Quantum-state tomography for optical polarization with arbitrary photon numbers

A scheme for quantum-state tomography is presented that can be performed for polarized light with an arbitrary photon-number distribution. The proposed method fills the gap between existing polarization-tomography schemes for single-photon states and for optical fields with very large photon numbers. It consists of an optical homodyne setup triggered by the outcome of a single-photon counting module. The configuration space of polarization is parametrized by Euler angles and by an additional interference parameter, whose assessment requires conditional homodyne measurements.

Figure 1

Sectors of the interference parameter $j$ in the density matrix of a state with maximally one photon [e.g., as given in Eq. (19)] with $|m,n\rangle ={|m\rangle }_{\mathrm{H}}\otimes {|n\rangle }_{\mathrm{V}}$.

Figure 2

Conditional balanced homodyne measurement: The wave plates realize the rotation by the angles $\Theta$, and the polarizing beam splitter (PBS) separates the V and H modes, where the V mode is detected by a single-photon counting module (SPCM). Conditioned on a no-count event in the V mode, the H mode is measured in the balanced homodyne scheme with a beam splitter (BS), phase-controlled local oscillator (LO), and two photodetectors (PD1, PD2).

Figure 3

Phase-space image of a single-photon state with a vacuum contribution, according to Eq. (19) with ${\psi }_0=0$. Dark red and dark blue correspond to the maximum positive value $+\frac{2}{3}$ and minimum negative value $-\frac{2}{3}$, respectively, of the phase-space distributions $\langle {\widehat{Q}}_j\left(\Theta \right)\rangle$.