reach the same temperature. Hence the data shown here were taken at 80 mK for [Dy$_{2}$Ti$_{2}$O$_{7}$]{} and 200 mK (and 80 mK when possible) for [Ho$_{2}$Ti$_{2}$O$_{7}$]{}.

[**Monopole density**]{}

We have phenomenologically estimated how the monopole density depends upon the rate of sample cooling, $dT/dt$ and the spin relaxation time $\tau(T) = 1/\nu(T)$, which is derived from the peaks in the imaginary component of the ac susceptibility. Differentiation of $\tau(T)$ to give $d\tau/dT$ and hence $dT/d\tau$, allows definition of an equilibrium cooling rate $dT/d\tau$, that gives the maximum cooling rate that may still maintain equilibrium. Fig. 2b compares $dT/dt$ and $dT/d \tau$ for both [Dy$_{2}$Ti$_{2}$O$_{7}$]{}  and [Ho$_{2}$Ti$_{2}$O$_{7}$]{}. It can be seen that after the AQP, $dT/dt$ for [Ho$_{2}$Ti$_{2}$O$_{7}$]{} crosses the equilibrium curve and goes out of equilibrium at $\approx 0.9$ K, and for [Dy$_{2}$Ti$_{2}$O$_{7}$]{} at $\approx 0.72$ K. The upper limit of the monopole density at low temperature can be estimated by equating it to the theoretical value at the crossing temperature: thus we find one monopole on approximately every $10^3$ tetrahedra for both [Ho$_{2}$Ti$_{2}$O$_{7}$]{} and [Dy$_{2}$Ti$_{2}$O$_{7}$]{}.

[**Spontaneous relaxation**]{}

We studied the effect of wait time $t_{\rm w}$ between the end of the avalanche quench and the application of the field 