content_image/425092.jpg content_image/425093.jpg content_image/425091.jpg content_image/425090.jpg content_image/425089.jpg content_image/425088.jpg content_image/425087.jpg content_image/425086.jpg content_image/425085.jpg
We start from (see Eq. 3) 

and 

where tan φ$_{+(}$$_{−}$$_{)}$ = − k X +( − ) k Y . The incident state { ψ$_{i}$ } is written a linear combination of { ψ$_{+}$ } and { ψ$_{−}$ } 

## Appendix A: Derivation of Eqs. 5a and 5b 

<img src='content_image/425092.jpg'>

## Supplementary information 

<img src='content_image/425093.jpg'>

<img src='content_image/425091.jpg'>

where [Ψ] ≡ [ { ψ$_{+}$ }{ ψ$_{−}$ } ]. After propagating from x = 0 to x = L , the final state is written as ( θ$_{+(}$$_{−}$$_{)}$ = k$_{X}$$_{+(}$$_{−}$$_{)}$L ) 

Hence we can write, { ψ$_{f}$ } = [ t ] { ψ$_{i}$ } , with 

where, 

Multiplying out the matrices leads to 

<img src='content_image/425090.jpg'>

<img src='content_image/425089.jpg'>

<img src='content_image/425088.jpg'>

<img src='content_image/425087.jpg'>

Setting φ$_{+}$ ≈ φ$_{−}$ ≡ φ (this amounts to ignoring non-orthogonality of ”+” and ”-” states), the expression simplifies to, 

Note that [ t ] can also be written as 

<img src='content_image/425086.jpg'>

<img src='content_image/425085.jpg'>

where Δ θ ≡ θ$_{+}$ − θ$_{−}$ = θ$_{L}$$_{/}$√$_{1}$ − s 2 and ˆ n is a unit vector in the direction of the effective magnetic field: ˆ n = cos φ ˆ x + sin φ ˆ y . This form is intuitively appealing showing the transmission [ t ] as a product of a simple phase-shift exp { i ( θ$_{+}$ + θ$_{−}$ ) / 2 } and a rotation around ˆ n by Δ θ . 

