Quantum-enhanced interferometry with large heralded photon-number states

Abstract

Quantum phenomena such as entanglement can improve fundamental limits on the sensitivity of a measurement probe. In optical interferometry, a probe consisting of $N$ entangled photons provides up to a $\sqrt{N}$ enhancement in phase sensitivity compared to a classical probe of the same energy. Here, we employ high-gain parametric down-conversion sources and photon-number-resolving detectors to perform interferometry with heralded quantum probes of sizes up to N = 8 (i.e. measuring up to 16-photon coincidences). Our probes are created by injecting heralded photon-number states into an interferometer, and in principle provide quantum-enhanced phase sensitivity even in the presence of significant optical loss. Our work paves the way towards quantum-enhanced interferometry using large entangled photonic states.

FIG. 1.. Interferometric scheme.

(a) Two type-II parametric down-conversion sources each produce orthogonally-polarized pairs of beams that are separated using polarizing beam splitters. By measuring one of the beams from each source with a photon-number-resolving detector, we herald a pair of photon-number states $|h_1,h_2\rangle$. We inject this probe into an interferometer and perform photon counting at the output to estimate the unknown phase difference ϕ. (b) Quantum Fisher information Q calculated for 8-photon (N = h₁ + h₂ = 8) probes inside the interferometer as a function of the signal transmissivity η_s which is assumed to be equal in both interferometer modes. Coloured curve in the main figure plots Q of the probe with the optimal Δ = |h₁ − h₂| for a given η_s, while the inset shows the full curves of each probe for η_s ∈ [0.4,0.75]. Our probe approximates the performance of the optimal state [black line] and surpasses that of the N00N state [dashed line] for efficiencies below ~ 90%. The grey filled region indicates performance below the shot-noise limit.

FIG. 2.. The weak gain regime.

(a) Rates measured with the probe $|1,1\rangle$: (s₁, s₂, h₁, h₂) = (2, 0, 1, 1) [blue], (1, 1, 1, 1) [orange], (0, 2, 1, 1) [green]. (b) Rates measured with the probe $|2,1\rangle$: (3, 0, 2, 1) [blue], (2, 1, 2, 1) [orange], (1,2,2,1) [green], (0,3,2,1) [red]. Error bars are one standard deviation assuming Poissonian counting statistics. Lines are a model fitted to ${\text{pr}}_{s_1,s_2,h_1,h_2}(\varphi )$. Bottom panels show the normalized Fisher information ${\tilde{\mathcal{F}}}_{h_1,h_2}(\varphi )$ calculated using two methods: (i) post-selecting on events where s₁ +s₂ = h₁ +h₂ [green] and (ii) using all events [red]. Line thicknesses show 1σ confidence intervals obtained by fitting 50 simulated data sets that are calculated with a Monte Carlo method. The dashed black line indicates the shot-noise limit.

FIG. 3.. The high gain regime.

(a) ${\tilde{\mathcal{F}}}_{h_1,h_2}(\varphi )$ of 8-photon probes (N = 8) parameterized by Δ = |h₁ − h₂|. Curves are calculated using the data and Eqs. (2) and (3) without post-selection. Probes with a larger Δ have a larger ${\tilde{\mathcal{F}}}_{h_1,h_2}(\varphi )$ and hence greater phase sensitivity due to their increased robustness to loss. Line thicknesses show 1σ confidence intervals obtained by fitting 50 simulated data sets that are calculated with a Monte Carlo method. (b) and (c) show a subset of rates for the probe with Δ = 8 and Δ = 0, respectively. Error bars are one standard deviation assuming Poissonian counting statistics. The lines are a model fitted to pr_s1,s2,h1,h2(ϕ).

FIG. 4.. Testing multiphoton interference.

Benefits of multiphoton interference using the probe $|3,2\rangle$. (a) Two sets of rates [blue: (5, 0, 3, 2), orange: (3, 2, 3, 2)] measured when the photons are injected inside the interferometer at the same time (data: circles, theory: bold lines) or at different times (data: crosses, theory: dashed line). In the latter case, the photons are well modelled by classical distinguishable particles. Error bars are one standard deviation assuming Poissonian counting statistics. (b) ${\tilde{\mathcal{F}}}_{3,2}(\varphi )$ shows a significant improvement in sensitivity in the former case (bold line) compared to the latter case (dashed line), demonstrating that multiphoton interference improved the sensitivity of our probe. Red shaded regions shows 1σ confidence intervals obtained by fitting 50 simulated data sets that are calculated with a Monte Carlo method.