Demonstrating an absolute quantum advantage in direct absorption measurement

ABSTRACT Engineering apparatus that harness quantum theory promises to offer practical advantages over current technology. A fundamentally more powerful prospect is that such quantum

technologies could out-perform any future iteration of their classical counterparts, no matter how well the attributes of those classical strategies can be improved. Here, for optical direct

absorption measurement, we experimentally demonstrate such an instance of an absolute advantage per photon probe that is exposed to the absorbative sample. We use correlated intensity

measurements of spontaneous parametric downconversion using a commercially available air-cooled CCD, a new estimator for data analysis and a high heralding efficiency photon-pair source. We

show this enables improvement in the precision of measurement, per photon probe, beyond what is achievable with an ideal coherent state (a perfect laser) detected with 100% efficient and

noiseless detection. We see this absolute improvement for up to 50% absorption, with a maximum observed factor of improvement of 1.46. This equates to around 32% reduction in the total

number of photons traversing an optical sample, compared to any future direct optical absorption measurement using classical light. SIMILAR CONTENT BEING VIEWED BY OTHERS NEAR-ULTRAVIOLET

PHOTON-COUNTING DUAL-COMB SPECTROSCOPY Article Open access 06 March 2024 ENHANCING SENSITIVITY IN ABSORPTION SPECTROSCOPY USING A SCATTERING CAVITY Article Open access 21 July 2021

THEORETICAL STUDIES ON QUANTUM IMAGING WITH TIME-INTEGRATED SINGLE-PHOTON DETECTION UNDER REALISTIC EXPERIMENTAL CONDITIONS Article Open access 29 March 2022 INTRODUCTION For successful

demonstration of any quantum technology proposed to exceed the performance of current techniques, it is important to identify under what conditions an advantage can arise and under what

situations if any classical schemes may actually regain advantage. How to maximise information gained from each photon used to probe an unknown sample is a central question for the

underpinning physics of metrology. By generating various quantum states of light, it is being demonstrated that one can improve upon the sensitivity and precision of current classical

measurement techniques1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16. Sub-shot noise detection of changes in optical phase have been demonstrated in interferometers using squeezed light and designed

for gravitational wave detection13, but the first successful detection of gravitational waves took place without the need of squeezed light larger interferometers with greater

sensitivity17. Therefore, it is still valuable for ad-hoc experiments to be performed to observe absolute advantages, specifically to show that the precision obtained through reported data

cannot be classically achieved — but we must do so with clear understanding of the conditions of the advantage. An approach that allows sub-shot-noise-limit (SNL) measurement of an unknown

sample’s transmission, the parameter measured in this work, is to use correlated beams of photons1, 12, 18,19,20. In particular, such techniques have been used in the context of imaging21,

22. However demonstrating unambiguous real-world enhancements through an ad hoc scheme using correlated photons has remained a challenge23. For such low-photon intensity measurements, a low

cost approach to achieve desired performance of a classical analogue measurement is often to simply increase the intensity of probe light to give rise to a shorter measurement time and a

greater signal to noise ratio. But it is still interesting, and may yet prove practical, to consider the information obtainable per photon exposure for whenever we face limits on the

brightness of optical probes. For example, where samples are altered or damaged by light, it is highly desirable to maximise information per unit of exposure16. Intensity measurement of an

idealised laser fluctuates with a Poisson distribution12, 24, 25. This is typically used to define the SNL in optical measurements—which defines the limit of precision in optically measuring

experimental parameters—and it can only be reached in classical experiments once all other sources of noise are removed. Accounting for information per-photon exposure (PPE) is particularly

important for benchmarking schemes considered for measuring photo-sensitive samples16. For measuring optical transmission, the proportion of photons that pass through a sample is used to

directly estimate absorption _α_, and so the minimum error in estimation, Δ2 _α_, is defined by the SNL. If we then fail to detect a subset of these photons—due to poor detection efficiency

or component loss—exposure must be increased to compensate, which in turn reduces the information gained PPE. The reduction of precision PPE can also manifest to varying degrees in different

strategies—for example, noise introduced by reduced detector efficiency can have a greater impact on schemes using two detectors than on schemes designed to just use one detector (Fig. 1).

It is also important to note that for measurement schemes where the precision itself varies with parameters of the measured sample it is possible for precision to be reduced, potentially

requiring either prior information about the optical sample or the addition of iterative feedback and control to ensure sub-shot-noise perfomance26,27,28. For example, the use of amplitude

squeezed light to measure reduced intensity fluctuations with homodyne detection29, 30 means precision is also dependent on optical phase introduced by the object being measured, leading to

a random amount of precision that can be above the SNL. The reduced noise of quantum-correlated twin-beams generated with spontaneous parametric downconversion (SPDC) of a laser can be

measured by recording correlated intensity using photodiodes1, 20, 31. For practical application, this is attractive because correlations between photon pairs are unaffected by optical phase

induced by a measured object. This regime of detection can be preferable to photon counting because it reduces the amount of recorded data that is associated with the measurement of

individual photons in the macroscopic states that are needed for most imaging and sensing scenarios. This technique can be transferred to detecting correlated photons altogether in the same

image of a CCD camera, to image sub-SNL noise fluctuation of spatial correlations from SPDC11. With the inclusion of a spatially absorbing sample, it has been shown that SPDC can be used to

suppress noise in imaging objects to a degree that out-performs classical laser-measurement using an equally efficient detection12. However, the performance reported in ref. 12 experimental

can now be exceeded by direct optical absorption measurement with classical light by 8% PPE (see Supplementary material) using currently available CCD camera efficiency of 95%32. To extend

such results to the claim that quantum correlated light can out perform any classical version of this measurement, it would need to be assumed that the undetected photons, were they

detected, would give more information about the sample than the information gained from detected photons. Removing such an assumption is analogous to removing a fair sampling assumption in

tests of Bell non-locality33, 34. Here, we report precision PPE in measuring absorption that is up to 45% beyond what is achievable with ideal direct optical absorption measurement using

classical light. We do this using a 90% efficient commercial air-cooled CCD, a new unbiased estimator for data analysis that maximises precision gained in our experiment and a high heralding

efficiency photon-pair source enabled by recent developments of efficient sources of entangled pairs of photons35. This enables observation of sub-SNL physics with quantum correlated light

without assuming fair sampling. It is also a direct comparison of current quantum experiments to the SNL of any future classical implementation directly illuminated by idealised Poisson

distributed light and recorded with 100% efficient noiseless detection. The absolute sub-shot-noise performance that we observe with a relatively simple experimental setup makes future use

of quantum correlated light for enhanced measurement practical for applications. EXPERIMENTAL SETUP Our experimental setup is presented in Fig. 2: a periodically poled potassium titanyl

phosphate (PPKTP) crystal designed for type-II phase matching is pumped with a continuous wave laser beam at 404 nm to generate photon pairs—the signal photons emitted in arm 1 have

wavelength 798 nm and the idler photons emitted in arm 2 have wavelength 818 nm, which is stabilised with the crystal temperature set to 24 °C. A dichroic mirror then reflects the pump and

transmits the down-converted photon pairs. Type-II phase matching ensures the signal and idler photons in each emitted photon pair are vertically and horizontally polarised respectively, and

they are separated with a polarisation beam-splitter (PBS) before being injected into two single mode optical fibres. The whole photon pair generation setup is compact (<20 cm × 60 cm)

and it is designed for high-efficiency collection into optical fibre: ~69% of the photons leaving the fibres are paired. The output of the two fibres are then imaged on a CCD camera, an

Andor iDus 416 cooled to −35 °C, originally developed for spectroscopy. The camera exhibits a device detection efficiency of _η_ _d_  = 90 ± 3% under these conditions, this value has been

experimentally characterised using calibrated detectors and is a lower bound according to manufacturer data. The varied absorption of a ‘sample’ is implemented by a PBS preceded by a

rotatable half-wave plate (HWP) and is intercalated between one of the fibres and the camera on the path of the beam (arm 1). THEORY When using one optical beam, absorption can be found

using the standard estimator $${\alpha }_{s}=1-{\eta }_{s}=1-\frac{{N}_{1}}{\langle {N}_{1p}\rangle },$$ (1) where _η_ _s_ is the transmission efficiency of the sample, _N_ 1 is the number

of photons detected on arm 1 with the sample and _N_ 1_p_ is the mean number of photons detected on arm 1 without the sample. After each experimental trial, we use an estimator that gives an

estimate of _α_ _s_ that is ideally close to 〈_α_ _s_ 〉 for each trial—the closer _α_ _s_ is to its actual mean value 〈_α_ _s_ 〉, the smaller the error on estimating absorption. As eq. (1)

shows, this estimation is limited by the evaluation of _N_ 1 which fluctuates according to the quantum nature of light and manifests as noise when estimating 〈_N_ 1〉. Photon pairs emitted

from two-mode SPDC fluctuate in number, but the fluctuation in total number of photons in each of the two output modes, _N_ 1 and _N_ 2, are correlated1, 20. Noise on measuring _N_ 1 can

therefore be suppressed using the knowledge gained by measuring _N_ 2. For example, if for a single trial _N_ 2 is measured to be more than its expectation value 〈_N_ 2〉, we know that the

intensity _N_ 1 will have exhibited the same fluctuation simultaneously and therefore that this value is also greater than its expectation value 〈_N_ 1〉. These sub-Poissonnian fluctuations

can be used to improve the estimation of the number of photons _N_ 1 after the sample therefore decrease statistical error in measuring optical absorption20. Due to loss in an optical

measurement however, uncorrelated single photons are also present in the experiment. Therefore, to maximise information available in our experiment, we have developed a new and unbiased

estimator that uses all detected photons that pass through a sample to estimate its absorption with quantum-enhanced noise-suppression PPE. This estimator is given by $${\alpha

}_{s}=1-\frac{{N}_{1}^{^{\prime} }+\delta E}{\langle {N}_{1p}\rangle }=1-\frac{{N}_{1}-k\delta {N}_{2}+\delta E}{\langle {N}_{1p}\rangle },$$ (2) where \({N}_{1}^{^{\prime} }\) is a

corrected estimate for _N_ 1 that is a function of: _δN_ 2, the deviation of _N_ 2 from its mean value 〈_N_ 2〉; _δE_, a small correction used to remove bias in the estimator, calibrated by

taking images without a sample; and _k_ = _CN_ 1, a correction factor with constant _C_ that encapsulates the sources of noise in our photon source obtained during the calibration phase. See

supplementary material, section I for more details on the derivation of _δE_ and _C_ and their measurement. In contrast to previous demonstrations, the new estimator eq. (2) is designed to

use all detected light that has passed through the sample—both the correlated light that provides access as Fock state behaviour36 and the classical contribution due to photons arriving on

the sample with no detected twin due to system loss. Note that this strategy does not need to measure the arrival of individual photons, it only needs to record the fluctuation of large

photon number. Therefore the main requirement for detection is high quantum efficiency. In contrast, the performance of a perfect direct classical scheme is given37 by $${{\rm{\Delta

}}}^{2}{\alpha }_{cl}=\frac{\mathrm{(1}-\alpha ){\eta }_{d}}{\langle {N}_{1p}\rangle },$$ (3) where we have increased the average detected intensity in the perfect classical model by a

factor of 1/_η_ _d_ in order to make a fair comparison with our experiment that uses detection efficiency _η_ _d_ . To quantify a quantum advantage, we compute a performance parameter Γ that

is a ratio of the minimum noise in a classical measurement Δ2 _α_ _cl_ to the measured variance of absorption estimates $${\rm{\Gamma }}=\frac{{{\rm{\Delta }}}^{2}{\alpha

}_{cl}}{{{\rm{\Delta }}}^{2}{\alpha }_{\exp }}\mathrm{.}$$ (4) Γ > 1 signifies a quantum advantage in precision PPE. RESULTS To characterise our setup, we recorded vertically binned

images using the setup presented on Fig. 2 without the PBS and HWP implementing the sample. This enables characterisation of 〈_N_ 2〉, 〈_N_ 1_p_ 〉, _k_, _δE_. We find the normalised variance

of the difference between the two beam intensities to be \(\sigma =\frac{{{\rm{\Delta }}}^{2}({N}_{1}-{N}_{2})}{{N}_{1}+{N}_{2}}=0.38\pm 0.02\), which is a witness of strong sub-shot noise

statistics since _σ_ = 0 corresponds to fully correlated statistics and _σ_ = 1 corresponds to two perfectly independent beams fluctuating at the SNL. For comparison to single photon

counting experiments, this corresponds to a Klyshko heralding efficiency of 62%38. Although the evaluation of the absorption could be done with just one acquisition of ~0.5 s, in practice we

measured 1000 acquisitions (10 series of 100 acquisitions) to estimate Δ2 _α_ exp which corresponds to a total acquisition time of 15 minutes. We repeated this process for each measured

value of absorption. We corrected the value of 〈_N_ 1_p_ 〉 for each set of 100 acquisitions using the mean value 〈_N_ 2〉 obtained during those same acquisitions to avoid bias in the

estimation of the absorbance due to the classical fluctuations of the source. For each image acquisition we make an estimation of _α_ _s_ using the estimator in eq. (2) as presented in Fig. 

3, we use the 100 acquisitions of the series to evaluate Δ2 _α_ exp and reproduce this for 10 series in order to evaluate the experimental uncertainty on the mean value of Δ2 _α_ exp. We can

see in Fig. 3(c), the mean absorption of the probability profiles of the blue and green curves overlap, indicating that our estimator eq. (2) is unbiased. As shown in Fig. 3 with the

example of measuring _α_ _s_  = 0.04, our estimator eq. (2) (shown in panel (a) gives a reduced dispersion of the estimation compared to the best direct classical scheme for the same number

of photons probing the sample, marked by the red theoretical curve in panel (c). We measure Δ2 _α_ exp over the range 0 < _α_ _s_  < 1 and compare in Fig. 4(a) to the ideal performance

of perfect classical direct detection using the precision ratio Γ (eq. (4)). For 0 < _α_ _s_  < 0.5 we observe Γ > 1, signifying an absolute quantum advantage. The highest

advantage that we see is a reduction in estimate variance by 46 ± 6% for _α_ _s_  = 5.99 · 10−3, which is distinguished by more than 7 standard deviations from the classical limit. This also

equates to an advantage of 63 ± 7% over a direct classical scheme measured with 90% detection efficiency, that corresponds to the efficiency of the CCD camera used in our experiment. Since

for both the reported strategy and the perfect classical strategy, the variance scales linearly with _N_ _p_ , it follows for an equal precision that the ratio between the total number of

probe photons used in the reported quantum measurement and the perfect classical strategy is given by 1/Γ. Figure 4(b) shows this reduced number of probe photons using our experimental data

compared to a perfect classical scheme achieving the same measurement precision. The highest effective reduction of probe photons we observe is 32 ± 3% (1.7 dB), which is also a reduction by

39 ± 3% (2.1 dB) compared to a classical method using 90% detection efficiency. DISCUSSION To conclude, we have demonstrated an absolute quantum advantage over the best classical optical

strategy for direct absorption measurement over the range 0 < _α_ < 0.5. This demonstrates that quantum-enhanced absorption measurement need not be confined to measuring weak

absorption. From a practical perspective, an absolute quantum advantage per photon flux is prevalent without the need of high temporal resolution of coincidence detection, data storage or

equivalently low photon flux requirements associated to individual photon counting. Therefore, future quantum advantage experiments can be performed with high photon flux such as from

high-brightness correlated photon sources2. In fact recent experiments using relatively bright (pulsed) SPDC light31 suggest a noise reduction factor up to 83.4% (_σ_ = 0.166) could be

achieved if detector noise and classical fluctuations can be suppressed. Similarly, by further optimising our collection efficiency, we expect to be able to report much higher precision and

greater noise reductions in future. Furthermore the technique reported here is of immediate use to obtain absolute quantum advantages in imaging12 and absorption spectroscopy37, in

particular when used with narrow linewidth photon pair generation2, 14. The relative simplicity of the scheme can also motivate analogous setups applied to high-energy optics—for example,

the demonstrations of X-ray parametric down-conversion39, 40 and the development of high efficiency direct hard X-ray detector arrays41 could be used to minimise radiation exposure in

medical imaging. _Note added_. During the review process of this manuscript, a correlated photon pair experiment applied to the context of absorption imaging was reported22 that reports

improvement over the previous realisation12 experimental, both in quantum noise reduction and in resolution. It is not claimed in that reference that an absolute quantum advantage is

obtained (there the quantum experiments are compared to classical experiments under the same conditions, including detector and collection efficiency), and we note that from the performance

metrics reported it is not possible to conclude if an absolute quantum advantage is obtained. The methods we have used in our manuscript could be used to address this, while the improved

estimator we have used could further improve such results. METHOD The photon pair source has been designed to optimise the collection of the spontaneous-down converted photons after their

emission in the crystal. First, focusing conditions were chosen to maximise mode matching between the down-conversion modes and the collection modes using both numerical calculation to

determine the initial design and improved experimentally by beam profiling. The mode matching has been estimated to be 80%. Another critical point is efficient removal of the pump beam by

using a combination of a dichroic mirror and a colour glass filter. The collection of the twin photons in mono-mode fibres also acts as spatial filtering for parasitic light from the UV

laser pump, that unlike the SPDC are not mode matched with the fibre modes. The transmission of the SPDC photons on each filter is 98%. To avoid fluorescence of the collection lenses due to

the UV laser pump, we used Fused Silica optics. All the optics, including the PPKTP crystal, have anti-reflection coating with less than 0.2% loss per surface. The fibres were coated only on

one end and so add an extra 4% loss. We can therefore predict a heralding efficiency, not taking into account the detector efficiency, of ~72% and a heralding efficiency of 65% when

including the 90% detector efficiency. This is compatible with the experimental results giving a 62% heralding efficiency with the camera. The final source exhibit ~120 k photon pairs

detected per second with the camera, with single-photon counts of ~200 k per second — the pump intensity was attenuated and optimised to have the highest signal possible while not saturating

the camera. We characterised the photon source using Excelitas Avalanche Photo-Diodes (APD) Model SPCM-800-14 that have a detection efficiency as certified by the manufacturer of >62%.

We measured an overall experimental heralding efficiency measured with these detectors of 42.5%. By comparing the heralding efficiency of the photon source measured with the camera (62%) we

can deduce a lower bound on the iDus 416 camera detection efficiency of 90 ± 3%. This is in agreement with the manufacturer data that quotes efficiency between 90% and 94% when imaging 800 

_nm_ and cooled to −35 °C. The two mono-mode beams imaged on the camera are acquired as fully vertically binned images. The data in grey scale level are then converted into number of photon

through the relation _N_ = _S_(_E_ _s_  − _E_ Off) where N is the number of photons, _E_ _s_ is the intensity signal in grey scale level obtained, _E_ Off is an electronic offset obtained

when no light is illuminating the sensor and _S_ = 0.71 photoelectrons per grey level is the sensitivity of the camera. The photon number values _N_ 1 and _N_ 2 are then obtained by

selecting some region of interest (ROI) delimited over 5 standard deviations of the beam profile, and by summing the intensity of the different pixels inside these ROIs. We used a readout

rate of 0.03 MHz in order to minimise the camera electronic noises, an exposure time of 0.5 s which lead to a frequency of read ~1 Hz. DATA AVAILABILITY The datasets generated during the

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grateful to E. Allen, H.V. Cable, C. Erven, A. Laing, E. Lantz, J. Mueller, J.L. O’Brien and M. Thompson for helpful discussions. This work was supported by EPSRC through the QUANTIC hub,

ERC, and the Centre for Nanoscience and Quantum Information (NSQI). J.G.R. acknowledges support from an EPSRC established career fellowship. J.C.F.M. acknowledges support from an EPSRC early

career fellowship. AUTHOR INFORMATION Author notes * Paul-Antoine Moreau Present address: School of Physics and Astronomy, University of Glasgow, Glasgow, G12 8QQ, UK * Siddarth K. Joshi

Present address: Institute for Quantum Optics and Quantum Information (IQOQI) Austrian Academy of Sciences, Boltzmanngasse 3, A-1090, Vienna, Austria AUTHORS AND AFFILIATIONS * Quantum

Engineering Technology Labs, H. H. Wills Physics Laboratory and Department of Electrical and Electronic Engineering, University of Bristol, Merchant Venturers Building, Woodland Road,

Bristol, BS8 1FD, UK Paul-Antoine Moreau, Javier Sabines-Chesterking, Rebecca Whittaker, Siddarth K. Joshi, Patrick M. Birchall, Alex McMillan, John G. Rarity & Jonathan C. F. Matthews

Authors * Paul-Antoine Moreau View author publications You can also search for this author inPubMed Google Scholar * Javier Sabines-Chesterking View author publications You can also search

for this author inPubMed Google Scholar * Rebecca Whittaker View author publications You can also search for this author inPubMed Google Scholar * Siddarth K. Joshi View author publications

You can also search for this author inPubMed Google Scholar * Patrick M. Birchall View author publications You can also search for this author inPubMed Google Scholar * Alex McMillan View

author publications You can also search for this author inPubMed Google Scholar * John G. Rarity View author publications You can also search for this author inPubMed Google Scholar *

Jonathan C. F. Matthews View author publications You can also search for this author inPubMed Google Scholar CONTRIBUTIONS J.C.F.M. and J.G.R. initiated and supervised this project.

J.C.F.M., J.G.R. and P.-A.M. conceived and designed the experiment. J.C.F.M., J.G.R. and S.K.J. conceived and designed the photon source. S.K.J., R.W., J.S.-C. and A.M. built and optimised

the source. P.-A.M. and J.S.-C. performed the experiment. P.-A.M., J.C.F.M., J.S.-C., P.M.B. and J.G.R. interpreted the results. All authors contributed to the manuscript. CORRESPONDING

AUTHORS Correspondence to Paul-Antoine Moreau or Jonathan C. F. Matthews. ETHICS DECLARATIONS COMPETING INTERESTS The authors declare that they have no competing interests. ADDITIONAL

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CITE THIS ARTICLE Moreau, PA., Sabines-Chesterking, J., Whittaker, R. _et al._ Demonstrating an absolute quantum advantage in direct absorption measurement. _Sci Rep_ 7, 6256 (2017).

https://doi.org/10.1038/s41598-017-06545-w Download citation * Received: 11 April 2017 * Accepted: 21 June 2017 * Published: 24 July 2017 * DOI: https://doi.org/10.1038/s41598-017-06545-w

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