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I understand the principle of the Bloch sphere. You write your state in the following way: $$|\Psi\rangle = \cos(\theta/2)|0\rangle+e^{i\varphi}\sin(\theta/2)|1\rangle.$$

The angles $\varphi,\theta$ are just relative phases. However for the case of describing spin-1/2 particles these angles are really related to the orientation of spin in real space.

I was wondering if the Poincaré sphere was the same for photons? Suppose that we have a pure photonic state, does the Poincaré sphere representation can be written as: $$|\Psi\rangle=\alpha(\theta,\varphi)|0\rangle+\beta(\theta,\varphi)|1\rangle,$$ with $\theta$ and $\varphi$ meaning something in real space for photons? Or is the Bloch sphere called the Poincaré sphere only for photon qubits?

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  • $\begingroup$ An analogous question has been asked here. There is no accepted answer, but the comment argues that they are the same with some links to back up the claim. $\endgroup$
    – AG47
    Commented Mar 23 at 16:44

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The situation for photons is in three ways different from the Bloch sphere:

  1. The Bloch sphere describes a spin-1/2 particle in a pure state, which has 100% polarization, whereas the Poincaré sphere is also sometimes used for partial polarization (but it's better to use a smaller radius in that case, as described by Wikipedia so then the sphere's surface still describes fully polarized states).
  2. The Bloch sphere describes a state's spin direction in 3-dimensional space, whereas the Poincaré sphere only describes polarization in the plane perpendicular to the propagation of the light. This is allowed because light always has transverse polarization, whereas particles with mass, like an electron can have spin in any direction, regardless of their movement.
  3. The polarization of a spin-1/2 particle is fully described by one direction. A spin-1 state can also be circularly, or in general elliptically polarized, which requires more information to describe.

So this seems rather different, even if we forget about 1) and only look at pure states. Nevertheless you are correct in your observation that some similarity exist. In fact the larger set of polarization directions for massive particles as mentioned in 2), is compensated by the extra information mentioned in 3) to include elliptical polarization for spin-1 particles. This leads to an equally large solution space for the massive spin-1/2 and massless spin-1 cases, allowing a similar description. (In both cases 2 complex coefficients exist, and we can leave total phase and normalization out of the description so we end up with 2 real degrees of freedom, mapped to the surface of a sphere.)

As should be clear now, for a massive spin-1 particle the state will contain more information, namely 4 real parameters instead of the 2 for the previous cases. We can see that in two ways:
A) Spin-1 states have 3 components, therefore 3 complex coefficients are present, which after ignoring of total phase and normalization still leave us with 4 real parameters.
B) Just like the massless photon, a massive spin-1 state must also allow elliptical polarization, described by two orthogonal axes and their length ratio. For the photon case with one polarization plane the axes choice is just one angle, but in 3D the first axis already needs 2 angles, and subsequently the other one still has one angle of freedom around the first one. Together with the length ratio that gives 4 real degrees of freedom for the massive spin-1 case.

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