The relativistic time-dependent Aharonov-Bohm effect in two spatial dimensions

1. The relativistic time-dependent Aharonov-Bohm effect in two spatial dimensions Athan Petridis Zachary Kertzman Drake University

2. The Aharonov-Bohm effect • The charged fermions interact directly with the e/m 4-potential, NOT the electric or magnetic fields. • Around an non-penetrable solenoid spinors diffract acquiring an extra phase. • Difficult to directly, experimentally confirm that it is not due to residual fields. • It is used by interferometry experiments. • Nanosecond pulse electron sources available.

3. The Dirac Equation • Relativistic quantum equation for spin-1/2 fermions, which are described by a 4-dimentional spinor Ψ. • Including an external scalar potential, V:

4. The Numerical Algorithm • The staggered leap-frog algorithm is applied in a spatial grid of bin-size Δx (in 1 dim) and with time step Δt: • The spatial derivatives are computed symmetrically. • Reflecting boundary conditions are applied on a very large grid (running stops before reflections occur if necessary). • Works well on a PC, using dynamic memory allocation.

5. Stability of the Algorithm • Use the norm as measure • The stability region is (d = spatial grid bin, Δt = time step) • Obtained via a standard stability analysis usingplane waves (for large component) probability 1 time

6. 1D Free Electron Propagation • The initial spinor is (N = normalization factor, m =1): • The probability density ρ=Ψ†Ψ at t=0 is Gaussian (s.d.=σ0). • As σ0→∞,Ψ becomes a positive energy plane wave, which for p=0 is a spin +1/2 eigenstate. • ρ(x,t) is shown next for σ0=1. The method is stable. The accuracy is of order 10-10 per bin (Δx=0.01, Δt=0.001).

8. Position Expectation Value • <x> vs time after subtraction of the drift velocity (red: σ0 = 0.5, p = 0.01; green: σ0 = 0.5, p=1.37; blue: σ0=1, p=0.01, purple: σ0 =1.5, p=1.0). • High-frequency (~2E) oscillations are observed (Zitterbewegung). • The effect has a non-linear dependence on σ0 and is maximized when 2 σ0 = λc (Compton wavelength) for given p. It increases with p.

9. Standard Deviation • Standard deviation of the probability density, σ, vs time (red: σ0 = 0.5, p = 0; green: σ0 = 0.5, p=1.0; blue: σ0 = 1.0, p = 0; purple: σ0 = 1.0, p = 1.0; light blue: σ0 = 1.5, p = 0; yellow: σ0 = 1.5, p = 1.0). • High-frequency oscillations are more pronounced as p increases and die out with time. • σ increases faster at lower p.

10. Spin, z-component • Expectation value of the z-component of the spin (perpendicular to the propagation direction) (red: σ0 = 0.5, p = 0.01; green: σ0 = 0.5, p = 1.37; blue: σ0 = 1.0, p = 0.01; purple: σ0 = 4.0, p = 0.0). • The high-frequency oscillations die out with time and are maximized at 2σ0 = λc. • Results agree with J. W. Braun, Q. Su, and R. Grobe, Phys. Rev. A59, 604 (1999).

11. Decay and Survival Probability • A decaying fermionic system can be described as a Dirac spinor initially set inside a potential well that tunnels through the potential walls. • Introduce constant potential: V(x). • In a given reference frame, the survival probability of the system is defined as

12. Finite Square Well Potential • The width, 2a, is set equal to 2 σ0 with σ0 = λc = 1.0. • Pinvs time (p = 0) for V = 0.1 (red), 1.0 (green), 1.5 (blue), and 2.0 (purple) [A] and V = 0.1, p = 0.01 (red), V = 2.2, p = 0.01 (green), V = 0.9, p = 0.1 (blue) and V = 2.2, p = 0.1 (purple) [B]. • Pin decays non-exponentially performing oscillations. This has also been observed in non-relativistic decays. • The relativistic case includes a sudden increase in Pin for V > 1 due to Klein-paradox. The effect of p is small. A B

13. Strong solenoid + dipole field • Minimal substitution: • Infinite solenoid vector potential (r >R): • Dipole (residual) vector potential (r>R): • Cylindrical electric potential V=const. (r<R) • A0=0.88, B0=A0/10, k=1.134, σ0= 5, R=4

21. Weak solenoid + dipole field • Same type of potential but weaker. • A0=0.01, B0=A0/10, k=1.134, σ0= 5, R=4. • The wavefunction diffracts.

28. Comparison: with/out dipole • Results without the dipole do not differ much visually from those with dipole. • For a strong field the difference is a fraction of a percept per point and for a weak field is of order one part in a million. • The difference has azimuthal asymmetry and is time-dependent.

31. Spin with/out dipole • The same observations can be made for the z- component of the spin distribution. • The difference is even more subtle.

32. Evolved spin in high field

33. Evolved spin in low field

34. Spin difference in high field

35. Spin difference in low field

36. Quantum Ring (Gaussian) • A “solid” ring of size comparable to the correlation length (spread, s.d.) of the initial spinor surrounds the solenoid. • An initial Gaussian spinor inside the ring spreads around the ring radially and azimuthally.

37. Initial spinor

38. Evolved spinor, small A

39. Quantum Ring (eigenfunction) • The initial spinor is an azimuthal eigenfunction. It spreads only radially. • In the presence of an A-field it is “turned” around. The energy distribution does not change. • The phase of the large component characteristically changes in the presence of the magnetic potential.

40. Initial spinor

41. Evolved spinor, A = 0

42. Evolved spinor, A = 0.88

43. Evolved large component phase, A = 0

44. Evolved large component phase, A = 0.88

45. Evolved large component phase difference between field and no-field cases

46. Perspectives • Coupled two-fermion systems. propagating in medium are studied. • Dynamic mass renormalization due to self-interaction is studied (interleaved Maxwell’s equations also used). • Strong interactions will be introduced.