Robust optical clock transitions in trapped ions using dynamical decoupling

Abstract

We present a novel method for engineering an optical clock transition that is robust against external field fluctuations and is able to overcome limits resulting from field inhomogeneities. The technique is based on the application of continuous driving fields to form a pair of dressed states essentially free of all relevant shifts. Specifically, the clock transition is robust to magnetic field shifts, quadrupole and other tensor shifts, and amplitude fluctuations of the driving fields. The scheme is applicable to either a single ion or an ensemble of ions, and is relevant for several types of ions, such as Ca 40 + , Sr 88 + , Ba 138 + and Lu 176 + . Taking a spherically symmetric Coulomb crystal formed by 400 Ca 40 + ions as an example, we show through numerical simulations that the inhomogeneous linewidth of tens of Hertz in such a crystal together with linear Zeeman shifts of order 10MHz are reduced to form a linewidth of around 1Hz. We estimate a twoorder-of-magnitude reduction in averaging time compared to state-of-the art single ion frequency references, assuming a probe laser fractional instability of 10 15 -. Furthermore, a statistical uncertainty reaching 2.9×10 −16 in 1s is estimated for a cascaded clock scheme in which the dynamically decoupled Coulomb crystal clock stabilizes the interrogation laser for an Al 27 + clock.- [3,22,23]. The statistical uncertainty can be improved by probing for longer times, ultimately limited by the excited clock state lifetime or the laser coherence time [24,25]. Alternatively, the number of probed ions can be increased.Recently, multi-ion clock schemes have been proposed to address this issue [26][27][28][29]. However, several challenges have to be overcome to maintain and transfer the small and very well characterizable systematic shifts achievable with single trapped ions to larger ion crystals. The oscillating rf field in Paul traps results in ac Stark and second order Doppler shifts through micromotion [30][31][32]. Furthermore, electric field gradients from the trapping fields and the surrounding ions couple to atomic quadrupole moments, resulting in an electric quadrupole shift (QPS). The effects of micromotion can be avoided by trapping strings of ions in a precisionmachined linear Paul trap with negligible excess micromotion from trap imperfections [28,33]. The QPS in such Q 0 m J J J m , J j å = =-[67]. Such an average will also eliminate the linear Zeeman shift, assuming the field does not change between the frequency measurements contributing to the average. We propose a novel dynamical decoupling scheme in which robustness to this type of shifts is achieved by the application of a detuned driving field, mixing all m J states to form dressed states with effective Q 0 J m , j = . While the cancellation scheme is general and applies to tensor shifts of arbitrary electronic states, we consider in the following the Hamiltonian of the D 2 5 2 states of e.g.Ca + ions 2 1 , 1 0 J J 2

Coherent Suppression of Tensor Frequency Shifts through Magnetic Field Rotation

“…Coherent suppression is another form of shift elimination that relies on dynamic modification of experimental parameters during the atomic state evolution, providing the advantages of a fixed clock sequence and shift suppression within a single cycle. For optical clocks, different types of dynamic decoupling schemes have been investigated in which sublevels of the clock states are coherently coupled during the dark time of a Ramsey interrogation for cancellation of various shifts [11][12][13]. Further examples of coherent suppression can be found in hyper-Ramsey spectroscopy dealing with light shifts [14], and entangled many-atom states designed to provide immunity from selected perturbations [15].…”

“…Here, the radio frequency trapping field and large static electric field gradients typically cause tensor frequency shifts that exceed the resolvable linewidth resulting in inhomogeneous broadening, thus preventing long coherent interrogation. In contrast to dynamic decoupling schemes [11][12][13], our method does not require any additional drive field that can also cause frequency shifts, but provides a straightforward solution for different optical clock systems seeking to mitigate tensor shifts.…”

Suppressing Inhomogeneous Broadening in a Lutetium Multi-ion Optical Clock

“…1. Relevant energy level diagram of a 176 Lu + ion showing: the 1 S0 ↔ 3 D1 clock transition at 848 nm, the 3 D1 ↔ 3 P0 transition at 646 nm used for Doppler cooling and detection, and optical pumping transitions at 350 nm, 622 nm, and 895 nm to initialize population to the 3 D1 level.suppress EQ shifts [17,18]. All three approaches are readily adaptable to 176 Lu + although the first approach requires EQ shifts to be mitigated in a linear ion crystal.In this work, high resolution microwave spectroscopy is used to characterize the effects of neighbouring ions in a three-ion crystal.…”

“…suppress EQ shifts [17,18]. All three approaches are readily adaptable to 176 Lu + although the first approach requires EQ shifts to be mitigated in a linear ion crystal.…”

“…Multi-ion operation is complicated by electric quadrupole (EQ) shifts arising from the Coulomb fields of neighbouring ions, excess-micromotion (EMM) shifts induced by the radio-frequency (rf) trapping field, and inhomogeneous magnetic fields. Efforts and proposals towards high-accuracy multi-ion optical clocks include (i) precision engineering of the ion trap to suppress EMM shifts [14], (ii) employing clock transitions with a negative differential scalar polarisability, ∆α 0 , to eliminate EMM shifts in a large ion crystal [15,16], and (iii) using dynamic decoupling or rf-dressed states to suppress EQ shifts [17,18]. All three approaches are readily adaptable to 176 Lu + although the first approach requires EQ shifts to be mitigated in a linear ion crystal.…”