Experimental retrodiction of trajectories of single photons in double interferometers

When a photon passes through an interferometer, quantum mechanics does not provide a clear answer as to its past. Quantum retrodiction is a quantitative theory, which endeavors to make statements about the past of a system based on present knowledge. Quantum retrodiction may be used to analyze the past of a photon, that is, its trajectory. Here we experimentally retrodict the trajectories of single photons in double interferometers by measuring the final state of the photon. A sequence of measurements is made on a photon to determine which path the photon followed, so a series of retrodiction of measurement results can be regarded as a photon trajectory. We obtain information about the partial trajectory and the entire trajectory of the photon. Furthermore, we also observe the effect of different measurements in the extraction of trajectory information. Our experiment highlights the application of retrodiction theory to the study of the photon's past, and provides potential application in quantum communications.

Figure 1

(a) Concept of retrodiction from measurement results. A quantum system is prepared in an initial state $|\mathrm{\Phi }\rangle$ at time $t=0$ and then subjected to a series of measurements {$M_1,M_2,...,M_N$}. Different measurement results correspond to different final states $|\mathrm{\Psi }\rangle$ of the system. We can retrodict the information of measurement results just from the final state of the system. (b) Sketch of our experimental setup. A qubit passes a Hadamard gate ($H$); a positive-operator-valued measure (POVM) is then performed to obtain information about which path ($+$ or $-$) the qubit has passed through. We denote the outcomes of the measurement by $\{+,-\}$. Next, the qubit passes through a second Hadamard gate, and we perform a second POVM. Finally, it passes through the last Hadamard gate. In regard to the pair of two-outcome measurements, the output state of the qubit is in four possible states $|{\mathrm{\Psi }}_{\text{out}}^{++}\rangle ,|{\mathrm{\Psi }}_{\text{out}}^{+-}\rangle ,|{\mathrm{\Psi }}_{\text{out}}^{-+}\rangle ,|{\mathrm{\Psi }}_{\text{out}}^{--}\rangle$. We measure the final state of the qubit to retrodict the trajectory of the qubit.

Figure 2

Experimental setup. In the single-photon source module, the single-photon source is generated by spontaneous parametric down-conversion in a type-II beta-barium-borate (BBO) crystal used under beamlike phase-matching conditions. The blue half-wave plates (HWPs) with an angle of $22.5^{\circ }$ perform the Hadamard gate operation. In the POVM modules, the red HWPs with an angle of ${45}^{\circ }$ and beam displacers (BDs) comprise the interferometric network to perform the POVM operations; the yellow HWPs with an angle of $0^{\circ }$ are inserted into the middle path to compensate the optical path difference between the up path and the down path; the POVM operation is changed by rotating the black HWPs. The state discrimination module is used to retrodict the POVM results. The state discrimination setups for determining the first POVM result, the second POVM result, and both are shown in Figs. 2(a)– 2(c), respectively.

Figure 3

(a) Experimental results for the relationship between different POVMs and retrodiction of the second measurement results; the blue line is the theoretical expectation, and the red dots are values from experimental data. (b) Comparison of $P\left({\rho }_{2+}\right)$ and $P\left({\rho }_{2+}\right|M_{\text{out}}=+)$. The blue line and blue circles represent $P\left({\rho }_{2+}\right|M_{\text{out}}=+)$ obtained using Bayes's theorem. The dash-dotted green line and green dots represent $P\left({\rho }_{2+}\right)$; the dashed pink line and pink stars represent $P\left({\rho }_{2-}\right)$.

Figure 4

(a) Experimental results for the relationship between different POVMs and retrodiction of the first measurement results. The blue line is the theoretical expectation; the red dots are values from experimental data. (b) A comparison of $P\left({\rho }_{1+}\right)$ and $P\left({\rho }_{1+}\right|M_{\text{out}}=+)$. The blue line and blue circles represent $P\left({\rho }_{1+}\right|M_{\text{out}}=+)$ obtained using Bayes's theorem. The dashed pink line is the theoretical expectation of $P\left({\rho }_{1+}\right)$ and $P\left({\rho }_{1-}\right)$. The pink stars and green dots are the experimental values of $P\left({\rho }_{1+}\right)$ and $P\left({\rho }_{1-}\right)$, respectively.

Figure 5

Experimental results for the relationship between different POVMs and retrodiction of both the first and second measurements. The blue line is the theoretical expectations; the red dots are values from experimental data.