The best cesium clocks are good to ~one second every 300 million years, now surpassed by optical lattice clocks.
from
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: "Optical “lattice clocks,” which embed ultracold ytterbium or strontium atoms in a matrix of laser light — think of eggs resting in an egg carton — are also achieving stunning levels of performance. Because lattice clocks allow scientists to probe thousands of atoms at once, they can perform precision measurements much faster than single-ion clocks. This allows scientists to validate the clock performance over shorter time periods and perform precision studies of the clock frequencies. NIST’s lattice clocks would not have gained or lost a second had they started running at the Big Bang, roughly 13.8 billion years ago."
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The 87Sr version of the optical lattice clock was used in the 'Quantum Sensor Breakthrough...' thread here, as well as:
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The NIST pages are intended for the general public and most papers on optical lattice clocks were behind paywalls in my quick search, but arXiv.org has some details of current research:
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Glen