The Aharonov-Bohm Effect

Michael Florian Wondrak The Aharonov-Bohm Effect 1 The Aharonov-Bohm Effect Michael Florian Wondrak wondrak@fias.uni-frankfurt.de Frankfurt Institute (FIAS) Institut für Theoretische Physik Goethe-Universität Frankfurt Journal Club in High Energy Physics February 1, 2016

Michael Florian Wondrak The Aharonov-Bohm Effect 2 The Aharonov-Bohm Effect Outline History Review of Electrodynamics and Quantum Mechanics Aharonov-Bohm Effect Berry Phase Interpretation

Michael Florian Wondrak The Aharonov-Bohm Effect 3 History History Theory: 1939: W. Franz 1949: W. Ehrenberg, R. E. Siday 1959: Y. Aharonov, D. Bohm Experiment: 1960: R. G. Chambers 1962: G. Möllenstedt, W. Bayh 1986: A. Tonomura et al. Y. Aharonov/D. Bohm, Phys. Rev. 115 (1959) 485 491; R.G. Chambers, Phys. Rev. Lett. 5 (1960) 3 5; W. Ehrenberg/R.E. Siday, Proc. Phys. Soc. B 62 (1949) 8 21; G. Möllenstedt/W. Bayh, Die Naturwissenschaften 49 (1962) 81; A. Tonomura et al., Phys. Rev. Lett. 56 (1986) 792 795

Michael Florian Wondrak The Aharonov-Bohm Effect 4 Review of Electrodynamics and Quantum Mechanics Formulation of Electrodynamics Electromagnetic effects can be described in terms of the electric and magnetic fields, E and B, e.g. the Lorentz force: ( F L = q E + v ) c B In the canonical formulation, electrodynamics is expressed using the potentials Φ and A: B = A E = Φ 1 c t A The potentials are defined up to a function Λ( r, t), i.e. they are gauge-dependent: A A = A + Λ Φ Φ = Φ 1 c tλ

Review of Electrodynamics and Quantum Mechanics Schrödinger Equation Quantum Mechanics is based on the canonical formulation. The time-dependent Schrödinger equation including electromagnetic effects reads: [ 1 2m ( i q c A( r, t)) 2 + q Φ( r, t) + V ( r, t) ] ψ( r, t) = i t ψ( r, t) Despite using electromagnetic potentials, the equation is invariant under gauge transformations. The wave function transforms as: ( ) iq ψ( r, t) ψ ( r, t) = exp Λ( r, t) ψ( r, t) Since ψ only obtains a space- and time-dependent phase factor, the probability density ψ 2 as well as other physical observables are not affected. Are electromagnetic potentials only a mathematical aid? Michael Florian Wondrak The Aharonov-Bohm Effect 5

Aharonov-Bohm Effect Setting Consider the following situation: In a region with vanishing electric and magnetic field, one can choose Φ = 0 A = Λ( r) since B = A = 0 vanishes. Λ( r) = r r 0 d s A( s) Michael Florian Wondrak The Aharonov-Bohm Effect 6

Michael Florian Wondrak The Aharonov-Bohm Effect 7 Aharonov-Bohm Effect Setting Performing a gauge transformation with Λ( r) leads to vanishing A: A = A + ( Λ) = 0 Φ = Φ 1 c t( Λ( r)) = 0 Thus, the wave functions in the presence and absence of the vector potential, ψ( r) and ψ ( r), are related by: ( ) iq ψ( r) = exp Λ( r, t) ψ ( r) ( ) iq r = exp d s A( s) ψ ( r) r 0

Aharonov-Bohm Effect Setting In the Aharonov-Bohm interference experiment, a homogeneous magnetic field B is produced by a thin and infinitely long solenoid. Thus, it exists only near the coil which itself is part of the middle blind. The region accessible to particles (e.g. electrons) exhibits only a non-zero vector potential A. To calculate the interference pattern of this double slit experiment, we solve the Schrödinger equation for each slit separately and superpose the wavefunctions afterwards. Michael Florian Wondrak The Aharonov-Bohm Effect 8

Michael Florian Wondrak The Aharonov-Bohm Effect 9 Aharonov-Bohm Effect Calculation Only considering the first slit to be open, we can trace back the wave function in the case of applied magnetic field ψ 1,B ( r) to the one without magnetic field ψ 1,0 ( r). Since in both cases B = 0, we can do so by a gauge transformation: ( ) iq ψ 1,B ( r) = ψ 1,0 ( r) exp d s A( s) C 1 Analogously for the second slit: ψ 2,B ( r) = ψ 2,0 ( r) exp where C j has to use the jth slit. ( iq ) d s A( s) C 2

Aharonov-Bohm Effect Calculation The complete wave function is: ψ B ( r) = ψ 1,B ( r) + ψ 2,B ( r) ( ) ( ) iq = ψ 1,0 ( r) exp d s A( s) iq + ψ 2,0 ( r) exp d s C 1 A( s) C ( ( ) ) 2 iq = ψ 1,0 ( r) exp d s A( s) + ψ 2,0 ( r) C ( 1 C ) 2 iq exp d s A( s) C ( ( 2 ) ) iq = ψ 1,0 ( r) exp d σ A + ψ 2,0 ( r) ( ) iq exp d s A( s) C ( ( 2 ) ) ( ) iq iq = ψ 1,0 ( r) exp Φ B + ψ 2,0 ( r) exp d s A( s) C 2 Michael Florian Wondrak The Aharonov-Bohm Effect 10

Michael Florian Wondrak The Aharonov-Bohm Effect 11 Aharonov-Bohm Effect Calculation Constructive interference occurs if on the screen ψ B ( r) 2 is maximal. Assuming cylindrical waves behind the slits, ψ j,0 = eikr j rj, the condition for constructive interference reads: kr 1 + q Φ B kr 2 = 2πn r 1 r 2 = λ 2π ( 2πn qφ ) B The magnetic flux Φ B shifts the interference pattern!

Michael Florian Wondrak The Aharonov-Bohm Effect 12 Aharonov-Bohm Effect Experimental Evidence R. G. Chambers, 1960:

Michael Florian Wondrak The Aharonov-Bohm Effect 13 Aharonov-Bohm Effect Experimental Evidence G. Möllenstedt, W. Bayh, 1962:

History Review Aharonov-Bohm Effect Berry Phase Interpretation Aharonov-Bohm Effect Experimental Evidence A. Tonomura et al., 1986: Michael Florian Wondrak The Aharonov-Bohm Effect 14

Michael Florian Wondrak The Aharonov-Bohm Effect 15 Aharonov-Bohm Effect Other Aharonov-Bohm Experiments In the Aharonov-Bohm Interference Experiment, the phase shift is determined by the magnetic flux Φ B which is a gauge-invariant quantity: ϕ = q Φ B All generalizations can be cast in the following formula: ( φ c dt A d r) ϕ = q A µ dx µ = q

Michael Florian Wondrak The Aharonov-Bohm Effect 16 Berry Phase Adiabatic Approximation For a time-dependent Hamilton operator, the time-dependent Schrödinger equation has to be employed: i d ψ(t) = H(t) ψ(t) dt Its eigenstates and eigenvalues are connected as follows: H(t) φ n (t) = E n (t) φ n (t) The eigenstates form a complete orthonormal basis for any fixed time t: φ n (t) φ m (t) = δ nm

Michael Florian Wondrak The Aharonov-Bohm Effect 17 Berry Phase Adiabatic Approximation A general wavefunction ψ(t) can be decomposed with respect to these eigenstates: ψ(t) = ( c n(t) exp i t dt ( E n t )) φ n(t) n 0 The coefficients c m (t) obey the differential equation: ċ m(t) + ( c n(t) φ m(t) φ n(t) exp i t dt ( ( E n t ) ( E m t ))) = 0 n 0 By differentiating the eigenvalue equation one obtains: ċ m(t) + c m(t) φ m(t) φ m(t) + c n(t) n m φm(t) Ḣ(t) φn(t) E n(t) E m(t) ( exp i t 0 dt ( E n ( t ) E m ( t ))) = 0

Berry Phase Adiabatic Approximation Applying the adiabatic approximation: 1 φ m (t) H(t) φ n (t) is small. 2 H(t) not degenerate within considered time span. ċ m (t) + c m (t) φ m (t) φ m (t) = 0 ( ) i c m (t) = c m (0) exp γ m(t) t γ m (t) = i dt φ m (t ) d 0 dt φ m(t ) In this approximation, no transition between different eigenstates occurs. Michael Florian Wondrak The Aharonov-Bohm Effect 18

Michael Florian Wondrak The Aharonov-Bohm Effect 19 Berry Phase Adiabatic Approximation Parametrization of the system by R = R (t): ( ) φ n (t) = n R ( ) ( ) ( ) ( ) H R n R = E n R n R If the system ψ(t) ( has been ) prepared in an eigenstate, ψ(t = 0) = n R(t = 0), its evolution is given by: ( ψ(t) = exp(iγ n (t)) exp i t ( dt ( E n R t ))) ( ) n R(t) 0

Berry Phase Berry Phase The phase γ n (t) can be parametrized by R, too: t ( ) γ n (t) = i dt n R d ( ) dt n R = i = i 0 t 0 R(t) R(0) C dt R(t) n ( R ) R n ( ) R dr ( ) n R ( ) R n R In case of periodic time evolution: γ n (C) = i dr ( ) n R ( ) R n R γ n (C) is called Berry phase or geometric phase. M. V. Berry, Proc. R. Soc. Lond. A 392 (1984) 45 57 Michael Florian Wondrak The Aharonov-Bohm Effect 20

Michael Florian Wondrak The Aharonov-Bohm Effect 21 Berry Phase Relation to the Aharonov-Bohm Effect Comparing the resulting difference in the Berry phase for the case of presence (γ n(c)) and absence (γ n (C)) of the magnetic field.

Michael Florian Wondrak The Aharonov-Bohm Effect 22 Berry Phase Relation to the Aharonov-Bohm Effect The relation between the wave functions is given by: ( r n ( ) R iq r ) = exp d s A( s) ( ) r n R This leads to: R γ n(c) γ n (C) = q C d s A( s) = q Φ B = ϕ Thus the Berry phase resembles the phase factor found above.

Michael Florian Wondrak The Aharonov-Bohm Effect 23 Interpretation Interpretation Role of: Potentials Lagrangians Locality

Michael Florian Wondrak The Aharonov-Bohm Effect 24 Literature Literature Hamilton, James: Aharonov-Bohm and other Cyclic Phenomena. In: Kühn, J. et al. (ed.). Springer Tracts in Modern Physics 139. Berlin Heidelberg: Springer-Verlag 1997. Peshkin, M./Tonomura, A: The Aharonov-Bohm Effect. In: Araki, H. et al. (ed.). Lecture Notes in Physics 340. Berlin Heidelberg: Springer-Verlag 1989. Reinhardt, Hugo: Quantenmechanik 2. Pfadintegralformulierung und Operatorformalismus. München: Odenbourg Wissenschaftsverlag GmbH 2013. Schwabl, Franz: Quantenmechanik. Eine Einführung. 7. Aufl. Berlin: Springer-Verlag 2007.