This is particularly interesting for improving the ease of use, as no realignment is necessary when changing the SUT; compared to Equation 1 , the third-order term in Equation 3 is cancelled owing to energy—time correlations This means that the quantum strategy allows data fitting using exactly one free parameter, namely, , which is an essential step towards absolute optical-property determination with high precision without systematic errors.

Finally, due to the use of a two-photon N 00 N state, double resolution of is achieved, enabling measurements on shorter samples and components compared to standard WLI, that is, down to the technologically interesting mm to cm scale.

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We used the same interferometer for all measurements and actively stabilised it using a reference laser and a piezoelectric transducer on one mirror in the reference arm additional details are provided in the methods section. For chromatic dispersion measurements using classical WLI, we used a state-of-the-art superluminescent diode. For the Q-WLI approach, the light source was a The single-photon spectrometer at the other output was made of a wavelength-tunable 0. Typical interference patterns for chromatic dispersion measurements using both methods are shown in Figure 2a and 2b.

With the Q-WLI setup, we found twice as many interference fringes for the same spectral bandwidth, which is a direct consequence of the doubled phase sensitivity of the two-photon N 00 N state. The results of the statistical data analysis are shown in Figure 3. This result is among the most precise reported to date in the literature 13 , 17 , 18 , 19 , 20 , 21 , Typical measurements acquired for inferring chromatic dispersion in a 1-m-long standard single-mode fibre. Red dots are data points; blue curves are appropriate fits to the data following Equations 1 and 3 , from which D is extracted.

Error bars assume Poissonian photon number statistics. For standard WLI, normalization was obtained by measuring two reference spectra. For Q-WLI, normalization was performed on the fly by counting non-zero arrival time difference coincidences. For more details, refer to the Supplementary Information.

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Histogram of inferred chromatic dispersion coefficients after repetitions with the same SUT for both standard blue and quantum-enhanced red measurements. Fits to the data assumed a normal distribution. In our two sets of data, we observed a difference of between the central values, which is larger than the deviation expected from statistical uncertainties.

Polarization mode dispersion can be excluded as it would introduce at most an offset of. Consequently, the difference in central values must originate from systematic errors. Because Q-WLI presents fewer sources of systematic errors, it is therefore natural to conclude that Q-WLI determines chromatic dispersion with absolute accuracy. It is therefore interesting to compare the achievable precision normalised to the number of transmitted photons.

Consequently, the standard and quantum methods achieve precisions of and , respectively. In other words, in addition to being more prone to systematic errors, the standard measurement requires times more photons to achieve the same precision as Q-WLI. Another advantage provided by Q-WLI lies in its straightforward device calibration. All of the optical components in the interferometer actually show small residual chromatic dispersion, and this undesired offset needs to be evaluated and subtracted from the data to avoid systematic errors. In both cases, this implies performing a measurement without any SUT.

This procedure is technically challenging, time-consuming, and might lead to additional systematic errors. At this point, Q-WLI demonstrates its ability for user-friendly operation. Even after removing the SUT, interference is observed without any interferometer realignment. Figure 4 shows the experimental results that we have obtained when measuring chromatic dispersion in our bare interferometer, that is, without the SUT.

For all of the data discussed above, except for the raw data in Figure 2a and 2b , we have subtracted the residual chromatic dispersion. Red dots, data points; the blue curve is an appropriate fit to the data following Equation 3. We have introduced and demonstrated the concept of spectrally resolved Q-WLI, exploiting energy—time entangled two-photon N 00 N states. Compared to standard measurements, the N 00 N state permits achieving a phase sensitivity higher by a factor of two.

More strikingly, this use of such quantum states of light reduces the number of free parameters for fitting experimental data from three to one, representing a major advantage for determining optical properties with high precision and absolute accuracy. In addition, our setup does not require a balanced interferometer for performing the measurement, which represents a significant time-saving advantage compared to standard WLI. This is of particular interest for device calibration and when measuring a large set of samples.

As an exemplary demonstration, we have applied our method to infer chromatic dispersion in a standard single-mode fibre, obtaining 2. We note that the sensitivity of our approach could be further doubled by using a double-pass configuration 18 ; this could achieve measurements on short samples, such as optical components and waveguide structures mm to cm length scale. Such measurements would also be of interest for medical applications where precise knowledge of chromatic dispersion in tissues is required to yield optimal image quality in optical coherence tomography From this perspective, the reduced number of photons required for quantum WLI is also highly interesting for measurements performed on photosensitive biological samples 32 , 33 , In optical telecommunication systems, by rotating the polarizations of the entangled photon pairs, our setup could be used for measuring polarization mode dispersion in optical components, which would lead to refinement of manufacturing processes.

Alternatively, quantum-inspired strategies may also prove to be suitable 36 , In summary, we believe that combining the fundamental and conceptual advantages enabled by quantum light is a very promising approach for the future development and improvement of applications requiring absolute and high-precision measurements of optical properties. Quantum-enhanced measurements: beating the standard quantum limit. Science ; : — De Broglie wavelength of a non-local four-photon state. Nature ; : — Super-resolving phase measurements with a multiphoton entangled state.

Beating the standard quantum limit with four-entangled photons. Entanglement-free Heisenberg-limited phase estimation.

Manipulation of multiphoton entanglement in waveguide quantum circuits. Nat Photonics ; 3 : — High-NOON states by mixing quantum and classical light. Advances in quantum metrology.

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Nat Photonics ; 5 : — Practical quantum realization of the ampere from the elementary charge. Phys Rev X ; 6 : Absolute detector quantum-efficiency measurements using correlated photons.

Metrologia ; 32 : — Group-delay measurement using the Fourier transform of an interferometric cross correlation generated by white light. Opt Lett ; 15 : — View Our Timetable.

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## Phase estimation in optical interferometry

Atomic, Molecular, and Optical Physics , vol. Atomic, Molecular, and Optical Physics , Vol. AU - Wiseman,H. AU - Breslin,J. PY - Y1 - N2 - Optimal N-photon two-mode input states for interferometric phase measurements were derived. Access to Document