Mode reconstruction of a light field by multiphoton statistics

We present a simple method to reconstruct the mode distribution of multimode classical and nonclassical optical fields using a single measurement of higher-order photon number correlation functions. Knowing the underlying number and structure of occupied modes of a light field plays a crucial role in minimizing loss and decoherence of quantum information. Typically, full characterization of the mode structure involves a series of several separate measurements in spatial, temporal, frequency, and polarization domains. We experimentally demonstrate reconstruction of up to three modes with excellent agreement and study the robustness of our method in experimentally realizable regimes.

Figure 1

Example case of one thermal mode, one single-photon mode, one Poissonian mode, and fixed ${\mu }_{\mathrm{total}}$. (a) Lines of constant $g^{\left(2\right)}$ (solid blue), $g^{\left(3\right)}$ (dashed purple), and $g^{\left(4\right)}$ (dotted dark blue) plotted against fractions of Poissonian and single-photon light. Bold line is $g^{\left(2\right)}=1$ and shaded area represents nominally nonclassical region ($g^{\left(2\right)}<1$). Contour line values through point 1: $g^{\left(2\right)}=1.0,g^{\left(3\right)}=1.2,g^{\left(4\right)}=2.2$; point 2: $g^{\left(2\right)}=1.0,g^{\left(3\right)}=1.0,g^{\left(4\right)}=1.0$; point 3: $g^{\left(2\right)}=0.5,g^{\left(3\right)}=0.3,g^{\left(4\right)}=0.2$; point 4: $g^{\left(2\right)}=1.4,g^{\left(3\right)}=2.9,g^{\left(4\right)}=8.0$. Inset is the mode structure of the four points. Bar heights are the mean photon numbers $\mu$ in the thermal mode (left, blue), single-photon mode (middle, red), and Poissonian mode (right, green). (b) Sensitivity gained by using an overdetermined set of six correlation functions compared to three. Color represents the ratio of the gradients for six- to threefold detection at each point in the space.

Figure 2

Source and detector configurations. SPDC: spontaneous parametric down conversion. APD: avalanche photodiode.

Figure 3

Bar heights are mean photon numbers $\mu$. Each group of three bars represents a single experiment with a particular set of mean photon numbers. Dark bars in back are measured mean photon numbers, ${\vec{\mu }}_{\mathrm{m}}$, and translucent bars overlaid are the reconstructed mean photon numbers, ${\vec{\mu }}_{\mathrm{r}}$. Noted above each plot is the set of input modes.

Figure 4

Bar heights are mean photon numbers $\mu$. Each group of three bars represents a single experiment with a particular set of mean photon numbers. Dark bars in back are measured mean photon numbers, ${\vec{\mu }}_{\mathrm{m}}$, and translucent bars overlaid are the reconstructed mean photon numbers, ${\vec{\mu }}_{\mathrm{r}}$. Blue bars are thermal modes and rightmost, green, crosshatched bar is Poissonian mode. Noted above each plot is the set of input modes.

Figure 5

Bar heights are mean photon numbers $\mu$. Each group of three bars represents a single experiment with a particular set of mean photon numbers. Dark bars in back are measured mean photon numbers, ${\vec{\mu }}_{\mathrm{m}}$, and translucent bars overlaid are the reconstructed mean photon numbers, ${\vec{\mu }}_{\mathrm{r}}$. Blue bars are thermal modes and rightmost, red, crosshatched bar is single-photon mode. Noted above each plot is the set of input modes.