longitudinal periodic modulation of the refractive index change plays the role of an ac field in temporal quantum systems, and Rabi oscillations are indicated by the resonant mode conversion. As far as we know, the investigation of optical Rabi oscillations thus far has been limited to the linear regime only, and the Rabi oscillation in nonlinear systems is still an open problem that needs to be explored. It is addressed in this paper.
Hence, the aim of this work is to investigate Rabi oscillations of azimuthons in weakly nonlinear waveguides that is accomplished by applying a weak longitudinally modulated periodic potential. Azimuthons are a special type of spatial solitons; they are azimuthally modulated vortex beams that exhibit steady angular rotation upon propagation [16]. Generally, azimuthons, especially the ones with higher-order angular momentum structures, are unstable in media with local Kerr or saturable nonlinearities. To overcome the instability drawback, a nonlocal nonlinearity is introduced, and recently published reports demonstrate that the stable propagation of azimuthons can indeed be obtained [17][18][19]. In addition, it was also reported that the spin-orbit-coupled Bose-Einstein condensates can support stable azimuthons as well [20]. However, the treatment of nonlocal nonlinearity and spin-orbit-coupled Bose-Einstein condensates is challenging in both theoretical modeling and experimental demonstration. Nevertheless, it has been confirmed that the weakly nonlinear waveguides [21] represent an ideal platform for the investigation of stable azimuthons [22,23], even with higher-order modal structures.
Following this path of inquiry, we first investigate Rabi oscillations of azimuthons in a circular waveguide and then in a square waveguide. Since in this nonlinear three-dimensional wave propagation problem no analytical solutions are known, necessarily the mode of inquiry will be predominantly numerical with some theoretical background. In the circular waveguide, the azimuthons will exhibit Rabi oscillation while rotating during propagation.
In the square waveguide, the behavior of the azimuthons is different in two aspects [15]: (i) azimuthons will rotate only if the corresponding Hamiltonian (energy) is bigger than a certain threshold value; (ii) azimuthons will be deformed during propagation. Hence, in this work we choose azimuthons with large enough energies to avoid wobbling motions in the square waveguide during propagation.

A. Theoretical analysis
The propagation of a light beam in a photonic waveguide can be described by the Schrödinger-like paraxial wave equation where Ψ(X, Y, Z) is the complex amplitude of the light beam, the quantities (X, Y ) and Z are the transverse and longitudinal coordinates, and k 0 = 2πn b /λ 0 with λ 0 being the wavelength. The other quantities in Eq. (1) are: µ 1 is the longitudinal modulation strength, δ is the longitudinal modulation frequency, n 2 is the nonlinear Kerr coefficient, n(X, Y ) is the linear refractive index distribution, and n b is the ambient index. In Eq. (1), the refractive index change includes two parts, which are |n − n b | (linear part) and n 2 |Ψ| 2 (nonlinear part). We would like to note that weakly nonlinear waveguides demand not only both the linear and nonlinear refractive index changes to be small in comparison with n b , but also the nonlinear part to be much smaller than the linear part. According to the relations x = X/r 0 , y = Y /r 0 , z = Z/(k 0 r 2 0 ), d = k 0 r 2 0 δ, and σ = sgn(n 2 ), with r 0 being determined by the real beam width, Eq. (1) can be rewritten into its dimensionless version with ψ = k 0 r 0 |n 2 |/n b Ψ and V (x, y) = k 2 0 r 2 0 [n(x, y) − n b ]/n b . Here, we will consider propagation in a deep circular potential V (x, y) = V 0 exp[−(x 2 + y 2 ) 5 /w 10 ] with w characterizing the potential width and V 0 the potential depth. To guarantee a weak nonlinearity, the potential should be deep enough. Thus, the potential is deep but the potential modulation is shallow. The parameter σ = 1 (σ = −1) corresponds to the focusing (defocusing) nonlinearity. In this work, we consider the focusing nonlinearity, i.e., we take σ = 1. There are large amounts of materials to be used to produce waveguides, and silica is one of the popular materials among them with typical parameters n b = 1.4, |n − n b | ≤ 9 × 10 −3 , and n 2 = 3 × 10 −16 cm 2 /W for light beams with wavelength ranging from visible to nearinfrared. Without loss of generality, we choose λ 0 = 800 nm in this work. Therefore if we choose V 0 = 500, the value of r 0 ≈ 25.0 µm can be obtained according to the relation adopted in Eq. (2). Indeed, such a value is reasonable for a multi-mode fiber [24]. According to the wavelength and r 0 , one knows that the diffraction length is ∼ 7 mm. Considering the group velocity dispersion coefficient is ∼ 35 fs 2 /mm at the wavelength λ 0 = 800 nm, the dispersion length is of the order of kilometers for a picosecond light beam, which is much longer than the propagation distance taken in this work. As a result, it is safely to neglect the temporal effect.
To start with, we consider the modes supported by the deep potential alone, therefore the nonlinear term and the longitudinal modulation in Eq. (2) are initially neglected.
The corresponding solution of the reduced linear Eq. (2) can be written as ψ(x, y, z) = u(x, y) exp(iβz), with u(x, y) being the stationary profile of the mode and β the propagation constant. Plugging this solution into the reduced Eq. (2), one obtains βu = 1 2 which is the linear steady-state eigenvalue problem of Eq. (2) with σ and µ equal to zero.
Equation (3) can be solved by utilizing the plane-wave expansion method, and the eigenstates supported by the deep potential V (x, y) can be easily obtained. In Fig. 1, the first-order as well as higher-order basic modes, degenerate dipole modes, degenerate quadrupole modes, degenerate hexapole modes, and degenerate octopole modes that can exist in the potential are displayed. Here, the "degenerate" means that all modes feature the same propagation constants, in the usual optical meaning. These linear modes will be used as the input modes of the more general nonlinear and modulated modes of the complete Eq. (3).
Therefore, to seek approximate azimuthons in weakly nonlinear waveguides, one takes the degenerate modes and makes a superposition of them, as the initial wave in Eq. (4), where A is an amplitude factor, 1 − B the azimuthal modulation depth, and and numerically propagate it, to obtain an output mode at arbitrary z. We would like to note that the inputs of the form (5) do not rotate in linear medium, since modes are degenerate. Rotation appears only when nonlinearity is added into model.
In Fig. 2, we display such approximate azimuthons with A = 0.4 and B = 0.5. One finds that the phase of azimuthons is nontrivial, displaying angular momentum and topological charge. For dipole azimuthons, the topological charge is ±1, while for quadrupole, hexapole, and octopole azimuthons, the values are ±2, ±3 and ±4, respectively. As expected, these azimuthons will rotate with a constant frequency ω during propagation when the nonlinear term in Eq. (2) is included. Therefore, the wave U (x, y, z) can be rewritten as U (r, θ − ωz) in polar coordinates, with r = x 2 + y 2 and θ being the azimuthal angle in the transverse plane (x, y). This fact allows for a bit of theoretical analysis.
After plugging Eq. (4) into Eq. (2) with µ = 0, multiplying by U * and ∂ θ U * respectively, and integrating over the transverse coordinates, one ends up with a linear system of equations −βP + ωL z + I + N = 0, where P = |U | 2 dxdy, L z = −i (−y∂ x U +x∂ y U )U * dxdy, P = |−y∂ x U +x∂ y U | 2 dxdy, Obviously, the quantities P and L z stand for the power and angular momentum of the beam, and P is the norm of the state ∂ θ U .
The integrals I and I are related to the diffraction mechanism of the system, while N and N account for the waveguide and nonlinearity. The angular frequency of the azimuthon during propagation can be obtained by directly solving Eq. (6), that is After these preliminaries, we are ready to address the Rabi oscillation of azimuthons. To this end we adopt the superposition of two azimuthons U m,n (x, y) exp(iβ m,n z) as an input we borrowed the bra-ket notation from quantum mechanics. Note that the orthogonality of azimuthon shapes is only valid in the weakly nonlinear regime. As a result, one obtains two coupled equations based on Eq. (9) i ∂c m ∂z where U m V U n = rU * m V U n drdθ, with the asterisk representing the conjugate operation. Based on Eq. (10), the period of Rabi oscillaiton can be obtained, as with We note that the azimuthon conversion happens at half of the period, i.e., at z R /2. Note also that the Rabi spatial frequency directly depends on the modulation strength µ.

B. Circular waveguide
We investigate the propagation of azimuthons in the circular weakly nonlinear waveguide by also including the longitudinal modulation, and the results are displayed in Fig. 3.
Without loss of generality, we choose the dipole and hexapole azimuthons, which are shown in Fig. 3(a) and 3(b), respectively. By taking the dipole azimuthon as an example [ Fig. 3(a) is expected to depend on the modulation strength µ, and here we set it to be µ ≈ 0.031, to make the period z R ∼ 60. As a consequence, one expects to see the second-order dipole azimuthon at a distance ∼ 30 after turning on the longitudinal modulation.
In Fig. 3(a), the propagation of the dipole azimuthon is exhibited as a 3D iso-surface plot, in which the longitudinal modulation exists only in the interval 30 ≤ z ≤ 90. When the propagation distance is smaller than z ≤ 30, one in fact observes the stable rotational propagation of the dipole azimuthon. The selected amplitude distributions at z = 0 and z = 30 are shown above the 3D iso-surface plots. In the interval 30 ≤ z ≤ 90, which is about one period of the Rabi oscillation, the oscillation between the dipole azimuthon and the second-order azimuthon is displayed, in which the dipole azimuthon completely switches to the second-order azimuthon at z = 60. Indeed, the corresponding amplitude distribution is same as that in Fig. 2(e) except for a rotation, and the reason is quite obvious-azimuthons rotate steadily during propagation. When the propagation distance reaches z = 90, the dipole azimuthon is recovered and the longitudinal modulation is also lifted at the same time.
Therefore, one observes a stable rotating dipole azimuthon in the interval 90 ≤ z ≤ 120, and the amplitude distributions at z = 90 and z = 120 which are dipole azimuthons explicitly, are shown above the iso-surface plot. The analogous propagation dynamics of the hexapole azimuthon is shown in Fig. 3(b), the setup of which is same as that of Fig. 3(a); it also clearly displayss the Rabi oscillation of a higher-order azimuthon.
Here, we would like to note that the Rabi oscillation is not feasible between arbitrary two azimuthons. Only azimuthons with similar structures (e.g., the dipole and higher-order dipole azimuthons) can switch into each other, and azimuthons with different symmetries (e.g., the dipole and quadrupole azimuthons) will not, on the account that the overlap integrals in general areexactly zero, U m V U n = 0. We would like to note that the Rabi oscillation between two modes with opposite symmetry is also possible if the potential is anti-symmetrically modulated in the transverse plane [25].
Generally, there is a frequency detuning = d−d between the real modulation frequency d and the resonant frequency d. Therefore, it is reasonable to have a look at the efficiency of the azimuthon conversion versus the detuning . However, one cannot obtain the direct efficiency via the projections of the field amplitude ψ on the targeting azimuthons due to the rotation of the azimuthons during propagation. But the efficiency can be reflected by the Rabi oscillation period z R -the bigger the value of z R the bigger the efficiency [13][14][15].
The dependence of z R on frequency detuning is shown in Fig. 3(c). As a result, one finds that the efficiency of the azimuthon conversion is the biggest at the resonant frequency, and it reduces with the growth of the frequency detuning . In Fig. 5(a), the hexapole azimuthon is obtained at z ∼ 37.1 (one half of the Rabi oscillation period, and also a quarter of the rotation period of the hexapole azimuthon), while in Fig. 5(b), the higher-order dipole azimuthon is obtained at z ∼ 36.6. When the propagation distance reaches one Rabi oscillation period, the dipole azimuthon is recovered with a small deformation. To show the azimuthon conversion more transparently, we also display the corresponding phase distributions. Evidently, there is only one phase singularity in the phase of the dipole azimuthon (the topological charge is 1), five singularities for the hexapole azimuthon (the topological charge is 3), and nine for the higher-order dipole azimuthon (the topological charge is again 1). As seen, the phase distributions are in accordance with the expectations and with those displayed in Fig. 4.

III. CONCLUSION
We investigated and demonstrated Rabi oscillations of azimuthons in weakly nonlinear waveguides with weak longitudinally periodic modulations. Based on the coupled mode theory, we find the period of the Rabi oscillation, which is affected by the modulation strength and also by the spatial symmetry of the azimuthon. The analysis is feasible for both circular and square waveguides, and can be extended to waveguides with other symmetries.
Based on the model taken in this work, switching between a vortex-carrying azimuthon and a multipole that is free-of-vortex will not happen. The reason is that the initial azimuthon is composed of two degenerated modes u 1,2 , which will switch into another two degenerated modes u 3,4 during propagation. So, the output is a composition of u 3,4 which carries vortex. However, if the potential is modulated transversely in a proper manner, such a switch becomes possible since both the longitudinal and transverse phase matching can be satisfied.