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Johnson TH, Yuan Y, Bao W, Clark SR, Foot C, Jaksch D. Hubbard Model for Atomic Impurities Bound by the Vortex Lattice of a Rotating Bose-Einstein Condensate. PHYSICAL REVIEW LETTERS 2016; 116:240402. [PMID: 27367366 DOI: 10.1103/physrevlett.116.240402] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/01/2016] [Indexed: 06/06/2023]
Abstract
We investigate cold bosonic impurity atoms trapped in a vortex lattice formed by condensed bosons of another species. We describe the dynamics of the impurities by a bosonic Hubbard model containing occupation-dependent parameters to capture the effects of strong impurity-impurity interactions. These include both a repulsive direct interaction and an attractive effective interaction mediated by the Bose-Einstein condensate. The occupation dependence of these two competing interactions drastically affects the Hubbard model phase diagram, including causing the disappearance of some Mott lobes.
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Affiliation(s)
- T H Johnson
- Centre for Quantum Technologies, National University of Singapore, 117543 Singapore
- Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, United Kingdom
- Keble College, University of Oxford, Parks Road, Oxford OX1 3PG, United Kingdom
| | - Y Yuan
- Beijing Computational Science Research Center, Beijing 100094, China
- Department of Mathematics, National University of Singapore, 119076 Singapore
- College of Mathematics and Computer Science, Synthetic Innovation Center for Quantum Effects and Applications, Hunan Normal University, Changsha, Hunan Province 410081, China
| | - W Bao
- Department of Mathematics, National University of Singapore, 119076 Singapore
| | - S R Clark
- Keble College, University of Oxford, Parks Road, Oxford OX1 3PG, United Kingdom
- Department of Physics, University of Bath, Claverton Down, Bath BA2 7AY, United Kingdom
| | - C Foot
- Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, United Kingdom
| | - D Jaksch
- Centre for Quantum Technologies, National University of Singapore, 117543 Singapore
- Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, United Kingdom
- Keble College, University of Oxford, Parks Road, Oxford OX1 3PG, United Kingdom
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Adhikari SK. Stable spatial and spatiotemporal optical soliton in the core of an optical vortex. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:042926. [PMID: 26565323 DOI: 10.1103/physreve.92.042926] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/07/2015] [Indexed: 06/05/2023]
Abstract
We demonstrate a robust, stable, mobile, two-dimensional (2D) spatial and three-dimensional (3D) spatiotemporal optical soliton in the core of an optical vortex, while all nonlinearities are of the cubic (Kerr) type. The 3D soliton can propagate with a constant velocity along the vortex core without any deformation. Stability of the soliton under a small perturbation is established numerically. Two such solitons moving along the vortex core can undergo a quasielastic collision at medium velocities. Possibilities of forming such a 2D spatial soliton in the core of a vortical beam are discussed.
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Affiliation(s)
- S K Adhikari
- Instituto de Física Teórica, Universidade Estadual Paulista, 01.140-070 São Paulo, São Paulo, Brazil
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Mallory K, Van Gorder RA. Stationary solutions for the nonlinear Schrödinger equation modeling three-dimensional spherical Bose-Einstein condensates in general potentials. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:013201. [PMID: 26274295 DOI: 10.1103/physreve.92.013201] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/04/2014] [Indexed: 06/04/2023]
Abstract
Stationary solutions for the cubic nonlinear Schrödinger equation modeling Bose-Einstein condensates (BECs) confined in three spatial dimensions by general forms of a potential are studied through a perturbation method and also numerically. Note that we study both repulsive and attractive BECs under similar frameworks in order to deduce the effects of the potentials in each case. After outlining the general framework, solutions for a collection of specific confining potentials of physical relevance to experiments on BECs are provided in order to demonstrate the approach. We make several observations regarding the influence of the particular potentials on the behavior of the BECs in these cases, comparing and contrasting the qualitative behavior of the attractive and repulsive BECs for potentials of various strengths and forms. Finally, we consider the nonperturbative where the potential or the amplitude of the solutions is large, obtaining various qualitative results. When the kinetic energy term is small (relative to the nonlinearity and the confining potential), we recover the expected Thomas-Fermi approximation for the stationary solutions. Naturally, this also occurs in the large mass limit. Through all of these results, we are able to understand the qualitative behavior of spherical three-dimensional BECs in weak, intermediate, or strong confining potentials.
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Affiliation(s)
- Kristina Mallory
- Division of Applied Mathematics, Brown University, 182 George Street, Providence, Rhode Island 02912, USA
| | - Robert A Van Gorder
- Oxford Centre for Industrial and Applied Mathematics, Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, United Kingdom
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Berman OL, Kezerashvili RY, Kolmakov GV, Pomirchi LM. Spontaneous formation and nonequilibrium dynamics of a soliton-shaped Bose-Einstein condensate in a trap. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:062901. [PMID: 26172766 DOI: 10.1103/physreve.91.062901] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/10/2015] [Indexed: 06/04/2023]
Abstract
The Bose-stimulated self-organization of a quasi-two-dimensional nonequilibrium Bose-Einstein condensate in an in-plane potential is proposed. We obtained the solution of the nonlinear, driven-dissipative Gross-Pitaevskii equation for a Bose-Einstein condensate trapped in an external asymmetric parabolic potential within the method of the spectral expansion. We found that, in sharp contrast to previous observations, the condensate can spontaneously acquire a solitonlike shape for spatially homogeneous pumping. This condensate soliton performs oscillatory motion in a parabolic trap and, also, can spontaneously rotate. Stability of the condensate soliton in the spatially asymmetric trap is analyzed. In addition to the nonlinear dynamics of nonequilibrium Bose-Einstein condensates of ultracold atoms, our findings can be applied to the condensates of quantum well excitons and cavity polaritons in semiconductor heterostructure, and to the condensates of photons.
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Affiliation(s)
- Oleg L Berman
- Physics Department, New York City College of Technology, The City University of New York, Brooklyn, New York 11201, USA
| | - Roman Ya Kezerashvili
- Physics Department, New York City College of Technology, The City University of New York, Brooklyn, New York 11201, USA
| | - German V Kolmakov
- Physics Department, New York City College of Technology, The City University of New York, Brooklyn, New York 11201, USA
| | - Leonid M Pomirchi
- Physics Department, New York City College of Technology, The City University of New York, Brooklyn, New York 11201, USA
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Mallory K, Van Gorder RA. Stationary solutions for the 2+1 nonlinear Schrödinger equation modeling Bose-Einstein condensates in radial potentials. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:023201. [PMID: 25215837 DOI: 10.1103/physreve.90.023201] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/06/2014] [Indexed: 06/03/2023]
Abstract
Stationary solutions for the 2+1 cubic nonlinear Schrödinger equation modeling Bose-Einstein condensates (BEC) in a small potential are obtained via a form of perturbation. In particular, perturbations due to small potentials which either confine or repel the BECs are studied, and under arbitrary piecewise continuous potentials, we obtain the general representation for the perturbation theory of radial BEC solutions. Numerical results are also provided for regimes where perturbative results break down (i.e., the large-potential regime). Both repulsive and attractive BECs are considered under this framework. Solutions for many specific confining potentials of physical relevance to experiments on BECs are provided in order to demonstrate the approach. We make several observations regarding the influence of the particular small potentials on the behavior of the BECs.
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Affiliation(s)
- Kristina Mallory
- Department of Mathematics, University of Central Florida, Orlando, Florida 32816-1364, USA
| | - Robert A Van Gorder
- Department of Mathematics, University of Central Florida, Orlando, Florida 32816-1364, USA
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Mallory K, Van Gorder RA. Stationary solutions for the 1+1 nonlinear Schrödinger equation modeling attractive Bose-Einstein condensates in small potentials. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:013204. [PMID: 24580353 DOI: 10.1103/physreve.89.013204] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/28/2013] [Indexed: 06/03/2023]
Abstract
Stationary solutions for the 1+1 cubic nonlinear Schrödinger equation (NLS) modeling attractive Bose-Einstein condensates (BECs) in a small potential are obtained via a form of nonlinear perturbation. The focus here is on perturbations to the bright soliton solutions due to small potentials which either confine or repel the BECs: under arbitrary piecewise continuous potentials, we obtain the general representation for the perturbation theory of the bright solitons. Importantly, we do not need to assume that the nonlinearity is small, as we perform a sort of nonlinear perturbation by allowing the zeroth-order perturbation term to be governed by a nonlinear equation. This is useful, in that it allows us to consider perturbations of bright solitons of arbitrary size. In some cases, exact solutions can be recovered, and these agree with known results from the literature. Several special cases are considered which involve confining potentials of specific relevance to BECs. We make several observations on the influence of the small potentials on the behavior of the perturbed bright solitons. The results demonstrate the difference between perturbed bright solitons in the attractive NLS and those results found in the repulsive NLS for dark solitons, as discussed by Mallory and Van Gorder, [Phys. Rev. E 88 013205 (2013)]. Extension of these results to more spatial dimensions is mentioned.
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Affiliation(s)
- Kristina Mallory
- Department of Mathematics, University of Central Florida, Orlando, Florida 32816-1364, USA
| | - Robert A Van Gorder
- Department of Mathematics, University of Central Florida, Orlando, Florida 32816-1364, USA
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Mallory K, Van Gorder RA. Stationary solutions for the 1+1 nonlinear Schrödinger equation modeling repulsive Bose-Einstein condensates in small potentials. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:013205. [PMID: 23944574 DOI: 10.1103/physreve.88.013205] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/09/2013] [Indexed: 06/02/2023]
Abstract
Stationary solutions for the 1+1 cubic nonlinear Schrödinger equation modeling repulsive Bose-Einstein condensates (BEC) in a small potential are obtained through a form of nonlinear perturbation. In particular, for sufficiently small potentials, we determine the perturbation theory of stationary solutions, by use of an expansion in Jacobi elliptic functions. This idea was explored before in order to obtain exact solutions [Bronski, Carr, Deconinck, and Kutz, Phys. Rev. Lett. 86, 1402 (2001)], where the potential itself was fixed to be a Jacobi elliptic function, thereby reducing the nonlinear ODE into an algebraic equation, (which could be easily solved). However, in the present paper, we outline the perturbation method for completely general potentials, assuming only that such potentials are locally small. We do not need to assume that the nonlinearity is small, as we perform a sort of nonlinear perturbation by allowing the zeroth-order perturbation term to be governed by a nonlinear equation. This allows us to consider even poorly behaved potentials, so long as they are bounded locally. We demonstrate the effectiveness of this approach by considering a number of specific potentials: for the simplest potentials, and we recover results from the literature, while for more complicated potentials, our results are new. Dark soliton solutions are constructed explicitly for some cases, and we obtain the known one-soliton tanh-type solution in the simplest setting for the repulsive BEC. Note that we limit our results to the repulsive case; similar results can be obtained for the attractive BEC case.
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Affiliation(s)
- Kristina Mallory
- Department of Mathematics, University of Central Florida, Orlando, FL 32816-1364, USA
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Zhang J, Xu Z, Wu X. Unified approach to split absorbing boundary conditions for nonlinear Schrödinger equations: Two-dimensional case. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 79:046711. [PMID: 19518384 DOI: 10.1103/physreve.79.046711] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/07/2009] [Indexed: 05/27/2023]
Abstract
This paper aims to design local absorbing boundary conditions (LABCs) for the two-dimensional nonlinear Schrödinger equations on a rectangle by extending the unified approach. Based on the time-splitting idea, the main process of the unified approach is to approximate the kinetic energy part by a one-way equation, unite it with the potential energy equation, and then obtain the well-posed and accurate LABCs on the artificial boundaries. In the corners, we use the (1,1)-Padé approximation to the kinetic term and also unite it with the nonlinear term to give some local corner boundary conditions. Numerical tests are given to verify the stable and tractable advantages of the method.
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Affiliation(s)
- Jiwei Zhang
- Department of Mathematics, Hong Kong Baptist University, Kowloon, Hong Kong, People's Republic of China.
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Zhang J, Xu Z, Wu X. Unified approach to split absorbing boundary conditions for nonlinear Schrödinger equations. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 78:026709. [PMID: 18850975 DOI: 10.1103/physreve.78.026709] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/14/2008] [Indexed: 05/26/2023]
Abstract
An efficient method is proposed for numerical solutions of nonlinear Schrödinger equations on an unbounded domain. Through approximating the kinetic energy term by a one-way equation and uniting it with the potential energy equation, absorbing boundary conditions are designed to truncate the unbounded domain, which are in nonlinear form and can perfectly absorb waves outgoing from the boundaries of the truncated computational domain. The stability of the induced initial boundary value problem defined on the computational domain is examined by a normal mode analysis. Numerical examples are given to illustrate the stable and tractable advantages of the method.
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Affiliation(s)
- Jiwei Zhang
- Department of Mathematics, Hong Kong Baptist University, Kowloon, Hong Kong, People's Republic of China.
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Palpacelli S, Succi S. Quantum lattice Boltzmann simulation of expanding Bose-Einstein condensates in random potentials. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 77:066708. [PMID: 18643398 DOI: 10.1103/physreve.77.066708] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/18/2008] [Indexed: 05/26/2023]
Abstract
The phenomenon of Anderson localization in expanding one-dimensional Bose-Einstein condensates is investigated by numerically solving the Gross-Pitaevskii equation with a random speckle potential. To this purpose, a quantum lattice Boltzmann (QLB) method is used, and compared with a standard Crank-Nicolson scheme. The QLB simulations show evidence of Anderson localization even for relatively low-energy condensates, with a healing length as large as one-tenth of the Thomas-Fermi length. Moreover, very long-time simulations, lasting up to 15 000 optical confinement periods, indicate that the Anderson localization degrades in time, although at a very slow pace. In particular, the inverse localization length is found to decay according to a t;{-1/3} law. This lends support to the idea that localized wave functions, although not strictly ground states, represent extremely long-lived metastable states of the expanding condensate.
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Affiliation(s)
- S Palpacelli
- Dipartimento di Matematica, Università Roma Tre, Largo San Leonardo Murialdo 1, 00146 Roma, Italy
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Palpacelli S, Succi S, Spigler R. Ground-state computation of Bose-Einstein condensates by an imaginary-time quantum lattice Boltzmann scheme. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 76:036712. [PMID: 17930366 DOI: 10.1103/physreve.76.036712] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/05/2007] [Revised: 09/03/2007] [Indexed: 05/25/2023]
Abstract
The multidimensional formulation of the quantum lattice Boltzmann (qLB) scheme is extended to the case of nonlinear quantum wave equations. More specifically, imaginary-time formulations of the qLB scheme are developed and applied to the numerical computation of the ground state of the Gross-Pitaevskii equation in one and two spatial dimensions. The calculation is validated through detailed comparison with other numerical methods, as well as with analytical results based on the Thomas-Fermi approximation. The linear scaling of the time-step size with the spatial mesh spacing, a distinctive feature of the present quantum kinetic approach, is also numerically demonstrated.
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Affiliation(s)
- S Palpacelli
- Dipartimento di Matematica, Università Roma Tre, Largo San Leonardo Murialdo 1, 00146, Roma, Italy
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Xu Z, Han H. Absorbing boundary conditions for nonlinear Schrödinger equations. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 74:037704. [PMID: 17025792 DOI: 10.1103/physreve.74.037704] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/24/2006] [Indexed: 05/12/2023]
Abstract
A local time-splitting method (LTSM) is developed to design absorbing boundary conditions for numerical solutions of time-dependent nonlinear Schrödinger equations associated with open boundaries. These boundary conditions are significant for numerical simulations of propagations of nonlinear waves in physical applications, such as nonlinear fiber optics and Bose-Einstein condensations. Numerical examples are implemented to demonstrate the attractive features of using the LTSM.
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Affiliation(s)
- Zhenli Xu
- Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, China.
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Dion CM, Cancès E. Spectral method for the time-dependent Gross-Pitaevskii equation with a harmonic trap. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 67:046706. [PMID: 12786528 DOI: 10.1103/physreve.67.046706] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/05/2002] [Indexed: 05/24/2023]
Abstract
We study the numerical resolution of the time-dependent Gross-Pitaevskii equation, a nonlinear Schrödinger equation used to simulate the dynamics of Bose-Einstein condensates. Considering condensates trapped in harmonic potentials, we present an efficient algorithm by making use of a spectral-Galerkin method, using a basis set of harmonic-oscillator functions, and the Gauss-Hermite quadrature. We apply this algorithm to the simulation of condensate breathing and scissor modes.
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Affiliation(s)
- Claude M Dion
- CERMICS, Ecole Nationale des Ponts et Chaussées, 6 & 8 avenue Blaise Pascal, Cité Descartes, Champs-sur-Marne, 77455 Marne-la-Vallée, France
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Adhikari SK. Numerical study of the coupled time-dependent Gross-Pitaevskii equation: application to Bose-Einstein condensation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 63:056704. [PMID: 11415042 DOI: 10.1103/physreve.63.056704] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/20/2000] [Indexed: 05/23/2023]
Abstract
We present a numerical study of the coupled time-dependent Gross-Pitaevskii equation, which describes the Bose-Einstein condensate of several types of trapped bosons at ultralow temperature with both attractive and repulsive interatomic interactions. The same approach is used to study both stationary and time-evolution problems. We consider up to four types of atoms in the study of stationary problems. We consider the time-evolution problems where the frequencies of the traps or the atomic scattering lengths are suddenly changed in a stable preformed condensate. We also study the effect of periodically varying these frequencies or scattering lengths on a preformed condensate. These changes introduce oscillations in the condensate, which are studied in detail. Good convergence is obtained in all cases studied.
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Affiliation(s)
- S K Adhikari
- Instituto de Física Teórica, Universidade Estadual Paulista, 01.405-900 São Paulo, São Paulo, Brazil
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