# Measuring the quantum state of light

The following scheme to measure linear polarization states (a single polarizing beam splitter and two photo counters) orientation (as $arctan \sqrt{\frac{v}{h}}$) of coherent light pulses cannot discriminate two states which make the same angle with the measuring bases. $\langle n \rangle$ is the average number os photons per pulse.

In the example, the $+45^o$ and $-45^o$ polarized pulses will result on the same output.

I'd like to know if it is OK to do the following setup and what would be change in precision of the polarization angle, if any:

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Of course you can build such a detection setup. The only question is what it will do.

To get a grip of things, let us consider three pairs of input states - horizontal/vertical (H/V), diagonal/anti-diagonal (45/-45 degrees, D/A), and circular left/right-handed (L/R) polarization. With H/V polarization on the HV detector, H is always detected by H detector and V by V detector, on the DA detector each gets split in half on both detectors. With D/A polarization, it is the other way round - D always ends on D detector but gets split on H/V detector. L/R light gets always split on both H/V and D/A discriminators.

More rigorously, assuming a definite polarization state of the pulse, this can be written in a superposition of H and V polarization as $|\psi\rangle=c_H|H\rangle+c_V|V\rangle$. Simultaneously, this can be written in the D/A basis as $|\psi\rangle=c_D|D\rangle+c_A|A\rangle$. Because these two bases are connected by $|D\rangle=(|H\rangle+|V\rangle)/\sqrt{2}$, $|A\rangle=(|H\rangle-|V\rangle)/\sqrt{2}$, there is a connection between coefficients in the two bases $$c_D+c_A = \sqrt{2}c_H$$ $$c_D-c_A = \sqrt{2}c_V.$$

By measuring the polarization, we obtain information about the modulus squared of these coefficients, $|c_i|^2$, where $i = H,V,D,A$. If we now express $c_{D,A}$ in terms of $c_{H,V}$, we get four equations for the coefficients $c_{H,V}$ together with $c_{H,V}^\ast$. This means that we can obtain full information about $c_{H,V}$ from the measurement.

Although this scheme can work in this way, the usual approach is (as far as I know) to use only one PBS and one pair of detectors and change the basis in which they operate. This is advantageous because you get more light on each detector so the SNR is better.

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Note that in traditional polarimetry this is called a "division of amplitude" linear Stokes polarimeter. The "linear" term is because you are not measuring any of the circular component. As you said the SNR will be worse... – daaxix Jan 6 '13 at 16:05
If you measure in both H/V and D/A bases, you measure $|c_H|^2$, $|c_V|^2$, $|c_D|^2=|c_H|^2+|c_V|^2+(c_Hc_V^\ast+c_H^\ast c_V)$, $|c_A|^2=|c_H|^2+|c_V|^2-(c_Hc_V^\ast+c_H^\ast c_V)$. (Note that the coefficients are complex.) Thus, you have four equations for four real numbers (real and imaginary parts of $c_H$, $c_V$) and can determine them completely. – Ondřej Černotík Jan 6 '13 at 20:06
@daaxix Note that this is not the measurement of the imaginary part of the electric field of the light (which would be completely impossible with our technology due to rapid oscillations) but the measurement of the imaginary parts of the amplitudes of the horizontally and vertically polarized components of the electric field. – Ondřej Černotík Jan 7 '13 at 8:49
I misread your second comment above and thought that you were claiming to be able to measure $C_L, C_R$, which you are not. – daaxix Jan 7 '13 at 14:49