Geometric phase acquired by a photon on the Poincare sphere I know that when a photon (spin 1) is parallely transported in the Poincare sphere in a closed loop, the geometric phase which it acquires is 'half' the solid angle subtended by the closed loop at the origin although photon is a spin 1 particle. Is this true for an electron also (spin 1/2) on the Poincare sphere? Is it even valid to talk of an electron associated with the Poincare sphere because Poincare sphere represents the polarization state of an EM field.
This question originated in my mind because when we subject an electron or any particle of spin '$s$' kept at the origin to a magnetic field constant in magnitude but varying direction, the associated geometric phase is '$s\Omega$', where '$\Omega$' is the solid angle subtended by the varying Magnetic field (a parameter of adiabatically varying Hamiltonian) at the origin in the real space.
 A: The Poincaré sphere as a parameter space is not exclusive to neither photons nor electrons. It can appear in the evolution of various physical systems; for example the dynamics among two hyperfine atomic states driven by a laser beam. Please see the following work by Viennot  for a general characterization of generic parametrization spaces.
One important property of the parameter spaces is that they are Kählerian, i.e., symplectic with a compatible complex structure. The computation of the symplectic form is very easily performed as a byproduct of the Berry phase computation as follows:
Suppose we have a family of Hamiltonians $H(R)$ parametrized by a parameter space $\mathcal{M} \ni R$. Suppose that $\psi(R)$ is a normalized eignevector corresponding to the eigenvalue $E(R)$ of $H(R)$:
$$H(R) \psi(R) = E(R) \psi(R)$$
Suppose that the parameters $R$ are varied such that the level $E(R)$ does not cross any other level; then the Berry connection:
$$A = \psi(R)^{\dagger} d \psi(R)$$
Let $P(R)$ be the projector on the state $\psi(R)$:
$$ P(R) = \psi(R) \psi(R) ^{\dagger}$$
Then the symplectic structure of the parameter space $\mathcal{M}$ is given by:
$$\omega = \mathrm{tr} \left(P(R) dP(R) \wedge dP(R)\right)$$
(It is not hard to check that $\omega$ is closed). When the parameter space is the Poincaré sphere, $\mathcal{M} = S^2$, the symplectic structure is always an integer multiple of the Poincaré sphere area element, irrespective of the Hamiltonian that we started from.
$$\omega = n \omega_{S^2}$$
(with $\omega_{S^2} = \sin \theta d\theta d\phi$ in spherical coordinates).
Thus only the parameter $n$ determines the how many multiples of the solid angle are equal to the Berry phase.
The symplectic form of any parameter space computed as above is integral, i.e., its flux through any two dimensional cycle divided by $4\pi$ is an integer). The reason for that is only then, the Berry connection will be a connection on a line bundle. This happens when the Dirac's quantization condition ($n\in \mathbb{Z}$ is satisfied).
The requirement of the Berry phase to be a bundle holonomy is very important; for example it is the deep reason behind the classification of topological insulators.
Now, in order to know what is the integer $n$ corresponding to a physical situation, one only needs to compute the eigenvector:
For example, for a spin $s$ electron in a magnetic field, the Hamiltonian is
$ H = \mathbf{\sigma} \cdot  \mathbf{B }$
Here $\mathbf{\sigma}$ are $2s+1 \times 2s+1$ matrices. If we take the eignevector corresponding to the state $(m_s, s)$, we get:
$n = 2 m_s$
Thus for the case of the electron $m_s = \pm \frac{1}{2}$, we get $n= \pm 1$.
I the case of the photon, the polarization dynamics takes place in the plane perpendicular to the direction of motion. The Hamiltonian is two dimensional. The explicit form of the Hamiltonian is given for example in: the paper  by Bliokh, Niv, Kleiner and Hasman:
$$H = S_z  \mathbf{A(p)}\cdot  \dot{\mathbf{p}}$$
($S_z$ is the third component of the Stokes vector, $\mathbf{A(p)}$ is the spin orbit interaction)
Thus in this case also, we have $m_s = \pm \frac{1}{2}$, we get $n= \pm 1$, even though the eigenvalues of $S_z$ (the helicities) are equal to $\pm 1$.
