MACROSCOPIC QUANTUM PHENOMENA IN SPIN- ORBIT COUPLED BOSE- EINSTEIN CONDENSATE AND THEIR IMPLICATIONS
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1 MACROSCOPIC QUANTUM PHENOMENA IN SPIN- ORBIT COUPLED BOSE- EINSTEIN CONDENSATE AND THEIR IMPLICATIONS SHIZHONG ZHANG INT, APRIL 2011
2 INTRODUCTION Solid State Physics: effects of magne3c field B (abelian gauge field)
3 INTRODUCTION Solid State Physics: effects of magne3c field B (abelian gauge field) de Haas- Van Alphen Effect
4 INTRODUCTION Solid State Physics: effects of magne3c field B (abelian gauge field) de Haas- Van Alphen Effect FQHE
5 INTRODUCTION Solid State Physics: effects of magne3c field B (abelian gauge field) de Haas- Van Alphen Effect FQHE effects of spin- orbit coupling (constant non- abelian gauge field)
6 INTRODUCTION Solid State Physics: effects of magne3c field B (abelian gauge field) de Haas- Van Alphen Effect FQHE effects of spin- orbit coupling (constant non- abelian gauge field) spin Hall effect
7 INTRODUCTION Solid State Physics: effects of magne3c field B (abelian gauge field) de Haas- Van Alphen Effect FQHE effects of spin- orbit coupling (constant non- abelian gauge field) topological insulator spin Hall effect
8 INTRODUCTION Solid State Physics: effects of magne3c field B (abelian gauge field) de Haas- Van Alphen Effect FQHE Fermions effects of spin- orbit coupling (constant non- abelian gauge field) topological insulator spin Hall effect
9 INTRODUCTION Cold atom Physics:
10 Cold atom Physics: INTRODUCTION 1. Bosons and Fermions can be studied equally easily;
11 Cold atom Physics: INTRODUCTION 1. Bosons and Fermions can be studied equally easily; 2. Abelian gauge field can be generated by rota3ng;
12 Cold atom Physics: INTRODUCTION 1. Bosons and Fermions can be studied equally easily; 2. Abelian gauge field can be generated by rota3ng; MIT ENS,JILA
13 Cold atom Physics: INTRODUCTION 1. Bosons and Fermions can be studied equally easily; 2. Abelian gauge field can be generated by rota3ng; MIT ENS,JILA A mω r symmetric gauge
14 Cold atom Physics: INTRODUCTION 1. Bosons and Fermions can be studied equally easily; 2. Abelian gauge field can be generated by rota3ng; MIT ENS,JILA A mω r symmetric gauge 3. Both abelian and non- abelian gauge field can be generated using light;
15 Cold atom Physics: INTRODUCTION 1. Bosons and Fermions can be studied equally easily; 2. Abelian gauge field can be generated by rota3ng; MIT ENS,JILA A mω r symmetric gauge 3. Both abelian and non- abelian gauge field can be generated using light; NIST
16 Cold atom Physics: INTRODUCTION 1. Bosons and Fermions can be studied equally easily; 2. Abelian gauge field can be generated by rota3ng; MIT ENS,JILA A mω r symmetric gauge 3. Both abelian and non- abelian gauge field can be generated using light; NIST
17 Cold atom Physics: INTRODUCTION 1. Bosons and Fermions can be studied equally easily; 2. Abelian gauge field can be generated by rota3ng; MIT ENS,JILA A mω r symmetric gauge 3. Both abelian and non- abelian gauge field can be generated using light; Bosons NIST
18 INTRODUCTION Yes! We can generate synthe3c gauge field in cold atoms. What then?
19 INTRODUCTION Yes! We can generate synthe3c gauge field in cold atoms. What then? Fundamental differences between bosons and fermions under the influence of the spin- orbit interacmon, or more generally gauge fields;
20 INTRODUCTION Yes! We can generate synthe3c gauge field in cold atoms. What then? Fundamental differences between bosons and fermions under the influence of the spin- orbit interacmon, or more generally gauge fields; Novel use of the spin- orbit coupled Bose- Einstein condensate as elasmc maser latce ; e.g., Peierls distormon etc.
21 INTRODUCTION Yes! We can generate synthe3c gauge field in cold atoms. What then? Fundamental differences between bosons and fermions under the influence of the spin- orbit interacmon, or more generally gauge fields; Novel use of the spin- orbit coupled Bose- Einstein condensate as elasmc maser latce ; e.g., Peierls distormon etc. Possibility of generamng fracmonal charge in the system;
22 INTRODUCTION Yes! We can generate synthe3c gauge field in cold atoms. What then? Fundamental differences between bosons and fermions under the influence of the spin- orbit interacmon, or more generally gauge fields; Novel use of the spin- orbit coupled Bose- Einstein condensate as elasmc maser latce ; e.g., Peierls distormon etc. Possibility of generamng fracmonal charge in the system; Analogous transport propermes in spin- orbit coupled boson system;
23 INTRODUCTION Yes! We can generate synthe3c gauge field in cold atoms. What then? Fundamental differences between bosons and fermions under the influence of the spin- orbit interacmon, or more generally gauge fields; Novel use of the spin- orbit coupled Bose- Einstein condensate as elasmc maser latce ; e.g., Peierls distormon etc. Possibility of generamng fracmonal charge in the system; Analogous transport propermes in spin- orbit coupled boson system; Many more possibilimes...
24 OUTLINE
25 OUTLINE I. General discussions on the creamon of synthemc gauge field; Generalized adiabamc principles;
26 OUTLINE I. General discussions on the creamon of synthemc gauge field; Generalized adiabamc principles; II. The NIST scheme, experimental findings and theoremcal explanamon;
27 OUTLINE I. General discussions on the creamon of synthemc gauge field; Generalized adiabamc principles; II. The NIST scheme, experimental findings and theoremcal explanamon; III. Peierls distormon as an example of how to make use of the spin- orbit coupled Bose- Einstein condensate;
28 OUTLINE I. General discussions on the creamon of synthemc gauge field; Generalized adiabamc principles; II. The NIST scheme, experimental findings and theoremcal explanamon; III. Peierls distormon as an example of how to make use of the spin- orbit coupled Bose- Einstein condensate; IV. Further direcmon in the field; vormces, fracmonal charge, transport and some speculamons;
29 References: Part I and II Tin- Lun Ho and Shizhong Zhang, arxiv: Part III Work with Onur Erten, Tin- Lun Ho, to appear. Part IV Work with Tin- Lun Ho, to appear.
30 GENERAL STRATEGY I CreaMng spamally varying internal eigenstates (adiabamc states); Gauge fields appear in the basis of these internal states.
31 GENERAL STRATEGY I CreaMng spamally varying internal eigenstates (adiabamc states); Gauge fields appear in the basis of these internal states. H = P2 2m δ αβ + W αβ (r)
32 GENERAL STRATEGY I CreaMng spamally varying internal eigenstates (adiabamc states); Gauge fields appear in the basis of these internal states. H = P2 2m δ αβ + W αβ (r) spin- f par3cles α, β =1, 2,, 2f +1 f f- 1 - f z
33 GENERAL STRATEGY I CreaMng spamally varying internal eigenstates (adiabamc states); Gauge fields appear in the basis of these internal states. H = P2 2m δ αβ + W αβ (r) We now find the local internal eigenstates of W αβ (r): spin- f par3cles α, β =1, 2,, 2f +1 f f- 1 - f z W αβ (r) = 2f+1 i=1 i (r)χ i (r; α)χ i(r; β) i index of the internal eigenstate, with its component given by α,β.
34 GENERAL STRATEGY I CreaMng spamally varying internal eigenstates (adiabamc states); Gauge fields appear in the basis of these internal states. H = P2 2m δ αβ + W αβ (r) We now find the local internal eigenstates of W αβ (r): spin- f par3cles α, β =1, 2,, 2f +1 f f- 1 - f z W αβ (r) = 2f+1 i=1 i (r)χ i (r; α)χ i(r; β) i index of the internal eigenstate, with its component given by α,β. r i r f
35 GENERAL STRATEGY I CreaMng spamally varying internal eigenstates (adiabamc states); Gauge fields appear in the basis of these internal states. H = P2 2m δ αβ + W αβ (r) We now find the local internal eigenstates of W αβ (r): spin- f par3cles α, β =1, 2,, 2f +1 f f- 1 - f z W αβ (r) = 2f+1 i=1 i (r)χ i (r; α)χ i(r; β) i index of the internal eigenstate, with its component given by α,β. r i 2π q r f
36 GENERAL STRATEGY I CreaMng spamally varying internal eigenstates (adiabamc states); Gauge fields appear in the basis of these internal states. H = P2 2m δ αβ + W αβ (r) We now find the local internal eigenstates of W αβ (r): spin- f par3cles α, β =1, 2,, 2f +1 f f- 1 - f z W αβ (r) = 2f+1 i=1 i (r)χ i (r; α)χ i(r; β) i index of the internal eigenstate, with its component given by α,β. r i The typical energy scale associated with the moving par3cle is group of states i=1,2,...,l, such that 2π q i (r) q r f q = 2 q 2 2m, and if a
37 GENERAL STRATEGY I CreaMng spamally varying internal eigenstates (adiabamc states); Gauge fields appear in the basis of these internal states. H = P2 2m δ αβ + W αβ (r) We now find the local internal eigenstates of W αβ (r): spin- f par3cles α, β =1, 2,, 2f +1 f f- 1 - f z W αβ (r) = 2f+1 i=1 i (r)χ i (r; α)χ i(r; β) i index of the internal eigenstate, with its component given by α,β. r i The typical energy scale associated with the moving par3cle is group of states i=1,2,...,l, such that 2π q i (r) q r f q = 2 q 2 2m, and if a These L- states will be mixed significantly during the par3cle mo3on and we thus obtain an effec3ve low energy manifold, characterized by χ i (r),i=1,2,...,l.
38 GENERAL STRATEGY II To see explicitly how gauge fields arise, transform to the local adiaba/c basis: (Generalized Adiaba/c Principle) H = P2 2m δ αβ + W αβ (r)
39 GENERAL STRATEGY II To see explicitly how gauge fields arise, transform to the local adiaba/c basis: (Generalized Adiaba/c Principle) f f- 1 z - f
40 GENERAL STRATEGY II To see explicitly how gauge fields arise, transform to the local adiaba/c basis: (Generalized Adiaba/c Principle) f f- 1 z q i =2f +1 - f i =2 i =1
41 GENERAL STRATEGY II To see explicitly how gauge fields arise, transform to the local adiaba/c basis: (Generalized Adiaba/c Principle) f f- 1 z q i =2f +1 - f i =2 i =1 Abelian
42 GENERAL STRATEGY II To see explicitly how gauge fields arise, transform to the local adiaba/c basis: (Generalized Adiaba/c Principle) f f- 1 z q i =2f +1 - f i =2 i =1 Abelian non- Abelian
43 GENERAL STRATEGY II To see explicitly how gauge fields arise, transform to the local adiaba/c basis: (Generalized Adiaba/c Principle) f f- 1 z q i =2f +1 - f i =2 i =1 Abelian non- Abelian The Hamiltonian: (F.Wilczek and A.Zee, PRL (1984))
44 GENERAL STRATEGY II To see explicitly how gauge fields arise, transform to the local adiaba/c basis: (Generalized Adiaba/c Principle) f f- 1 z q i =2f +1 - f i =2 i =1 Abelian non- Abelian The Hamiltonian: (F.Wilczek and A.Zee, PRL (1984)) H = P2 2m δ + W αβ αβ (r)
45 GENERAL STRATEGY II To see explicitly how gauge fields arise, transform to the local adiaba/c basis: (Generalized Adiaba/c Principle) f f- 1 z q i =2f +1 - f i =2 i =1 Abelian non- Abelian The Hamiltonian: (F.Wilczek and A.Zee, PRL (1984)) H = 2f+1 (P + A)2 2m + i i (r) r; ir; i
46 GENERAL STRATEGY II To see explicitly how gauge fields arise, transform to the local adiaba/c basis: (Generalized Adiaba/c Principle) f f- 1 z q i =2f +1 - f i =2 i =1 Abelian non- Abelian The Hamiltonian: (F.Wilczek and A.Zee, PRL (1984)) H = 2f+1 (P + A)2 2m + i i (r) r; ir; i A ij = iχ i (r) rχ j (r) Gauge potenmal (Berry connecmon)
47 GENERAL STRATEGY II To see explicitly how gauge fields arise, transform to the local adiaba/c basis: (Generalized Adiaba/c Principle) f f- 1 z q i =2f +1 - f i =2 i =1 Abelian non- Abelian The Hamiltonian: (F.Wilczek and A.Zee, PRL (1984)) H = 2f+1 (P + A)2 2m + i i (r) r; ir; i A ij = iχ i (r) rχ j (r) Gauge potenmal (Berry connecmon) i (r) Space- dependent Zeeman potenmal
48 GENERAL STRATEGY II To see explicitly how gauge fields arise, transform to the local adiaba/c basis: (Generalized Adiaba/c Principle) f f- 1 z q i =2f +1 - f i =2 i =1 Abelian non- Abelian The Hamiltonian: (F.Wilczek and A.Zee, PRL (1984)) H = 2f+1 (P + A)2 2m + i i (r) r; ir; i A ij = iχ i (r) rχ j (r) Gauge potenmal (Berry connecmon) i (r) Space- dependent Zeeman potenmal If different components of A ij do not commute Non- abelian!
49 NIST SCHEME A specific realizamon in NIST of the Generalized Adiaba/c Principle:
50 NIST SCHEME A specific realizamon in NIST of the Generalized Adiaba/c Principle: m = 1 m =0 m =1 ẑ ŷ kop + qˆx kop ω op + ω 87 Rb ω op ˆx
51 NIST SCHEME A specific realizamon in NIST of the Generalized Adiaba/c Principle: Denote the internal hyperfine- Zeeman states as F y =1,0,- 1 m = 1 m =0 m =1 ẑ ŷ kop + qˆx kop ω op + ω 87 Rb ω op ˆx
52 NIST SCHEME A specific realizamon in NIST of the Generalized Adiaba/c Principle: Denote the internal hyperfine- Zeeman states as F y =1,0,- 1 m = 1 m =0 m =1 ẑ ŷ kop + qˆx kop ω op + ω 87 Rb ω op ˆx W (t) = Ω y F y + λf 2 y Ω R 2 e i(qx ωt) F + + h.c.
53 NIST SCHEME A specific realizamon in NIST of the Generalized Adiaba/c Principle: Denote the internal hyperfine- Zeeman states as F y =1,0,- 1 kop + qˆx ω op + ω m =1 m = 1 m =0 ẑ 87 Rb ŷ kop ω op ˆx Ω y = Ω o + Gy G is the field gradient along y- direc3on λ Ω R quadra3c Zeeman effect Two- photon Rabi frequency W (t) = Ω y F y + λf 2 y Ω R 2 e i(qx ωt) F + + h.c.
54 NIST SCHEME A specific realizamon in NIST of the Generalized Adiaba/c Principle: Denote the internal hyperfine- Zeeman states as F y =1,0,- 1 kop + qˆx ω op + ω m =1 m = 1 m =0 ẑ 87 Rb ŷ kop ω op ˆx Ω y = Ω o + Gy G is the field gradient along y- direc3on λ Ω R quadra3c Zeeman effect Two- photon Rabi frequency W (t) = Ω y F y + λf 2 y Ω R 2 e i(qx ωt) F + + h.c. This Hamiltonian describes both abelian and non- abelian gauge fields (spin- orbit coupling is a par3cular case of constant non- abelian gauge field)
55 EFFECTIVE HAMILTONIAN I To derive the effecmve Hamiltonian, go to the rotamng frame: H = P2 2m + e iqxf y ( Ω y F y + λfy 2 Ω R F z )e iqxf y Ω y = Ω 0 ω + Gy m =1 m =0 m = 1
56 EFFECTIVE HAMILTONIAN I To derive the effecmve Hamiltonian, go to the rotamng frame: H = P2 2m + e iqxf y ( Ω y F y + λfy 2 Ω R F z )e iqxf y Ω y = Ω 0 ω + Gy To simplify the discussion, set G=0
57 EFFECTIVE HAMILTONIAN I To derive the effecmve Hamiltonian, go to the rotamng frame: H = P2 2m + e iqxf y ( Ω y F y + λfy 2 Ω R F z )e iqxf y Ω y = Ω 0 ω + Gy To simplify the discussion, set G=0 Space- independent!
58 EFFECTIVE HAMILTONIAN I To derive the effecmve Hamiltonian, go to the rotamng frame: H = P2 2m + e iqxf y ( Ω y F y + λfy 2 Ω R F z )e iqxf y Ω y = Ω 0 ω + Gy To simplify the discussion, set G=0 Space- independent! m = 1 m =0 m =1
59 EFFECTIVE HAMILTONIAN I To derive the effecmve Hamiltonian, go to the rotamng frame: H = P2 2m + e iqxf y ( Ω y F y + λfy 2 Ω R F z )e iqxf y Ω y = Ω 0 ω + Gy To simplify the discussion, set G=0 Space- independent! Selng G=0 and Choose: ω = λ + Ω o We obtain two degenerate states! m = 1 m =0 m =1
60 EFFECTIVE HAMILTONIAN I To derive the effecmve Hamiltonian, go to the rotamng frame: H = P2 2m + e iqxf y ( Ω y F y + λfy 2 Ω R F z )e iqxf y Ω y = Ω 0 ω + Gy To simplify the discussion, set G=0 Space- independent! m = 1 Selng G=0 and Choose: ω = λ + Ω o We obtain two degenerate states! 2λ Ω R Ω R m =1 m =0
61 EFFECTIVE HAMILTONIAN I To derive the effecmve Hamiltonian, go to the rotamng frame: H = P2 2m + e iqxf y ( Ω y F y + λfy 2 Ω R F z )e iqxf y Ω y = Ω 0 ω + Gy To simplify the discussion, set G=0 Space- independent! m = 1 Selng G=0 and Choose: ω = λ + Ω o We obtain two degenerate states! 2λ Ω R Ω R q m =1 m =0
62 EFFECTIVE HAMILTONIAN I To derive the effecmve Hamiltonian, go to the rotamng frame: H = P2 2m + e iqxf y ( Ω y F y + λfy 2 Ω R F z )e iqxf y Ω y = Ω 0 ω + Gy To simplify the discussion, set G=0 Space- independent! m = 1 Selng G=0 and Choose: ω = λ + Ω o We obtain two degenerate states! 2λ Ω R Ω R q q λ SU(2) non- abelian gauge field m =1 m =0
63 EFFECTIVE HAMILTONIAN I To derive the effecmve Hamiltonian, go to the rotamng frame: H = P2 2m + e iqxf y ( Ω y F y + λfy 2 Ω R F z )e iqxf y Ω y = Ω 0 ω + Gy To simplify the discussion, set G=0 Space- independent! 2λ m = 1 Ω R Ω R m =1 m =0 q Selng G=0 and Choose: ω = λ + Ω o We obtain two degenerate states! q λ SU(2) non- abelian gauge field λ Ω R Can neglect the coupling to m=- 1 state and we obtain an system with two internal states!
64 EFFECTIVE HAMILTONIAN I cont. Incidentally, one obtain the abelian field in the following way: ( Ω 0 + ω Gy)F y + λf 2 y Ω R F z Again, set: ω = λ + Ω o m = 1 m =0 m =1
65 EFFECTIVE HAMILTONIAN I cont. Incidentally, one obtain the abelian field in the following way: ( Ω 0 + ω Gy)F y + λf 2 y Ω R F z Again, set: ω = λ + Ω o
66 EFFECTIVE HAMILTONIAN I cont. Incidentally, one obtain the abelian field in the following way: ( Ω 0 + ω Gy)F y + λf 2 y Ω R F z Again, set: ω = λ + Ω o But now, we choose: Ω R λ, q
67 EFFECTIVE HAMILTONIAN I cont. Incidentally, one obtain the abelian field in the following way: ( Ω 0 + ω Gy)F y + λf 2 y Ω R F z Again, set: ω = λ + Ω o But now, we choose: Ω R λ, q m = 1 m =0 m =1
68 EFFECTIVE HAMILTONIAN I cont. Incidentally, one obtain the abelian field in the following way: ( Ω 0 + ω Gy)F y + λf 2 y Ω R F z Again, set: ω = λ + Ω o But now, we choose: Ω R λ, q Ω R
69 EFFECTIVE HAMILTONIAN I cont. Incidentally, one obtain the abelian field in the following way: ( Ω 0 + ω Gy)F y + λf 2 y Ω R F z Again, set: ω = λ + Ω o But now, we choose: Ω R λ, q Ω R q
70 EFFECTIVE HAMILTONIAN I cont. Incidentally, one obtain the abelian field in the following way: ( Ω 0 + ω Gy)F y + λf 2 y Ω R F z Again, set: ω = λ + Ω o But now, we choose: Ω R λ, q Ω R q
71 EFFECTIVE HAMILTONIAN I cont. Incidentally, one obtain the abelian field in the following way: e iqxf y ( Ω 0 + ω Gy)F y + λ 2 F 2 y Ω R F z e iqxf y y 2π q x Weiran Li and Tin- Lun Ho, to appear.
72 EFFECTIVE HAMILTONIAN I cont. Incidentally, one obtain the abelian field in the following way: e iqxf y ( Ω 0 + ω Gy)F y + λ 2 F 2 y Ω R F z e iqxf y y 2π q x Weiran Li and Tin- Lun Ho, to appear.
73 EFFECTIVE HAMILTONIAN I cont. Incidentally, one obtain the abelian field in the following way: e iqxf y ( Ω 0 + ω Gy)F y + λ 2 F 2 y Ω R F z e iqxf y y 2π q x Weiran Li and Tin- Lun Ho, to appear.
74 EFFECTIVE HAMILTONIAN I cont. Incidentally, one obtain the abelian field in the following way: e iqxf y ( Ω 0 + ω Gy)F y + λ 2 F 2 y Ω R F z e iqxf y y 2π q x Weiran Li and Tin- Lun Ho, to appear.
75 EFFECTIVE HAMILTONIAN II Within the lowest two states, the effec3ve single par3cle Hamiltonian is: H = 2 2M i + ˆxq Ω R 2 Ω R 2 0
76 EFFECTIVE HAMILTONIAN II Within the lowest two states, the effec3ve single par3cle Hamiltonian is: H = 2 2M i + ˆxq Ω R 2 Ω R 2 0 Can be regarded as a linear combina3on of Rashba and Dresselhaus spin- orbit coupling!
77 EFFECTIVE HAMILTONIAN II Within the lowest two states, the effec3ve single par3cle Hamiltonian is: H = 2 2M i + ˆxq Ω R 2 Ω R 2 0 E 1 (p) Can be regarded as a linear combina3on of Rashba and Dresselhaus spin- orbit coupling! E 0 (p) k 0 k 0 k p + q 2
78 EFFECTIVE HAMILTONIAN II Within the lowest two states, the effec3ve single par3cle Hamiltonian is: H = 2 2M i + ˆxq Ω R 2 Ω R 2 0 E 1 (p) Can be regarded as a linear combina3on of Rashba and Dresselhaus spin- orbit coupling! Symmetry properties χ n = e iγ e iqx (τ 1 ) nm χ m E 0 (p) Tuning parameter: sin θ 4MΩ R 2q 2 k 0 k 0 k p + q 2 k o = q 2 cos θ
79 EFFECTIVE HAMILTONIAN II Within the lowest two states, the effec3ve single par3cle Hamiltonian is: H = 2 2M i + ˆxq Ω R 2 Ω R 2 0 E 1 (p) Can be regarded as a linear combina3on of Rashba and Dresselhaus spin- orbit coupling! Symmetry properties χ n = e iγ e iqx (τ 1 ) nm χ m E 0 (p) Tuning parameter: sin θ 4MΩ R 2q 2 k 0 k 0 k p + q 2 k o = q 2 cos θ χ (p +) = icos θ 2 sin θ 2 χ (p +) = isin θ 2 cos θ 2 χ (p +) χ (p +) =sinθ = 0
80 STRUCTURE OF THE CONDENSATE I
81 STRUCTURE OF THE CONDENSATE I E 1 (p) E 0 (p) k 0 k 0 k p + q 2
82 STRUCTURE OF THE CONDENSATE I What is the Gross- Pitaevskii ansatz? E 1 (p) E 0 (p) k 0 k 0 k p + q 2
83 STRUCTURE OF THE CONDENSATE I What is the Gross- Pitaevskii ansatz? E 1 (p) Φ m (x) =A + χ (p +) m (x)+a χ (p ) m (x) Our task is to fix the two complex coefficients. For that we need the full GP energy func3onal. E 0 (p) k 0 k 0 k p + q 2
84 STRUCTURE OF THE CONDENSATE I What is the Gross- Pitaevskii ansatz? E 1 (p) Φ m (x) =A + χ (p +) m (x)+a χ (p ) m (x) Our task is to fix the two complex coefficients. For that we need the full GP energy func3onal. ˆK = E 0 (p) k 0 k 0 k p + q 2 ˆφ mh mn ˆφn ˆn mg mnˆn n +(V µ)ˆn
85 STRUCTURE OF THE CONDENSATE I What is the Gross- Pitaevskii ansatz? E 1 (p) Φ m (x) =A + χ (p +) m (x)+a χ (p ) m (x) Our task is to fix the two complex coefficients. For that we need the full GP energy func3onal. ˆK = E 0 (p) k 0 k 0 k p + q 2 ˆφ mh mn ˆφn ˆn mg mnˆn n +(V µ)ˆn Note: index m and n correspond to the original spin states m=1 and m=0, rather than the two degenerate states at +(- ) k 0 g mn is the interac3on matrix between different spin states. For later convenience, we define the following parameters: g g 11 + g 00 2 α g 10 g β g 11 g 00 g
86 STRUCTURE OF THE CONDENSATE II g g 11 + g 00 2 α g 10 g β g 11 g 00 g Φ m (x) =A + χ (p +) m (x)+a χ (p ) m (x) β β c II p + I (p +,p ) α c α β c III p
87 STRUCTURE OF THE CONDENSATE II g g 11 + g 00 2 α g 10 g β g 11 g 00 g Φ m (x) =A + χ (p +) m (x)+a χ (p ) m (x) β β c II p + I (p +,p ) α c α β c III p
88 STRUCTURE OF THE CONDENSATE II g g 11 + g 00 2 α g 10 g β g 11 g 00 g β Φ m (x) =A + χ (p +) m (x)+a χ (p ) m (x) Region I: both components β c II p + I (p +,p ) α c α β c III p
89 STRUCTURE OF THE CONDENSATE II g g 11 + g 00 2 α g 10 g β g 11 g 00 g β Φ m (x) =A + χ (p +) m (x)+a χ (p ) m (x) Region I: both components Region II & III: single components. β c II p + I (p +,p ) α c α β c III p
90 STRUCTURE OF THE CONDENSATE II g g 11 + g 00 2 α g 10 g β g 11 g 00 g β Φ m (x) =A + χ (p +) m (x)+a χ (p ) m (x) Region I: both components Region II & III: single components. β c II p + α c = 2 tan2 θ 2 + tan 2 θ > 0 I (p +,p ) α c α β c = cos θ(2 tan 2 θ) β c III p
91 STRUCTURE OF THE CONDENSATE II g g 11 + g 00 2 α g 10 g β g 11 g 00 g Φ m (x) =A + χ (p +) m (x)+a χ (p ) m (x) Region I: both components Region II & III: single components. α c = 2 tan2 θ 2 + tan 2 θ > 0 β c = cos θ(2 tan 2 θ)
92 STRUCTURE OF THE CONDENSATE II g g 11 + g 00 2 α g 10 g β g 11 g 00 g Φ m (x) =A + χ (p +) m (x)+a χ (p ) m (x) Region I: both components Region II & III: single components. α c = 2 tan2 θ 2 + tan 2 θ > 0 β c = cos θ(2 tan 2 θ) Note: non- zero density modula3on develops as a result of the fact:
93 STRUCTURE OF THE CONDENSATE II g g 11 + g 00 2 α g 10 g β g 11 g 00 g Φ m (x) =A + χ (p +) m (x)+a χ (p ) m (x) Region I: both components Region II & III: single components. α c = 2 tan2 θ 2 + tan 2 θ > 0 β c = cos θ(2 tan 2 θ) Note: non- zero density modula3on develops as a result of the fact: χ (p +) χ (p +) =sinθ = 0
94 MORE REFERENCES ON SPIN- ORBIT COUPLE BECS: Discussions on spin- orbit coupled BEC in the literatures, for example: T.Stanescu, B.Anderson and V. Galitski, PRA 78, (2008) Jonas Larson and Eric Sjoqvist, PRA 79, (2009) Chunji Wang, Chao Gao,Chao- Ming Jian and Hui Zhai, PRL 105, (2010) Congjun Wu and Ian Mondragon- Shem, arxiv: S.- K. Yip, arxiv: Other effects related to abelian/non- abelian gauge fields: Jaksch&Zoller, Lewenstein, Reseckas et al., Gerbier&Dalibard......
95 SO, WHAT CAN WE DO WITH IT?
96 ELASTIC MATTER LATTICE
97 ELASTIC MATTER LATTICE We have a Bose- Einstein Condensate with density modula3ons: Φ 0 (x) =A + e iθ ++ik 0 x χ + (k 0 )+A e iθ ik 0 z χ (k 0 )
98 ELASTIC MATTER LATTICE We have a Bose- Einstein Condensate with density modula3ons: Boson density given by: Φ 0 (x) =A + e iθ ++ik 0 x χ + (k 0 )+A e iθ ik 0 z χ (k 0 ) n (0) B (x) =n 0 +2A + A sin θ cos(2k 0 x + ϕ)
99 ELASTIC MATTER LATTICE We have a Bose- Einstein Condensate with density modula3ons: Boson density given by: Φ 0 (x) =A + e iθ ++ik 0 x χ + (k 0 )+A e iθ ik 0 z χ (k 0 ) n (0) B (x) =n 0 +2A + A sin θ cos(2k 0 x + ϕ) Sliding phase cf. CDW
100 ELASTIC MATTER LATTICE We have a Bose- Einstein Condensate with density modula3ons: Boson density given by: Φ 0 (x) =A + e iθ ++ik 0 x χ + (k 0 )+A e iθ ik 0 z χ (k 0 ) n (0) B (x) =n 0 +2A + A sin θ cos(2k 0 x + ϕ) Sliding phase cf. CDW n B (x) x
101 ELASTIC MATTER LATTICE We have a Bose- Einstein Condensate with density modula3ons: Boson density given by: Φ 0 (x) =A + e iθ ++ik 0 x χ + (k 0 )+A e iθ ik 0 z χ (k 0 ) n (0) B (x) =n 0 +2A + A sin θ cos(2k 0 x + ϕ) Sliding phase cf. CDW n B (x) This looks just like the intensity field of light in the op3cal lalce! Unlike the usual op3cal lalce, the poten3al can respond to external perturba3ons and support its own dynamics! Possibility of observing Peierls distor3on in the system! x
102 PEIERLS DISTORTION I
103 PEIERLS DISTORTION I Peierls (1950s,1930s according to wiki) showed that 1D coupled electron- lalce system leads to spontaneous distor3on (dimeriza3on) of the lalce (cf. Jone s theory of Bismuth)!
104 PEIERLS DISTORTION I Peierls (1950s,1930s according to wiki) showed that 1D coupled electron- lalce system leads to spontaneous distor3on (dimeriza3on) of the lalce (cf. Jone s theory of Bismuth)! t ij 1 2 i j N
105 PEIERLS DISTORTION I Peierls (1950s,1930s according to wiki) showed that 1D coupled electron- lalce system leads to spontaneous distor3on (dimeriza3on) of the lalce (cf. Jone s theory of Bismuth)! t ij 1 2 i j N
106 PEIERLS DISTORTION I Peierls (1950s,1930s according to wiki) showed that 1D coupled electron- lalce system leads to spontaneous distor3on (dimeriza3on) of the lalce (cf. Jone s theory of Bismuth)! t ij 1 2 i j N ϕ i ϕ j 1 2 i j N
107 PEIERLS DISTORTION I Peierls (1950s,1930s according to wiki) showed that 1D coupled electron- lalce system leads to spontaneous distor3on (dimeriza3on) of the lalce (cf. Jone s theory of Bismuth)! t ij 1 2 i j N ϕ i ϕ j 1 2 i j N Consider the following dimeriza3on pavern: ϕ i =( 1) i δ
108 PEIERLS DISTORTION I Peierls (1950s,1930s according to wiki) showed that 1D coupled electron- lalce system leads to spontaneous distor3on (dimeriza3on) of the lalce (cf. Jone s theory of Bismuth)! t ij 1 2 i j N ϕ i ϕ j 1 2 i j N Consider the following dimeriza3on pavern: ϕ i =( 1) i δ
109 PEIERLS DISTORTION I Peierls (1950s,1930s according to wiki) showed that 1D coupled electron- lalce system leads to spontaneous distor3on (dimeriza3on) of the lalce (cf. Jone s theory of Bismuth)! t ij 1 2 i j N ϕ i ϕ j 1 2 i j N Consider the following dimeriza3on pavern: Peierls showed that the energy gain in the electronic system is given by: ϕ i =( 1) i δ δ 2 log δ
110 PEIERLS DISTORTION I Peierls (1950s,1930s according to wiki) showed that 1D coupled electron- lalce system leads to spontaneous distor3on (dimeriza3on) of the lalce (cf. Jone s theory of Bismuth)! t ij 1 2 i j N ϕ i ϕ j 1 2 i j N Consider the following dimeriza3on pavern: Peierls showed that the energy gain in the electronic system is given by: ϕ i =( 1) i δ δ 2 log δ Cost of elas3city energy is given by: δ 2
111 PEIERLS DISTORTION I Peierls (1950s,1930s according to wiki) showed that 1D coupled electron- lalce system leads to spontaneous distor3on (dimeriza3on) of the lalce (cf. Jone s theory of Bismuth)! t ij 1 2 i j N ϕ i ϕ j 1 2 i j N Consider the following dimeriza3on pavern: Peierls showed that the energy gain in the electronic system is given by: ϕ i =( 1) i δ δ 2 log δ Cost of elas3city energy is given by: polyacetylene δ 2
112 PEIERLS DISTORTION II
113 PEIERLS DISTORTION II E 1 (p) E 0 (p) k 0 k 0 k p + q 2
114 PEIERLS DISTORTION II E 1 (p) E 0 (p) k 0 k 0 k p + q 2
115 PEIERLS DISTORTION II E 1 (p) Ini3al configura3on: Φ 0 (x) =A + e iθ ++ik 0 x χ + (k 0 )+A e iθ ik 0 z χ (k 0 ) E 0 (p) n (0) B (x) =n 0 +2A + A sin θ cos(2k 0 x + ϕ) k 0 k 0 k p + q 2
116 PEIERLS DISTORTION II E 1 (p) Ini3al configura3on: Φ 0 (x) =A + e iθ ++ik 0 x χ + (k 0 )+A e iθ ik 0 z χ (k 0 ) E 0 (p) n (0) B (x) =n 0 +2A + A sin θ cos(2k 0 x + ϕ) k 0 k 0 k p + q 2
117 PEIERLS DISTORTION II E 1 (p) Ini3al configura3on: Φ 0 (x) =A + e iθ ++ik 0 x χ + (k 0 )+A e iθ ik 0 z χ (k 0 ) E 0 (p) n (0) B (x) =n 0 +2A + A sin θ cos(2k 0 x + ϕ) k 0 k 0 k p + q 2 Condensate structure: Φ 1 (x) =A + e iθ ++ik 0 z χ k0 (x)+a e iθ ik 0 z χ k0 (z)+b χ 0 (x) n (1) B = n 0 + const. cos(2k 0 x + θ + θ ) + const. cos(k 0 x + ϕ )
118 PEIERLS DISTORTION II E 1 (p) Ini3al configura3on: Φ 0 (x) =A + e iθ ++ik 0 x χ + (k 0 )+A e iθ ik 0 z χ (k 0 ) E 0 (p) k 0 k 0 k p + q 2 n (0) B (x) =n 0 +2A + A sin θ cos(2k 0 x + ϕ) Condensate structure: Φ 1 (x) =A + e iθ ++ik 0 z χ k0 (x)+a e iθ ik 0 z χ k0 (z)+b χ 0 (x) n (1) B = n 0 + const. cos(2k 0 x + θ + θ ) + const. cos(k 0 x + ϕ )
119 PEIERLS DISTORTION II E 1 (p) Ini3al configura3on: Φ 0 (x) =A + e iθ ++ik 0 x χ + (k 0 )+A e iθ ik 0 z χ (k 0 ) E 0 (p) k 0 k 0 k p + q 2 n (0) B (x) =n 0 +2A + A sin θ cos(2k 0 x + ϕ) Condensate structure: Φ 1 (x) =A + e iθ ++ik 0 z χ k0 (x)+a e iθ ik 0 z χ k0 (z)+b χ 0 (x) n (1) B = n 0 + const. cos(2k 0 x + θ + θ ) + const. cos(k 0 x + ϕ ) energy cost for bosons: = E 0 (p = 0) E 0 (p = k 0 ) E B = B 2
120 PEIERLS DISTORTION III The fermionic sector: sees an external poten3al: within mean field: V(x) =g bf n (1) B (x) resonant regime? To determine the size of the effect, we solve the band structure in the poten3al. α g bf w 2 2πv F β g bf n B δ The size of the gap can be calculated in the perturba3ve limit (cf. BCS) = 2 F e 1 αβ b = 2 F g bf n b w e 1 αβ
121 PEIERLS DISTORTION IV
122 FRACTIONAL CHARGE As is well- known in the polyacetylene research: 1. Soliton excita3ons of the lalce: E(- δ)= E(δ) ϕ i =( 1) i δ f i
123 FRACTIONAL CHARGE As is well- known in the polyacetylene research: 1. Soliton excita3ons of the lalce: E(- δ)= E(δ) ϕ i =( 1) i δ f i 2. fermion frac3onaliza3on around the soliton.
124 FRACTIONAL CHARGE As is well- known in the polyacetylene research: 1. Soliton excita3ons of the lalce: E(- δ)= E(δ) ϕ i =( 1) i δ f i 2. fermion frac3onaliza3on around the soliton.
125 FRACTIONAL CHARGE As is well- known in the polyacetylene research: 1. Soliton excita3ons of the lalce: E(- δ)= E(δ) ϕ i =( 1) i δ f i 2. fermion frac3onaliza3on around the soliton
126 FRACTIONAL CHARGE As is well- known in the polyacetylene research: 1. Soliton excita3ons of the lalce: E(- δ)= E(δ) ϕ i =( 1) i δ f i 2. fermion frac3onaliza3on around the soliton
127 FRACTIONAL CHARGE As is well- known in the polyacetylene research: 1. Soliton excita3ons of the lalce: E(- δ)= E(δ) ϕ i =( 1) i δ f i 2. fermion frac3onaliza3on around the soliton
128 FRACTIONAL CHARGE As is well- known in the polyacetylene research: 1. Soliton excita3ons of the lalce: E(- δ)= E(δ) ϕ i =( 1) i δ f i 2. fermion frac3onaliza3on around the soliton Can be used as a qubit. We have demonstrate all the necessary one qubit opera3on and CNOT gate!
129 WHAT HAVE WE LEARNED?
130 SUMMARY A general way of looking at light induced abelian/non- abelian gauge fields (N.B. fermions) i =2f +1 q i =2 i =1
131 SUMMARY A general way of looking at light induced abelian/non- abelian gauge fields (N.B. fermions) i =2f +1 q i =2 i =1 y 2π q x
132 SUMMARY A general way of looking at light induced abelian/non- abelian gauge fields (N.B. fermions) i =2f +1 y y q i =2 i =1 y c y c I A ± =0 x c II A + =0,A =0 III A =0,A + =0 x 2π q x
133 FUTURE DIRECTIONS 1. Crossover from abelian to non- abelian gauge field; mel3ng of the vortex lalce; 2. Vor3ces in a spin- orbit coupled Bose- Einstein condensate; 3. Transport proper3es of bosons with spin- orbit interac3ons; 4. Maver lalce in high dimensions; electron- phonon system (BO approx.)
134 CONCLUSIONS 1. We discuss the general route to construct abelian/non- abelian gauge field. Example: NIST experiment; 2. We worked out the phase diagram of the NIST experiment; 3. The condensate will develop appropriate density (spin) modula3on in the presence of spin- orbit coupling 4. The possibility of genera3ng Peierls distor3on and as a result, frac3onalized fermion;
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