Abstract
Quantum metrology aims to realise new sensors operating at the ultimate limit of precision measurement. However, optical loss, the complexity of proposed metrology schemes and interferometric instability each prevent the realisation of practical quantumenhanced sensors. To obtain a quantum advantage in interferometry using these capabilities, new schemes are required that tolerate realistic device loss and sample absorption. We show that losstolerant quantum metrology is achievable with photoncounting measurements of the generalised multiphoton singlet state, which is readily generated from spontaneous parametric downconversion without any further state engineering. The power of this scheme comes from coherent superpositions, which give rise to rapidly oscillating interference fringes that persist in realistic levels of loss. We have demonstrated the key enabling principles through the fourphoton coincidence detection of outcomes that are dominated by the fourphoton singlet term of the fourmode downconversion state. Combining stateoftheart quantum photonics will enable a quantum advantage to be achieved without using postselection and without any further changes to the approach studied here.
Introduction
Quantum shot noise represents a hard limit for the precision of all sensors that do not harness quantum resources. Photonic quantum metrology^{1} promises to surpass the shotnoise limit (SNL, $\mathrm{\Delta}\varphi \sim 1/\surd N$) by using quantum states of light that exhibit entanglement,^{2} discord^{3} or squeezing^{4} to suppress statistical fluctuation in opticalphase estimation. This will ultimately enable greater precision in measuring experimental quantities of interest, such as distance, birefringence, angle or sample concentration. Metrology in the lowphotonflux regime is pursued to gain maximal information while minimising detrimental effects from probe light—for example, in biological sensing.^{5,6} This is complementary to the objectives of gravitywave astronomy schemes that use squeezed light and require high circulating laser power (watts to kilowatts) in kmscale interferometry for subSNL performance.^{7}
A widespread objective of quantum metrology has been to engineer instances of ‘NOON’ states,^{2,8–12} which are pathentangled states of N photons across two modes $\frac{1}{\surd 2}\left(N\u30090\u3009+0\u3009N\u3009\right)$. They offer both superresolution (Nfold decrease in fringe period) and supersensitivity (enhanced precision towards the Heisenberg limit—Δϕ~1/N), and they are the optimal state for lowflux sensing in the lossless regime. The current record in size of NOONlike states is five photons using postselection^{13} and four photons using ancillaryphoton detection.^{14} Key components for this architecture have been demonstrated in integrated optics, including state generation and manipulation,^{15} microfluidics^{5} and photon detection.^{16} Ultimately, this could enable practical deployment of quantumenhanced sensors outside of the quantum optics laboratory. However, to date, unavoidable optical loss hampers quantum advantage and can actually lead to worse precision than by just using a bright laser.^{2} Moreover, loss will be present in any practical scenario, including absorbance in measured samples and nonunit efficiency detectors. Consequently, revised scaling laws of precision with photon flux have been derived,^{17} along with optimised superposition states of fixed photon number, numerically for small photon number^{18,19} and analytically for large.^{17}
Here, we demonstrate the underpinning principles of a practical—losstolerant—scheme for subshotnoise interferometry, illustrated in Figure 1. This scheme is designed to use the full fourmode multiphoton state naturally occurring in a nonlinear optical process known as typeII spontaneous parametric downconversion (SPDC), which generates a coherent superposition of correlated photonnumber states.^{20–22} This scheme can exhibit a remarkably high tolerance to loss by harnessing all detected photonnumber correlations arising from the superposition of all generated photonnumber components in the generalised singlet state. We greatly simplify the theoretical analysis of a prior version proposed in ref. 23 using the positiveoperatorvalued measurement (POVM) formalism; this enables application of arbitrary photonnumbercounting methods, including the multiplexed detection scheme used in our experiment here. The scheme can tolerate up to 40% total loss using photon counting without postselection (Figure 2). To motivate developing such a photoncountingbased system, the typeII SPDC state can be studied in terms of photonnumber measurement outcomes in isolation: nphoton detection events can be registered from any photonnumber component of the entire multiphoton state ⩾n, under all loss. Here, we measure the fourfold detection events arising from all SPDC contributions of four or more photons, which is the smallest nontrivial measurement subset that displays the singlet behaviour with more than one photon in the sensing interferometer of the scheme (and thereby capable of exhibiting quantum advantage). When accounting for all photons passing through the phase shift ϕ in our experiment, the total loss in our experiment of ∼83% prohibits us from observing precision that surpasses the SNL. However, as evidence of the potential of this scheme to provide a quantum advantage, we investigate experimentally the correlations of the subset of fourfold detection events, which in principle enable a quantum advantage of up to 45% in the meansquared error for estimating phase for our measured experimental parameters, and we find that the dynamics we observe is in good agreement with theoretical prediction.
Our approach addresses serious challenges faced by discretevariable photoncounting schemes that are designed to operate with a deterministically generated or heralded fixed number of photons. The only system that has demonstrated quantum interference of more than two photons is SPDC, and complex experimental techniques comprising optical delay, fast switching and auxiliary photon detection are required to generate fixed photonnumber states. This includes approaches to losstolerant quantum metrology, such as Holland and Burnett states.^{24–27} To perform experiments with n photons, postselection is commonly used to ignore components of fewer photons (<n), whereas terms associated with higherphoton number (>n) are treated as noise. This is particularly problematic for quantum metrology, where all photons passing through the sample need to be accounted for, and unwanted photonnumber components are detrimental to measurement precision. In contrast, for the scheme presented here, higherphoton terms contribute to enhancement of phase sensitivity of the fourfold detection events.
The Scheme: metrology using typeII SPDC
Our demonstration (Figure 1b) can be treated in three stages: (i) a source of multiphoton entangled light, (ii) unitary rotation with an unknown parameter ϕ to be estimated on the sensing path a, equivalent to interferometry and (iii) correlated photoncounting measurement.
(i) The source is based on pulsed noncolinear typeII SPDC^{20,21}, which generates entanglement across four modes—two spatial paths (a, b) and two polarisations (h,v) (see Materials and methods). In the ideal case, for which all experimental imperfections are neglected, the state generated is a fourmode squeezed state^{23} that is the superposition of photonnumber states that are symmetrical across a and b
where τ is an interaction parameter that corresponds to the parametric gain, and the modes are listed in order (a_{h}, a_{v}, b_{h}, b_{v}). For example, the fourphoton term $2,0,0,2\u3009$ comprises two horizontally polarised photons in path a and two vertically polarised photons in path b. Note that we have omitted normalisation. This state has the property that each term indexed by n corresponds to an entangled state having a total of 2n photons, and maps onto the singlet state that represents two spin $\frac{n}{2}$ systems in the Schwinger representation.^{28} When τ is small, as is the case for our experiment with τ=0.061, $\mathrm{P}\mathrm{D}\mathrm{C}\u3009$ is dominated by the $n=1$ term, which enables postselection of the twophoton entangled state^{1} $\frac{1}{\sqrt{2}}\left({H\u3009}_{a}{V\u3009}_{b}{V\u3009}_{a}{H\u3009}_{b}\right)$, which is now common in quantum optics, as ref. 20. For larger τ, the photon intensity grows as ∼2sinh^{2} τ (ref. 21). The symmetry and correlation properties of $\mathrm{P}\mathrm{D}\mathrm{C}\u3009$ have been the subject of several investigations, with experimental evidence reported for entanglement between ∼100 photons,^{29} with possible applications proposed outside of metrology.^{21,30} However, the potential for increasing temporal indistinguishability when using optical cavities to increase gain τ needs to be considered in future developments of the scheme investigated here.
The fourmode squeezed state (1) that we require is a coherent superposition of multiphoton Fockstate terms; this is distinct to an undesirable statistical mixture of photon pairs. To avoid this mixture in the state generation, it is important to use a pulsed laser pump that has a pulse duration shorter than the coherence length of the photon pairs generated. The detector time bandwidth is longer than the coherence envelope of the generated photons in our experiment. However, with our interference filters (<3 nm bandwidth), we estimate a coherence time of >700 fs, whereas the jitter on the Perkin & Elmer singlephoton counting module (SPCM, Excelitas Technologies corp., Waltham, MA, USA) that we use is of the order picosecond. However, the pulses of our pump laser (which we estimate to be <160fs long, after SHG of 85 fs Ti:sapph) act as an effective gate:^{11} the pulses are separated by 12 ns, and our detectors and coincidence logic can resolve well within this window. Temporal modes can introduce large amounts of distinguishability between photons generated in SPDC when the pulse length of the pump laser is longer than the coherence length of the photons generated. This is not the case in our experiment. Evidence that we have achieved this is given by the close agreement between theory and experiment of our multiphoton interference fringes in Figure 3. As highlighted by, for example, ref.11, the quantum interference of statistically mixed photon pairs in different temporal modes differs from that of photonnumber Fock states.
(ii) The rotation we consider is
where ϕ is the parameter we wish to estimate with quantumenhanced precision. This operator maps exactly to rotations of any twolevel quantum system, including the relative phase shift in an interferometer. We implement U_{a}(ϕ) using a halfwaveplate in the sensing path a, operating on modes {a_{h}, a_{v}}, for which ϕ is four times the waveplate’s rotation angle.
(iii) Finally, photons in each of the four modes (a_{h}, a_{v}, b_{h}, b_{v}) are detected with numberresolving photodetection—the original proposal^{23} assumed fully photonnumberresolving detectors that implement projections onto all Fock states. We approximate numberresolving detection using a multiplexed method^{25,31} using readily available components. We use four 1×4 optical fibre splitters with 16 ‘bucket detector’ avalanche photodiode SPCM (APDs). Each APD has two possible outcomes: no detection event (‘0’) for a vacuum projection and a detection event (‘1’) for the detection of one or more photons, with nominal ∼60% efficiency. We use a 16channel coincidence counting system that records all possible combinations of multiphoton detection events occurring coincidentally across the 16 APDs.
Our approach uses a superposition of all photonnumber singlet states and can in principle achieve Heisenberg scaling^{23} in a similar manner to NOON states. More importantly, this state surpasses the shotnoise limit despite a realistic level of loss that would otherwise preclude any quantum advantage when using NOON states. The original proposal^{23} predicts a quantum advantage in the presence of up to 50% total system loss using a theoretically optimal (but experimentally challenging) measurement scheme, without postselection. Figure 2 illustrates new results on the subSNL performance of photon counting on the TypeII SPDC scheme in the presence of loss. Intuition for the loss tolerance in this scheme can be gained by considering the effect of losing a single photon from one of the modes: each singlet component transforms into a state that closely approximates another singlet of lower photon number.^{21} As loss decreases, detection outcomes of equal photon numbers detected in the sensing and reference arm become the dominant contribution (See Figure 4 and the associated discussion below). Experimentally, we observe subSNL phase sensitivity in the fourphoton coincidence detection subspace of our experiment, without postselecting zero loss or assuming a fixed photon number in our theoretical analysis. This supports the loss tolerance expected from detecting N>4 from any higherphoton number components of typeII SPDC.^{23}
POVM theory
A powerful method to simplify calculating measurement outcome probabilities for our experiment is to use the POVM formalism.^{32,33} All photoncounting operations correspond to POVM elements E_{r}, which are diagonal in the Fockstate basis $\left\{c\u3009\right\}$:
where r and c denote the detection pattern and the photon number, respectively. The weights w_{r}(c) are nonnegative and satisfy ∑_{r}w_{r}(c)=1. The probability of r detection events is given by ${\mathit{P}}_{r}=\text{tr}\left(\rho {\stackrel{\u02c6}{E}}_{r}\right)$, where ρ is the density matrix of any state input to the measurement setup. For the perfectly numberresolving case, the only nonzero POVM weight is when c=r and w_{r}(r)=1. However, with multiplexed detection, all weights w_{r}(c) with c⩾r can be nonzero. For example, for a twophoton state incident on one of our multiplexed detectors, there is a probability of 1/4 that both photons go to the same APD causing one detection event (w_{1}(2)=1/4) and a probability of 3/4 for two detection events (w_{2}(2)=3/4). The entire table of the POVM weights for our multiplexed system is illustrated in Figure 5. The method to compute these weightings stems from ref. 34 and is explained in the Supplementary Information.
Multiplexed detector POVMs are applied to each of the modes a_{h}, a_{v}, b_{h} and b_{v} to compute the probability for a detection outcome r=(r_{ah}, r_{av}, r_{bh}, r_{bv}), given a phase rotation ϕ:
where c=(c_{ah}, c_{av}, c_{bh}, c_{bv}) is the photon number for each mode and p_{c}(ϕ) corresponds to the probability for a measurement outcome of a perfect projection ${c}_{ah},{c}_{av},{c}_{bh},{c}_{bv}\u3009\u3008{c}_{ah},{c}_{av},{c}_{bh},{c}_{bv}$. From equation (1), rotation on modes a_{h} and a_{v} yields the probability to detect c according to
where photon number for the two paths is denoted by c_{a}=c_{ah}+c_{av} and c_{a}=c_{bh}+c_{bv}, and where the Wignerd matrix element ${d}_{m\prime ,m}^{j}\left(\varphi \right)={\left\u3008j+m\prime ,jm\prime U\left(\varphi \right)j+m,jm\u3009\right}^{2}$ describes the rotation amplitudes on two separate modes populated by number states,^{28} and is conveniently represented as a cosine Fourier series.^{23}
We incorporate the total circuit and detector efficiency (η) into the POVM elements via an adjustment of w_{r}(c). We use a standard loss model for which the mode in question is coupled via a hypothetical beamsplitter to an ancillary mode, initially in the vacuum state, which is traced out at the end. We assume that losses are polarisation independent, and therefore all loss that can arise in our setup commutes with U_{a}(ϕ). We model the singlephoton detectors in each multiplexed photoncounting array with the same efficiency η_{d}, and hence detector loss can be incorporated as a loss channel with efficiency η_{d} to the combined POVM; this loss commutes with fibre splitters and can be considered as part of the combined system efficiency. The effect of system efficiency η can be incorporated into the multiplexed POVM by the linear transformation:
The weights w_{r}(c) are altered correspondingly as illustrated in Figure 5.
Results
In the reported setup, the typical total rates for detecting two photons in coincidence and four photons in coincidence across the two paths are ∼17 k per second and ∼2 per second, respectively, making our setup suitable for observing subSNL precision for the subset of fourfold events. Note that, according to our value of τ=0.061, the rate for the generation of fourphoton singlets at the sources is less compared with that for twophoton singlets by a factor 3tanh^{2}(0.061)/2~0.006; however, the measured rates of four and twophoton coincidences are modified from this because of the effects of losses, as well as limited resolution for the multiplexed photonnumber counting for discriminating Fock states $1\u3009$ and $2\u3009$.
We plot in Figure 3 all nine possible fourphoton detection patterns r of two photons in the reference path and two photons in the sensing path as a function of ϕ, measured simultaneously by the setup in Figure 1b. For comparison, we plot these data together with theoretical curves P_{r}(ϕ), normalising to the total counts collected at each ϕ. These theoretical curves use the measured experimental parameters of τ=0.061, and lumped collection/detection efficiencies of η_{a}=0.23 and η_{b}=0.12 in the sensing and reference paths, respectively (a geometric average of 83.2% loss), assuming otherwise perfect U_{a}(ϕ) and photon interference. Parameters τ, η_{a} and η_{b} are extracted from our setup as follows. The probability of generating one pair of photons in SPDC is computed via equation (1) and given by
We sum the normalised detection rates of pairs of detected single photons that are not part of a coincidence event with ⩾1 other photon event or with no events elsewhere in the detection scheme:
Summing all four (normalised) twofold coincidences yields
This is then solved for η_{b} and η_{a}, then for p, and hence τ via a cubic equation. These relations are assumed to be constant with respect to the phase rotation ϕ and taken as the ϕaverage over experimental data.
The asymmetry in η_{a} and η_{b} arises from the different spectral width of the extraordinary and ordinary light on, respectively, paths a and b, passing through identical spectral filtering.^{35} The setup is robust to this, as the state symmetry is preserved despite η_{a}≠η_{b}, provided loss is polarisation insensitive.^{21} From the data presented in Figure 3, we extract the probability distributions p_{i}(ϕ) as leastsquare fits from each data set, and normalise such that ∑_{i}p_{i}(ϕ)=1.
Statistical information about ϕ can be extracted from the frequencies of each output detection pattern and quantified using Fisher information $\mathcal{J}\left(\varphi \right)$, (ref. 36). We compute the Fisher information of our demonstration using two methods, both plotted in Figure 6. The first (solid black line) is directly computed using the experimentally extracted p_{i}(ϕ) in the relation $\mathcal{J}\left(\varphi \right)={\sum}_{i=1}^{9}{p}_{i}{\left(\varphi \right)}^{1}{\left(\mathrm{d}{p}_{i}\left(\varphi \right)/\mathrm{d}\varphi \right)}^{2}$, with error estimated using a MonteCarlo simulation that assumes Poissondistributed noise on the fourphoton detection rates. The second method is to obtain the variance Δ^{2}ϕ_{j} of ℳ maximumlikelihood estimates {ϕ_{j}}, each using $\mathcal{N}$ photons, and to evaluate the relation ${\mathcal{J}}_{\mathrm{M}\mathrm{L}}=1/\left(N\times {\Delta}^{2}{\varphi}_{j}\right)$. Note that maximumlikelihood estimation saturates the Cramér–Rao bound and loses any bias as data are accumulated, and it is practical for characterising an unknown phase when p_{i}(ϕ) are characterised. We simulate $\mathcal{M}=10,000$ maximumlikelihood estimates for a discrete set of waveplate settings, and for each estimate we sample $\mathcal{N}=1,000$ times from p_{i}(ϕ). This number of samplings ensures unbiased and efficient estimation.^{36} Computed values of ${\mathcal{J}}_{\mathrm{M}\mathrm{L}}\left(\varphi \right)$ are then plotted (circles) in Figure 6, showing close agreement with $\mathcal{J}\left(\varphi \right)$.
We also plot in Figure 6 theoretical Fisher information computed from the POVM description of our multiplexed detection system, by taking into account the SPDC gain parameter τ and the total circuit and detector efficiency η of our setup. We find general agreement of the main features between theory and experiment ($\mathcal{J}$ and ${\mathcal{J}}_{\mathrm{M}\mathrm{L}}$), whereas the discrepancy is attributed to imperfect waveplate rotations and imperfect indistinguishability (mode overlap) of multiphoton states across the four modes.
Figure 6 also shows the shotnoise limit for two photons passing through the measured phase, computed on the basis of the average photon number in the sensing path for the fourfold detection events. For our experiments τ<0.1, which bounds the Fisher information for the target path and is computed to lie in the range 2.01±0.01. The shaded region displays the quantum advantage over the shotnoise limit for the postselected subset of fourfold detection events—the maximum advantage, over detecting the same intensity as for our experiment in arm a but for classical interferometry, achieved in our experiment is 28.2±2.4% at ϕ=3.91±0.06 rad. The theoretical maximum advantage that can be achieved by the scheme with our τ and η parameters is 45%. This is possible because of the low rate of components with six photons or more compared with four photons for the preloss SPDC state (0.5%), and a quantum advantage is predicted theoretically for τ⩽0.250 with our experimental values for loss.
An important feature of the theory and experiment curves in Figure 6 is the troughs in $\mathcal{J}$ (similar features were presented elsewhere, e.g., the Supplementary Information for ref. 25), occurring about points where some or all of the fringes in Figure 3 have minima or maxima. In contrast, when all experimental imperfections are absent, $\mathcal{J}$ is predicted to be independent of phase rotation—a common feature of metrology schemes using photonnumbercounting measurement.^{37} The definition of $\mathcal{J}\left(\varphi \right)$ reveals points of instability when the numerator ${\mathrm{d}p}_{i}\left(\varphi \right)/\mathrm{d}\varphi $ vanishes but p_{i}(ϕ) does not—this will arise even with very small experiment imperfections that lead to interference fringes with visibility <1. A solution is to incorporate a reference phase in conjunction with a feedback protocol to optimise the estimation of an unknown phase.^{25} The symmetry of the generalised singlet state at the heart of this scheme enables a control phase to be placed on the reference path as opposed to the sensing path in the traditional manner. We demonstrate the feasibility of the former by repeating our experiment with a controllable reference phase rotation (θ in Figure 1b) placed in the reference path b that shifts the regions of maximal sensitivity with respect to the phase in the sensing path—see Supplementary Information for fourphoton interference fringes and corresponding Fisher information. This may find practical application where the reference phase has to be separated from the sensing path. Furthermore, the reference path could be used for heralding to maximise the Fisher information per photon passing through the unknown sample using fast switching^{38} of the sensing path conditioned on detection events at the reference path. Using heralding and perfect photonnumberresolving detection, the entire downconversion state can achieve quantum advantage with the τ value from our experiment (see Supplementary Information).
Finally, we return to a detailed theoretical analysis of the contribution to phase sensitivity of subsets of outcomes (r_{a}, r_{b}). The symmetry property of $\mathrm{P}\mathrm{D}\mathrm{C}\u3009,$ namely ${U}_{a}\otimes {U}_{b}\mathrm{P}\mathrm{D}\mathrm{C}\u3009=\mathrm{P}\mathrm{D}\mathrm{C}\u3009$ for arbitrary unitary rotation U of polarisation modes, is preserved under polarisationinsensitive photon loss. This maps onto the requirement of polarisationinsensitive loss in our setup—we denote the loss rates in path a (or b) by $1{\eta}_{a\left(b\right)}$. The symmetry property dictates the general form of the typeII SPDC state post loss, as well as key features of the detection probabilities (after polarisation rotation of a_{h}, a_{v}) when outcomes are grouped according to total detections in each arm (r_{a}, r_{b}) (refs 21,23). Consequently, we consider renormalised probabilities ${P}_{r}^{\prime}\left(\varphi \right)={P}_{r}\left(\varphi \right)/{P}_{\left({r}_{a},{r}_{b}\right)}$, where $r=\left({r}_{ah},{r}_{av},{r}_{bh},{r}_{bv}\right)$ and ${P}_{\left({r}_{a},{r}_{b}\right)}$ denote the total probability for detections in the subset (r_{a},r_{b}). ${P}_{\left({r}_{a},{r}_{b}\right)}$ is independent of the rotation angle when fully numberresolving detectors are used. Only when the total number of photons detected in each of the sensing and reference paths are equal (r_{a}=r_{b}) does ${P}_{r}^{\prime}\left(\varphi \right)$ have a singlet contribution (the n_{a}=r_{a}=r_{b} singlet term in equation (1)), whereas all other contributions arise because of loss from singlets with greater photon number (and reduce the achievable precision). The effects of loss on ${P}_{r}^{\prime}\left(\varphi \right)$ are parameterised by $\Xi =\sqrt{\left(1{\eta}_{a}\right)\left(1{\eta}_{b}\right)}\mathrm{tanh}\tau $: Ξ increases with higher losses and higher values for τ (which increase the rate of singlet production at highphoton numbers), and it is reduced if the losses are predominantly in one arm. To achieve subshot noise precision without postselection, values of τ and η_{a,b} must be chosen to achieve simultaneously a low value of Ξ, high weighting for ${P}_{\left({r}_{a},{r}_{b}\right)}$ with r_{a}=r_{b}>1 and high average photon number. The components with r_{a}=r_{b} provide supersensitive precision when Ξ is small: this enabling principle of our scheme can be demonstrated using postselection when high values for τ and η_{a (b)} are not achieved, as in our current setup. We give illustrative examples in Figure 4.
Discussion
We have demonstrated the key enabling principles of a promising technique for realising practical quantumenhanced sensors that is robust to loss and designed to use a photon source based on current technology. Our experimental results demonstrate the validity of our theoretical predictions concerning phase supersensitivity for those detection events that correspond to multiphoton singlet states. Our theory in turn predicts that supersensitive precision is achievable without postselection in an experimental regime that is achievable with nearterm and stateoftheart componentry, whereas detection events that have contributions from singlet components behave as in our experiment. Our approach is amenable to nearterm implementation in an integrated architecture with onchip interference of downconversion,^{15} onchip^{16} detectors and integration with microfluidic channels.^{5} This would enable inherently a stable path encoding to measure very small optical path lengths—other bulk optical techniques could be used to achieve this to convert polarisation to path, such as in ref. 39. We note that our scheme does not require optical delays or GHz switching. This is in contrast to other quantum technology schemes that rely on generating a fixed number of photons and therefore require greater architecture complexity—e.g., reliance on heralded photon sources that comprise fast switching, lowloss optical delay, highspeed detection and microelectronic integration. Our demonstration now shifts the emphasis for practical quantum metrology onto developing and using lowloss circuitry and highefficiency photon detection; 95% efficiency transition edge sensor^{40} and 93% efficient superconducting nanowire detectors operating in the infrared^{41} have recently been reported. For a given efficiency η, the gain parameter τ in the downconversion process also dictates the level of precision the scheme can achieve. As circuit loss is reduced, it would be beneficial to increase τ to the values (τ>1) studied in ref. 23; enhancing SPDC with a cavity^{42} may be a promising approach to achieve this, enabling future experimental work to examine larger photon detection subsets.
Materials and methods
Horizontally polarised 404nm laser pulses (estimated to be <160fs long) are generated by upconversion (SHG) of a TiSapphire laser system (85 fs pulse length, 80MHz repetition rate). This is focused to a waist of 50 μm within a 2mmthick nonlinear β barium borate crystal, phase matched for typeII SPDC, to ideally generate the state $PDC\u3009$ at the intersection of the ordinary (o) and extraordinary (e) cones of photons^{20} in paths a′ and b′ of Figure 1b. Spatial and temporal walkoff between e and o light is compensated^{20} with one halfwaveplate (optic axis at 45° to the vertical) and one 1 mmthick β barium borate crystal in each of the two paths a and b. The spectral width of ordinary and extraordinary light generated in typeII SPDC differs, leading to spectral correlation of the two polarisations. Setting one waveplate to 90° and aligning the two paths onto a PBS separates the e and o light, sending all e light onto output a and all o light onto output b, ref. 43. This removes spectralpath correlation in the PDC state, leaving only polarisation entanglement across paths a and b, and thus erasing polarisationdependent loss in the sensing path and the reference path of the setup.
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Acknowledgements
We are grateful for financial support from EPSRC, ERC, NSQI, NRF (SG), MOE (SG) and ARC CQC2T. During the writing of this article, J.C.F.M. was supported by a Leverhulme Trust Early Career Fellowship. X.Z. acknowledges support from University of Bristol, the National Young 1000 Talents Plan and Natural Science Foundation of Guangdong (2016A030312012). G.J.P. acknowledges support from the Benjamin Meaker Visiting Fellowship and from the ARC Future Fellowship. J.L.O'B. acknowledges a Royal Society Wolfson Merit Award and a Royal Academy of Engineering Chair in Emerging Technologies.
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Matthews, J., Zhou, X., Cable, H. et al. Towards practical quantum metrology with photon counting. npj Quantum Inf 2, 16023 (2016). https://doi.org/10.1038/npjqi.2016.23
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