Source: http://aoot.osa.org/oe/abstract.cfm?uri=oe-25-26-32401
Timestamp: 2019-04-21 14:30:34+00:00

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In a theoretical and numerical analysis, we report resonant mode conversions and Rabi oscillations in the fractional Schrödinger equation through the longitudinal modulation of the transverse potential. As specific systems of interest, we select eigenmodes of the transverse Gaussian and periodic potentials. In the Gaussian potential, we find that an increasing number of eigenmodes can be supported as the Lévy index α is reduced from 2 to 1, and that the conversion distance between the first and third eigenmodes first decreases and then increases. In the periodic potential, we obtain a cascade conversion between the neighboring eigenmodes because the parity of eigenmodes remains the same. We also find that the conversion distances between the first and second eigenmodes, as well as between the second and third eigenmodes, decrease monotonously, while that between the first and third eigenmodes first decreases and then increases with increasing α. In addition, we find that for a certain α, these conversion distances can be equal to each other.
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Fig. 1 (a)–(c) Eigenmodes of the Gaussian potential, corresponding to α = 2, 1.5 and 1, respectively. The blue, green, red, magenta, and cyan curves indicate the first, second, third, fourth, and fifth modes, respectively. All the modes are shifted vertically, to show the corresponding energy levels in the potential. (d)–(f) Resonant mode conversions between the first and third modes, corresponding to α = 2, 1.5 and 1, respectively. (g) Resonant mode conversion among the first, third and fifth modes, corresponding to α = 1.
Fig. 2 (a)–(c) Relation between the conversion period zcr13 and the detuning δ, corresponding to α = 2, 1.5 and 1, respectively. (d) Relation between the conversion period zcr13/2 and the Lévy index α at different longitudinal modulation depths µ. (e) Relation between the conversion period zcr13 and the longitudinal modulation depth µ, for different Lévy indices α.
Fig. 3 (a)–(c) Band structure in the first Brillouin zone, corresponding to α = 2, 1.5 and 1, respectively. The black, blue and red curves represent the first, second and third bands. (d)–(f) Eigenmodes corresponding to α = 2, 1.5 and 1, respectively. The black, blue and red modes are corresponding to the black, blue and red dots (kx = 0.55) in the band structure.
Fig. 4 (a) Cascading mode conversion at α = 2. (b) Weight of the eigenmode during propagation. (c)&(d) and (e)&(f) are same as (a)&(b), but for α = 1.5 and α = 1, respectively. Black, blue and red curves in (b), (d) and (f) represent the weights of the first, second and third eigenmodes, respectively. The other parameter: µ = 0.04.
Fig. 5 (a) Conversion distance between different modes. (b) Weight of the eigenmode during propagation. The blue curve refers to the left y axis, while the red curves (solid and dashed curves are zcr23 and zcr13, respectively) refer to the right y axis. (b)–(d) Mode conversion between the first and third eigenmodes with α = 2, 1.5 and 1, respectively. (e) Eigenmodes at α ≈ 1.242. The setup is as in Fig. 3(b). The other parameter: µ = 0.04.
(2) ψ ( x , z ) = ϕ ( x ) exp ( i β z ) .
(3) − β ϕ = 1 2 ( − ∂ 2 ∂ x 2 ) α / 2 ϕ + p R ( x ) ϕ .
(4) P m = 1 d 0 ∫ d 0 V ( x ) exp ( − i K m x ) d x .
(4) ∑ n [ β + 1 2 | k + K n | α ] c n exp [ i ( k + K n ) x ] + ∑ m , n P m c n exp [ i ( k + K n + K m ) x ] = 0 .
(10) Ω x = μ p 2 ⋅ 〈 ϕ m R ϕ n 〉 〈 ϕ m ϕ m 〉 〈 ϕ n ϕ n 〉 .

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