Pushing the cold-molecule OH constraint on the evolution of the fine-structure constant below the micro-precision level, in the laboratory and with redshifted OH lines

Hudson, Lewandowski, Sawyer, and Ye (Phys. Rev. Lett. 96, 143004, 2006) used a Stark decelerator to measure the OH ground-state Lambda-doublet microwave transitions to 4 Hz and 12 Hz, improving the 30-year-old benchmark by an order of magnitude, and proposed that comparing these laboratory frequencies with redshifted OH 18 cm lines from interstellar sources would constrain the fractional change of the fine-structure constant alpha at the level of about 1 ppm over a 10-billion-year look-back time. The task is to assess, taking the later literature as the evidentiary basis, how far below 10^-6 the OH route can go and what laboratory and metrological developments that requires. The micro-precision astrophysical milestone itself has been reached: using the 2006 rest frequencies, Kanekar et al. (2012) obtained Delta(alpha)/alpha = (-1.7 +/- 1.4) x 10^-6 at redshift 0.765 (with auxiliary assumptions on mu and g_p), and Kanekar, Ghosh, and Chengalur (2018) obtained (-1.0 +/- 1.3) x 10^-6 on mu*alpha^2 from conjugate 18 cm satellite lines, in both cases through absorption systems rather than megamaser emission, with inter-species velocity systematics rather than laboratory frequencies as the dominant error source. The surviving open problem is how far below 10^-6 the OH-line method can be pushed, given the scarcity of suitable redshifted sources and the velocity systematics of the comparison, together with the laboratory-side developments still to be realized: the reflection-free, long-interrogation, sub-Hz cold-molecule spectroscopy and molecular frequency standard that would enable a molecular-atomic clock intercomparison sensitive to alpha rather than only to the mass ratio.

Background

The fine-structure constant $\alpha = e^2/\hbar c$ sets the strength of the electromagnetic interaction, and many theories that attempt to unify gravity with the other fundamental forces predict that $\alpha$ (and other dimensionless constants such as the proton-to-electron mass ratio $\mu = m_p/m_e$ and the proton g-factor $g_p$, the dimensionless magnetic moment of the proton) drift slowly over cosmological time. Distant-quasar optical spectra give one class of tests, but they use spatially diverse absorbers and are sensitive to relative Doppler shifts, so independent confirmation is valuable.

The hydroxyl radical OH has a special role here. In its electronic and rovibrational ground state, OH is a $^2\Pi$ molecule whose unpaired electron carries one unit of orbital angular momentum; the resulting 'Lambda-doubling' splits each rotational level into two nearly degenerate opposite-parity states separated by only about 1600-1720 MHz (the '18 cm' lines). Coupling to the hydrogen nuclear spin ($I = 1/2$) further splits each component into hyperfine levels with total angular momentum $F = 1$ and $F = 2$, giving the four famous astronomical maser lines at 1612, 1665, 1667, and 1720 MHz. Darling (Phys. Rev. Lett. 91, 011301, 2003) showed that the sum and the difference of the hyperfine-conserving ($\Delta F = 0$) transition frequencies depend on different powers of $\alpha$ (approximately $\alpha^{0.4}$ and $\alpha^4$), so that comparing redshifted OH line frequencies with laboratory rest frequencies constrains $\Delta\alpha/\alpha$ over the look-back time to the source; because several lines with different constant-dependence arise from one localized cloud, real changes can be separated from plain Doppler shifts. Chengalur and Kanekar (Phys. Rev. Lett. 91, 241302, 2003) and Kanekar, Chengalur, and Ghosh (Phys. Rev. Lett. 93, 051302, 2004) demonstrated the technique on redshifted OH absorption systems, and Darling (2004) applied it to the 'conjugate' 1612/1720 MHz satellite lines toward PKS 1413+135 (the two satellite lines are called conjugate because they share the same inverted upper levels, so the 1720 MHz emission and the 1612 MHz absorption have identical line shapes; comparing them therefore cancels plain Doppler shifts). The laboratory bottleneck was precision: the previous best measurement of the ground-state Lambda-doublet (ter Meulen and Dymanus, beam-maser spectroscopy) had stood for over thirty years at roughly 100 Hz accuracy, comparable to the change expected from the interesting constraints.

A Stark decelerator removes this bottleneck. Polar molecules in a supersonic beam can be slowed by an array of time-switched electric electrodes (exploiting the Stark shift, the energy shift of a polar molecule in an electric field), yielding slow, cold packets whose long field-free interrogation enables narrow microwave lines. Hudson, Lewandowski, Sawyer, and Ye (Phys. Rev. Lett. 96, 143004, 2006) combined a Stark decelerator with Rabi and Ramsey microwave spectroscopy (Rabi spectroscopy drives the transition with a single oscillatory field, while Ramsey spectroscopy drives it with two separated oscillatory fields so that the linewidth is set by the separation rather than the pulse length, the basis of atomic fountain clocks) to measure the $F'=2 \to F=2$ transition at $(1\,667\,358\,996 \pm 4)$ Hz and the $F'=1 \to F=1$ transition at $(1\,665\,401\,803 \pm 12)$ Hz, improvements of twenty-five-fold and ten-fold. They reported that the results were limited only by statistical uncertainty, with conservative upper bounds on Zeeman, stray-electric-field, cavity, and velocity-related systematic shifts; they identified a decoherence mechanism caused by microwave radiation leaking from the cavity and reflecting off the decelerator, to be eliminated in future designs; and they proposed, as the natural continuation, a molecular fountain clock and a comparison of a molecular clock against an atomic clock whose frequency depends differently on the fundamental constants. The companion astrophysical requirement is data: OH megamasers (cosmically amplified OH maser emission in distant infrared-luminous galaxies) or redshifted OH absorption systems observed with uncertainties and systematics that make the laboratory-level accuracy meaningful. The field's plans for extending the astrophysical side of this program point to the next generation of radio arrays, notably the Square Kilometre Array (the planned next-generation radio telescope array). The question is how much of this coupled program later literature has delivered and what precisely remains open.

Problem Statement

The problem, as posed by its source and still live after twenty years of follow-up, is to carry the cold-molecule OH program for testing the evolution of the fine-structure constant to completion: to determine how precisely the comparison of laboratory OH Lambda-doublet frequencies (and those of successor cold radicals) with redshifted OH 18 cm astronomical lines can constrain $\Delta\alpha/\alpha$, and what experimental developments that requires. Concretely, the program has three coupled components: First, on the astrophysical side, obtaining redshifted OH 18 cm line measurements from (mega)maser or conjugate-satellite absorption sources with uncertainties and systematics controlled well enough to exploit hertz-level laboratory rest frequencies, which includes identifying, discovering, and long-duration-observing suitable sources, and reaching and then surpassing the micro-precision ($10^{-6}$) level for $\Delta\alpha/\alpha$, or its combinations with $\mu$ and $g_p$, over cosmological look-back times; the $10^{-6}$ level has been reached by Kanekar et al. (2012, 2018), so the open question is whether the method can be pushed meaningfully below $10^{-6}$, given cloud-to-cloud velocity systematics and the scarcity of known systems. Second, on the laboratory side, reducing the systematic budget of cold-molecule microwave spectroscopy (Zeeman, Doppler, cavity-related shifts, and the microwave-reflection decoherence identified in the source paper) below the statistical limit and extending the interrogation time, through reflection-free microwave delivery and fountain-scale free flight, to deliver sub-hertz OH (or successor-radical) frequencies, improving on the 4 Hz and 12 Hz values of 2006. Third, in metrology, realizing a molecular microwave frequency standard and a molecular-atomic clock intercomparison whose different dependence on the fundamental constants provides a laboratory test of present-day drift of $\alpha$ (as the earlier SF6-versus-Cs comparison did for $\mu$). These components are scientifically linked: better laboratory rest frequencies and longer interrogation are what make the astronomical comparison meaningful, and the clock intercomparison is the laboratory endpoint of the same constant-dependence physics. The substance of the problem is how far below $10^{-6}$ the OH route can go and why, together with the enabling laboratory and metrological results.

References

1. Eric R. Hudson, H. J. Lewandowski, Brian C. Sawyer, and Jun Ye, 'Cold Molecule Spectroscopy for Constraining the Evolution of the Fine Structure Constant', Phys. Rev. Lett. 96, 143004 (2006), DOI 10.1103/PhysRevLett.96.143004, https://doi.org/10.1103/PhysRevLett.96.143004
2. Jeremy Darling, 'Methods for Constraining Fine Structure Constant Evolution with OH Microwave Transitions', Phys. Rev. Lett. 91, 011301 (2003), DOI 10.1103/PhysRevLett.91.011301, https://doi.org/10.1103/PhysRevLett.91.011301
3. J. J. Ter Meulen and A. Dymanus, 'Beam-Maser Measurements of the Ground-State Transition Frequencies of OH', Astrophys. J. 172, L21 (1972), DOI 10.1086/180882, https://doi.org/10.1086/180882
4. J. N. Chengalur and N. Kanekar, 'Constraining the Variation of Fundamental Constants using 18 cm OH Lines', Phys. Rev. Lett. 91, 241302 (2003), DOI 10.1103/PhysRevLett.91.241302, https://doi.org/10.1103/PhysRevLett.91.241302
5. N. Kanekar, J. N. Chengalur, and T. Ghosh, 'Conjugate 18 cm OH satellite lines at a cosmological distance', Phys. Rev. Lett. 93, 051302 (2004), DOI 10.1103/PhysRevLett.93.051302, https://doi.org/10.1103/PhysRevLett.93.051302
6. Jeremy Darling, 'A Laboratory for Constraining Cosmic Evolution of the Fine-Structure Constant: Conjugate 18 Centimeter OH Lines toward PKS 1413+135 at z = 0.24671', Astrophys. J. 612, 58 (2004), DOI 10.1086/422450, https://doi.org/10.1086/422450
7. N. Kanekar, G. I. Langston, J. T. Stocke, C. L. Carilli, and K. M. Menten, 'Constraining Fundamental Constant Evolution with H I and OH Lines', Astrophys. J. Lett. 746, L16 (2012), DOI 10.1088/2041-8205/746/2/L16, https://doi.org/10.1088/2041-8205/746/2/L16
8. N. Kanekar, T. Ghosh, and J. N. Chengalur, 'Stringent Constraints on Fundamental Constant Evolution Using Conjugate 18 cm Satellite OH Lines', Phys. Rev. Lett. 120, 061302 (2018), DOI 10.1103/PhysRevLett.120.061302, https://doi.org/10.1103/PhysRevLett.120.061302 (arXiv:1801.07688, https://arxiv.org/abs/1801.07688)
9. S. Truppe, R. J. Hendricks, S. K. Tokunaga, H. J. Lewandowski, M. G. Kozlov, Christian Henkel, E. A. Hinds, and M. R. Tarbutt, 'A search for varying fundamental constants using hertz-level frequency measurements of cold CH molecules', Nat. Commun. 4, 2600 (2013), DOI 10.1038/ncomms3600, https://doi.org/10.1038/ncomms3600 (arXiv:1308.1496, https://arxiv.org/abs/1308.1496)
10. Cunfeng Cheng, Aernout P. P. van der Poel, Paul Jansen, Marina Quintero-Perez, Thomas E. Wall, Wim Ubachs, and Hendrick L. Bethlem, 'Molecular Fountain', Phys. Rev. Lett. 117, 253201 (2016), DOI 10.1103/PhysRevLett.117.253201, https://doi.org/10.1103/PhysRevLett.117.253201
11. A. Shelkovnikov, R. J. Butcher, C. Chardonnet, and A. Amy-Klein, 'Stability of the Proton-to-Electron Mass Ratio', Phys. Rev. Lett. 100, 150801 (2008), DOI 10.1103/PhysRevLett.100.150801, https://doi.org/10.1103/PhysRevLett.100.150801
12. M. G. Kozlov and S. A. Levshakov, 'Microwave and submillimeter molecular transitions and their dependence on fundamental constants', Ann. Phys. (Berlin) 525, 452 (2013), DOI 10.1002/andp.201300010, https://doi.org/10.1002/andp.201300010 (arXiv:1304.4510, https://arxiv.org/abs/1304.4510)
13. M. S. Safronova, D. Budker, D. DeMille, Derek F. Jackson Kimball, A. Derevianko, and Charles W. Clark, 'Search for New Physics with Atoms and Molecules', Rev. Mod. Phys. 90, 025008 (2018), DOI 10.1103/RevModPhys.90.025008, https://doi.org/10.1103/RevModPhys.90.025008
14. Ren-Zhi Su and Min-Feng Gu, 'Measuring the spatio-temporal variation of fundamental physical constants using the OH sum rules', Astron. Astrophys. (2026), DOI 10.1051/0004-6361/202661874, https://doi.org/10.1051/0004-6361/202661874 (arXiv:2609.13962, https://arxiv.org/abs/2609.13962)
15. Zheng Zheng, Di Li, Elaine M. Sadler, James R. Allison, and Ningyu Tang, 'A pilot search for extragalactic OH absorption with FAST', Mon. Not. R. Astron. Soc. 499, 3085 (2020), DOI 10.1093/mnras/staa3033, https://doi.org/10.1093/mnras/staa3033
16. K. M. Hess, H. Roberts, H. Dénes, B. Adebahr, J. Darling, E. A. K. Adams, W. J. G. de Blok, A. Kutkin, et al., 'Apertif view of the OH megamaser IRAS 10597+5926: OH 18 cm satellite lines in wide-area H I surveys', Astron. Astrophys. 647, A193 (2021), DOI 10.1051/0004-6361/202040019, https://doi.org/10.1051/0004-6361/202040019 (arXiv:2101.05818, https://arxiv.org/abs/2101.05818)