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Scattering of bright soliton by 𝒫𝒯 -symmetric potential well and barrier is investigated numerically. The numerical results show that increasing strength of real part of 𝒫𝒯 -symmetric potential can cause repeated reflection, transmission, and trapping regions due to energy exchange between the kinetic energy of incoming soliton and the internal modes of the potential structure. In addition, the width of the scattering windows increases with increase of the imaginary part of the 𝒫𝒯 -symmetric potential. The results obtained can be exploited in the development of interferometry as well as for diodes devices in information technology.
© 2014 Optical Society of America
1. Introduction
The investigation of various physical systems subject to parity-time (𝒫𝒯) symmetric potentials has drawn a lot of interest in the physics community in the last several years. The 𝒫𝒯 -symmetric principle in quantum theory was suggested by Bender and co-workers [1–3] who demonstrated that systems described by 𝒫𝒯 -symmetric non-Hermitian Hamiltonians may have purely real eigenvalues provided that the strength of the anti-Hermitian part of the Hamiltonian does not exceed a certain critical value. The investigation of the consequences of having a 𝒫𝒯 -symmetric potential was then immediately extended to several other fields embrassing a variety of physical phenomena such as plasmonics [4] and Bose-Einstein condensates [5,6]. However, in real life applications it was found that optics can offer a rich floor where 𝒫𝒯 -symmetry can be realized and controlled experimentally [7, 8]. Interesting phenomena were observed such as for example the giant wave amplification [9], power oscillations [10] and unidirectional invisibility [11]. Usually the complex refractive index of the optical one-dimensional system is written as n(x) = nR(x) + inI(x), where nR(x) is the index guiding part and nI(x) is the gain/loss profile of the complex refractive index, respectively, and x is the transverse wave propagation direction. Therefore, a 𝒫𝒯 -symmetric effective optical potential n(x) is required to have a symmetric real part nR(x) = nR(−x) and antisymmetric imaginary part nI(x) = −nR(−x). The theoretical research on 𝒫𝒯 -symmetric optical potentials was extended by the incorporation of nonlinearities [12]which are inherent in optical systems due to the Kerr effect.
Numerical investigation of scattering of soliton by defects had been reported in [13,14] where reflection, transmission, and trapping regions as a function of the potential strength was examined. Other studies has been recently completed for gap solitons (GSs) in a periodic nonlinear Schrodinger equation in the presence of localized defects that exhibit 𝒫𝒯 -symmetry [15]. In this case the quantum reflection from wells has a resonant character, where the internal mode of solitons and local impurity modes are interacting.
In this letter we investigate soliton scattering through a complex 𝒫𝒯 -symmetric Gaussian potentials numerically by setting in motion a single soliton toward the scattering potential well and barrier. Recently [16] the scattering of soliton by 𝒫𝒯 -symmetric potential has been considered. Under special conditions, the author found numerically that the soliton can have a unidirectional flow giving rise to a diode type of effect. Her we show that the existence of repeated reflections transmission and trapping regions observed for increasing strength of the even part (potential well case) of the 𝒫𝒯 -symmetric potential. In addition, when the real part becomes a barrier, the resonance in the transmission can also occurs. The paper is organized as follows. In Sec. II we introduce the model, governing equations, and statement of the problem. Section III we perform numerical simulations for the solitons scattering through a 𝒫𝒯 -Symmetric potential while Sec. IV is dedicated to the variational analysis of the model. The paper is concluded in Section V with a few remarks.
2. Formulation of the problem
We consider the traditional normalized nonlinear Schrödiger (NLS) with 𝒫𝒯 -symmetric potential, referred also as the Groos-Pitaevskii equation in the dynamics of a weakly interacting BEC at zero temperature:
where and V0 and W0 are real-valued constants corresponding to the depth or amplitude of the real and imaginary parts of the potential, respectively. It has been reported that solitons are stable in the presence of the above Gaussian PT-symmetric complex potentials [17]. We also note that the properties of the beam in PT-symmetric potential (2) have been studied using the variational approach [18].3. Numerical results
For the numerical examination of the scattering process, the initial position of the soliton x0 is considerably away from the potential well so that there is no interaction between them at t = 0. When the soliton is set in motion with initial velocity v0 a collision between soliton and potential arises, if the soliton is allowed to interact with the potential for a sufficiently long time then we may have full transmission (T), trapping (L), reflection (R) or combination of these states. In an attempt to satisfy this condition, we recall the following standard conservation law, R + T + L = 1 to explain, in principle, any nonlinear scattering problem. These parameters are determined at times after the scattering from the potential well, according to the relations
where h denotes position on the x-axis at which the influence of the potential is negilgeable V(h) ∼ 0, and is the norm of the soliton wavefunction. We perform numerical simulations using the split-step Fourier method with Fourier modes 2048. The periodic boundary conditions are assumed, and the time step is 0.001 in our units. For the numerical investigation we use, as an initial condition, the homogeneous solution of Eq. (1) with A = 1 and x0 = −12 and v0 is the initial velocity.We consider first the scattering of single soliton by complex 𝒫𝒯 -symmetric Gaussian potential with V0 > 0 (well) and W0 > 0 (energy loss). Figure 1 shows the reflection, transmission and trapping for various potential depths V0 with a fixed W0 = 0.5 and initial velocity v0 = 0.35 for single soliton after collision with a complex 𝒫𝒯 -symmetric Gaussian potential. The figure shows that the reflection of the soliton smoothly increases to maximum as V0 increases and suddenly dropped to zero when V0 = 4.5 while the transmission jumps to values close to unity. A further increase in V0 will widen the reflection window. However, their is no region of complete trapping but when V0 = 0.6, the maximum trapping of soliton reached ∼ 90%.
Fig. 1 Dependence of the reflection (solid), transmission (dot) and trapping (dashed) on the strength of V0 for soliton incoming initial velocity v=0.35 and the strength of W0 is 0.5.
Different scenarios are illustrated in Fig. 2 by spatiotemporal trajectories of the soliton hitting the complex 𝒫𝒯 -symmetric Gaussian potential. An example of trapping is displayed in Fig. 2(a) for V0 = 0.6. Figure 2(b) with V0 = 15.1 shows an example of complete reflection, and full transmission is observed in Fig. 2(c) for V0 = 16.3. Figure 3, shows the remarkable feature of unidirectional flow that for a fixed strength V0 = 4, the windows of reflection get narrower when the imaginary part of the PT-symmetric potential W0 is increased. In particular, one can see that when W0 = 0.1 the soliton scattering is dominated by reflections when the incident velocity is below ∼ 0.42 but this threshold reduces to incident velocity ∼ 0.40 for W0 = 0.5.
Fig. 2 Density plot of soliton scattering through 𝒫𝒯 -symmetric potential from left to right for different strength of v0. (a) V0 = 0.6 (trapping) (b) V0 = 15.1 (reflection) (c) V0 = 16.3 (transmission). The parameters set is as follows: v0 = 0.35, x0 = −12 and W0 = 0.5.
Fig. 3 Reflection coefficient of scattering soliton with different values of W0 and fixed value of V0 = 4. The parameter set is as follows: x0 = −12 and v0 = 0.35.
We now focus on the interaction of the soliton with V0 < 0 (Wall). In contrast to the above case, now the soliton can only be either reflected or transmitted, depending on the relation between its initial kinetic energy and the strength of W0 and V0. In Fig. 4, the reflection and transmission windows are shown for W0 = |0.5| and V0 = −1. One of the noticeable features of the scattered solition is the reduction of the width of the total reflection regions as if W0 < 0. The soliton moving from the left collides with the potential, will encounter gain and then loss if W0 > 0 while the situation is reversed if W0 < 0. Thus, the reflection and transmission windows depends on the order of sequences of pumping and damping regions encountered. By decreasing the strength of the imaginary part of 𝒫𝒯 -symmetric, one also observes that the transmission region is enhanced as shown in Fig. 5. Exciting results are obtained when the transmission resonances are obtained for W0 > 0 as shown in Fig. 6. Two resonances are shown in the figure occuring around the values of v0 = 1.51 and v0 = 1.55.
Fig. 4 Dependence of the reflection and transmission on the the incident velocity for different values of strength W0 where the strength of real part of PT-symmetric is fixed V0 = −1.
Fig. 5 Transmission window as a function of incident velocity for different values of strength W0 where the strength of real part of PT-symmetric is fixed V0 = −1.
Fig. 6 Transmission window as a function of incident velocity. The parameter set is as follows: x0 = −12, V0 = −1 and W0 = |1.0|.
4. Variational method
We present in this section a variational approach shown unidirectional flow will be obtained when the real part of the complex 𝒫𝒯 symmetric Gaussian potential is negative. Our ansatz wavefucntion takes the form
where A(t), ξ(t), v(t), a(t),b(t) and ϕ(t) are time-dependent variational parameters corresponding to the amplitude, center-of mass position, center-of mass speed, width, chirp, and phase, respectively. The quadratic chirp employed here is rather simpler than the one used by [19,20]. While the latter leads to better agreement with the numerical results [21], our choice will be sufficient, at least qualitatively, and confirms the above explanation for the existence of the unidirectional flow. According the the literatures [22, 23], the conventional variational approach can be modified to allow for dissipative terms like the imaginary part of 𝒫𝒯 -symmetric potential. The Euler-Lagrange equations for parameter ζ = ζ(A, ξ, v, a, b, ϕ) is where R = −iW(x)ψ is dissipative term and L is the averaged conservative Lagrangian that is given by By substituting the trial solution (5) into Eq. (7), Lagrangian can be written as follows where . Applying the modified Euler-Lagrange equation (6) and introducing , we arrive at the evolution equationsTo verify the importance of 𝒫𝒯 -symmetric potential in accounting for the unidirectional flow process, we solve the equations of motion for constant amplitude A = 1, imaginary strength W0 = 0.2, and initial velocity v0 = 0.25 with different values of V0. We present in Fig. 7 the full numerical calculations of the time evolution of the centre-of-mass position. We can observe different regimes for the soliton scattering with complex 𝒫𝒯 symmetric Gaussian potentials: refection, transmission and trapping.
Fig. 7 For a fixed value of W0 = 0.2, we present numerical solution of the evolution equations for different values of V0. The parameter set is as follows: v0 = 0.25, ξ0 = −12, A = 1, a0 = 1.0, and .
Further insight on the mechanism of unidirectional flow can be obtained by calculating the effective force and potential experienced by the soliton during the scattering process. Here we interpret the force, F, through the acceleration ξ(ẗ), and the hence potential is given by U = − ∫(ξẗ)dξ. In Fig. 8, we plot numerically both quantities for the two cases of W0 with fixed V0 = −1 and initial velocity v0 = 0.25. Figures 8(a) and 8(b) correspond to the cases when W0 = −2 and W0 = 2 respectively. These figures shows that the profiles of the force and potential vary considerably depending on the order in which the dissipative and amplifying parts of the potential are encountered. Finally, the lagrangian approach exposes a common theoretical description of scattering process and constitutes an approximate approach so no quantitative agreements with the direct numerical simulations of the scattering soltion is predicted.
Fig. 8 The effective Force F (solid) and effective potential U (dot) as function of center-mass postion ξ. The parameter set for both figures is as follows: V0 = −1, v0 = 0.25, A = 1, a0 = 1.0, and , (a) W0 = −2 and (b) W0 = 2 respectively.
5. Conclusions
We have examined the scattering of single soliton with complex 𝒫𝒯 symmetric Gaussian potentials. In case of attractive real part, we notice the existences of regions of complete transmission or reflections as well as regions of partial trapping of the soliton by the potential. We found numerically that resonant states of soliton occur with potential barrier (real part of 𝒫𝒯 -symmetric potential). We assume that the obtained results can be exciting and very beneficial for the construction of interferometry as well as diodes devices in information processing.
Acknowledgments
The author acknowledges the support of the King Fahd University of Petroleum and Minerals under Projects No. RG1217-1 and No. RG1217-2 and the Saudi Center for Theoretical Physics (SCTP).
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