Scattering amplitude and two-body loss of ultracold alkaline-earth atoms in a shaking synthetic magnetic field

The idea of manipulating the interaction between ultracold fermionic alkaline-earth (like) atoms via a laser-induced periodical synthetic magnetic field was proposed in Kanász-Nagy et al (2018 Phys. Rev. B 97, 155156). In that work, it was shown that in the presence of the shaking synthetic magnetic field, two atoms in ¹S₀ and ³P₀ states experience a periodical interaction in a rotated frame, and the effective inter-atomic interaction was approximated as the time-averaged operator of this time-dependent interaction. This technique is supposed to be efficient for ¹⁷³Yb atoms which have a large natural scattering length. Here we examine this time-averaging approximation and derive the rate of the two-body loss induced by the shaking of the synthetic magnetic field, by calculating the zero-energy inter-atomic scattering amplitude corresponding to the explicit periodical interaction. We find that for the typical cases with shaking angular frequency λ of the synthetic magnetic field being of the order of (2π) kHz, the time-averaging approximation is applicable only when the shaking amplitude is small enough. Moreover, the two-body loss rate increases with the shaking amplitude, and is of the order of 10⁻¹⁰ cm³ · s⁻¹ or even larger when the time-averaging approximation is not applicable. Our results are helpful for the quantum simulations with ultracold gases of fermionic alkaline-earth (like) atoms.