Multimode metrology via scattershot sampling

Scattershot photon sources are known to have useful properties for optical quantum computing and boson-sampling purposes, in particular for scaling to large numbers of photons. This paper investigates the application of these scattershot sources towards the metrological task of estimating an unknown phase shift. In this regard, we introduce three different scalable multimode interferometers and quantify their quantum Fisher information performance using scattershot sources with arbitrary system sizes. We show that two of the interferometers need the probing photons to be in certain input configurations to beat the classical shot-noise precision limit, while the remaining interferometer has the necessary symmetry which allows it to always beat the classical limit no matter the input configuration. However, we can prove that all three interferometers give the same amount of information on average, which can be shown to beat the classical precision limit. We also perform Monte Carlo simulations to compare the interferometers in different experimentally relevant regimes, as a function of the number of samples.

Figure 1

(a) A two-mode squeezer, characterized by a $\chi$ squeezing parameter, has a chance of creating pairs of photons via spontaneous parametric down-conversion of pump photons. This means if we detect $n_i$ photons in the red upper mode, we expect to find $n_i$ photons in the blue lower mode. (b) By stacking an $m$ array of $\chi$ strength squeezers and heralding photodetectors (here for $m=4$), we can create a scattershot source which allows us to quickly perform metrology experiments with good potential for scaling to higher modes.

Figure 2

(a) Standard metrology experiment to determine $\varphi$, using a Mach-Zehnder interferometer, photon inputs, and photon-counting detectors. (b) We explore the natural multimode version of this standard setup, where we investigate interferometers with phase shifts applied to half of the $m$ modes. (c) One multimode interferometer of this type is made from stacking together $m/2$ copies of Mach-Zehnder interferometers. However, the amount of information gathered about $\varphi$ depends on which ports the photons are injected into. (d) We will show that we can create a symmetrizing transformation $T_m$, which results in all mode being treated equally, just like in the standard two-mode metrology experiment.

Figure 3

(a) The upper graph is the cumulative probability distribution of ${10}^4$ random walkers. Each individual walker tracks the total Fisher information difference $\mathrm{\Delta }{\mathcal{F}}_{\text{tot}}\left(k\right)={\mathcal{F}}_{\text{tot}}^{\left(\text{sep}\right)}\left(k\right)-{\mathcal{F}}_{\text{tot}}^{\left(\text{sym}\right)}\left(k\right)$, between the separable $Y_m^{\left(\text{sep}\right)}$ and symmetric interferometers $Y_m^{\left(\text{sym}\right)}$, as more random samples are generated. The lower graph distills the probability that the separable experiment gave more information $\mathbb{P}[{\mathcal{F}}_{\text{tot}}^{\left(\text{sep}\right)}\left(k\right)>{\mathcal{F}}_{\text{tot}}^{\left(\text{sym}\right)}\left(k\right)]$, for a given sample size $k$; the maximum sample size in which $Y_m^{\left(\text{sym}\right)}$ still has an advantage $k_{\text{adv}}$ is indicated. The shaded regions with labels $\ge 1, \ge 2$, etc., refer to the number of quantum-enhanced measurements required by the separable interferometer to beat the symmetric interferometer. These graphs were constructed using a system with $m=2^{16}$ modes and samples from a scattershot source with $\chi \approx 0.0247$ squeezers. (b) Same graphs as in (a) but with more modes $m=2^{18}$ and weaker squeezing $\chi \approx 0.0107$. (c) Photon statistics of the samples, which show that these particular $m$ and $\chi$ parameters translate to an average of $\langle n\rangle =40$ photons for (a) and $\langle n\rangle =30$ photons for (b). We plot an analytically derived expression for the maximum sample size advantage of the symmetric interferometer $k_{\text{adv}}$, which shows that it (d) increases linearly as we increase the number of modes $m$ in the system, but (e) decreases inversely as we increase the average number of photons $\langle n\rangle$.

Figure 4

One possible decomposition of the four-mode symmetrizing transformation. The blue $V_2$ beam splitters have reflectivities of ${\eta }_1=1/\sqrt{50+20\sqrt{6}}\approx 0.10, {\eta }_2=\sqrt{2/5}\approx 0.63, {\eta }_3=1/\sqrt{251-100\sqrt{6}}\approx 0.41$, and ${\eta }_4=1/\sqrt{300+120\sqrt{6}}\approx 0.04$, while the red $\pi$ phase shifters induce a relative $-1$ phase difference in the indicated modes.

Figure 5

Example of how a particular mode permutation $M_4$ can be used to make input invariant statements. At the first step, the arbitrary input state is simply permuted $M_4^{-1}{Y}_4^{\left(\text{sym}\right)}{M}_4|n_1{n}_2{n}_3{n}_4\rangle =M_4^{-1}{Y}_4^{\left(\text{sym}\right)}|n_3{n}_1{n}_2{n}_4\rangle$. Then it must be recognized that $M_4^{-1}{Y}_4^{\left(\text{sym}\right)}|n_3{n}_1{n}_2{n}_4\rangle$ and $Y_4^{\left(\text{sym}\right)}|n_3{n}_1{n}_2{n}_4\rangle$ have the same number measurement outcomes but at different modes, which means they have the same Fisher information as its a sum over all outcomes. If there is a self-similarity transformation $Y_4^{\left(\text{sym}\right)}=M_4^{-1}{Y}_4^{\left(\text{sym}\right)}{M}_4$, then it follows that $Y_4^{\left(\text{sym}\right)}|n_1{n}_2{n}_3{n}_4\rangle$ and $Y_4^{\left(\text{sym}\right)}|n_3{n}_1{n}_2{n}_4\rangle$ also have the same Fisher information.

Figure 6

(a) Similar to Fig. 3, where the upper graph is the cumulative probability distribution of ${10}^4$ random walkers that individually track the running total QFI difference $\mathrm{\Delta }{\mathcal{F}}_{\text{tot}}\left(k\right)={\mathcal{F}}_{\text{tot}}^{\left(\text{sep}\right)}\left(k\right)-{\mathcal{F}}_{\text{tot}}^{\left(\text{sym}\right)}\left(k\right)$ as more random samples are taken. The lower graph is the probability that the separable experiment gave more information $\mathbb{P}[{\mathcal{F}}_{\text{tot}}^{\left(\text{sep}\right)}\left(k\right)>{\mathcal{F}}_{\text{tot}}^{\left(\text{sym}\right)}\left(k\right)]$, for a given sample size $k$. This was done using a small system with $m=2^5$ modes and random samples from a scattershot source with $\chi =0.25$ squeezers. (b) Same graphs as in (a) but with more modes $m=2^6$ and weaker squeezing $\chi =0.125$. (c) Photon statistics of the samples.

Figure 7

(a) Similar to Fig. 3, where the upper graph is the cumulative probability distribution of ${10}^4$ random walkers that individually track the running total QFI difference $\mathrm{\Delta }{\mathcal{F}}_{\text{tot}}^{\prime }\left(k\right)={\mathcal{F}}_{\text{tot}}^{\left(\text{uni}\right)}\left(k\right)-{\mathcal{F}}_{\text{tot}}^{\left(\text{sym}\right)}\left(k\right)$ as more random samples are taken. The lower graph is the probability that the uniform experiment gave more information $\mathbb{P}[{\mathcal{F}}_{\text{tot}}^{\left(\text{uni}\right)}\left(k\right)>{\mathcal{F}}_{\text{tot}}^{\left(\text{sym}\right)}\left(k\right)]$, for a given sample size $k$. This was done using a system with $m=2^{16}$ modes and samples from a scattershot source with $\chi \approx 0.0247$ squeezers. (b) Same graphs as in (a) but with more modes $m=2^{18}$ and weaker squeezing $\chi \approx 0.0107$. (c) Photon statistics of the samples, which show that these particular $m$ and $\chi$ parameters translate to an average of $\langle n\rangle =40$ photons for (a) and $\langle n\rangle =30$ photons for (b).