Geometry of the Berry Phase

Transcription

1 ... a concise µ-seminar exposition... October 11, Řež

2

3 The problem formulation What is given:

4 The problem formulation What is given: separable Hilbert space H

5 The problem formulation What is given: separable Hilbert space H space of parameters Q (structureness object, e.g. manifold)

6 The problem formulation What is given: separable Hilbert space H space of parameters Q (structureness object, e.g. manifold) smooth family of self-adjoint Hamiltonians {Ĥ(q)} q Q in H

7 The problem formulation What is given: separable Hilbert space H space of parameters Q (structureness object, e.g. manifold) smooth family of self-adjoint Hamiltonians {Ĥ(q)} q Q in H simplification: spectra Ĥ(q) are discrete and nondegenerate

8 The problem formulation What is given: separable Hilbert space H space of parameters Q (structureness object, e.g. manifold) smooth family of self-adjoint Hamiltonians {Ĥ(q)} q Q in H simplification: spectra Ĥ(q) are discrete and nondegenerate equivalently, there a collection of smooth maps {Φ m } Φ m : q { m(q), E m (q)}, where Ĥ(q) m(q) = E m (q) m(q)

9 The problem formulation What is given: separable Hilbert space H space of parameters Q (structureness object, e.g. manifold) smooth family of self-adjoint Hamiltonians {Ĥ(q)} q Q in H simplification: spectra Ĥ(q) are discrete and nondegenerate equivalently, there a collection of smooth maps {Φ m } Φ m : q { m(q), E m (q)}, where Ĥ(q) m(q) = E m (q) m(q) Remark I.: (gauge transformation in given eigenvalue sector) Φ m Φ m : q { m(q) = e iαm(q) m(q), E m (q)}

10 Geometrical overview - fiber bundle perspective Q H ˆ m(q) {H, Ĥ(q)} ˆ m(q ) {H, Ĥ(q )} ˆ n(q) ˆ k(q) ˆ n(q ) ˆ k(q ) q q Q

11 The problem formulation continues... What is still given:

12 The problem formulation continues... What is still given: a circuit (smooth loop) C in the parameter space Q C : t [0, T] q C (t) Q, such that q C (0) = q C (T)

13 The problem formulation continues... What is still given: a circuit (smooth loop) C in the parameter space Q C : t [0, T] q C (t) Q, such that q C (0) = q C (T) an initial condition = n-th eigenstate n(q C (t = 0)) H

14 The problem formulation continues... What is still given: a circuit (smooth loop) C in the parameter space Q C : t [0, T] q C (t) Q, such that q C (0) = q C (T) an initial condition = n-th eigenstate n(q C (t = 0)) H What is the task:

15 The problem formulation continues... What is still given: a circuit (smooth loop) C in the parameter space Q C : t [0, T] q C (t) Q, such that q C (0) = q C (T) an initial condition = n-th eigenstate n(q C (t = 0)) H What is the task: to find a solution Ψ(q C (t = T)) of the Schrödinger equation: i t Ψ(t) = Ĥ(q C (t)) Ψ(t) + initial condition

16

17 General procedure

18 General procedure circuit C specifies a subensemble: {q, t m(q C ), E m (q C )}

19 General procedure circuit C specifies a subensemble: {q, t m(q C ), E m (q C )} ansatz:

20 General procedure circuit C specifies a subensemble: {q, t m(q C ), E m (q C )} ansatz: Ψ = m B m (t) exp{ i t 0 E m (q C (τ))dτ} m(q C (t))

21 General procedure circuit C specifies a subensemble: {q, t m(q C ), E m (q C )} ansatz: Ψ = m B m (t) exp{ i t 0 E m (q C (τ))dτ} m(q C (t)) self-adjointness of the family of Hamiltonians {Ĥ(q)} q Q

22 General procedure circuit C specifies a subensemble: {q, t m(q C ), E m (q C )} ansatz: Ψ = m B m (t) exp{ i t 0 E m (q C (τ))dτ} m(q C (t)) self-adjointness of the family of Hamiltonians {Ĥ(q)} q Q set of equations: d dt {Ĥ(q C ) m(q C ) } = d dt {E m(q C ) m(q C ) }

23 General procedure circuit C specifies a subensemble: {q, t m(q C ), E m (q C )} ansatz: Ψ = m B m (t) exp{ i t 0 E m (q C (τ))dτ} m(q C (t)) self-adjointness of the family of Hamiltonians {Ĥ(q)} q Q set of equations: d dt {Ĥ(q C ) m(q C ) } = d dt {E m(q C ) m(q C ) } result in a system of the first-order diff. eq. for {B m (t)}: Ḃ m = B m m ṁ k m B k m d dtĥ(t) k E k E m exp{ i t 0 (E k E m )dτ}

24 General procedure circuit C specifies a subensemble: {q, t m(q C ), E m (q C )} ansatz: Ψ = m B m (t) exp{ i t 0 E m (q C (τ))dτ} m(q C (t)) self-adjointness of the family of Hamiltonians {Ĥ(q)} q Q set of equations: d dt {Ĥ(q C ) m(q C ) } = d dt {E m(q C ) m(q C ) } result in a system of the first-order diff. eq. for {B m (t)}: Ḃ m = B m m ṁ k m B k m d dtĥ(t) k E k E m exp{ i together with set of initial conditions: {B m (t = 0) = δ mn } t 0 (E k E m )dτ}

25 Adiabatic regime solution

26 Adiabatic regime solution d by definition: adiabatic regime m dtĥ(t) k 0 for m k

27 Adiabatic regime solution d by definition: adiabatic regime m dtĥ(t) k 0 for m k physically it is equivalent to the requirement: mk d dtĥ(q C) Ĥ(q C ), 1 mk = 1 min q C { E m (q C ) E k (q C ) }

28 Adiabatic regime solution d by definition: adiabatic regime m dtĥ(t) k 0 for m k physically it is equivalent to the requirement: mk d dtĥ(q C) Ĥ(q C ), 1 mk = 1 min q C { E m (q C ) E k (q C ) } adiabatic theorem: Ψ(t) = e iγn(t) n(q C (t)), where γ n (t) = 1 t E n (τ)dτ + Im n(q) d dq n(q) dq =: γd n + γ G n 0 C(t)

29 Adiabatic regime solution d by definition: adiabatic regime m dtĥ(t) k 0 for m k physically it is equivalent to the requirement: mk d dtĥ(q C) Ĥ(q C ), 1 mk = 1 min q C { E m (q C ) E k (q C ) } adiabatic theorem: Ψ(t) = e iγn(t) n(q C (t)), where γ n (t) = 1 t 0 E n (τ)dτ + Im C(t) n(q) d dq n(q) dq =: γd n + γ G n Remark II.: for almost 50 years was γ G n ignored (!)

30 Adiabatic regime solution d by definition: adiabatic regime m dtĥ(t) k 0 for m k physically it is equivalent to the requirement: mk d dtĥ(q C) Ĥ(q C ), 1 mk = 1 min q C { E m (q C ) E k (q C ) } adiabatic theorem: Ψ(t) = e iγn(t) n(q C (t)), where γ n (t) = 1 t 0 E n (τ)dτ + Im C(t) n(q) d dq n(q) dq =: γd n + γ G n Remark II.: for almost 50 years was γ G n ignored (!) Born-Fock gauge fixing: n(q) ñ(q) = e iγg n n(q)

31 Few facts...

32 Few facts... M. Berry: it fails globally, if fundamental group π 1 (C) {e}

33 Few facts... M. Berry: it fails globally, if fundamental group π 1 (C) {e} some fundamental groups: π 1 (point) = {e} π 1 (line) = {e} π 1 (cirle) = Z π 1 ( ) = Z Z/{a 3 b 2 }

34 Few facts... M. Berry: it fails globally, if fundamental group π 1 (C) {e} some fundamental groups: π 1 (point) = {e} π 1 (line) = {e} π 1 (cirle) = Z π 1 ( ) = Z Z/{a 3 b 2 } how to observe a geometric phase(?):

35 Few facts... M. Berry: it fails globally, if fundamental group π 1 (C) {e} some fundamental groups: π 1 (point) = {e} π 1 (line) = {e} π 1 (cirle) = Z π 1 ( ) = Z Z/{a 3 b 2 } how to observe a geometric phase(?): interference experiments

36 Few facts... M. Berry: it fails globally, if fundamental group π 1 (C) {e} some fundamental groups: π 1 (point) = {e} π 1 (line) = {e} π 1 (cirle) = Z π 1 ( ) = Z Z/{a 3 b 2 } how to observe a geometric phase(?): interference experiments Ψ(0) = a n n +a m m Ψ(T) = a n e iγn n +a m e iγm m

37 Few facts... M. Berry: it fails globally, if fundamental group π 1 (C) {e} some fundamental groups: π 1 (point) = {e} π 1 (line) = {e} π 1 (cirle) = Z π 1 ( ) = Z Z/{a 3 b 2 } how to observe a geometric phase(?): interference experiments Ψ(0) = a n n +a m m Ψ(T) = a n e iγn n +a m e iγm m measurement of Ψ(T)  Ψ(T) for some observable  a n 2 n  n + a m 2 m  m + 2Re{a m a n n  m e i(γm γn) } }{{}

38 Few facts... M. Berry: it fails globally, if fundamental group π 1 (C) {e} some fundamental groups: π 1 (point) = {e} π 1 (line) = {e} π 1 (cirle) = Z π 1 ( ) = Z Z/{a 3 b 2 } how to observe a geometric phase(?): interference experiments Ψ(0) = a n n +a m m Ψ(T) = a n e iγn n +a m e iγm m measurement of Ψ(T)  Ψ(T) for some observable  a n 2 n  n + a m 2 m  m + 2Re{a m a n n  m e i(γm γn) } }{{} contributes only if [Â,Ĥ(T)] 0

39

40 Quintessences - geometrical prerequisites

41 Quintessences - geometrical prerequisites in the adiabatic regime system stays in the n-th eigenstate

42 Quintessences - geometrical prerequisites in the adiabatic regime system stays in the n-th eigenstate n-th spectral bundle (subset of the trivial vector bundle Q H) L n := {(q, Ψ ) Q H, Ψ Ψ = 1 and Ĥ(q) Ψ = E n (q) Ψ }

43 Quintessences - geometrical prerequisites in the adiabatic regime system stays in the n-th eigenstate n-th spectral bundle (subset of the trivial vector bundle Q H) L n := {(q, Ψ ) Q H, Ψ Ψ = 1 and Ĥ(q) Ψ = E n (q) Ψ } L n Q H inherits connection, similarly as Σ surface R 3

44 Quintessences - geometrical prerequisites in the adiabatic regime system stays in the n-th eigenstate n-th spectral bundle (subset of the trivial vector bundle Q H) L n := {(q, Ψ ) Q H, Ψ Ψ = 1 and Ĥ(q) Ψ = E n (q) Ψ } L n Q H inherits connection, similarly as Σ surface R 3 RLC connection uses ambient space parallelism and metric:

45 Quintessences - geometrical prerequisites in the adiabatic regime system stays in the n-th eigenstate n-th spectral bundle (subset of the trivial vector bundle Q H) L n := {(q, Ψ ) Q H, Ψ Ψ = 1 and Ĥ(q) Ψ = E n (q) Ψ } L n Q H inherits connection, similarly as Σ surface R 3 RLC connection uses ambient space parallelism and metric: {R 3, (..)} w(q) w(q ) Σ surface q q w RLC (q )

46 Quintessences - geometrical prerequisites

47 Quintessences - geometrical prerequisites Berry-Simon connection (Berry-Barry) follows the same idea:

48 Quintessences - geometrical prerequisites Berry-Simon connection (Berry-Barry) follows the same idea: L n Σ surface and Q H {R 3, (..)}

49 Quintessences - geometrical prerequisites Berry-Simon connection (Berry-Barry) follows the same idea: L n Σ surface and Q H {R 3, (..)} picture of the Berry-Barry parallel transport:

50 Quintessences - geometrical prerequisites Berry-Simon connection (Berry-Barry) follows the same idea: L n Σ surface and Q H {R 3, (..)} picture of the Berry-Barry parallel transport: H L n (q ) H L n (q) Ψ(q ) BB Ψ(q) q Ψ(q ) q

51 Gauge potential A µ

52 Gauge potential A µ choose a local gauge (section) Φ n : q Q ( q, n(q) ) L n

53 Gauge potential A µ choose a local gauge (section) Φ n : q Q ( q, n(q) ) L n what is a parallel state-mate of n(q) after an infinitesimal parameter translation q q = q + δv?

54 Gauge potential A µ choose a local gauge (section) Φ n : q Q ( q, n(q) ) L n what is a parallel state-mate of n(q) after an infinitesimal parameter translation q q = q + δv? the answer:

55 Gauge potential A µ choose a local gauge (section) Φ n : q Q ( q, n(q) ) L n what is a parallel state-mate of n(q) after an infinitesimal parameter translation q q = q + δv? the answer: n(q ) BB = n(q ) n(q) n(q ) = e iδϕ n(q

56 Gauge potential A µ choose a local gauge (section) Φ n : q Q ( q, n(q) ) L n what is a parallel state-mate of n(q) after an infinitesimal parameter translation q q = q + δv? the answer: n(q ) BB = n(q ) n(q) n(q ) = e iδϕ n(q the gauge potential arises on parameter space: δϕ = A µ (q) δv µ = i n(q) q µn(q) δv µ

57 Gauge potential A µ choose a local gauge (section) Φ n : q Q ( q, n(q) ) L n what is a parallel state-mate of n(q) after an infinitesimal parameter translation q q = q + δv? the answer: n(q ) BB = n(q ) n(q) n(q ) = e iδϕ n(q the gauge potential arises on parameter space: δϕ = A µ (q) δv µ = i n(q) q µn(q) δv µ Remark III.: n(q) q µn(q) is pure imaginary quantity, thus A = A µ (q)dq µ = Im n(q) q µn(q) dq µ

58 More realistic perspective U(1) S 1 L n n(q) δv n(q ) BB δϕ Φ n n(q ) q δv q = q + δv Q

59 Berry phase as (an)holonomy of A µ

60 Berry phase as (an)holonomy of A µ gauge transformation: ( q, n(q) ) ( q, e iαn(q) n(q) )

61 Berry phase as (an)holonomy of A µ gauge transformation: ( q, n(q) ) ( q, e iαn(q) n(q) ) A µ A µ µ α n F µν := µ A ν ν A µ F µν

62 Berry phase as (an)holonomy of A µ gauge transformation: ( q, n(q) ) ( q, e iαn(q) n(q) ) A µ A µ µ α n F µν := µ A ν ν A µ F µν if the curvature (electromagnetic stress tensor) F µν 0, then BB transport is path dependent

63 Berry phase as (an)holonomy of A µ gauge transformation: ( q, n(q) ) ( q, e iαn(q) n(q) ) A µ A µ µ α n F µν := µ A ν ν A µ F µν if the curvature (electromagnetic stress tensor) F µν 0, then BB transport is path dependent phase acquired over a circuit C is the (an)holonomy of A µ, ie.

64 Berry phase as (an)holonomy of A µ gauge transformation: ( q, n(q) ) ( q, e iαn(q) n(q) ) A µ A µ µ α n F µν := µ A ν ν A µ F µν if the curvature (electromagnetic stress tensor) F µν 0, then BB transport is path dependent phase acquired over a circuit C is the (an)holonomy of A µ, ie. γn G = A µ dq µ 1 = 2 F µνdq µ dq ν, where Σ = C C Σ }{{}

65 Berry phase as (an)holonomy of A µ gauge transformation: ( q, n(q) ) ( q, e iαn(q) n(q) ) A µ A µ µ α n F µν := µ A ν ν A µ F µν if the curvature (electromagnetic stress tensor) F µν 0, then BB transport is path dependent phase acquired over a circuit C is the (an)holonomy of A µ, ie. γn G = A µ dq µ 1 = 2 F µνdq µ dq ν, where Σ = C C Σ }{{} magnetic flux through Σ

66 Conclusion

67 Conclusion unfortunately there is not time to discuss some example

68 Conclusion unfortunately there is not time to discuss some example in the case, when spectra of {Ĥ(q)} are degenerate, there is an analog of the geometric phase called Wilczek-Zee phase

69 Conclusion unfortunately there is not time to discuss some example in the case, when spectra of {Ĥ(q)} are degenerate, there is an analog of the geometric phase called Wilczek-Zee phase in classical mechanics, there is as well in adiabatic regime an analog of the Berry phase called Hannay phase (or angel)

70 a čo povedať fakt úplne na záver?

71 a čo povedať fakt úplne na záver? možno za všetkých povestná teta Kateřina z Jirotkovho Saturnina:

72 a čo povedať fakt úplne na záver? možno za všetkých povestná teta Kateřina z Jirotkovho Saturnina: A strýček opravdu na leccos přišel. Tak například zjistil při pokusu, který měl velmi vzrušující průběh, že lít vodu do kyseliny je blbost, a vůbec mu nevadilo, že tento poznatek, korektněji vyjádřený, mohl získat z učebnice chemie pro nižší třídy škol středních, aniž by si byl při tom popálil prsty a zánovní vestu.

73 a čo povedať fakt úplne na záver? možno za všetkých povestná teta Kateřina z Jirotkovho Saturnina: A strýček opravdu na leccos přišel. Tak například zjistil při pokusu, který měl velmi vzrušující průběh, že lít vodu do kyseliny je blbost, a vůbec mu nevadilo, že tento poznatek, korektněji vyjádřený, mohl získat z učebnice chemie pro nižší třídy škol středních, aniž by si byl při tom popálil prsty a zánovní vestu. thanks for your attention!