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Abstract
Quantum metrology promises a great enhancement in measurement precision that beyond the possibilities of classical physics. We demonstrate a Hong-Ou-Mandel sensor that acts as a photonic frequency inclinometer for ultrasensitive tilt angle measurement within a wide range of tasks, ranging from the determination of mechanical tilt angles, the tracking of rotation/tilt dynamics of light-sensitive biological and chemical materials, or in enhancing the performance of optical gyroscope. The estimation theory shows that both a wider single-photon frequency bandwidth and a larger difference frequency of color-entangled states can increase its achievable resolution and sensitivity. Building on the Fisher information analysis, the photonic frequency inclinometer can adaptively determine the optimum sensing point even in the presence of experimental nonidealities.
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1. Introduction
The precise estimation of a physical quantity is an essential tool in many research areas and practical applications. One particular interest is that high precision inclinometers are engaging lots of attention in scientific facilities for fundamental research. For example, the inclinometers are used in the advanced Laser Interferometer Graviational Wave Observatory to correct the tilt coupling of the inertial sensor in the seismic isolation system [1,2]. Additionally, in the extremely sensitive and stable Large Ring Laser Gyroscopy, the local tilts are carefully measured and corrected to cancel the tilt effect in the rotation measurement [3,4]. Moreover, the precise inclinometers are essential requisites in measuring the crustal deformation and are widely used in the field of volcanology, seismology and geophysics [5–7]. In addition to the static metrology, the long-term slow tilt fluctuations need to be monitored and corrected according to its corresponding real-time feedbacks [1–4]. As a direct consequence, a sensitive and stable quantum inclinometer will inspire common interest in geoscience and in atomic, molecular and optical physics. However, the sensitivities and resolutions of current quantum inclinometers cannot meet the requirements of these applications [8,9]. Thus, enhancing the sensitivity and resolution of quantum inclinometer is a matter of utmost importance and urgency.
In the context of quantum metrology, quantum interference promises to enhance sensing technologies that also beyond the possibilities of classical physics [10–12]. Hong-Ou-Mandel (HOM) interference, a typical example of such quantum phenomena, has been used to measure optical delays between different paths, whose precision can reach few-femtosecond scale [13]. While great precision in HOM measurement requires that photons contain many frequencies, namely a large bandwidth, biphoton beat note has been proposed that suffices to achieve great precision with a larger scaling that is relevant to the difference frequency of discrete color entangled states [14]. Compared to the schemes that map angle variations to interference patterns for precise measurement of tilt or rotation angles [15–18], this HOM interferometry exploits spatial quantum beats, which thus has no requirement for a spatial resolution detector and achieves great advantages in reducing the experimental complexity and inherent stability. However, how to use HOM interferometry for the precise measurement of tilt angle has been explored relatively little.
Here, we demonstrate a photonic frequency inclinometer that is used to detect the inclination-induced time delays in a piece of transparent glass with smooth surface. By tilting this transparent glass placed in one arm of HOM interferometer [19], the manifestation of fourth-order spatial beating measurement would produce a temporal delay that is determined by the tilt angle [14]. We explore the sensitivity limits as functions of the single-photon frequency bandwidth $\sigma$, the difference frequency of color-entangled states $\Delta$ and the thickness of the used transparent glass $d$, and obtain an angular uncertainty scaling $\propto 1/d\sqrt {\Delta ^2+4\sigma ^2}$. Building on analyzing the Fisher information to determine the optimum sensing points, an optimized photonic frequency inclinometer can be adaptively implemented in a practical experiment [13]. As such, our work has the potential to open up an alternative route towards unconditional angular supersensitivity.
Finally, we identify four main advantages associated with our photonic frequency inclinometer. (i) The suitable frequency entangled state, that provides an enhancement in the metrology resolution and precision, can be readily obtained with comparatively little technological effort [20,21]. For example, our experiment employs the crossed-crystal configuration for preparing tunable frequency entangled photons, and thus the difference frequency can be controlled by tuning the phase matching temperature of nonlinear crystals. (ii) The photonic frequency inclinometer is dependent on the temporal measurement rather than a high-resolution camera, whose precision can reach attosecond that is scanned by using a simple and convenient translation stage. This indicates that our photonic frequency inclinometer is extremely likely to become commercial in the near future. (iii) Our photonic frequency inclinometer focuses on the angle-dependent time delay, which enables a wide range of applications. Besides for the determination of mechanical tilt angles as shown in this work, the performance of optical gyroscope can be enhanced with using our scheme, since the tilt fluctuation need to be monitored and corrected according to its corresponding real-time feedbacks or it may affect the sensitivity and accuracy of optical gyros [22,23]. Moreover, since our approach makes the single photons to transmit through a transparent sample, this maybe used in the tracking of rotation/tilt dynamics of light-sensitive biological and chemical materials [24,25]. (iv) As a direct result of inherent stability of HOM interference measurement, our photonic frequency inclinometer promises great robustness against deleterious noise and experimental imperfections, which has the potential to be widely used in the remote sensing applications [26,27].
2. Photonic frequency inclinometer
Let us consider an experimental configuration for photonic frequency inclinometer as shown in Fig. 1. In our setup, frequency-entangled photons are prepared by using spontaneous parametric down conversion (SPDC) process in two crossed nonlinear crystals designed for type-II quasi-phase matching. They are arranged in sequence and oriented with a relative inclination of $90^{\circ }$ along their common propagation axis, resulting in equal probability amplitudes for $\lvert V,\omega _p \rangle \rightarrow \lvert V,\omega _s \rangle \lvert H,\omega _i \rangle$ and $\lvert H,\omega _p \rangle \rightarrow \lvert H,\omega _s \rangle \lvert V,\omega _i \rangle$ with a diagonal-polarized pumping state $\lvert D \rangle =(\lvert H \rangle +\lvert V \rangle )/\sqrt {2}$. Since we use quasi-phase matching nonlinear crystals to prepare the frequency-entangled photons, the non-degenerated mode can be readily obtained by manipulating the temperature of the crystals, namely the two well-separated center frequency bins can be achieved. Since the frequency difference of down-converted photons typically exceeds that of the pump laser, frequency entanglement arises quite naturally as a consequence of energy conservation, which works as the probe in photonic frequency inclinometer metrology. Hence the two-dimensional discrete color-entangled state can be written as [20,21]
where $\omega _{1/2}$ represent two well separated single-photon frequency bins. As shown in the inset of Fig. 1, the tilt of a piece of transparent glass would introduce an angle-dependent time delay as where $d$ is the thickness of this transparent glass, $n$ represents its refractive index, $c$ is the speed of light in vacuum, and $\theta$ represents the tilt angle. Thus the frequency entanglement is transformed into
Fig. 1. Experimental setup of photonic frequency inclinometer for angular measurement. HWP: half-wave plate; ppKTP: periodically poled potassium titanyl phosphate nonlinear crystal; LP: longpass filter; PBS: polarizing beam splitter; BS: beam splitter; SPDM: single photon detection module; AND: coincidence counts. The inset illustrates the optical axis layout by tilting the transparent glass that would introduce the angle-dependent time delay.
An estimator is a function of the experimental data that allows us to infer the value of the unknown tilt angle using a particular statistical model for the probability distribution of measurement outcomes. Typically, for any such estimator, classical estimation theory states that the standard deviation is lower bounded by the Cram$\acute{e}$r-Rao bound as [29–31]
where $N$ represents the number of independent experimental trials, and the Fisher information $F_\theta$ reveals the information that can be obtained in an individual experimental trial, and readsEvaluating the Fisher information for this set of probabilities, we find that its maximal value is achieved in ideal case (${\gamma }=0$, $\alpha =1$) as
where $a=dn/c$, $\eta _1=n/\sqrt {n^2-sin(\theta )^2}-1$ is relevant to the relative time delay introduced by tilting the transparent glass with an angle of $\theta$, and $\eta _2=exp(-2a^2\sigma ^2\eta _1^2)$ represents the corresponding change in interference visibility, and $\eta _3$ readsAdditionally, we note that the upper bound for photonic frequency inclinometer is achieved in ideal case at position of $\tau _{\theta =\frac {\pi }{4}}\rightarrow 0$ as
which indicates that the achievable Fisher information would scale with $d^2(\Delta ^2+4\sigma ^2)$ for a particular test sample.Accordingly, the measurement precision of photonic frequency inclinometer can be enhanced by several methods: (i) Increasing the single-photon frequency bandwidth $\sigma$ or the difference frequency $\Delta$ as proved in several recent works [14,28]. (ii) Increasing the thickness $d$ or the refractive index $n$ of the used transparent glass such that the ratio of angle-to-length transfer is magnified. (iii) Adaptively finding the optimum sensing positions by analyzing the Fisher information in practical applications because the optimal angular position of angle-to-length transfer and optimal temporal position of HOM sensor would vary with the parameters $\theta$ and $\tau$ [13,14]. (iv) Increasing the value of $N$ by optimizing the experimental components, such as increasing the detection efficiency or decreasing the photon loss rate.
3. Experimental implementation
In order to demonstrate the viability principle of employing our photonic frequency inclinometer, we perform a proof-of-concept experiment in which we estimate the tilt angle that determines the relative time delay. Photon pairs are typically prepared by using SPDC process in a nonlinear crystal. Specifically, the tunable frequency entanglement is readily generated by balanced SPDC emission in two crossed type-II periodically poled potassium titanyl phosphate (ppKTP) crystals with a diagonal-polarized pumping state $\lvert D \rangle =(\lvert H \rangle +\lvert V \rangle )/\sqrt {2}$, wherein the optical axis of the second crystal is rotated by $90^{\circ }$ with respect to the first crystal [21,28]. Thus, this configuration enables equal probability amplitudes for $\lvert V,\omega _p \rangle \rightarrow \lvert V,\omega _s \rangle \lvert H,\omega _i \rangle$ and $\lvert H,\omega _p \rangle \rightarrow \lvert H,\omega _s \rangle \lvert V,\omega _i \rangle$, which can be written as (1). These frequency entangled photons are then separated by using a polarizing beam splitter, and routed to the input ports of a beam splitter, which constitutes a HOM interferometer. For the observation of nonclassical interference fringes that are caused by biphoton beating, we only identify the coincidence events that paired photons are detected simultaneously at opposite spatial modes as a direct result of anti-bunching effect.
In our experiment, we use a piece of transparent glass with refractive index of $\sim 1.47$ as the test sample. The tilt of this transparent glass would change the path length that single photons are transmitted. Additionally, for the trade off between measurement precision and dynamic range, we set $\tau _{\theta =0}\rightarrow 0$ in our proof-of-principle experiment. As shown in Fig. 2, the HOM signals are measured as a function of relative time delays for various frequency detunings by using the transparent glass with thickness of 0.6 mm. By fitting the experimental measurement results to the theoretical prediction as demonstrated in (5), we are able to estimate the single-photon frequency bandwidth to be $\sim$1.67 THz, which corresponds to a bandwidth in wavelength of $\sim$3.65 nm and a coherence time of 0.54 ps. However, Fig. 2(a,b) shows that the resolution of HOM sensor using non-degenerate photons is severely limited by the single-photon frequency bandwidth, which is a typical notorious challenge in HOM-based quantum metrology [13].
Fig. 2. Experimental observation of HOM interference signals as functions of tilt angle and relative time delay, by using the transparent glass with thickness of 0.6 mm. The HOM interference signals are recorded for (a-b) $\Delta =0\;\textrm {THz}$, (c-d) $\Delta =24.2\;\textrm {THz}$, (e-f) $\Delta =40.1\;\textrm {THz}$.
To tackle this issue, we use a HOM sensor based on frequency entanglement to further enhance the resolution. As shown in Fig. 1, the quasi phase matching conditions in the orthogonally oriented non-linear crystals can be tuned by controlling their temperature. More specifically, since the center wavelengths of frequency-entangled photons are related to the phase-matching temperature of nonlinear crystals, we can readily prepare tunable frequency-entangled photon pairs in our proof-of-principle experiment. Namely, the difference frequency $\Delta$ of discrete color entanglement is determined by the temperature of nonlinear crystals. As calculated in Eq. (5), the HOM interference patterns manifest themselves as the periodic oscillation within a coherence Gaussian envelope, where the period of oscillation is dependent on the difference frequency $\Delta$. Building on the Fisher information analysis as expressed in Eq. (10), the measurement precision of tilt angle would be enhanced with the increase of the difference frequency $\Delta$. These intriguing oscillation in HOM interference pattern emerges in the experimental observations as shown in Fig. 2(c-f). It is obvious that the angular resolution is improved by speeding up the oscillation. By fitting these interference fringes to the normalized coincidence probabilities as shown in (5), we are able to extract that the maximal difference frequency we have measured in our experiment is $\sim$ 40 THz at temperature of 75$^{\circ }$ C, which is about 25 times the single-photon frequency bandwidth. We note that this difference frequency can be larger by increasing the phase-matching temperature, or using quantum frequency conversion [32,33].
Additionally, the Fisher information (8) indicates that the angular resolution can be enhanced by increasing the thickness of the used transparent glass. The experimental detail of using different thickness of the transparent glass is shown in Supplement 1, and the experimental results show remarkable fitness with the theoretical prediction. It’s worth noting that the thickness and refractive index of the used transparent glass is relevant to the measurement efficiency, which is reversely imposes the ultimate limit on the resolution and precision. Therefore, a trade-off between the signal-to-noise ratio and the Fisher information obtained in an individual experimental trial is of great concern for achieving the optimal angular resolution and precision.
To demonstrate the viability of photonic frequency inclinometer for estimating the tilt angle, we set the relative time delay as $\tau =0$ in the case of $\theta =0$ that corresponds to the situation when the single photons transmit through the transparent glass perpendicularly. The HOM signals as a function of tilt angles are demonstrated in Fig. 3(a-c). Thereinto, the oscillation period of HOM interference fringes is also dependent on the thickness of the transparent glass as shown in Fig. 4.(a). The corresponding theoretical prediction and experimental measurement of Fisher information with respect to tilt angles are demonstrated in Fig. 3(d-f). In our experiment, the achievable minimum uncertainty is shown in Fig. 4.(b), which is comparable to [18], and can be readily enhanced by further enlarging the frequency difference.
Fig. 3. HOM interference signals as functions of tilt angles for various difference frequencies as (a) $\Delta =0\;\textrm {THz}$, (b) $\Delta {\approx } 23.3\;\textrm {THz}$, and (c) $\Delta {\approx } 38.3\;\textrm {THz}$. Normalized detection rate: actual coincidence count rate/baseline, in which baseline stands for coincidence count rate at large delay points. Theoretical bound and experimental measurement of Fisher information for various difference frequencies as (d) $\Delta =0\;\textrm {THz}$, (e) $\Delta {\approx } 23.3\;\textrm {THz}$, and (f) $\Delta {\approx }38.3\;\textrm {THz}$. $F_e$: experimental measurement of Fisher information; $F_i$: ideal bound of Fisher information.
Fig. 4. (a) Theoretical prediction of HOM signals as functions of the tilt angle and thickness of the used transparent glass. (b) The minimum uncertainty that obtained in our experiment, where the number of independent experimental trials $N{\approx }3000$.
4. Discussion
We have reported a scheme to measure tilt angles by using photonic frequency inclinometer that promises a great enhancement in resolution and precision. Notably, this precision scales with the single-photon frequency bandwidth, the difference frequency of color-entangled photons, the thickness of the used transparent glass and the total number of independent experimental trials. Thus, by further improving the experimental implementations, the achievable sensitivity can be pushed with minimal experimental complexity. For example, the single-photon frequency bandwidth can be increased by using shorter nonlinear crystal, the difference frequency of color-entangled photons can be enlarged by increasing the phase-matching temperature, and the total number of experimental trials can be increased by improving the system efficiency and decreasing the photon losses.
In our proof-of-principle experiment, the achievable precision and resolution is similar to those reported in [34,35]. However, our work provides alternative routes toward enhancing the measurement precision by concise yet efficient approaches. For example, in a recent work, a detuning exceeding 1000 THz is experimentally achievable, which is approximately 25 times the difference frequency prepared in our experiment and corresponds to a minimal uncertainty of $0.001^{\circ }$ for merely 3000 photon pairs [36]. Additionally, multiphoton interference has been widely used quantum metrology [11,18]. Inspired by this technique, the measurement precision of our photonic frequency inclinometer can be further enhanced.
Apart from the measurement of determination of mechanical tilt angles, our approach may have a range of potential applications in enhancing the performance of optical gyroscope or in the tracking of rotation/tilt dynamics of light-sensitive biological and chemical materials since our approach makes the single photons to transmit through a transparent sample. In summary, we anticipate that immediate applications of this photonic frequency inclinometer in more interesting prospects can improve the precision for measuring tilt or rotation angles to a supersensitivity.
Funding
Program for New Century Excellent Talents in University (NCET-13-0495); Natural Science Foundation of Fujian Province of China for Distinguished Young Scientists (2015J06002); Natural Science Foundation of Fujian Province (2020J05004); Fundamental Research Funds for the Central Universities at Xiamen University (20720190057, 20720210096); National Natural Science Foundation of China (12004318, 12034016, 61975169).
Disclosures
The authors declare no conflicts of interest.
Data Availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
References
1. F. Matichard, B. Lantz, R. Mittleman, et al., “Seismic isolation of advanced ligo: Review of strategy, instrumentation and performance,” Classical Quantum Gravity 32(18), 185003 (2015). [CrossRef]
2. B. P. Abbott, R. Abbott, T. Abbott, et al., “Observation of gravitational waves from a binary black hole merger,” Phys. Rev. Lett. 116(6), 061102 (2016). [CrossRef]
3. K. U. Schreiber and J.-P. R. Wells, “Invited review article: Large ring lasers for rotation sensing,” Rev. Sci. Instrum. 84(4), 041101 (2013). [CrossRef]
4. K. U. Schreiber, T. Klügel, J.-P. R. Wells, R. B. Hurst, and A. Gebauer, “How to detect the chandler and the annual wobble of the earth with a large ring laser gyroscope,” Phys. Rev. Lett. 107(17), 173904 (2011). [CrossRef]
5. J. J. Dvorak and D. Dzurisin, “Volcano geodesy: The search for magma reservoirs and the formation of eruptive vents,” Rev. Geophys. 35(3), 343–384 (1997). [CrossRef]
6. L. N. Medina, D. F. Arcos, and M. Battaglia, “Twenty years (1990-2010) of geodetic monitoring of galeras volcano (colombia) from continuous tilt measurements,” J. Volcanol. Geotherm. Res. 344, 232–245 (2017). [CrossRef]
7. Y. Ito, K. Obara, K. Shiomi, S. Sekine, and H. Hirose, “Slow earthquakes coincident with episodic tremors and slow slip events,” Science 315(5811), 503–506 (2007). [CrossRef]
8. W.-J. Xu, M.-K. Zhou, M.-M. Zhao, K. Zhang, and Z.-K. Hu, “Quantum tiltmeter with atom interferometry,” Phys. Rev. A 96(6), 063606 (2017). [CrossRef]
9. J. Liu, W.-J. Xu, C. Zhang, Q. Luo, Z.-K. Hu, and M.-K. Zhou, “Sensitive quantum tiltmeter with nanoradian resolution,” Phys. Rev. A 105(1), 013316 (2022). [CrossRef]
10. S. Witte, R. T. Zinkstok, W. Ubachs, W. Hogervorst, and K. S. Eikema, “Deep-ultraviolet quantum interference metrology with ultrashort laser pulses,” Science 307(5708), 400–403 (2005). [CrossRef]
11. Z.-E. Su, Y. Li, P. P. Rohde, H.-L. Huang, X.-L. Wang, L. Li, N.-L. Liu, J. P. Dowling, C.-Y. Lu, and J.-W. Pan, “Multiphoton interference in quantum fourier transform circuits and applications to quantum metrology,” Phys. Rev. Lett. 119(8), 080502 (2017). [CrossRef]
12. V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5(4), 222–229 (2011). [CrossRef]
13. A. Lyons, G. C. Knee, E. Bolduc, T. Roger, J. Leach, E. M. Gauger, and D. Faccio, “Attosecond-resolution hong-ou-mandel interferometry,” Sci. Adv. 4(5), eaap9416 (2018). [CrossRef]
14. Y. Chen, M. Fink, F. Steinlechner, J. P. Torres, and R. Ursin, “Hong-ou-mandel interferometry on a biphoton beat note,” npj Quantum Inf. 5(1), 43 (2019). [CrossRef]
15. M. Ikram and G. Hussain, “Michelson interferometer for precision angle measurement,” Appl. Opt. 38(1), 113–120 (1999). [CrossRef]
16. Y. Wu, H. Cheng, and Y. Wen, “High-precision rotation angle measurement method based on a lensless digital holographic microscope,” Appl. Opt. 57(1), 112–118 (2018). [CrossRef]
17. V. D’ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4(1), 2432 (2013). [CrossRef]
18. M. Hiekkamäki, F. Bouchard, and R. Fickler, “Photonic angular superresolution using twisted n00n states,” Phys. Rev. Lett. 127(26), 263601 (2021). [CrossRef]
19. C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59(18), 2044–2046 (1987). [CrossRef]
20. Y. Chen, S. Ecker, S. Wengerowsky, L. Bulla, S. K. Joshi, F. Steinlechner, and R. Ursin, “Polarization entanglement by time-reversed hong-ou-mandel interference,” Phys. Rev. Lett. 121(20), 200502 (2018). [CrossRef]
21. Y. Chen, S. Ecker, J. Bavaresco, T. Scheidl, L. Chen, F. Steinlechner, M. Huber, and R. Ursin, “Verification of high-dimensional entanglement generated in quantum interference,” Phys. Rev. A 101(3), 032302 (2020). [CrossRef]
22. M. R. Grace, C. N. Gagatsos, Q. Zhuang, and S. Guha, “Quantum-enhanced fiber-optic gyroscopes using quadrature squeezing and continuous-variable entanglement,” Phys. Rev. Appl. 14(3), 034065 (2020). [CrossRef]
23. M. Fink, F. Steinlechner, J. Handsteiner, J. P. Dowling, T. Scheidl, and R. Ursin, “Entanglement-enhanced optical gyroscope,” New J. Phys. 21(5), 053010 (2019). [CrossRef]
24. M. A. Taylor and W. P. Bowen, “Quantum metrology and its application in biology,” Phys. Rep. 615, 1–59 (2016). [CrossRef]
25. K. Lin, I. Tutunnikov, J. Ma, J. Qiang, L. Zhou, O. Faucher, Y. Prior, I. S. Averbukh, and J. Wu, “Spatiotemporal rotational dynamics of laser-driven molecules,” Adv. Photonics 2(02), 1 (2020). [CrossRef]
26. B. Ndagano, H. Defienne, D. Branford, Y. D. Shah, A. Lyons, N. Westerberg, E. M. Gauger, and D. Faccio, “Quantum microscopy based on hong–ou–mandel interference,” Nat. Photonics 16(5), 384–389 (2022). [CrossRef]
27. A. E. Ulanov, I. A. Fedorov, D. Sychev, P. Grangier, and A. Lvovsky, “Loss-tolerant state engineering for quantum-enhanced metrology via the reverse hong–ou–mandel effect,” Nat. Commun. 7(1), 11925 (2016). [CrossRef]
28. S. Ramelow, L. Ratschbacher, A. Fedrizzi, N. K. Langford, and A. Zeilinger, “Discrete tunable color entanglement,” Phys. Rev. Lett. 103(25), 253601 (2009). [CrossRef]
29. C. W. Helstrom, “Quantum detection and estimation theory,” J. Stat. Phys. 1(2), 231–252 (1969). [CrossRef]
30. A. Fujiwara and H. Nagaoka, “Quantum fisher metric and estimation for pure state models,” Phys. Lett. A 201(2-3), 119–124 (1995). [CrossRef]
31. S. Pirandola, B. R. Bardhan, T. Gehring, C. Weedbrook, and S. Lloyd, “Advances in photonic quantum sensing,” Nat. Photonics 12(12), 724–733 (2018). [CrossRef]
32. M. Bock, P. Eich, S. Kucera, M. Kreis, A. Lenhard, C. Becher, and J. Eschner, “High-fidelity entanglement between a trapped ion and a telecom photon via quantum frequency conversion,” Nat. Commun. 9(1), 1998 (2018). [CrossRef]
33. T. Walker, K. Miyanishi, R. Ikuta, H. Takahashi, S. Vartabi Kashanian, Y. Tsujimoto, K. Hayasaka, T. Yamamoto, N. Imoto, and M. Keller, “Long-distance single photon transmission from a trapped ion via quantum frequency conversion,” Phys. Rev. Lett. 120(20), 203601 (2018). [CrossRef]
34. B.-O. Guan, H.-Y. Tam, and S.-Y. Liu, “Temperature-independent fiber bragg grating tilt sensor,” IEEE Photonics Technol. Lett. 16(1), 224–226 (2004). [CrossRef]
35. S. Prakash, S. Singh, and S. Rana, “Automated small tilt-angle measurement using lau interferometry,” Appl. Opt. 44(28), 5905–5909 (2005). [CrossRef]
36. C. P. Lualdi, K. A. Meier, S. J. Johnson, and P. G. Kwiat, “Quantum-enhanced interferometry via extreme non-degenerate energy-entangled photons,” in Frontiers in Optics, (Optical Society of America, 2021), pp. FTh6D–2.
