Self-trapped spatially localized states in combined linear-nonlinear periodic potentials
Jin-Cheng Shi1,2,3, Jian-Hua Zeng1,2()
1. State Key Laboratory of Transient Optics and Photonics, Xi’an Institute of Optics and Precision Mechanics of CAS, Xi’an 710119, China 2. University of Chinese Academy of Sciences, Beijing 100049, China 3. Key Laboratory for Physical Electronics and Devices of the Ministry of Education & Shaanxi Key Lab of Information Photonic Technique, Xi’an Jiaotong University, Xi’an 710049, China
We analyze the existence and stability of two kinds of self-trapped spatially localized gap modes, gap solitons and truncated nonlinear Bloch waves, in one- and two-dimensional optical or matter-wave media with self-focusing nonlinearity, supported by a combination of linear and nonlinear periodic lattice potentials. The former is found to be stable once placed inside a single well of the nonlinear lattice, it is unstable otherwise. Contrary to the case with constant self-focusing nonlinearity, where the latter solution is always unstable, here, we demonstrate that it nevertheless can be stabilized by the nonlinear lattice since the model under consideration combines the unique properties of both the linear and nonlinear lattices. The practical possibilities for experimental realization of the predicted solutions are also discussed.
D. E. Pelinovsky, Localization in Periodic Potential: From Schrödinger Operators to the Gross–Pitaevskii Equation, Cambridge: Cambridge University Press, 2011
C. Kittel, Introduction to Solid State Physics, 8th Ed., New York: John Wiley & Sons, 2005
5
J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, Princeton: Princeton University Press, 2008
6
C. Markos, J. C. Travers, A. Abdolvand, B. J. Eggleton, and O. Bang, Hybrid photonic-crystal fiber, Rev. Mod. Phys. 89(4), 045003 (2017) https://doi.org/10.1103/RevModPhys.89.045003
7
J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices, Nature 422(6928), 147 (2003) https://doi.org/10.1038/nature01452
8
F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Discrete solitons in optics, Phys. Rep. 463(1–3), 1 (2008) https://doi.org/10.1016/j.physrep.2008.04.004
9
I. L. Garanovich, S. Longhi, A. A. Sukhorukov, and Y. S. Kivshar, Light propagation and localization in modulated photonic lattices and waveguides, Phys. Rep. 518(1–2), 1 (2012) https://doi.org/10.1016/j.physrep.2012.03.005
J. Shi and J. Zeng, Asymmetric localized states in periodic potentials with a domain-wall-like Kerr nonlinearity, J. Phys. Commun. 3(3), 035003 (2019) https://doi.org/10.1088/2399-6528/ab07d1
12
L. Zeng and J. Zeng, Gap-type dark localized modes in a Bose–Einstein condensate with optical lattices, Adv. Photon. 1(04), 046004 (2019) https://doi.org/10.1117/1.AP.1.4.046004
13
F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Discrete solitons in optics, Phys. Rep. 463(1–3), 1 (2008) https://doi.org/10.1016/j.physrep.2008.04.004
P. G. Kevrekidis, D. J. Frantzeskakis, and R. Carretero-González (Eds.), Emergent Nonlinear Phenomena in Bose–Einstein Condensates, Berlin: Springer, 2008 https://doi.org/10.1007/978-3-540-73591-5
16
Z. Chen, M. Segev, and D. N. Christodoulides, Optical spatial solitons: Historical overview and recent advances, Rep. Prog. Phys. 75(8), 086401 (2012) https://doi.org/10.1088/0034-4885/75/8/086401
17
B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, Bragg grating solitons, Phys. Rev. Lett. 76(10), 1627 (1996) https://doi.org/10.1103/PhysRevLett.76.1627
18
A. Szameit, Y. V. Kartashov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, and L. Torner, Observation of two-dimensional surface solitons in asymmetric waveguid arrays, Phys. Rev. Lett. 98(17), 173903 (2007) https://doi.org/10.1103/PhysRevLett.98.173903
D. Mandelik, R. Morandotti, J. S. Aitchison, and Y. Silberberg, Gap solitons in waveguide arrays, Phys. Rev. Lett. 92(9), 093904 (2004) https://doi.org/10.1103/PhysRevLett.92.093904
21
O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, Conical diffraction and gap solitons in honeycomb photonic lattices, Phys. Rev. Lett. 98(10), 103901 (2007) https://doi.org/10.1103/PhysRevLett.98.103901
22
A. S. Desyatnikov, E. A. Ostrovskaya, Y. S. Kivshar, and C. Denz, Composite band-gap solitons in nonlinear optically induced lattices, Phys. Rev. Lett. 91(15), 153902 (2003) https://doi.org/10.1103/PhysRevLett.91.153902
23
E. A. Ostrovskaya, J. Abdullaev, M. D. Fraser, A. S. Desyatnikov, and Y. S. Kivshar, Self-localization of polariton condensates in periodic potentials, Phys. Rev. Lett. 110(17), 170407 (2013) https://doi.org/10.1103/PhysRevLett.110.170407
24
E. A. Cerda-Méndez, D. Sarkar, D. N. Krizhanovskii, S. S. Gavrilov, K. Biermann, M. S. Skolnick, and P. V. Santos, Exciton–polariton gap solitons in two-dimensional lattices, Phys. Rev. Lett. 111(14), 146401 (2013) https://doi.org/10.1103/PhysRevLett.111.146401
25
D. Tanese, H. Flayac, D. Solnyshkov, A. Amo, A. Lemaître, E. Galopin, R. Braive, P. Senellart, I. Sagnes, G. Malpuech, and J. Bloch, Polariton condensation in solitonic gap states in a one-dimensional periodic potential, Nat. Commun. 4(1), 1749 (2013) https://doi.org/10.1038/ncomms2760
26
B. Eiermann, Th. Anker, M. Albiez, M. Taglieber, P. Treutlein, K. P. Marzlin, and M. K. Oberthaler, Bright Bose–Einstein gap solitons of atoms with repulsive interaction, Phys. Rev. Lett. 92(23), 230401 (2004) https://doi.org/10.1103/PhysRevLett.92.230401
27
Y. Sivan, G. Fibich, and M. I. Weinstein, Waves in nonlinear lattices: Ultrashort optical pulses and Bose–Einstein condensates, Phys. Rev. Lett. 97, 193902 (2006) https://doi.org/10.1103/PhysRevLett.97.193902
28
J. Belmonte-Beitia, V. M. Pérez-García, V. Vekslerchik, and P. J. Torres, Lie symmetries and solitons in nonlinear systems with spatially inhomogeneous nonlinearities, Phys. Rev. Lett. 98(6), 064102 (2007) https://doi.org/10.1103/PhysRevLett.98.064102
29
J. Zeng and B. A. Malomed, Stabilization of onedimensional solitons against the critical collapse by quintic nonlinear lattices, Phys. Rev. A 85(2), 023824 (2012) https://doi.org/10.1103/PhysRevA.85.023824
30
X. Gao and J. Zeng, Two-dimensional matter-wave solitons and vortices in competing cubic-quintic nonlinear lattices, Front. Phys. 13(1), 130501 (2018) https://doi.org/10.1007/s11467-017-0697-3
31
J. Shi, J. Zeng, and B. A. Malomed, Suppression of the critical collapse for one-dimensional solitons by saturable quintic nonlinear lattices, Chaos 28(7), 075501 (2018) https://doi.org/10.1063/1.5015933
32
L. Zeng and J. Zeng, One-dimensional solitons in fractional Schrödinger equation with a spatially periodical modulated nonlinearity: Nonlinear lattice, Opt. Lett. 44(11), 2661 (2019) https://doi.org/10.1364/OL.44.002661
R. Carretero-González, D. J. Frantzeskakis, and P. G. Kevrekidis, Nonlinear waves in Bose–Einstein condensates: Physical relevance and mathematical techniques, Nonlinearity 21(7), R139 (2008) https://doi.org/10.1088/0951-7715/21/7/R01
35
D. L. Machacek, E. A. Foreman, Q. E. Hoq, P. G. Kevrekidis, A. Saxena, D. J. Frantzeskakis, and A. R. Bishop, Statics and dynamics of an inhomogeneously nonlinear lattice, Phys. Rev. E 74(3), 036602 (2006) https://doi.org/10.1103/PhysRevE.74.036602
36
R. Carretero-González, P. G. Kevrekidis, B. A. Malomed, and D. J. Frantzeskakis, Three-dimensional nonlinear lattices: From oblique vortices and octupoles to discrete diamonds and vortex cubes, Phys. Rev. Lett. 94(20), 203901 (2005) https://doi.org/10.1103/PhysRevLett.94.203901
37
V. Achilleos, G. Theocharis, P. G. Kevrekidis, N. I. Karachalios, F. K. Diakonos, and D. J. Frantzeskakis, Stationary states of a nonlinear Schrödinger lattice with a harmonic trap, J. Math. Phys. 52(9), 092701 (2011) https://doi.org/10.1063/1.3625953
38
P. G. Kevrekidis, D. J. Frantzeskakis, R. Carretero-González, B. A. Malomed, and A. R. Bishop, Discrete solitons and vortices on anisotropic lattices, Phys. Rev. E 72(4), 046613 (2005) https://doi.org/10.1103/PhysRevE.72.046613
39
L. Salasnich, A. Cetoli, B. A. Malomed, F. Toigo, and L. Reatto, Bose–Einstein condensates under a spatially modulated transverse confinement, Phys. Rev. A 76(1), 013623 (2007) https://doi.org/10.1103/PhysRevA.76.013623
40
Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Soliton modes, stability, and drift in optical lattices with spatially modulated nonlinearity, Opt. Lett. 33(15), 1747 (2008) https://doi.org/10.1364/OL.33.001747
41
Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Powerdependent shaping of vortex solitons in optical lattices with spatially modulated nonlinear refractive index, Opt. Lett. 33(19), 2173 (2008) https://doi.org/10.1364/OL.33.002173
42
H. Sakaguchi and B. A. Malomed, Solitons in combined linear and nonlinear lattice potentials, Phys. Rev. A 81(1), 013624 (2010) https://doi.org/10.1103/PhysRevA.81.013624
43
J. Zeng and B. A. Malomed, Two-dimensional solitons and vortices in media with incommensurate linear and nonlinear lattice potentials, Phys. Scr. T 149, 014035 (2012) https://doi.org/10.1088/0031-8949/2012/T149/014035
Y. V. Bludov, V. A. Brazhnyi, and V. V. Konotop, Delocalizing transition in one-dimensional condensates in optical lattices due to inhomogeneous interactions, Phys. Rev. A 76(2), 023603 (2007) https://doi.org/10.1103/PhysRevA.76.023603
46
Z. Rapti, P. G. Kevrekidis, V. V. Konotop, and C. K. R. T. Jones, Solitary waves under the competition of linear and nonlinear periodic potentials, J. Phys. A Math. Theor. 40(47), 14151 (2007) https://doi.org/10.1088/1751-8113/40/47/008
47
J. Belmonte-Beitia, V. V. Konotop, V. M. Perez-García, and V. E. Vekslerchik, Localized and periodic exact solutions to the nonlinear Schrödinger equation with spatially modulated parameters: Linear and nonlinear lattices, Chaos Solitons Fractals 41(3), 1158 (2009) https://doi.org/10.1016/j.chaos.2008.04.057
48
Th. Anker, M. Albiez, R. Gati, S. Hunsmann, B. Eiermann, A. Trombettoni, and M. K. Oberthaler, Nonlinear self-trapping of matter waves in periodic potentials, Phys. Rev. Lett. 94(2), 020403 (2005) https://doi.org/10.1103/PhysRevLett.94.020403
49
T. J. Alexander, E. A. Ostrovskaya, and Y. S. Kivshar, Self-trapped nonlinear matter waves in periodic potentials, Phys. Rev. Lett. 96(4), 040401 (2006) https://doi.org/10.1103/PhysRevLett.96.040401
50
Y. Zhang and B. Wu, Composition relation between gap solitons and bloch waves in nonlinear periodic systems, Phys. Rev. Lett. 102(9), 093905 (2009) https://doi.org/10.1103/PhysRevLett.102.093905
51
F. H. Bennet, T. J. Alexander, F. Haslinger, A. Mitchell, D. N. Neshev, and Y. S. Kivshar, Observation of nonlinear self-trapping of broad beams in defocusing waveguide arrays, Phys. Rev. Lett. 106(9), 093901 (2011) https://doi.org/10.1103/PhysRevLett.106.093901
52
C. Bersch, G. Onishchukov, and U. Peschel, Optical gap solitons and truncated nonlinear Bloch waves in temporal lattices, Phys. Rev. Lett. 109(9), 093903 (2012) https://doi.org/10.1103/PhysRevLett.109.093903
53
J. Wang, J. Yang, T. J. Alexander, and Y. S. Kivshar, Truncated-Bloch-wave solitons in optical lattices, Phys. Rev. A 79(4), 043610 (2009) https://doi.org/10.1103/PhysRevA.79.043610
54
Z. Shi, J. Wang, Z. Chen, and J. Yang, Linear instability of two-dimensional low-amplitude gap solitons near band edges in periodic media, Phys. Rev. A 78(6), 063812 (2008) https://doi.org/10.1103/PhysRevA.78.063812
55
J. Zeng and B. A. Malomed, Two-dimensional intra-band solitons in lattice potentials with local defects and selffocusing nonlinearity, J. Opt. Soc. Am. B 30(7), 1786 (2013) https://doi.org/10.1364/JOSAB.30.001786
56
N. Dror and B. A. Malomed, Stability of two-dimensional gap solitons in periodic potentials: Beyond the fundamental modes, Phys. Rev. E 87(6), 063203 (2013) https://doi.org/10.1103/PhysRevE.87.063203
57
H. Sakaguchi and B. A. Malomed, Solitons in combined linear and nonlinear lattice potentials, Phys. Rev. A 81(1), 013624 (2010) https://doi.org/10.1103/PhysRevA.81.013624
58
P. J. Y. Louis, E. A. Ostrovskaya, C. M. Savage, and Y. S. Kivshar, Bose–Einstein condensates in optical lattices: Bandgap structure and solitons, Phys. Rev. A 67(1), 013602 (2003) https://doi.org/10.1103/PhysRevA.67.013602
Z. Shi and J. Yang, Solitary waves bifurcated from Blochband edges in two-dimensional periodic media, Phys. Rev. E 75(5), 056602 (2007) https://doi.org/10.1103/PhysRevE.75.056602
T. Mayteevarunyoo, B. A. Malomed, B. B. Baizakov, and M. Salerno, Matter-wave vortices and solitons in anisotropic optical lattices, Physica D 238(15), 1439 (2009) https://doi.org/10.1016/j.physd.2008.07.024
63
M. Vakhitov, and A. Kolokolov, Stationary solutions of the wave equation in a medium with nonlinearity saturation, Radiophys. Quantum Electron. 16(7), 783 (1973) https://doi.org/10.1007/BF01031343
64
V. A. Brazhnyi and V. V. Konotop, Theory of nonlinear matter waves in optical lattice, Mod. Phys. Lett. B 18(14), 627 (2004) https://doi.org/10.1142/S0217984904007190
65
O. Morsch and M. Oberthaler, Dynamics of Bose– Einstein condensates in optical lattices, Rev. Mod. Phys. 78(1), 179 (2006) https://doi.org/10.1103/RevModPhys.78.179
66
R. Yamazaki, S. Taie, S. Sugawa, and Y. Takahashi, Submicron spatial modulation of an interatomic interaction in a Bose–Einstein condensate, Phys. Rev. Lett. 105(5), 050405 (2010) https://doi.org/10.1103/PhysRevLett.105.050405
67
M. Yan, B. J. DeSalvo, B. Ramachandhran, H. Pu, and T. C. Killian, Controlling condensate collapse and expansion with an optical Feshbach resonance, Phys. Rev. Lett. 110(12), 123201 (2013) https://doi.org/10.1103/PhysRevLett.110.123201
68
L. W. Clark, L. C. Ha, C. Y. Xu, and C. Chin, Quantum dynamics with spatiotemporal control of interactions in a stable Bose–Einstein condensate, Phys. Rev. Lett. 115(15), 155301 (2015) https://doi.org/10.1103/PhysRevLett.115.155301
69
S. Ghanbari, T. D. Kieu, A. Sidorov, and P. Hannaford, Permanent magnetic lattices for ultracold atoms and quantum degenerate gases, J. Phys. At. Mol. Opt. Phys. 39(4), 847 (2006) https://doi.org/10.1088/0953-4075/39/4/009
70
O. Romero-Isart, C. Navau, A. Sanchez, P. Zoller, and J. I. Cirac, Superconducting vortex lattices for ultracold atoms, Phys. Rev. Lett. 111(14), 145304 (2013) https://doi.org/10.1103/PhysRevLett.111.145304
P. G. Kevrekidis, G. Theocharis, D. J. Frantzeskakis, and B. A. Malomed, Feshbach resonance management for Bose–Einstein condensates, Phys. Rev. Lett. 90(23), 230401 (2003) https://doi.org/10.1103/PhysRevLett.90.230401
Y. Li, Z. Fan, Z. Luo, Y. Liu, H. He, J. Lü, J. Xie, C. Huang, and H. Tan, Cross-symmetry breaking of twocomponent discrete dipolar matter-wave solitons, Front. Phys. 12(5), 124206 (2017) https://doi.org/10.1007/s11467-017-0702-x
75
R. Zhong, Z. Chen, C. Huang, Z. Luo, H. Tan, B. A. Malomed, and Y. Li, Self-trapping under two-dimensional spin–orbit coupling and spatially growing repulsive nonlinearity, Front. Phys. 13(4), 130311 (2018) https://doi.org/10.1007/s11467-018-0778-y
