system
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human
Can someone explain the spontaneity of time symmetry breaking in the Lukin and Monroe time crystal experiments?

As I understand it:
In both experiments, there was a drive (lasers in Monroe's, microwave pulses in Lukin's) that causes a change in spin state of a system (Monroe: a chain of ytterbium ions; Lukin: defects in a diamond). In both cases, the spin states of these systems revert back to their original state. However, the drives were periodically applied, and the resulting spin state oscillation did not match the periodicity of the drive, but oscillated at 1/2 or 1/3 the frequency.

(I may have misunderstood this - I'm a biologist, not a physicist. Please correct me if so).

What I don't understand is - if the periodicity of these systems is a result of the periodicity of the drive, and not a product of the systems themselves under specific conditions, why is this *spontaneous* breaking of time symmetry?
gpt
A priori, continuous time translation symmetry is broken by the drive to a discrete time symmetry with period T. To say that the remaining discrete time translation symmetry is spontaneously broken is to say that the Hamiltonian (which governs the evolution of the system) still has time translation symmetry with period T (i.e. H(t+T)=H(t)), but the *state* of the system only obeys a lower symmetry, in this case |\psi(t+NT)>=l\psi(t)>. We then would say that the time translation symmetry with period T was spontaneously broken to a time translation symmetry with period NT.