Precisely resolving two-photon energy-time entanglement by dual-channel Fabry-Pérot interferometry

Methods for precisely resolving the continuous-variable energy-time entanglement of paired photons have always been sought in quantum optics. The Fabry-Pérot interferometer provides such a distinguished opportunity if the single-photon pulse's self-interference is carefully avoided. A dual-channel Fabry-Pérot interferometer is proposed and studied with the focus put upon higher-order quantum interference effects. When the two channels are properly set up, the interferometer is capable of resolving the energy-time entanglement in detail, similar to how the usual Fabry-Pérot interferometer can resolve classical light's spectrum. A variation form of the dual-channel Fabry-Pérot interferometry is also discussed.

Figure 1

Schematic of the equal-arm dual-channel Fabry-Pérot interferometer. Abbreviations in the graph: mirror, M; quarter–wave plate, QWP; polarization beam splitter, PBS. The source emits an energy-time entangled photon pair traveling separately along the two symmetric arms, where the polarizations are assumed to be fixed. The positive direction of each arm's local coordinate system is along the $k$ vector of the photons propagating in that arm. The polarizations of the emitted photons are such that they make it through when they first hit the PBS. In both arms the transmissions and the reflections are monitored by SPCM. The parts in the dashed box are individual Fabry-Pérot interferometers. They have similar cavity lengths $d_L\approx d_R$ in the sense that $Q|d_L-d_R|$ is small compared to the individual photon pulse length $(c$ times the single-photon coherence time), where $Q$ is the quality factor of the Fabry-Pérot interferometers. Here the Fabry-Pérot interferometers are composed of planar mirrors for the ease of derivation, while in real experiments concave mirrors would be considered to make them work in the stable resonator regime. If the experimental results of only one arm are collected, the photon counting result would look purely normal as a single-photon pulse experiment, and no knowledge about the other arm could be gained.

Figure 2

Numerical simulation results of the transmission and reflection coincidence rates for the interferometer in Fig. 1. All rates are normalized to the coincidence rate without inserting the two Fabry-Pérot interferometers. The rates are plotted as a function of the Fabry-Pérot cavity lengths $d_{L,R}$, where the horizontal axis is $k_L{d}_{L}mod\pi$ and the vertical axis is $k_R{d}_{R}mod\pi$ for all the four plots. (a) transmission coincidence for $T=0.5$, (b) transmission coincidence for $T=0.2$, (c) reflection coincidence for $T=0.5$, and (d) reflection coincidence for $T=0.2$.

Figure 3

A numerical simulation of the experimental process to retrieve energy-time entanglement information according to Eq. (26). For this simulation the cavity lengths of the dual-channel Fabry-Pérot interferometer are fixed at $d$, while the mirror transmission coefficient is set to $T=0.2$. For both (a) and (b), the horizontal axis is ${\omega }_{1}d/cmod\pi$, while the vertical axis is ${\omega }_{1}d/cmod\pi$. (a) Plot of $\sigma ({\omega }_1,{\omega }_2)$, whose form is assumed to be Gaussian. (b) Plot of $|{\sigma }^{\prime }({\omega }_1,{\omega }_2)|$ as the result of the interaction with the dual-channel Fabry-Pérot interferometer for the $m=0$ case of transmission coincidences, with a particular frequency-shifting choice of ${\omega }_L$ and ${\omega }_R$. By shifting the frequencies of ${\omega }_L$ and ${\omega }_R$ to other values, $|{\sigma }^{\prime }({\omega }_1,{\omega }_2)|$ will become the portion of $\left|\sigma \right({\omega }_1,{\omega }_2\left)\right|$ around a different line: ${\omega }_{1}d/c+{\omega }_{2}d/c=\text{const}$. In this numerical example, the phase of $\sigma ({\omega }_1,{\omega }_2)$ is assumed to be uniformly zero, while in reality this is not necessarily the case, and therefore an additional phase retrieval technique will be required.

Figure 4

Dual-channel interferometry realized via two polarization modes in a single Fabry-Pérot interferometer. HWP stands for half-wave plate. A pair of photons with orthogonal polarizations are combined, sent through the same cavity in the same spatial mode, and then split towards the SPCMs after they transmit. It will become clear that the dual channels can be realized by many methods, such as spatial separation, different polarizations, or even different spatial modes. However, because only one physical Fabry-Pérot interferometer is actually used and the settings for the two photons cannot be independently adjusted, the demonstration of entanglement or the violation of LHV would be difficult in this particular setup.

Figure 5

Numerical simulation of the transmission coincidence rates for the interferometer in Fig. 4 with different values of $T$. The rate is plotted as a function of the cavity length $d$, in logarithmic scale. All rates are normalized to the coincidence rate without inserting the Fabry-Pérot interferometer.