Technical note realistic FOMs

The figure of merit

A single photon source is characterized by the the single-photon purity $g^{(2)}(0)$, the indistinguishability $\mathcal{I}$, and the collection efficiency $\eta_\mathrm{tot} = \beta \times \eta_\mathrm{cav \rightarrow wg} \times \eta_{\mathrm{wg \rightarrow SMF}}$ where $\beta=\frac{\Gamma_\mathrm{1D}}{\Gamma_\mathrm{1D} + \Gamma^\prime}$ is the proportion of photons emitted into the cavity mode, $\eta_\mathrm{cav\rightarrow wg}$ is the coupling efficiency of the cavity to the waveguide, and $\eta_\mathrm{wg\rightarrow SMF}$ is the coupling efficiency of the waveguide to a single-mode fiber.

We want to make the case that $\eta_\mathrm{tot}$ is the only technologically challenging figure of merit. $g^{(2)}(0)$ for DBT is already exceptional in free-space and so is $\mathcal{I}$. In a cavity, they will only be better. For realistic applications, the only bottle-neck is $\eta_\mathrm{tot}$.

• $\mathcal{I}$ for DBT in free-space was measured to be $0.89\pm0.05$ under CW excitation and $0.78\pm0.04$ with pulsed excitation [article].
• DBATT was measured to have $\mathcal{I}=0.96$ without a cavity.

We have already demonstrated $\beta=0.44$. With a ten-fold increase in cooperativity (achievable with a high-Q cavity design) we could achieve $\beta\approx 0.9$. Commercial contractors could obtain $\eta_\mathrm{wg \rightarrow SMF}=0.9$ easily or even as high as $\eta_\mathrm{wg \rightarrow SMF}=0.95$ with specialized techniques. We have experimentally measured $\eta_\mathrm{cav\rightarrow wg}]\approx 1$ already. With specialized fabrication, this could be achieved even after integration and with a high Q cavity. We can therefore expect $\eta_\mathrm{cav\rightarrow wg}\approx 0.9$. This gives a realistic $\eta_\mathrm{tot}=0.7$ in the near-term.

With state-of-the-art cavities ($Q=100,000$) at our current mode volums, we chould achieve $\beta=90\%$. It has been shown that the DWFC factor of terrylene can be as high as 80% on hBN [article]. This means that, with improved material design and synthesis, we could triple the DWFC factor of DBT, giving $\beta\approx95\%$. Specialized fiber-waveguide coupling techniques, such as evanescent coupling, could achieve coupling efficiencies near 99%. This would put $\eta_\mathrm{tot}$ at 95%.

To summarize, | | Current demonstrations | With commercially-available technology | With state-of-the-art technology | With future technology | | --- | --- | --- | --- | --- | | $g^{(2)}(0)$ | $<10^{-3}$ | negligible | negligible | negligible | | $\mathrm{I}$ | $0.88$ | $0.97$ | ? | $>0.999$ | | $\beta$ | 0.44 | 0.9 | 0.96 | $>0.96$ | | $\eta_\mathrm{tot}$ | ? | 0.7 | 0.95 | ? |

Note: The cavity proposed in this paper may be able to give a coupling efficiency as high as 97% article`

We can compare this with the requirements for different applications.

Quantum repeater protocol: article