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Ciurla D, Forgács P, Lukács Á, Romańczukiewicz T. Negative radiation pressure in Bose-Einstein condensates. Phys Rev E 2024; 109:014228. [PMID: 38366411 DOI: 10.1103/physreve.109.014228] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/07/2023] [Accepted: 01/02/2024] [Indexed: 02/18/2024]
Abstract
In two-component nonlinear Schrödinger equations, the force exerted by incident monochromatic plane waves on an embedded dark soliton and on dark-bright-type solitons is investigated, both perturbatively and by numerical simulations. When the incoming wave is nonvanishing only in the orthogonal component to that of the embedded dark soliton, its acceleration is in the opposite direction to that of the incoming wave. This somewhat surprising phenomenon can be attributed to the well-known negative effective mass of the dark soliton. When a dark-bright soliton, whose effective mass is also negative, is hit by an incoming wave nonvanishing in the component corresponding to the dark soliton, the direction of its acceleration coincides with that of the incoming wave. This implies that the net force acting on it is in the opposite direction to that of the incoming wave. This rather counterintuitive effect is a yet another manifestation of negative radiation pressure exerted by the incident wave, observed in other systems. When a dark-bright soliton interacts with an incoming wave in the component of the bright soliton, it accelerates in the opposite direction; hence the force is pushing it now. We expect that these remarkable effects, in particular the negative radiation pressure, can be experimentally verified in Bose-Einstein condensates.
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Affiliation(s)
- Dominik Ciurla
- Institute of Theoretical Physics, Jagiellonian University, Łojasiewicza 11, 30-348 Kraków, Poland
| | - Péter Forgács
- Wigner RCP RMI, POB 49, 1525 Budapest, Hungary
- Institut Denis-Poisson, UMR No. 7013, CNRS, Université de Tours, Parc de Grandmont, 37200 Tours, France
| | - Árpád Lukács
- Wigner RCP RMI, POB 49, 1525 Budapest, Hungary
- Department of Mathematical Sciences, Durham University, Stockton Road, Durham DH1 3LE, United Kingdom
| | - Tomasz Romańczukiewicz
- Institute of Theoretical Physics, Jagiellonian University, Łojasiewicza 11, 30-348 Kraków, Poland
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2
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Che WJ, Liu C, Akhmediev N. Fundamental and second-order dark soliton solutions of two- and three-component Manakov equations in the defocusing regime. Phys Rev E 2023; 107:054206. [PMID: 37329094 DOI: 10.1103/physreve.107.054206] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/15/2023] [Accepted: 04/10/2023] [Indexed: 06/18/2023]
Abstract
We present exact multiparameter families of soliton solutions for two- and three-component Manakov equations in the defocusing regime. Existence diagrams for such solutions in the space of parameters are presented. Fundamental soliton solutions exist only in finite areas on the plane of parameters. Within these areas, the solutions demonstrate rich spatiotemporal dynamics. The complexity increases in the case of three-component solutions. The fundamental solutions are dark solitons with complex oscillating patterns in the individual wave components. At the boundaries of existence, the solutions are transformed into plain (nonoscillating) vector dark solitons. The superposition of two dark solitons in the solution adds more frequencies in the patterns of oscillating dynamics. These solutions admit degeneracy when the eigenvalues of fundamental solitons in the superposition coincide.
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Affiliation(s)
- Wen-Juan Che
- School of Physics, Northwest University, Xi'an 710127, China
| | - Chong Liu
- School of Physics, Northwest University, Xi'an 710127, China
- Department of Fundamental and Theoretical Physics, Research School of Physics, The Australian National University, Canberra, ACT 2600, Australia
- Shaanxi Key Laboratory for Theoretical Physics Frontiers, Xi'an 710127, China
- Peng Huanwu Center for Fundamental Theory, Xi'an 710127, China
| | - Nail Akhmediev
- Department of Fundamental and Theoretical Physics, Research School of Physics, The Australian National University, Canberra, ACT 2600, Australia
- Arts & Sciences Division, Texas A&M University at Qatar, Doha, Qatar
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3
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Bera J, Batin AQ, Ghosh S, Malomed B, Roy U. Generation of higher harmonics in dipolar Bose-Einstein condensates trapped in periodically modulated potentials. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2023; 381:20220075. [PMID: 36842989 DOI: 10.1098/rsta.2022.0075] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/21/2022] [Accepted: 09/26/2022] [Indexed: 06/18/2023]
Abstract
We consider a quasi-one-dimensional Bose-Einstein condensate with contact and long-range dipolar interactions, under the action of the time-periodic modulation applied to the harmonic-oscillator and optical-lattice trapping potentials. The modulation results in generation of a variety of harmonics in oscillations of the condensate's width and centre-of-mass coordinate. These include multiple and combinational harmonics, represented by sharp peaks in the system's spectra. Approximate analytical results are produced by the variational method, which are verified by systematic simulations of the underlying Gross-Pitaevskii equation. This article is part of the theme issue 'New trends in pattern formation and nonlinear dynamics of extended systems'.
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Affiliation(s)
- Jayanta Bera
- Department of Physics, C. V. Raman Global University, Bhubaneswar 752054, Odisha, India
| | - Abdul Q Batin
- Department of Physics, Indian Institute of Technology Patna,Patna 801106, Bihar, India
| | - Suranjana Ghosh
- Department of Physics, Indian Institute of Science Education and Research Kolkata, Mohanpur 741246, West Bengal, India
| | - Boris Malomed
- Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, and Center for Light-Matter Interaction, Tel Aviv University, P.O.B. 39040, Ramat Aviv, Tel Aviv, Israel
- Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica, Chile
| | - Utpal Roy
- Department of Physics, Indian Institute of Technology Patna,Patna 801106, Bihar, India
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4
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Tabi CB, Wamba E, Nare E, Kofané TC. Interplay between spin-orbit couplings and residual interatomic interactions in the modulational instability of two-component Bose-Einstein condensates. Phys Rev E 2023; 107:044206. [PMID: 37198763 DOI: 10.1103/physreve.107.044206] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/31/2022] [Accepted: 03/21/2023] [Indexed: 05/19/2023]
Abstract
The nonlinear dynamics induced by the modulation instability (MI) of a binary mixture in an atomic Bose-Einstein condensate (BEC) is investigated theoretically under the joint effects of higher-order residual nonlinearities and helicoidal spin-orbit (SO) coupling in a regime of unbalanced chemical potential. The analysis relies on a system of modified coupled Gross-Pitaevskii equations on which the linear stability analysis of plane-wave solutions is performed, from which an expression of the MI gain is obtained. A parametric analysis of regions of instability is carried out, where effects originating from the higher-order interactions and the helicoidal spin-orbit coupling are confronted under different combinations of the signs of the intra- and intercomponent interaction strengths. Direct numerical calculations on the generic model support our analytical predictions and show that the higher-order interspecies interaction and the SO coupling can balance each other suitably for stability to take place. Mainly, it is found that the residual nonlinearity preserves and reinforces the stability of miscible pairs of condensates with SO coupling. Additionally, when a miscible binary mixture of condensates with SO coupling is modulationally unstable, the presence of residual nonlinearity may help soften such instability. Our results finally suggest that MI-induced formation of stable solitons in mixtures of BECs with two-body attraction may be preserved by the residual nonlinearity even though the latter enhances the instability.
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Affiliation(s)
- Conrad Bertrand Tabi
- Department of Physics and Astronomy, Botswana International University of Science and Technology, Private Mail Bag 16, Palapye, Botswana
| | - Etienne Wamba
- Faculty of Engineering and Technology, University of Buea, P.O. Box 63, Buea, Cameroon
- STIAS, Wallenberg Research Centre, Stellenbosch University, Stellenbosch 7600, South Africa
| | - Emmanual Nare
- Department of Physics and Astronomy, Botswana International University of Science and Technology, Private Mail Bag 16, Palapye, Botswana
| | - Timoléon Crépin Kofané
- Department of Physics and Astronomy, Botswana International University of Science and Technology, Private Mail Bag 16, Palapye, Botswana
- Laboratory of Mechanics, Department of Physics, Faculty of Science, University of Yaoundé I, P.O. Box 812, Yaoundé, Cameroon
- Centre d'Excellence Africain en Technologies de l'Information et de la Communication, University of Yaoundé I, Yaoundé, Cameroon
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5
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Zhang AX, Hu XW, Jiang YF, Liang JC, Zhang Y, Zhang W, Xue JK. Localization and spin dynamics of spin-orbit-coupled Bose-Einstein condensates in deep optical lattices. Phys Rev E 2021; 104:064215. [PMID: 35030834 DOI: 10.1103/physreve.104.064215] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/02/2021] [Accepted: 12/09/2021] [Indexed: 11/07/2022]
Abstract
We analytically and numerically discuss the dynamics of two pseudospin components Bose-Einstein condensates (BECs) with spin-orbit coupling (SOC) in deep optical lattices. Rich localized phenomena, such as breathers, solitons, self-trapping, and diffusion, are revealed and strongly depend on the strength of the atomic interaction, SOC, Raman detuning, and the spin polarization (i.e., the initial population difference of atoms between the two pseudospin components of BECs). The critical conditions for the transition of localized states are derived analytically. Based on the critical conditions, the detailed dynamical phase diagram describing the different dynamical regimes is derived. When the Raman detuning satisfies a critical condition, localized states with a fixed initial spin polarization can be observed. When the critical condition is not satisfied, we use two quenching methods, i.e., suddenly and linearly quenching Raman detuning from the soliton or breather state, to discuss the spin dynamics, phase transition, and wave packet dynamics by numerical simulation. The sudden quenching results in a damped oscillation of spin polarization and transforms the system to a new polarized state. Interestingly, the linear quenching of Raman detuning induces a controllable phase transition from an unpolarized phase to an expected polarized phase, while the soliton or breather dynamics is maintained.
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Affiliation(s)
- Ai-Xia Zhang
- College of Physics and Electronic Engineering, Northwest Normal University, Lanzhou 730070, China
| | - Xiao-Wen Hu
- College of Physics and Electronic Engineering, Northwest Normal University, Lanzhou 730070, China
| | - Yan-Fang Jiang
- College of Physics and Electronic Engineering, Northwest Normal University, Lanzhou 730070, China
| | - Jun-Cheng Liang
- College of Physics and Electronic Engineering, Northwest Normal University, Lanzhou 730070, China
| | - Ying Zhang
- College of Physics and Electronic Engineering, Northwest Normal University, Lanzhou 730070, China
| | - Wei Zhang
- College of Physics and Electronic Engineering, Northwest Normal University, Lanzhou 730070, China
| | - Ju-Kui Xue
- College of Physics and Electronic Engineering, Northwest Normal University, Lanzhou 730070, China
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6
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Wang W. Systematic vector solitary waves from their linear limits in one-dimensional n-component Bose-Einstein condensates. Phys Rev E 2021; 104:014217. [PMID: 34412218 DOI: 10.1103/physreve.104.014217] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/23/2020] [Accepted: 07/09/2021] [Indexed: 11/07/2022]
Abstract
We systematically construct a series of vector solitary waves in harmonically trapped one-dimensional three-, four-, and five-component Bose-Einstein condensates. These stationary states are continued in chemical potentials from the analytically tractable low-density linear limit of respective states, as independent linear quantum harmonic oscillator states, to the high-density nonlinear Thomas-Fermi regime. A systematic interpolation procedure is proposed to achieve this sequential continuation via a trajectory in the multidimensional space of the chemical potentials. The Bogoliubov-de Gennes spectral analysis shows that all of the states considered herein can be fully stabilized in suitable chemical potential intervals in the Thomas-Fermi regime. Finally, we present some typical SU(n)-rotation-induced and driving-induced dynamics. This method can be extended to higher dimensions and shows significant promise for finding a wide range of solitary waves ahead.
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Affiliation(s)
- Wenlong Wang
- College of Physics, Sichuan University, Chengdu 610065, China
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7
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Qin YH, Zhao LC, Yang ZQ, Ling L. Multivalley dark solitons in multicomponent Bose-Einstein condensates with repulsive interactions. Phys Rev E 2021; 104:014201. [PMID: 34412198 DOI: 10.1103/physreve.104.014201] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/12/2021] [Accepted: 06/09/2021] [Indexed: 06/13/2023]
Abstract
We obtain multivalley dark soliton solutions with asymmetric or symmetric profiles in multicomponent repulsive Bose-Einstein condensates by developing the Darboux transformation method. We demonstrate that the width-dependent parameters of solitons significantly affect the velocity ranges and phase jump regions of multivalley dark solitons, in sharp contrast to scalar dark solitons. For double-valley dark solitons, we find that the phase jump is in the range [0,2π], which is quite different from that of the usual single-valley dark soliton. Based on our results, we argue that the phase jump of an n-valley dark soliton could be in the range [0,nπ], supported by our analysis extending up to five-component condensates. The interaction between a double-valley dark soliton and a single-valley dark soliton is further investigated, and we reveal a striking collision process in which the double-valley dark soliton is transformed into a breather after colliding with the single-valley dark soliton. Our analyses suggest that this breather transition exists widely in the collision processes involving multivalley dark solitons. The possibilities for observing these multivalley dark solitons in related Bose-Einstein condensates experiments are discussed.
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Affiliation(s)
- Yan-Hong Qin
- School of Physics, Northwest University, Xi'an 710127, China
- Shaanxi Key Laboratory for Theoretical Physics Frontiers, Xi'an 710127, China
| | - Li-Chen Zhao
- School of Physics, Northwest University, Xi'an 710127, China
- NSFC-SPTP Peng Huanwu Center for Fundamental Theory, Xi'an 710127, China
- Shaanxi Key Laboratory for Theoretical Physics Frontiers, Xi'an 710127, China
| | - Zeng-Qiang Yang
- Department of Physics, School of Arts and Sciences, Shaanxi University of Science and Technology, Xi'an 710021, China
| | - Liming Ling
- School of Mathematics, South China University of Technology, Guangzhou 510640, China
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8
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Otlaadisa P, Tabi CB, Kofané TC. Modulation instability in helicoidal spin-orbit coupled open Bose-Einstein condensates. Phys Rev E 2021; 103:052206. [PMID: 34134292 DOI: 10.1103/physreve.103.052206] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/15/2020] [Accepted: 04/08/2021] [Indexed: 06/12/2023]
Abstract
We introduce a vector form of the cubic complex Ginzburg-Landau equation describing the dynamics of dissipative solitons in the two-component helicoidal spin-orbit coupled open Bose-Einstein condensates (BECs), where the addition of dissipative interactions is done through coupled rate equations. Furthermore, the standard linear stability analysis is used to investigate theoretically the stability of continuous-wave (cw) solutions and to obtain an expression for the modulational instability gain spectrum. Using direct simulations of the Fourier space, we numerically investigate the dynamics of the modulational instability in the presence of helicoidal spin-orbit coupling. Our numerical simulations confirm the theoretical predictions of the linear theory as well as the threshold for amplitude perturbations.
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Affiliation(s)
- Phelo Otlaadisa
- Department of Physics and Astronomy, Botswana International University of Science and Technology, Private Mail Bag 16, Palapye, Botswana
| | - Conrad Bertrand Tabi
- Department of Physics and Astronomy, Botswana International University of Science and Technology, Private Mail Bag 16, Palapye, Botswana
| | - Timoléon Crépin Kofané
- Department of Physics and Astronomy, Botswana International University of Science and Technology, Private Mail Bag 16, Palapye, Botswana
- Laboratory of Mechanics, Department of Physics, Faculty of Science, University of Yaoundé I, P.O. Box 812, Yaoundé, Cameroon
- Centre d'Excellence Africain en Technologies de l'Information et de la Communication, University of Yaoundé I, Yaoundé, Cameroon
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9
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Abstract
We study the reflection and transmission properties of matter wave solitons impinging on localized scattering potentials in one spatial dimension. By mean field analysis we identify regimes where the solitons behave more like waves or more like particles as a result of the interplay between the dispersive wave propagation and the attractive interactions between the atoms. For a bright soliton propagating together with a dark soliton void in a two-species Bose-Einstein condensate, we find different reflection and transmission properties of the dark and the bright components.
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10
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Farolfi A, Trypogeorgos D, Mordini C, Lamporesi G, Ferrari G. Observation of Magnetic Solitons in Two-Component Bose-Einstein Condensates. PHYSICAL REVIEW LETTERS 2020; 125:030401. [PMID: 32745386 DOI: 10.1103/physrevlett.125.030401] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/22/2019] [Revised: 02/11/2020] [Accepted: 06/11/2020] [Indexed: 06/11/2023]
Abstract
We experimentally investigate the dynamics of spin solitary waves (magnetic solitons) in a harmonically trapped, binary superfluid mixture. We measure the in situ density of each pseudospin component and their relative local phase via an interferometric technique we developed and as such, fully characterize the magnetic solitons while they undergo oscillatory motion in the trap. Magnetic solitons exhibit nondispersive, dissipationless longtime dynamics. By imprinting multiple magnetic solitons in our ultracold gas sample, we engineer binary collisions between solitons of either the same or opposite magnetization and map out their trajectories.
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Affiliation(s)
- A Farolfi
- INO-CNR BEC Center and Dipartimento di Fisica, Università di Trento, and Trento Institute for Fundamental Physics and Applications, INFN, 38123 Povo, Italy
| | - D Trypogeorgos
- INO-CNR BEC Center and Dipartimento di Fisica, Università di Trento, and Trento Institute for Fundamental Physics and Applications, INFN, 38123 Povo, Italy
| | - C Mordini
- INO-CNR BEC Center and Dipartimento di Fisica, Università di Trento, and Trento Institute for Fundamental Physics and Applications, INFN, 38123 Povo, Italy
| | - G Lamporesi
- INO-CNR BEC Center and Dipartimento di Fisica, Università di Trento, and Trento Institute for Fundamental Physics and Applications, INFN, 38123 Povo, Italy
| | - G Ferrari
- INO-CNR BEC Center and Dipartimento di Fisica, Università di Trento, and Trento Institute for Fundamental Physics and Applications, INFN, 38123 Povo, Italy
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11
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Chai X, Lao D, Fujimoto K, Hamazaki R, Ueda M, Raman C. Magnetic Solitons in a Spin-1 Bose-Einstein Condensate. PHYSICAL REVIEW LETTERS 2020; 125:030402. [PMID: 32745412 DOI: 10.1103/physrevlett.125.030402] [Citation(s) in RCA: 9] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/02/2020] [Accepted: 05/26/2020] [Indexed: 06/11/2023]
Abstract
Vector solitons are a type of solitary or nonspreading wave packet occurring in a nonlinear medium composed of multiple components. As such, a variety of synthetic systems can be constructed to explore their properties, from nonlinear optics to ultracold atoms, and even in metamaterials. Bose-Einstein condensates have a rich panoply of internal hyperfine levels, or spin components, which make them a unique platform for exploring these solitary waves. However, existing experimental work has focused largely on binary systems confined to the Manakov limit of the nonlinear equations governing the soliton behavior, where quantum magnetism plays no role. Here we observe, using a "magnetic shadowing" technique, a new type of soliton in a spinor Bose-Einstein condensate, one that exists only when the underlying interactions are antiferromagnetic and which is deeply embedded within a full spin-1 quantum system. Our approach opens up a vista for future studies of "solitonic matter" whereby multiple solitons interact with one another at deterministic locations.
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Affiliation(s)
- X Chai
- School of Physics, Georgia Institute of Technology, 837 State Street, Atlanta, Georgia 30332, USA
| | - D Lao
- School of Physics, Georgia Institute of Technology, 837 State Street, Atlanta, Georgia 30332, USA
| | - Kazuya Fujimoto
- Institute for Advanced Research, Nagoya University, Nagoya 464-8601, Japan
- Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan
| | - Ryusuke Hamazaki
- Department of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
- Nonequilibrium Quantum Statistical Mechanics RIKEN Hakubi Research Team, RIKEN Cluster for Pioneering Research (CPR), RIKEN iTHEMS, Wako, Saitama 351-0198, Japan
| | - Masahito Ueda
- Department of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
- Institute for Physics of Intelligence, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
- RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan
| | - C Raman
- School of Physics, Georgia Institute of Technology, 837 State Street, Atlanta, Georgia 30332, USA
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12
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A Dual-Species Bose-Einstein Condensate with Attractive Interspecies Interactions. CONDENSED MATTER 2020. [DOI: 10.3390/condmat5010021] [Citation(s) in RCA: 15] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
Abstract
We report on the production of a 41 K- 87 Rb dual-species Bose–Einstein condensate with tunable interspecies interaction and we study the mixture in the attractive regime; i.e., for negative values of the interspecies scattering length a 12 . The binary condensate is prepared in the ground state and confined in a pure optical trap. We exploit Feshbach resonances for tuning the value of a 12 . After compensating the gravitational sag between the two species with a magnetic field gradient, we drive the mixture into the attractive regime. We let the system evolve both in free space and in an optical waveguide. In both geometries, for strong attractive interactions, we observe the formation of self-bound states, recognizable as quantum droplets. Our findings prove that robust, long-lived droplet states can be realized in attractive two-species mixtures, despite the two atomic components possibly experiencing different potentials.
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13
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Plestid R, O'Dell DHJ. Balancing long-range interactions and quantum pressure: Solitons in the Hamiltonian mean-field model. Phys Rev E 2019; 100:022216. [PMID: 31574683 DOI: 10.1103/physreve.100.022216] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/09/2019] [Indexed: 06/10/2023]
Abstract
The Hamiltonian mean-field (HMF) model describes particles on a ring interacting via a cosine interaction, or equivalently, rotors coupled by infinite-range XY interactions. Conceived as a generic statistical mechanical model for long-range interactions such as gravity (of which the cosine is the first Fourier component), it has recently been used to account for self-organization in experiments on cold atoms with long-range optically mediated interactions. The significance of the HMF model lies in its ability to capture the universal effects of long-range interactions and yet be exactly solvable in the canonical ensemble. In this work we consider the quantum version of the HMF model in one dimension and provide a classification of all possible stationary solutions of its generalized Gross-Pitaevskii equation (GGPE), which is both nonlinear and nonlocal. The exact solutions are Mathieu functions that obey a nonlinear relation between the wave function and the depth of the mean-field potential, and we identify them as bright solitons. Using a Galilean transformation these solutions can be boosted to finite velocity and are increasingly localized as the mean-field potential becomes deeper. In contrast to the usual local GPE, the HMF case features a tower of solitons, each with a different number of nodes. Our results suggest that long-range interactions support solitary waves in a novel manner relative to the short-range case.
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Affiliation(s)
- Ryan Plestid
- Department of Physics and Astronomy, McMaster University, 1280 Main St. W., Hamilton, Ontario, Canada L8S 4M1
- Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario, Canada N2L 2Y5
| | - D H J O'Dell
- Department of Physics and Astronomy, McMaster University, 1280 Main St. W., Hamilton, Ontario, Canada L8S 4M1
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14
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Qin YH, Zhao LC, Ling L. Nondegenerate bound-state solitons in multicomponent Bose-Einstein condensates. Phys Rev E 2019; 100:022212. [PMID: 31574652 DOI: 10.1103/physreve.100.022212] [Citation(s) in RCA: 12] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/15/2019] [Indexed: 06/10/2023]
Abstract
We investigate nondegenerate bound-state solitons systematically in multicomponent Bose-Einstein condensates, through developing the Darboux transformation method to derive exact soliton solutions analytically. In particular, we show that bright solitons with nodes correspond to the excited bound states in effective quantum wells, in sharp contrast to the bright solitons and dark solitons reported before (which usually correspond to ground state and free state, respectively). We further demonstrate that bound-state solitons with nodes are induced by incoherent superposition of solitons in different components. Moreover, we reveal that the interactions between these bound-state solitons are usually inelastic, caused by the incoherent interactions between solitons in different components and the coherent interactions between solitons in the same component. Additionally, the detailed spectral stability analysis demonstrates the stability of nondegenerate bound-state solitons. The bound-state solitons can be used to study many different physical problems, such as beating dynamics, spin-orbit coupling effects, quantum fluctuations, and even quantum entanglement states.
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Affiliation(s)
- Yan-Hong Qin
- School of Physics, Northwest University, Xi'an 710127, China
- Shaanxi Key Laboratory for Theoretical Physics Frontiers, Xi'an 710127, China
| | - Li-Chen Zhao
- School of Physics, Northwest University, Xi'an 710127, China
- Shaanxi Key Laboratory for Theoretical Physics Frontiers, Xi'an 710127, China
| | - Liming Ling
- School of Mathematics, South China University of Technology, Guangzhou 510640, China
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15
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Wan NS, Li YE, Xue JK. Solitons in spin-orbit-coupled spin-2 spinor Bose-Einstein condensates. Phys Rev E 2019; 99:062220. [PMID: 31330691 DOI: 10.1103/physreve.99.062220] [Citation(s) in RCA: 11] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/20/2018] [Indexed: 11/07/2022]
Abstract
We investigate the different types of matter-wave solitons in spin-orbit-coupled spin-2 spinor Bose-Einstein condensates. Using mean-field theory and adopting the multiscale perturbation method, the original five-component Gross-Pitaevskii spin-orbit-coupled spin-2 spinor Bose-Einstein condensate model can be reduced to a single effective nonlinear Schrödinger equation, which allows us to find analytical soliton solutions of this system. In this way, for different regimes of the spin-orbit coupling, Raman coupling, and interatomic interactions, we find approximate bright and dark soliton solutions. Particularly, the type of solitons depends on the dispersion properties of the system. When the lowest-energy band has a single-well structure, we find there only exist positive mass bright or dark solitons due to the dispersion coefficient of effective nonlinear Shrödinger equation always positive. However, when the lowest-energy band has a double-well structure, there will appear positive (negative) mass bright or dark solitons because the sign of the dispersion coefficient can be positive (negative) under different momentum. We employ direct numerical simulation of the original five-component Gross-Pitaevskii equations to confirm the analytical results.
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Affiliation(s)
- Nian-Sheng Wan
- College of Physics and Electronic Engineering, Northwest Normal University, Lanzhou 730070, China
| | - Yu-E Li
- College of Physics and Electronic Engineering, Northwest Normal University, Lanzhou 730070, China
| | - Ju-Kui Xue
- College of Physics and Electronic Engineering, Northwest Normal University, Lanzhou 730070, China
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16
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Hu X, Guo J, Shao GD, Song YF, Yoo SW, Malomed BA, Tang DY. Observation of incoherently coupled dark-bright vector solitons in single-mode fibers. OPTICS EXPRESS 2019; 27:18311-18317. [PMID: 31252776 DOI: 10.1364/oe.27.018311] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/09/2019] [Accepted: 06/03/2019] [Indexed: 06/09/2023]
Abstract
We report experimental observation of incoherently coupled dark-bright vector solitons in single-mode fibers. Properties of the vector solitons accord well with those predicted by the respective systems of incoherently coupled nonlinear Schrödinger equations. To our knowledge, this is the first experimental observation of temporal incoherently coupled dark-bright solitons in single-mode fibers.
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Fujimoto K, Hamazaki R, Ueda M. Flemish Strings of Magnetic Solitons and a Nonthermal Fixed Point in a One-Dimensional Antiferromagnetic Spin-1 Bose Gas. PHYSICAL REVIEW LETTERS 2019; 122:173001. [PMID: 31107065 DOI: 10.1103/physrevlett.122.173001] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/13/2018] [Indexed: 06/09/2023]
Abstract
Thermalization in a quenched one-dimensional antiferromagnetic spin-1 Bose gas is shown to proceed via a nonthermal fixed point through annihilation of Flemish-string bound states of magnetic solitons. A possible experimental situation is discussed.
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Affiliation(s)
- Kazuya Fujimoto
- Department of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
| | - Ryusuke Hamazaki
- Department of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
| | - Masahito Ueda
- Department of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
- RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan
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18
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Affiliation(s)
- Mati Aharonyan
- Department of Physics and Center for Quantum Entanglement Science and Technology (QUEST), Bar-Ilan University, Ramat Gan, Israel
| | - Emanuele G. Dalla Torre
- Department of Physics and Center for Quantum Entanglement Science and Technology (QUEST), Bar-Ilan University, Ramat Gan, Israel
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19
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Spin current generation and relaxation in a quenched spin-orbit-coupled Bose-Einstein condensate. Nat Commun 2019; 10:375. [PMID: 30670693 PMCID: PMC6343014 DOI: 10.1038/s41467-018-08119-4] [Citation(s) in RCA: 16] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/21/2018] [Accepted: 12/14/2018] [Indexed: 11/27/2022] Open
Abstract
Understanding the effects of spin-orbit coupling (SOC) and many-body interactions on spin transport is important in condensed matter physics and spintronics. This topic has been intensively studied for spin carriers such as electrons but barely explored for charge-neutral bosonic quasiparticles (including their condensates), which hold promises for coherent spin transport over macroscopic distances. Here, we explore the effects of synthetic SOC (induced by optical Raman coupling) and atomic interactions on the spin transport in an atomic Bose-Einstein condensate (BEC), where the spin-dipole mode (SDM, actuated by quenching the Raman coupling) of two interacting spin components constitutes an alternating spin current. We experimentally observe that SOC significantly enhances the SDM damping while reducing the thermalization (the reduction of the condensate fraction). We also observe generation of BEC collective excitations such as shape oscillations. Our theory reveals that the SOC-modified interference, immiscibility, and interaction between the spin components can play crucial roles in spin transport. Spin-orbit coupling is interesting for fundamental understanding of spin transport and quench dynamics. Here the authors demonstrate spin-current generation and its relaxation in spin-orbit-coupled Bose-Einstein condensates of Rb atoms in different spin states.
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20
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Zhao LC. Beating effects of vector solitons in Bose-Einstein condensates. Phys Rev E 2018; 97:062201. [PMID: 30011505 DOI: 10.1103/physreve.97.062201] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/07/2018] [Indexed: 06/08/2023]
Abstract
We study the beating effects of solitons in multicomponent coupled Bose-Einstein condensate systems. Our analysis indicates that the period of beating behavior is determined by the energy eigenvalue difference in the effective quantum well induced by solitons, and the beating pattern is determined by the eigenstates of a quantum well, which are involved in the beating behavior. We show that the beating solitons correspond to linear superpositions of eigenstates in some quantum wells, and the correspondence relations are identical for solitons in both an attractive interaction and a repulsive interaction condensate. This provides a possible way to understand the beating effects of solitons for attractive and repulsive interaction cases in a unified way, based on the knowledge of quantum eigenstates. Moreover, our results demonstrate many different beating patterns for solitons in multicomponent coupled condensates, in sharp contrast to the beating dark soliton reported before. The beating behavior can be used to test the eigenvalue differences in certain quantum wells, and more abundant beating patterns are expected to exist in more component-coupled systems.
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Affiliation(s)
- Li-Chen Zhao
- School of Physics, Northwest University, Xi'an 710069, China and Shaanxi Key Laboratory for Theoretical Physics Frontiers, Xi'an 710069, China
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21
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Fava E, Bienaimé T, Mordini C, Colzi G, Qu C, Stringari S, Lamporesi G, Ferrari G. Observation of Spin Superfluidity in a Bose Gas Mixture. PHYSICAL REVIEW LETTERS 2018; 120:170401. [PMID: 29756820 DOI: 10.1103/physrevlett.120.170401] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/13/2017] [Indexed: 06/08/2023]
Abstract
The spin dynamics of a harmonically trapped Bose-Einstein condensed binary mixture of sodium atoms is experimentally investigated at finite temperature. In the collisional regime the motion of the thermal component is shown to be damped because of spin drag, while the two condensates exhibit a counterflow oscillation without friction, thereby providing direct evidence for spin superfluidity. Results are also reported in the collisionless regime where the spin components of both the condensate and thermal part oscillate without damping, their relative motion being driven by a mean-field effect. We also measure the static polarizability of the condensed and thermal parts and we find a large increase of the condensate polarizability with respect to the T=0 value, in agreement with the predictions of theory.
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Affiliation(s)
- Eleonora Fava
- INO-CNR BEC Center and Dipartimento di Fisica, Università di Trento, 38123 Povo, Italy
| | - Tom Bienaimé
- INO-CNR BEC Center and Dipartimento di Fisica, Università di Trento, 38123 Povo, Italy
| | - Carmelo Mordini
- INO-CNR BEC Center and Dipartimento di Fisica, Università di Trento, 38123 Povo, Italy
- Trento Institute for Fundamental Physics and Applications, INFN, 38123 Povo, Italy
| | - Giacomo Colzi
- INO-CNR BEC Center and Dipartimento di Fisica, Università di Trento, 38123 Povo, Italy
- Trento Institute for Fundamental Physics and Applications, INFN, 38123 Povo, Italy
| | - Chunlei Qu
- INO-CNR BEC Center and Dipartimento di Fisica, Università di Trento, 38123 Povo, Italy
| | - Sandro Stringari
- INO-CNR BEC Center and Dipartimento di Fisica, Università di Trento, 38123 Povo, Italy
- Trento Institute for Fundamental Physics and Applications, INFN, 38123 Povo, Italy
| | - Giacomo Lamporesi
- INO-CNR BEC Center and Dipartimento di Fisica, Università di Trento, 38123 Povo, Italy
- Trento Institute for Fundamental Physics and Applications, INFN, 38123 Povo, Italy
| | - Gabriele Ferrari
- INO-CNR BEC Center and Dipartimento di Fisica, Università di Trento, 38123 Povo, Italy
- Trento Institute for Fundamental Physics and Applications, INFN, 38123 Povo, Italy
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22
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Helm JL, Billam TP, Rakonjac A, Cornish SL, Gardiner SA. Spin-Orbit-Coupled Interferometry with Ring-Trapped Bose-Einstein Condensates. PHYSICAL REVIEW LETTERS 2018; 120:063201. [PMID: 29481231 DOI: 10.1103/physrevlett.120.063201] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/10/2017] [Indexed: 06/08/2023]
Abstract
We propose a method of atom interferometry using a spinor Bose-Einstein condensate with a time-varying magnetic field acting as a coherent beam splitter. Our protocol creates long-lived superpositional counterflow states, which are of fundamental interest and can be made sensitive to both the Sagnac effect and magnetic fields on the sub-μG scale. We split a ring-trapped condensate, initially in the m_{f}=0 hyperfine state, into superpositions of internal m_{f}=±1 states and condensate superflow, which are spin-orbit coupled. After interrogation, the relative phase accumulation can be inferred from a population transfer to the m_{f}=±1 states. The counterflow generation protocol is adiabatically deterministic and does not rely on coupling to additional optical fields or mechanical stirring techniques. Our protocol can maximize the classical Fisher information for any rotation, magnetic field, or interrogation time and so has the maximum sensitivity available to uncorrelated particles. Precision can increase with the interrogation time and so is limited only by the lifetime of the condensate.
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Affiliation(s)
- J L Helm
- The Dodd-Walls Centre for Photonic and Quantum Technologies, Department of Physics, University of Otago, Dunedin 9016, New Zealand
| | - T P Billam
- Joint Quantum Center (JQC) Durham-Newcastle, School of Mathematics, Statistics and Physics, Newcastle University, Newcastle upon Tyne NE1 7RU, United Kingdom
| | - A Rakonjac
- Joint Quantum Center (JQC) Durham-Newcastle, Department of Physics, Durham University, Durham DH1 3LE, United Kingdom
| | - S L Cornish
- Joint Quantum Center (JQC) Durham-Newcastle, Department of Physics, Durham University, Durham DH1 3LE, United Kingdom
| | - S A Gardiner
- Joint Quantum Center (JQC) Durham-Newcastle, Department of Physics, Durham University, Durham DH1 3LE, United Kingdom
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23
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Bersano TM, Gokhroo V, Khamehchi MA, D'Ambroise J, Frantzeskakis DJ, Engels P, Kevrekidis PG. Three-Component Soliton States in Spinor F=1 Bose-Einstein Condensates. PHYSICAL REVIEW LETTERS 2018; 120:063202. [PMID: 29481230 DOI: 10.1103/physrevlett.120.063202] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/23/2017] [Revised: 11/14/2017] [Indexed: 06/08/2023]
Abstract
Dilute-gas Bose-Einstein condensates are an exceptionally versatile test bed for the investigation of novel solitonic structures. While matter-wave solitons in one- and two-component systems have been the focus of intense research efforts, an extension to three components has never been attempted in experiments. Here, we experimentally demonstrate the existence of robust dark-bright-bright (DBB) and dark-dark-bright solitons in a multicomponent F=1 condensate. We observe lifetimes on the order of hundreds of milliseconds for these structures. Our theoretical analysis, based on a multiscale expansion method, shows that small-amplitude solitons of these types obey universal long-short wave resonant interaction models, namely, Yajima-Oikawa systems. Our experimental and analytical findings are corroborated by direct numerical simulations highlighting the persistence of, e.g., the DBB soliton states, as well as their robust oscillations in the trap.
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Affiliation(s)
- T M Bersano
- Washington State University, Department of Physics & Astronomy, Pullman, Washington 99164 USA
| | - V Gokhroo
- Washington State University, Department of Physics & Astronomy, Pullman, Washington 99164 USA
| | - M A Khamehchi
- Washington State University, Department of Physics & Astronomy, Pullman, Washington 99164 USA
| | - J D'Ambroise
- Department of Mathematics, Computer & Information Science, State University of New York (SUNY) College at Old Westbury, Westbury, New York 11568, USA
| | - D J Frantzeskakis
- Department of Physics, National and Kapodistrian University of Athens, Panepistimiopolis, Zografos, Athens 15784, Greece
| | - P Engels
- Washington State University, Department of Physics & Astronomy, Pullman, Washington 99164 USA
| | - P G Kevrekidis
- Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003, USA
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24
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Ivanov SK, Kamchatnov AM, Congy T, Pavloff N. Solution of the Riemann problem for polarization waves in a two-component Bose-Einstein condensate. Phys Rev E 2018; 96:062202. [PMID: 29347287 DOI: 10.1103/physreve.96.062202] [Citation(s) in RCA: 19] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/14/2017] [Indexed: 11/07/2022]
Abstract
We provide a classification of the possible flows of two-component Bose-Einstein condensates evolving from initially discontinuous profiles. We consider the situation where the dynamics can be reduced to the consideration of a single polarization mode (also denoted as "magnetic excitation") obeying a system of equations equivalent to the Landau-Lifshitz equation for an easy-plane ferromagnet. We present the full set of one-phase periodic solutions. The corresponding Whitham modulation equations are obtained together with formulas connecting their solutions with the Riemann invariants of the modulation equations. The problem is not genuinely nonlinear, and this results in a non-single-valued mapping of the solutions of the Whitham equations with physical wave patterns as well as the appearance of interesting elements-contact dispersive shock waves-that are absent in more standard, genuinely nonlinear situations. Our analytic results are confirmed by numerical simulations.
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Affiliation(s)
- S K Ivanov
- Institute of Spectroscopy, Russian Academy of Sciences, Troitsk, Moscow 108840, Russia.,Moscow Institute of Physics and Technology, Institutsky lane 9, Dolgoprudny, Moscow region 141700, Russia
| | - A M Kamchatnov
- Institute of Spectroscopy, Russian Academy of Sciences, Troitsk, Moscow 108840, Russia.,Moscow Institute of Physics and Technology, Institutsky lane 9, Dolgoprudny, Moscow region 141700, Russia
| | - T Congy
- LPTMS, CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - N Pavloff
- LPTMS, CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
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25
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Ancilotto F, Levy D, Pimentel J, Eloranta J. First Observation of Bright Solitons in Bulk Superfluid ^{4}He. PHYSICAL REVIEW LETTERS 2018; 120:035302. [PMID: 29400543 DOI: 10.1103/physrevlett.120.035302] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/02/2017] [Revised: 09/21/2017] [Indexed: 06/07/2023]
Abstract
The existence of bright solitons in bulk superfluid ^{4}He is demonstrated by time-resolved shadowgraph imaging experiments and density functional theory (DFT) calculations. The initial liquid compression that leads to the creation of nonlinear waves is produced by rapidly expanding plasma from laser ablation. After the leading dissipative period, these waves transform into bright solitons, which exhibit three characteristic features: dispersionless propagation, negligible interaction in a two-wave collision, and direct dependence between soliton amplitude and the propagation velocity. The experimental observations are supported by DFT calculations, which show rapid evolution of the initially compressed liquid into bright solitons. At high amplitudes, solitons become unstable and break down into dispersive shock waves.
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Affiliation(s)
- Francesco Ancilotto
- Dipartimento di Fisica e Astronomia "Galileo Galilei" and CNISM, Università di Padova, via Marzolo 8, 35122 Padova, Italy and CNR-IOM Democritos, via Bonomea, 265-34136 Trieste, Italy
| | - David Levy
- Department of Physics and Astronomy, California State University, Northridge, Northridge, California 91330, USA
| | - Jessica Pimentel
- Department of Chemistry and Biochemistry, California State University, Northridge, Northridge, California 91330, USA
| | - Jussi Eloranta
- Department of Chemistry and Biochemistry, California State University, Northridge, Northridge, California 91330, USA
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26
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Qin YH, Zhao LC, Yang ZY, Yang WL. Several localized waves induced by linear interference between a nonlinear plane wave and bright solitons. CHAOS (WOODBURY, N.Y.) 2018; 28:013111. [PMID: 29390636 DOI: 10.1063/1.5004491] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
We investigate linear interference effects between a nonlinear plane wave and bright solitons, which are admitted by a pair-transition coupled two-component Bose-Einstein condensate. We demonstrate that the interference effects can induce several localized waves possessing distinctive wave structures, mainly including anti-dark solitons, W-shaped solitons, multi-peak solitons, Kuznetsov-Ma like breathers, and multi-peak breathers. Specifically, the explicit conditions for them are clarified by a phase diagram based on the linear interference properties. Furthermore, the interactions between these localized waves are discussed. The detailed analysis indicates that the soliton-soliton interaction induced phase shift brings the collision between these localized waves which can be inelastic for solitons involving collision and can be elastic for breathers. These characters come from the fact that the profile of solitons depends on the relative phase between bright solitons and a plane wave, and the profile of breathers does not depend on the relative phase. These results would motivate more discussions on linear interference between other nonlinear waves. Specifically, the solitons or breathers obtained here are not related to modulational instability. The underlying reasons are discussed in detail. In addition, possibilities to observe these localized waves are discussed in a two species Bose-Einstein condensate.
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Affiliation(s)
- Yan-Hong Qin
- School of Physics, Northwest University, Xi'an 710069, China
| | - Li-Chen Zhao
- School of Physics, Northwest University, Xi'an 710069, China
| | - Zhan-Ying Yang
- School of Physics, Northwest University, Xi'an 710069, China
| | - Wen-Li Yang
- Shaanxi Key Laboratory for Theoretical Physics Frontiers, Xi'an 710069, China
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27
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Nath D, Gao Y, Babu Mareeswaran R, Kanna T, Roy B. Bright-dark and dark-dark solitons in coupled nonlinear Schrödinger equation with PT-symmetric potentials. CHAOS (WOODBURY, N.Y.) 2017; 27:123102. [PMID: 29289045 DOI: 10.1063/1.4997534] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
We explore different nonlinear coherent structures, namely, bright-dark (BD) and dark-dark (DD) solitons in a coupled nonlinear Schrödinger/Gross-Pitaevskii equation with defocusing/repulsive nonlinearity coefficients featuring parity-time ( PT)-symmetric potentials. Especially, for two choices of PT-symmetric potentials, we obtain the exact solutions for BD and DD solitons. We perform the linear stability analysis of the obtained coherent structures. The results of this linear stability analysis are well corroborated by direct numerical simulation incorporating small random noise. It has been found that there exists a parameter regime which can support stable BD and DD solitons.
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Affiliation(s)
- Debraj Nath
- Department of Mathematics, Vivekananda College, Kolkata 700063, India
| | - Yali Gao
- School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, China
| | - R Babu Mareeswaran
- PG and Research Department of Physics, Bishop Heber College, Tiruchirappalli 620017, Tamil Nadu, India
| | - T Kanna
- PG and Research Department of Physics, Bishop Heber College, Tiruchirappalli 620017, Tamil Nadu, India
| | - Barnana Roy
- Physics and Applied Mathematics Unit, Indian Statistical Institute, Kolkata 700 108, India
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28
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Kim JH, Seo SW, Shin Y. Critical Spin Superflow in a Spinor Bose-Einstein Condensate. PHYSICAL REVIEW LETTERS 2017; 119:185302. [PMID: 29219587 DOI: 10.1103/physrevlett.119.185302] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/08/2017] [Indexed: 06/07/2023]
Abstract
We investigate the critical dynamics of spin superflow in an easy-plane antiferromagnetic spinor Bose-Einstein condensate. Spin-dipole oscillations are induced in a trapped condensate by applying a linear magnetic field gradient and we observe that the damping rate increases rapidly as the field gradient increases above a certain critical value. The onset of dissipation is found to be associated with the generation of dark-bright solitons due to the modulation instability of the counterflow of two spin components. Spin turbulence emerges as the solitons decay because of their snake instability. We identify another critical point for spin superflow, in which transverse magnon excitations are dynamically generated via spin-exchanging collisions, which leads to the transient formation of axial polar spin domains.
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Affiliation(s)
- Joon Hyun Kim
- Department of Physics and Astronomy, and Institute of Applied Physics, Seoul National University, Seoul 08826, Korea
| | - Sang Won Seo
- Department of Physics and Astronomy, and Institute of Applied Physics, Seoul National University, Seoul 08826, Korea
- Center for Correlated Electron Systems, Institute for Basic Science, Seoul 08826, Korea
| | - Y Shin
- Department of Physics and Astronomy, and Institute of Applied Physics, Seoul National University, Seoul 08826, Korea
- Center for Correlated Electron Systems, Institute for Basic Science, Seoul 08826, Korea
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29
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Stability and Dynamics of Dark-Bright Soliton Bound States Away from the Integrable Limit. APPLIED SCIENCES-BASEL 2017. [DOI: 10.3390/app7040388] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/17/2022]
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30
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Wang W, Kevrekidis PG. Two-component dark-bright solitons in three-dimensional atomic Bose-Einstein condensates. Phys Rev E 2017; 95:032201. [PMID: 28415293 DOI: 10.1103/physreve.95.032201] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/19/2016] [Indexed: 06/07/2023]
Abstract
In the present work, we revisit two-component Bose-Einstein condensates in their fully three-dimensional (3D) form. Motivated by earlier studies of dark-bright solitons in the 1D case, we explore the stability of these structures in their fully 3D form in two variants. In one the dark soliton is planar and trapping a planar bright (disk) soliton. In the other case, a dark spherical shell soliton creates an effective potential in which a bright spherical shell of atoms is trapped in the second component. We identify these solutions as numerically exact states (up to a prescribed accuracy) and perform a Bogolyubov-de Gennes linearization analysis that illustrates that both structures can be dynamically stable in suitable intervals of sufficiently low chemical potentials. We corroborate this finding theoretically by analyzing the stability via degenerate perturbation theory near the linear limit of the system. When the solitary waves are found to be unstable, we explore their dynamical evolution via direct numerical simulations which, in turn, reveal wave forms that are more robust. Finally, using the SO(2) symmetry of the model, we produce multi-dark-bright planar or shell solitons involved in pairwise oscillatory motion.
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Affiliation(s)
- Wenlong Wang
- Department of Physics and Astronomy, Texas A&M University, College Station, Texas 77843-4242, USA
| | - P G Kevrekidis
- Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003-4515, USA
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31
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Charalampidis EG, Kevrekidis PG, Frantzeskakis DJ, Malomed BA. Vortex-soliton complexes in coupled nonlinear Schrödinger equations with unequal dispersion coefficients. Phys Rev E 2016; 94:022207. [PMID: 27627298 DOI: 10.1103/physreve.94.022207] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/23/2015] [Indexed: 11/07/2022]
Abstract
We consider a two-component, two-dimensional nonlinear Schrödinger system with unequal dispersion coefficients and self-defocusing nonlinearities, chiefly with equal strengths of the self- and cross-interactions. In this setting, a natural waveform with a nonvanishing background in one component is a vortex, which induces an effective potential well in the second component, via the nonlinear coupling of the two components. We show that the potential well may support not only the fundamental bound state, but also multiring excited radial state complexes for suitable ranges of values of the dispersion coefficient of the second component. We systematically explore the existence, stability, and nonlinear dynamics of these states. The complexes involving the excited radial states are weakly unstable, with a growth rate depending on the dispersion of the second component. Their evolution leads to transformation of the multiring complexes into stable vortex-bright solitons ones with the fundamental state in the second component. The excited states may be stabilized by a harmonic-oscillator trapping potential, as well as by unequal strengths of the self- and cross-repulsive nonlinearities.
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Affiliation(s)
- E G Charalampidis
- Department of Mathematics and Statistics, University of Massachusetts Amherst, Amherst, Massachusetts 01003-4515, USA
| | - P G Kevrekidis
- Department of Mathematics and Statistics, University of Massachusetts Amherst, Amherst, Massachusetts 01003-4515, USA
| | - D J Frantzeskakis
- Department of Physics, University of Athens, Panepistimiopolis, Zografos, Athens 15784, Greece
| | - B A Malomed
- Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel and Laboratory of Nonlinear-Optical Informatics, ITMO University, St. Petersburg 197101, Russia
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32
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Qu C, Pitaevskii LP, Stringari S. Magnetic Solitons in a Binary Bose-Einstein Condensate. PHYSICAL REVIEW LETTERS 2016; 116:160402. [PMID: 27152776 DOI: 10.1103/physrevlett.116.160402] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/19/2016] [Indexed: 06/05/2023]
Abstract
We study solitary waves of polarization (magnetic solitons) in a two-component Bose gas with slightly unequal repulsive intra- and interspin interactions. In experimentally relevant conditions we obtain an analytical solution which reveals that the width and the velocity of magnetic solitons are explicitly related to the spin healing length and the spin sound velocity of the Bose mixture, respectively. We calculate the profiles, the energy, and the effective mass of the solitons in the absence of external fields and investigate their oscillation in a harmonic trap where the oscillation period is calculated as a function of the oscillation amplitude. The stability of magnetic solitons in two dimensions and the conditions for their experimental observation are also briefly discussed.
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Affiliation(s)
- Chunlei Qu
- INO-CNR BEC Center and Dipartimento di Fisica, Università di Trento, 38123 Povo, Italy
| | - Lev P Pitaevskii
- INO-CNR BEC Center and Dipartimento di Fisica, Università di Trento, 38123 Povo, Italy
- Kapitza Institute for Physical Problems RAS, Kosygina 2, 119334 Moscow, Russia
| | - Sandro Stringari
- INO-CNR BEC Center and Dipartimento di Fisica, Università di Trento, 38123 Povo, Italy
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33
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Manikandan K, Muruganandam P, Senthilvelan M, Lakshmanan M. Manipulating localized matter waves in multicomponent Bose-Einstein condensates. Phys Rev E 2016; 93:032212. [PMID: 27078349 DOI: 10.1103/physreve.93.032212] [Citation(s) in RCA: 24] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/14/2015] [Indexed: 11/07/2022]
Abstract
We analyze vector localized solutions of two-component Bose-Einstein condensates (BECs) with variable nonlinearity parameters and external trap potentials through a similarity transformation technique which transforms the two coupled Gross-Pitaevskii equations into a pair of coupled nonlinear Schrödinger equations with constant coefficients under a specific integrability condition. In this analysis we consider three different types of external trap potentials: a time-independent trap, a time-dependent monotonic trap, and a time-dependent periodic trap. We point out the existence of different interesting localized structures; namely, rogue waves, dark- and bright-soliton rogue waves, and rogue-wave breatherlike structures for the above three cases of trap potentials. We show how the vector localized density profiles in a constant background get deformed when we tune the strength of the trap parameter. Furthermore, we investigate the nature of the trajectories of the nonautonomous rogue waves. We also construct the dark-dark rogue wave solution for the repulsive-repulsive interaction of two-component BECs and analyze the associated characteristics for the three different kinds of traps. We then deduce single-, two-, and three-composite rogue waves for three-component BECs and discuss the correlated characteristics when we tune the strength of the trap parameter for different trap potentials.
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Affiliation(s)
- K Manikandan
- Centre for Nonlinear Dynamics, Bharathidasan University, Tiruchirappalli 620024, Tamilnadu, India
| | - P Muruganandam
- Department of Physics, Bharathidasan University, Tiruchirappalli 620024, Tamilnadu, India
| | - M Senthilvelan
- Centre for Nonlinear Dynamics, Bharathidasan University, Tiruchirappalli 620024, Tamilnadu, India
| | - M Lakshmanan
- Centre for Nonlinear Dynamics, Bharathidasan University, Tiruchirappalli 620024, Tamilnadu, India
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Ling L, Zhao LC. Integrable pair-transition-coupled nonlinear Schrödinger equations. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:022924. [PMID: 26382492 DOI: 10.1103/physreve.92.022924] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/20/2015] [Indexed: 06/05/2023]
Abstract
We study integrable coupled nonlinear Schrödinger equations with pair particle transition between components. Based on exact solutions of the coupled model with attractive or repulsive interaction, we predict that some new dynamics of nonlinear excitations can exist, such as the striking transition dynamics of breathers, new excitation patterns for rogue waves, topological kink excitations, and other new stable excitation structures. In particular, we find that nonlinear wave solutions of this coupled system can be written as a linear superposition of solutions for the simplest scalar nonlinear Schrödinger equation. Possibilities to observe them are discussed in a cigar-shaped Bose-Einstein condensate with two hyperfine states. The results would enrich our knowledge on nonlinear excitations in many coupled nonlinear systems with transition coupling effects, such as multimode nonlinear fibers, coupled waveguides, and a multicomponent Bose-Einstein condensate system.
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Affiliation(s)
- Liming Ling
- School of Mathematics, South China University of Technology, 510640, Guangzhou, China
| | - Li-Chen Zhao
- Department of Physics, Northwest University, 710069, Xi'an, China
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Wang W, Kevrekidis PG. Transitions from order to disorder in multiple dark and multiple dark-bright soliton atomic clouds. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:032905. [PMID: 25871170 DOI: 10.1103/physreve.91.032905] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/22/2014] [Indexed: 06/04/2023]
Abstract
We have performed a systematic study quantifying the variation of solitary wave behavior from that of an ordered cloud resembling a "crystalline" configuration to that of a disordered state that can be characterized as a soliton "gas." As our illustrative examples, we use both one-component, as well as two-component, one-dimensional atomic gases very close to zero temperature, where in the presence of repulsive interatomic interactions and of a parabolic trap, a cloud of dark (dark-bright) solitons can form in the one- (two-) component system. We corroborate our findings through three distinct types of approaches, namely a Gross-Pitaevskii type of partial differential equation, particle-based ordinary differential equations describing the soliton dynamical system, and Monte Carlo simulations for the particle system. We define an "empirical" order parameter to characterize the order of the soliton lattices and study how this changes as a function of the strength of the "thermally" (i.e., kinetically) induced perturbations. As may be anticipated by the one-dimensional nature of our system, the transition from order to disorder is gradual without, apparently, a genuine phase transition ensuing in the intermediate regime.
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Affiliation(s)
- Wenlong Wang
- Department of Physics, University of Massachusetts, Amherst, Massachusetts 01003, USA
| | - P G Kevrekidis
- Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003-4515, USA and Center for Nonlinear Studies and Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87544, USA
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36
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Charalampidis EG, Kevrekidis PG, Frantzeskakis DJ, Malomed BA. Dark-bright solitons in coupled nonlinear Schrödinger equations with unequal dispersion coefficients. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:012924. [PMID: 25679698 DOI: 10.1103/physreve.91.012924] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/03/2014] [Indexed: 06/04/2023]
Abstract
We study a two-component nonlinear Schrödinger system with equal, repulsive cubic interactions and different dispersion coefficients in the two components. We consider states that have a dark solitary wave in one component. Treating it as a frozen one, we explore the possibility of the formation of bright-solitonic structures in the other component. We identify bifurcation points at which such states emerge in the bright component in the linear limit and explore their continuation into the nonlinear regime. An additional analytically tractable limit is found to be that of vanishing dispersion of the bright component. We numerically identify regimes of potential stability, not only of the single-peak ground state (the dark-bright soliton), but also of excited states with one or more zero crossings in the bright component. When the states are identified as unstable, direct numerical simulations are used to investigate the outcome of the instability development. Although our principal focus is on the homogeneous setting, we also briefly touch upon the counterintuitive impact of the potential presence of a parabolic trap on the states of interest.
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Affiliation(s)
- E G Charalampidis
- School of Civil Engineering, Faculty of Engineering, Aristotle University of Thessaloniki, Thessaloniki 54124, Greece and Department of Mathematics and Statistics, University of Massachusetts Amherst, Amherst, Massachusetts 01003-4515, USA
| | - P G Kevrekidis
- Department of Mathematics and Statistics, University of Massachusetts Amherst, Amherst, Massachusetts 01003-4515, USA
| | - D J Frantzeskakis
- Department of Physics, University of Athens, Panepistimiopolis, Zografos, Athens 15784, Greece
| | - B A Malomed
- Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel
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37
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Xu Y, Mao L, Wu B, Zhang C. Dark solitons with Majorana fermions in spin-orbit-coupled Fermi gases. PHYSICAL REVIEW LETTERS 2014; 113:130404. [PMID: 25302874 DOI: 10.1103/physrevlett.113.130404] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/14/2014] [Indexed: 06/04/2023]
Abstract
We show that a single dark soliton can exist in a spin-orbit-coupled Fermi gas with a high spin imbalance, where spin-orbit coupling favors uniform superfluids over nonuniform Fulde-Ferrell-Larkin-Ovchinnikov states, leading to dark soliton excitations in highly imbalanced gases. Above a critical spin imbalance, two topological Majorana fermions without interactions can coexist inside a dark soliton, paving a way for manipulating Majorana fermions through controlling solitons. At the topological transition point, the atom density contrast across the soliton suddenly vanishes, suggesting a signature for identifying topological solitons.
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Affiliation(s)
- Yong Xu
- Department of Physics, The University of Texas at Dallas, Richardson, Texas 75080, USA
| | - Li Mao
- Department of Physics, The University of Texas at Dallas, Richardson, Texas 75080, USA and School of Physics and Technology, Wuhan University, Wuhan 430072, China
| | - Biao Wu
- International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China and Collaborative Innovation Center of Quantum Matter, Beijing 100871, China
| | - Chuanwei Zhang
- Department of Physics, The University of Texas at Dallas, Richardson, Texas 75080, USA
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38
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Zhao LC, Xin GG, Yang ZY. Rogue-wave pattern transition induced by relative frequency. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:022918. [PMID: 25215810 DOI: 10.1103/physreve.90.022918] [Citation(s) in RCA: 16] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/17/2014] [Indexed: 06/03/2023]
Abstract
We revisit a rogue wave in a two-mode nonlinear fiber whose dynamics is described by two-component coupled nonlinear Schrödinger equations. The relative frequency between two modes can induce different rogue wave patterns transition. In particular, we find a four-petaled flower structure rogue wave can exist in the two-mode coupled system, which possesses an asymmetric spectrum distribution. Furthermore, spectrum analysis is performed on these different type rogue waves, and the spectrum relations between them are discussed. We demonstrate qualitatively that different modulation instability gain distribution can induce different rogue wave excitation patterns. These results would deepen our understanding of rogue wave dynamics in complex systems.
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Affiliation(s)
- Li-Chen Zhao
- Department of Physics, Northwest University, Xi'an 710069, China
| | - Guo-Guo Xin
- Department of Physics, Northwest University, Xi'an 710069, China
| | - Zhan-Ying Yang
- Department of Physics, Northwest University, Xi'an 710069, China
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39
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Ling L, Guo B, Zhao LC. High-order rogue waves in vector nonlinear Schrödinger equations. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:041201. [PMID: 24827185 DOI: 10.1103/physreve.89.041201] [Citation(s) in RCA: 19] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/18/2014] [Indexed: 06/03/2023]
Abstract
We study the dynamics of high-order rogue waves (RWs) in two-component coupled nonlinear Schrödinger equations. We find that four fundamental rogue waves can emerge from second-order vector RWs in the coupled system, in contrast to the high-order ones in single-component systems. The distribution shape can be quadrilateral, triangle, and line structures by varying the proper initial excitations given by the exact analytical solutions. The distribution pattern for vector RWs is more abundant than that for scalar rogue waves. Possibilities to observe these new patterns for rogue waves are discussed for a nonlinear fiber.
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Affiliation(s)
- Liming Ling
- Department of Mathematics, South China University of Technology, Guangzhou 510640, China
| | - Boling Guo
- Institute of Applied Physics and Computational Mathematics, Beijing 100088, China
| | - Li-Chen Zhao
- Department of Physics, Northeast University, Xi'an 710069, China
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40
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Hamner C, Zhang Y, Chang JJ, Zhang C, Engels P. Phase winding a two-component Bose-Einstein condensate in an elongated trap: experimental observation of moving magnetic orders and dark-bright solitons. PHYSICAL REVIEW LETTERS 2013; 111:264101. [PMID: 24483796 DOI: 10.1103/physrevlett.111.264101] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/25/2013] [Indexed: 06/03/2023]
Abstract
We investigate the phase winding dynamics of a harmonically trapped two-component BEC subject to inhomogeneous Rabi oscillations between two pseudospin components. While the single-particle dynamics can be explained by mapping the system to a two-component Bose-Hubbard model, nonlinearities due to the interatomic repulsion lead to new effects observed in the experiments: In the presence of a linear magnetic field gradient, a qualitatively stable moving magnetic order that is similar to antiferromagnetic order is observed after critical winding is achieved. We also demonstrate how the phase winding can be used to generate copious dark-bright solitons in a two-component BEC, opening the door for new experimental studies of these nonlinear features.
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Affiliation(s)
- C Hamner
- Department of Physics and Astronomy, Washington State University, Pullman, Washington 99164, USA
| | - Yongping Zhang
- Department of Physics and Astronomy, Washington State University, Pullman, Washington 99164, USA
| | - J J Chang
- Department of Physics and Astronomy, Washington State University, Pullman, Washington 99164, USA
| | - Chuanwei Zhang
- Department of Physics and Astronomy, Washington State University, Pullman, Washington 99164, USA
| | - P Engels
- Department of Physics and Astronomy, Washington State University, Pullman, Washington 99164, USA
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41
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Zhao LC, Liu J. Rogue-wave solutions of a three-component coupled nonlinear Schrödinger equation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 87:013201. [PMID: 23410451 DOI: 10.1103/physreve.87.013201] [Citation(s) in RCA: 35] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/07/2012] [Revised: 12/09/2012] [Indexed: 06/01/2023]
Abstract
We investigate rogue-wave solutions in a three-component coupled nonlinear Schrödinger equation. With certain requirements on the backgrounds of components, we construct a multi-rogue-wave solution that exhibits a structure like a four-petaled flower in temporal-spatial distribution, in contrast to the eye-shaped structure in one-component or two-component systems. The results could be of interest in such diverse fields as Bose-Einstein condensates, nonlinear fibers, and superfluids.
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Affiliation(s)
- Li-Chen Zhao
- Science and Technology Computation Physics Laboratory, Institute of Applied Physics and Computational Mathematics, Beijing 100088, China
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42
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Sabbatini J, Zurek WH, Davis MJ. Phase separation and pattern formation in a binary Bose-Einstein condensate. PHYSICAL REVIEW LETTERS 2011; 107:230402. [PMID: 22182069 DOI: 10.1103/physrevlett.107.230402] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/29/2011] [Revised: 10/09/2011] [Indexed: 05/31/2023]
Abstract
The miscibility-immiscibility phase transition in binary Bose-Einstein condensates (BECs) can be controlled by a coupling between the two components. Here we propose a new scheme that uses coupling-induced pattern formation to test the Kibble-Zurek mechanism (KZM) of topological-defect formation in a quantum phase transition. For a binary BEC in a ring trap we find that the number of domains forming the pattern scales as a function of the coupling quench rate with an exponent as predicted by the KZM. For a binary BEC in an elongated harmonic trap we find a different scaling law due to the transition being spatially inhomogeneous. We perform a "quantum simulation" of the harmonically trapped system in a ring trap to verify the scaling exponent.
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Affiliation(s)
- Jacopo Sabbatini
- The University of Queensland, School of Mathematics and Physics, Queensland 4072, Australia.
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