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By Amandine Aftalion
Because the first experimental success of Bose-Einstein condensates (BEC) in 1995 and the award of the Nobel Prize for Physics in 2001, the houses of those gaseous quantum fluids were the focal point of overseas curiosity in condensed subject physics. This monograph is devoted to the mathematical modeling of a few particular experiments which show vortices and to a rigorous research of positive aspects rising experimentally.In distinction to a classical fluid, a quantum fluid equivalent to a Bose-Einstein condensate can rotate in basic terms in the course of the nucleation of quantized vortices past a few severe speed. There are fascinating regimes: one on the subject of the serious speed, the place there's just one vortex that has a really designated form; and one other one at excessive rotation values, for which a dense lattice is observed.One of the foremost gains regarding superfluidity is the life of those vortices. We tackle this factor mathematically and derive info on their form, quantity and site. within the dilute restrict of the experiments, the condensate is definitely defined by way of a median box conception and a macroscopic wave functionality fixing the so-called Gross-Pitaevskii equation. The mathematical instruments hired are strength estimates, Gamma convergence, and homogenization recommendations. We turn out life of recommendations that experience homes in keeping with the experimental observations. Open difficulties with regards to fresh experiments are presented.The paintings can function a reference for mathematical researchers and theoretical physicists drawn to superfluidity and quantum condensates, and will additionally supplement a graduate seminar in elliptic PDEs or modeling of actual experiments.
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Additional info for Vortices in Bose-Einstein Condensates
Sample text
As before, we are interested in the number and location of vortices. In this setting, we expect a vortex lattice of characteristic spacing of order 1 and a condensate of characteristic size R such that R 4 is proportional to 1/(1 − ). The ball B R is the region where the wave function is dominant. The results that we will present deal with an upper bound for the energy, the lower bound remaining an open issue. 5. Let be a lattice, Q its unit cell. Assume that V = |Q| > π . 17) j∈ ∩B R with A R chosen such that ψ R |ψ R (z)| ∗ = 1.
18. Let α 1/2 − α , 3(N R,α + 1) δα = and for k = 0, . . , 3N R,α + 2 we consider αk = α 1/2 − kδα , Ik = [ε αk , ε αk+1 ], and Ck = B(x0 , ε αk+1 ) \ B(x0 , ε αk ). Then there is some k0 ∈ {1, . . , 3N R,α + 1} such that ε Ck0 ∩ ∪ Nj=1 B(x j , λ R ε) = ∅. 61) Indeed, since Nε ≤ N R,α and 2λ R ε < |Ik | for small ε, the union of Nε intervals of length 2λ R ε, ε (|xi − x0 | − λ R ε, |xi − x0 | + λ R ε), ∪ Nj=1 cannot intersect all the intervals Ik of disjoint interior, for 1 ≤ k ≤ 3N R,α + 1.
45) implies π ρTF ( pi )|di | |log ε| i, di >0 +π ω0 ω1 | pi |2 − (C0 + ρTF (| pi |))log|log ε| 2 2 ρTF ( pi )|di ||log ε| i, di <0 + 2 ρTF 1 ρTF |∇v|2 + (1 − |v|2 )2 ≤ O(|log ε|−1 ). 47) 44 3 Two-Dimensional Model for a Rotating Condensate If ω1 < −2C0 /ρ0 , similar arguments as in the previous case yield that there are no vortices. 3rd case: ω0 = ω0∗ . If ω1 is too large, we are going to prove a bound on the number of vortices. We let I˜∗ = {i ∈ I∗ : pi ∈ Bε } , N∗ = |di |, i∈ I˜∗ and I˜− = {i ∈ I− : pi ∈ Bε } , N− = |di |.