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## Mathematical models and numerical methods for Bose-Einstein condensation (2014)

Citations: | 1 - 1 self |

### Citations

445 |
BoseEinstein Condensation
- Pitaevskii, Stringari
- 2003
(Show Context)
Citation Context ...46, 58, 59, 65, 66, 67]: i~∂tψ = δE(ψ) δψ = [ − ~ 2 2m ∇2 + V (x) +Ng|ψ|2 ] ψ, x ∈ R3, t > 0. (2.8) In most BEC experiments, the trapping potential has been taken as the harmonic oscillator potential =-=[3, 7, 28, 39, 67]-=- V (x) = m 2 ( ω2xx 2 + ω2yy 2 + ω2zz 2 ) , x = (x, y, z)T ∈ R3, (2.9) where ωx, ωy and ωz are the trap frequencies in x-, y- and z-direction, respectively. Without loss of generality, we assume that ... |

349 | Observation of Bose-Einstein condensation in a dilute atomic vapor
- Anderson, Ensher, et al.
- 1995
(Show Context)
Citation Context ...ion, Gross-Pitaevskii equation, nonlinear Schrödinger equation, ground state, dynamics, numerical methods. 1. Introduction The achievement of Bose-Einstein condensation (BEC) of dilute gases in 1995 =-=[3, 28, 39]-=- marked the beginning of a new era in atomic, molecular and optical (AMO) physics and quantum optics. In fact, the phenomenon known as BEC was predicted by Einstein in 1924 [40, 41] based on the ideas... |

325 |
The Nonlinear Schrödinger Equation: Self-Focusing and
- Sulem, Sulem
- 1999
(Show Context)
Citation Context ...ii) E(ψ0) > 0 and Im (∫ Rd ψ0(x) (x · ∇ψ0(x)) dx ) < − √ E(ψ0)d ‖xψ0‖L2. If there is no external potential in the GPE (2.25), i.e. V (x) ≡ 0, then the momentum and angular momentum are also conserved =-=[4, 7, 70]-=-. The GPE (2.25) admits the plane wave solution as ψ(x, t) = Aei(k·x−ωt), where the time frequency ω, amplitude A and spatial wave number k satisfy the following dispersion relation [4, 7, 70]: ω = |k... |

301 |
Theory of Bose-Einstein condensation in trapped gases
- Dalfovo, Giorgini, et al.
- 1999
(Show Context)
Citation Context ... as long as several minutes [37, 52]. For better understanding of the long history towards the BEC and its physical study, we refer to the Nobel lectures [37, 52] and several review papers in physics =-=[38, 56, 65, 67]-=-. The experimental advances in BEC [3, 28, 39] have spurred great excitement in the AMO community and condense matter community as well as computational and applied mathematics community. Since 1995, ... |

277 | Bose-Einstein Condensation in a Gas of Sodium Atoms
- Davis, Mewes, et al.
- 1995
(Show Context)
Citation Context ...ion, Gross-Pitaevskii equation, nonlinear Schrödinger equation, ground state, dynamics, numerical methods. 1. Introduction The achievement of Bose-Einstein condensation (BEC) of dilute gases in 1995 =-=[3, 28, 39]-=- marked the beginning of a new era in atomic, molecular and optical (AMO) physics and quantum optics. In fact, the phenomenon known as BEC was predicted by Einstein in 1924 [40, 41] based on the ideas... |

235 |
Bose-Einstein Condensation in Dilute Gases
- Pethick, Smith
- 2002
(Show Context)
Citation Context ...h two components can be well described by the macroscopic wave function Ψ := Ψ(x, t) = (ψ1(x, t), ψ2(x, t)) T whose evolution is governed by the following 3D coupled GrossPitaevskii equations (CGPEs) =-=[5, 7, 8, 48, 60, 65, 67, 73]-=- for x ∈ R3 and t > 0 as i~∂tψ1 = [ − ~ 2 2m ∇2 + V (x) + i~k̃0 2m ∂x + ~δ̃ 2 +Ng11|ψ1|2 +Ng12|ψ2|2 ] ψ1 + ~Ω̃ 2 ψ2, i~∂tψ2 = [ − ~ 2 2m ∇2 + V (x)− i~k̃0 2m ∂x − ~δ̃ 2 +Ng21|ψ1|2 +Ng22|ψ2|2 ] ψ2 + ~Ω... |

229 | Homogenization limits and Wigner transforms - Gérad, Markowich, et al. - 1997 |

172 |
Many-body physics with ultracold gases
- Bloch, Dalibard, et al.
- 2008
(Show Context)
Citation Context ...ality, we assume that ωx ≤ ωy ≤ ωz throughout the paper. For other trapping potentials used in BEC experiments, such as box potential, double-well potential and optical lattice potential, we refer to =-=[7, 26, 65, 66, 67, 68]-=- and references therein. The derivation of the GPE (2.8) from the linear Schrödinger equation (2.3) for a BEC (or a system of N identical particles) based on mean field approximation – dimension redu... |

161 |
Evidence of Bose-Einstein condensation in an atomic gas with attractive interactions, Phys
- Bradley, Sackett, et al.
- 1995
(Show Context)
Citation Context ...ion, Gross-Pitaevskii equation, nonlinear Schrödinger equation, ground state, dynamics, numerical methods. 1. Introduction The achievement of Bose-Einstein condensation (BEC) of dilute gases in 1995 =-=[3, 28, 39]-=- marked the beginning of a new era in atomic, molecular and optical (AMO) physics and quantum optics. In fact, the phenomenon known as BEC was predicted by Einstein in 1924 [40, 41] based on the ideas... |

139 |
Structure of a quantized vortex in boson systems
- Gross
- 1961
(Show Context)
Citation Context ...ng and guiding the experiments. The goal of this paper is to offer a short survey on mathematical models and theories as well as numerical methods for BEC based on the Gross-Pitaevskii equation (GPE) =-=[7, 46, 65, 66, 67]-=-. The paper is organized as follows. In section 2, we present the GPE for BEC based on the mean field approximation. Ground Models and Methods for Bose-Einstein Condensation 3 states and their computa... |

139 | Bosons in a trap: A rigorous derivation of the Gross-Pitaevskii energy functional, Phys
- Lieb, Seiringer, et al.
- 2000
(Show Context)
Citation Context ...l(ΨN ) δΨN = HNΨN (x1, . . . ,xN , t), (2.3) where i = √−1 denotes the imaginary unit and t is time. For a BEC, all particles are in the same quantum state and we can formally take the Hartree ansatz =-=[7, 42, 46, 58, 59, 65, 66, 67]-=- ΨN (x1, . . . ,xN , t) ≈ N∏ j=1 ψ(xj , t), (2.4) with the normalization for the single-particle wave function ψ := ψ(x, t) as ‖ψ(·, t)‖2 := ∫ R3 |ψ(x, t)|2 dx = 1, (2.5) 4 Weizhu Bao where x = (x, y,... |

111 |
Plancks Gesetz und Lichtquantenhypothese
- Bose
- 1924
(Show Context)
Citation Context ...he beginning of a new era in atomic, molecular and optical (AMO) physics and quantum optics. In fact, the phenomenon known as BEC was predicted by Einstein in 1924 [40, 41] based on the ideas of Bose =-=[27]-=- concerning photons: In a system of bosons obeying Bose statistics under the assumption that it is in equilibrium at temperature T and chemical potential µ, Einstein [40, 41] derived the so-called Bos... |

111 |
Quantentheorie des einatomigen idealen Gases. Zweite Abhandlung
- Einstein
- 1925
(Show Context)
Citation Context ...e gases in 1995 [3, 28, 39] marked the beginning of a new era in atomic, molecular and optical (AMO) physics and quantum optics. In fact, the phenomenon known as BEC was predicted by Einstein in 1924 =-=[40, 41]-=- based on the ideas of Bose [27] concerning photons: In a system of bosons obeying Bose statistics under the assumption that it is in equilibrium at temperature T and chemical potential µ, Einstein [4... |

108 | Numerical solution of the Gross-Pitaevskii equation for Bose-Einstein condensation
- Bao, Jaksch, et al.
(Show Context)
Citation Context ...2 := ∫ R |χ(z̃)|2 dz̃ = 1 admits distinct orthonormal eigenfunctions χk(z̃) with corresponding eigenvalues µk for k = 0, 1, . . . In fact, they form an orthonormal basis of L2(R) and can be chosen as =-=[7, 14, 25, 65, 66, 67]-=- µk = k + 1 2 , χk(z̃) = 1 pi1/4 √ 2k k! e−z̃ 2/2Hk(z̃), z̃ ∈ R, k = 0, 1, 2, . . . , (2.19) with Hk(z̃) the standard Hermite polynomial of degree k. Thus (χ ε k(z), µ ε k) for k ≥ 0 are orthonormal e... |

92 |
Bose-Einstein condensation in the alkali gases: Some fundamental concepts
- Leggett
- 2001
(Show Context)
Citation Context ... as long as several minutes [37, 52]. For better understanding of the long history towards the BEC and its physical study, we refer to the Nobel lectures [37, 52] and several review papers in physics =-=[38, 56, 65, 67]-=-. The experimental advances in BEC [3, 28, 39] have spurred great excitement in the AMO community and condense matter community as well as computational and applied mathematics community. Since 1995, ... |

85 | Computing the ground state solution of Bose-Einstein condensates by a normalized gradient flow
- Bao, Du
- 2004
(Show Context)
Citation Context ...action energy Eint(ψ) defined as Ekin(ψ) = ∫ Rd 1 2 |∇ψ|2dx, Eint(ψ) = ∫ Rd β 2 |ψ|4dx, Epot(ψ) = ∫ Rd V (x)|ψ|2dx. 3. Ground states To find the stationary state of the GPE (2.25) for a BEC, we write =-=[7, 12, 65, 66, 67]-=- ψ(x, t) = φ(x) e−iµt, x ∈ Rd, t ≥ 0, (3.1) where µ is the chemical potential of the condensate and φ(x) is a function independent of time. Substituting (3.1) into (2.25) gives the following for (µ, φ... |

85 | Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate
- Erdős, Schlein, et al.
- 2010
(Show Context)
Citation Context ...l(ΨN ) δΨN = HNΨN (x1, . . . ,xN , t), (2.3) where i = √−1 denotes the imaginary unit and t is time. For a BEC, all particles are in the same quantum state and we can formally take the Hartree ansatz =-=[7, 42, 46, 58, 59, 65, 66, 67]-=- ΨN (x1, . . . ,xN , t) ≈ N∏ j=1 ψ(xj , t), (2.4) with the normalization for the single-particle wave function ψ := ψ(x, t) as ‖ψ(·, t)‖2 := ∫ R3 |ψ(x, t)|2 dx = 1, (2.5) 4 Weizhu Bao where x = (x, y,... |

56 | states and dynamics of multicomponent Bose-Einstein condensates, Multiscale Model
- Bao, Ground
(Show Context)
Citation Context ...h two components can be well described by the macroscopic wave function Ψ := Ψ(x, t) = (ψ1(x, t), ψ2(x, t)) T whose evolution is governed by the following 3D coupled GrossPitaevskii equations (CGPEs) =-=[5, 7, 8, 48, 60, 65, 67, 73]-=- for x ∈ R3 and t > 0 as i~∂tψ1 = [ − ~ 2 2m ∇2 + V (x) + i~k̃0 2m ∂x + ~δ̃ 2 +Ng11|ψ1|2 +Ng12|ψ2|2 ] ψ1 + ~Ω̃ 2 ψ2, i~∂tψ2 = [ − ~ 2 2m ∇2 + V (x)− i~k̃0 2m ∂x − ~δ̃ 2 +Ng21|ψ1|2 +Ng22|ψ2|2 ] ψ2 + ~Ω... |

55 | Theory of ultracold atomic fermi gases
- Giorgini, Pitaevskii, et al.
(Show Context)
Citation Context ...of the system. For describing the dynamics of a BEC, it is natural to consider the linearized behavior of small perturbations around its ground state φg with chemical potential µg and take the ansatz =-=[38, 43, 45, 67]-=- ψ(x, t) = e−iµet [ φg(x) + u(x)e −iωt − v(x)eiωt ] , x ∈ Rd, t > 0, (4.10) where the Bogoliubov amplitudes u(x) and v(x) are treated as small and ω ∈ C to be determined. Substituting (4.10) into (2.2... |

53 |
Nobel lecture: BoseEinstein condensation in a dilute gas, the first 70 years and some recent experiments, Rev. Mod. Phys
- Cornell, Wieman
- 2002
(Show Context)
Citation Context ...h a diameter of 10–15 µm, or cigar-shaped with about 15 µm in diameter and 300 µm in length. The full cooling cycle that produce a condensate may take from a few seconds to as long as several minutes =-=[37, 52]-=-. For better understanding of the long history towards the BEC and its physical study, we refer to the Nobel lectures [37, 52] and several review papers in physics [38, 56, 65, 67]. The experimental a... |

53 | On the uniqueness of solutions to the GrossPitaevskii hierarchy
- Klainerman, Machedon
(Show Context)
Citation Context ...; and for dynamics, Yau et al. [42] established the validity of the GPE (2.8) as an approximation for (2.3), which inspired great interests in the study on dynamics for such many body system recently =-=[35, 36, 54]-=-. The above GPE (2.8) is a very simple equation, which is very convenient for mathematical analysis and numerical calculations, and in the case of the BEC alkali gases, appears to give a rather good q... |

51 | Vortices in a Bose-Einstein Condensate
- Matthews, Anderson, et al.
- 1999
(Show Context)
Citation Context ...with an internal atomic Josephon junction (JJ) and an spin-orbit coupling term. 5.1. For rotating BEC. At temperatures T much smaller than the critical temperature Tc, following the mean field theory =-=[1, 2, 7, 31, 43, 57, 62, 69]-=-, a BEC in the rotational frame is well described by the macroscopic wave function ψ := ψ(x, t), whose evolution is governed by the GPE with an angular momentum rotation term i~∂tψ = [ − ~ 2 2m ∇2 + V... |

48 | Semi-Classical Analysis for Nonlinear Schrödinger Equations, World Scientific
- Carles
- 2008
(Show Context)
Citation Context ...of the wave function, respectively, inserting it into the GPE (4.13) and separating real and imaginary parts, we obtain the transport and Hamilton-Jacobi equations for density and phase, respectively =-=[7, 32, 44]-=- ∂tρ ε + div (ρε ∇Sε) = 0, x ∈ Rd, t > 0, ∂tS ε + 1 2 |∇Sε|2 + ρε + V (x) = ε 2 2 1√ ρε ∆ √ ρε. (4.17) Furthermore, defining the quantum velocity uε = ∇Sε and current Jε = ρε uε, we get from (4.17) th... |

47 | Derivation of the Gross-Pitaevskii equation for rotating Bose gases
- Lieb, Seiringer
(Show Context)
Citation Context ...with an internal atomic Josephon junction (JJ) and an spin-orbit coupling term. 5.1. For rotating BEC. At temperatures T much smaller than the critical temperature Tc, following the mean field theory =-=[1, 2, 7, 31, 43, 57, 62, 69]-=-, a BEC in the rotational frame is well described by the macroscopic wave function ψ := ψ(x, t), whose evolution is governed by the GPE with an angular momentum rotation term i~∂tψ = [ − ~ 2 2m ∇2 + V... |

44 |
Efficient and spectrally accurate numerical methods for computing ground and first excited states in Bose-Einstein condensates
- Bao, Chern, et al.
(Show Context)
Citation Context ...ng on β such that the truncation error can be negligible! 3.3. Numerical methods. Various numerical methods for computing the ground state φg in (3.5) have been proposed and studied in the literature =-=[7, 11, 12, 22, 34, 64]-=-. Among them, one of the most efficient and simple methods is the following gradient flow with discrete normalization (GFDN) [7, 12]. Choose a time step τ := ∆t > 0 and denote time steps as tn = nτ fo... |

44 | A fourth-order time-splitting Laguerre-Hermite pseudospectral method for BoseEinstein condensates
- Bao, Shen
(Show Context)
Citation Context ...), ġ(t) = V (xc(t)) = 1 2 xc(t) · (Λxc(t)), t > 0, (4.9) with initial data w(0) = w0 and g(0) = g0. 4.2. Numerical methods. Various numerical methods have been proposed and studied in the literature =-=[4, 7, 14, 20, 34, 64]-=- for computing the dynamics of the GPE (2.25) with (4.1). Among them, one of the most efficient and accurate as well as simple methods is the following time-splitting sine pseudospectral (TSSP) method... |

44 | Nobel lecture: When atoms behave as waves: Bose-Einstein condensation and the atom laser
- Ketterle
- 2002
(Show Context)
Citation Context ...h a diameter of 10–15 µm, or cigar-shaped with about 15 µm in diameter and 300 µm in length. The full cooling cycle that produce a condensate may take from a few seconds to as long as several minutes =-=[37, 52]-=-. For better understanding of the long history towards the BEC and its physical study, we refer to the Nobel lectures [37, 52] and several review papers in physics [38, 56, 65, 67]. The experimental a... |

42 | Observation of vortex lattices - Abo-Shaeer, Raman, et al. - 2001 |

38 |
Rotating trapped Bose-Einstein condensates
- Fetter
- 2009
(Show Context)
Citation Context ...995, numerous efforts have been devoted to the studies of ultracold atomic gases and various kinds of condensates of dilute gases have been produced for both bosonic particles and fermionic particles =-=[38, 43, 56]-=-. In this rapidly growing research area, mathematical models and analysis as well as numerical simulation have been playing an important role in understanding the theoretical part of BEC and predictin... |

34 | Spinor Bose condensates in optical traps.
- Ho
- 1998
(Show Context)
Citation Context ...anding of ground states and dynamics of BEC. The research in this area is still very active and highly demanded due to the latest experimental and/or technological advances in BEC, such as spinor BEC =-=[18, 22, 47, 51]-=-, BEC with damping terms [15] or impurities [50] or random potentials [63], degenerate Fermi gas [45], Rydberg gas [53], spinorbit-coupled BEC [60], BEC at finite temperature [72], etc. These achievem... |

31 |
symmetric and central vortex states in rotating Bose-Einstein condensates
- Bao, Wang, et al.
(Show Context)
Citation Context ...dition (2.5). Under the harmonic potential (2.9), similarly to the nondimensionalization in section 2.2 and dimension reduction in 2.3 from 3D to 2D when ωz ≫ max{ωx, ωy} for a disk-shaped condensate =-=[2, 7, 13, 23]-=-, we can obtain the following dimensionless GPE with an angular momentum rotation term in d-dimensions (d = 2, 3): i ∂tψ = [ −1 2 ∇2 + V (x)− ΩLz + β|ψ|2 ] ψ, x ∈ Rd, t > 0, (5.2) where Ω = Ω̃/ωx, β =... |

28 | Mathematical theory and numerical methods for Bose-Einstein condensation
- Bao, Cai
(Show Context)
Citation Context ...h two components can be well described by the macroscopic wave function Ψ := Ψ(x, t) = (ψ1(x, t), ψ2(x, t)) T whose evolution is governed by the following 3D coupled GrossPitaevskii equations (CGPEs) =-=[5, 7, 8, 48, 60, 65, 67, 73]-=- for x ∈ R3 and t > 0 as i~∂tψ1 = [ − ~ 2 2m ∇2 + V (x) + i~k̃0 2m ∂x + ~δ̃ 2 +Ng11|ψ1|2 +Ng12|ψ2|2 ] ψ1 + ~Ω̃ 2 ψ2, i~∂tψ2 = [ − ~ 2 2m ∇2 + V (x)− i~k̃0 2m ∂x − ~δ̃ 2 +Ng21|ψ1|2 +Ng22|ψ2|2 ] ψ2 + ~Ω... |

28 |
Three Dimensional Simulation of Jet Formation in Collapsing Condensates
- Bao, Jaksch, et al.
(Show Context)
Citation Context ...EC. The research in this area is still very active and highly demanded due to the latest experimental and/or technological advances in BEC, such as spinor BEC [18, 22, 47, 51], BEC with damping terms =-=[15]-=- or impurities [50] or random potentials [63], degenerate Fermi gas [45], Rydberg gas [53], spinorbit-coupled BEC [60], BEC at finite temperature [72], etc. These achievements have brought great chall... |

28 |
The λ-Phenomenon of Liquid Helium and the Bose-Einstein Degeneracy
- London
- 1938
(Show Context)
Citation Context ...riginal prediction was for a noninteracting gas and did not receive much attention in a long time. After the observation of superfluidity in liquid 4He below the λ temperature (2.17K) in 1938, London =-=[61]-=- suggested that despite the strong interatomic interactions BEC was indeed occurring in this system and was responsible for the superfluid properties. This suggestion has stood the test of time and is... |

25 | The physics of dipolar bosonic quantum gases
- Lahaye, CMenotti, et al.
- 2009
(Show Context)
Citation Context ... BEC. At temperature T much smaller than the critical temperature Tc, a dipolar BEC is well described by the macroscopic wave function ψ := ψ(x, t) whose evolution is governed by the following 3D GPE =-=[6, 7, 10, 24, 55, 71]-=- i~∂tψ = [ − ~ 2 2m ∇2 + V (x) +Ng|ψ|2 +NCdd ( Vdip ∗ |ψ|2 )] ψ, x ∈ R3, t > 0, where Cdd = µ0µ 2 dip/3 with µ0 the vacuum magnetic permeability and µdip the permanent magnetic dipole moment, ψ satisf... |

25 |
Dynamics of trapped Bose gases at finite temperature
- Zaremba, Nikuni, et al.
- 1999
(Show Context)
Citation Context ...inor BEC [18, 22, 47, 51], BEC with damping terms [15] or impurities [50] or random potentials [63], degenerate Fermi gas [45], Rydberg gas [53], spinorbit-coupled BEC [60], BEC at finite temperature =-=[72]-=-, etc. These achievements have brought great challenges to AMO community, condensed matter community, and computational and applied mathematics community for modeling, simulating and understanding var... |

24 | Efficient numerical methods for computing ground states and dynamics of dipolar Bose-Einstein condensates
- Bao, Cai, et al.
(Show Context)
Citation Context ... BEC. At temperature T much smaller than the critical temperature Tc, a dipolar BEC is well described by the macroscopic wave function ψ := ψ(x, t) whose evolution is governed by the following 3D GPE =-=[6, 7, 10, 24, 55, 71]-=- i~∂tψ = [ − ~ 2 2m ∇2 + V (x) +Ng|ψ|2 +NCdd ( Vdip ∗ |ψ|2 )] ψ, x ∈ R3, t > 0, where Cdd = µ0µ 2 dip/3 with µ0 the vacuum magnetic permeability and µdip the permanent magnetic dipole moment, ψ satisf... |

23 |
Vortices in Bose Einstein condensates
- Aftalion
(Show Context)
Citation Context ...with an internal atomic Josephon junction (JJ) and an spin-orbit coupling term. 5.1. For rotating BEC. At temperatures T much smaller than the critical temperature Tc, following the mean field theory =-=[1, 2, 7, 31, 43, 57, 62, 69]-=-, a BEC in the rotational frame is well described by the macroscopic wave function ψ := ψ(x, t), whose evolution is governed by the GPE with an angular momentum rotation term i~∂tψ = [ − ~ 2 2m ∇2 + V... |

22 |
Particle-inspired scheme for the GrossPitaevski equation: An application to Bose-Einstein condensation
- Cerimele, Pistella, et al.
(Show Context)
Citation Context ...ng on β such that the truncation error can be negligible! 3.3. Numerical methods. Various numerical methods for computing the ground state φg in (3.5) have been proposed and studied in the literature =-=[7, 11, 12, 22, 34, 64]-=-. Among them, one of the most efficient and simple methods is the following gradient flow with discrete normalization (GFDN) [7, 12]. Choose a time step τ := ∆t > 0 and denote time steps as tn = nτ fo... |

22 |
Vortex lines in an imperfect Bose gas, Soviet Phys
- Pitaevskii
- 1961
(Show Context)
Citation Context ...ng and guiding the experiments. The goal of this paper is to offer a short survey on mathematical models and theories as well as numerical methods for BEC based on the Gross-Pitaevskii equation (GPE) =-=[7, 46, 65, 66, 67]-=-. The paper is organized as follows. In section 2, we present the GPE for BEC based on the mean field approximation. Ground Models and Methods for Bose-Einstein Condensation 3 states and their computa... |

20 | The nonlinear Schrödinger equation with a strongly anisotropic harmonic potential
- Abdallah, Méhats, et al.
(Show Context)
Citation Context ...2 := ∫ R |χ(z̃)|2 dz̃ = 1 admits distinct orthonormal eigenfunctions χk(z̃) with corresponding eigenvalues µk for k = 0, 1, . . . In fact, they form an orthonormal basis of L2(R) and can be chosen as =-=[7, 14, 25, 65, 66, 67]-=- µk = k + 1 2 , χk(z̃) = 1 pi1/4 √ 2k k! e−z̃ 2/2Hk(z̃), z̃ ∈ R, k = 0, 1, 2, . . . , (2.19) with Hk(z̃) the standard Hermite polynomial of degree k. Thus (χ ε k(z), µ ε k) for k ≥ 0 are orthonormal e... |

19 |
Coherent dynamics of vortex formation in trapped Bose-Einstein condensates
- Caradoc-Davis, Ballagh, et al.
- 1999
(Show Context)
Citation Context |

18 | Optimal error estimates of finite difference methods for the GrossPitaevskii equation with angular momentum rotation
- Bao, Cai
(Show Context)
Citation Context ...tification is still missing. For more results on the ground state of the rotating BEC (5.2) and efficient and accurate numerical methods for simulation, such as BEFD [7, 23] or BEFP [11], we refer to =-=[2, 7, 9, 23, 43, 69]-=- and references therein. Similarly, for the well-posedness of the Cauchy problem of (5.2) with the initial data (4.1) and its dynamical properties as well as efficient and accurate numerical methods, ... |

18 |
Theoretical progress in many-body physics with ultracold dipolar gases, Phys. Rep
- Baranov
- 2008
(Show Context)
Citation Context ... BEC. At temperature T much smaller than the critical temperature Tc, a dipolar BEC is well described by the macroscopic wave function ψ := ψ(x, t) whose evolution is governed by the following 3D GPE =-=[6, 7, 10, 24, 55, 71]-=- i~∂tψ = [ − ~ 2 2m ∇2 + V (x) +Ng|ψ|2 +NCdd ( Vdip ∗ |ψ|2 )] ψ, x ∈ R3, t > 0, where Cdd = µ0µ 2 dip/3 with µ0 the vacuum magnetic permeability and µdip the permanent magnetic dipole moment, ψ satisf... |

18 | Derivation of the cubic NLS and Gross-Pitaevskii hierarchy from many body dynamics in d = 3 based on space time norms - Chen, Pavlović |

18 |
Spin-orbit-coupled Bose-Einstein condensates, Nature 471
- Lin, Jiménez-Garcia, et al.
- 2011
(Show Context)
Citation Context |

16 | On the rigorous derivation of the 3D cubic nonlinear Schrödinger equation with a quadratic trap
- Chen
(Show Context)
Citation Context ...; and for dynamics, Yau et al. [42] established the validity of the GPE (2.8) as an approximation for (2.3), which inspired great interests in the study on dynamics for such many body system recently =-=[35, 36, 54]-=-. The above GPE (2.8) is a very simple equation, which is very convenient for mathematical analysis and numerical calculations, and in the case of the BEC alkali gases, appears to give a rather good q... |

15 | Dynamics of rotating Bose-Einstein condensates and its efficient and accurate numerical computation
- Bao, Du, et al.
(Show Context)
Citation Context ...square of the condensate width δα(t) = ∫ Rd α2|ψ(x, t)|2dx with α = x, y or z, and angular momentum expectation 〈Lz〉(t) = ∫ Rd ψ(x, t)Lzψ(x, t) dx with Lz = −i (x∂y − y∂x) when d = 2, 3. Then we have =-=[7, 13]-=- Lemma 4.3 (Angular momentum expectation [7, 13]). For any initial data ψ0(x) in (4.1), when γy = 1 in (2.26), i.e. the trapping potential is radially/cylindrically symmetric in 2D/3D, then the angula... |

15 | On the Gross–Pitaevskii equation for trapped dipolar quantum gases, Nonlinearity 21 - Carles, Markowich, et al. - 2008 |

15 |
Uniting BoseEinstein condensates in optical resonators
- Jaksch, Gardiner, et al.
(Show Context)
Citation Context |

15 | Gross-Pitaevskii theory of the rotating Bose gas
- Seiringer
(Show Context)
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14 | Dynamics of rotating two-component Bose-Einstein condensates and its efficient computation, Phys
- Zhang, Bao, et al.
(Show Context)
Citation Context |

13 | Computing ground states of spin-1 Bose-Einstein condensates by the normalized gradient flow
- Bao, Lim
- 2008
(Show Context)
Citation Context ...anding of ground states and dynamics of BEC. The research in this area is still very active and highly demanded due to the latest experimental and/or technological advances in BEC, such as spinor BEC =-=[18, 22, 47, 51]-=-, BEC with damping terms [15] or impurities [50] or random potentials [63], degenerate Fermi gas [45], Rydberg gas [53], spinorbit-coupled BEC [60], BEC at finite temperature [72], etc. These achievem... |

13 |
Mean-field regime of trapped dipolar Bose-Einstein condensates in one and two dimensions, Phys
- Cai, Rosenkranz, et al.
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Citation Context ...y to the nondimensionalization in section 2.2 and dimension reduction in 2.3 from 3D to 2D when ωz ≫ max{ωx, ωy} for a disk-shaped condensate and to 1D when ωz = ωy ≫ ωx for a cigar-shaped condensate =-=[6, 7, 30]-=-, by using the decomposition of contact and long-range (or repulsive and attractive) parts of the DDI (5.7) [10, 30] Udip(x) = 3 4pi|x|3 ( 1− 3(x · n) 2 |x|2 ) = −δ(x)− 3∂nn ( 1 4pi|x| ) , x ∈ R3, (5.... |

12 | The physics of dipolar bosonic quantum gases, Rep. Prog. Phys - Lahaye, Menotti, et al. - 2009 |

11 | Gross-Pitaevskii-Poisson equations for dipolar Bose-Einstein condensate with anisotropic confinement
- Bao, N, et al.
(Show Context)
Citation Context ...] i∂tψ2 = [ −1 2 ∆⊥ + V2(x⊥) + κ √ γz 2pi |ψ2|2 ] ψ2, x⊥ ∈ R2, t > 0. (2.23) The above dimension reduction from 3D to 2D is mathematically and rigorously justified in the very weak interaction regime =-=[6, 25]-=-, i.e. κ = O(ε) = O(1/ √ γz) as ε→ 0+. However, for the strong interaction regime, i.e. κ = O(1) and ε→ 0+, it is very challenging. The key difficulty is due to that the energy associated to the 2D GP... |

11 |
Numerical methods for atomic quantum gases with applications to Bose-Einstein condensates and to ultracold fermions, Phys. Rep
- Minguzzi, Succi, et al.
- 2004
(Show Context)
Citation Context ...ng on β such that the truncation error can be negligible! 3.3. Numerical methods. Various numerical methods for computing the ground state φg in (3.5) have been proposed and studied in the literature =-=[7, 11, 12, 22, 34, 64]-=-. Among them, one of the most efficient and simple methods is the following gradient flow with discrete normalization (GFDN) [7, 12]. Choose a time step τ := ∆t > 0 and denote time steps as tn = nτ fo... |

11 |
Symmetry breaking and self-trapping of a dipolar Bose-Einstein condensate in a double-well potential, Phys. Rev. A 79
- Xiong, Gong, et al.
- 2009
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10 | Computational methods for the dynamics of the nonlinear Schrödinger/Gross-Pitaevskii equations, preprint
- Antoine, Bao, et al.
- 2013
(Show Context)
Citation Context ...ii) E(ψ0) > 0 and Im (∫ Rd ψ0(x) (x · ∇ψ0(x)) dx ) < − √ E(ψ0)d ‖xψ0‖L2. If there is no external potential in the GPE (2.25), i.e. V (x) ≡ 0, then the momentum and angular momentum are also conserved =-=[4, 7, 70]-=-. The GPE (2.25) admits the plane wave solution as ψ(x, t) = Aei(k·x−ωt), where the time frequency ω, amplitude A and spatial wave number k satisfy the following dispersion relation [4, 7, 70]: ω = |k... |

10 | Spinor Bose condensates in optical traps, Phys - Ho - 1998 |

7 | Ground states of two-component Bose-Einstein condensates with an internal atomic Josephson junction
- Bao, Cai
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7 | A simple and efficient numerical method for computing the dynamics of rotating Bose-Einstein condensates via a rotating Lagrangian coordinate
- Bao, Marahrens, et al.
(Show Context)
Citation Context ... α =∫ Rd α2|ψ0(x)|2dx and δ(1)α = 2 ∫ Rd α Im ( ψ0∂αψ0 ) dx for α = x or y. Thus δx in 1D and δr in 2D are periodic functions with frequency doubling the trapping frequency. Lemma 4.5 (Center-of-mass =-=[7, 13, 19]-=-). For any initial data ψ0(x) in (4.1), the dynamics of the center-of-mass satisfies the following second-order ODE ẍc(t) + Λxc(t) = 0, t ≥ 0, (4.6) with the following initial data xc(0) = x (0) c = ... |

5 | A mass and magnetization conservative and energy-diminishing numerical method for computing ground state of spin-1 Bose-Einstein condensates
- Bao, Wang
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4 |
A generalized Laguerre-Fourier-Hermite pseudospectral method for computing the dynamics of rotating Bose-Einstein condensates
- Bao, Li, et al.
(Show Context)
Citation Context ...arly, for the well-posedness of the Cauchy problem of (5.2) with the initial data (4.1) and its dynamical properties as well as efficient and accurate numerical methods, such as TSADI [21] or TSGLFHP =-=[17]-=-, we refer to [4, 7, 43, 69] and references therein. Here we present a different formulation of the GPE (5.2) under the rotating Lagrangian coordinates so that the angular momentum rotation term will ... |

4 |
Spinor Bose-Einstein condensates, Phys
- Kawaguchi, Ueda
(Show Context)
Citation Context ...anding of ground states and dynamics of BEC. The research in this area is still very active and highly demanded due to the latest experimental and/or technological advances in BEC, such as spinor BEC =-=[18, 22, 47, 51]-=-, BEC with damping terms [15] or impurities [50] or random potentials [63], degenerate Fermi gas [45], Rydberg gas [53], spinorbit-coupled BEC [60], BEC at finite temperature [72], etc. These achievem... |

3 | Fast and accurate evaluation of nonlocal Coulomb and dipole-dipole interactions via the nonuniform FFT
- Jiang, Greengard, et al.
(Show Context)
Citation Context ...xistence and uniqueness as well as nonexistence of the ground state of the dipolar BEC (5.9) and efficient and accurate numerical methods for simulation, such as BESP [10] or BEFP with nonuniform FFT =-=[49]-=-, we refer to [6, 7, 10] and references therein. Similarly, for the well-posedness of the Cauchy problem of (5.9) with the initial data (4.1) and its dynamical properties as well as efficient and accu... |

3 |
Subdiffusive spreading of a BoseEinstein condensate in random potentials, Phys. Rev. A 86 (2012), article 053612
- Min, Li, et al.
(Show Context)
Citation Context ...ctive and highly demanded due to the latest experimental and/or technological advances in BEC, such as spinor BEC [18, 22, 47, 51], BEC with damping terms [15] or impurities [50] or random potentials =-=[63]-=-, degenerate Fermi gas [45], Rydberg gas [53], spinorbit-coupled BEC [60], BEC at finite temperature [72], etc. These achievements have brought great challenges to AMO community, condensed matter comm... |

3 | Self-trapping of Bose-Einstein condensates expanding in shallow optical lattices, Phys. Rev. A 77 (2008) article 063607 - Rosenkranz, Jaksch, et al. |

3 | states of two-component Bose-Einstein condensates with an internal atomic Josephson junction - Ground |

2 | Self-trapping of impurities in Bose-Einstein condensates: Strong attractive and repulsive coupling, EPL 82 - Bruderer, Bao, et al. - 2008 |

2 |
Three-body bound states in dipole-dipole interacting Rydberg atoms
- Kiffner, Li, et al.
- 2013
(Show Context)
Citation Context ...xperimental and/or technological advances in BEC, such as spinor BEC [18, 22, 47, 51], BEC with damping terms [15] or impurities [50] or random potentials [63], degenerate Fermi gas [45], Rydberg gas =-=[53]-=-, spinorbit-coupled BEC [60], BEC at finite temperature [72], etc. These achievements have brought great challenges to AMO community, condensed matter community, and computational and applied mathemat... |

2 | error estimates of finite difference methods for the GrossPitaevskii equation with angular momentum rotation - Optimal |

1 |
Dimension reduction for the anistropic BoseEinstein condensates in the strong interaction regime
- Bao, Treust, et al.
- 2014
(Show Context)
Citation Context ...e key difficulty is due to that the energy associated to the 2D GPE (2.23) is unbounded in this regime. Recently, by using a proper re-scaling, the dimension reduction is justified in this regime too =-=[16]-=-. Models and Methods for Bose-Einstein Condensation 7 Similarly, when γz ≫ 1 and γy ≫ 1 (⇔ ωz ≫ ωx and ωy ≫ ωx), i.e. cigarshaped condensate with strong confinement in the (y, z)-plane [3, 28, 39, 65,... |

1 |
Breathing oscillations of a trapped impurity in a Bose gas, EPL 98 (2012), article 26001. Mathematical models and numerical methods for Bose-Einstein condensation 995
- Johnson, Bruderer, et al.
(Show Context)
Citation Context ... this area is still very active and highly demanded due to the latest experimental and/or technological advances in BEC, such as spinor BEC [18, 22, 47, 51], BEC with damping terms [15] or impurities =-=[50]-=- or random potentials [63], degenerate Fermi gas [45], Rydberg gas [53], spinorbit-coupled BEC [60], BEC at finite temperature [72], etc. These achievements have brought great challenges to AMO commun... |

1 | des einatomigen idealen gases, zweite abhandlung, Sitzungsberichte der Preussischen Akademie der Wissenschaften 1 - Quantentheorie - 1925 |