Editing Magnetic flux quantum

{{Technical|date=April 2024}}{{short description|Quantized unit of magnetic flux}}

The [[magnetic flux]], represented by the symbol {{math|'''Φ'''}}, threading some contour or loop is defined as the magnetic field {{math|'''B'''}} multiplied by the loop area {{math|'''S'''}}, i.e. {{math|1='''Φ''' = '''B''' ⋅ '''S'''}}. Both {{math|'''B'''}} and {{math|'''S'''}} can be arbitrary, meaning that the flux {{math|'''Φ'''}} can be as well but increments of flux can be quantized. The wave function can be multivalued as it happens in the [[Aharonov–Bohm effect]] or quantized as in [[superconductors]]. The unit of quantization is therefore called '''magnetic flux quantum'''.

== Dirac magnetic flux quantum ==
The first to realize the importance of the flux quantum was Dirac in his publication on monopoles<ref>{{cite journal |last=Dirac |first=Paul |author-link=Paul Dirac |title=Quantised Singularities in the Electromagnetic Field |journal=Proceedings of the Royal Society A |location=London |volume=133 |page=60 |year=1931 |issue=821 |doi=10.1098/rspa.1931.0130 |bibcode=1931RSPSA.133...60D |url=http://rspa.royalsocietypublishing.org/content/133/821/60 |url-access=subscription }}</ref>
{| class="wikitable"
|+ Dirac magnetic flux quantum<ref name="Kittel2">{{cite book|title=[[Introduction to Solid State Physics]]|author=C. Kittel|publisher=Wiley & Sons|year=1953–1976|isbn=978-0-471-49024-1|pages=281}}</ref> 
|-
! scope="col" style="width: 50%;" | [[International System of Quantities|ISQ]]
! scope="col" style="width: 50%;"| [[CGS units]]
|-
| <math>\Phi_0 = \frac{2\pi \hbar }{q}= \frac{h }{q}</math>
| <math>\Phi_0 = \frac{2\pi \hbar c}{q}= \frac{h c}{q}</math>
|}
The phenomenon of flux quantization was predicted first by [[Fritz London]] then within the [[Aharonov–Bohm effect]] and later discovered experimentally in superconductors (see ''{{slink|#Superconducting magnetic flux quantum}}'' below).

== Superconducting magnetic flux quantum ==
{| class="wikitable" style="float: right;"
! colspan=2 | CODATA values
! Units
|-
| {{math|Φ}}<sub>0</sub> || {{physconst|Phi0|unit=no}} || [[Weber (unit)|Wb]] 
|-
| {{math|''K''}}<sub>J</sub> || {{physconst|KJ|unit=no}} || [[Hertz|Hz]]/[[volt|V]]
|-
|}
If one deals with a superconducting ring<ref>{{cite journal | url=https://www.nature.com/articles/nphys813 | doi=10.1038/nphys813 | title=Magnetic flux periodicity of h/E in superconducting loops | date=2008 | last1=Loder | first1=F. | last2=Kampf | first2=A. P. | last3=Kopp | first3=T. | last4=Mannhart | first4=J. | last5=Schneider | first5=C. W. | last6=Barash | first6=Y. S. | journal=Nature Physics | volume=4 | issue=2 | pages=112–115 | arxiv=0709.4111 | bibcode=2008NatPh...4..112L }}</ref> (i.e. a closed loop path in a [[superconductor]]) or a hole in a bulk [[superconductor]], the magnetic flux threading such a hole/loop is quantized. 

The (superconducting) '''magnetic flux quantum''' {{nowrap|{{math|1=Φ<sub>0</sub> = ''h''/(2''e'')}} ≈ {{physconst|Phi0}}}} is a combination of fundamental physical constants: the [[Planck constant]] {{math|''h''}} and the [[electron charge]] {{math|''e''}}. 
Its value is, therefore, the same for any superconductor.

To understand this definition in the context of the Dirac flux quantum one shall consider that the effective quasiparticles active in a superconductors are [[Cooper pairs]] with an effective charge of 2 electrons {{math|1=''q'' = 2''e''}}.

The phenomenon of flux quantization was first discovered in superconductors experimentally by B. S. Deaver and W. M. Fairbank<ref name=Deaver:1961:FluxQuantum /> and, independently, by R. Doll and M. Näbauer,<ref name=Doll:1961:FluxQuantum /> in 1961. The quantization of magnetic flux is closely related to the [[Little–Parks effect]],<ref>{{Cite journal|last=Parks|first=R. D.|date=1964-12-11|title=Quantized Magnetic Flux in Superconductors: Experiments confirm Fritz London's early concept that superconductivity is a macroscopic quantum phenomenon|url=https://www.science.org/doi/10.1126/science.146.3650.1429|journal=Science|language=en|volume=146|issue=3650|pages=1429–1435|doi=10.1126/science.146.3650.1429|issn=0036-8075|pmid=17753357|s2cid=30913579|url-access=subscription}}</ref> but was predicted earlier by [[Fritz London]] in 1948 using a [[Phenomenology (particle physics)|phenomenological model]].<ref>{{Cite book|url=https://books.google.com/books?id=VNxEAAAAIAAJ|title=Superfluids: Macroscopic theory of superconductivity|last=London|first=Fritz|date=1950|publisher=John Wiley & Sons|pages=152 (footnote)|language=en}}</ref><ref name=":0" />

The inverse of the flux quantum, {{math|1/Φ<sub>0</sub>}}, is called the '''Josephson constant''', and is denoted {{math|''K''}}<sub>J</sub>. It is the constant of proportionality of the [[Josephson effect]], relating the [[potential difference]] across a Josephson junction to the [[frequency]] of the irradiation. {{anchor|KJ-1990}}The Josephson effect is very widely used to provide a standard for high-precision measurements of potential difference, which (from 1990 to 2019) were related to a fixed, [[conventional electrical unit|conventional value]] of the Josephson constant, denoted {{math|''K''}}<sub>J-90</sub>.  With the [[2019 revision of the SI]], the Josephson constant has an exact value of {{math|''K''}}<sub>J</sub> = {{val|483597.84841698|end=...|u=GHz⋅V{{sup|−1}}}}.<ref>{{cite web|url=https://www.bipm.org/utils/en/pdf/si-mep/MeP-a-2018.pdf|title=''Mise en pratique'' for the definition of the ampere and other electric units in the SI|publisher=[[BIPM]] |archive-url=https://web.archive.org/web/20210308034514/https://www.bipm.org/utils/en/pdf/si-mep/MeP-a-2018.pdf |archive-date=2021-03-08 |url-status=dead}}</ref>

== Derivation of the superconducting flux quantum ==
The following physical equations use SI units. In CGS units, a factor of {{math|''c''}} would appear.

The superconducting properties in each point of the [[superconductor]] are described by the ''complex'' quantum mechanical wave function {{math|Ψ('''r''', ''t'')}} – the superconducting order parameter. As with any complex function, {{math|Ψ}} can be written as {{math|1=Ψ = Ψ<sub>0</sub>''e''<sup>''iθ''</sup>}}, where {{math|Ψ<sub>0</sub>}} is the amplitude and {{math|''θ''}} is the phase. Changing the phase {{math|''θ''}} by {{math|2''πn''}} will not change {{math|Ψ}} and, correspondingly, will not change any physical properties. However, in the superconductor of non-trivial topology, e.g. superconductor with the hole or superconducting loop/cylinder, the phase {{mvar|θ}} may continuously change from some value {{math|''θ''<sub>0</sub>}} to the value {{math|''θ''<sub>0</sub> + 2''πn''}} as one goes around the hole/loop and comes to the same starting point. If this is so, then one has {{mvar|n}} magnetic flux quanta trapped in the hole/loop,<ref name=":0">{{Cite web| url=https://feynmanlectures.caltech.edu/III_21.html |title=The Feynman Lectures on Physics Vol. III Ch. 21: The Schrödinger Equation in a Classical Context: A Seminar on Superconductivity, Section 21-7: Flux quantization | website=feynmanlectures.caltech.edu| access-date=2020-01-21}}</ref> as shown below:

Per [[minimal coupling]], the [[current density]] of [[Cooper pair|Cooper pairs]] in the superconductor is:
<math display="block">\mathbf J = \frac{1}{2m} \left[\left(\Psi^* (-i\hbar\nabla) \Psi - \Psi (-i\hbar\nabla) \Psi^*\right) - 2q \mathbf{A} |\Psi|^2 \right] .</math>
where {{math|1=''q'' = 2''e''}} is the charge of the Cooper pair.
The wave function is the [[Ginzburg–Landau theory|Ginzburg–Landau order parameter]]:
<math display="block">\Psi(\mathbf{r})=\sqrt{\rho(\mathbf{r})} \, e^{i\theta(\mathbf{r})}.</math>

Plugged into the expression of the current, one obtains:
<math display="block">\mathbf{J} = \frac{\hbar}{m} \left(\nabla{\theta}- \frac{q}{\hbar} \mathbf{A}\right)\rho.</math>

Inside the body of the superconductor, the current density '''J''' is zero, and therefore
<math display="block">\nabla{\theta} = \frac{q}{\hbar} \mathbf{A}.</math>

Integrating around the hole/loop using [[Stokes' theorem]] and {{math|1=∇ × '''A''' = '''B'''}} gives:
<math display="block">\Phi_B = \oint\mathbf{A}\cdot d\mathbf{l} = \frac{\hbar}{q} \oint\nabla{\theta}\cdot d\mathbf{l}.</math>

Now, because the order parameter must return to the same value when the integral goes back to the same point, we have:<ref> R. Shankar, "Principles of Quantum Mechanics", eq. 21.1.44</ref>
<math display="block">\Phi_B=\frac{\hbar}{q} 2\pi = \frac{h}{2e}.</math>

Due to the [[Meissner effect]], the magnetic induction {{math|'''B'''}} inside the superconductor is zero. More exactly, magnetic field {{math|'''H'''}} penetrates into a superconductor over a small distance called [[London penetration depth|London's magnetic field penetration depth]] (denoted {{math|''λ''<sub>L</sub>}} and usually ≈&nbsp;100&nbsp;nm). The screening currents also flow in this {{math|''λ''<sub>L</sub>}}-layer near the surface, creating magnetization {{math|'''M'''}} inside the superconductor, which perfectly compensates the applied field {{math|'''H'''}}, thus resulting in {{math|1='''B''' = 0}} inside the superconductor.

The magnetic flux frozen in a loop/hole (plus its {{math|''λ''<sub>L</sub>}}-layer) will always be quantized. However, the value of the flux quantum is equal to {{math|Φ<sub>0</sub>}} only when the path/trajectory around the hole described above can be chosen so that it lays in the superconducting region without screening currents, i.e. several {{math|''λ''<sub>L</sub>}} away from the surface. There are geometries where this condition cannot be satisfied, e.g. a loop made of very thin ({{math|≤ ''λ''<sub>L</sub>}}) superconducting wire or the cylinder with the similar wall thickness. In the latter case, the flux has a quantum different from {{math|Φ<sub>0</sub>}}.

The flux quantization is a key idea behind a [[SQUID]], which is one of the most sensitive [[magnetometer]]s available.

Flux quantization also plays an important role in the physics of [[type II superconductor]]s. When such a superconductor (now without any holes) is placed in a magnetic field with the strength between the first critical field {{math|'''H'''<sub>c1</sub>}} and the second critical field {{math|'''H'''<sub>c2</sub>}}, the field partially penetrates into the superconductor in a form of [[Abrikosov vortex|Abrikosov vortices]]. The [[Abrikosov vortex]] consists of a normal core – a cylinder of the normal (non-superconducting) phase with a diameter on the order of the {{math|''ξ''}}, the [[superconducting coherence length]]. The normal core plays a role of a hole in the superconducting phase. The magnetic field lines pass along this normal core through the whole sample. The screening currents circulate in the  {{math|''λ''<sub>L</sub>}}-vicinity of the core and screen the rest of the superconductor from the magnetic field in the core. In total, each such [[Abrikosov vortex]] carries one quantum of magnetic flux {{math|Φ<sub>0</sub>}}.

== Measuring the magnetic flux ==
Prior to the [[2019 revision of the SI]], the magnetic flux quantum was measured with great precision by exploiting the [[Josephson effect]]. When coupled with the measurement of the [[von Klitzing constant]] {{math|1=''R''<sub>K</sub> = ''h''/''e''<sup>2</sup>}}, this provided the most accurate values of the [[Planck constant]] {{math|''h''}} obtained until 2019. This may be counterintuitive, since {{math|''h''}} is generally associated with the behaviour of microscopically small systems, whereas the quantization of magnetic flux in a superconductor and the [[quantum Hall effect]] are both [[emergent phenomena]] associated with [[thermodynamics|thermodynamically]] large numbers of particles.

As a result of the [[2019 revision of the SI]], the Planck constant {{math|''h''}} has a fixed value {{nowrap|1={{math|''h''}} = {{physconst|h|after=,}}}} which, together with the definitions of the [[second]] and the [[metre]], provides the official definition of the [[kilogram]]. Furthermore, the [[elementary charge]] also has a fixed value of {{nowrap|1={{math|''e''}} = {{physconst|e}}}} to define the [[ampere]]. Therefore, both the Josephson constant {{math|1=''K''<sub>J</sub> = 2''e''/''h''}} and the von Klitzing constant {{math|1=''R''<sub>K</sub> = ''h''/''e''<sup>2</sup>}} have fixed values, and the Josephson effect along with the von Klitzing quantum Hall effect becomes the primary ''mise en pratique''<ref>{{cite web |url=https://www.bipm.org/en/publications/mises-en-pratique/ |title=BIPM – mises en pratique|website=www.bipm.org|access-date=2020-01-21}}</ref> for the definition of the ampere and other electric units in the SI.

== See also ==
* [[Aharonov–Bohm effect]]
* [[Brian Josephson]]
* [[Committee on Data for Science and Technology]]
* [[Domain wall (magnetism)]]
* [[Flux pinning]]
* [[Ginzburg–Landau theory]]
* [[Husimi Q representation]]
* [[Macroscopic quantum phenomena]]
* [[Magnetic domain]]
* [[Magnetic monopole]]
* [[Quantum vortex]]
* [[Topological defect]]
* [[von Klitzing constant]]

== References ==
{{reflist
|refs =
<ref name=Deaver:1961:FluxQuantum>{{cite journal |last1=Deaver|first1=Bascom|last2=Fairbank |first2 = William |title = Experimental Evidence for Quantized Flux in Superconducting Cylinders |journal=Physical Review Letters |date=July 1961 |volume=7 |issue=2 |pages=43–46 |doi = 10.1103/PhysRevLett.7.43 |bibcode = 1961PhRvL...7...43D }}</ref> 
<ref name=Doll:1961:FluxQuantum>{{cite journal |last1=Doll |first1=R.|last2=Näbauer |first2 = M. |title = Experimental Proof of Magnetic Flux Quantization in a Superconducting Ring |journal=Physical Review Letters |date=July 1961 |volume=7 |issue=2 |pages=51–52 |doi = 10.1103/PhysRevLett.7.51 |bibcode = 1961PhRvL...7...51D }}</ref> 
}}

== Further reading ==
* [[Aharonov–Bohm effect]] and flux quantization in superconductors {{citation |title=(physics stackexchange) |url=https://physics.stackexchange.com/questions/34990/aharonov-bohm-effect-and-flux-quantization-in-superconductors }}
* David tong lectures: {{citation |title=Quantum hall effect |url=https://www.damtp.cam.ac.uk/user/tong/qhe/qhe.pdf }}

[[Category:Superconductivity]]
[[Category:Quantum magnetism]]
[[Category:Metrology]]
[[Category:Physical constants]]