The 2017 Florence experiment of Rosi et al. performed the first test of the weak equivalence principle (WEP) with atoms prepared in a coherent superposition of two internal energy eigenstates, constraining the off-diagonal element r of the mass-energy operators to |r| <= 5e-8 and the diagonal (classical) Eötvös ratios for the two hyperfine levels to about 1e-9. The experiment left open whether quantum-WEP violations exist below this level, whether the uncontrolled preparation phase of the superposition can be mastered so that r is measured deterministically instead of through excess phase noise, and whether the test can be extended to internal states with much larger energy separation, such as the optically separated clock states of strontium. This problem asks for a next-generation experiment that extends the quantum WEP test beyond the 2017 envelope along one or both of these axes, with the dominant AC Stark and second-order Zeeman systematics reduced toward the 1e-10 level. As of August 2026, no published improvement of the |r| bound, no preparation-phase-controlled measurement, and no optical-gap superposition-state test at sensitivity competitive with the 2017 test are known; the first such platform demonstration with optically separated strontium clock states is three to four orders of magnitude less sensitive, so the problem is open.

Background

The Einstein equivalence principle (EEP) is a foundational pillar of general relativity; its weak form, the weak equivalence principle (WEP), states the universality of free fall: test bodies of different composition fall with the same acceleration in a gravitational field. Classical tests compare the free-fall accelerations of two bodies A and B through the Eötvös ratio, ηA−B=2(aA−aB)/(aA+aB)\eta_{A-B}=2(a_A-a_B)/(a_A+a_B), with torsion-balance experiments reaching parts in 101310^{13}.

In quantum theory the situation is subtler, because the internal energy of a body is described by an operator that can place a body in a coherent superposition of different energy eigenvalues. Since mass and energy are equivalent, the rest, inertial, and gravitational mass-energy of a quantum body should themselves be operators, and Zych and Brukner (2018) formulated a quantum version of the EEP requiring the equivalence of these three operators. In the two-dimensional subspace spanned by internal eigenstates ∣1⟩|1\rangle and ∣2⟩|2\rangle, a violation theory introduces diagonal parameters r1r_1, r2r_2 (the classical WEP sector, probed by comparing free fall of the two eigenstates) and an off-diagonal parameter rr coupling ∣1⟩↔∣2⟩|1\rangle\leftrightarrow|2\rangle, which has no classical analogue: probing it requires atoms in coherent superpositions of different internal energies. All EEP tests before 2017 were sensitive only to the diagonal elements.

Rosi et al. (2017) reported the first test of this quantum WEP. A Bragg-pulse atom interferometer in a gravity-gradiometer configuration — two vertical Mach–Zehnder interferometers about 30 cm apart interrogated simultaneously by the same Bragg lasers (beam splitters and mirrors realized by coherent two-photon scattering that change momentum, not internal state) at a magic detuning where the two hyperfine levels ∣1⟩|1\rangle and ∣2⟩|2\rangle of 87^{87}Rb experience identical optical coupling — compared the free fall of atoms prepared in ∣1⟩|1\rangle, in ∣2⟩|2\rangle, and in their coherent superposition. They measured the diagonal Eötvös ratios η1−2=(1.0±1.4)×10−9\eta_{1-2}=(1.0\pm1.4)\times10^{-9} and η1−s=(3.3±2.9)×10−9\eta_{1-s}=(3.3\pm2.9)\times10^{-9}, improving previous hyperfine-level tests by almost two orders of magnitude, and bounded the off-diagonal element by attributing all excess phase noise of the superposition-state interferometer to a WEP violation, obtaining ∣r∣≤5×10−8|r|\le5\times10^{-8}.

Three limitations define the open problem. First, the dominant systematic uncertainty is the AC Stark shift (a light-induced energy shift proportional to laser intensity, here spatially varying because of diffraction of the Bragg beams) at 2.6×10−92.6\times10^{-9} in δg/g\delta g/g, followed by the second-order Zeeman shift (a magnetic-field-induced level shift quadratic in the field strength, here caused by magnetic-field inhomogeneities across the atomic cloud) at 0.6×10−90.6\times10^{-9}; the paper proposes higher-power and suitably shaped beams and an active light-shift compensation scheme to cut the leading error by more than an order of magnitude, pushing the test toward the 10−1010^{-10} level. Second, the preparation phase γ\gamma of the superposition state is uncontrolled and varies randomly shot-to-shot, so a violation entering through rr would appear as excess phase noise with zero mean rather than a deterministic phase shift; only an upper bound on ∣r∣|r|, not a direct measurement, is therefore possible. Third, the energy gap probed is the GHz-scale hyperfine splitting, whereas several candidate theories motivating WEP violations predict effects growing with the internal-energy difference; the paper identifies optically separated levels, for example as in strontium, as a feasible aim for near-future experiments — the 1S0^{1}S_{0} and 3P0^{3}P_{0} clock states that define optical atomic clocks being the natural example — which requires beam splitters that coherently manipulate two states of an optical transition identically, via magic-wavelength Bragg pulses or single-photon clock-transition pulses.

Problem Statement

Extend the quantum test of the weak equivalence principle beyond the envelope set by Rosi et al. (2017), whose single overarching objective is to determine whether WEP-violating mass-energy operator effects exist for atoms in coherent superpositions of internal-energy eigenstates at levels below the current limits — either because they are smaller than the present sensitivity or because they appear only at larger internal-energy separations than the hyperfine gap. This objective is pursued along two complementary, explicitly linked axes: (1) sensitivity — perform a superposition-state atom-interferometric WEP test that tightens the bound on the off-diagonal mass-operator element below ∣r∣≤5×10−8|r|\le5\times10^{-8}, ideally by controlling the preparation phase γ\gamma so that an rr-induced violation appears as a deterministic phase shift that is measured directly rather than bounded through excess phase noise, which in turn requires reducing the dominant systematics (AC Stark shifts from Bragg-beam intensity gradients, currently 2.6×10−92.6\times10^{-9} in δg/g\delta g/g, targeted for reduction by more than an order of magnitude via higher-power and suitably shaped beams and light-shift compensation; second-order Zeeman shifts, currently 0.6×10−90.6\times10^{-9}, via magnetic shielding and characterization) toward the 10−1010^{-10} accuracy level; and (2) energy gap — perform the equivalent quantum WEP test with internal states separated by optical rather than hyperfine energy (for example the clock states of strontium, using magic-wavelength or single-photon clock-transition beam splitters that treat the two states identically) at a sensitivity at least comparable to the 2017 hyperfine-state test, i.e. with diagonal Eötvös-ratio sensitivity at the 10−910^{-9} level or better. Completion is recognizable by a quantitative experimental design and sensitivity/systematics analysis that projects the improvement required on either axis, with its statistical and systematic error budget, or by the published experimental result that realizes it, while preserving the genuinely quantum character of the test (atoms in coherent superposition of different internal-energy eigenstates): a tightened or direct bound on rr, a first competitive optical-gap superposition-state test, or a detected violation. Either outcome directly addresses the source question of whether quantum-WEP violations exist below ∣r∣∼5×10−8|r|\sim5\times10^{-8} or at internal-energy separations larger than the hyperfine splitting.

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