Beyond coincidence in Hong-Ou-Mandel interferometry

The Hong-Ou-Mandel effect provides a mechanism to determine the distinguishability of a photon pair by measuring the bunching rates of two photons interfering at a beam splitter. Of particular interest is the distinguishability in time, which can be used to probe a time delay. Photon detectors themselves give some timing information, however, while that resolution may dwarf that of an interferometric technique, typical analyses reduce the interference to a binary event, neglecting temporal information in the detector. By modeling detectors with number and temporal resolution we demonstrate a greater precision than coincidence rates or temporal data alone afford. Moreover, the additional information can allow simultaneous estimation of a time delay alongside calibration parameters, opening up the possibility of calibration-free protocols and approaching the precision of the quantum Cramér-Rao bound.

Figure 1

(a) A schematic of our protocol. Photons are generated via spontaneous parametric down conversion. In the top arm, the photon passes through a sample and is delayed by ${\delta }_s$. An adjustable delay ${\delta }_a$ is placed in the bottom arm. Both photons then interfere at the beam splitter, with a relative time delay $\delta ={\delta }_s-{\delta }_a$. Detectors D1 and D2 are placed behind the beam splitter output ports. Each detection event is then assigned to a time bin and checked for coincidence or bunching. (b) How we model the photon modes either side of the beam splitter in Sec. 3. In the bottom arm, an additional orthogonal mode ${\widehat{b}}^{†}$ is present to capture differences other than temporal between the photons. A mixture of modes ${\widehat{c}}^{†}$ and ${\widehat{d}}^{†}$ leave both beam splitter ports.

Figure 2

Possible outputs for a pair of photons incident on the different types of detector we consider. First, standard (non-time-resolving) bucket detectors: each clicks only once whether one or more photons arrive. Then, bucket detectors with time-resolving capabilities: each detector click is now associated with a specific time bin, giving some precision as to when a photon arrives. As a consequence of detector dead time if two photons arrive at the same detector then it only clicks once, for the time bin when the earlier photon arrived. Then, number-resolving detectors without time-resolving capabilities: the detector can now differentiate between one or two photons arriving. Finally, number-resolving detectors with time-resolving capabilities: now, it can indicate when each photon arrived, up to the precision of our time bins.

Figure 3

Comparison of the relative information $I_{\mathrm{rel}}=F_{\delta }/H_{\delta }^{2\mathrm{ph}}$, the classical Fisher information as a fraction of the two-photon conditioned quantum Fisher information, for our protocol configurations as we vary the delay $\delta$. We contrast the basic HOM protocol with bucket detectors against versions with number-resolving (NR-HOM) and time-resolving (TR-HOM) detectors, as well as both enhancements simultaneously (NRTR-HOM). Finally, no-HOM is a time of flight protocol that solely measures arrival times. We set $\alpha =0.9$ and $\gamma =0.4$, and the time bin widths $T=5/\sigma$ on the left, and $T=1/\sigma$ on the right. All HOM protocols feature peaks in the information equally spaced around $\delta =0$. The no-HOM protocol displays (nonsinusoidal) oscillation in the relative information with period equal to $T$. The amplitude of this oscillation reduces with low $T$, and can be seen in more detail in the insets. The right plot shows the benefits of reduced $T$ (better resolution) by the increased information.

Figure 4

The relative information $I_{\mathrm{rel}}$ for time-resolving protocols as we vary time bin size $T$. We set $\gamma =0.4$, and $\delta$ is chosen to maximize the information; with $\alpha =0.9$ (as in Fig. 3) in the top panel and $\alpha =0.99$ in the bottom panel. The no-HOM protocol performs poorly at low time resolution, but around $T=1/\sigma$ overtakes TR-HOM and achieves parity with NRTR-HOM. As $T\to 0$, no-HOM and NRTR-HOM approach the QFI limit. Increased visibility extends the range across which a HOM-based protocol is advantageous.

Figure 5

The relative information $I_{\mathrm{rel}}$ as we vary $\delta$ and $\alpha$ for the two extreme HOM protocols: standard HOM (bottom surface) vs NRTR-HOM (top surface). The two characteristic peaks grow and move inwards at higher visibility; merging to a single peak at the origin that saturates the two-photon conditioned QFI for perfect visibility $\alpha =1$ as shown in the inset. At large $\delta$, background time bin information manifests itself in the NRTR-HOM surface not falling to zero. Other parameters are $\gamma =0.4$ and $T=5/\sigma$.

Figure 6

Left: $F_{\delta }$ as a function of $\gamma$ and $T$. The standard HOM case (bottom surface) is independent of $T$ and thus only depends on $\gamma$. Here, $\alpha =0.9$ and $\delta$ is chosen to maximise information at each point. Both protocols naturally perform better at lower $\gamma$, and NRTR-HOM (top surface) approaches $H_{\delta }^{2\mathrm{ph}}$ as $T\to 0$. Center: $I_{\mathrm{rel}}$ as a function of $\gamma$. $\delta$ is chosen to maximise information at each point. For NRTR-HOM, the relative information remains constant as $\gamma$ is varied, but for standard HOM the information decreases at higher $\gamma$ (see text). $T=5/\sigma$ for the top plot, $T=1/\sigma$ for the bottom plot, $\alpha =0.9$ is used in both. Right: the relative information $I_{\mathrm{rel}}$, taking $\gamma =0.4$, and varying $\delta$ and $T$. Standard HOM (bottom surface) is again independent of T. At vanishingly small $T$, NRTR-HOM (top surface) achieves the QFI limit for sufficiently large $\delta$.

Figure 7

The determinant of the Fisher information matrix for parameters $\delta$ and $\alpha$ for $\alpha =0.9$ and $\gamma =0.4$. Top: for fixed $\delta =0.2/\sigma$ and varying time bin size $T$. At higher time resolution (smaller $T$) the determinant increases, and the benefit of number-resolving detectors become more pronounced. Bottom: for fixed $T=5/\sigma$ and varying $\delta$. The two peaks in the determinant represent the optimal points for estimating $\delta$ and $\alpha$ simultaneously.

Figure 8

The relative information $I_{\mathrm{rel}}$ for our non-time-resolving protocols. We set $\gamma =0.4$, choose $\delta$ to maximize information, and vary $\alpha$. As we approach $\alpha =1$, the two information peaks converge, leading to a single peak at $\delta =0$ where $F_{\delta }=H_{\delta }^{2\mathrm{ph}}$ and we have reached the QFI limit.

Figure 9

Comparison of the CFI $F_{\alpha }$ for our various protocol configurations, as we vary the delay $\delta$. We set $\alpha =0.9,\gamma =0.4,T=5/\sigma$. The information peaks at $\delta =0$, where coincidence and bunching rates vary most rapidly with $\alpha$. The benefits of time resolution are small, both bucket curves closely overlap, as do both number-resolving curves, and at $\delta =0$ the difference in information between time-resolving and non-time-resolving protocols vanishes.

Figure 10

Comparison of the CFI $F_{\sigma }$ for our various protocol configurations, as we vary the delay $\delta$. We set $\alpha =0.9, \gamma =0.4, T=5/\sigma$. These follow a similar shape to the plots of $F_{\delta }$ in Fig. 3; the main differences being that the peaks are narrower, more spread out, and for TR-HOM and NRTR-HOM $F_{\sigma }$ does not dip all the way to zero at the origin. With the inset we can see that the no-HOM information follows a similar oscillatory pattern, with maxima and minima in the same positions but an overall more squared curve.

Figure 11

Comparison of the CFI $F_{\gamma }$ for our various protocol configurations, as we vary the delay $\delta$. We set $\alpha =0.9$ and $\gamma =0.4$. No-HOM is functionally identical to NR(TR)-HOM in regard to $F_{\gamma }$, as the photons are always directed to separate detectors. In these cases the information is constant. For (TR-)HOM, the information is reduced somewhat due to the bunching or loss ambiguity. It tends to a constant for large $\delta$, but further dips slightly near the origin where bunching is more likely.