Scalable Hong-Ou-Mandel manifolds in quantum-optical ring resonators

Quantum information processing, from cryptography to computation, based upon linear quantum-optical circuit elements relies heavily on the ability offered by the Hong-Ou-Mandel (HOM) effect to “route” photons from separate input modes into one of two common output modes. Specifically, the HOM effect accomplishes the path entanglement of two photons at a time such that no coincidences are observed in the output modes of a system exhibiting the effect. In this paper, we prove, in principle, that by operating a specific nanophotonic device properly, one can conditionally “bunch” coincident input photons in a way that is more configurable than with an ordinary 50:50 beam splitter, while maintaining the inherent scalability of such an on-chip device.

Figure 1

Schematic comparison of (a) a beam splitter with (b) a dual coupled ring resonator. The system shown in (b) represents our prototype for a scalable QIP circuit element. The inset to the right reminds the reader of the action of the evanescent directional coupler. The phase convention here follows that implied by Yariv in Ref. [16].

Figure 2

A cartoonlike representation of modeling a singly coupled lossy ring resonator (a) using a dually coupled, lossless ring resonator and drop port (b). Losses in (a) are modeled by tracing over mode $l$ in (b). This figure serves as a rationale for our choices of $f$ (“fictitious”) and $l$ (“loss”) for mode labels and operators in the system we analyze in this work.

Figure 3

Surface plot of the probability $P$(1,1) for output photon coincidence for the case in which two incident photons are coincident on the ring along modes $a$ and $f$. The Hong-Ou-Mandel effect is the zeros of this surface plot. The level set of zeros forms a continuous manifold in this case; it is a one-dimensional HOMM. In this figure, as in Figs. 4 and 5, we display the same result over the ranges (a) $\theta \in \left[0,2\pi \right]$ and (b) $\theta \in \left[-\pi ,\pi \right]$. We do this merely for ease of visualization and reference; the two plots in each figure so displayed convey the same information.

Figure 4

Parametric plot of a one-dimensional HOMM based upon the wHOMMC. The point(s) labeled $C$ is the one for which the wHOMMC breaks down. Away from point(s) $C$, the parametric curve shown in this figure lies along the level set of zeros in Fig. 3. Points $A$ and $B$ are included to accompany the discussion in Sec. 4. In (a) $\theta \in \left[0,2\pi \right]$ and in (b) $\theta \in \left[-\pi ,\pi \right]$.

Figure 5

Surface plot of the two-dimensional HOMM for the case of an unbalanced dually coupled ring resonator. In (a) $\theta \in \left[0,2\pi \right]$ and in (b) $\theta \in \left[-\pi ,\pi \right]$. The higher dimension of the HOMM for this case suggests even more opportunity for robustness of the HOM along the lines of that described for the one-dimensional case discussed explicitly in the paper.

Figure 6

Plots of $P$(1,1) for several values of $\tau =\eta \in \mathbb{R}$, corresponding to balanced operation of the dually driven ring resonator. The HOM dips are apparent, provided they occur. Clearly, both the existence and resolution of such and HOM dip are dependent on the coupling parameters for the system. This suggests the existence of an “optimal” operating region for an on-chip device made from dually coupled rings. In (b) we demonstrate that the pragmatically important case involving 3 dB couplers is safely within this “optimal” region, however, heuristically defined at this point.