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Quantum technologies have received substantial attention as a means to improve the resolution and precision of metrological tasks by reducing statistical errors due to quantum noise1-8. Far less attention has been given to their ability to reduce systematic errors. However, statistical and systematic errors are of equal importance in any measurement, and the latter are typically more difficult to characterise. Notable examples of quantum-improved measurements are the combination of multiple fundamental electronic quantum effects for a more accurate definition of the ampere9 and quantum-correlated 'twin photon beams' in establishing absolute and universal optical power standards10. In this letter we demonstrate a new use of quantum optics to reduce systematic errors in the technologically prominent application of spectrally resolved white-light interferometry (WLI). WLI is used for precise measurements of chromatic dispersion, that is, the second derivative of the wavelength-dependent optical phase. Classical WLI, however, requires precise interferometer equalization11, 12 and is influenced by third-order dispersion13, 14. This leads to systematic errors that are difficult to account for.
We eliminate these drawbacks by inferring chromatic dispersion using energy–time entangled photon pairs and coincidence counting to measure spectral correlation functions. In addition, we exploit photon–number correlations to achieve a twofold resolution enhancement. Our results demonstrate that this new strategy outperforms the precision and accuracy of previous quantum15, 16 and state-of-the-art techniques11, 12. Moreover, because our approach is essentially alignment-free, it enables the use of the same interferometer in a user-friendly manner for analysing a wide variety of different optical materials in terms of type, optical properties, length, etc.
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The standard scheme for WLI is shown in Figure 1a. The emission of a white-light source is directed to an interferometer in which the reference arm is free-space (with well-known optical properties) and the other arm comprises the sample under test (SUT). Recombining both arms at the output beam splitter leads to an interference pattern for which the intensity follows I∝1+cos(φ(λ)), with $\phi \left(\lambda \right) = \frac{{2\pi }}{\lambda }\left({n\left(\lambda \right){L_{\rm{s}}} - {L_{\rm{r}}}} \right)$. Here, λ represents the wavelength, Lr and Ls are the physical lengths of the reference arm and the SUT, respectively, and n(λ) is the effective refractive index of the SUT. It is worth noting that interference is observed only when the interferometer is precisely balanced to within the larger of the coherence length of the white-light source and the coherence length imposed by the resolution of the spectrometer, which is typically on the order of microns to millimetres11, 12. In this case, the phase term reads (more details are given in the Supplementary Information):
$$ \phi \left( \lambda \right) \approx 2\pi {L_{\rm{s}}}\left( {\frac{{1{{\rm{d}}^2}n}}{{2{\rm{d}}{\lambda ^2}}}{|_{{\lambda _0}}} \cdot \frac{{{{\left( {\mathit{\Delta} \lambda } \right)}^2}}}{{{\lambda _0} + \mathit{\Delta} \lambda }} + \frac{{1{{\rm{d}}^3}n}}{{6{\rm{d}}{\lambda ^3}}}{|_{{\lambda _0}}} \cdot \frac{{{{\left( {\mathit{\Delta} \lambda } \right)}^3}}}{{{\lambda _0} + \mathit{\Delta} \lambda }}} \right) + {\phi _{{\rm{off}}}} $$ (1) Fig. 1
Typical experimental set-ups. (a) Standard spectrally resolved WLI. (b) Quantum WLI. BS, beam splitter.Here, λ0 represents the stationary phase point, that is, the wavelength at which the absolute phase difference between the two interferometer arms is exactly zero. In standard WLI, λ0 is extracted experimentally by identifying the symmetry point of the observed interferogram11, 12. Additionally, Δλ=λ−λ0, and φoff is a constant offset phase. Provided that Ls is precisely known, the optical material parameters $\frac{{{d^2}n}}{{d{\lambda ^2}}}{|_{{\lambda _0}}}$ and $\frac{{{{\rm{d}}^3}n}}{{{\rm{d}}{\lambda ^3}}}{|_{{\lambda _0}}}$ can be extracted from a fit to the data as a function of Δλ. It is noteworthy that the three free parameters, that is, ${{\lambda _0}}$, $\frac{{{{\rm{d}}^2}n}}{{{\rm{d}}{\lambda ^2}}}{|_{{\lambda _0}}}$ and $\frac{{{{\rm{d}}^3}n}}{{{\rm{d}}{\lambda ^3}}}{|_{{\lambda _0}}}$, are usually all interdependent in a non-trivial manner such that uncertainties in one parameter systematically induce uncertainties in the others. In fact, the high number of fitting parameters required and the necessity to re-equilibrate the interferometer for every new SUT are the main limiting factors of this technique13, 14.
However, more accurate optical measurements are eagerly demanded in almost all fields involving optics. A special focus is made on the optical parameter $\frac{{{{\rm{d}}^2}n}}{{{\rm{d}}{\lambda ^2}}}{|_{{\lambda _0}}}$, as it is directly related to the chromatic dispersion coefficient $D = - \frac{{{\lambda _0}}}{c} \cdot \frac{{{{\rm{d}}^2}n}}{{{\rm{d}}{\lambda ^2}}}{|_{{\lambda _0}}}$, where c is the speed of light13, 17-23. Chromatic dispersion causes optical pulse broadening, and more accurate measurements of D would have significant repercussions for optimising today's telecommunication networks, developing new-generation pulsed lasers and amplifiers, and designing novel linear and nonlinear optical components and circuits, as well as for assessing the properties of biological tissues.
Standard WLI
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Figure 1b depicts the new experimental schematic dedicated to spectrally resolved quantum WLI (Q-WLI) intended to overcome the above issues. The quantum white-light source is composed of a continuous-wave pump laser and a nonlinear crystal in which energy–time entangled photon pairs are generated through spontaneous parametric downconversion24, 25. This process obeys the conservation of the energy, that is, $\frac{1}{{{\lambda _{\rm{p}}}}} = \frac{1}{{{\lambda _1}}} + \frac{1}{{{\lambda _2}}}$. Here, λp, 1, 2 respectively represent the wavelengths in vacuum of the pump laser photons and the individual photons for each generated pair. Another implication of the conservation of the energy is that the degenerate vacuum wavelength of the emission spectrum is λ*=2λp. We send the paired photons to the interferometer; however, as opposed to standard WLI, we now intentionally unbalance it. This provides us with two advantages: first, we avoid single-photon interference, and second, we obtain a means to distinguish events in which the two photons take opposite paths (strongly delayed arrival times at the interferometer's outputs) or the same path (near-zero arrival time difference)24. We postselect the latter events by considering only two-photon coincidence detection events in which both the single-photon detector (SPD) and the single-photon-sensitive spectrometer fire simultaneously. Our goal is now to observe quantum interference between these two-photon contributions, which necessitates that they be coherent and indistinguishable. Coherence is ensured by operating the interferometer at a path-length difference that is shorter than the coherence length of the pump laser (~100 m) such that the photon pair contributions are in phase26. Indistinguishability concerns mainly the temporal envelope of the photon pair wave packet, which is distorted from its original shape by the dispersion-induced temporal walk-off between the individual photons in the SUT. For standard fibres, this means that path-length differences up to ~10 m are acceptable27.
Thus, provided that the interferometer is operated in these conditions, near-zero arrival time coincidence detection results in a two-photon N00N state:
$$ \left| \psi \right\rangle = \frac{{{{\left| 2 \right\rangle }_{\rm{r}}}{{\left| 0 \right\rangle }_{\rm{s}}} + {e^{{\rm{i}}{\phi _{N00N}}}}{{\left| 0 \right\rangle }_{\rm{r}}}{{\left| 2 \right\rangle }_{\rm{s}}}}}{{\sqrt 2 }} $$ (2) Here, the ket vectors, indexed by s and r, indicate the number of photons in the reference and SUT arms, respectively, and φN00N=φ(λ1)+φ(λ2). We obtain the spectral dependence of φN00N by computing φ(λ1) and φ(λ2) according to Equation (1) and respecting the conservation of the energy during the downconversion process:
$$ {\phi _{N00N}} \approx \frac{{{{\rm{d}}^2}n}}{{{\rm{d}}{\lambda ^2}}}{|_{{\lambda ^*}}} \cdot \frac{{\pi {L_{\rm{s}}} \cdot {{\left( {\mathit{\Delta} \lambda } \right)}^2}}}{{\frac{{{\lambda ^*}}}{2} + \mathit{\Delta} \lambda }} + {\phi _{{\rm{off}}}} $$ (3) Here, ${\phi _{{\rm{off}}}}{\rm{ = }}\frac{{4\pi \left({n\left({{\lambda ^*}} \right){L_{\rm{s}}} - {L_{\rm{r}}}} \right)}}{{{\lambda ^*}}}$ is an offset term, and Δλ=λ−λ*. The phase-dependent two-photon coincidence rate R is then R∝1+cos(φN00N). In the past, numerous studies have investigated the term φoff, as it allows measuring optical phase shifts at constant wavelengths with double resolution compared to the standard approach28-30.
We access here, for the first time, the wavelength-dependent term in Equation (3) by recording R as a function of Δλ; that is, the two-photon coincidence rate is measured as a function of the paired-photons' wavelengths.
This leads to several pertinent purely quantum-enabled features. Due to the use of an energy–time entangled two-photon N00N state, the required precision of equilibrating the interferometer is ~10 m instead of microns to millimetres in standard WLI11-14. 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 $\frac{{{{\rm{d}}^3}n}}{{{\rm{d}}{\lambda ^3}}}{|_{{\lambda ^*}}}$ in Equation (3) is cancelled owing to energy–time correlations16. Furthermore, the wavelength at which chromatic dispersion is measured, λ*, need not be extracted from the data, as it is exactly twice the wavelength of the continuous-wave pump laser, λp, and can therefore be known with extremely high accuracy. This means that the quantum strategy allows data fitting using exactly one free parameter, namely, $\frac{{{{\rm{d}}^2}n}}{{{\rm{d}}{\lambda ^2}}}{|_{{\lambda ^*}}}$, 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 N00N state, double resolution of $\frac{{{{\rm{d}}^2}n}}{{{\rm{d}}{\lambda ^2}}}{|_{{\lambda ^*}}}$ 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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To benchmark standard and quantum approaches, we used a 1-m long SMF28e fibre from Corning as the SUT. 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). This ensured that φoff remained constant.
For chromatic dispersion measurements using classical WLI, we used a state-of-the-art superluminescent diode. At the output of the interferometer we measured an average spectral intensity of ~125 pW nm−1 from 1450 to 1650 nm. Interferograms were recorded using a spectrometer from Anritsu (model MS9710B, Atsugi-shi, Japan) with 0.1 s integration time and 0.5 nm resolution, which are standard parameters for this kind of measurement11, 12.
For the Q-WLI approach, the light source was a 780.246 nm laser pumping a type-0 periodically poled lithium niobate waveguide. We stabilised the laser wavelength against the $F = 2 \to F' = 2 \times 3$ hyperfine crossover transition in atomic 87Rb such that λp and λ* were known with a precision of the order of 1 fm. The quasi-phase matching in the periodically poled lithium niobate waveguide was engineered to generate energy–time entangled photon pairs around the degenerate wavelength of λ*=1560.493 nm with a bandwidth of ~140 nm25. To detect the paired photons, we used an InGaAs SPD (IDQ 220) at one interferometer output. The single-photon spectrometer at the other output was made of a wavelength-tunable 0.5 nm bandpass filter followed by another InGaAs SPD (IDQ 230). To avoid saturation of these detectors, the spectral intensity at the interferometer output was reduced to ~25 fW nm−1, which was partially compensated by increasing the integration time to 8 s.
All measurements were repeated 100 times on the same SUT to infer the statistical accuracy of both WLI and Q-WLI approaches.