Optical atomic clocks based on trapped single ions like this, and also those based on lattices of neutral atoms do not provide a continuous clock signal.

They are used together with a laser (which is a component included in a so-called frequency comb, which acts as a frequency divider between the hundreds of THz of the optical signal and some hundreds of MHz or a few GHz of a clock signal that can be counted with a digital counter; that digital counter could be used as a date and time clock, except that you would need more such optical clocks, to guard against downtime; the present optical clocks do not succeed to operate for very long times before needing a reset because the trapped ion has been lost from the trap or the neutral atoms have been lost from the optical lattice; therefore you need many of them to implement a continuous time scale).

The laser is the one that provides a continuous signal. In this case the laser produces infrared light in the same band as the lasers used for optical fiber communications, and it is based on glass doped with erbium and ytterbium. The frequency of the laser is adjusted to match some resonance frequency of the trapped ion (in this case a submultiple of the frequency, because the frequency of the transition used in the aluminum ion is very high, in ultraviolet). For very short time intervals, when it cannot follow the reference frequency, because that must be filtered of noise, the stability of the laser frequency is determined by a resonant cavity made of silicon (which is transparent for the infrared light of the laser), which is cooled at a very low temperature, in order to improve its quality factor.

So this is similar to the behavior of the clock of a computer, which for long time intervals has the stability of the clocks used by the NTP servers used by it for synchronization, but for short time intervals it has the stability of its internal quartz oscillator.

This new optical atomic clock has the lowest ever uncertainty for the value of its reference frequency, but being a trapped single ion clock it has a higher noise than the clocks based on lattices of neutral atoms (because those can use thousands of atoms instead of one ion), so its output signal must be averaged over long times (e.g. many days) in order to reach the advertised accuracy.

For short averaging times, e.g. of one second, its accuracy is about a thousand times worse than the best attainable (however, its best accuracy is so high that even when averaged for a few seconds it is about as good as the best microwave clocks based on cesium or hydrogen).

Thank you for that very excellent reply.

Is there really an absolute reference point to measure time other than the big bang or something?

We can both agree that the Big Bang happened 13.8 billion years ago but that's all, we'll still disagree about the timing of everything else. Not even the CMB can be used as an universal rest frame. I'm not a cosmologist though.

Relativity. Every path starting at the big bang to the present has its own unique clock.

These clocks can measure the difference in the flow of time between your head and your feet (and quite a lot more accurate than that)

No, we do not have an absolute reference point for time. Our best models of physics are time-translation symmetric (this is equivalent to conversation of energy via Noether's theorem), so absolute time cannot have any measurable effect. There is, however, a human made reference time scale (International Atomic Time, TAI) which is generated by the BIPM using (mostly GPS-based) comparisons of the local time scales of many national labs.

The clock signals can be counted and are accurate over long periods, it is not just a rate that drifts.

Being able to count trillions of ticks is entirely possible in clocks or rotary encoders, just nobody bothers to do so on rotary encoders very often.