Discrete quantum effect observed in superconductors with a small insulating layer, a device known as a Josephson junction.
To understand the behaviour effect, it is important to look at the Josephson equations consider the following Josephson effect regimes separately:
A good summary from Wikipedia by physicist Andrew Whitaker:
at a junction of two superconductors, a current will flow even if there is no drop in voltage; that when there is a voltage drop, the current should oscillate at a frequency related to the drop in voltage; and that there is a dependence on any magnetic field
Bibliography:
 www.youtube.com/watch?v=cnZ6exn2CkE "Superconductivity: Professor Brian Josephson". Several random excerpts from Cambridge people talking about the Josephson effect
In 1962 Brian Josephson published his inaugural paper predicting the effect as Section "Possible new effects in superconductive tunnelling (1963, Prediction of the Josephson effect)".
In 1963 Philip W. Anderson and John M. Rowell published their paper that first observed the effect as Section "Possible new effects in superconductive tunnelling (1963, Prediction of the Josephson effect)".
Some golden notes can be found at True Genius: The Life and Science of John Bardeen page 224 and around. Philip W. Anderson commented:
We were all  Josephson, Pippard and myself, as well as various other people who also habitually sat at the Mond tea and participated in the discussions of the next few weeks  very much puzzled by the meaning of the fact that the current depends on the phase
As part of the course Anderson had introduced the concept of broken symmetry in superconductors. Josephson "was fascinated by the idea of broken symmetry, and wondered whether there could be any way of observing it experimentally."
The inaugural that predicted the Josephson effect.
Published on Physics Letters, then a new journal, before they split into Physics Letters A and Physics Letters B. True Genius: The Life and Science of John Bardeen mentions that this choice was made rather than the more prestigious Physical Review Letters because they were not yet so confident about the results.
Paper by Philip W. Anderson and John M. Rowell that first (?) experimentally observed the Josephson effect.
Paywalled by the American Physical Society as of 2023 at: journals.aps.org/prl/abstract/10.1103/PhysRevLett.10.230
TODO understand the graphs in detail.
A reconstruction of their circuit in Ciro's ASCII art circuit diagram notation TODO:
DCR_10XG
There are not details of the physical construction of course. Reproducibility lol.
This is what happens when you apply a DC voltage across a Josephson junction.
It is called "AC effect" because when we apply a DC voltage, it produces an alternating current on the device.
By looking at the Josephson equations, we see that $V(t)=k$ a positive constant, then $φ$ just increases linearly without bound.
Therefore, from the first equation:
we see that the current will just vary sinusoidally between $±I_{c}$.
$I(t)=I_{c}sin(φ(t))$
This meas that we can use a Josephson junction as a perfect voltage to frequency converter.
Wikipedia mentions that this frequency is $484GHz/mV$, so it is very very high, so we are not able to view individual points of the sine curve separately with our instruments.
Also it is likely not going to be very useful for many practical applications in this mode.
An IV curve can also be seen at: Figure "Electron microscope image of a Josephson junction its IV curve".
If you shine microwave radiation on a Josephson junction, it produces a fixed average voltage that depends only on the frequency of the microwave. TODO how is that done more preciesely? How to you produce and inject microwaves into the thing?
It acts therefore as a perfect frequency to voltage converter.
The Wiki page gives the formula: en.wikipedia.org/wiki/Josephson_effect#The_inverse_AC_Josephson_effect You get several sinusoidal harmonics, so the output is not a perfect sine. But the infinite sum of the harmonics has a fixed average voltage value.
And en.wikipedia.org/wiki/Josephson_voltage_standard#Josephson_effect mentions that the effect is independent of the junction material, physical dimension or temperature.
All of the above, compounded with the fact that we are able to generate microwaves with extremely precise frequency with an atomic clock, makes this phenomenon perfect as a Volt standard, the Josephson voltage standard.
TODO understand how/why it works better.
Two equations derived from first principles by Brian Josephson that characterize the device, somewhat like an IV curve:
where:
$I(t)=I_{c}sin(φ(t))dtdφ(t) =ℏ2eV(t) $
 $I_{c}$: Josephson current
 $φ$: the Josephson phase, a function $R→R$ defined by the second equation plus initial conditions
 $V(t)$: input voltage of the system
 $I(t)$: current across the junction, determined by the input voltage
Note how these equations are not a typical IV curve, as they are not an instantaneous dependency between voltage and current: the history of the voltage matters! Or in other words, the system has an internal state, represented by the Josephson phase at a given point in time.
To understand them better, it is important to look at some important cases separately:
 AC Josephson effect: V is a fixed DC voltage
Maximum current that can flow across a Josephson junction, as can be directly seen from the Josephson equations.
Is a fixed characteristic value of the physical construction of the junction.
A function $R→R$ defined by the second of the Josephson equations plus initial conditions.
It represents an internal state of the junction.
A device that exhibits the Josephson effect.
TODO is there any relationship between this and the Josephson effect?
Experimental observation published as Experimental Evidence for Quantized Flux in Superconducting Cylinders.
This appears to happen to any superconducting loop, because the superconducting wave function has to be continuous.
Video "Superconducting Qubit by NTT SCL (2015)" suggests that anything in between gets cancelled out by a superposition of current in both directions.
Paywalled at: journals.aps.org/prl/abstract/10.1103/PhysRevLett.7.43
The first published experimental observation of the magnetic flux quantum.
The paper that follows it in the journal is also of interest, "Theoretical Considerations Concerning Quantized Magnetic Flux In Superconducting Cylinders" by N. Byers and C. N. Yang, it starts:
In a recent experiment, the magnetic flux through a superconducting ring has been found to be quantized in units of ch/2e. Quantization in twice this unit has been briefly discussed by London' and by Onsager. ' Onsager' has also considered the possibility of quantization in units ch/2e due to pairs of electrons forming quasibosons.So there was some previous confusion about the flux quantum due to the presence of Cooper pairs or not.
Dumping the fitures at: archive.org/details/experimentalevidenceforquantizedfluxinsuperconductingcylinders One day we can also dump the paper scans when it goes into the public domain in 2056! Public domain scientific paper by year.
The inverse of the magnetic flux quantum.
As mentioned in True Genius: The Life and Science of John Bardeen page 224, the idea of symmetry breaking was a major motivation in Josephson's study of the Josephson effect.
 the basis for the most promising 2019 quantum computing implementation: superconducting quantum computer
 Josephson voltage standard: the most practical/precise Volt standard, which motivated the definition of the ampere in the 2019 redefinition of the SI base units
 SQUID devices, which are:
 very precise magnetometer
 the basis for superconducting quantum computers
The most practical/precise volt standard.
It motivated the definition of the ampere in the 2019 redefinition of the SI base units
Quick NIST article about it: www.nist.gov/newsevents/news/2013/04/primaryvoltagestandardwholeworld (archive)
The wiki page en.wikipedia.org/wiki/Josephson_voltage_standard contains amazing schematics of the device, apparently made by the US Government.
Can be used as a very precise magnetometer.
There are high temperature yttrium barium copper oxide ones that work on liquid nitrogen.
Two parallel Josephson junctions.
In Ciro's ASCII art circuit diagram notation:

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