Experimental polarization state tomography using optimal polarimeters

We report on the experimental implementation of a polarimeter based on a scheme known to be optimal for obtaining the polarization vector of ensembles of spin-$\frac{1}{2}$ quantum systems and the alignment procedure for this polarimeter. We also show how to use this polarimeter to estimate the polarization state for identically prepared ensembles of single photons and photon pairs and extend the method to obtain the density matrix for generic multiphoton states. State reconstruction and performance of the polarimeter is illustrated by actual measurements on identically prepared ensembles of single photons and polarization entangled photon pairs.

Figure 1

Four (reduced) Stokes vectors in the Poincaré sphere that form a tetrahedron define the optimal POVM operators used for polarization state estimation. The tetrahedron gives the largest volume encompassable by a vector quartet in the sphere, making it the optimal measurement when using four POVM’s.

Figure 2

Practical implementation of the tetrahedron polarimeter that achieves the ideal instrument matrix. Each detector $b_j$ is associated to the tetrahedron vector ${\vec{b}}_j$. The partially polarizing beam splitter (PPBS) separates incoming light according to polarization, and quartz plates remove unwanted phase shifts. Light leaving the PPBS is passed through wave plates and polarizing beam splitters (PBS) to be projected on two different bases ($\pm 45°$ basis for transmitted light and the circular basis for reflected light).

Figure 3

Scheme for estimating the polarization state of an ensemble of $N$-correlated photons using $N$ polarimeters $(P1,P2,\dots ,PN)$. A multiple-coincidence circuit identifies the $4^N$ possible coincidence combinations. For photon pairs $(N=2)$, two polarimeters are used, giving 16 possible coincidence combinations. Several copies of the state are processed, giving a coincidence pattern used in estimating the polarization state of the ensemble.

Figure 4

Instrument response of the polarimeter to linearly polarized light. The data points show the variation in relative intensity at each detector with respect to the angle of the half-wave plate (HWP) in the polarization state preparation. The solid lines show the expected intensity modulation for an ideal device for each HWP setting [(13)], scaled for appropriate detector efficiencies. Error bars are smaller than the point markers. Panel (a) shows the relative intensity at detector 2 without compensation plates. Panels (b) and (c) are taken with compensation for phase shifts. The oscillation in (a) is out of phase and also of lower amplitude compared to the phase-corrected behavior of detector 2 in panel (b). Plots in (c) shows a lower amplitude because light in the reflected arm is not projected on a linear polarization basis.

Figure 5

(Color online) A set of polarization states $\left({\vec{S}}_i\right)$ equally distributed over the Poincaré sphere surface was generated; photons from each of these states were sent to the polarimeter, from which an estimated state $\left({\vec{S}}_e\right)$ is obtained. The figure shows the fidelity of the estimated state to the prepared state, $\frac{1}{2}({\vec{S}}_e\bullet {\vec{S}}_i)$. It is roughly constant over the Poincaré sphere, showing that the polarimeter is an unbiased polarization state estimator.

Figure 6

(Color online) The density matrix of a Bell state $\mid {\Psi }^{+}⟩⟨{\Psi }^{+}\mid$ obtained by linear reconstruction from photon pairs.