than the tunnelling matrix element $\Delta E$. The monopolar fields may add a longitudinal component that takes the spin off the resonance condition (Fig. 1c,d) but in addition may add a transverse component that amplifies $\Delta E$: together these lead to a suppression and dispersion of the monopole mobility.

In the following, we will demonstrate experimentally that hyperfine interactions (Fig. 1e) play a significant role in bringing monopoles back to their resonance condition, enabling dynamics at very low temperatures ($T~<~0.35$ K).

Results {#results .unnumbered}
=======

[**Samples**]{}

                      $I$                                                                            
  -- ------------------------------------- ------ -------- -------- --------- ---------------------- --
                      7/2                   4.17    0.3     0.034      5.7     1$\times$10$^{-5}  $  
         $\approx$ 19 $\%$, $^{161}$Dy      5/2    -0.48             -0.0039                         
         $\approx$ 25 $\%$, $^{163}$Dy      5/2     0.67             0.0054                          
      $\approx$ 66 $\%$, $^{\rm other}$Dy    0       0                  0                            
                       0                     0       0        0                                      
                      5/2                   0.67   0.0599   0.0054                                   

To investigate the effect of nuclear spins on the magnetic relaxation in spin ice, we studied four spin ice samples: [Ho$_{2}$Ti$_{2}$O$_{7}$]{}, with $I = 7/2$ and three [Dy$_{2}$Ti$_{2}$O$_{7}$]{} samples spanning a range of nuclear spin composition from $I=0$ to $I=5/2$. Details of nuclear spins and hyperfine parameters are given in Table 1. Ho$^{3+}$ is a non-Kramers ion with intrinsically fast dynamics owing to the possibility of transverse terms 