Observation of the Spin Hall Effect of Light via Weak Measurements

1. Observation of the Spin Hall Effect of Light via Weak Measurements OnurHosten* and Paul Kwiat Department of Physics, University of Illinois at Urbana – Champaign, Urbaana, IL 61801, USA Science Vol. 319 Page 787 Feb. 8th 2008

2. Abstract • Spin-dependent displacement of photons passing through interface • Use pre-selection and post-selection technique to measure ~1 angstrom displacement

3. Outline What is a weak measurement? To get interaction Hamiltonian at the interface

4. Why spin–dependent displacement For paraxial beam: From Snell’s law: We have:

5. Calculation • The eigenstates of reflection and refraction are the linear s – and p – polarization states. For an arbitrary wave vector : • Under the action of the interface, the eigenstates evolve as: Fresnel transmission coefficients

6. Calculation • Eigenstates in x(I,T)-y-z(I,T) coordinates: • Initial states as the linear combination of s-|p- states:

7. Calculation • Horizontal and vertical states evolves as: where: • If we use: • We have:

8. Calculation • We can get the interaction Hamiltonian from the initial and final states: • If : • Therefore,

9. Calculation • Assume initial state to be horizontal: • The final state will evolve as: • Therefore the final displacement will be: Fig 1 c

10. Back surface of the prism • Normal incident to the back surface will not cause any reflection. • The range of the angle between the front and back surfaces of the variable angle prism is 0° to 36.4°. Therefore for incident angle larger than 64°, they have to consider the influence of back surface reflection of the prism. • Modified results: Fig 4

11. General polarization dependence of the displacement • Input polarization state: • Displacement:

12. How to Measure

13. Pre-selection and post-selection • In a quantum measurement, an observable of a system is first coupled to a meter, and then the meter is read out, from which information about the observable is acquired. • First coupling(pre-selection) with: • Observable and its eigenstates: • Reading out(post-selection) with:

14. Pre-selection and post-selection • Initial state after pre-selection: • Under , initial state evolves as : • Read out after post-selection: with:

15. Real and imaginary weak value • is real: meter read out(displacement) is • is imaginary: the momentum distribution of the meter is altered as: • Assume initial momentum wave function is real and evolves with the free Hamiltonian, then we have: L2

16. imaginary weak value • Assume for initial meter state: • Final state: • And: with: assuming:

17. Experiment Setup

18. Achieve normal incidence to the back of the prism • The SHEL takes place at the front surface of a variable angle prism (VAP) for various incidence angles, with the back surface adjusted to be at normal incidence to avoid secondary Hall shifts (because there is obviously no Hall shift at normal incidence). The VAP is constructed by attaching two BK7 round wedge prisms together with the surface tension of a thin layer of index-matching fluid and is mounted loosely to avoid any stress-induced birefringence. The prisms can rotate with respect to each other, and the entire assembly can rotate around three orthogonal axes, allowing the desired surface orientations.

19. Two polarizers • Pre-selection and post-selection states: • The variable: • Therefore: • Some value: (times larger)

20. Results

21. Thank you!