Oscillating quadrupole effects in high-precision metrology

Abstract

The influence of oscillating quadrupole fields on atomic energy levels is examined theoretically and general expressions for the quadrupole matrix elements are given. The results are relevant to any ion-based clock in which one of the clock states supports a quadrupole moment. Clock shifts are estimated for 176 Lu + and indicate that coupling to the quadrupole field would not be a limitation to clock accuracy at the 10 −19 level. Nevertheless, a method is suggested that would allow this shift to be calibrated. This method utilises a resonant quadrupole coupling that enables the quadrupole moment of the atom to be measured. A proof-of-principle demonstration is given using 138 Ba + , in which the quadrupole moment of the D 5/2 state is estimated to be Θ = 3.229(89)ea 2 0 .In a recent paper [1], the effects of oscillating magnetic fields in high precision metrology were explored. In that work it was shown that magnetic fields driven by the oscillating potential of a Paul trap could have a significant influence on high precision measurements and optical atomic clocks. Given that the oscillating potential itself provides a strong quadrupole field, it is of interest to consider the effects this might have on energy levels supporting a non-zero quadrupole moment.The interaction of external electric-field gradients with the quadrupole moment of the atom is described by tensor operators of rank two [2]. As for ac magnetic fields, the interaction couples levels primarily within the same fine-structure manifold. However, the rank-2 operators provide a coupling between levels having ∆F = 0, ±1, ±2 and ∆m = 0, ±1, ±2. In this paper, a general expression for the interaction matrix elements is derived and the various level shifts and effects that can occur are considered. These results can be readily applied to any system. For the purposes of illustration, fractional frequency shifts for three clock transitions of 176 Lu + are estimated for experimentally relevant parameter values. I. THEORYThe notations and conventions used here follow that used in [2]. The principal-axis (primed) frame (x , y , z ) is one in which the electric potential in the neighbourhood of the atom has the simple form Φ(x , y , z ) = A(x 2 + y 2 − 2z 2 ) + (x 2 − y 2 ), (1) while a laboratory (unprimed) frame (x, y, z) is one in which the magnetic field is oriented along the z axis. Using this form of the potential, the time-dependent potential associated with an ideal linear Paul trap has A = 0 and that for the ideal quadrupole trap has = 0, with the time-dependence provided by a cos(Ω rf t) factor, where Ω rf is the trap drive frequency.In the principal-axis frame, the spherical components of ∇E (2) are.(2) and the interaction H Q has the simple formStates | JF m defined in the principal-axis frame and states | JF µ defined in the laboratory frame are related bywith the inverse relationwhere ω denotes a set of Euler angles {α, β, γ} taking the principal-axis frame to the laboratory frame defined with the same convention used in [2]. Specifically, start...

Precision measurements on the $^{138}$Ba$^+$ $6s^2S_{1/2}-5d^2D_{5/2}$ clock transition

“…where C 5 2 ,± 1 2 = − 4 5 , Θ(J) = 3.229(89) ea 2 0 [30] is the D 5/2 quadrupole moment, ω z is the axial confinement frequency, m is the atomic mass, e is the elementary charge, and β = π − φ is the angle between the applied magnetic field and the trap axis. Note that our geometry has α = π/4, where α is the second Euler angle used in [29].…”