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Ground State Patterns of Spin-1 Bose-Einstein condensation via Γ-convergence Theory Tien-Tsan Shieh joint work with I-Liang Chern and Chiu-Fen Chou National Center of Theoretical Science December 19, 2015 Tien-Tsan Shieh joint work with I-Liang Chern and Chiu-Fen Chou (National Center of Theoretical Science) Ground State Patterns of Spin- 1 Bose-Einstein condensation via Γ -convergence Theory December 19, 2015 1 / 41

Outline Introduction to Spin-1 Bose-Einstein condensate Thomas-Fermi Approximation of the spin-1 BEC Γ-convergence result of the spin-1 BEC Tien-Tsan Shieh joint work with I-Liang Chern and Chiu-Fen Chou (National Center of Theoretical Science) Ground State Patterns of Spin- 1 Bose-Einstein condensation via Γ -convergence Theory December 19, 2015 2 / 41

The Bose-Einstein condensation (BEC) In 1925 Einstein and Bose predicted a new state of matter for very dilute Boson gas which tend to occupy the state of the lowest energy at very low temperature and behave as a coherent matter wave. Tien-Tsan Shieh joint work with I-Liang Chern and Chiu-Fen Chou (National Center of Theoretical Science) Ground State Patterns of Spin- 1 Bose-Einstein condensation via Γ -convergence Theory December 19, 2015 3 / 41

Realization of BEC BECs were realized in lab by E. Cornell, W. Ketterle and C. Wieman (1995). Tien-Tsan Shieh joint work with I-Liang Chern and Chiu-Fen Chou (National Center of Theoretical Science) Ground State Patterns of Spin- 1 Bose-Einstein condensation via Γ -convergence Theory December 19, 2015 4 / 41

The mean field model for BEC N particle system: wave function Ψ N ( x 1 , · · · , x N , t ), Hamiltonian: N � � − � 2 � � ∇ 2 H N = j + V ( x j ) + V int ( x j − x k ) , 2 M a j =1 1 ≤ j < k ≤ N Ultracold and dilute gases, the mean field approximation: V int ( x j − x k ) ≈ g δ ( x j − x k ) Hartree ansatz: all boson particles are in the same quantum state N � Ψ N ( x 1 , · · · , x N , t ) = ψ ( x j , t ) . j =1 Tien-Tsan Shieh joint work with I-Liang Chern and Chiu-Fen Chou (National Center of Theoretical Science) Ground State Patterns of Spin- 1 Bose-Einstein condensation via Γ -convergence Theory December 19, 2015 5 / 41

The Gross-Pitaeviskii equation Hamiltonian: � 2 |∇ ψ | 2 + V ( x ) | ψ | 2 + β 2 | ψ | 4 , β = gN H = 2 M a � Energy E [ ψ ] = H dx . Gross-Pitaevskii equation: i � ∂ t ψ = δ E /δψ ∗ . i � ∂ t ψ = − � 2 ∇ 2 ψ + V ( x ) ψ + β | ψ | 2 ψ 2 M a ψ wave function � 3 V ( x ) trap potential: V ( x ) = 1 i =1 ω 2 i x 2 i . 2 Interaction: repulsive if β > 0, attractive if β < 0. Tien-Tsan Shieh joint work with I-Liang Chern and Chiu-Fen Chou (National Center of Theoretical Science) Ground State Patterns of Spin- 1 Bose-Einstein condensation via Γ -convergence Theory December 19, 2015 6 / 41

One-, multi-component and spinor BECs One-component BECs: atoms with a single quantum state are trapped. E.g. Using magnetic trap Two-component BECs: mixture of two different species of bosons. E.g. two isotopes of the same elements, or two different elements Spinor BECs: mixture of different hyperfine states of the same isotopes. E.g. Spin-1 atoms using optical trap. There are 3 hyperfine states m F = 1 , 0 , − 1 Tien-Tsan Shieh joint work with I-Liang Chern and Chiu-Fen Chou (National Center of Theoretical Science) Ground State Patterns of Spin- 1 Bose-Einstein condensation via Γ -convergence Theory December 19, 2015 7 / 41

Spinor BECs Spin-1 atom has 3 hyperfine states: m F = 1 , 0 , − 1. Vector order parameter Ψ = ( ψ 1 , ψ 0 , ψ − 1 ). Associate with a spinor Ψ, the spin vector F = Ψ † F Ψ ∈ R 3 , which is just like a magnetic dipole moment. F = ( F x , F y , F z ) is the spin-1 Pauli operator:  0 1 0   0 − 1 0   1 0 0  1 i  , F y =  , F z =  . F x = √ 1 0 1 √ 1 0 − 1 0 0 0    2 2 0 1 0 0 1 0 0 0 − 1 Tien-Tsan Shieh joint work with I-Liang Chern and Chiu-Fen Chou (National Center of Theoretical Science) Ground State Patterns of Spin- 1 Bose-Einstein condensation via Γ -convergence Theory December 19, 2015 8 / 41

G-P equation for spin-1 BECs Hamiltonian: � 2 |∇ Ψ | 2 + V ( x ) | Ψ | 2 + c n 2 | Ψ | 4 + c s 2 | Ψ † F Ψ | 2 H = 2 M a | Ψ | 2 · | Ψ | 2 : spin-independent interaction | Ψ † F Ψ | 2 : spin-spin interaction (spin-exchange). � The total energy E [Ψ] = H dx . The G-P equation i � ∂ t Ψ = δ E δ Ψ † Tien-Tsan Shieh joint work with I-Liang Chern and Chiu-Fen Chou (National Center of Theoretical Science) Ground State Patterns of Spin- 1 Bose-Einstein condensation via Γ -convergence Theory December 19, 2015 9 / 41

Physical parameters � 2 + c n + c s |∇ Ψ | 2 + V ( x ) | Ψ | 2 2 | Ψ | 4 2 | Ψ † F Ψ | 2 H = 2 M a � �� � � �� � � �� � � �� � H pot H n H s H kin interaction > 0 < 0 c n spin-independent repulsive attractive c s spin-exchange antiferromagnetic ferromagnetic c n c s 87 Rb 7.793 -0.0361 ferromagnetic 23 Na 15.587 0.4871 anti-ferromagnetic Tien-Tsan Shieh joint work with I-Liang Chern and Chiu-Fen Chou (National Center of Theoretical Science) Ground State Patterns of Spin- 1 Bose-Einstein condensation via Γ -convergence Theory December 19, 2015 10 / 41

Spinor BEC in uniform magnetic field Under an uniform magnetic field, we have to consider additional Zeeman shift energy in the Hamiltonian. Hamiltonian H = H kin + H pot + H n + H s + H Zee Zeeman shift energy: Suppose magnetic field B ˆ z , 1 � H Zee = E j ( B ) n j j = − 1 = q ( n 1 + n − 1 ) + p ( n 1 − n − 1 ) + E 0 n where n j = | ψ j | 2 and p = 1 2( E − 1 − E 1 ) ≈ − µ B B 2 2( E − 1 + E 1 − 2 E 0 ) ≈ µ 2 B B 2 q = 1 4 E hfs Tien-Tsan Shieh joint work with I-Liang Chern and Chiu-Fen Chou (National Center of Theoretical Science) Ground State Patterns of Spin- 1 Bose-Einstein condensation via Γ -convergence Theory December 19, 2015 11 / 41

Gauge invariants and conservation laws Energy � E [Ψ] = ( H kin + H pot + H n + H s + H Zee ) dx Gauge invariant: energy is invariant under transform Ψ → e i φ R z ( α )Ψ This leads to two conservation laws: ◮ Total number of atoms � ( | ψ 1 | 2 + | ψ 0 | 2 + | ψ − 1 | 2 ) dx = N ◮ Total magnetization � ( | ψ 1 | 2 − | ψ − 1 | 2 ) dx = M Tien-Tsan Shieh joint work with I-Liang Chern and Chiu-Fen Chou (National Center of Theoretical Science) Ground State Patterns of Spin- 1 Bose-Einstein condensation via Γ -convergence Theory December 19, 2015 12 / 41

The ground state problem � � min E [Ψ] subject to n ( x ) dx = N , m ( x ) dx = M . � E [Ψ] = H dx H = H kin + H pot + H n + H s + H Zee n j = | ψ j | 2 , n = n 1 + n 0 + n − 1 m = n 1 − n − 1 Set u j = | ψ j | , j = 1 , 0 , − 1. 1 � � 2 |∇ u j | 2 + c n � 2 | u | 4 + V ( x ) | u | 2 E [ u ] = 2 M a R 3 j = − 1 + c s � 0 ( u 1 − sgn ( c s ) u − 1 ) 2 + ( u 2 − 1 ) 2 � 2 u 2 1 − u 2 2 � � q ( u 2 1 + u 2 + − 1 ) dx + E 0 N + pM . Tien-Tsan Shieh joint work with I-Liang Chern and Chiu-Fen Chou (National Center of Theoretical Science) Ground State Patterns of Spin- 1 Bose-Einstein condensation via Γ -convergence Theory December 19, 2015 13 / 41

The goal of this project Show the phase separation do occur in the Ground states Give a complete phase diagram Characterize the patterns of the Ground states The problem could be formulated as a problem in calculus of variation. �� 1 � 2 |∇ u j | 2 + c n 2 | u | 4 + V ( x ) | u | 2 � inf 2 M a R 3 j = − 1 + c s 0 ( u 1 − sgn ( c s ) u − 1 ) 2 + ( u 2 2 u 2 1 − u 2 − 1 ) 2 � � 2 � p ( u 2 1 − u 2 − 1 ) + q ( u 2 1 + u 2 � � + − 1 ) dx subject to the constrains � � R 3 u 2 1 + u 2 0 + u 2 R 3 u 2 1 − u 2 − 1 dx = N , − 1 dx = M . � 2 In particular, we are interesting in the case that ǫ = 2 M i ≪ 1. The problem becomes a singular perturbation problem. Tien-Tsan Shieh joint work with I-Liang Chern and Chiu-Fen Chou (National Center of Theoretical Science) Ground State Patterns of Spin- 1 Bose-Einstein condensation via Γ -convergence Theory December 19, 2015 14 / 41

The corresponding nonlinear eigenvalue problem The corresponding Euler-Lagrange equation is � − � 2 � u 1 + c 2 ( n 1 + n 0 − n − 1 ) u 1 + c 2 u ∗ 2 ( µ + λ ) u 1 = 2 m ∆ + V ( x ) + q + c 0 n − 1 u 0 � − � 2 � u 0 + 2 c 2 ( n 1 − n − 1 ) u 0 + c 2 u 1 u − 1 u ∗ µ u 0 = 2 m ∆ + V ( x ) + c 0 n 0 � − � 2 � u − 1 + c 2 ( n − 1 + n 0 − n 1 ) u − 1 + c 2 u ∗ 1 u 2 ( µ − λ ) u − 1 = 2 m ∆ + V ( x ) + q + c 0 n 0 We denote that n 1 = | u 1 | 2 , n 0 = | u 0 | 2 , n − 1 = | u − 1 | 2 . Here, µ and λ are the two Lagrange multipliers corresponding to the two constraints. Tien-Tsan Shieh joint work with I-Liang Chern and Chiu-Fen Chou (National Center of Theoretical Science) Ground State Patterns of Spin- 1 Bose-Einstein condensation via Γ -convergence Theory December 19, 2015 15 / 41