Vector solitons in nonparity-time-symmetric complex potentials

Xing Zhu; Yingji He

doi:10.1364/OE.26.026511

1. Introduction

In the past ten years, the parity-time (PT) symmetry and solitons in PT-symmetric complex potentials have been widely investigated both theoretically [1–17] and experimentally [18–23]. A PT-symmetric Hamiltonian with a complex potential can also have entirely real spectrum and the potential must fulfill the condition: V(x) = V*(-x) [24,25], where the superscript * represents complex conjugation.

The non-PT-symmetric complex potentials [V(x)≠V*(-x)] can also have completely real spectra [26,27]. A method for constructing non-PT-symmetric complex potentials with entirely real spectra was given [28], the phase transition can also occur by tuning the parameter of the real part of the non-PT-symmetric complex potentials to a critical threshold. Classes of new non-PT-symmetric complex potentials were reported [29], above phase transition point, a pair of complex eigenvalues bifurcate out from an eigenvalue which is in the interior of the eigenvalue continuous spectrum.

Continuous families of solitons supported by non-PT-symmetric complex potentials were first reported in [30]. In the non-PT-symmetric single-hump complex potential, solitons can be stable. Later, the numerical finding of above reference have been explained [31]. Moreover, solitons are also stable in the non-PT-symmetric double-hump complex waveguides. Solitons in χ⁽²⁾ media with non-PT-symmetric complex potentials were investigated [32], these solitons can be stable. The main result which is contrast with solitons in χ⁽²⁾ media with PT-symmetric potentials was reported. The form V(x) = g²(x) + ig’(x) is argued to be the only one-dimensional (1D) non-PT-symmetric complex potentials in which soliton families can exist and solitons can also exist in the non-PT-symmetric complex potentials with saturable nonlinearity [33]. The stability of soliton families in 1D non-PT-symmetric complex potentials were carefully investigated [34]. The surprising finding is the appearance of the eigenvalues of the linear-stability operators of these solitons is in quartets. Below the phase transition, fundamental solitons bifurcating from the first discrete eigenvalue are stable and dipole solitons bifurcating from the second discrete eigenvalue are stable in the low power region but unstable in the high power region. Above the phase transition, the fundamental soliton family, which also bifurcates from the first discrete eigenvalue are unstable in the low power region but stable in the high power region. Moreover, the non-PT-symmetric complex potentials can be realized physically in a coherent atomic medium which consists of a cold three-level atomic gas and this system can be support stable optical solitons [35]. However, vector solitons in non-PT-symmetric complex potentials have not been studied yet.

In this work, we investigate the existence and stability of vector solitons supported by 1D non-PT-symmetric complex potentials. The propagation constants of the two components of the vector solitons are different. Below the phase transition, vector solitons are stable in the low power region and unstable in the high power region. The existence and stable regions can be changed significantly by tuning the real part of the non-PT-symmetric complex potentials. Above the phase transition, there are two continuous stable domain in the existence region. The two components of these vector solitons exhibit asymmetry and the transverse power flow of the two components of these vector solitons in non-PT-symmetric potentials is also studied. Our result first demonstrates that the vector solitons can be stable in the optical potentials with gain and loss when the phase transition occurs.

2. The theoretical model

Two mutually incoherent light beams propagating in a non-PT-symmetric complex potential with focusing Kerr nonlinearity can be described by the coupled normalized 1D nonlinear Schrödinger Eqs [3,15,34].

i \frac{U_{1, 2}}{\partial z} + \frac{\partial^{2} U_{1, 2}}{\partial x^{2}} + V (x) U_{1, 2} + ({| U_{1} |}^{2} + {| U_{2} |}^{2}) U_{1, 2} = 0.

Here U₁ and U₂ are the complex light field amplitudes of the two components. The normalized transverse and longitudinal coordinates are represented by x and z, respectively. The form of the non-PT-symmetric complex potentials in the paper is [28,34]

V (x) = g^{2} (x) + 2 c_{0} g (x) + i g_{x} (x) .

Where c₀ and g(x) are real constant and function, respectively. we take

g (x) = \tan h 2 (x + 2) - \tanh (x - 2) .

In order to obtain the spectrum, we introduce the Schrödinger Eq.

[\frac{\partial^{2}}{\partial x^{2}} + V (x)] ψ = λ ψ .

Here λ and ψ are the eigenvalue and eigenfunction, respectively. We numerically solve Eq. (4) by using Fourier collocation method [36]. There is a critical threshold

c_{0}^{t h} = - 0.3666

. When

c_{0} > c_{0}^{t h}

, the spectrum of the non-PT-symmetric complex potential is entirely real. For

c_{0} \leq c_{0}^{t h}

, the spectrum will be partially complex. When c₀ = 0.2, c₀ = 0, and c₀ = −0.4, the 1D non-PT-symmetric complex potentials [Eqs. (2) and (3)] are shown in Figs. 1(a)-1(c), respectively. Figures 1(d)-1(f) are the corresponding spectra.

Fig. 1 (a)-(c) are the 1D non-PT-symmetric complex potentials for c₀ = 0.2, c₀ = 0, and c₀ = −0.4. (d)-(f) are the corresponding spectra.

Download Full Size | PDF

We seek the vector soliton solutions in the form of $U_{1, 2} (x, z) = q_{1, 2} (x) e^{i μ_{1, 2} z}$ . Where µ_1,2 are the real-valued propagation constants of the two components of vector solitons, and q_1,2 are complex localized functions in x satisfying the following coupled Eqs.

\frac{\partial^{2} q_{1, 2}}{\partial x^{2}} + V (x) q_{1, 2} + ({| q_{1} |}^{2} + {| q_{2} |}^{2}) q_{1, 2} - μ_{1, 2} q_{1, 2} = 0.

We use a method that developed from the modified squared-operator iteration method [37] to solve Eqs. (5) numerically and thus obtain the stationary vector soliton solutions. The total and partial powers of vector solitons are defined by

P = \int_{- \infty}^{\infty} ({| q_{1} |}^{2} + {| q_{2} |}^{2}) d x

,

P_{1} = \int_{- \infty}^{\infty} {| q_{1} |}^{2} d x

, and

P_{2} = \int_{- \infty}^{\infty} {| q_{2} |}^{2} d x

. For µ₁ = µ₂, the simplest vector solitons are found in the form of

q_{1} = q \cos ϕ

and

q_{2} = q \sin ϕ

, where q and ϕ are the solution of scalar soliton and arbitrary projection angle, respectively [15,38]. The most interesting situation of vector solitons is the nondegenerated case, when µ₁≠µ₂ [15,38]. In the work, we study the vector solitons with µ₁≠µ₂.

The linear stability of vector solitons in non-PT-symmetric complex potentials is checked by adding perturbations g_1,2(x) and t_1,2(x) [12]

U_{1, 2} (x, z) = e^{i μ_{1, 2} z} [q_{1, 2} (x) + g_{1, 2} (x) e^{δ z} + t_{1, 2}^{*} (x) e^{δ^{*} z}] .

Here

| g_{1, 2} |, | t_{1, 2} | < < | q_{1, 2} |

and δ is the growth rate. By taking Eqs. (6) into Eqs. (1) and linearizing, the perturbations g_1,2(x) and t_1,2(x) obey the following linear stability eigenvalues Eqs.

{\begin{cases} g_{1} δ = i [(- μ_{1} + \frac{\partial^{2}}{\partial x^{2}} + V + 2 {| q_{1} |}^{2} + {| q_{2} |}^{2}) g_{1} + q_{1}^{2} t_{1} + q_{1} q_{2}^{*} g_{2} + q_{1} q_{1} t_{2}], \\ t_{1} δ = i [- (^{q_{1}^{2}) *} g_{1} + (μ_{1} - \frac{\partial^{2}}{\partial x^{2}} - V^{*} - 2 {| q_{1} |}^{2} - {| q_{2} |}^{2}) t_{1} - q_{1}^{*} q_{2}^{*} g_{2} - q_{1}^{*} q_{2} t_{2}], \\ g_{2} δ = i [q_{1}^{*} q_{2} g_{1} + q_{1} q_{2} t_{1} + (- μ_{2} + \frac{\partial^{2}}{\partial x^{2}} + V + {| q_{1} |}^{2} + 2 {| q_{2} |}^{2}) g_{2} + q_{2}^{2} t_{2}], \\ t_{2} δ = i [- q_{1}^{*} q_{2}^{*} g_{1} - q_{1} q_{2}^{*} t_{1} - (^{q_{2}^{2}) *} g_{2} + (μ_{2} - \frac{\partial^{2}}{\partial x^{2}} - V^{*} - {| q_{1} |}^{2} - 2 {| q_{2} |}^{2}) t_{2}] . \end{cases}

Equations (7) can be solved numerically by using the Fourier collocation method [36]. If there exists a complex value δ with Re(δ)>0, the vector soliton cannot be linearly stable; otherwise, it can be linearly stable.

3. The numerical results

First, we take c₀ = 0.2, which is below the phase transition. Fix µ₁ = 6, a vector soliton family which the first component is the fundamental mode and the second component is the out-of-phase dipole mode can exist in the region of 2.90≤µ₂≤4.90. The total power increases with the increase of the propagation constant of the second component of the vector solitons, as depicted in Fig. 2(a). As µ₂ increases, Fig. 2(b) shows that the power of the first component decreases and the power of the second component also increases. In the low power region, vector solitons are stable, the stable region is 2.90≤µ₂≤4.02. However, in the high power region (4.03≤µ₂≤4.90), vector solitons cannot be stable. The max[Re(δ)]>0 in the region [as depicted in Fig. 2(c)] demonstrates that vector solitons are unstable. We also study the parameters $S_{1, 2} = \frac{i}{2} (q_{1, 2} \frac{\partial q_{1, 2}^{*}}{\partial x} - q_{1, 2}^{*} \frac{\partial q_{1, 2}}{\partial x})$ associated with the transverse power-flow densities [3] in the two components. For µ₂ = 3.9, Figs. 2(d) and 2(e) are the profiles of the first and second components. From Figs. 2(f) and 2(g), the transverse power-flow densities (S_1,2) are everywhere negative in the two components. It indicates that the power flow from gain to loss regions always in one direction in the two components. The vector soliton is stable, the stable propagations of the two perturbed components [the direct simulations of Eqs. (1) are added random noises with 5% amplitudes of the two components] are shown in Figs. 2(h) and 2(i), respectively. We take µ₂ = 4.3 as the unstable case in the high power region. The amplitude distributions of the two components of the vector soliton which exhibit the asymmetry are depicted in Figs. 3(a) and 3(b), respectively. Figures 3(c) and 3(d) show the unstable propagations of the two perturbed components.

Fig. 2 For c₀ = 0.2 and µ₁ = 6. (a) and (b) are the total and partial powers of the vector solitons. (c) is the max[Re(δ)] versus the propagation constant of the second component (µ₂). The profiles of the two component when µ₂ = 3.9 are depicted in (d) and (e), respectively. (f) and (g) are the transverse power-flow densities of the two components. (h) and (i) exhibit the stable propagations of the two perturbed components.

Download Full Size | PDF

Fig. 3 (a) and (b) are the amplitude distributions of the two components of the vector soliton for µ₂ = 4.3. (c) and (d) are the corresponding unstable propagations of the two perturbed components. The other parameters are c₀ = 0.2 and µ₁ = 6.

Download Full Size | PDF

The parameter c₀ can significantly affect the existence and stability of vector solitons. For c₀ = 0 and µ₁ = 6, the existence region is 2.34≤µ₂≤5.12, which is wider than the case of c₀ = 0.2. Vector solitons are also stable in the low power region but unstable in the high power region. The stable region is 2.34≤µ₂≤4.70, it is also wider than c₀ = 0.2. The profile of the vector soliton of µ₂ = 4.3 is shown in Fig. 4(d) and the corresponding stable propagations of the two perturbed components are depicted in Figs. 4(f) and 4(g). Figure 4(e) shows the profile of the vector soliton of µ₂ = 4.8. The vector soliton is unstable, as demonstrated in Figs. 4(h) and 4(i).

Fig. 4 When c₀ = 0 and µ₁ = 6. (a) and (b) are the total and partial powers diagrams. (c) is the max[Re(δ)] versus µ₂. (d) and (e) are the profiles of vector solitons for µ₂ = 4.3 and µ₂ = 4.8, respectively. (f) and (g) show the stable propagations of the two perturbed components when µ₂ = 4.3. The unstable propagations of the two perturbed components for µ₂ = 4.8 are depicted in (h) and (i).

Download Full Size | PDF

Above the phase transition, vector solitons can also be stable and the stability becomes more complicated. We choose c₀ = −0.4, which is above the phase transition. For µ₁ = 6, the existence region of the vector solitons is 1.07≤µ₂≤5.44. There are two continuous stable regions in the existence domain. In the regions of 2.19≤µ₂≤3.28 and 3.55≤µ₂≤3.65, vector solitons are stable. For µ₂ = 3.0, Fig. 5(e) depicts the profile of the vector soliton. The stable propagations of the two perturbed components are shown in Figs. 5(h) and 5(i), respectively. We take µ₂ = 3.6 as the stable case in the second stable region. The profile of the vector soliton is shown in Fig. 6(d) and Figs. 6(e) and 6(f) exhibit the stable propagations of the two perturbed components. In the three regions of 1.07≤µ₂≤2.18, 3.29≤µ₂≤3.54, and 3.66≤µ₂≤5.44, vector solitons cannot be stable. In the three unstable regions, max[Re(δ)]>0, as depicted in Fig. 5(c). We also take µ₂ = 1.65, µ₂ = 3.43, and µ₂ = 3.9 as the three cases in the unstable regions. The profiles of the three unstable vector solitons are shown in Fig. 5(d) and Figs. 6(a) and 6(g), respectively. Figures 5(f) and 5(g), Figs. 6(b) and 6(c), and Figs. 6(h) and 6(i) depict the unstable propagations of the perturbed components of the three vector solitons.

Fig. 5 (a)-(c) are the total power, partial power, and max[Re(δ)] versus µ₂, respectively. (d) and (e) are the vector soliton profiles of µ₂ = 1.65 and µ₂ = 3.0, respectively. (f)-(i) are the corresponding unstable and stable propagations of the perturbed components of the two vector solitons. The other parameters are c₀ = −0.4 and µ₁ = 6.

Download Full Size | PDF

Fig. 6 For c₀ = −0.4 and µ₁ = 6. (a)-(c) are the profile of the vector soliton and the unstable propagations of the two perturbed components when µ₂ = 3.43. The vector soliton profile and the stable propagations of the two perturbed components for µ₂ = 3.6 are depicted in (d)-(f). (g) is the vector soliton profile when µ₂ = 3.9. (h) and (i) show the corresponding unstable propagations of the two perturbed components.

Download Full Size | PDF

Moreover, if the self and cross phase modulations terms are unequal, continuous families of vector solitons with µ₁ = µ₂ can also be stable below and above the phase transition. However, the continuous families of vector solitons with µ₁≠µ₂ cannot be found. Therefore the above properties of vector solitons (µ₁≠µ₂) cannot exist.

4. Conclusion

In conclusion, we have studied the existence and stability of vector solitons supported by 1D non-PT-symmetric complex potentials. The two components of these vector solitons exhibit the asymmetry. Below the phase transition, vector soliton are stable in the low power region. The existence and stable regions are significantly affected by a constant of the real part of the non-PT-symmetric complex potentials. Above the phase transition, the stability of vector solitons are more complicated, stable vector solitons exist in two continuous regions. Our result first shows that vector solitons can be stable in complex optical potentials above phase transition.

Funding

National Natural Science Foundation of China (NSFC) (Grant Nos. 11774068, 11547212, and 61675001); Guangdong Province Nature Foundation of China (Grant No. 2017A030311025); Guangdong Province Education Department Foundation of China (Grant No. 2014KZDXM059).

References

1. R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32(17), 2632–2634 (2007). [CrossRef] [PubMed]

2. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric optical lattices,” Phys. Rev. A 81(6), 063807 (2010). [CrossRef]

3. Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100(3), 030402 (2008). [CrossRef] [PubMed]

4. X. Zhu, H. Wang, L.-X. Zheng, H. Li, and Y.-J. He, “Gap solitons in parity-time complex periodic optical lattices with the real part of superlattices,” Opt. Lett. 36(14), 2680–2682 (2011). [CrossRef] [PubMed]

5. F. K. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zezyulin, “Solitons in PT-Symmetric nonlinear lattices,” Phys. Rev. A 83(4), 041805 (2011). [CrossRef]

6. R. Driben and B. A. Malomed, “Stability of solitons in parity-time-symmetric couplers,” Opt. Lett. 36(22), 4323–4325 (2011). [CrossRef] [PubMed]

7. Z. Shi, X. Jiang, X. Zhu, and H. Li, “Bright spatial solitons in defocusing Kerr media with PT-symmetric potentials,” Phys. Rev. A 84(5), 053855 (2011). [CrossRef]

8. N. V. Alexeeva, I. V. Barashenkov, A. A. Sukhorukov, and Y. S. Kivshar, “Optical solitons in PT-symmetric nonlinear couplers with gain and loss,” Phys. Rev. A 85(6), 063837 (2012). [CrossRef]

9. V. Achilleos, P. G. Kevrekidis, D. J. Frantzeskakis, and R. Carretero-González, “Dark solitons and vortices in PT-symmetric nonlinear media: from spontaneous symmetry breaking to nonlinear PT phase transitions,” Phys. Rev. A 86(1), 013808 (2012). [CrossRef]

10. D. A. Zezyulin and V. V. Konotop, “Nonlinear modes in finite-dimensional PT-symmetric systems,” Phys. Rev. Lett. 108(21), 213906 (2012). [CrossRef] [PubMed]

11. Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Lattice solitons in PT-symmetric mixed linear-nonlinear optical lattices,” Phys. Rev. A 85(1), 013831 (2012). [CrossRef]

12. S. Nixon, L. Ge, and J. Yang, “Stability analysis for solitons in PT-symmetric optical lattices,” Phys. Rev. A 85(2), 023822 (2012). [CrossRef]

13. F. C. Moreira, F. K. Abdullaev, V. V. Konotop, and A. V. Yulin, “Localized modes in χ⁽²⁾ media with PT-symmetric localized potential,” Phys. Rev. A 86(5), 053815 (2012). [CrossRef]

14. C. Li, H. Liu, and L. Dong, “Multi-stable solitons in PT-symmetric optical lattices,” Opt. Express 20(15), 16823–16831 (2012).

15. Y. V. Kartashov, “Vector solitons in parity-time-symmetric lattices,” Opt. Lett. 38(14), 2600–2603 (2013). [CrossRef] [PubMed]

16. X. Zhu, H. Wang, H. Li, W. He, and Y. He, “Two-dimensional multipeak gap solitons supported by parity-time-symmetric periodic potentials,” Opt. Lett. 38(15), 2723–2725 (2013). [CrossRef] [PubMed]

17. P. Li, C. Dai, R. Li, and Y. Gao, “Symmetric and asymmetric solitons supported by a PT-symmetric potential with saturable nonlinearity: bifurcation, stability and dynamics,” Opt. Express 26(6), 6949–6961 (2018). [CrossRef] [PubMed]

18. A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009). [CrossRef] [PubMed]

19. C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6(3), 192–195 (2010). [CrossRef]

20. A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488(7410), 167–171 (2012). [CrossRef] [PubMed]

21. H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346(6212), 975–978 (2014). [CrossRef] [PubMed]

22. M. Wimmer, A. Regensburger, M.-A. Miri, C. Bersch, D. N. Christodoulides, and U. Peschel, “Observation of optical solitons in PT-symmetric lattices,” Nat. Commun. 6(1), 7782 (2015). [CrossRef] [PubMed]

23. Z. Zhang, Y. Zhang, J. Sheng, L. Yang, M.-A. Miri, D. N. Christodoulides, B. He, Y. Zhang, and M. Xiao, “Observation of parity-time symmetry in optically induced atomic lattices,” Phys. Rev. Lett. 117(12), 123601 (2016). [CrossRef] [PubMed]

24. C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998). [CrossRef]

25. C. M. Bender, “Making sense of non-Hermitian Hamiltonians,” Rep. Prog. Phys. 70(6), 947–1018 (2007). [CrossRef]

26. F. Cannata, G. Junker, and J. Trost, “Schrödinger operators with complex potential but real spectrum,” Phys. Lett. A 246(3–4), 219–226 (1998). [CrossRef]

27. M.-A. Miri, M. Heinrich, and D. N. Christodoulides, “Supersymmetry-generated complex optical potentials with real spectra,” Phys. Rev. A 87(4), 043819 (2013). [CrossRef]

28. S. Nixon and J. Yang, “All-real spectra in optical systems with arbitrary gain-and-loss distributions,” Phys. Rev. A (Coll. Park) 93(3), 031802 (2016). [CrossRef]

29. J. Yang, “Classes of non-parity-time-symmetric optical potentials with exceptional-point-free phase transitions,” Opt. Lett. 42(20), 4067–4070 (2017). [CrossRef] [PubMed]

30. E. N. Tsoy, I. M. Allayarov, and F. Kh. Abdullaev, “Stable localized modes in asymmetric waveguides with gain and loss,” Opt. Lett. 39(14), 4215–4218 (2014). [CrossRef] [PubMed]

31. V. V. Konotop and D. A. Zezyulin, “Families of stationary modes in complex potentials,” Opt. Lett. 39(19), 5535–5538 (2014). [CrossRef] [PubMed]

32. F. C. Moreira and S. B. Cavalcanti, “Localized modes in χ⁽²⁾ media with non-PT-symmetric complex localized potentials,” Phys. Rev. A (Coll. Park) 94(4), 043818 (2016). [CrossRef]

33. S. D. Nixon and J. Yang, “Bifurcation of soliton families from linear modes in non-PT-symmetric complex potentials,” Stud. Appl. Math. 136(4), 459–483 (2016). [CrossRef]

34. J. Yang and S. Nixon, “Stability of soliton families in nonlinear Schrödinger equations with non-parity-time-symmetric complex potentials,” Phys. Lett. A 380(45), 3803–3809 (2016). [CrossRef]

35. C. Hang, G. Gabadadze, and G. Huang, “Realization of non-PT-symmetric optical potentials with all-real spectra in a coherent atomic system,” Phys. Rev. A (Coll. Park) 95(2), 023833 (2017). [CrossRef]

36. J. Yang, Nonlinear Waves in Integrable and Nonintegrable Systems (SIAM, 2010).

37. J. Yang and T. I. Lakoba, “Universally-convergent squared-operator iteration methods for solitary waves in general nonlinear wave equations,” Stud. Appl. Math. 118(2), 153–197 (2007). [CrossRef]

38. Z. Xu, Y. V. Kartashov, and L. Torner, “Stabilization of vector soliton complexes in nonlocal nonlinear media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(5), 055601 (2006). [CrossRef] [PubMed]

Vector solitons in nonparity-time-symmetric complex potentials

Abstract

1. Introduction

2. The theoretical model

3. The numerical results

4. Conclusion

Funding

References

Cited By

Figures (6)

Equations (7)

Optics Express