Open Ionizing opened 3 months ago
Consider the degenerated states psi1 and psi2 form another two states psi1' and psi2' with spin polarization purely along the z axis.
psi1
psi2
psi1'
psi2'
z
The transformation matrix is unitary and takes the form
U = [[ cos(phi), exp(i*theta) * sin(phi)], ...... (1) [-exp(-i*theta) * sin(phi), cos(phi)]]
where theta \in [0, 2pi) and phi \in [0, pi/2]. We want that the state cos(phi) psi1 + exp(i*theta) sin(phi) psi2 has the property <sigma_z> = 1:
theta \in [0, 2pi)
phi \in [0, pi/2]
cos(phi) psi1 + exp(i*theta) sin(phi) psi2
<sigma_z> = 1
<sigma_z> = cos(phi)^2 sigma_z11 + sin(phi)^2 sigma_z22 + 2 sin(phi) cos(phi) Re(exp(-i*theta) sigma_z21)
To maximize <sigma_z>, theta should satisfy
<sigma_z>
exp(i*theta) = sigma_z21 / |sigma_z21| <sigma_z> = cos(phi)^2 + sin(phi)^2 + 2sin(phi) cos(phi) |sigma_z21| = sigma_z11 (cos(phi)^2 - sin(phi)^2) + 2 sin(phi) cos(phi) |sigma_z21| = sigma_z11 + sin(2phi) |sigma_z21|
Assume sigma_z11 < 0, the maximum value is taken at tan(2phi) = |sigma_z21| / sigma_z11, which means
sigma_z11 < 0
tan(2phi) = |sigma_z21| / sigma_z11
exp(i*theta) = sigma_z21 / |sigma_z21| ......(2) phi = atan( |sigma_z21| / sigma_z11 ) / 2 ......(3)
Substitute (2) and (3) into (1) and the re-organized states with pure spin polarizations can be obtained via [psi1'; psi2'] = matmul(U, [psi1; psi2]).
[psi1'; psi2'] = matmul(U, [psi1; psi2])
Credit: Zhi Li @ USTC
How to do unitary transformation
Consider the degenerated states
psi1
andpsi2
form another two statespsi1'
andpsi2'
with spin polarization purely along thez
axis.The transformation matrix is unitary and takes the form
where
theta \in [0, 2pi)
andphi \in [0, pi/2]
. We want that the statecos(phi) psi1 + exp(i*theta) sin(phi) psi2
has the property<sigma_z> = 1
:To maximize
<sigma_z>
, theta should satisfyAssume
sigma_z11 < 0
, the maximum value is taken attan(2phi) = |sigma_z21| / sigma_z11
, which meansSubstitute (2) and (3) into (1) and the re-organized states with pure spin polarizations can be obtained via
[psi1'; psi2'] = matmul(U, [psi1; psi2])
.Credit: Zhi Li @ USTC