In the case of the ALP-wind linear coupling, the field acting on the 1H-13C spins induces an energy shift equal to the one produced by a magnetic field with amplitudeBALP,zeff(t)=23gaNNγH+γC2cρDM×sin (ωDMtk·r+φ)v·e^z(8)where ωDMmDMc2/ℏ is the ALP Compton frequency, kmDMv/ℏ is the wave vector (v is the relative velocity), mDM is the rest mass of the ALP, φ is an unknown phase, and e^z is the axis along which the leading DC magnetic field is applied.

It is theoretically possible that interaction of nuclear spins with ∇a can be suppressed (33, 46), in which case the dominant axion wind interaction, referred to as the quadratic wind coupling, is related to a2. In the case of the ALP-wind quadratic coupling, the equivalent magnetic field amplitude isBquad,zeff(t)=43c2gquad2γH+γCρDMωDM×sin (2ωDMt2k·r+φ)v·e^z(9)where gquad, having dimensions of inverse energy, parameterizes the ALP quadratic coupling strength to nuclear spins.

There are two possible interactions of dark photons with nuclear spins that can be detected with CASPEr-ZULF: the coupling of the dark electric field to the dark EDM (dEDM) and the coupling of the dark magnetic field to the dark magnetic dipole moment (dMDM). The equivalent magnetic field amplitudes areBdEDMeff(t)=23gdEDMγC+γH2c3ρDM×cos (ωDMt+φ)ε·e^z (10)andBdMDMeff(t)=23gdMDMγC+γHvc2c3ρDM×cos (ωDMt+φ)ε·e^z(11)with coupling constants gdEDM and gdMDM (having dimensions of inverse energy) and dark photon field polarization ε.

The experimental sensitivity to real magnetic fields then directly translates to sensitivity to the coupling constants gaNN, gquadgdEDM, and gdMDM. Inverting Eqs. 8 to 11 yields the corresponding conversion factors from magnetic field to the dark matter coupling constantsδgaNN(ω)[1.3×108GeV1T]δB(ω)(12)δgquad(ω)[190GeV1Trad/s]ω·δB(ω)(13)δgdEDM(ω)[1.3×105GeV1T]δB(ω)(14)δgdMDM(ω)[1.3×108GeV1T]δB(ω)(15)

Here, we have used γC/2π = 10.70 MHz.T−1 and γH/2π = 42.57 MHz.T−1, v ≈ 10−3c, and ρDM ≈ 0.4 GeV/cm3. The full derivation of these expressions is given in section S2.3.

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