* Borel (1942) E. Borel, “Sur l’emploi du théorème de bernoulli pour faciliter le calcul d’une infinité de coefficients. application au problème de l’attente à un guichet,” C. R. Acad. Sci. 214, 452–456 (1942).
  * Tanner (1961) J. C. Tanner, “A derivation of the borel distribution,” Biometrika 48, 222–224 (1961).
  * Foss  _et al._ (2013) S. Foss, D. Korshunov, and S. Zachary,  _An Introduction to Heavy-Tailed and Subexponential Distributions_ (Springer, 2013).
  * Zhang  _et al._ (2013) Y. R. Zhang, G. R. Jin, J. P. Cao, W. M. Liu, and H. Fan, “Unbounded quantum fisher information in two-path interferometry with finite photon number,” J. Phys. A: Math. Theor. 46, 035302 (2013).
  * Boixo  _et al._ (2007) S. Boixo, S. T. Flammia, C. M. Caves, and J. M. Geremia, “Generalized limits for single-parameter quantum estimation,” Phys. Rev. Lett. 98, 090401 (2007).
  * Boixo  _et al._ (2008) S. Boixo, A. Datta, M. J. Davis, S. T. Flammia, A. Shaji, and C. M. Caves, “Quantum metrology: Dynamics versus entanglement,” Phys. Rev. Lett. 101, 040403 (2008).
  * Choi and Sundaram (2008) S. Choi and B. Sundaram, “Bose-einstein condensate as a nonlinear ramsey interferometer operating beyond the heisenberg limit,” Phys. Rev. A 77, 053613 (2008).
  * Roy and Braunstein (2008) S. M. Roy and S. L. Braunstein, “Exponentially enhanced quantum metrology,” Phys. Rev. Lett. 100, 220501 (2008).
  * Woolley  _et al._ (2008) M. J. Woolley, G. J. Milburn, and C. M. Caves, “Nonlinear quantum metrology using coupled nanomechanical resonators,” New J. Phys. 10, 125018 (2008).
  * Napolitano and Mitchell (2010) M. Napolitano and M. W. Mitchell, “Nonlinear metrology with a quantum interface,” New J. Phys. 12, 093016 (2010).
  * Rams  _et al._ (2018) M. M. Rams, P. Sierant, O. Dutta, P. Horodecki, and J. Zakrzewski, “At the limits of criticality-based quantum metrology: Apparent super-heisenberg scaling revisited,” Phys. Rev. X 8, 021022 (2018).
  * Napolitano  _et al._ (2011) M. Napolitano, M. Koschorreck, B. Dubost, N. Behbood, R. J. Sewell, and M. W. Mitchell, “Interaction-based quantum metrology showing scaling beyond the heisenberg limit,” Nature 471, 486–489 (2011).
  * Zwierz  _et al._ (2010) M. Zwierz, C. A. Pérez-Delgado, and P. Kok, “General optimality of the heisenberg limit for quantum metrology,” Phys. Rev. Lett. 105, 180402 (2010).
  * Luis and Rodil (2013) A. Luis and A. Rodil, “Alternative measures of uncertainty in quantum metrology: Contradictions and limits,” Phys. Rev. A 87, 034101 (2013).
  * Luis (2013) A. Luis, “Signal detection without finite-energy limits to quantum resolution,” Ann. Phys. 331, 1–8 (2013).
  * Tsang (2012) M. Tsang, “Ziv-zakai error bounds for quantum parameter estimation,” Phys. Rev. Lett. 108, 230401 (2012).
  * Giovannetti and Maccone (2012) V. Giovannetti and L. Maccone, “Sub-heisenberg estimation strategies are ineffective,” Phys. Rev. Lett. 108, 210404 (2012).
  * Berry  _et al._ (2012) D. W. Berry, M. J. W. Hall, M. Zwierz, and H. M. Wiseman, “Optimal heisenberg-style bounds for the average performance of arbitrary phase estimates,” Phys. Rev. A 86, 053813 (2012).
  * Hall  _et al._ (2012) M. J. W. Hall, D. W. Berry, M. Zwierz, and H. M. Wiseman, “Universality of the heisenberg limit for estimates of random phase shifts,” Phys. Rev. A 85, 041802 (2012).
  * Hall and Wiseman (2012a) M. J. W. Hall and H. M. Wiseman, “Heisenberg-style bounds for arbitrary estimates of shift parameters including prior information,” New J. Phys. 14, 033040 (2012a).
  * Hall and Wiseman (2012b) M. J. W. Hall and H. M. Wiseman, “Does nonlinear metrology offer improved resolution? answers from quantum information theory,” Phys. Rev. X 2, 041006 (2012b).
  * Jarzyna and Demkowicz-Dobrzański (2015) M. Jarzyna and R. Demkowicz-Dobrzański, “True precision limits in quantum metrology,” New J. Phys. 17, 013010 (2015).
  * Ziv and Zakai (1969) J. Ziv and M. Zakai, “Some lower bounds on signal parameter estimation,” IEEE Trans. Inform. Theor. IT-15, 386–391 (1969).
  * Seidman (1970) L. P. Seidman, “Performance limitations and error calculations for parameter estimation,” Proc. IEEE 58, 644–652 (1970).
  * Chazan  _et al._ (1975) D. Chazan, M. Zakai, and J. Ziv, “Improved lower bounds on signal parameter estimation,” IEEE Trans. Inform. Theor. IT-21, 90–93 (1975).
  * Bellini and Tartara (1974) S. Bellini and G. Tartara, “Bounds on error in signal parameter estimation,” IEEE Trans. Commun. COM-22, 340–342 (1974).
  * Weinstein (1988) E. Weinstein, “A general class of lower bounds in parameter estimation,” IEEE Trans. Inform. Theor. 34, 388–342 (1988).
  * Górecki  _et al._ (2020) W. Górecki, R. Demkowicz-Dobrzański, H. M. Wiseman, and D. W. Berry, “$\pi$-corrected heisenberg limit,” Phys. Rev. Lett. 124, 030501 (2020).
  * Friis  _et al._ (2017) N. Friis, D. Orsucci, M. Skotiniotis, P. Sekatski, V. Dunjko, H. J. Briegel, and W. Dür, “Flexible resources for quantum metrology,” New J. Phys. 19, 063044 (2017).
  * Rubio  _et al._ (2018) J. Rubio, P. Knott, and J. Dunningham, “Non-asymptotic analysis of quantum metrology protocols beyond the cramér-rao bound,” J. Phys. Commun. 2, 015027 (2018).
  * Rubio and Dunningham (2019) J. Rubio and J. Dunningham, “Quantum metrology in the presence of limited data,” New J. Phys. 21, 043037 (2019).
  * Rubio and Dunningham (2020) J. Rubio and J. Dunningham, “Bayesian multiparameter quantum metrology with limited data,” Phys. Rev. A 101, 032114 (2020).
  * Lumio  _et al._ (2018) A. Lumio, E. Polino, A. S. Rab, G. Milani, N. Spagnolo, N. Wiebe, and F. Sciarrino, “Experimental phase estimation enhanced by machine learning,” Phys. Rev. Applied 10, 044033 (2018).
  * Cimini  _et al._ (2019) V. Cimini, I. Gianani, N. Spagnolo, F. Leccese, F. Sciarrino, and M. Barbieri, “Calibration of quantum sensors by neural networks,” Phys. Rev. Lett. 123, 230502 (2019).
  * Demkowicz-Dobrzański  _et al._ (2020) R. Demkowicz-Dobrzański, W. Górecki, and M. Guţă, “Multi-parameter estimation beyond quantum fisher information,” J. Phys. A: Math. Theor. 53, 363001 (2020).
  * Lee  _et al._ (2016) C. Lee, F. Dieleman, J. Lee, C. Rockstuhl, S. A. Maier, and M. Tame, “Quantum plasmonic sensing: Beyond the shot-noise and diffraction limit,” ACS Photonics 3, 992–999 (2016).
  * Zuloaga  _et al._ (2009) J. Zuloaga, E. Prodan, and P. Nordlander, “Quantum description of the plasmon resonances of a nanoparticle dimer,” Nano Lett. 9, 887–891 (2009).
  * Savage  _et al._ (2012) K. J. Savage, M. M. Hawkeye, R. Esteban, A. G. Borisov, J. Aizpurua, and J. J. Baumberg, “Revealing the quantum regime in tunelling plasmonics,” Nature 491, 574–577 (2012).
  * Halperin (1986) W. P. Halperin, “Quantum size effects in metal particles,” Rev. Mod. Phys. 58, 533–606 (1986).
  * Halas  _et al._ (2011) N. J. Halas, S. Lal, W.-S. Chang, S. Link, and P. Nordlander, “Plasmons in strongly coupled metallic nanostructures,” Chem. Rev. 111, 3913–3961 (2011).
  * Scholl  _et al._ (2012) J. A. Scholl, A. L. Koh, and J. A. Dionne, “Quantum plasmon resonances of individual metallic nanoparticles,” Nature 483, 421–427 (2012).
  * Gonçalves  _et al._ (2020a) P. A. D. Gonçalves, T. Christensen, N. M. R. Peres, A.-P. Jauho, I. Epstein, F. H. L. Koppens, M. Soljačić, and N. Asger Mortensen, “Quantum surface-response of metals revealed by acoustic graphene plasmons,” arXiv , 2008.07613 (2020a).
  * Tame  _et al._ (2008) M. S. Tame, C. Lee, J. Lee, D. Ballester, M. Paternostro, A. V. Zayats, and M. Kim, “Single-photon excitation of surface plasmon polaritons,” Phys. Rev. Lett. 101, 190504 (2008).
  * Ballester  _et al._ (2009) D. Ballester, M. S. Tame, C. Lee, J. Lee, , and M. Kim, “Long-range surface plasmon-polariton-excitation at the quantum level,” Phys. Rev. A 79, 053845 (2009).
  * Lee  _et al._ (2017) J.-S. Lee, T. Huynh, S.-Y. Lee, K.-G. Lee, J. Lee, M. Tame, C. Rockstuhl, and C. Lee, “Quantum noise reduction in intensity-sensitive surface-plasmon-resonance sensors,” Phys. Rev. A 96, 033833 (2017).
  * Hofmann (2009) H. F. Hofmann, “All path-symmetric pure states achieve their maximal phase sensitivity in conventional two-path interferometry,” Phys. Rev. A 79, 033822 (2009).
  * Varcoe  _et al._ (2000) B. T. H. Varcoe, S. Brattke, M. Weidinger, and H. Walther, “Preparing pure photon number states of the radiation field,” Nature 403, 743–746 (2000).
  * Bertet  _et al._ (2002) P. Bertet, S. Osnaghi, P. Milman, A. Auffeves, P. Maioli, M. Brune, J. M. Raimond, and S. Haroche, “Generating and probing a two-photon fock state with a single atom in a cavity,” Phys. Rev. Lett. 88, 143601 (2002).
  * Varcoe  _et al._ (2004) Benjamin T H Varcoe, Simon Brattke, and Herbert Walther, “The creation and detection of arbitrary photon number states using cavity qed,” New J. Phys. 6, 97–97 (2004).
  * Waks  _et al._ (2006) E. Waks, E. Diamanti, and Y. Yamamoto, “Generation of photon number states,” New J. Phys. 8, 4 (2006).
  * Clausen  _et al._ (2001) J. Clausen, H. Hansen, L. Knöll, J. Mlynek, and D.-G. Welsch, “Conditional quantum-state engineering in repeated 2-photon down-conversion,” Appl. Phys. B 72, 43–50 (2001).
  * Sanaka (2005) Kaoru Sanaka, “Linear optical extraction of photon-number fock states from coherent states,” Phys. Rev. A 71, 021801 (2005).
  * Ou (2006) Z. Y. Ou, “Temporal distinguishability of an $n$-photon state and its characterization by quantum interference,” Phys. Rev. A 74, 063808 (2006).
  * Deléglise  _et al._ (2008) S. Deléglise, I. Dotsenko, C. Sayrin, J. Bernu, M. Brune, J.-M. Raimond, and S. Haroche, “Reconstruction of non-classical cavity field states with snapshots of their decoherence,” Nature 455, 510–514 (2008).
  * Wang  _et al._ (2008) H. Wang, M. Hofheinz, M. Ansmann, R. C. Bialczak, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, J. Wenner, A. N. Cleland, and J. M. Martinis, “Measurement of the decay of fock states in a superconducting quantum circuit,” Phys. Rev. Lett. 101, 240401 (2008).
  * Hofheinz  _et al._ (2008) M. Hofheinz, E. Weig, M. Ansmann, R. C. Bialczak, E. Lucero, M. Neeley, A. Oconnell, H. Wang, J. M. Martinis, and A. Cleland, “Generation of fock states in a superconducting quantum circuit,” Nature 454, 310–314 (2008).
  * Zhou  _et al._ (2012) X. Zhou, I. Dotsenko, B. Peaudecerf, T. Rybarczyk, C. Sayrin, S. Gleyzes, J. M. Raimond, M. Brune, and S. Haroche, “Field locked to a fock state by quantum feedback with single photon corrections,” Phys. Rev. Lett. 108, 243602 (2012).
  * Uria  _et al._ (2020) M. Uria, P. Solano, and C. Hermann-Avigliano, “Deterministic generation of large fock states,” Phys. Rev. Lett. 125, 093603 (2020).
  * Cosacchi  _et al._ (2020) M. Cosacchi, J. Wiercinski, T. Seidelmann, M. Cygorek, A. Vagov, D. E. Reiter, and V. M. Axt, “On-demand generation of higher-order fock states in quantum-dot–cavity systems,” Phys. Rev. Research 2, 033489 (2020).
  * Groiseau  _et al._ (2020) Caspar Groiseau, Alexander E. J. Elliott, Stuart J. Masson, and Scott Parkins, “Deterministic single-atom source of quasi-superradiant $n$-photon pulses,” arXiv , 2012.00246 (2020).
  * Zapletal  _et al._ (2020) P. Zapletal, T. Darras, H. Le Jeannic, A. Cavaillès, G. Guccione, J. Laurat, and R. Filip, “Experimental fock-state capability of non-ideal single-photon states,” arXiv , 2012.08544 (2020).
  * Zhao  _et al._ (2020) Y. Zhao, Y. Peng, X. Hu, F. Xia, and Q. Zhao, “Beating the shot-noise limit with optical fiber quantum sensors for salinity measurement,” Sensor. Actuat. B-Chem. 320, 128353 (2020).
  * Barer and Tkaczyk (1954) R. Barer and S. Tkaczyk, “Refractive index of concentrated protein solutions,” Nature 173, 821–822 (1954).
  * Huck  _et al._ (2009) A. Huck, S. Smolka, P. Lodahl, A. S. Sørensen, A. Boltasseva, J. Janousek, and U. L. Andersen, “Demonstration of quadrature-squeezed surface plasmons in a gold waveguide,” Phys. Rev. Lett. 102, 246802 (2009).
  * Peng  _et al._ (2020) Y. Peng, Y. Zhao, X. Hu, and Y. Yang, “Optical fiber quantum biosensor based on surface plasmon polaritons for the label-free measurement of protein,” Sensor. Actuat. B-Chem. 316, 128097 (2020).
  * Razavi  _et al._ (2009) M. Razavi, I. Sollner, E. Bocquillon, C. Couteau, R. Laflamme, and G. Weihs, “Characterizing heralded single-photon sources with imperfect measurement devices,” J. Phys. B: At. Mol. Opt. Phys. 42, 114013 (2009).
  * Bashkansky  _et al._ (2014) M. Bashkansky, I. Vurgaftman, A. C. R. Pipino, and J. Reintjes, “Significance of heralding in spontaneous parametric down-conversion,” Phys. Rev. A 90, 053825 (2014).
  * Lounis and Orrit (2005) B. Lounis and M. Orrit, “Single-photon sources,” Rep. Prog. Phys. 68, 1129–1179 (2005).
  * Aharonovich  _et al._ (2016) I. Aharonovich, D. Englund, and M. Toch, “Solid-state single-photon emitters,” Nat. Photonics 10, 631–641 (2016).
  * Berchera and Degiovanni (2019) I. R. Berchera and I. P. Degiovanni, “Quantum imaging with sub-poissonian light: Challenges and perspectives in optical metrology,” Metrologia 56, 024001 (2019).
  * Chu  _et al._ (2017) X.-L. Chu, S. Götzinger, and V. Sandoghdar, “A single molecule as a high-fidelity photon gun for producing intensity-squeezed light,” Nat. Photonics 11, 58–62 (2017).
  * Grosso  _et al._ (2017) G. Grosso, H. Moon, B. Lienhard, S. Ali, D. K. Efetov, M. M. Furchi, P. Jarillo-Herrero, M. J. Ford, I. Aharonovich, and D. Englund, “Tunable and high-purity room temperature single-photon emission from atomic defects in hexagonal boron nitride,” Nat. Commun. 8, 1–8 (2017).
  * Wang  _et al._ (2019b) H. Wang, Y.-M. He, T.-H. Chung, H. Hu, Y. Yu, S. Chen, X. Ding, M.-C. Chen, J. Qin, X. Yang, R-Z. Liu, Z.-C. Duan, J.-P. Li, S. Gerhardt, K. Winkler, J. Jurkat, L.-J. Wang, N. Gregersen, Y.-H. Huo, Q. Dai, S. Yu, S. Höfling, C.-Y. Lu, and J.-W. Pan, “Towards optimal single-photon sources from polarized microcavities,” Nat. Photonics 13, 770–775 (2019b).
  * Schnabel  _et al._ (2010) R. Schnabel, N. Mavalvala, D. E. McClelland, and P. K. Lam, “Quantum metrology for gravitational wave astronomy,” Nat. Commun. 1, 121 (2010).
  * Grote  _et al._ (2013) H. Grote, K. Danzmann, K. L. Dooley, R. Schnabel, J. Slutsky, and H. Vahlbruch, “First long-term application of squeezed states of light in a gravitational-wave observatory,” Phys. Rev. Lett. 110, 181101 (2013).
  * Taylor  _et al._ (2013) M. A. Taylor, J. Janousek, V. Daria, J. Knittel, B. Hage, H.-A. Bachor, and W. P. Bowen, “Biological measurement beyond the quantum limit,” Nat. Photonics 7, 229–233 (2013).
  * Hoff  _et al._ (2013) U. B. Hoff, G. I. Harris, L. S. Madsen, H. Kerdoncuff, M. Lassen, B. M. Nielsen, W. P. Bowen, and U. L. Andersen, “Quantum-enhanced micromechanical displacement sensitivity,” Opt. Lett. 38, 1413–1415 (2013).
  * Treps  _et al._ (2003) N. Treps, N. Grosse, W. P. Bowen, C. Fabre, H.-A. Bachor, and P. K. Lam, “A quantum laser pointer,” Science 301, 940–943 (2003).
  * Treps  _et al._ (2002) N. Treps, U. Andersen, B. Buchler, P. K. Lam, A. Maitre, H.-A. Bachor, and C. Fabre, “Surpassing the standard quantum limit for optical imaging using nonclassical multimode light,” Phys. Rev. Lett. 88, 203601 (2002).
  * Wolfgramm  _et al._ (2010) F. Wolfgramm, A. Cerè, F. A. Beduini, A. Predojević, M. Koschorreck, and M. W. Mitchell, “Squeezed-light optical magnetometry,” Phys. Rev. Lett. 105, 053601 (2010).
  * Otterstrom  _et al._ (2014) N. Otterstrom, R. C. Pooser, and B. J. Lawrie, “Nonlinear optical magnetometry with accessible in situ optical squeezing,” Opt. Lett. 39, 6533–6536 (2014).
  * Fan  _et al._ (2015) W. Fan, B. J. Lawrie, and R. C. Pooser, “Quantum plasmonic sensing,” Phys. Rev. A 92, 053812 (2015).
  * Holtfrerich  _et al._ (2016) M. W. Holtfrerich, M. Dowran, R. Davidson, B. J. Lawrie, R. C. Pooser, and A. M. Marino, “Toward quantum plasmonic networks,” Optica 3, 985–988 (2016).
  * Dowran  _et al._ (2018) M. Dowran, A. Kumar, B. J. Lawrie, R. C. Pooser, and A. M. Marino, “Quantum-enhanced plasmonic sensing,” Optica 5, 628–633 (2018).
  * Rodrigo  _et al._ (2010) S. G. Rodrigo, O. Mahboub, A. Degiron, C. Genet, F. J. García-Vidal, L. Martín-Moreno, and Thomas W. Ebbesen, “Holes with very acute angles: A new paradigm of extraordinary optical transmission through strongly localized modes,” Opt. Express 18, 23691–23697 (2010).
  * Lin and Roberts (2011) L. Lin and A. Roberts, “Light transmission through nanostructured metallic films: Coupling between surface waves and localized resonances,” Opt. Express 19, 2626–2633 (2011).
  * Blanchard-Dionne  _et al._ (2011) A.-P. Blanchard-Dionne, L. Guyot, S. Patskovsky, R. Gordon, and M. Meunier, “Intensity based surface plasmon resonance sensor using a nanohole rectangular array,” Opt. Express 19, 15041–15046 (2011).
  * Kaya  _et al._ (2013) S. Kaya, J.-C. Weeber, F. Zacharatos, K. Hassan, T. Bernardin, B. Cluzel, J. Fatome, and C. Finot, “Photo-thermal modulation of surface plasmon polariton propagation at telecommunication wavelengths,” Opt. Express 21, 22269–22284 (2013).
  * Robinson  _et al._ (2014) H. D. Robinson, B. A. Magill, X. Guo, R. L. Reyes, E. M. See, R. M. Davis, and W. L. Santos, “Two-photon activation of o-nitrobenzyl ligands bound to gold surfaces,” Proc. SPIE 9163, 916336 (2014).
  * Bartczak  _et al._ (2011) D. Bartczak, O. L. Muskens, T. M. Millar, T. Sanchez-Elsner, and A. G. Kanaras, “Laser-induced damage and recovery of plasmonically targeted human endothelial cells,” Nano Lett. 11, 1358 (2011).
  * Kalashnikov  _et al._ (2014) D. A. Kalashnikov, Z. Pan, A. I. Kuznetsov, and L. A. Krivitsky, “Quantum spectroscopy of plasmonic nanostructures,” Phys. Rev. X 4, 011049 (2014).
  * Altewischer  _et al._ (2002) E. Altewischer, M. V. Exter, and J. P. Woerdman, “Plasmon-assisted transmission of entangled photons,” Nature 418, 304–306 (2002).
  * Magaña-Loaiza and Boyd (2019) O. S. Magaña-Loaiza and R. W. Boyd, “Quantum imaging and information,” Rep. Prog. Phys. 82, 124401 (2019).
  * Moreau  _et al._ (2019) P.-A. Moreau, E. Toninelli, T. Gregory, and M. J. Padgett, “Imaging with quantum states of light,” Nat. Rev. Phys. 1, 367–380 (2019).
  * Kumar  _et al._ (2017) A. Kumar, H. Nunley, and A. M. Marino, “Observation of spatial quantum correlations in the macroscopic regime,” Phys. Rev. A 95, 053849 (2017).
  * Kumar and Marino (2019) A. Kumar and A. M. Marino, “Spatial squeezing in bright twin beams generated with four-wave mixing: Constraints on characterization with an electron-multiplying charge-coupled-device camera,” Phys. Rev. A 100, 063828 (2019).
  * Li  _et al._ (2020b) F. Li, T. Li, and G. S. Agarwal, “Temporal quantum noise reduction acquired by an electron-multiplying charge-coupled-device camera,” Opt. Express 28, 37538–37545 (2020b).
  * Bozhevolnyi  _et al._ (2006) S. I. Bozhevolnyi, V. S. Volkov, E. Devaux J.-Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature 440, 508–511 (2006).
  * Tassin  _et al._ (2012) P. Tassin, T. Koschny, M. Kafesaki, and C. M. Soukoulis, “A comparison of graphene, superconductors and metals as conductors for metamaterials and plasmonics,” Nat. Photonics 6, 259–264 (2012).
  * Grigorenko  _et al._ (2012) A. Grigorenko, M. Polini, and K. Novoselov, “Graphene plasmonics,” Nat. Photonics 6, 749–758 (2012).
  * Ozbay (2006) E. Ozbay, “Plasmonics: Merging photonics and electronics at nanoscale dimensions,” Science 311, 189–193 (2006).
  * Sorger  _et al._ (2012) V. J. Sorger, R. F. Oulton, R.-M. Ma, and X. Zhang, “Toward integrated plasmonic circuits,” MRS Bulletin 37, 728–738 (2012).
  * Fang and Sun (2015) Y. Fang and M. Sun, “Nanoplasmonic waveguides: Towards applications in integrated nanophotonic circuits,” Light Sci. Appl 4, e294 (2015).
  * Salaita  _et al._ (2007) K. Salaita, Y. Wang, and C. A. Mirkin, “Applications of dip-pen nanolithography,” Nat. Nanotechnol. 2, 145–155 (2007).
  * Leggett (2012) G. J. Leggett, “Light-directed nanosynthesis: Near-field optical approaches to integration of the top-down and bottom-up fabrication paradigms,” Nanoscale 4, 1840–1855 (2012).
  * Heeres  _et al._ (2013) R. W. Heeres, L. P. Kouwenhoven, and V. Zwiller, “Quantum interference in plasmonic circuits,” Nat. Nanotechnol. 8, 719–722 (2013).
  * Martino  _et al._ (2014) G. Di Martino, Y. Sonnefraud, M. S. Tame, S. Kéna-Cohen, F. Dieleman, Ş. K. Özdemir, M. S. Kim, and S. A. Maier, “Observation of quantum interference in the plasmonic hong-ou-mandel effect,” Phys. Rev. Applied 1, 034004 (2014).
  * Fakonas  _et al._ (2014) J. S. Fakonas, H. Lee, Y. A. Kelaita, and H. A. Atwater, “Two-plasmon quantum interference,” Nat. Photonics 8, 317–320 (2014).
  * Cai  _et al._ (2014) Y.-J. Cai, M. Li, X.-F. Ren, C.-L. Zou, X. Xiong, H.-L. Lei, B.-H. Liu, G.-P. Guo, and G.-C. Guo, “High visibility on-chip quantum interference of single surface plasmons,” Phys. Rev. Applied 2, 014004 (2014).
  * Fujii  _et al._ (2014) G. Fujii, D. Fukuda, and S. Inoue, “Direct observation of bosonic quantum interference of surface plasmon polaritons using photon-number-resolving detectors,” Phys. Rev. B 90, 085430 (2014).
  * Fakonas  _et al._ (2015) J. S. Fakonas, A. Mitskovets, and H. A. Atwater, “Path entanglement of surface plasmons,” New J. Phys. 17, 023002 (2015).
  * Vest  _et al._ (2018) B. Vest, I. Shlesinger, M.-C. Dheur, E. Devaux, J.-J. Greffet, G. Messin, and F. Marquier, “Plasmonic interferences of two-particle n00n states,” New J. Phys. 20, 053050 (2018).
  * Dieleman  _et al._ (2017) F. Dieleman, M. S. Tame, Y. Sonnefraud, M. S. Kim, and S. A. Maier, “Experimental verification of entanglement generated in a plasmonic system,” Nano Lett. 17, 7455–7461 (2017).
  * Nagata  _et al._ (2007) T. Nagata, R. Okamoto, J. L. O’Brien, K. Sasaki, and S. Takeuchi, “Beating the standard quantum limit with four-entangled photons,” Science 316, 726–729 (2007).
  * Thekkadath  _et al._ (2020) G. S. Thekkadath, M. E. Mycroft, B. A. Bell, C. G. Wade, A. Eckstein, D. S. Phillips, R. B. Patel, A. Buraczewski, A. E. Lita, T. Gerrits, S. W. Nam, M. Stobińska, A. I. Lvovsky, and I. A. Walmsley, “Quantum-enhanced interferometry with large heralded photon-number states,” npj Quantum Inf. 6, 89 (2020).
  * Kok  _et al._ (2002) P. Kok, H. Lee, and J. P. Dowling, “Creation of large-photon-number path entanglement conditioned on photodetection,” Phys. Rev. A 65, 052104 (2002).
  * Lawrie  _et al._ (2019) B. J. Lawrie, P. D. Lett, A. M. Marino, and R. C. Pooser, “Quantum sensing with squeezed light,” ACS Photonics 6, 1307–1318 (2019).
  * Hümmer  _et al._ (2013) T. Hümmer, F. J. García-Vidal, L. Martín-Moreno, and D. Zueco, “Weak and strong coupling regimes in plasmonic qed,” Phys. Rev. B 87, 115419 (2013).
  * Kristensen and Hughes (2014) P. T. Kristensen and S. Hughes, “Modes and mode volumes of leaky optical cavities and plasmonic nanoresonators,” ACS Photonics 1, 2–10 (2014).
  * Lodahl  _et al._ (2015) P. Lodahl, S. Mahmoodian, and S. Stobbe, “Interfacing single photons and single quantum dots with photonic nanostructures,” Rev. Mod. Phys. 87, 347–400 (2015).
  * Lawrie  _et al._ (2012) B. J. Lawrie, K.-W. Kim, D. P. Norton, and Jr. R. F. Haglund, “Plasmon-exciton hybridization in zno quantum-well al nanodisc heterostructures,” Nano Lett. 12, 6152–6157 (2012).
  * Gonçalves  _et al._ (2020b) P. A. D. Gonçalves, T. Christensen, N. Rivera, A.-P. Jauho, N. Asger Mortensen, and M. Soljačić, “Plasmon-emitter interaction at the nanoscale,” Nat. Commun. 11, 366 (2020b).
  * Buckley  _et al._ (2012) S. Buckley, K. Rivoire, and J. Vučković, “Engineered quantum dot single-photon sources,” Rep. Prog. Phys. 75, 126503 (2012).
  * Aharonovich  _et al._ (2011) I. Aharonovich, S. Castelletto, D. A. Simpson, C.-H. Su, A. D. Greentree, and S. Prawer, “Diamond-based single-photon emitters,” Rep. Prog. Phys. 74, 076501 (2011).
  * Hatef  _et al._ (2013) A. Hatef, S. M. Sadeghi, É. Boulais, and M. Meunier, “Quantum dot-metallic nanorod sensors via exciton-plasmon interaction,” Nanotechnology 24, 015502 (2013).
  * Sadeghi  _et al._ (2013a) S. M. Sadeghi, B. Hood, K. D. Patty, and C.-B. Mao, “Theoretical investigation of optical detection and recognition of single biological molecules using coherent dynamics of exciton-plasmon coupling,” J. Phys. Chem. C 117, 17344–17351 (2013a).
  * Sadeghi  _et al._ (2013b) S. M. Sadeghi, A. Hatef, and M. Meunier, “Quantum detection and ranging using exciton-plasmon coupling in coherent nanoantennas,” Appl. Phys. Lett. 102, 203113 (2013b).
  * Qian  _et al._ (2020) Z. Qian, J. Ren, F. Zhang, X. Duan, Q. Gong, and Y. Gu, “Nanoscale quantum plasmon sensing based on strong photon-exciton coupling,” Nanotechnology 31, 125001 (2020).
  * Nechepurenko  _et al._ (2018) I. A. Nechepurenko, E. S. Andrianov, A. A. Zyablovsky, A. V. Dorofeenko, A. A. Pukhov, and Yu. E. Lozovik, “Absorption sensor based on graphene plasmon quantum amplifier,” Phys. Rev. B 98, 075411 (2018).
  * Kongsuwan  _et al._ (2019) N. Kongsuwan, X. Xiong, P. Bai, J.-B. You, C. E. Png, L. Wu, and O. Hess, “Quantum plasmonic immunoassay sensing,” Nano Lett. 19, 5853–5861 (2019).
  * Mauranyapin  _et al._ (2017) N. P. Mauranyapin, L. S. Madsen, M. A. Taylor, M. Waleed, and W. P. Bowen, “Evanescent single-molecule biosensing with quantum-limited precision,” Nat. Photonics 11, 477–481 (2017).
  * Nefedkin  _et al._ (2020) Nikita Nefedkin, Andrea Alú, and Alex Krasnok, “Quantum embedded superstates,” arXiv , 2007.13863 (2020).
  * Chen  _et al._ (2010b) Y. Chen, T. R. Nielsen, N. Gregersen, P. Lodahl, and J. Mørk, “Finite-element modeling of spontaneous emission of a quantum emitter at nanoscale proximity to plasmonic waveguides,” Phys. Rev. B 81, 125431 (2010b).
  * Jahromi (2018) H. Rangani Jahromi, “Parameter estimation in plasmonic qed,” Opt. Commun. 411, 119–125 (2018).
  * Balasubramanian  _et al._ (2008) G. Balasubramanian, I. Chan, R. Kolesov, M. Al-Hmoud, J. Tisler, C. Shin, C. Kim, A. Wojcik, P. R. Hemmer, A. Krueger, T. Hanke, A. Leitenstorfer, R. Bratschitsch, F. Jelezko, and J. Wrachtrup, “Nanoscale imaging magnetometry with diamond spins under ambient conditions,” Nature 455, 648–651 (2008).
  * Schirhagl  _et al._ (2014) R. Schirhagl, K. Chang, M. Loretz, and C. L. Degen, “Nitrogen-vacancy centers in diamond: Nanoscale sensors for physics and biology,” Annu. Rev. Phys. Chem. 65, 83–105 (2014).
  * Dolde  _et al._ (2011) F. Dolde, H. Fedder, M. W. Doherty, T. Nöbauer, F. Rempp, G. Balasubramanian, T. Wolf, F. Reinhard, L. C. L. Hollenberg, F. Jelezko, and J. Wrachtrup, “Electric-field sensing using single diamond spins,” Nat. Phys. 7, 459–463 (2011).
  * Shalaginov  _et al._ (2020) M. Y. Shalaginov, S. I. Bogdanov, A. S. Lagutchev, A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “On-chip single-layer integration of diamond spins with microwave and plasmonic channels,” ACS Photonics 7, 2018–2026 (2020).
  * Kim  _et al._ (2020) Laura Kim, Hyeongrak Choi, Matthew Trusheim, and Dirk Englund, “Absorption-based diamond spin microscopy on a plasmonic quantum metasurface,” arXiv , 2011.04885 (2020).
  * Arrad  _et al._ (2014) G. Arrad, Y. Vinkler, D. Aharonov, and A. Retzker, “Increasing sensing resolution with error correction,” Phys. Rev. Lett. 112, 150801 (2014).
  * Kessler  _et al._ (2014) E. M. Kessler, I. Lovchinsky, A. O. Sushkov, and M. D. Lukin, “Quantum error correction for metrology,” Phys. Rev. Lett. 112, 150802 (2014).
  * Cohen  _et al._ (2016) L. Cohen, Y. Pilnyak, D. Istrati, A. Retzker, and H. S. Eisenberg, “Demonstration of quantum error correction for enhanced sensitivity of photonic measurements,” Phys. Rev. A 94, 012324 (2016).
  * Layden  _et al._ (2019) D. Layden, S. Zhou, P. Cappellaro, and L. Jiang, “Ancilla-free quantum error correction codes for quantum metrology,” Phys. Rev. Lett. 122, 040502 (2019).
  * Shettell  _et al._ (2021) Nathan Shettell, William J. Munro, Damian Markham, and Kae Nemoto, “Practical limits of error correction for quantum metrology,” arXiv , 2101.02823 (2021).
  * Wang  _et al._ (2019c) W. Wang, Y. Wu, Y. Ma, W. Cai, L. Hu, X. Mu, Y. Xu, Z.-J. Chen, H. Wang, Y. P. Song, H. Yuan, C.-L. Zou, L.-M. Duan, and L. Sun, “Heisenberg limited single-mode quantum metrology,” Nat. Commun. 10, 4382 (2019c).
  * Shettell and Markham (2020) N. Shettell and D. Markham, “Graph states as a resource for quantum metrology,” Phys. Rev. Lett. 124, 110502 (2020).
  * Ouyang  _et al._ (2019) Y. Ouyang, N. Shettell, and D. Markham, “Robust quantum metrology with explicit symmetric states,” arXiv , 1908.02378 (2019).
  * Consales  _et al._ (2020b) Marco Consales, Giuseppe Quero, Sara Spaziani, Maria Principe, Alberto Micco, Vincenzo Galdi, Antonello Cutolo, and Andrea Cusano, “Metasurface-enhanced lab-on-fiber biosensors,” Laser Photonics Rev. 14, 2000180 (2020b).
  * Rodrigo  _et al._ (2015) D. Rodrigo, O. Limaj, D. Janner, D. Etezadi, F. Javier García de Abajo, V. Pruneri, and H. Altug, “Mid-infrared plasmonic biosensing with graphene,” Science 349, 165–168 (2015).
  * Heydari and Samiei (2020) M. B. Heydari and M. H. V. Samiei, “A short review of plasmonic graphene-based resonators: Recent advances and prospects,” arXiv , 2011.14767 (2020).
  * Das  _et al._ (2015) S. Das, J. A. Robinson, M. Dubey, H. Terrones, and M. Terrones, “Beyond graphene: Progress in novel two-dimensional materials and van der waals solids,” Annu. Rev. Mater. Res. 45, 20.1–20.27 (2015).
  * Zeng  _et al._ (2018) M. Zeng, Y. Xiao, J. Liu, K. Yang, and L. Fu, “Exploring two-dimensional materials toward the next-generation circuits: From monomer design to assembly control,” Chem. Rev. 118, 6236–6296 (2018).
  * Mello and Kubota (2002) L. D. Mello and L. T. Kubota, “Review of the use of biosensors as analytical tools in the food and drink industries,” Food Chem. 77, 237–256 (2002).
  * Kauranen and Zayats (2012) M. Kauranen and A. V. Zayats, “Nonlinear plasmonics,” Nat. Photonics 6, 737–748 (2012).
  * Raza  _et al._ (2015) S. Raza, S. I. Bozhevolnyi, M. Wubs, and N. A. Mortensen, “Nonlocal optical response in metallic nanostructures,” J. Phys.: Condens. Matter 27, 183204 (2015).
  * Zhan  _et al._ (2018) C. Zhan, X.-J. Chen, J. Yi, J.-F. Li, D.-Y. Wu, and Z.-Q. Tian, “From plasmon-enhanced molecular spectroscopy to plasmon-mediated chemical reactions,” Nat. Rev. Chem. 2, 216–230 (2018).
  * Matsumoto (2002) K. Matsumoto, “A new approach to the cramér-rao-type bound of the pure-state model,” J. Phys. A: Math. Gen. 35, 3111–3123 (2002).
  * Uhlmann (1976) A. Uhlmann, “The “transition probability” in the state space of a $*$-algebra,” Rep. Math. Phys. 9, 273–279 (1976).
  * Jozsa (1994) R. Jozsa, “Fidelity for mixed quantum states,” J. Mod. Opt. 41, 2315–2323 (1994).
  * Braunstein and van Loock (2005) S. L. Braunstein and P. van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77, 513–577 (2005).
  * Weedbrook  _et al._ (2012) C. Weedbrook, S. Pirandola, R. Garcia-Patron, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621 (2012).
  * Marian and Marian (2012) P. Marian and T. A. Marian, “Uhlmann fidelity between two-mode gaussian states,” Phys. Rev. A 86, 022340 (2012).
  * Giedke and Cirac (2002) G. Giedke and J. I. Cirac, “Characterization of gaussian operations and distillation of gaussian states,” Phys. Rev. A 66, 032316 (2002).
  * Porat and Friedlander (1986) B. Porat and B. Friedlander, “Computation of the exact information matrix of gaussian time series with stationary random components,” IEEE Trans. Acoust. Speech Signal Process. 34, 118–130 (1986).


# Saturated absorption spectroscopy of buffer-gas-cooled Barium monofluoride
molecules

Wenhao Bu  Yuhe Zhang  Qian Liang  Tao Chen  Interdisciplinary Center of
Quantum Information, State Key Laboratory of Modern Optical Instrumentation,
and Zhejiang Province Key Laboratory of Quantum Technology and Device of
Physics Department, Zhejiang University, Hangzhou 310027, China  Bo Yan
<EMAIL_ADDRESS>Interdisciplinary Center of Quantum Information, State
Key Laboratory of Modern Optical Instrumentation, and Zhejiang Province Key
Laboratory of Quantum Technology and Device of Physics Department, Zhejiang
University, Hangzhou 310027, China

###### Abstract

We report an experimental investigation on the Doppler-free saturated
absorption spectroscopy of buffer-gas-cooled Barium monofluoride (BaF)
molecules in a 4 K cryogenic cell. The obtained spectra with a resolution of
19 MHz, much smaller than previously observed in absorption spectroscopy,
clearly resolve the hyperfine transitions. Moreover, we use these high-
resolution spectra to fit the hyperfine splittings of excited $A(v=0)$ state
and find the hyperfine splitting of the laser-cooling-relevant
$A^{2}\Pi_{1/2}(v=0,J=1/2,+)$ state is about 18 MHz, much higher than the
previous theoretically predicted value. This provides important missing
information for laser cooling of BaF molecules.

##  I introduction

Cold molecules, due to the additional internal degrees of freedom and the
long-range dipolar interactions, provide novel potential applications and
sophisticated understandings in controlled chemistry reactions Krems (2008);
Ospelkaus  _et al._ (2010); Hu  _et al._ (2019); Liu and Luo (2021), precision
measurement Safronova  _et al._ (2018); Cairncross and Ye (2019); Cairncross
_et al._ (2017); Altuntaş  _et al._ (2018); Kozyryev and Hutzler (2017); Lim
_et al._ (2018); Vutha  _et al._ (2018), quantum simulation and computation
DeMille (2002); Rabl  _et al._ (2006); Andre  _et al._ (2006), and many-body
physics Wang  _et al._ (2006); Buechler  _et al._ (2007). In the last decade,
various cooling techniques have been developed to achieve dense cold molecular
samples, for example, buffer gas cooling Kantrowitz and Grey (1951); Hutzler
_et al._ (2012), Stark and Zeeman deceleration van de Meerakker  _et al._
(2012), and assembling molecules from two ultracold atoms Ni  _et al._ (2008).
Despite these efforts, producing cold molecule samples is limited to several
kinds of species and remains an ongoing challenge.

Direct laser cooling as an alternative route, mainly for those molecules with
a quasi-cycling transition Di Rosa (2004); Stuhl  _et al._ (2008), has been
largely developed to extend the coolable molecular species. In recent years,
the molecules, SrF Barry  _et al._ (2014), CaF Truppe  _et al._ (2017);
Anderegg  _et al._ (2017) and YO Collopy  _et al._ (2018), have been
successfully loaded into three-dimensional magneto-optical traps with
temperatures at few millikelvin, followed by deep laser cooling to achieve
higher phase-space densities and longer coherence time Cheuk  _et al._ (2018);
Caldwell  _et al._ (2019, 2020) taking advantage of effective trapping in
conservative potentials McCarron  _et al._ (2018); Williams  _et al._ (2018);
Anderegg  _et al._ (2018) and grey molasses technique Ding  _et al._ (2020).
Specifically, an optical tweezer array of CaF molecules and further studies of
cold collisions of two single molecules have been reported recently Anderegg
_et al._ (2019); Cheuk  _et al._ (2020). Moreover, one-dimensional laser
cooling of YbF Lim  _et al._ (2018) and polyatomic molecules, such as, SrOH
Kozyryev  _et al._ (2017), CaOH Baum  _et al._ (2020), YbOH Augenbraun  _et
al._ (2020) and CaOCH3 Mitra  _et al._ (2020), have also been realized, making
the quantum state manipulation and potential applications be feasible. Besides
the above mentioned ones, other species including MgF Xu  _et al._ (2016); Yan
_et al._ (2022), BaH Tarallo  _et al._ (2016), TlF Norrgard  _et al._ (2017),
AlF Truppe  _et al._ (2019) and AlCl Daniel  _et al._ (2021) have also
attracted great interests.

Figure 1: (color online) A schematic illustration of the experimental setup.
The three layers from outside to inside are vacuum chamber, $30~{}{\rm K}$
shield and $4~{}{\rm K}$ shield, respectively. The BaF molecules are produced
by laser ablation with the Nd: YAG laser in a cell, which is attached to a
$4~{}{\rm K}$ cryogenic head to perform effective buffer gas cooling. The
relative position of the BaF2 target and the three laser beams for the
spectroscopy detections are also shown. The propagation directions of the
probe, pump, and reference beams are indicated by the arrows.

The BaF molecule which can be potentially laser-cooled Chen  _et al._ (2016)
has also received continuous attentions in recent years Bu  _et al._ (2017);
Chen  _et al._ (2017); Aggarwal  _et al._ (2018); Cournol  _et al._ (2018);
Albrecht  _et al._ (2020); Liang  _et al._ (2021). A precise measurement of
the relevant transitions is consequently highly required. Usually, the main
cooling transition is chosen as $X^{2}\Sigma_{1/2}(v=0,N=1,-)\to
A^{2}\Pi_{1/2}(v=0,J=1/2,+)$. The hyperfine structure of the
$X^{2}\Sigma_{1/2}(v=0,N=1,-)$ state has been recognized with high-resolution
microwave spectroscopy Ernst  _et al._ (1986). But the hyperfine structure of
the excited $A^{2}\Pi_{1/2}(v=0,J=1/2,+)$ state has not yet been resolved.

The resolution of the BaF absorption spectroscopy is limited by the Doppler
broadening Bu  _et al._ (2017); Albrecht  _et al._ (2020). One way to get a
better resolution is performing the saturated absorption spectrum, which has
been widely used in atomic cases Hänsch  _et al._ (1971); Wieman and Hänsch
(1976); Nakayama (1985). This has also been applied for YbF Skoff  _et al._
(2009), where a resolution of 30 MHz has been reported. Here we extend such a
method to the BaF molecule, and further resolve the unknown hyperfine
splittings of the excited states.

The paper is organized as follows. In Sec.II, a brief description of the
experimental setup for the saturated absorption spectroscopy is given. In
Sec.III, we analyze the observed spectra for hyperfine transitions in
different branchings and fit them with multi-peak Gaussian function to yield
the relevant parameters. Especially, for the main laser-cooling transition
$X^{2}\Sigma_{1/2}(v=0,N=1,-)\leftrightarrow A^{2}\Pi_{1/2}(v=0,J=1/2,+)$, the
fitting shows the hyperfine splitting of the excited state is about $18~{}{\rm
MHz}$. Finally, we conclude in Sec. IV.

##  II experimental setup

To overcome the Doppler broadening limit and get more knowledge of the laser-
cooling relevant transition, we set up a saturated absorption spectroscopy
experiment for BaF molecules. The previous experiment on YbF in Ref. Skoff
_et al._ (2009) used this technique with a temperature of 10 K in a liquid
helium cryostat. Our experiment here is performed with buffer-gas cooled BaF
molecules, under a lower temperature at 4 K produced by a pulse-tube cryostat.

The cryogenic apparatus is built around a pulse tube refrigerator machine
(Sumitomo, ${\rm SRP-082B-F70H}$), which has two temperature stages:
$30~{}{\rm K}$ and $4~{}{\rm K}$. As shown in Fig. 1 (a more detailed drawing
of the cryogenic apparatus could be found in Ref. Bu  _et al._ (2017)), the
cell in which laser ablation is performed is installed below the 4 K cryogenic
head, and surrounded by the 30 K and 4 K shieldings to block the blackbody
radiation. The buffer gas is injected into the cell from the inlet on the back
face. The BaF molecules are produced by ablating the BaF2 target with a 532-nm
pulsed ND: YAG laser (the repetition rate of the pulse is $1~{}{\rm Hz}$).

To reduce the systematic errors, we use three laser beams in our experiment,
denoted as the pump, probe, and reference beams in Fig. 1, respectively. These
beams are split from an external-cavity semiconductor laser (ECDL), all with
diameters of $\sim 1~{}{\rm mm}$. The intensities of the probe and reference
beams are set to be nearly equal, at $\sim 20~{}{\rm mW/cm^{2}}$; while the
pump beam is $\sim 80~{}{\rm mW/cm^{2}}$. We let the probe and the reference
beams parallelly enter the cell, while the pump beam counter-propagates from
the opposite direction and overlaps with the probe laser in the cell region.
The laser frequency is stabilized by a transfer cavity with the linewidth less
than $2~{}{\rm MHz}$ Wang  _et al._ (2018). The absorption signals of the
probe and the reference beams are monitored by two photodiodes; see Fig. 1. We
scan the laser frequency with a step of $1~{}{\rm MHz}$, ranging typically
several hundreds of ${\rm MHz}$. To obtain a high signal-to-noise ratio, we
average the signals by repeating the scanning hundreds of times.

##  III results and analysis

Figure 2: (color online) The saturated absorption spectroscopy for the main
laser cooling transition, $X(v=0,N=1,-)\leftrightarrow A(v=0,J=1/2,+)$, of the
BaF molecule. (a) shows the Doppler-broadened absorption spectra (red dots,
from the reference beam) and saturated absorption spectra (blue dots, from the
probe beam). The solid lines are Gaussian fittings to the two spectra,
respectively. (b) The Doppler-free saturated absorption spectra. The gray
solid line is the fitted curve. Each peak in our model has been plotted
individually, the color is matched with that in the inset. The positions of
each hyperfine level in the $X(v=0,N=1,-)$ are marked by the vertical black
dashed lines. The inset shows the corresponding transitions and their
branching ratios. The transition $F=1(J=3/2)\to F^{\prime}=1$ is marked by
gray, for its branching ratio is negligible.

Absorption signal: A typical time-dependent absorption signal monitored by the
photodiode is shown in the inset of Fig. 2. The ablation laser fires at
$t=0~{}{\rm ms}$, and the absorption signal initially rapidly increases and
then exponentially decays. The absorption fraction $A$ is estimated by making
a time average of the signal curve from $1.5~{}{\rm ms}$ to $10~{}{\rm ms}$
[the shadow region in the inset of Fig. 2(a)]. We use this time window to
avoid the influence of transient heating of the buffer gas by ablation laser.
Then the strength of the absorption can be described by the optical depth
$\alpha$ which is related to $A$ as

$\alpha=-\ln(1-A)$ (1)

By scanning the laser frequency, we record the $\alpha$’s for both the
reference beam and the probe beam as shown in Fig. 2(a).

Laser cooling relevant transition: The absorption spectra for the two beams
near-resonant with the main laser cooling transition
[$X(v=0,N=1,-)\leftrightarrow A(v=0,J^{\prime}=1/2,+)$] are shown in Fig.
2(a). Both of them are Doppler-broadened, and only the fine structure of the
ground state can be resolved (the two peaks), similar to our previous
observation Bu  _et al._ (2017). The difference lies in that the spectra from
the probe beam show saturated absorption features, i.e., the hole-burning
phenomena at the positions of each hyperfine transition. We first rescale the
spectra from the probe beam, and then make a subtraction with the two spectra
sets to get the Doppler-free saturated absorption spectra in Fig. 2(b), where
the hyperfine sublevels are resolved.

The width of the Doppler-broadened spectra of the reference beam in Fig. 2(a)
reflects the translational temperatures of the buffer-gas-cooled molecules. We
fit the spectra with a four-peak Gaussian function [the peak positions are
fixed with the values in Fig. 2(b)]. The averaged full width at half maximum
(FWHM) is about $54(4)~{}{\rm MHz}$, which corresponds to a temperature of
$\sim 6.5~{}{\rm K}$. It is higher than the cell temperature at $4~{}{\rm K}$,
which might be caused by the initial heating of the buffer gas when the
ablation laser fires. A similar phenomenon also has been observed in other
buffer gas experiments Barry  _et al._ (2011).

Next, let us analyze the saturated absorption spectra in Fig. 2(b). According
to the selection rules, there are six allowed transitions in our sub-doppler
spectrum, and the branch ratios are taken from Chen  _et al._ (2019); Albrecht
_et al._ (2020), as shown in Table 1. Here, no bias magnetic field is applied
in the cell (the earth’s magnetic field may exist, but small), so we make an
assumption that the Zeeman substates are degenerate. With this, the normalized
branching ratios are shown in Fig. 2(b).

Table 1:  Calculated hyperfine branching ratios for decays from $|A,J^{\prime}=1/2,+\rangle$ state to $|X,N=1,-\rangle$ state, also see Chen  _et al._ (2019); Albrecht  _et al._ (2020). |  |  |  $F^{\prime}=0~{}$ |  | $F^{\prime}=1$ |
---|---|---|---|---|---|---
$J$ | $F$ | $m_{F}$ | $m^{\prime}_{F}=0$ | $m^{\prime}_{F}=-1$ | $m^{\prime}_{F}=0$ | $m^{\prime}_{F}=1$
1/2 | 0 | 0 | 0 | 2/9 | 2/9 | 2/9
|  | $-1$ | 0.1282 | 0.2493 | 0.2493 | 0
1/2 | 1 | 0 | 0.1282 | 0.2493 | 0 | 0.2493
|  | 1 | 0.1282 | 0 | 0.2493 | 0.2493
|  | -1 | 0.2051 | 0.0007 | 0.0007 | 0
3/2 | 1 | 0 | 0.2051 | 0.0007 | 0 | 0.0007
|  | 1 | 0.2051 | 0 | 0.0007 | 0.0007
|  | -2 | 0 | 1/6 | 0 | 0
|  | -1 | 0 | 1/12 | 1/12 | 0
3/2 | 2 | 0 | 0 | 1/36 | 1/9 | 1/36
|  | 1 | 0 | 0 | 1/12 | 1/12
|  | 2 | 0 | 0 | 0 | 1/6

We set the hyperfine splitting between $F^{\prime}=1$ and $F^{\prime}=0$ of
the $A(v=0,J^{\prime}=1/2,+)$ to be $\Delta$, as shown in the inset of Fig.
2(b). The hyperfine splittings of the $X(v=0,N=1,-)$ are taken from the high-
resolution microwave spectra Ernst  _et al._ (1986). Besides, we assumed that
the width of each Gaussian peak is the same, and the relative height of each
peak is proportional to its corresponding branch ratio, so the fitting formula
can be expressed as:

$\begin{split}\eta[&0.3739e^{-[(f-95)/w]^{2}}+0.1667e^{-[(f-67)/w]^{2}}+\\\
&0.1538e^{-[(f+23+\Delta)/w]^{2}}+0.2083e^{-[(f+57)/w]^{2}}+\\\
&0.0962e^{-[(f-95+\Delta)/w]^{2}}+0.0011e^{-[(f+23)/w]^{2}}]\end{split}$ (2)

But because one of the transition (
$|J=3/2,~{}F=1\rangle\to|F^{\prime}=1\rangle$ ) has very small branch ratio,
it is actually more like a five-Gaussian function. The best-fitting parameters
are: $w=11.4(6)$, $\Delta=17.7(2.1)$. The fitting curve and the experimental
data agree quite well as shown in Fig. 2(b).

The parameter $w=11.4(6)$ corresponds to an FWHM of $19(1){\rm MHz}$ for the
transitions observed in our sub-Doppler spectrum. There are several potential
broadening mechanisms in our experiment that contribute to this line-width.
The main one is the power broadening from the strong pump beam, which can be
estimated by the formula, $\Gamma\sqrt{1+s_{p}}$, with the $A^{2}\Pi_{1/2}$
state linewidth $\Gamma/2\pi=2.84~{}{\rm MHz}$ and $s_{p}$ the saturation
parameter. In our experiment, $s_{p}\approx 45$ leads to a broadening of $\sim
18~{}{\rm MHz}$, in agreement with our observation. The collisional broadening
also plays a role, although less important, in our experiment. A He flow-rate
of 2 SCCM corresponds to a density of $1.5\times 10^{15}~{}{\rm cm^{-3}}$ in
the cell, and the collisional cross-section has been measured at an order of
$10^{-14}~{}{\rm cm^{2}}$ Bu  _et al._ (2017); Albrecht  _et al._ (2020). With
these, we estimate the collisional broadening to be $\sim 6~{}{\rm MHz}$.
Other negligible broadening mechanisms include the residual Doppler-broadening
due to imperfect alignment of the probe and the pump beams, Zeeman broadening
from the Earth’s magnetic field (less than $1~{}{\rm MHz}$), and the transit
time broadening.

The parameter $\Delta=17.7(2.1)$ gives the hyperfine splitting of the the
$A(v=0,J^{\prime}=1/2,+)$ state. The key reason we can resolve such a small
splitting lies in that the transition from
$|J=3/2,~{}F=1\rangle\to|F^{\prime}=1\rangle$ is almost forbidden, making the
peak of $|J=3/2,~{}F=1\rangle\to|F^{\prime}=0\rangle$ well isolated, as shown
in Fig. 2 (b). Unlike other monofluorides, such as CaF and SrF, the upper-
state hyperfine splittings of which are only a few MHz, this value of BaF is
relatively large. Such a large splitting definitely affects the laser cooling
efficiency and should be carefully considered in the preparation of the
relevant laser frequencies and cooling scheme. The cooling laser used in our
BaF Doppler cooling experiment has already considered such splitting Zhang
_et al._ (2022).

Figure 3: (color online) The hyperfine splitting $\Delta$ of states in
$A^{2}\Pi_{1/2}(v=0)$ measurement: (a) the schematic illustration for the
$P_{1}$ and ${}^{Q}R_{12}$ branch transitions, which we used to measure the
splitting $\Delta$. (b) the typical data measured by the $P_{1}$ branch
[$X(v=0,J=N+1/2)\leftrightarrow A(v=0,J^{\prime}=N-1/2)$], the black dashed
lines marks the positions of the ground states. (c) the typical data measured
by ${}^{Q}R_{12}$ branch [$X(v=0,J=N-1/2)\leftrightarrow
A(v=0,J^{\prime}=N+1/2)$], the positions of the ground states is also marked
by the black dashed lines. (d) Distribution of the measured $\Delta$ in
different $J^{\prime}$ state, the red and blue dots are obtained from
$P_{1}(N)$ and ${}^{Q}R_{12}(N)$ branches, respectively. The solid lines show
the fitting curves accordingly.

Other transitions with high $J^{\prime}$: We further extend our measurement to
the hyperfine splittings $\Delta$ of states with a higher value of
$J^{\prime}$ in $A^{2}\Pi_{1/2}(v=0)$. We measure the $P_{1}$ and
${}^{Q}R_{12}$ branch transitions, then use a similar fitting method as
described above, we could obtain the $\Delta$’s for each $J^{\prime}$ state.
The transitions of these two branches are illustrated in Fig. 3(a). According
to the illustration and the selection rules, there are three allowed hyperfine
transitions for a $P_{1}$ transition ($J\to J^{\prime}=J-1$). We consequently
set the fitting function as a three-Gaussian model

$\begin{split}\eta\\{r_{N+1,N}&e^{-(f/w)^{2}}+r_{N,N}e^{-[(f-\Delta_{0})/w]^{2}}+\\\
&r_{N,N-1}e^{-[(f-\Delta_{0}+\Delta)/w]^{2}}\\}\end{split}$ (3)

Similarly, the ${}^{Q}R_{12}$ branch spectrum should be fitted with the
formula:

$\begin{split}\eta\\{r_{N-1,N}&e^{-(f/w)^{2}}+r_{N,N}e^{-[(f-\Delta_{0})/w]^{2}}+\\\
&r_{N,N+1}e^{-[(f-\Delta_{0}-\Delta)/w]^{2}}\\}\end{split}$ (4)

where $r_{ij}$ is the branching ratio of the transition from $F=i$ to
$F^{\prime}=j$, and $\Delta_{0}$ is the relative position of the hyperfine
level of the ground state. We set the state $F=N+1$ and $F=N-1$ to be at the
zero position for $P_{1}$ and ${}^{Q}R_{12}$ branches, respectively. The
$r_{ij}$ and $f$ can be calculated by using the molecular constants of BaF
measured in Ryzlewicz and Törring (1980); Ernst  _et al._ (1986). The other
parameters $\eta,w$, and $\Delta$ are fitting parameters, and here we are only
concerned about $\Delta$.

Fig. 3(b) and Fig. 3(c) show the typical data measured in our experiment,
corresponding to one transition line for $P_{1}$ branch and ${}^{Q}R_{12}$
branch, respectively. Note that we only observed two transitions in the
experiment for both branches, although we have mentioned above that there
should be three transitions. This is because the branching ratio of $r_{N,N}$
in both $P_{1}$ and $Q^{R}_{12}$ transitions is negligible. For sufficiently
large $J^{\prime}$, there is a quasi-selection rule
$F^{\prime}-F=J^{\prime}-J$, which makes the transitions reduce to two. For
the two transitions observed in the spectra, due to the hyperfine splitting
$\Delta$ of the excited state, the distance between the two observed
transition peaks is shifted from the interval of the ground states by
$\Delta$. As shown in Fig. 3(b) and Fig. 3(c), the distance of two peaks in
$P_{1}$ branch is smaller than the interval of the ground states by $\Delta$,
while oppositely, for ${}^{Q}R{12}$ branch, it becomes larger by $\Delta$.

In Fig. 3(d) we plot the measured $\Delta$. The hyperfine splittering of BaF
of $A^{2}\Pi_{1/2}$ is Sauer  _et al._ (1999)

$\Delta=\left(\frac{A_{\parallel}}{2J^{\prime}+1}+\eta
A_{\perp}\right)\frac{(2J^{\prime}+1)^{2}}{8J^{\prime}(J^{\prime}+1)}$ (5)

where $\eta=-1$ for $J^{\prime}=N-1/2$ states and $\eta=+1$ for
$J^{\prime}=N+1/2$ states. The best-fit gives $A_{\parallel}=60.4(5.4)~{}$MHz
and $A_{\perp}=-1.4(1.0)$ MHz, and the fitting curves are shown in Fig. 3(d),

Finally, let us discuss the origin of the uncertainty, i.e., the error bars in
Fig. 3(d). The main source is the nonlinear effect of the PZT of the Fabry-
Perot cavity. As described in Ref. Wang  _et al._ (2018); Bu  _et al._ (2016),
we developed a transfer cavity lock system for BaF laser-cooling experiment.
We determine the frequency of our laser from its absorption position in the
Fabry-Perot cavity. When we transform the position to frequency, we assumed
that their relationship is linear. However, there is a slight nonlinear effect
for the PZT of the Fabry-Perot cavity, which makes the linear correspondence
not as accurate as expected. For the measurements of $P_{1}$ and
${}^{Q}R_{12}$ branches, the interval between two peaks is about 30MHz. By
using the results from Ref. Bu  _et al._ (2016), we declare the error due to
the nonlinear effect of PZT is about 1MHz. Another systematic error is the
linewidth of our locked laser. We have measured it in our previous work Wang
_et al._ (2018), which is less than 1MHz. Therefore we estimate that the error
caused by the laser linewidth is 1MHz. The fitting error is 68 % confidence
interval provided by the fit function and already included in Fig. 3(d).
Errors caused by other mechanisms should be negligible in our experiment.

##  IV conclusion

To conclude, we have employed the Doppler-free saturated absorption scheme to
measure the hyperfine splittings of each $J^{\prime}$ in the excited $A(v=0)$
state of the BaF molecule. The information of the hyperfine splittings
provides fundamental foundations for future laser cooling experiments on BaF
molecules. And the knowledge of the hyperfine splittings of the higher
$J^{\prime}$ ($J^{\prime}>1/2$) states is useful to gain more information
about the molecular structures for the BaF molecule.

###### Acknowledgements.

We thank Prof. Eric Hessels for useful discussions, and acknowledge the
support from the National Key Research and Development Program of China under
Grant No. 2018YFA0307200, the National Natural Science Foundation of China
under Grant Nos. U21A20437 and 12074337, Natural Science Foundation of
Zhejiang Province under Grant No. LR21A040002, Zhejiang Province Plan for
Science and technology No. 2020C01019, and the Fundamental Research Funds for
the Central Universities under No.2021FZZX001-02.

## References

  * Krems (2008) R. V. Krems, Phys. Chem. Chem. Phys. 10, 4079 (2008).
  * Ospelkaus  _et al._ (2010) S. Ospelkaus, K.-K. Ni, D. Wang, M. H. G. de Miranda, B. Neyenhuis, G. Quéméner, P. S. Julienne, J. L. Bohn, D. S. Jin, and J. Ye, Science 327, 853 (2010).
  * Hu  _et al._ (2019) M.-G. Hu, Y. Liu, D. D. Grimes, Y.-W. Lin, A. H. Gheorghe, R. Vexiau, N. Bouloufa-Maafa, O. Dulieu, T. Rosenband, and K.-K. Ni, Science 366, 1111 (2019).
  * Liu and Luo (2021) Y. Liu and L. Luo, Front. Phys. 16, 42300 (2021).
  * Safronova  _et al._ (2018) M. S. Safronova, D. Budker, D. DeMille, D. F. J. Kimball, A. Derevianko, and C. W. Clark, Rev. Mod. Phys. 90, 025008 (2018).
  * Cairncross and Ye (2019) W. B. Cairncross and J. Ye, Nat. Rev. Phys. 1, 510 (2019).
  * Cairncross  _et al._ (2017) W. B. Cairncross, D. N. Gresh, M. Grau, K. C. Cossel, T. S. Roussy, Y. Ni, Y. Zhou, J. Ye, and E. A. Cornell, Phys. Rev. Lett. 119, 153001 (2017).
  * Altuntaş  _et al._ (2018) E. Altuntaş, J. Ammon, S. B. Cahn, and D. DeMille, Phys. Rev. Lett. 120, 142501 (2018).
  * Kozyryev and Hutzler (2017) I. Kozyryev and N. R. Hutzler, Phys. Rev. Lett. 119, 133002 (2017).
  * Lim  _et al._ (2018) J. Lim, J. R. Almond, M. A. Trigatzis, J. A. Devlin, N. J. Fitch, B. E. Sauer, M. R. Tarbutt, and E. A. Hinds, Phys. Rev. Lett. 120, 123201 (2018).
  * Vutha  _et al._ (2018) A. C. Vutha, M. Horbatsch, and E. A. Hessels, Phys. Rev. A 98, 032513 (2018).
  * DeMille (2002) D. DeMille, Phys. Rev. Lett. 88, 067901 (2002).
  * Rabl  _et al._ (2006) P. Rabl, D. DeMille, J. M. Doyle, M. D. Lukin, R. J. Schoelkopf, and P. Zoller, Phys. Rev. Lett. 97, 033003 (2006).
  * Andre  _et al._ (2006) A. Andre, D. Demille, J. M. Doyle, M. D. Lukin, S. E. Maxwell, P. Rabl, R. J. Schoelkopf, and P. Zoller, Nat. Phys. 2, 636 (2006).
  * Wang  _et al._ (2006) D.-W. Wang, M. D. Lukin, and E. Demler, Phys. Rev. Lett. 97, 180413 (2006).
  * Buechler  _et al._ (2007) H. P. Buechler, A. Micheli, and P. Zoller, Nat. Phys. 3, 726 (2007).
  * Kantrowitz and Grey (1951) A. Kantrowitz and J. Grey, Review of Scientific Instruments 22, 328 (1951).
  * Hutzler  _et al._ (2012) N. R. Hutzler, H.-I. Lu, and J. M. Doyle, Chem. Rev. 112, 4803 (2012).
  * van de Meerakker  _et al._ (2012) S. Y. T. van de Meerakker, H. L. Bethlem, N. Vanhaecke, and G. Meijer, Chem. Rev. 112, 4828 (2012).
  * Ni  _et al._ (2008) K.-K. Ni, S. Ospelkaus, M. H. G. de Miranda, A. Peer, B. Neyenhuis, J. J. Zirbel, S. Kotochigova, P. S. Julienne, D. S. Jin, and J. Ye, Science 322, 231 (2008).
  * Di Rosa (2004) M. Di Rosa, Eur. Phys. J. D 31, 395 (2004).
  * Stuhl  _et al._ (2008) B. K. Stuhl, B. C. Sawyer, D. Wang, and J. Ye, Phys. Rev. Lett. 101, 243002 (2008).
  * Barry  _et al._ (2014) J. F. Barry, D. J. McCarron, E. B. Norrgard, M. H. Steinecker, and D. DeMille, Nature 512, 286 (2014).
  * Truppe  _et al._ (2017) S. Truppe, H. J. Williams, M. Hambach, L. Caldwell, N. J. Fitch, E. A. Hinds, S. B. E., and M. R. Tarbutt, Nat. Phys. 13, 1173 (2017).
  * Anderegg  _et al._ (2017) L. Anderegg, B. L. Augenbraun, E. Chae, B. Hemmerling, N. R. Hutzler, A. Ravi, A. Collopy, J. Ye, W. Ketterle, and J. M. Doyle, Phys. Rev. Lett. 119, 103201 (2017).
  * Collopy  _et al._ (2018) A. L. Collopy, S. Ding, Y. Wu, I. A. Finneran, L. Anderegg, B. L. Augenbraun, J. M. Doyle, and J. Ye, Phys. Rev. Lett. 121, 213201 (2018).
  * Cheuk  _et al._ (2018) L. W. Cheuk, L. Anderegg, B. L. Augenbraun, Y. Bao, S. Burchesky, W. Ketterle, and J. M. Doyle, Phys. Rev. Lett. 121, 083201 (2018).
  * Caldwell  _et al._ (2019) L. Caldwell, J. A. Devlin, H. J. Williams, N. J. Fitch, E. A. Hinds, B. E. Sauer, and M. R. Tarbutt, Phys. Rev. Lett. 123, 033202 (2019).
  * Caldwell  _et al._ (2020) L. Caldwell, H. J. Williams, N. J. Fitch, J. Aldegunde, J. M. Hutson, B. E. Sauer, and M. R. Tarbutt, Phys. Rev. Lett. 124, 063001 (2020).
  * McCarron  _et al._ (2018) D. J. McCarron, M. H. Steinecker, Y. Zhu, and D. DeMille, Phys. Rev. Lett. 121, 013202 (2018).
  * Williams  _et al._ (2018) H. J. Williams, L. Caldwell, N. J. Fitch, S. Truppe, J. Rodewald, E. A. Hinds, B. E. Sauer, and M. R. Tarbutt, Phys. Rev. Lett. 120, 163201 (2018).
  * Anderegg  _et al._ (2018) L. Anderegg, B. L. Augenbraun, Y. Bao, S. Burchesky, L. W. Cheuk, W. Ketterle, and J. M. Doyle, Nat. Phys. 14, 890 (2018).
  * Ding  _et al._ (2020) S. Ding, Y. Wu, I. A. Finneran, J. J. Burau, and J. Ye, Phys. Rev. X 10, 021049 (2020).
  * Anderegg  _et al._ (2019) L. Anderegg, L. W. Cheuk, Y. Bao, S. Burchesky, W. Ketterle, K.-K. Ni, and J. M. Doyle, Science 365, 1156 (2019).
  * Cheuk  _et al._ (2020) L. W. Cheuk, L. Anderegg, Y. Bao, S. Burchesky, Y. S., W. Ketterle, K.-K. Ni, and J. M. Doyle, Phys. Rev. Lett. 125, 043401 (2020).
  * Kozyryev  _et al._ (2017) I. Kozyryev, L. Baum, K. Matsuda, B. L. Augenbraun, L. Anderegg, A. P. Sedlack, and J. M. Doyle, Phys. Rev. Lett. 118, 173201 (2017).
  * Baum  _et al._ (2020) L. Baum, N. B. Vilas, C. Hallas, B. L. Augenbraun, S. Raval, D. Mitra, and J. M. Doyle, Phys. Rev. Lett. 124, 133201 (2020).
  * Augenbraun  _et al._ (2020) B. L. Augenbraun, Z. D. Lasner, A. Frenett, H. Sawaoka, C. Miller, T. C. Steimle, and J. M. Doyle, New Journal of Physics 22, 022003 (2020).
  * Mitra  _et al._ (2020) D. Mitra, N. B. Vilas, C. Hallas, L. Anderegg, B. Augenbraun, L. Baum, C. Miller, S. Raval, and J. M. Doyle, Science 369, 1366 (2020).
  * Xu  _et al._ (2016) L. Xu, Y. Yin, B. Wei, Y. Xia, and J. Yin, Phys. Rev. A 93, 013408 (2016).
  * Yan  _et al._ (2022) K. Yan, R. X. Gu, D. Wu, J. Wei, Y. Xia, and J. P. Yin, Front. Phys. 17, 42502 (2022).
  * Tarallo  _et al._ (2016) M. G. Tarallo, G. Z. Iwata, and T. Zelevinsky, Phys. Rev. A 93, 032509 (2016).
  * Norrgard  _et al._ (2017) E. B. Norrgard, E. R. Edwards, D. J. McCarron, M. H. Steinecker, D. DeMille, S. S. Alam, S. K. Peck, N. S. Wadia, and L. R. Hunter, Phys. Rev. A 95, 062506 (2017).
  * Truppe  _et al._ (2019) S. Truppe, S. Marx, S. Kray, M. Doppelbauer, S. Hofsäss, H. C. Schewe, N. Walter, J. Pérez-Ríos, B. G. Sartakov, and G. Meijer, Phys. Rev. A 100, 052513 (2019).
  * Daniel  _et al._ (2021) J. R. Daniel, C. Wang, K. Rodriguez, B. Hemmerling, T. N. Lewis, C. Bardeen, A. Teplukhin, and B. K. Kendrick, Phys. Rev. A 104, 012801 (2021).
  * Chen  _et al._ (2016) T. Chen, W. Bu, and B. Yan, Phys. Rev. A 94, 063415 (2016).
  * Bu  _et al._ (2017) W. Bu, T. Chen, G. Lv, and B. Yan, Phys. Rev. A 95, 032701 (2017).
  * Chen  _et al._ (2017) T. Chen, W. Bu, and B. Yan, Phys. Rev. A 96, 053401 (2017).
  * Aggarwal  _et al._ (2018) P. Aggarwal, H. Bethlem, A. Borschevsky, M. Denis, K. Esajas, Y. Hao, S. Hoekstra, K. Jungmann, T. B. Meijknecht, M. Mooij, R. Timmermans, A. Touwen, W. Ubachs, S. Vermeulen, L. Willmann, Y. Yin, and A. Zapara, Eur. Phys. J. D 72, 197 (2018).
  * Cournol  _et al._ (2018) A. Cournol, P. Pillet, H. Lignier, and D. Comparat, Phys. Rev. A 97, 031401 (2018).
  * Albrecht  _et al._ (2020) R. Albrecht, M. Scharwaechter, T. Sixt, L. Hofer, and T. Langen, Phys. Rev. A 101, 013413 (2020).
  * Liang  _et al._ (2021) Q. Liang, T. Chen, W. H. Bu, Y. H. Zhang, and B. Yan, Front. Phys. 16, 32501 (2021).
  * Ernst  _et al._ (1986) W. E. Ernst, J. Kandler, and T. Törring, J. Chem. Phys. 84, 4769 (1986).
  * Hänsch  _et al._ (1971) T. W. Hänsch, M. D. Levenson, and A. L. Schawlow, Phys. Rev. Lett. 26, 946 (1971).
  * Wieman and Hänsch (1976) C. Wieman and T. W. Hänsch, Phys. Rev. Lett. 36, 1170 (1976).
  * Nakayama (1985) S. Nakayama, J. Opt. Soc. Am. B 2, 1431 (1985).
  * Skoff  _et al._ (2009) S. M. Skoff, R. J. Hendricks, C. D. J. Sinclair, M. R. Tarbutt, J. J. Hudson, D. M. Segal, B. E. Sauer, and E. A. Hinds, New Journal of Physics 11, 123026 (2009).
  * Wang  _et al._ (2018) D. Wang, W. Bu, D. Xie, T. Chen, and B. Yan, J. Opt. Soc. Am. B 35, 1658 (2018).
  * Barry  _et al._ (2011) J. F. Barry, E. S. Shuman, and D. DeMille, Phys. Chem. Chem. Phys. 13, 18936 (2011).
  * Chen  _et al._ (2019) T. Chen, W. Bu, and B. Yan, Phys. Rev. A 100, 029901 (2019).
  * Zhang  _et al._ (2022) Y. Zhang, Z. Zeng, Q. Liang, W. Bu, and B. Yan, Phys. Rev. A 105, 033307 (2022).
  * Ryzlewicz and Törring (1980) C. Ryzlewicz and T. Törring, Chem. Phys. 51, 329 (1980).
  * Sauer  _et al._ (1999) B. E. Sauer, S. B. Cahn, M. G. Kozlov, G. D. Redgrave, and E. A. Hinds, J. Chem. Phys. 110, 8424 (1999).
  * Bu  _et al._ (2016) W. Bu, M. Liu, D. Xie, and B. Yan, Review of Scientific Instruments 87, 096102 (2016).