Rabi oscillation of azimuthons in weakly nonlinear waveguides
Kaichao Jin, Yongdong Li, Feng Li, Milivoj R. Belić, Yanpeng Zhang, Yiqi Zhang
AAPS/123-QED
Rabi oscillation of azimuthons in weakly nonlinear waveguides
Kaichao Jin , , Yongdong Li , Feng Li , MilivojR. Beli´c , Yanpeng Zhang , and Yiqi Zhang , , ∗ Key Laboratory for Physical Electronics and Devices of the Ministry of Education,Xi’an Jiaotong University, Xi’an 710049, China Guangdong Xi’an Jiaotong University Academy, Foshan 528300, China Science Program, Texas A&M University at Qatar, P. O. Box 23874 Doha, Qatar (Dated: July 24, 2020)
Abstract
Rabi oscillation, an inter-band oscillation, depicts the periodic flopping between two states thatbelong 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 therefractive index change of the system. However, the Rabi oscillation of nonlinear states has yet tobe discussed. We report Rabi oscillations of azimuthons—spatially modulated vortex solitons—inweakly nonlinear waveguides with different symmetries, both numerically and theoretically. Theperiod of Rabi oscillation can be determined by applying the coupled mode theory, which largelydepends on the modulation strength. Whether the Rabi oscillation between two states can beobtained or not is determined by the spatial symmetry of the azimuthons and the modulatingpotential. In this paper we succeeded in obtaining the Rabi oscillation of azimuthons in the weaklynonlinear waveguides with different symmetries. Our results not only enrich the Rabi oscillationphenomena, but also provide a new avenue in the study of pattern formation and spatial fieldmanipulation in nonlinear optical systems. ∗ Corresponding author: [email protected] a r X i v : . [ phy s i c s . op ti c s ] J u l . INTRODUCTION The Rabi oscillation originated in quantum mechanics [1], but by now is much investigatedin a variety of optical and photonic systems that include fibers [2, 3], multimode waveguides[4–6], coupled waveguides [7], waveguide arrays [8–10], and two-dimensional modal structures[11, 12]. Recently, Rabi oscillations of topological edge states [13, 14] and modes in fractionalSchr¨odinger equation [15] were also reported. Rabi oscillations are inter-band oscillationsthat require an ac field to be applied as an external periodic potential. In optics, thelongitudinal periodic modulation of the refractive index change plays the role of an ac fieldin temporal quantum systems, and Rabi oscillations are indicated by the resonant modeconversion. As far as we know, the investigation of optical Rabi oscillations thus far hasbeen limited to the linear regime only, and the Rabi oscillation in nonlinear systems is stillan 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 weaklynonlinear waveguides that is accomplished by applying a weak longitudinally modulatedperiodic potential. Azimuthons are a special type of spatial solitons; they are azimuthallymodulated vortex beams that exhibit steady angular rotation upon propagation [16]. Gen-erally, azimuthons, especially the ones with higher-order angular momentum structures,are unstable in media with local Kerr or saturable nonlinearities. To overcome the in-stability drawback, a nonlocal nonlinearity is introduced, and recently published reportsdemonstrate that the stable propagation of azimuthons can indeed be obtained [17–19]. Inaddition, it was also reported that the spin-orbit-coupled Bose-Einstein condensates cansupport stable azimuthons as well [20]. However, the treatment of nonlocal nonlinearity andspin-orbit-coupled Bose-Einstein condensates is challenging in both theoretical modeling andexperimental demonstration. Nevertheless, it has been confirmed that the weakly nonlin-ear 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 acircular waveguide and then in a square waveguide. Since in this nonlinear three-dimensionalwave propagation problem no analytical solutions are known, necessarily the mode of in-quiry will be predominantly numerical with some theoretical background. In the circularwaveguide, the azimuthons will exhibit Rabi oscillation while rotating during propagation.2n 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 acertain threshold value; (ii) azimuthons will be deformed during propagation. Hence, in thiswork we choose azimuthons with large enough energies to avoid wobbling motions in thesquare waveguide during propagation.
II. RESULTSA. Theoretical analysis
The propagation of a light beam in a photonic waveguide can be described by theSchr¨odinger-like paraxial wave equation i ∂∂Z
Ψ + 12 k (cid:18) ∂ ∂X + ∂ ∂Y (cid:19) Ψ + k n n b | Ψ | Ψ + k n ( X, Y ) − n b n b [1 + µ cos( δZ )] Ψ = 0 , (1)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 = 2 πn b /λ with λ being thewavelength. The other quantities in Eq. (1) are: µ (cid:28) δ is the longitudinal modulation frequency, n 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 | Ψ | (nonlinear part). We would like to note that weakly nonlinear waveguides demand not onlyboth the linear and nonlinear refractive index changes to be small in comparison with n b , butalso the nonlinear part to be much smaller than the linear part. According to the relations x = X/r , y = Y /r , z = Z/ ( k r ), d = k r δ , and σ = sgn( n ), with r being determinedby the real beam width, Eq. (1) can be rewritten into its dimensionless version i ∂∂z ψ + 12 (cid:18) ∂ ∂x + ∂ ∂y (cid:19) ψ + σ | ψ | ψ + V [1 + µ cos( dz )] ψ = 0 , (2)with ψ = k r (cid:112) | n | /n b Ψ and V ( x, y ) = k r [ n ( x, y ) − n b ] /n b . Here, we will consider propa-gation in a deep circular potential V ( x, y ) = V exp[ − ( x + y ) /w ] with w characterizingthe potential width and V the potential depth. To guarantee a weak nonlinearity, thepotential should be deep enough. Thus, the potential is deep but the potential modula-tion 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 = 1.3 a) (b) (c) (d) (e) y x ¢ ¡ ! ¡! FIG. 1. (a) Basic modes. (b) Degenerate dipole modes. (c) Degenerate quadrupole modes. (d)Degenerate hexapole modes. (e) Degenerate octopole modes. First row: first-order modes. Secondrow: second-order modes. Third row: third-order modes. The panels are shown in the window − ≤ x ≤ − ≤ y ≤
2. Other parameters: V = 500 and w = 1. There are large amounts of materials to be used to produce waveguides, and silica is oneof the popular materials among them with typical parameters n b = 1 . | n − n b | ≤ × − ,and n = 3 × − cm / W for light beams with wavelength ranging from visible to near-infrared. Without loss of generality, we choose λ = 800 nm in this work. Therefore ifwe choose V = 500, the value of r ≈ . µ m can be obtained according to the relationadopted in Eq. (2). Indeed, such a value is reasonable for a multi-mode fiber [24]. Accordingto the wavelength and r , one knows that the diffraction length is ∼ ∼
35 fs / mm at the wavelength λ = 800 nm, thedispersion length is of the order of kilometers for a picosecond light beam, which is muchlonger than the propagation distance taken in this work. As a result, it is safely to neglectthe temporal effect.To start with, we consider the modes supported by the deep potential alone, there-fore 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 propaga-tion constant. Plugging this solution into the reduced Eq. (2), one obtains βu = 12 (cid:18) ∂ ∂x + ∂ ∂y (cid:19) u + V u, (3)which is the linear steady-state eigenvalue problem of Eq. (2) with σ and µ equal to zero.4quation (3) can be solved by utilizing the plane-wave expansion method, and the eigenstatessupported by the deep potential V ( x, y ) can be easily obtained. In Fig. 1, the first-order aswell as higher-order basic modes, degenerate dipole modes, degenerate quadrupole modes,degenerate hexapole modes, and degenerate octopole modes that can exist in the potentialare displayed. Here, the “degenerate” means that all modes feature the same propagationconstants, in the usual optical meaning. These linear modes will be used as the input modesof the more general nonlinear and modulated modes of the complete Eq. (3).Therefore, to seek approximate azimuthons in weakly nonlinear waveguides, one takesthe degenerate modes and makes a superposition of them, as the initial wave ψ ( x, y ) = A [ u ( x, y ) + iBu ( x, y )] exp( iβz ) , (4)in Eq. (4), where A is an amplitude factor, 1 − B the azimuthal modulation depth, and u , ( x, y ) are the degenerate linear modes (see examples in Fig. 1). Thus, we set thetransverse profile of the azimuthon at the initial place as U ( x, y, z = 0) = A [ u ( x, y ) + iBu ( x, y )] (5)and numerically propagate it, to obtain an output mode at arbitrary z . We would liketo note that the inputs of the form (5) do not rotate in linear medium, since modes aredegenerate. Rotation appears only when nonlinearity is added into model.In Fig. 2, we display such approximate azimuthons with A = 0 . B = 0 .
5. One findsthat the phase of azimuthons is nontrivial, displaying angular momentum and topologicalcharge. For dipole azimuthons, the topological charge is ±
1, while for quadrupole, hexapole,and octopole azimuthons, the values are ± ± ±
4, respectively. As expected, theseazimuthons will rotate with a constant frequency ω during propagation when the nonlinearterm 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 = (cid:112) x + y and θ being the azimuthal angle in the transverseplane ( 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 , − βL z + ωP (cid:48) + I (cid:48) + N (cid:48) = 0 , (6)5 a) (b) (c) (d)(e) (f) (g) (h) + p - p + p - p FIG. 2. Amplitude and phase of the first-order (a-d) and second-order (e-h) azimuthons constructedfrom the degenerate dipoles (a,e), quadrupoles (b,f), hexapoles (c,g), and octopoles (d,h). Thepanels are shown in the window − ≤ x ≤ − ≤ y ≤
2. Other parameters: A = 0 . B = 0 . where P = (cid:82)(cid:82) | U | dxdy, L z = − i (cid:82)(cid:82) ( − y∂ x U + x∂ y U ) U ∗ dxdy, P (cid:48) = (cid:82)(cid:82) |− y∂ x U + x∂ y U | dxdy,I = (cid:82)(cid:82) U ∗ ∆ ⊥ U dxdy, N = (cid:82)(cid:82) [ σ | U | + V ] | U | dxdy, I (cid:48) = i (cid:82)(cid:82) ( − y∂ x U ∗ + ∂ y U ∗ )∆ ⊥ U dxdy,N (cid:48) = i (cid:82)(cid:82) ( σ | U | + V )( − y∂ x U ∗ + x∂ y U ∗ ) U dxdy.
Obviously, the quantities P and L z standfor the power and angular momentum of the beam, and P (cid:48) is the norm of the state ∂ θ U .The integrals I and I (cid:48) are related to the diffraction mechanism of the system, while N and N (cid:48) account for the waveguide and nonlinearity. The angular frequency of the azimuthonduring propagation can be obtained by directly solving Eq. (6), that is ω = P ( I (cid:48) + N (cid:48) ) − L z ( I + N ) L z − P P (cid:48) . (7)After these preliminaries, we are ready to address the Rabi oscillation of azimuthons. Tothis end we adopt the superposition of two azimuthons U m,n ( x, y ) exp( iβ m,n z ) as an input ψ = c m ( z ) U m ( x, y ) exp( iβ m z ) + c n ( z ) U n ( x, y ) exp( iβ n z ) , (8)where c m,n ( z ) are the slowly varying complex amplitudes of the azimuthons, and the mod-ulation frequency is d = β m − β n . Plugging Eq. (8) into Eq. (2), and without considering6he nonlinear term (but still on the level of analysis of Eq. (3)), one obtains i ∂c m ∂z U m exp( idz ) + 12 µc m V U m [1 + exp(2 idz )]+ i ∂c n ∂z U n + 12 µc n V U n [exp( idz ) + exp( − idz )] = 0 . (9)Since the azimuthons are constructed based on Eq. (5), they satisfy the relation (cid:104) U m , U n (cid:105) (cid:54) = 0if m = n and (cid:104) U m , U n (cid:105) = 0 if m (cid:54) = n , thus forming a complete set of eigenstates. Here,we borrowed the bra-ket notation from quantum mechanics. Note that the orthogonality ofazimuthon shapes is only valid in the weakly nonlinear regime. As a result, one obtains twocoupled equations based on Eq. (9) i ∂c m ∂z + 12 µ (cid:104) U m V U n (cid:105)(cid:104) U m U m (cid:105) c n = 0 ,i ∂c n ∂z + 12 µ (cid:104) U n V U m (cid:105)(cid:104) U n U n (cid:105) c m = 0 , (10)where (cid:104) U m V U n (cid:105) = (cid:82)(cid:82) 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 z R = π | Ω R | (11)with Ω R = µ (cid:104) U m V U n (cid:105) (cid:112) (cid:104) U m U m (cid:105)(cid:104) U n U n (cid:105) . (12)We note that the azimuthon conversion happens at half of the period, i.e., at z R /
2. Notealso 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 waveguideby 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 shownin Fig. 3(a) and 3(b), respectively. By taking the dipole azimuthon as an example [Fig. 3(a)],we want to see whether the Rabi oscillation between the dipole azimuthon [Fig. 2(a)] and itscorresponding second-order dipole azimuthon [Fig. 2(e)] can be established. So, to induceresonance, we set the modulation frequency to be the difference between the eigenvalues ofthe two modes, which is d ≈ .
2. As shown by Eq. (12), the period of the Rabi oscillation7 - y - x z - y - x z z = 0 z = 30 z = 60 z = 90 z = 120 z = 0 z = 30 z = 60 z = 90 z = 120 (a) (b) (c) ‘ FIG. 3. Rabi transition of a dipole (a) and a hexapole (b). In each case, the propagation is shownby the iso-surface plot, above which amplitude distributions at selected distances are shown. Inboth cases, the weak longitudinally periodic modulation exists in the region 30 ≤ z ≤
90 with d ≈ . µ ≈ .
031 in (a), and d ≈ .
36 and µ ≈ .
014 in (b). (c) Rabi oscillation period z R versus frequency detuning (cid:96) . is expected to depend on the modulation strength µ , and here we set it to be µ ≈ . z R ∼
60. As a consequence, one expects to see the second-order dipoleazimuthon 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-surfaceplot, in which the longitudinal modulation exists only in the interval 30 ≤ z ≤
90. Whenthe propagation distance is smaller than z ≤
30, one in fact observes the stable rotational8 p - p + p - p + p - p (a) (b)(c) (d)(e) (f) FIG. 4. (a) Rabi transition of a deformed dipole. (b) Amplitude and phase of the azimuthonbased on the dipole in (a). (c) Transition of a deformed hexapole. (d) Amplitude and phase ofthe azimuthon based on the hexapole in (c). (e) Transition of a deformed higher-order dipole. (f)Amplitude and phase of the azimuthon based on the higher-order dipole in (e). The panels areshown in the window − ≤ x ≤ − ≤ y ≤
2. Other parameters: A = 0 . B = 0 . 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 isabout one period of the Rabi oscillation, the oscillation between the dipole azimuthon and thesecond-order azimuthon is displayed, in which the dipole azimuthon completely switches tothe second-order azimuthon at z = 60. Indeed, the corresponding amplitude distribution issame as that in Fig. 2(e) except for a rotation, and the reason is quite obvious—azimuthonsrotate steadily during propagation. When the propagation distance reaches z = 90, thedipole 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 ≤ z = 90 and z = 120 which are dipole azimuthons explicitly,are shown above the iso-surface plot. The analogous propagation dynamics of the hexapoleazimuthon is shown in Fig. 3(b), the setup of which is same as that of Fig. 3(a); it alsoclearly displayss the Rabi oscillation of a higher-order azimuthon.Here, we would like to note that the Rabi oscillation is not feasible between arbitrarytwo azimuthons. Only azimuthons with similar structures (e.g., the dipole and higher-orderdipole 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 overlap9ntegrals in general areexactly zero, (cid:104) U m V U n (cid:105) = 0. We would like to note that the Rabioscillation between two modes with opposite symmetry is also possible if the potential isanti-symmetrically modulated in the transverse plane [25].Generally, there is a frequency detuning (cid:96) = d − d (cid:48) between the real modulation frequency d (cid:48) and the resonant frequency d . Therefore, it is reasonable to have a look at the efficiencyof the azimuthon conversion versus the detuning (cid:96) . However, one cannot obtain the directefficiency via the projections of the field amplitude ψ on the targeting azimuthons due tothe rotation of the azimuthons during propagation. But the efficiency can be reflected bythe Rabi oscillation period z R — the bigger the value of z R the bigger the efficiency [13–15].The dependence of z R on frequency detuning (cid:96) is shown in Fig. 3(c). As a result, one findsthat the efficiency of the azimuthon conversion is the biggest at the resonant frequency, andit reduces with the growth of the frequency detuning (cid:96) . C. Square waveguide
Now, we investigate the azimuthon transition in a square waveguide, which is generatedby the potential in Eq. (2) of the form V ( x, y ) = V exp[ − ( x + y ) /w )] . Again, we solvefor the linear eigenmodes supported by the deep square waveguide, by using the plane-waveexpansion method. Connected with the geometry of the potential, the amplitude distribu-tions of the linear modes are more complex than those in the regular circular waveguide,therefore we denote them as the deformed modes. In Fig. 4, we display three kinds ofdeformed modes and the corresponding azimuthons, which will transform mutually, becauseof the relation (cid:104) U m V U n (cid:105) (cid:54) = 0.Different from the azimuthons in circular waveguides, azimuthons in square waveg-uides rotate conditionally. Due to the symmetry of the square waveguide, a rotatingazimuthon will be deformed, i.e., its profile changes. Without considering nonlinearity,linear superposition of two degenerated modes [e.g., dipoles in Fig. 4(a), and labelledas u , ( x, y )] is also dipole solution of the square potential, as u (cid:48) , ( x, y ) = [ u ( x, y ) ± u ( x, y )] / (cid:113)(cid:82)(cid:82) | u ( x, y ) ± u ( x, y ) | dxdy . It has been found that if an azimuthon in a squarewaveguide can rotate it should meet the condition: H ( ψ ) > H ( u (cid:48) , ) with H representing theHamiltonian [22]. Even though the wobbling azimuthon can be established in our numericalsimulations, we are more interested in rotating azimuthons, so we set again A = 0 . p - p + p - p (b) z = 0 z = 0 (a) z = 37.1 z = 74.2 z = 36.6 z = 73.2 FIG. 5. (a) Transition between dipole and hexapole azimuthons with d ≈ . µ ≈ . d ≈ . µ ≈ . B = 0 . µ , to make the Rabi oscillation period almostequal to one half of the target azimuthon rotation period, since the modulation frequency d is determined beforehand. Numerical simulations reveal that the rotation periods of thehexapole azimuthon and the higher-order dipole azimuthon are ∼ . ∼ .
4, re-spectively. Therefore, according to Eq. (12), the modulation strength µ for the two casesshould be ∼ .
085 and ∼ . z ∼ . z ∼ .
6. When thepropagation distance reaches one Rabi oscillation period, the dipole azimuthon is recoveredwith a small deformation. To show the azimuthon conversion more transparently, we alsodisplay the corresponding phase distributions. Evidently, there is only one phase singularityin the phase of the dipole azimuthon (the topological charge is 1), five singularities forthe hexapole azimuthon (the topological charge is 3), and nine for the higher-order dipoleazimuthon (the topological charge is again 1). As seen, the phase distributions are inaccordance with the expectations and with those displayed in Fig. 4.
III. CONCLUSION
We investigated and demonstrated Rabi oscillations of azimuthons in weakly nonlinearwaveguides with weak longitudinally periodic modulations. Based on the coupled modetheory, we find the period of the Rabi oscillation, which is affected by the modulationstrength and also by the spatial symmetry of the azimuthon. The analysis is feasible for bothcircular 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 azimuthonand a multipole that is free-of-vortex will not happen. The reason is that the initial az-imuthon is composed of two degenerated modes u , , which will switch into another twodegenerated modes u , during propagation. So, the output is a composition of u , whichcarries vortex. However, if the potential is modulated transversely in a proper manner, sucha switch becomes possible since both the longitudinal and transverse phase matching canbe satisfied. ACKNOWLEDGEMENTS
This work was supported by Guangdong Basic and Applied Basic Research Foundation(2018A0303130057), National Natural Science Foundation of China (U1537210, 11534008,11804267), and Fundamental Research Funds for the Central Universities (xzy012019038,xzy022019076). M.R.B acknowledges support from the NPRP 11S-1126-170033 project fromthe Qatar national Research Fund, while K.C.J. and Y.Q.Z. acknowledge the computational12esources provided by the HPC platform of Xi’an Jiaotong University. [1] I. I. Rabi,
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