in the single-ion spin Hamiltonian, while Dy$^{3+}$, being a Kramers ion, has intrinsically much slower dynamics. However, it should be noted that, at low temperature, bulk relaxation is slower in [Ho$_{2}$Ti$_{2}$O$_{7}$]{} than in [Dy$_{2}$Ti$_{2}$O$_{7}$]{}, owing to its larger $|\mu|$ and hence much smaller monopole density (See Supplementary Fig. 1).

[**Thermal protocol**]{}

![Controlled cooling of spin ice below its freezing temperature. How the temperature of the samples varies during and after the AQP: (a) The applied field (black) during an AQP, and the temperatures measured by a small thermometer glued directly on top of the samples (schematically shown in (b)) vs log time for [Ho$_{2}$Ti$_{2}$O$_{7}$]{} (HTO, red) and $^{\rm nat}$[Dy$_{2}$Ti$_{2}$O$_{7}$]{} (DTO, blue). The inset shows a zoom of the first 6 seconds vs time. (b) Comparison of the sample cooling rates $dT/dt$ as a function of temperature after the AQP for [Ho$_{2}$Ti$_{2}$O$_{7}$]{} and $^{\rm nat}$[Dy$_{2}$Ti$_{2}$O$_{7}$]{}  from the data in (a) to the Òequilibrium cooling rateÓ $dT/d \tau$ extracted from ac susceptibility data for the two samples (see Supplementary Fig. 6). The cooling rate for [Ho$_{2}$Ti$_{2}$O$_{7}$]{}  crosses the equilibrium rate at $\sim 0.9$ K, and $^{\rm nat}$[Dy$_{2}$Ti$_{2}$O$_{7}$]{}  at 0.72 K.[]{data-label="two"}](figure2.pdf){width="7.5cm"}

In previous experiments we have accurately manipulated the monopole density in [Dy$_{2}$Ti$_{2}$O$_{7}$]{} 