Zelevinsky, Kotochigova, and Ye [Phys. Rev. Lett. 100, 043201 (2008)] proposed a model-independent laboratory test of present-day time variation of the proton-electron mass ratio μ=mp/me\mu=m_p/m_e by Raman spectroscopy of ro-vibrational energy intervals in ultracold (even-isotope) Sr2 molecules trapped in a zero-differential-Stark-shift ('magic') optical lattice, with an initially projected sensitivity of about 10−1410^{-14}/year and a stated improvement path of at least a factor of ten. The proposal left open a set of linked experimental requirements: experimental determination of the excited-state resonance positions and strengths that fix the magic-lattice frequency crossings and detunings; validation of the model-based molecular inputs (Franck-Condon factors, transition dipole moments, excited-state lifetimes, dynamic polarizabilities, decay and photon-scattering rates); control of Raman-laser intensity and power balancing sufficient to keep Stark-shift-induced inaccuracy below the sub-Hz level; and demonstration of molecule production and transfer efficiencies adequate for the assumed signal-to-noise ratio. The enabling milestones have been substantially achieved (magic molecular lattice trapping in 2019, a terahertz vibrational molecular clock with total systematic uncertainty 4.6×10−144.6\times10^{-14} and a 13-digit absolute vibrational frequency in 2023, blackbody-radiation shifts evaluated to the 10−1610^{-16} level), but a published constraint on Δμ/μ\Delta\mu/\mu from the lattice-confined Sr2 vibrational clock, together with the improvements needed to reach ∼10−15\sim10^{-15}/year, has not been reported.

Background

The proton-electron mass ratio μ=mp/me\mu=m_p/m_e is a dimensionless fundamental constant whose possible slow time variation is predicted by many unified and dark-energy models and is tested both astronomically and in the laboratory. Molecules are natural probes: their vibrational and rotational energies are set mainly by nuclear dynamics, so a fractional change Δμ/μ\Delta\mu/\mu shifts a molecular transition frequency ν\nu by Δν/ν=Kμ Δμ/μ\Delta\nu/\nu=K_\mu\,\Delta\mu/\mu, where the dimensionless sensitivity coefficient KμK_\mu (called κ\kappa in the source paper) is of order one for vibrational intervals and can be computed reliably from the molecular potential. This is complementary to hyperfine-based atomic clocks, whose frequencies mix sensitivities to μ\mu, the fine-structure constant α\alpha, and the Rydberg constant, so that separating a μ\mu drift requires theoretical modeling. Before the source paper, laboratory limits came from atomic-clock comparisons (about 6×10−156\times10^{-15}/year) and molecular spectroscopy (e.g. an SF6 beam experiment near 3×10−143\times10^{-14}/year), while astrophysical constraints from NH3 and H2 quasar absorption spectra disagreed at the 10−1510^{-15}-10−1610^{-16}/year level and relied on cosmological modeling. The source paper proposed an ultracold-molecule route: photoassociation (PA, the formation of molecules by laser excitation of colliding atom pairs into a bound excited state that decays into ground-state vibrational levels) produces vibrationally excited Sr2, and two-color Raman spectroscopy, with the two Raman lasers tied by a femtosecond optical frequency comb (a laser source whose repetition rate links optical frequencies to radio frequencies), measures energy spacings between weakly bound and deeply bound vibrational levels of the X 1Σg+X\,{}^1\Sigma_g^+ ground potential, whose 30 THz depth makes the fractional error δν/ν\delta\nu/\nu small for a fixed absolute frequency uncertainty δν\delta\nu. The molecules are confined in a one-dimensional optical lattice (a standing-wave light trap) operated at a 'magic' frequency where the dynamic polarizabilities of the two vibrational states cross, so that the differential ac Stark shift of the clock transition vanishes - the same zero-differential-Stark-shift technique that underlies neutral-atom optical lattice clocks. Sr2 from even strontium isotopes was chosen because the J=0J=0 ground state lacks hyperfine and magnetic structure. The paper's model-based estimates relied on ab initio transition dipole moments and Franck-Condon factors (the squared vibrational wavefunction overlap that sets the strength of a vibronic transition), with estimated lattice photon-scattering rates of order 1-4 s−1^{-1}, Raman-laser background Stark shifts of order 100 Hz (requiring intensity control to below 1% for a sub-Hz measurement), and a collisional relaxation rate that 'must be determined experimentally'. The paper projected a sub-Hz frequency measurement supporting a ∼10−14\sim10^{-14}/year test of Δμ/μ\Delta\mu/\mu, with future improvements in precision by at least a factor of ten. A companion theory paper [Kotochigova, Zelevinsky, and Ye, Phys. Rev. A 79, 012504 (2009)] computed the two-photon dipole moments, excited-state lifetimes, and Stark shifts needed for the same program. In the following years this proposal developed into the 'molecular lattice clock' research program: coherent Raman manipulation of lattice-trapped Sr2 (McGuyer et al., New J. Phys. 17, 055004 (2015); Kondov et al., Nat. Phys. 15, 1118 (2019)), and subsequently a fully characterized terahertz vibrational clock (Leung et al., Phys. Rev. X 13, 011047 (2023)) with blackbody-radiation (BBR, the thermal radiation field of the apparatus) shifts evaluated by Iritani et al. [Phys. Rev. Lett. 131, 263201 (2023)]. Understanding the remaining question requires this context: the hardware and spectroscopy now exist, and the question is what must still be done to complete the originally proposed Δμ/μ\Delta\mu/\mu measurement and its sensitivity upgrade.

Problem Statement

Complete the proposed precision test of present-day time variation of the proton-electron mass ratio μ=mp/me\mu=m_p/m_e using ultracold even-isotope Sr2 molecules confined in a zero-differential-Stark-shift ('magic') optical lattice. The single overarching objective is a model-independent laboratory constraint on Δμ/μ\Delta\mu/\mu, initially at the ∼10−14\sim10^{-14}/year level and improved toward ∼10−15\sim10^{-15}/year or better, obtained by comparing the clock frequency of a Raman-driven vibrational transition of lattice-confined Sr2 against an atomic frequency standard over time. Achieving it requires, as scientifically linked components of one experimental program: (1) experimentally determining the spectral locations and strengths of the excited-state resonances that set the magic-lattice frequency crossings and the required lattice detunings for the chosen vibrational level pairs, together with the model-based molecular inputs they depend on (Franck-Condon factors, transition dipole moments, excited-state lifetimes, dynamic polarizabilities, spontaneous-decay and photon-scattering rates, and collisional relaxation rates); (2) controlling the Raman-laser intensities and power balancing to the level (order 0.1%-1%, depending on the level pair) that keeps Stark-shift-induced inaccuracy at or below the sub-Hz level needed for the projected precision; (3) demonstrating the molecule production, coherent population transfer, and interrogation conditions (ensemble size, linewidth, signal-to-noise ratio, uptime) that support the projected precision and the factor-of-ten-or-more improvement beyond the initial ∼10−14\sim10^{-14}/year sensitivity; and (4) executing and reporting the long-term frequency-ratio measurement that constitutes the Δμ/μ\Delta\mu/\mu constraint itself, namely a model-independent bound on Δμ/μ\Delta\mu/\mu with its uncertainty budget, together with the design of the long-duration frequency-ratio campaign and of the path to ∼10−15\sim10^{-15}/year or better, the campaign being resolved through its design and projected outcome rather than through physical execution.

References

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