## Multi-Ion Clock

Our main research goal is the implementation of a multi-ion clock to provide an improved stability for ion-clock systems. Currently we are seeking to implement clock operation over a small number of ions with N on the order of 10. Although this may seem only a modest improvement over a single ion system, it would still improve integration time by a factor of N.

The challenge for clock operation on multiple ions concerns inhomogeneous broadening. For a single ion, various clock shifts can be well calibrated and/or averaged away. For many ions, these effects can distort the line-shape and give a frequency shift that depends non-trivially on both the distribution of individual shifts across the ion crystal, and the interrogation time. As the interrogation time increases, shifts are better resolved, which increases line-shape distortion and hence the resulting clock shift. Provided the broadening is much smaller than the width of the individual line-shape function, the shift is linear in the skewness of the distribution and cubic in both the width of the distribution and the interrogation time. The cubic dependence on the distribution compromises shift cancellation through hyperfine averaging.

The key factors affecting inhomogeneous broadening are:

**Magnetic field gradients**: Spatial inhomogeneity will result in a differential shift between ions. Fortunately lutetium has one of the lowest field dependences amongst all clock candidates. At a typical operating field of 0.1 mT, the maximum sensitivity of any single transition involved in clock operation is about 1.4 kHz/G. For a string of ions occupying a 50 μm length would then have a maximum differential shift of 1 Hz for a gradient of 1.4 mT/m. This is rather trivial to achieve and, moreover, the resulting distribution would be symmetric.

**AC magnetic fields**: Fields driven by rf currents in the electrodes have a spatial variation. However for a well-designed linear Paul trap, we would expect this to be fairly uniform along the trap axis. Moreover the shift itself can be expected to be rather small at least for the macroscopic traps we currently use.

**Quadrupole shifts**: For the macroscopic traps we currently use, the quadrupole shift from the trap confinement will be very homogeneous over a small linear chain of ions. Hence the only significant broadening will be from electric field gradients generated from neighboring ions. These effects can be suppressed by having the magnetic field at a specific angle to the crystal. This can be tested using microwave spectroscopy on the

^{3}D

_{1}levels and we expect to suppress this shift to better than 10

^{-19}under realistic conditions.