Scalable multiphoton quantum metrology with neither pre- nor post-selected measurements

The quantum theory of electromagnetic radiation predicts statistical fluctuations for diverse sources of light.^1–4 Indeed, the underlying amplitude and phase fluctuations of photons establish a quantum limit for the uncertainty of optical measurements made using classical states of light.⁵ These fundamental properties of light have enabled scientists to identify the shot-noise limit (SNL) as the classical limit for precision measurements.^6–8 Remarkably, states of light characterized by nonclassical statistical properties can, in principle, surpass the SNL.^5,9–16 This possibility has triggered an enormous interest in the experimental demonstration of quantum-enhanced measurements.^17–20 Unfortunately, efficient and scalable schemes for quantum metrology have remained elusive due to stringent technical requirements. Nevertheless, the possibility of estimating small physical parameters with greater sensitivity has important implications for diverse quantum technologies ranging from quantum communication to quantum information processing.^21–26 

Over the past three decades, there has been a strong impetus to employ quantum multiphoton states, as well as diverse measurement schemes, to demonstrate estimation of phase shifts with sensitivities that surpass the SNL.^27,28 Hitherto, N00N states, an example of multiphoton path-entangled states, have been extensively used to demonstrate super-resolution in quantum metrology.^28–35 Despite the relevance that these quantum states of light have for quantum metrology, their generation is extremely difficult, particularly when the N00N state involves more than two photons.²⁸ These challenges have motivated the exploration of conditional schemes that rely on pre- or post-selective measurements.^32–40 In experiments with “pre-selection,” researchers have increased the resolution of phase measurements by conditioning experiments on the observation of specific photon detections that herald the presence of a desired quantum state before the state interacts with the sample to be measured.^32,33,37 Conversely, implementations of optical phase estimation based on “post-selection” have been demonstrated by conditioning the outcomes of experiments on specific detection events after the probe state has passed through the sample.^34–41 Unfortunately, both protocols discard photons that represent valuable resources for interrogation of phase elements, and in practice pre- and post-selected schemes often succeed with low probability. Thus far, all demonstrations of quantum metrology for phase estimation have relied on either pre- or post-selected measurements or have lacked photon-number resolution.^{27,28,32–41} Nevertheless, theoretical studies suggest that counting of resources only when pre- or post-selection is successful may lead to enhanced sensitivities beyond the SNL.^42–44 

In 2017, Slussarenko and co-workers demonstrated the first N00N state-based quantum metrology protocol that did not rely on post-selection.²⁷ That protocol relied on the generation of a high-fidelity N00N state with N = 2 photons. Sharing similarities with other schemes that rely on N00N states, the loss of a single photon in this protocol removes all phase information encoded in the N00N state.³⁹ This vulnerability results in higher uncertainties when using N00N states than when using two-mode squeezed vacuum (TMSV) state for the estimation of optical phase shifts.^39,45,46 In addition, higher-order photon generation in spontaneous parametric downconversion (SPDC) induces errors in protocols that rely on specific photon events to generate perfect N00N states.²⁷ Despite the importance of conditional measurements for quantum technologies,⁴⁷ the possibility of using every photon produced by quantum sources to perform quantum parameter estimation constitutes one of the main goals of quantum optics. Consequently, the first demonstration of scalable multiphoton quantum metrology using photon counting detectors with neither pre- nor post-selected measurements represents an important progression in the field of quantum photonics. Here, we demonstrate a scalable scheme that utilizes all the detected photons from a two-mode squeezed vacuum state to perform unconditional quantum-enhanced phase estimation with an overall system efficiency of 80%. We define a scalable optical phase measurement protocol to be one whose Fisher information per photon (that interacts with the sample being measured) exceeds the standard quantum limit and increases with the mean photon number. This technique relies on an efficient source of SPDC and transition edge sensors (TESs) that enable the implementation of photon-number-resolving detection. We demonstrate multiphoton quantum enhancement in the estimation of 62% of the complete range of optical phases, without conditional measurements.

Our experimental setup is depicted in Fig. 1(a). We use a Ti:sapphire laser that is spatially filtered by a 20 μm pinhole and then focused to pump a type-II periodically poled potassium titanyl phosphate (ppKTP) crystal with a length of 2 mm and a poling period of 46.2 μm. We produce a spectrally pure TMSV state with orthogonal polarization H and V at 1550 nm through a process of spontaneous parametric downconversion.⁶⁵ Ideally, the TMSV state is described by $|z⟩=1−|z|2∑n=0∞zn|n⟩s|n⟩i$⁠, where n denotes the number of photons in the signal (s) and idler (i) modes, and z represents the squeezing parameter. The parameter z depends on the nonlinear properties of the crystal, as well as the pump mode and pump power. The filtered signal and idler photons pass through a rotated KTP crystal that compensates for temporal differences between signal and idler photons. Our protocol for phase estimation uses a common-path Mach–Zehnder interferometer (MZI). Here, the two paths of a conventional MZI are replaced with two polarization modes H and V. The phase $ϕ$ that we estimate in our experiment is controlled by a half-wave plate (HWP). Finally, signal and idler photons interfere at a polarizing beam splitter (PBS). The transmitted and reflected photons are then collected by single-mode fibers, and detected by TESs with photon-number resolution.

In order to illustrate the fundamental differences between our scalable protocol for multiphoton quantum metrology and previous schemes based on click detection, we introduce operators that describe the measurements performed by a TES and a superconducting nanowire single-photon detector (SNSPD). The latter is used to implement click detection. First, we define the ideal TES's positive-operator valued measure (POVM) corresponding to the detection of n photons as $Πn=|n⟩⟨n|$⁠. Similarly, we describe the ideal click detection operator associated with a SNSPD as $Π0=|0⟩⟨0|$ and $Πc=I−Π0$⁠. Here, $Π0$ represents a vacuum detection, and $Πc$ represents a click detection. As shown below, our TES-based protocol is more sensitive than the SNSPD-based protocol and scales to larger photon numbers.^27,40,41,48 This particular feature makes our protocol scalable.

We demonstrate this feature by analyzing the Fisher information obtained through the implementation of photon-number-resolving and click detection. For this purpose, we maximized the Fisher information over phase angles that lead to the best phase sensitivity using both detection schemes, while keeping mean photon number, losses, and the photon-number resolution of our TESs fixed. Here, we have assumed that our TESs can resolve up to ten photons. However, it is worth mentioning that TESs can efficiently resolve up to 100 photons at 1550 nm with high fidelities.^49,50 In Figs. 2(a) and 2(b), we plot the quantum Fisher information (black line) as well as the classical Fisher information for photon-number-resolving (red) and click (blue) detection. The quantum Fisher information under lossy condition is a known result,⁴⁵ and classical Fisher information can be calculated using the theoretical model described in the supplementary material. In this case, Fisher information is normalized with respect to the mean photon number interacting with the sampling being measured. This metric enables one to quantify the Fisher information gained per photon, as well as the performance of the protocols with respect to the SNL. These plots show that the Fisher information obtained through photon-number-resolving detection is always higher than click detection. Interestingly, in the lossless region, the performance of our metrology protocol approaches the quantum Fisher information. This characteristic feature differentiates our multiphoton quantum metrology protocol from previous schemes.^27,40

Moreover, in Fig. 2(c), we report an estimation of the ratio of the maximum classical Fisher information of our experiment $max(fTES)$ with respect to $max(fSNSPD)$⁠. The results in Fig. 2(c) indicate that our protocol outperforms metrology schemes based on click detection.^27,40 This advantage scales with the mean photon number of the interrogating photons. Additionally, the inset plot shows the percentage of phases that can be estimated with sub-SNL performance for click and photon-number-resolved detection. Figure 2 demonstrates the superior robustness of our protocol against losses with respect to those relying on click detection schemes. Naturally, losses and the limited resolution of the TES diminishes the sensitivity of quantum protocols for multiphoton metrology. Nevertheless, the latter limitation can be alleviated by multiplexing multiple TESs.⁵¹ Finally, we note that one can also achieve quasi-photon-number-resolved detection by multiplexing multiple single-photon detectors and beam splitters. However, this approach will inevitably introduce multiple losses in the experiment, therefore degrading the performance of the metrology device. Such systems only work best when the number of photons is much lower than the number of multiplexed detectors.

To surpass the SNL with multiphoton states and demonstrate the scalability of our protocol, we use an entangled multiphoton source. To increase the overall coupling efficiency, a weakly focused pump and small collection waist were employed.^52,66 In turn, our source has one of the highest overall efficiencies (∼80%) reported.^53,54 In order to characterize the performance of our experimental setup, we use calibrated SNSPDs. These detectors are used to measure the Hong–Ou–Mandel visibility of $v=99.36(5)%$ for our common-path interferometer. Note that for the HOM measurement, we rotate the HWP by 45° at zero delay. We also determine the overall detection efficiencies (source to SNSPD) of the system. This in turn allows us to estimate the overall source-to-fiber efficiencies for each of the arms. In our experiment, we implement photon-number-resolving detection with TESs that were characterized through quantum detector tomography.⁵⁵ We estimate the efficiencies of our TESs from the tomography data shown in Fig. 1(b). The details of the tomography method can be found in the supplementary material. Furthermore, we estimate the overall system detection efficiencies (source to TES) for each of the outputs of our experiment. All the relevant efficiencies are listed in Table I.

As shown in Figs. 3(a) and 3(b), the TESs enable us to probe all the possible multiphoton interference events, up to four photons. The black dotted line in Fig. 3(a) shows that the total number of detected photons does not depend on the phase. We omit the vacuum events and single-photon events in the plot, which are nearly constant for varying phase angles and therefore do not contribute significantly to the ability to detect changes in this parameter (see supplementary material). This means that these events do not contribute to the sensitivity of our phase estimation measurements. Moreover, in Fig. 3(b) we plot the traces produced by three- and four-particle interference. The photon-number resolution of our protocol enables the identification of the complex multiparticle interactions that take place in our experiment. Every photon count conveys relevant information regarding the phase that we aim to estimate. This information is not available in quantum metrology protocols that rely on click detectors, which can distinguish zero from one or more photons but otherwise lack photon-number resolution. Furthermore, our scheme enables the estimation of a broad range of phases with sensitivities that surpass the SNL. To compare our results to those that would be obtained without photon-number-resolving detectors, we reconstruct interference structures obtained with click detectors in Fig. 3(c).

To quantify the performance of our metrology device, we use the classical Fisher information $f=∑i(∂ ln pi∂ϕ)2pi$⁠, where the probability p_i includes all possible detection results for an experimental trial, i.e., $i∈{00,01,10,11,…}$⁠; so $f$ quantifies the information gained in each trial. Each p_i is estimated using the data given in Fig. 3. Probabilities that do not vary with $ϕ$⁠, such as the probability p₀₀ of a vacuum event, contribute zero for the corresponding term in the sum defining $f$⁠. As shown in Fig. 4, we have calculated the classical Fisher information $fTES$ and $fSNSPD$⁠. We also calculate the SNL given by $fSNL=n¯$⁠, the average number of photons generated per experimental trial, derived from our experimental results. The average photon number $n¯=3.631(14)×10−3$ is estimated by fitting our experimental data to the theoretical model of our setup. Such an approach takes all losses into account, including the mode mismatch at the generation of entangled photons, fiber-coupling loss as well as the loss of our TESs. The details of the calculation can be found in the supplementary material. The blue line represents the Fisher information estimated for a detection scheme with photon-number resolution. As a comparison, we plot the Fisher information using the click detector data presented in Fig. 3(c), which shows lower Fisher information at all angles. These results confirm the fundamental difference between the implementation of the operator Π_n and $Πc$ as discussed in Ref. 31. It is also worth noticing that the click detector measurement barely reaches the SNL represented by the purple dashed line, whereas the measurement with TESs surpasses the SNL for an impressively broad range of 62% of the phase angles. It is worth highlighting the robustness of our protocol against losses. Remarkably, our scheme for quantum metrology allows for phase estimation with sensitivities that surpass the SNL, even in dissipative environments with up to 28% loss—meaning that in our specific experimental realization, inserting a sample with up to about 11% loss would still have surpassed the SNL at a phase angle of $ϕ=π/4$⁠. This prediction is obtained based on our theoretical model outlined in the supplementary material. These results demonstrate the superior performance of our metrology scheme utilizing photon-number-resolved detection. In contrast to other schemes that rely on N00N states or conditional measurements, the sensitivity of our technique is improved through the generation and detection of high-order photon pairs. This unique feature of our protocol makes it scalable.

Over the past three decades, scientists and engineers have developed technology and explored novel quantum states of light, as well as quantum measurement schemes, with the aim of demonstrating scalable and unconditional quantum protocols. For the first time, we demonstrated the possibility of using photon-number resolution detectors with quantum sources of light to perform scalable estimation of optical phase shifts with sensitivities that surpass the SNL over a wide range. The broad-peaked Fisher information in our metrology experiment enables one to exceed the shot-noise limit for almost 62% of the phase space even in the presence of losses. This feature of our technique is important for quantum-enhanced phase measurement applications such as quantum imaging.^56,57 Our multiphoton quantum metrology protocol increases Fisher information per photon compared to the click detection protocol. This possibility has shown important implications for measuring and analyzing sensitive properties of delicate samples.^58,59 Furthermore, our technique is relevant for other quantum technologies that rely on nonclassical multiphoton interference, such as boson sampling and quantum networks.^22,60–64