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{{Quantum book backlink|Condensed matter and solid-state physics}}

'''Quantum superconductivity''' is the study of [[superconductivity]] as a macroscopic [[quantum mechanics|quantum]] phenomenon. In a superconductor, electrical resistance vanishes and magnetic fields are expelled from the interior by the [[Meissner effect]]. These effects arise because electrons form a coherent quantum state, often described as a charged quantum fluid.

Superconductivity is one of the best-known examples of a '''macroscopic quantum phenomenon''', together with [[superfluidity]], the [[Josephson effect]], the [[quantum Hall effect]], and [[Bose–Einstein condensate]]s. Such systems show quantum behaviour not merely at the atomic scale but across macroscopic distances.<ref>D.R. Tilley and J. Tilley, ''Superfluidity and Superconductivity'', Adam Hilger, Bristol and New York, 1990</ref><ref>{{cite journal|last1=Jaeger|first1=Gregg|title=What in the (quantum) world is macroscopic?|journal=American Journal of Physics|date=September 2014|volume=82|issue=9|pages=896–905|doi=10.1119/1.4878358|bibcode=2014AmJPh..82..896J}}</ref>
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[[File:Flux lines in a superconductor01.jpg|400px]]
<div style="font-size:90%; text-align:center;">
Magnetic flux lines penetrating a type-II superconductor. The field enters in quantized vortex lines while superconductivity remains between the lower and upper critical fields.
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== Macroscopic quantum state ==

A superconducting state is described by a macroscopic wave function,

:<math>\Psi = \Psi_0 \exp(i\varphi)</math>

where <math>\Psi_0</math> is the amplitude and <math>\varphi</math> is the phase. When a quantum state is occupied by a very large number of particles, the wave function describes not only probability but also a macroscopic density and phase-coherent flow.<ref>Fritz London, ''Superfluids'', Wiley, 1954–1964.</ref><ref>{{Cite journal |last1=Gavroglu |first1=K. |last2=Goudaroulis |first2=Y. |title=Understanding macroscopic quantum phenomena: The history of superfluidity 1941–1955 |journal=Annals of Science |volume=45 |issue=4 |pages=367 |year=1988 |doi=10.1080/00033798800200291}}</ref>

The particle-flow density can be related to the phase of the wave function. For a condensate, the velocity is

:<math>\vec{v}_s=\frac{1}{m}\left(\frac{h}{2\pi}\vec{\nabla}\varphi-q\vec{A}\right)</math>

where <math>m</math> is the particle mass, <math>q</math> is the charge, and <math>\vec{A}</math> is the magnetic vector potential.<ref>{{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 21-5: Superconductivity|website=feynmanlectures.caltech.edu|access-date=2020-01-12}}</ref>

This equation shows the central idea: a classical observable such as flow velocity is controlled by the phase of a quantum wave function.

== Relation to superfluidity ==

Superconductivity is closely related to [[superfluidity]], but it is not the same phenomenon. Superfluidity is frictionless mass flow, while superconductivity is resistance-free charge flow.

In [[superfluid helium]], the particles are neutral, so <math>q=0</math>. The velocity of the superfluid is then determined only by the phase gradient:

:<math>\vec{v}_s = \frac{1}{m}\frac{h}{2\pi}\vec{\nabla}\varphi.</math>

Because the wave function must be single-valued, circulation is quantized:

:<math>\oint \vec{v}_s \cdot \mathrm{d}\vec{s} = \frac{h}{m}n.</math>

This leads to quantized vortices in rotating superfluids. Superconductors show a closely related phenomenon, but because the condensate is charged, magnetic flux becomes quantized.

For this reason, superconductivity is often described as a '''charged superfluid'''.

== Cooper pairs ==

In conventional superconductors, the particles forming the macroscopic quantum state are [[Cooper pair]]s. A Cooper pair consists of two electrons bound together by an effective attraction mediated by lattice vibrations, or [[phonon]]s.

Although electrons are [[fermion]]s, a Cooper pair behaves as a composite [[boson]]. The paired electrons can therefore occupy a collective quantum state. This coherent state is responsible for zero electrical resistance and the Meissner effect.

For a Cooper pair,

:<math>m = 2m_e</math>

and

:<math>q = -2e</math>

where <math>m_e</math> is the electron mass and <math>e</math> is the elementary charge.<ref>{{cite book|author=M. Tinkham|title=Introduction to Superconductivity |publisher=McGraw-Hill |year=1975}}</ref>

== Flux quantization ==

Because the superconducting wave function is single-valued, magnetic flux in a superconducting loop is quantized. The basic flux quantum is

:<math>\Phi_0=\frac{h}{2e} = 2.067833758 \times 10^{-15}\,\mathrm{Wb}.</math>

This value is extremely small, yet it is fundamental in superconducting circuits, vortex physics, and precision measurement.

Flux quantization is one of the clearest macroscopic manifestations of quantum mechanics. It shows that a superconducting ring behaves as a single coherent quantum object.

== Type-I and Type-II superconductors ==

Superconductors are commonly divided into two types.

'''Type-I superconductors''' expel magnetic fields completely below a critical magnetic field. When the applied field exceeds this critical value, superconductivity is destroyed abruptly.

'''Type-II superconductors''' have two critical fields. Between the lower critical field <math>H_{c1}</math> and the upper critical field <math>H_{c2}</math>, magnetic flux penetrates the superconductor in quantized vortex tubes. The material remains superconducting around these vortices.

This mixed state was explained by [[Alexei Abrikosov]] using [[Ginzburg–Landau theory]]. In a type-II superconductor, the vortices can form an ordered triangular lattice, now called the [[Abrikosov vortex]] lattice.<ref>{{Cite journal|last1=Landau|first1=Lev Davidovich|last2=Ginzburg|first2=Vitaly L|date=1950|title=On the theory of superconductivity|url=https://cds.cern.ch/record/486430|journal=Zh. Eksp. Teor. Fiz.|volume=20}}</ref><ref>{{Cite journal |last=Abrikosov |first=Alexei A. |date=2004-07-19 |title=Type H Superconductors and the Vortex Lattice |journal=ChemPhysChem |volume=5 |issue=7 |pages=924–929 |doi=10.1002/cphc.200400138 |pmid=15298378 |doi-access=free}}</ref>

== Josephson effect ==

The [[Josephson effect]] occurs when two superconductors are separated by a thin insulating barrier or weak link. Even without a voltage, a supercurrent can tunnel through the barrier.

The DC Josephson relation is

:<math>i_s = i_1\sin(\Delta\varphi)</math>

where <math>\Delta\varphi</math> is the phase difference between the two superconductors.<ref>{{cite journal|author=B.D. Josephson|title=Possible new effects in superconductive tunneling|journal=Phys. Lett.|volume=1|pages=251–253|year=1962|doi=10.1016/0031-9163(62)91369-0|issue=7|bibcode=1962PhL.....1..251J}}</ref>

If a voltage is applied, the phase difference changes in time:

:<math>V=\frac{1}{2\pi}\frac{h}{2e}\frac{\mathrm{d}\Delta\varphi}{\mathrm{d}t}.</math>

This gives an alternating supercurrent with frequency

:<math>\nu = \frac{2eV}{h} = \frac{V}{\Phi_0}.</math>

Josephson junctions are essential in superconducting electronics, [[SQUID]]s, voltage standards, and superconducting quantum circuits.

== SQUIDs ==

A [[SQUID]] is a superconducting quantum interference device. It usually consists of a superconducting loop interrupted by one or more Josephson junctions. Because the total flux through the loop is quantized, the current through the SQUID depends sensitively on the applied magnetic field.

This makes SQUIDs among the most sensitive magnetometers known. They are used in physics, materials science, geophysics, and biomedical measurements.

== Superconductivity and phonons ==

In conventional superconductors, phonons provide the attractive interaction that binds electrons into Cooper pairs. This electron–phonon mechanism is central to [[BCS theory]]. The isotope effect, in which the superconducting critical temperature depends on ionic mass, provides evidence for the role of lattice vibrations in conventional superconductivity.

Not all superconductors are fully explained by simple electron–phonon coupling. High-temperature superconductors, heavy-fermion superconductors, and unconventional superconductors involve more complex mechanisms.

== Quantum gases and mechanical systems ==

Macroscopic quantum behaviour is not limited to superconductors and superfluid helium. Dilute atomic gases cooled to nanokelvin temperatures can form [[Bose–Einstein condensate]]s, in which many atoms occupy a single quantum state.<ref>{{cite journal|author1=Anderson, M.H.|author2=Ensher, J.R.|author3=Matthews, M.R.|author4=Wieman, C.E.|author5=Cornell, E.A.|title=Observation of Bose–Einstein Condensation in a Dilute Atomic Vapor|journal=Science|volume=269|pages=198–201|year=1995|doi=10.1126/science.269.5221.198|pmid=17789847|issue=5221|bibcode=1995Sci...269..198A|doi-access=free}}</ref>

Macroscopic quantum states have also been demonstrated in engineered mechanical systems. In quantum optomechanics and circuit electromechanics, mechanical resonators can be cooled close to their quantum ground state and controlled at the level of individual phonons.<ref>{{Cite journal |last1=Aspelmeyer |first1=Markus |last2=Kippenberg |first2=Tobias J. |last3=Marquardt |first3=Florian |title=Cavity optomechanics |journal=Reviews of Modern Physics |volume=86 |issue=4 |pages=1391–1452 |year=2014 |doi=10.1103/RevModPhys.86.1391 |arxiv=1303.0733 |bibcode=2014RvMP...86.1391A}}</ref><ref>{{Cite journal |first1=A. D. |last1=O’Connell |first2=M. |last2=Hofheinz |first3=M. |last3=Ansmann |first4=R. C. |last4=Bialczak |first5=M. |last5=Lenander |first6=E. |last6=Lucero |first7=M. |last7=Neeley |first8=D. |last8=Sank |first9=H. |last9=Wang |first10=M. |last10=Weides |first11=J. |last11=Wenner |first12=John M.|last12=Martinis |first13=A. N. |last13=Cleland |title=Quantum ground state and single-phonon control of a mechanical resonator |journal=Nature |volume=464 |issue=7289|pages=697–703 |year=2010 |doi=10.1038/nature08967 |bibcode=2010Natur.464..697O |pmid=20237473}}</ref>

== Importance ==

Quantum superconductivity is important because it connects microscopic quantum mechanics to directly observable macroscopic effects. It provides examples of phase coherence, flux quantization, quantum tunneling, collective excitations, and topological defects in real materials.

Applications include superconducting magnets, magnetic sensors, quantum interference devices, particle accelerators, MRI systems, and superconducting qubits for quantum computing.

=See also=
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}

== References ==
{{reflist|3}}

{{Author|Harold Foppele}}
{{Sourceattribution|Physics:Quantum Superconductivity|1}}

Physics:Quantum Superconductivity - Revision history

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