Geometric phases and spin-orbit effects

Transcription

1 Geometric phases and spin-orbit effects Lecture 1 Alexander Shnirman (KIT, Karlsruhe)

2 Outline Geometric phases (Abelian and non-abelian) Spin manipulation through non-abelian phases a) Toy model; b) Moving quantum dots Spin decay due to random geometric phase Spin pumping

3 Geometric phases

4 Adiabatic Theorem H(t) - changes slowly Discrete Spectrum with no crossings

5 Adiabatic Theorem H(t) - changes slowly Discrete Spectrum with no crossings

6 Adiabatic Theorem H(t) - changes slowly Discrete Spectrum with no crossings Diagonalize at each t H(t) Ψ n (t) = E n (t) Ψ n (t)

7 Adiabatic Theorem H(t) - changes slowly Discrete Spectrum with no crossings Diagonalize at each t H(t) Ψ n (t) = E n (t) Ψ n (t) Theorem : the system stays in Ψ n (t)

8 Adiabatic Theorem H(t) - changes slowly Discrete Spectrum with no crossings Diagonalize at each t H(t) Ψ n (t) = E n (t) Ψ n (t) Theorem : the system stays in Ψ n (t) U(t) n Ψ n (t)n For example n Ψ n (t = 0) H d (t) =U (t)h(t)u(t) = n E n nn - diagonal

9 Time-dependent gauge transformations

10 Time-dependent gauge transformations i d dt Ψ = H(t) Ψ Schrödinger equation

11 Time-dependent gauge transformations i d dt Ψ = H(t) Ψ Schrödinger equation Φ = W (t) Ψ Gauge transformation

12 Time-dependent gauge transformations i d dt Ψ = H(t) Ψ Schrödinger equation Φ = W (t) Ψ Gauge transformation New Schrödinger equation i d dt Φ = iẇ (t) Ψ + iw d dt Ψ i d dt Φ = iẇw 1 Φ + WHW 1 Φ

13 Time-dependent gauge transformations i d dt Ψ = H(t) Ψ Schrödinger equation Φ = W (t) Ψ Gauge transformation New Schrödinger equation i d dt Φ = iẇ (t) Ψ + iw d dt Ψ i d dt Φ = iẇw 1 Φ + WHW 1 Φ New Hamiltonian H = iẇw + WHW

14 Time-dependent gauge transformations i d dt Ψ = H(t) Ψ Schrödinger equation Φ = W (t) Ψ Gauge transformation New Schrödinger equation i d dt Φ = iẇ (t) Ψ + iw d dt Ψ i d dt Φ = iẇw 1 Φ + WHW 1 Φ New Hamiltonian H = iẇw + WHW Alternatively U = W 1 = W Ψ = U(t) Φ H = iu U + U HU

15 Berry Phases vs. Landau-Zener Transitions H(t) Ψ n (t) = E n (t) Ψ n (t) U(t) n Ψ n (t)n For example n Ψ n (t = 0)

16 Berry Phases vs. Landau-Zener Transitions H(t) Ψ n (t) = E n (t) Ψ n (t) U(t) n Ψ n (t)n For example n Ψ n (t = 0) H = U HU iu U = n E n nn i ml mψ m t Ψ l l

17 Berry Phases vs. Landau-Zener Transitions H(t) Ψ n (t) = E n (t) Ψ n (t) U(t) n Ψ n (t)n For example n Ψ n (t = 0) H = U HU iu U = n E n nn i ml mψ m t Ψ l l Perturbative logic

18 Berry Phases vs. Landau-Zener Transitions H(t) Ψ n (t) = E n (t) Ψ n (t) U(t) n Ψ n (t)n For example n Ψ n (t = 0) H = U HU iu U = n E n nn i ml mψ m t Ψ l l Perturbative logic iu U = i diag n nψ n t Ψ n n Berry Phases M. Berry, 1984

19 Berry Phases vs. Landau-Zener Transitions H(t) Ψ n (t) = E n (t) Ψ n (t) U(t) n Ψ n (t)n For example n Ψ n (t = 0) H = U HU iu U = n E n nn i ml mψ m t Ψ l l Perturbative logic iu U = i nψ n t Ψ n n diag n iu U = i mψ m t Ψ l l off diag m=l Berry Phases M. Berry, 1984 Landau-Zener Transitions Landau, Zener, Stückelberg, Majorana, 1932

20 Berry Phases H = U HU iu U = n E n nn i ml mψ m t Ψ l l iu U = i diag n nψ n t Ψ n n

21 Berry Phases H = U HU iu U = n E n nn i ml mψ m t Ψ l l iu U = i diag n nψ n t Ψ n n H = diag n (E n + δe n ) nn δe n = iψ n t Ψ n real!

22 Berry Phases H = U HU iu U = n E n nn i ml mψ m t Ψ l l iu U = i diag n nψ n t Ψ n n H = diag n (E n + δe n ) nn δe n = iψ n t Ψ n real! Berry Phase t t Φ n,berry = dt δe n (t )=i Ψ n dψ n t 0 t 0

23 Example: spin-1/2 in time-dependent magnetic field

24 Example: spin-1/2 in time-dependent magnetic field

25 Example: spin-1/2 in time-dependent magnetic field b = Ω b b B B

26 Example: spin-1/2 in time-dependent magnetic field b = Ω b b B B H(t) = 1 2 B(t) σ U HU = 1 2 B σz

27 Example: spin-1/2 in time-dependent magnetic field b = Ω b b B B H(t) = 1 2 B(t) σ U HU = 1 2 B σz iu U = 1 2 Ω σ Φ /,Berry = 1 2 Ω dt

28 Example: spin-1/2 in time-dependent magnetic field Φ /,Berry = 1 2 Ω dt Φ /,Berry = 1 2 Ω cos θdt = 1 2 ϕ cos θdt Φ /,Berry = 1 2 cos θdϕ Solid angle? up to a phase of π

29 Gauge (non-)invariance Berry Phase Φ n,berry = t t 0 dt δe n (t )=i t t 0 Ψ n dψ n δe n = iψ n t Ψ n

30 Gauge (non-)invariance Berry Phase Φ n,berry = t t 0 dt δe n (t )=i t t 0 Ψ n dψ n δe n = iψ n t Ψ n Gauge transformation Ψ n (t) e iϕ(t) Ψ n (t) δe n δe n + ϕ

31 Gauge (non-)invariance Berry Phase Φ n,berry = t t 0 dt δe n (t )=i t t 0 Ψ n dψ n δe n = iψ n t Ψ n Gauge transformation Ψ n (t) e iϕ(t) Ψ n (t) δe n δe n + ϕ Only if periodicity imposed: H(t 0 + T )=H(t 0 ) Ψ n (t 0 + T ) = Ψ n (t 0 ), i.e.,ϕ(t 0 + T )=ϕ(t 0 ) Φ n,berry (T )=i t 0 +T t 0 Ψ n dψ n Gauge invariant

32 Spin-1/2 again Φ /,Berry = 1 2 cos θdϕ Solid angle? up to a phase of π

33 Spin-1/2 again Φ /,Berry = 1 2 cos θdϕ Solid angle? up to a phase of π Solution of the problem

34 Spin-1/2 again Φ /,Berry = 1 2 cos θdϕ Solid angle? up to a phase of π Solution of the problem Diagonalizing transformation H(t) = 1 2 B(t) σ U HU = 1 2 B σz

35 Spin-1/2 again Solid angle? up to a phase of π Φ /,Berry = 1 2 cos θdϕ Solution of the problem Diagonalizing transformation H(t) = 1 2 B(t) σ U HU = 1 2 B σz U(t) =e iϕ(t) 2 σ z e iθ 2 σ y U(t 0 + T )=e iπσ z e iθ 2 σ y ϕ(t 0 + T )=2π

36 Spin-1/2 again Solid angle? up to a phase of π Φ /,Berry = 1 2 cos θdϕ Solution of the problem Diagonalizing transformation H(t) = 1 2 B(t) σ U HU = 1 2 B σz U(t) =e iϕ(t) 2 σ z e iθ 2 σ y U(t 0 + T )=e iπσ z e iθ 2 σ y We have to gauge out the phase of π ϕ(t 0 + T )=2π Φ /,Berry = 1 2 (cos θ 1)dϕ

37 Iterative approach M. Berry, 1987 H(t) Ψ n (t) = E n (t) Ψ n (t) U 1 (t) n Ψ n (t)n H 1 = U 1 HU 1 iu 1 U 1 = n E n nn i ml mψ m t Ψ l l Next iteration H 1 (t) Ψ 1 n(t) = E 1 n(t) Ψ 1 n(t) U 2 (t) n Ψ 1 n(t)n H 2 = U 2 H 1U 2 iu 2 U 2 = n E 1 n nn i ml mψ 1 m t Ψ 1 l l And so on... a) Berry phase with high precision b) Landau-Zener - not always

38 Evolution in parameter space Hamiltonian controlled by parameters H(t) =H(χ (t))

39 Evolution in parameter space Hamiltonian controlled by parameters H(t) =H(χ (t)) Φ n,berry = i t Ψ n t Ψ n dt = i t Ψ n χ Ψ n χ dt = i t Ψ n χ Ψ n dχ t 0 t 0 t 0

40 Evolution in parameter space Hamiltonian controlled by parameters H(t) =H(χ (t)) Φ n,berry = i t Ψ n t Ψ n dt = i t Ψ n χ Ψ n χ dt = i t Ψ n χ Ψ n dχ t 0 Φ n,berry = A n (χ )dχ C A n (χ ) Ψ n i χ Ψ n t 0 M. Berry, 1984 t 0

41 Evolution in parameter space Hamiltonian controlled by parameters H(t) =H(χ (t)) Φ n,berry = i t Ψ n t Ψ n dt = i t Ψ n χ Ψ n χ dt = i t Ψ n χ Ψ n dχ t 0 Φ n,berry = A n (χ )dχ C A n (χ ) Ψ n i χ Ψ n t 0 M. Berry, 1984 t 0

42 Evolution in parameter space Hamiltonian controlled by parameters H(t) =H(χ (t)) Φ n,berry = i t Ψ n t Ψ n dt = i t Ψ n χ Ψ n χ dt = i t Ψ n χ Ψ n dχ t 0 Φ n,berry = A n (χ )dχ C A n (χ ) Ψ n i χ Ψ n t 0 M. Berry, 1984 t 0 Vector potential in parameter space (cf. AB phase in r-space) χ Φ n,berry = A n ds = B n ds

43 How to make Berry phase observable? Superposition A n + B m Ae i R E n dt e i R An dχ n + Be i R E m dt e i R Am dχ m Interference t An,L (χ )dχ t An,R (χ )dχ t 0 t 0

44 Non-Abelian geometric phases Degenerate subspace (e.g. spin) H(t) =H(χ (t)) Ψ σ n(t) = Ψ σ n(χ (t))

45 Non-Abelian geometric phases Degenerate subspace (e.g. spin) H(t) =H(χ (t)) Ψ σ n(t) = Ψ σ n(χ (t))

46 Non-Abelian geometric phases Degenerate subspace (e.g. spin) H(t) =H(χ (t)) Ψ σ n(t) = Ψ σ n(χ (t)) Evolution operator in subspace n: U n = e i R E n dt U σ,σ n U σ,σ n = Pe i R Ân (χ )dχ A σ,σ n (χ )=Ψ σ n i χ Ψ σ n

47 Non-Abelian geometric phases Degenerate subspace (e.g. spin) H(t) =H(χ (t)) Ψ σ n(t) = Ψ σ n(χ (t)) Evolution operator in subspace n: U n = e i R E n dt U σ,σ n U σ,σ n = Pe i R Ân (χ )dχ A σ,σ n (χ )=Ψ σ n i χ Ψ σ n Non-Abelian phases are directly observable

48 Example: Semiclassical spin 1/2 A: Spin-orbit interaction momentum dependent magnetic field (B ext =0) Rashba Dresselhaus B: Semiclassical picture: electron moves a distance dr in time dt the spin is rotated by U[dr], independent of dt ( geometric ) W. A. Coish, V. N. Golovach, J. C. Egues, D. Loss. Physica Status Solidi (b) 243, 3658 (2006)

49 Geometric spin manipulation (toy model)

50 Toy model: (2 orbital states + spin) 2 orbital states + spin + spin-orbit coupling H = 1 2 τ z τ y ( b SO σ )+ 1 2 [Z(t)τ z + X(t)τ x ] The only form of SO allowed by time reversal symmetry bso strength and direction of SO coupling

51 Specific example: single electron in a double quantum dot X(t) 2 orbital states: H orb = 1 2 τ z [Z(t)τ z + X(t)τ x ] H SO = α(p y σ x p x σ y )+β( p x σ x + p y σ y ) 1 + Z(t) 2 τ y ( b SO σ ) Naive projection: in reality more orbital states involved b SO,x = i0 αp y βp x 1 b SO,y = i0 βp y αp x 1 b SO,z =0

52 Berry phase of the orbital pseudo-spin For two projections ± of the spin along bso H ± = 1 2 τ z ± 1 2 τ y b SO [Z(t)τ z + X(t)τ x ] z h ±,x (t) =X(t) h ±,y (t) =±b SO h ±,z (t) = + Z(t) ϕ θ ϕ b SO y x Ground (and excited) states 2-fold degenerate due to spin (Kramers degeneracy) E 0+ = E 0 = h+ (t) = 2 h (t)

53 Opposite Berry phases for opposite spin projections H ± = 1 2 τ z ± 1 2 τ y b SO [Z(t)τ z + X(t)τ x ] Φ Berry = 1 2 cos θdϕ z Conclusion We can rotate the spin θ around b SO by varying ϕ ϕ y orbital parameters X and Z x Abelian: b SO - const.

54 Toy model for non-abelian manipulations H = 1 2 τ z τ y bso (X, Z,... ) σ [Z(t)τ z + X(t)τ x ] Direction of SO coupling bso varies