# What are the polarization states of the photons in a polarized and unpolarized light?

The photons are completely polarized, i.e their polarization states can be expressed as $a|R\rangle+b|L\rangle$, where $|R\rangle$ and $|L\rangle$ are two helicity eigenstates of the photon. For example, the $|R\rangle$ photon is right circularly polarized and the $|H\rangle=\frac{1}{\sqrt{2}}(|R\rangle+|L\rangle)$ is horizontally linearly polarized photon.

Is the polarized light a pure state and the unpolarized light a statistical mixture of photons with different polarizations?

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## 1 Answer

Yes, a photon in a polarized light is found in a pure state such as $|H\rangle$, $|V\rangle$, $|L\rangle$, $|R\rangle$, or any complex linear combination of them. A photon in (completely) unpolarized light is described by the density matrix $$\rho = \frac{1}{2} \left( |L\rangle \langle L| + |R\rangle \langle R| \right) = \frac{1}{2} \left( |H\rangle \langle H| + |V\rangle \langle V| \right)$$ Note that you omitted the relationship for the vertically polarized state, $|V\rangle = i(|R\rangle - |L\rangle)/\sqrt{2}$, up to an overall sign which is a convention (well, the whole phase including $i$ is physically inconsequential, so it doesn't matter at all but one must be self-consistent with the conventions).

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Your density matrix corresponds to a completely unpolarized light. Partially unpolarized light density matrix is expressed via projection operators with different "weights" $w_n$ (probabilities), not 1/2. –  Vladimir Kalitvianski Nov 7 '11 at 10:37
Thnx Lubos. This density matrix is for monochromatic unpolarized EM wave. What if it's not monochromatic? Photons in each mode have the same density matrix? –  Andyk Nov 7 '11 at 12:36
Dear @ANKU, to discuss frequency (and/or direction), you need to extend the Hilbert space by tensor-multiplying it with the space of different $\vec k$. Again, one may have pure states and mixed states with respect to frequencies, too. The totally non-monochromatic, unpolarized light is given by the density matrix like mine, but it would also have extra labels $\vec k$ in all the bra,ket states and one would integrated over some interval of $\vec k$. However, one may also have pure states relatively to the position/frequencies. There are many states, pure and mixed; what's your exact question? –  Luboš Motl Nov 7 '11 at 22:24
Where can I read about this? I would like to see the exact density matrix for totally non-monochromatic unpolarized light. And thanks again! Your answer helped me already. –  Andyk Nov 7 '11 at 23:30
Dear @Andyk, after 1.5 years. There is nothing such as "the" totally monochromatic light - the density matrix representing this "utopia" can't exist. It would probably mean that all frequencies are represented equally but there are infinitely many possible frequencies so the probability of each would have to be zero for the total probability to be one - the density matrix couldn't be normalized to 1. Any density matrix must always have preferences for "some" intervals of frequencies. –  Luboš Motl Jun 30 '13 at 17:25