Neither of the definitions pre-suppose the existence of any advanced technology in principle. But in practice advanced technology is required to make the required measurements. Here's my stab at constructing minimal "primary standards" for the existing SI system. I copy the current (April 2, 2022) definitions of the NIST SI units from https://physics.nist.gov/cuu/Units/current.html
Making a ruler that measures 1 m:
SI definitions discussion
The definiiton of the meter is
The meter, symbol m, is the SI unit of length. It is defined by taking the fixed numerical value of the speed of light in vacuum c to be 299 792 458 when expressed in the unit m s-1, where the second is defined in terms of ΔνCs.
What this tells us is that $1 \text{ m} = (c) * (1\text{ s}) / 299792458$ and $1 \text{ s}$ is defined by
The second, symbol s, is the SI unit of time. It is defined by taking the fixed numerical value of the cesium frequency ΔνCs, the unperturbed ground-state hyperfine transition frequency of the cesium 133 atom, to be 9 192 631 770 when expressed in the unit Hz, which is equal to s-1.
This means that to realize the meter it is necessary to realize the second. We can see that the second is defined by
$$
1 \text{ s} = 1/ \Delta\nu_{\text{Cs}}
$$
Where $\Delta \nu_{\text{Cs}}$ is the frequency of the Cs ground state hyperfine transition. From this discussion we see that to realize the meter we must have a realization of the second which means we must have a realization of the ground state hyperfine transition frequency in Cs. This means that if we want a primary standard for the meter we need Cs atoms and we need a way to measure it's ground state hyperfine transition frequency. This means that we need an atomic clock.
How do atomic clocks work?
You may find this answer helpful in undestanding how atomic clock work. The basic idea of an atomic clock is you have individually seperated (in a thermal or ultracold gas) Cs atoms. You shine microwaves onto the Cs atoms. As you tune the frequency of the microwaves you will see that there is one frequency where the microwaves are most efficiently absorbed by the Cs atoms. As you tune the microwaves off resonance the absorption probability decreases. The shape of the absorption as a function of frequency might look like the graph below.
An atomic clock works by (1) shining microwaves through a sample of atoms (2) measuring the absorption as a function of frequency and then (3) setting the microwaves to be as close to resonance as possible given the uncertainty of the measurement of the absorption function. Once this calibration of the microwave frequency has been performed, the oscillation of the microwaves serves as your "clock". According to the SI definition of the second, we would realize the second by (1) performing the above calibration on the microwave generator and then (2) counting oscillations of the microwaves until we count to 9,192,631,770. In short: we perform microwave spectroscopy of the Cs ground state hyperfine transition and then count oscillation of microwaves calibrated according to that spectroscopy. State of the art atomic clocks are state of the art because they can perform the most accurate and precise spectroscopy in the world.
Can you "DIY" an atomic clock?
Supplies: For this, you need a sample of Cs and a source of microwaves and a microwave detector. The first hit on google shows Cs vapor cells available for ~$600. If you want to DIY your own Cs vapor cell you're going to have to learn a lot of skills that I am not familiar with. Chemists might know how to make the vapor cells and source Cs but I don't. You can also buy the required microwave electronics directly, but, a dedicated electronics hobbyist could build the microwave generator, horn, and antenna/sensor from basic components themselves. I would recommend studying designs for commercial Cs atomic clocks to see the "easiest" way to do things as they probably utilize some nice tricks that make the job much easier.
Realizing the SI m
Approach (1): Time of Flight Measurement
Above we saw that $1\text{ m} = (c) * (1\text{ s})$. We also know that if an object is moving with velocity $v$ for time $t$ then the distance travelled is $d = vt$. Since electromagnetic radiation travels at constant speed with $v=c$ this means that $1 \text{ m}$ is the distance that electromagnetic radiation travels in $1 \text{ s}$.
This informs one measurement approach. We can setup some source of electromagnetic waves and setup a reflector for those waves distance $d$ away. We then setup a sensor for the radiation close to the source. We then time how long it takes the waves to travel to the reflector and back. We then know the distance between the source and the reflector is $d = ct/2$.
To measure the duration we need to use our reference atomic clock. The experiment would look something like this:
- Run atomic spectrosocpy to calibrate the atomic clock.
- Simultaneously trigger the generation of your electromagnetic radiation and counting of periods of the atomic clock's microwaves.
- When the sensor senses that your electromagnetic radiation has returned stop counting the ticks from the atomic clock.
We would then have
$$
d = c t = c N / \Delta \nu_{\text{Cs}} = (N) * (c) * (1\text{ s}) = (N) * (1\text{ m})
$$
where $N$ is the number of periods measured. Depending on how good you want to get at electronics, it will probably be reasonable to measure times of flight no shorter than ~10 $\mu\text{s}$. This corresponds to about $N\approx 9000$ periods of the atomic clock and a distance of $d\approx 1.5 \text{ km}$.
You have now generated a reference kilometer (likely with pretty good accuracy). To convert this to a reference for a single meter you would likely need to use some trig and surveying techniques. My guess is that the surveying would introduce some of the largest inaccuracies of your realization of the meter.
Above I left it vague as to the nature of the electromagnetic radiation that you are actually using for your time of flight measurement. In practice that radiation could be light and a regular mirror in which case you would need a source of light such as a laser that can be electronically triggered to generate light at a precice moment in time (for synchronization with the clock) as well as a photodetector whose readout can be precisely timed. Alternatively, if you're already a microwave electronics guru from building an atomic clock, you could use a microwave horn, reflector, and receiver/antenna for your experiment.
If you want to relax the constraint of electronic triggering you could do something like the old school speed of light measurements to accurately measure a much larger distance. However, in this case you will need to do more extensive surveying to get a realization of a single meter. One the other hand, if you want to get better at electronics you can try to measure faster and faster times (perhaps down to 10-50 ns) to reduce the amount of dividing down/surverying you need to do to realize the single meter.
Supplies: Atomic clock, triggerable EM source, timeable EM sensor, EM reflector, syncrhonization electronics.
Making a 1 kg reference mass
SI definitions discussion
The definition of the kg is
The kilogram, symbol kg, is the SI unit of mass. It is defined by taking the fixed numerical value of the Planck constant h to be 6.626 070 15 × 10-34 when expressed in the unit J s, which is equal to kg m2 s-1, where the meter and the second are defined in terms of c and ΔνCs.
This tells us that
$$
1 \text{ kg} = (h) * (1 \text{ s}) * (1 \text{ m})^{-2} * (6.62607015 \times 10^{34})
$$
Here $h$ is Planck's constant and 1 s and 1 m are defined as above.
YIKES!!! this definition tells us that to realize the kg we must have a realization of Planck's constant $h$, the fundamental unit of quantum mechanics! Frankly that is just going to make things tricky experimentally because we need a measurement that refers the mass of a macroscopic object (1 kg big) to some quantum system whose dynamics depend on $h$.
How to relate mass to $h$?
This is done by NIST using the https://www.nist.gov/si-redefinition/kilogram-kibble-balance. High level, the idea is to get an electronic voltage $V$ which is somehow calibrated in a way related to Planck's constant. This voltage is then used to generate magnetic fields which levitate a mass in earth's graviational field. The voltage is tuned to find the exact voltage when the weight of the mass is balanced by the magnetic force. If accleration due to earth's gravitational field is known then the mass of the object can be related to the balancing voltage which is in turn related to $h$.
Can you "DIY" a Kibble balance?
I am much less familiar with Kibble balances than with atomic clocks, I'm learning as I go. But here is my proposal for a homemade Kibble balance.
Getting a Voltage that is Related to $h$.
I found a nice method to get a voltage related to h on youtube. There are two steps involving an LED. The LED emits light of a certain color which has some optical wavelength $\lambda$. Note that, according to the electromagnetic theory of light, $\lambda = c / \nu$ where $c$ is the speed of light and $\nu$ is the frequency of the electromagnetic radiation.
The first step is to hook up the LED to a voltmeter and adjust the voltmeter's voltage until the LED lights up. Because the LED emits individual photons with energy $E = h\nu$ in a way related to (according to the theory of solid state physics) the photoelectric effect, it will not light up in $V = e E = e h \nu$ and $e$ is the charge of the electron.
... this section is a work in progress ...
Old version
Regarding the kg: I just looked this up as a refresher so this is fresh in my brain. It might not be that bad. I watched how to measure Planck's constant. Basically you get an LED and measure its wavelength (using your special ruler you made in the previous step) using a diffraction grating (you'll need to be able to make a grating with sub-wavelength spacing; I'm not sure how this can be done easily). Then you turn up a voltage supply until the LED turns on. Because you know the wavelength of the light, Planck's constant (it is defined) and the speed of light you now know (within the accuracy of your wavelength measurement which will depend on the accuracy of your ruler) the voltage coming out of your voltage supply. You could then use some multiplying or dividing techniques to get calibrated larger voltages.
I guess you then need to figure out how to use that voltage to support a mass. This is how the Kibble balance works (https://en.wikipedia.org/wiki/Kibble_balance, https://www.nist.gov/si-redefinition/kilogram-kibble-balance). In the kibble balance I guess the voltage is used to drive electromagnets that levitate a coil that supports the mass. Yikes, it's going to be hard to get calibrated currents through all those coils. Maybe you win by making all coils out of the same material so you know the resistance is the same for all of them, then you know that all currents in the system (and magnetic fields) are directly proportional to your voltages and you can get the currents and B-fields to cancel out leaving you just with calibrated voltages.
Finally, when you balance the mass you'll have mg so you also need to measure g. But that should be pretty easy, just drop something and time how long it takes.
Phew, I think it could be done. Would be interesting to see a challenge of like, measure 1 m and 1 kg to 10% or 1% precision or something according to the SI definition and do it for under $3k or something.
Edit: I plan to clean this answer up in a bit. For now I wanted to add that the most technologically advanced part of this might be the diffraction grating. Next is probably the LED because it relies on semiconductor technology (even though it’s inexpensive and common place). Then the Cs cell or the microwave generators and sensors depending on if you are more comfortable with E&M or chemistry. Next is probably the mechanical complexity of the Kibble balance then finally all of the advanced experimental methodology/theory including the surveying techniques.
