A mathematically minded model of atomic clocks

We have reached a point where timekeeping IQ has become standard for the function of everyday systems. Advances in laser spectroscopy and atom manipulation methods have driven the pursuit for better performing atomic clocks, clocks which, in their states of excitation and through their atomic resonance with the cesium or rubidium atoms, interact with electromagnetic radiation to create quantified fluctuations. Though a neutrality thwart can arise once we consider that, although our intentions for improvement are clear, the solutions to our problems are not. And the problems come in handfuls. But a situational hiccup rarely prevents the intervention of the universal constant itself, mathematics.
The sort of problems I’m referring to can be understood by delving into the functions that characterize an Atomic Clock. The focus of our discussion will be based on the accuracy of a timekeeping system. Accuracy relates to the level of frequency synchronization in several copies of a single clock. Note that accuracy does not equate to resolution.
Accuracy depends on certain factors. Here’s a few:
Firstly, the temperature of the atoms. In colder states, atoms move more slowly because the average motion energy (kinetic energy) is decreased. And vice versa for the hotter states. This is because molecules at higher temperatures have more energy and can vibrate faster. The change in friction tends to result in shortened or elongated probe pulses. These thermal results extend to more than just velocity. Cesium based clocks, for example, cannot endure certain conditions because it uses one of the five elemental metals that melts upon nearing to room temperature. It is, therefore, crucial to the functionality of the clock to note the effects of climate conductivity. In this case, we can use the thermal equilibrium formula:
[mcΔT]1 + [mcΔT]2 = 0


m = mass,
C = specific heat capacity (a constant which differs for different substances. For instance, water: C = 4.184 Joule per gram Celsius),
ΔT = the difference in temperature (hot temperature – cold temperature).
If you have two objects with a varying temperature, thermal energy will be transferred from the highest temperature to the lowest temperature. This is based on the conservation of momentum, and since momentum is conserved as it transfers, the objects will eventually reach thermal equilibrium. This quantity equity allows us to find the point in which the temperature of systems reach uniformity.
Secondly, Special Relativity. Relativity concludes that time moves relative to the observer. This is because gravitational fields affect time. These relativistic effects can cause clock drift, the desynchronizing of a fistful of clocks as a result of time dilation. But time dilation isn’t just a theory, its applications extend to functions like the ones in global positioning satellites, astronaut aging, and pulsar timing. Dilation requires these applications to consider the effects of mass and motion. We can use the Schwarzschild solution to the Einstein field equations to show the gravitational field outside an objects mass. Such is the following;
dtE2 = (1-2GMi / ric2) dt2/c – (1-2GMi / ric2) -1dx2 + dy2 + dz2 / c2

The Schwarzschild metric, also known as the Schwarzschild vacuum or solution, solves Einstein’s field equations by incorporating spherical symmetry. Where:

dtE is an axiom to proper time (the recording capacity of an atomic clock),
dtc is an axiom to the coordinate time, tc,
dx, dy, dz are axioms in the x, y, z coordinates of the clock’s position,
-GMi / ri symbolizes the sum of Newtonian potentials.
The coordinate velocity (celerity) is the coordinate distance over the time derived by an observer. This could be the time between two events measured by any observer. The one with the proper time belongs to the one who passes through both events. This is shown by:

v2 = dx2+dy2+dz2/dt2/c

This is the time coordinate of relativist theory. It used in a standard coordinate system to represent the length of a worldline, that is, the time interval derived from two events compared in a set of coordinates. An inertial observer measures this time with a clock of their own and a particular perspective on operation simultaneity, this might differ from the observations of a differing source. We would read this in a hypothetical clock, one far from the pull of gravitational mass, the “coordinate clock”. The numerical relationship between rate of coordinate and proper time for clock whose velocity points in the direction of the point and the object is:

dtE2/dtc = √ 1+ 2U/c2 – v2/c2 + (c2/2U+1) -1VII2/c2 = √1- (β2+β2e+β2II β2e /1-β2e)


VII is radial velocity,
Ve = √ 2GMi / ri is the escape speed,
β= v/c, βe = ve/c and βII = vII/c are velocities (as percentages to speed of light)
U= √ GMi / ri is the Newtonian Potential.
The Schwarzschild metric can allow us to derive the effects gravity has on space-time. This becomes useful once we take into consideration how the main function of clocks is timekeeping, and a warp in time can persuade it into existential nihilism.

Thirdly, frequency and linewidth of transition. The frequency in the sound of a clock is based on the vibrations caused by the mechanism that runs it, which, in our case, is atoms. The smaller the wavelength and higher the frequency, the more precise the timekeeping is. This is because the time intervals are considered in smaller rates, this allows for a more precise register. Linewidth, otherwise known as spontaneous emission, is the process is which a quantum system emits a photon after decelerating its state of excitation. Without the desired states of both, a system cannot prosper.

To calculate wave period (the time between oscillations), we need to recall the frequency equation, that is:

f = 1/t


f is frequency,
t is period.
To calculate frequency (the number of oscillations we have per unit time), we use the following frequency formula:

f = v/Λ

v is the wave speed,
Λ is the wavelength of the wave.
To define linewidth, we should consider the effects of quantum excitations. When the atom is in an excited state, E2, it may decay to a lower state, E1. This causes the difference between the two states to release a photon with location displacement ω and energy ħω:

E2 – E1 = ħω

E2 is an atom in an excited state,
E1 is atom decay to lower state,
ω is location displacement,
Ħ is a version of Planck’s constant.
We are usually unaware of the systems that, once lost, would make our world banal. This especially true for systems that have surpassed their purpose as commodities. And as dependency on atomic clocks intensifies, so does investment in the perfection of their mechanism. Accomplishing this becomes particularly difficult when the external factors that perform incongruencies in the process are numerous. But failure becomes affordable once we abide by the idea that every solution began as a problem unsolved. This allows us to consider that, by using things such as numerical tools, we can get one second closer to reaching a cosmic ditto.

Featured image retrieved from: https://www.mpq.mpg.de/4743142/laser-spectroscopy
1. https://www.omnicalculator.com/physics/frequency
2. https://jila.colorado.edu/news-events/articles/most-stable-clock-world
3. https://en.wikipedia.org/wiki/Spontaneous_emission
4. https://citeseerx.ist.psu.edu/viewdoc/download?doi=
5. https://physicsworld.com/a/atomic-clocks-feel-the-heat/
6. https://en.wikipedia.org/wiki/Atomic_clock
7. https://www.thalesgroup.com/en/worldwide/news/atomic-clocks-and-importance-being-time
8. https://www.nasa.gov/feature/jpl/what-is-an-atomic-clock

Estefania Olaiz

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Marina T.
Marina T.
1 month ago

Excellent article and well-explained topic.

1 month ago

Interesting information

Jack N
Jack N
1 month ago

Certainly a very interesting and complicated topic. A note on mathematical typesetting: LaTeX is an incredibly useful way to make your mathematics more readable. Keep up the good work.

Carolina González
Carolina González
1 month ago

Ohhh, very interesting article!!, Soooo…what will be the margin of error?

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