Interplay of nonreciprocity and nonlinearity on mean-field energy and dynamics of a Bose–Einstein condensate in a double-well potential

doi:10.1007/s11467-021-1133-2

Frontiers of Physics

2022, Vol. 17

Issue (4): 42503 https://doi.org/10.1007/s11467-021-1133-2

本期目录

Interplay of nonreciprocity and nonlinearity on mean-field energy and dynamics of a Bose–Einstein condensate in a double-well potential

Yi-Piao Wu¹, Guo-Qing Zhang^1,², Cai-Xia Zhang^1,²(

), Jian Xu³(

), Dan-Wei Zhang^1,²(

)

¹. Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials, School of Physics and Telecommunication Engineering, South China Normal University, Guangzhou 510006, China
². Guangdong-Hong Kong Joint Laboratory of Quantum Matter, Frontier Research Institute for Physics, South China Normal University, Guangzhou 510006, China
³. College of Electronics and Information Engineering, Guangdong Ocean University, Zhanjiang 524088, China

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Abstract：

We investigate the mean-field energy spectrum and dynamics in a Bose–Einstein condensate in a double-well potential with non-Hermiticity from the nonreciprocal hopping, and show that the interplay of nonreciprocity and nonlinearity leads to exotic properties. Under the two-mode and mean-field approximations, the nonreciprocal generalization of the nonlinear Schrödinger equation and Bloch equations of motion for this system are obtained. We analyze the $P T$ phase diagram and the dynamical stability of fixed points. The reentrance of $P T$ -symmetric phase and the reformation of stable fixed points with increasing the nonreciprocity parameter are found. Besides, we uncover a linear selftrapping effect induced by the nonreciprocity. In the nonlinear case, the self-trapping oscillation is enhanced by the nonreciprocity and then collapses in the $P T$ -broken phase, and can finally be recovered in the reentrant $P T$ -symmetric phase.

Key words： Bose–Einstein condensate non-Hermitian physics nonlinear dynamics parity–time symmetry

收稿日期: 2021-11-01 出版日期: 2021-12-16

Corresponding Author(s): Cai-Xia Zhang,Jian Xu,Dan-Wei Zhang

引用本文:

. [J]. Frontiers of Physics, 2022, 17(4): 42503.
Yi-Piao Wu, Guo-Qing Zhang, Cai-Xia Zhang, Jian Xu, Dan-Wei Zhang. Interplay of nonreciprocity and nonlinearity on mean-field energy and dynamics of a Bose–Einstein condensate in a double-well potential. Front. Phys. , 2022, 17(4): 42503.

链接本文:

https://academic.hep.com.cn/fop/CN/10.1007/s11467-021-1133-2
https://academic.hep.com.cn/fop/CN/Y2022/V17/I4/42503

1	C. M. Bender , Making sense of non-Hermitian Hamiltonians, Rep. Prog. Phys. 70 (6), 947 (2007) https://doi.org/10.1088/0034-4885/70/6/R03
2	R. El-Ganainy , K. G. Makris , M. Khajavikhan , Z. H. Musslimani , S. Rotter , and D. N. Christodoulides , NonHermitian physics and PT symmetry, Nat. Phys. 14 (1), 11 (2018) https://doi.org/10.1038/nphys4323
3	M. A. Miri and A. Alù , Exceptional points in optics and photonics, Science 363 (6422), eaar7709 (2019) https://doi.org/10.1126/science.aar7709
4	Y. Ashida , Z. Gong , and M. Ueda , Non-Hermitian physics, Adv. Phys. 69 (3), 249 (2020) https://doi.org/10.1126/science.aar7709
5	E. J. Bergholtz , J. C. Budich , and F. K. Kunst , Exceptional topology of non-Hermitian systems, Rev. Mod. Phys. 93 (1), 015005 (2021) https://doi.org/10.1103/RevModPhys.93.015005
6	C. M. Bender and S. Boettcher , Real spectra in nonHermitian Hamiltonians having PT symmetry, Phys. Rev. Lett. 80 (24), 5243 (1998) https://doi.org/10.1103/PhysRevLett.80.5243
7	W. D. Heiss , Exceptional points of non-Hermitian operators, J. Phys. Math. Gen. 37 (6), 2455 (2004) https://doi.org/10.1088/0305-4470/37/6/034
8	W. D. Heiss , The physics of exceptional points, J. Phys. A Math. Theor. 45 (44), 444016 (2012) https://doi.org/10.1088/1751-8113/45/44/444016
9	L. Pan , S. Chen , and X. Cui , High-order exceptional points in ultracold Bose gases, Phys. Rev. A 99 (1), 011601 (2019) https://doi.org/10.1103/PhysRevA.99.011601
10	Z. Gong , Y. Ashida , K. Kawabata , K. Takasan , S. Higashikawa , and M. Ueda , Topological phases of nonHermitian systems, Phys. Rev. X 8 (3), 031079 (2018) https://doi.org/10.1103/PhysRevX.8.031079
11	K. Kawabata , T. Bessho , and M. Sato , Classification of exceptional points and non-Hermitian topological semimetals, Phys. Rev. Lett. 123 (6), 066405 (2019) https://doi.org/10.1103/PhysRevLett.123.066405
12	N. Hatano and D. R. Nelson , Localization transitions in non-Hermitian quantum mechanics, Phys. Rev. Lett. 77 (3), 570 (1996) https://doi.org/10.1103/PhysRevLett.77.570
13	S. Yao and Z. Wang , Edge states and topological invariants of non-Hermitian systems, Phys. Rev. Lett. 121 (8), 086803 (2018) https://doi.org/10.1103/PhysRevLett.121.086803
14	F. K. Kunst , E. Edvardsson , J. C. Budich , and E. J. Bergholtz , Biorthogonal bulk–boundary correspondence in non-Hermitian systems, Phys. Rev. Lett. 121 (2), 026808 (2018) https://doi.org/10.1103/PhysRevLett.121.026808
15	L. Jin and Z. Song , Bulk–boundary correspondence in a non-Hermitian system in one dimension with chiral inversion symmetry, Phys. Rev. B 99 (8), 081103 (2019) https://doi.org/10.1103/PhysRevB.99.081103
16	S. Longhi , Topological phase transition in non-Hermitian quasicrystals, Phys. Rev. Lett. 122 (23), 237601 (2019) https://doi.org/10.1103/PhysRevLett.122.237601
17	H. Jiang , L. J. Lang , C. Yang , S. L. Zhu , and S. Chen , Interplay of non-Hermitian skin effects and Anderson localization in nonreciprocal quasiperiodic lattices, Phys. Rev. B 100 (5), 054301 (2019) https://doi.org/10.1103/PhysRevB.100.054301
18	D. W. Zhang , L. Z. Tang , L. J. Lang , H. Yan , and S. L. Zhu , Non-Hermitian topological Anderson insulators, Sci. China Phys. Mech. Astron. 63 (6), 267062 (2020) https://doi.org/10.1007/s11433-020-1521-9
19	X. W. Luo and C. Zhang , Non-Hermitian disorder-induced topological insulators, arXiv: 1912.10652v1 (2019)
20	L. Z. Tang , L. F. Zhang , G. Q. Zhang , and D. W. Zhang , Topological Anderson insulators in two-dimensional nonHermitian disordered systems, Phys. Rev. A 101 (6), 063612 (2020) https://doi.org/10.1103/PhysRevA.101.063612
21	H. Liu , Z. Su , Z. Q. Zhang , and H. Jiang , Topological Anderson insulator in two-dimensional non-Hermitian systems, Chin. Phys. B 29 (5), 050502 (2020) https://doi.org/10.1088/1674-1056/ab8201
22	Q. B. Zeng and Y. Xu , Winding numbers and generalized mobility edges in non-Hermitian systems, Phys. Rev. Research 2 (3), 033052 (2020) https://doi.org/10.1103/PhysRevResearch.2.033052
23	L. Li , C. H. Lee , S. Mu , and J. Gong , Critical nonHermitian skin effect, Nat. Commun. 11 (1), 5491 (2020) https://doi.org/10.1038/s41467-020-18917-4
24	D. W. Zhang , Y. L. Chen , G. Q. Zhang , L. J. Lang , Z. Li , and S. L. Zhu , Skin superfluid, topological Mott insulators, and asymmetric dynamics in an interacting non-Hermitian Aubry–André–Harper model, Phys. Rev. B 101 (23), 235150 (2020) https://doi.org/10.1103/PhysRevB.101.235150
25	T. Liu , J. J. He , T. Yoshida , Z. L. Xiang , and F. Nori , NonHermitian topological Mott insulators in one-dimensional fermionic superlattices, Phys. Rev. B 102 (23), 235151 (2020) https://doi.org/10.1103/PhysRevB.102.235151
26	Z. Xu , S. Chen , Z. Xu , and S. Chen , Topological Bose– Mott insulators in one-dimensional non-Hermitian superlattices, Phys. Rev. B 102 (3), 035153 (2020) https://doi.org/10.1103/PhysRevB.102.035153
27	T. Helbig , T. Hofmann , S. Imhof , M. Abdelghany , T. Kiessling , L. W. Molenkamp , C. H. Lee , A. Szameit , M. Greiter , and R. Thomale , Generalized bulk–boundary correspondence in non-Hermitian topolectrical circuits, Nat. Phys. 16 (7), 747 (2020) https://doi.org/10.1038/s41567-020-0922-9
28	M. Ezawa , Electric-circuit simulation of the Schrödinger equation and non-Hermitian quantum walks, Phys. Rev. B 100 (16), 165419 (2019) https://doi.org/10.1103/PhysRevB.100.165419
29	L. Xiao , T. Deng , K. Wang , G. Zhu , Z. Wang , W. Yi , and P. Xue , Non-Hermitian bulk–boundary correspondence in quantum dynamics, Nat. Phys. 16 (7), 761 (2020) https://doi.org/10.1038/s41567-020-0836-6
30	Z. Yu and S. Fan , Complete optical isolation created by indirect interband photonic transitions, Nat. Photonics 3 (2), 91 (2009) https://doi.org/10.1038/nphoton.2008.273
31	M. S. Kang , A. Butsch , and P. S. J. Russell , Reconfigurable light-driven opto-acoustic isolators in photonic crystal fibre, Nat. Photonics 5 (9), 549 (2011) https://doi.org/10.1038/nphoton.2011.180
32	L. Bi , J. Hu , P. Jiang , D. H. Kim , G. F. Dionne , L. C. Kimerling , and C. A. Ross , On-chip optical isolation in monolithically integrated non-reciprocal optical resonators, Nat. Photonics 5 (12), 758 (2011) https://doi.org/10.1038/nphoton.2011.270
33	L. Fan , J. Wang , L. T. Varghese , H. Shen , B. Niu , Y. Xuan , A. M. Weiner , and M. Qi , An all-silicon passive optical diode, Science 335 (6067), 447 (2012) https://doi.org/10.1126/science.1214383
34	S. A. R. Horsley , J. H. Wu , M. Artoni , and G. C. La Rocca , Optical nonreciprocity of cold atom Bragg mirrors in motion, Phys. Rev. Lett. 110 (22), 223602 (2013) https://doi.org/10.1103/PhysRevLett.110.223602
35	B. Peng , Ş. K. Özdemir , F. Lei , F. Monifi , M. Gianfreda , G. L. Long , S. Fan , F. Nori , C. M. Bender , and L. Yang , Parity–time-symmetric whispering-gallery microcavities., Nat. Phys. 10 (5), 394 (2014) https://doi.org/10.1038/nphys2927
36	Y. P. Wang , J. W. Rao , Y. Yang , P. C. Xu , Y. S. Gui , B. M. Yao , J. Q. You , and C. M. Hu , Nonreciprocity and unidirectional invisibility in cavity magnonics, Phys. Rev. Lett. 123 (12), 127202 (2019) https://doi.org/10.1103/PhysRevLett.123.127202
37	Y. Zhao , J. Rao , Y. Gui , Y. Wang , and C. M. Hu , Broadband nonreciprocity realized by locally controlling the Magnon’s radiation, Phys. Rev. Appl. 14 (1), 014035 (2020) https://doi.org/10.1103/PhysRevApplied.14.014035
38	Z. Shen , Y. L. Zhang , Y. Chen , C. L. Zou , Y. F. Xiao , X. B. Zou , F. W. Sun , G. C. Guo , and C. H. Dong , Experimental realization of optomechanically induced nonreciprocity, Nat. Photonics 10 (10), 657 (2016) https://doi.org/10.1038/nphoton.2016.161
39	F. Ruesink , M. A. Miri , A. Alù , and E. Verhagen , Nonreciprocity and magnetic-free isolation based on optomechanical interactions, Nat. Commun. 7 (1), 13662 (2016) https://doi.org/10.1038/ncomms13662
40	N. R. Bernier , L. D. Tóth , A. Koottandavida , M. A. Ioannou , D. Malz , A. Nunnenkamp , A. K. Feofanov , and T. J. Kippenberg , Nonreciprocal reconfigurable microwave optomechanical circuit, Nat. Commun. 8 (1), 604 (2017) https://doi.org/10.1038/s41467-017-00447-1
41	K. Fang , J. Luo , A. Metelmann , M. H. Matheny , F. Marquardt , A. A. Clerk , and O. Painter , Generalized nonreciprocity in an optomechanical circuit via synthetic magnetism and reservoir engineering, Nat. Phys. 13 (5), 465 (2017) https://doi.org/10.1038/nphys4009
42	G. A. Peterson , F. Lecocq , K. Cicak , R. W. Simmonds , J. Aumentado , and J. D. Teufel , Demonstration of effcient nonreciprocity in a microwave optomechanical circuit, Phys. Rev. X 7 (3), 031001 (2017) https://doi.org/10.1103/PhysRevX.7.031001
43	H. Xu , L. Jiang , A. A. Clerk , and J. G. E. Harris , Nonreciprocal control and cooling of phonon modes in an optomechanical system, Nature 568 (7750), 65 (2019) https://doi.org/10.1038/s41586-019-1061-2
44	W. Gou , T. Chen , D. Xie , T. Xiao , T. S. Deng , B. Gadway , W. Yi , and B. Yan , Tunable nonreciprocal quantum transport through a dissipative Aharonov–Bohm ring in ultracold atoms, Phys. Rev. Lett. 124 (7), 070402 (2020) https://doi.org/10.1103/PhysRevLett.124.070402
45	D. W. Zhang , Y. Q. Zhu , Y. X. Zhao , H. Yan , and S. L. Zhu , Topological quantum matter with cold atoms, Adv. Phys. 67 (4), 253 (2018) https://doi.org/10.1080/00018732.2019.1594094
46	Y. V. Kartashov , B. A. Malomed , and L. Torner , Solitons in nonlinear lattices, Rev. Mod. Phys. 83 (1), 247 (2011) https://doi.org/10.1103/RevModPhys.83.247
47	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
48	B. Wu and Q. Niu , Nonlinear Landau–Zener tunneling, Phys. Rev. A 61 (2), 023402 (2000) https://doi.org/10.1103/PhysRevA.61.023402
49	J. Liu , L. Fu , B. Y. Ou , S. G. Chen , D. I. Choi , B. Wu , and Q. Niu , Theory of nonlinear Landau–Zener tunneling, Phys. Rev. A 66 (2), 023404 (2002) https://doi.org/10.1103/PhysRevA.66.023404
50	J. Liu , B. Wu , and Q. Niu , Nonlinear evolution of quantum states in the adiabatic regime, Phys. Rev. Lett. 90 (17), 170404 (2003) https://doi.org/10.1103/PhysRevLett.90.170404
51	M. E. Kellman and V. Tyng , Bifurcation effects in coupled Bose–Einstein condensates, Phys. Rev. A 66 (1), 013602 (2002) https://doi.org/10.1103/PhysRevA.66.013602
52	A. P. Hines , R. H. McKenzie , and G. J. Milburn , Quantum entanglement and fixed-point bifurcations, Phys. Rev. A 71 (4), 042303 (2005) https://doi.org/10.1103/PhysRevA.71.042303
53	I. Siddiqi , R. Vijay , F. Pierre , C. M. Wilson , L. Frunzio , M. Metcalfe , C. Rigetti , R. J. Schoelkopf , M. H. Devoret , D. Vion , and D. Esteve , Direct observation of dynamical bifurcation between two driven oscillation states of a Josephson junction, Phys. Rev. Lett. 94 (2), 027005 (2005) https://doi.org/10.1103/PhysRevLett.94.027005
54	T. Zibold , E. Nicklas , C. Gross , and M. K. Oberthaler , Classical bifurcation at the transition from Rabi to Josephson dynamics, Phys. Rev. Lett. 105 (20), 204101 (2010) https://doi.org/10.1103/PhysRevLett.105.204101
55	C. Lee , L. B. Fu , and Y. S. Kivshar , Many-body quantum coherence and interaction blockade in Josephson-linked Bose–Einstein condensates, Europhys. Lett. 81 (6), 60006 (2008) https://doi.org/10.1209/0295-5075/81/60006
56	C. Lee , Universality and anomalous mean-field breakdown of symmetry-breaking transitions in a coupled twocomponent Bose–Einstein condensate, Phys. Rev. Lett. 102 (7), 070401 (2009) https://doi.org/10.1103/PhysRevLett.102.070401
57	C. Lee , J. Huang , H. Deng , H. Dai , and J. Xu , Nonlinear quantum interferometry with Bose condensed atoms, Front. Phys. 7 (1), 109 (2012) https://doi.org/10.1007/s11467-011-0228-6
58	A. Burchianti , C. Fort , and M. Modugno , Josephson plasma oscillations and the Gross–Pitaevskii equation: Bogoliubov approach versus two-mode model, Phys. Rev. A 95 (2), 023627 (2017) https://doi.org/10.1103/PhysRevA.95.023627
59	S. Martínez-Garaot , G. Pettini , and M. Modugno , Nonlinear mixing of Bogoliubov modes in a bosonic Josephson junction, Phys. Rev. A 98 (4), 043624 (2018) https://doi.org/10.1103/PhysRevA.98.043624
60	D. W. Zhang , L. B. Fu , Z. D. Wang , and S. L. Zhu , Josephson dynamics of a spin–orbit-coupled Bose–Einstein condensate in a double-well potential, Phys. Rev. A 85 (4), 043609 (2012) https://doi.org/10.1103/PhysRevA.85.043609
61	D. W. Zhang , Z. D. Wang , and S. L. Zhu , Relativistic quantum effects of Dirac particles simulated by ultracold atoms, Front. Phys. 7 (1), 31 (2012) https://doi.org/10.1007/s11467-011-0223-y
62	W. Y. Wang , J. Lin , and J. Liu , Cyclotron dynamics of a Bose–Einstein condensate in a quadruple-well potential with synthetic gauge fields, Front. Phys. 16 (5), 52502 (2021) https://doi.org/10.1007/s11467-021-1078-5
63	A. Smerzi , S. Fantoni , S. Giovanazzi , and S. R. Shenoy , Quantum coherent atomic tunneling between two trapped Bose–Einstein condensates, Phys. Rev. Lett. 79 (25), 4950 (1997) https://doi.org/10.1103/PhysRevLett.79.4950
64	S. Raghavan , A. Smerzi , S. Fantoni , and S. R. Shenoy , Coherent oscillations between two weakly coupled Bose–Einstein condensates: Josephson effects, ff oscillations, and macroscopic quantum self-trapping, Phys. Rev. A 59 (1), 620 (1999) https://doi.org/10.1103/PhysRevA.59.620
65	M. Albiez , R. Gati , J. Fölling , S. Hunsmann , M. Cristiani , and M. K. Oberthaler , Direct observation of tunneling and nonlinear self-trapping in a single bosonic Josephson junction, Phys. Rev. Lett. 95 (1), 010402 (2005) https://doi.org/10.1103/PhysRevLett.95.010402
66	M. Abbarchi , A. Amo , V. G. Sala , D. D. Solnyshkov , H. Flayac , L. Ferrier , I. Sagnes , E. Galopin , A. Lemaĭtre , G. Malpuech , and J. Bloch , Macroscopic quantum selftrapping and Josephson oscillations of exciton polaritons, Nat. Phys. 9 (5), 275 (2013) https://doi.org/10.1038/nphys2609
67	V. V. Konotop , J. Yang , and D. A. Zezyulin , Nonlinear waves in PT-symmetric systems, Rev. Mod. Phys. 88 (3), 035002 (2016) https://doi.org/10.1103/RevModPhys.88.035002
68	E. M. Graefe , H. J. Korsch , and A. E. Niederle , Mean-field dynamics of a non-Hermitian Bose–Hubbard dimer, Phys. Rev. Lett. 101 (15), 150408 (2008) https://doi.org/10.1103/PhysRevLett.101.150408
69	E. M. Graefe , U. Günther , H. J. Korsch , and A. E. Niederle , A non-HermitianPT symmetric Bose–Hubbard model: Eigenvalue rings from unfolding higher-order exceptional points, J. Phys. A Math. Theor. 41 (25), 255206 (2008) https://doi.org/10.1088/1751-8113/41/25/255206
70	D. Witthaut , F. Trimborn , and S. Wimberger , Dissipationinduced coherence and stochastic resonance of an open two-mode Bose–Einstein condensate, Phys. Rev. A 79 (3), 033621 (2009) https://doi.org/10.1103/PhysRevA.79.033621
71	E. M. Graefe and C. Liverani , Mean-field approximation for a Bose–Hubbard dimer with complex interaction strength, J. Phys. A Math. Theor. 46 (45), 455201 (2013) https://doi.org/10.1088/1751-8113/46/45/455201
72	E. M. Graefe , H. J. Korsch , and A. E. Niederle , Quantum-classical correspondence for a non-Hermitian Bose–Hubbard dimer, Phys. Rev. A 82 (1), 013629 (2010) https://doi.org/10.1103/PhysRevA.82.013629
73	E. M. Graefe , Stationary states of a PT symmetric twomode Bose–Einstein condensate, J. Phys. A Math. Theor. 45 (44), 444015 (2012) https://doi.org/10.1088/1751-8113/45/44/444015
74	H. Cartarius and G. Wunner , Model of a PT-symmetric Bose–Einstein condensate in a δ-function double-well potential, Phys. Rev. A 86 (1), 013612 (2012) https://doi.org/10.1103/PhysRevA.86.013612
75	D. Dast D. Haag H. Cartarius G. Wunner R. Eichler , and J. Main , A Bose–Einstein condensate in a PT-symmetric double well, Fortschr. Phys. 61 (2-3), 124 (2013) https://doi.org/10.1002/prop.201200080
76	D. Dast , D. Haag , H. Cartarius , J. Main , and G. Wunner , Eigenvalue structure of a Bose–Einstein condensate in a PT-symmetric double well, J. Phys. A Math. Theor. 46 (37), 375301 (2013) https://doi.org/10.1088/1751-8113/46/37/375301
77	F. Single , H. Cartarius , G. Wunner , and J. Main , Coupling approach for the realization of a PT-symmetric potential for a Bose–Einstein condensate in a double well, Phys. Rev. A 90 (4), 042123 (2014) https://doi.org/10.1103/PhysRevA.90.042123
78	R. Fortanier , D. Dast , D. Haag , H. Cartarius , J. Main , G. Wunner , and R. Gutöhrlein , Dipolar Bose–Einstein condensates in a PT-symmetric double-well potential, Phys. Rev. A 89 (6), 063608 (2014) https://doi.org/10.1103/PhysRevA.89.063608
79	D. Dast , D. Haag , H. Cartarius , J. Main , and G. Wunner , Bose–Einstein condensates with balanced gain and loss beyond mean-field theory, Phys. Rev. A 94 (5), 053601 (2016) https://doi.org/10.1103/PhysRevA.94.053601
80	D. Haag , D. Dast , H. Cartarius , and G. Wunner PTsymmetric gain and loss in a rotating Bose–Einstein condensate, Phys. Rev. A 97 (3), 033607 (2018) https://doi.org/10.1103/PhysRevA.97.033607
81	Y. Zhang , Z. Chen , B. Wu , T. Busch , and V. V. Konotop , Asymmetric loop spectra and unbroken phase protection due to nonlinearities in PT-symmetric periodic potentials, Phys. Rev. Lett. 127 (3), 034101 (2021) https://doi.org/10.1103/PhysRevLett.127.034101
82	B. Wu and Q. Niu , Landau and dynamical instabilities of the superflow of Bose–Einstein condensates in optical lattices, Phys. Rev. A 64 (6), 061603 (2001) https://doi.org/10.1103/PhysRevA.64.061603
83	B. Wu and Q. Niu , Superfluidity of Bose–Einstein condensate in an optical lattice: Landau–Zener tunnelling and dynamical instability, New J. Phys. 5, 104 (2003) https://doi.org/10.1088/1367-2630/5/1/104
84	A. P. Seyranian and A. A. Mailybaev , Multiparameter Stability Theory with Mechanical Applications, World Scientific, 2003 https://doi.org/10.1142/5305

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