Overcomplete quantum tomography of a path-entangled two-photon state

Path-entangled $N$-photon states are key resources for quantum enhanced metrology and quantum imaging, as well as quantum computation. However, the quantum tomography of path-entangled indistinguishable photons is still in its infancy. We propose and implement a quantum tomographical method to characterize path-entangled two-photon NOON states, which can be extended to arbitrary $N$. To access both the populations and the coherences of the path-encoded density matrix, a single ancilla spatial mode is introduced, and photon correlations are performed as a function of a single phase within an interferometer. We characterize a two-photon state generated through the Hong-Ou-Mandel interference of indistinguishable single photons emitted by a semiconductor quantum dot and show that an overcomplete data set reveals spatial coherences that could be otherwise hidden due to limited statistics. We finally extract the truly indistinguishable part of the density matrix and identify the main origin for the reduced degree of entanglement.

Figure 1

(a) Schematic of the experimental setup used to generate the path-entangled two-photon state. LP stands for linear polarizer, HWP for half-wave plate, BS for beam splitter, PBS for polarizing beam splitter, and PC for polarization control, which is composed by a half-wave plate and a quarter-wave plate. The shutter on the lower line is used to measure the phase in the tomography setup every 10 s. (b) Schematic of the tomography setup. In our implementation ${\text{BS}}_1$ has a splitting ratio of 40:60 while that of ${\text{BS}}_2$ is 50:50. (c) Mode diagram corresponding to the setup in (b), where $\varphi$ is the phase shift between the two arms of the interferometer and the additional beam splitters ${\eta }_i$ describe the experimental losses. (d) Measurement of the second-order correlation function $g^{\left(2\right)}\left(t\right)$ of the single-photon source. (e) Measurement of the mean wave packet overlap of two photons from the source. This same histogram provides the correlation rate $R_{0,1}$ as defined in Sec. 3.

Figure 2

Expected results from the measurements presented in the text for different input states. Solid lines denote calculated correlation rates for the ideal two-photon state ${|\psi \rangle }^{\mathrm{in}}=\frac{1}{\sqrt{2}}\left(\right|2,0\rangle +|0,2\rangle )$. Dotted lines denote calculated correlation rates for the mixture ${\rho }^{\text{in}}=\frac{1}{2}|2,0\rangle \langle 2,0|+\frac{1}{2}|0,2\rangle \langle 0,2|$. In this case $R_{3,4}=R_{3,3}$ is independent on $\varphi$. Dashed lines denote calculated correlation rates corresponding to the state ${|\psi \rangle }^{\mathrm{in}}=\frac{cos\left(\theta \right)}{\sqrt{2}}|2,0\rangle +sin\left(\theta \right)e^{i\frac{\pi }{4}}|1,1\rangle -\frac{cos\left(\theta \right)}{\sqrt{2}}|0,2\rangle$, with $\theta =0.2$.

Figure 3

(a) Single-photon rate giving access to the phase $\varphi$ in the tomography setup, measured as a function of time on path 3 when blocking the lower path in Fig. 1. (b) Corresponding histogram of the acquisition time periods as a function of 20 phase bins. (c) Real and imaginary parts of the density matrix deduced from a set of nine measurements using the linear inversion tomography for $({\varphi }_1,{\varphi }_2)=(0,\frac{5\pi }{19})$. (d) Fidelity to the maximally entangled NOON state deduced from the maximum-likelihood method with nine measurements as a function of $({\varphi }_1,{\varphi }_2)$.

Figure 4

(a) Normalized coincidence rates as a function of $\varphi$. Symbols denote measurements. Dotted lines denote calculated coincidence rates for the state deduced from linear tomography for $({\varphi }_1,{\varphi }_2)=(0,\frac{\pi }{4})$ [see Fig. 3]. Solid lines denote coincidence rates calculated for the state deduced from the overcomplete set of 79 measurements. (b) Real and (c) imaginary part of the density matrix deduced from an overcomplete set of 79 measurements. (d) Mean ${\chi }^2$ between the measured correlation rates over the full $\varphi$ range and the expected ones for the state deduced for each phase couple (${\varphi }_1,{\varphi }_2$) using the maximum-likelihood method. (e) Mean ${\chi }^2$ as a function of an index representing the 200 phase couples (${\varphi }_1,{\varphi }_2$). The horizontal dashed line shows the mean ${\chi }^2$ value of 1.7 obtained for the state reconstructed with the overcomplete data set.

Figure 5

The $4\times 4$ density matrix with the distinguishable and indistinguishable parts of the two-photon state.