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{{short description|Quantum physical phenomenon}}
{{technical|date=September 2018}}
[[File:NISTvoltChip.jpg|thumb|Josephson junction array chip developed by the [[NIST|National Institute of Standards and Technology]] as a standard volt]]

In physics, the '''Josephson effect''' is a phenomenon that occurs when two [[superconductors]] are placed in proximity, with some barrier or restriction between them. The effect is named after the British physicist [[Brian Josephson]], who predicted in 1962 the mathematical relationships for the current and voltage across the weak link.<ref name="possibleNewEffects">{{cite journal |author=Josephson |first=B. D. |year=1962 |title=Possible new effects in superconductive tunnelling |journal=Physics Letters |volume=1 |issue=7 |pages=251–253 |bibcode=1962PhL.....1..251J |doi=10.1016/0031-9163(62)91369-0}}</ref><ref name="Joe">{{cite journal |author=Josephson |first=B. D. |year=1974 |title=The discovery of tunnelling supercurrents |journal= Reviews of Modern Physics|volume=46 |issue=2 |pages=251–254 |bibcode=1974RvMP...46..251J |doi=10.1103/RevModPhys.46.251 |s2cid=54748764|url=https://www.europhysicsnews.org/10.1051/epn/19740503001/pdf }}
:Also in {{cite journal |doi=10.1051/epn/19740503001 |title=The Discovery of Tunnelling Supercurrents |date=1974 |last1=Josephson |first1=B. D. |journal=Europhysics News |volume=5 |issue=3 |pages=1–5 |bibcode=1974ENews...5c...1J }}
</ref> It is an example of a [[macroscopic quantum phenomenon]], where the effects of quantum mechanics are observable at ordinary, rather than atomic, scale. The Josephson effect has many practical applications because it exhibits a precise relationship between different physical measures, such as voltage and frequency, facilitating highly accurate measurements.

The Josephson effect produces a current, known as a [[supercurrent]], that flows continuously without any voltage applied, across a device known as a '''Josephson junction''' (JJ).{{Clarification needed|reason=Does this mean that if the circuit is closed, it pushes the current in a circle? Obviously it can't push current forward forever into a piece of metal that doesn't connect to the beginning again, because a voltage would build up resisting the current.|date=December 2024}} These consist of two or more superconductors coupled by a weak link. The weak link can be a thin insulating barrier (known as a [[superconducting tunnel junction|superconductor–insulator–superconductor junction]], or S-I-S), a short section of non-superconducting metal (S-N-S), or a physical constriction that weakens the superconductivity at the point of contact (S-c-S).

Josephson junctions have important applications in [[Quantum circuit|quantum-mechanical circuits]], such as [[SQUID]]s, [[Superconducting quantum computing|superconducting qubits]], and [[RSFQ]] digital electronics. The [[NIST]] standard for one [[volt]] is achieved by [[Josephson voltage standard|an array of 20,208 Josephson junctions in series]].<ref>Steven Strogatz, ''Sync: The Emerging Science of Spontaneous Order'', Hyperion, 2003.</ref>

==History==
The DC Josephson effect had been seen in experiments prior to 1962,<ref>{{Cite web |last=Josephson |first=Brian D. |date=December 12, 1973 |title=The Discovery of Tunneling Supercurrents (Nobel Lecture) |url=https://www.nobelprize.org/prizes/physics/1973/josephson/lecture/}}</ref> but had been attributed to "super-shorts" or breaches in the insulating barrier leading to the direct conduction of electrons between the superconductors.

In 1962, Brian Josephson became interested into superconducting tunneling. He was then 23 years old and a second-year graduate student of [[Brian Pippard]] at the [[New Museums Site|Mond Laboratory]] of the [[University of Cambridge]]. That year, Josephson took a many-body theory course with [[Philip W. Anderson]], a [[Bell Labs]] employee on sabbatical leave for the 1961–1962 academic year. The course introduced 
Josephson to the idea of broken symmetry in superconductors, and he "was fascinated by the idea of broken symmetry, and wondered whether there could be any way of observing it experimentally". Josephson studied the experiments by [[Ivar Giaever]] and Hans Meissner, and theoretical work by Robert Parmenter. Pippard initially believed that the tunneling effect was possible but that it would be too small to be noticeable, but Josephson did not agree, especially after Anderson introduced him to a preprint of "Superconductive Tunneling" by Cohen, Falicov, and Phillips about the superconductor-barrier-normal metal system.<ref>{{cite journal |title=Superconductive Tunneling |first1=M. H. |last1=Cohen |first2=L. M. |last2=Falicov |first3=J. C. |last3=Phillips |date=15 April 1962 |journal=Physical Review Letters |volume=8 |issue=8 |pages=316–318 |doi=10.1103/PhysRevLett.8.316 |bibcode=1962PhRvL...8..316C |url=https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.8.316}}</ref>{{r|trueGenius|pp=223-224}}

Josephson and his colleagues were initially unsure about the validity of Josephson's calculations. Anderson later remembered:
<blockquote>
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.
</blockquote>
After further review, they concluded that Josephson's results were valid. Josephson then submitted "Possible new effects in superconductive tunnelling" to ''[[Physics Letters]]'' in June 1962{{r|possibleNewEffects}}. The newer journal ''Physics Letters'' was chosen instead of the better established ''[[Physical Review Letters]]'' due to their uncertainty about the results. [[John Bardeen]], by then already Nobel Prize winner, was initially publicly skeptical of Josephson's theory in 1962, but came to accept it after further experiments and theoretical clarifications.<ref name="trueGenius">{{cite book |url=https://archive.org/details/truegeniuslifesc0000hodd |url-access=registration |title=True Genius: The Life and Science of John Bardeen |publisher=Joseph Henry Press |page=[https://archive.org/details/truegeniuslifesc0000hodd/page/117 117] |isbn=9780309084086 |last1=Daitch |first1=Vicki |last2=Hoddeson |first2=Lillian |year= 2002 }}</ref>{{rp|pp=222–227}} See also: {{section link|John Bardeen|Josephson effect controversy}}.

In January 1963, Anderson and his [[Bell Labs]] colleague John Rowell submitted the first paper to ''Physical Review Letters'' to claim the experimental observation of Josephson's effect "Probable Observation of the Josephson Superconducting Tunneling Effect".<ref>{{cite journal |last1=Anderson |first1=P. W. |last2=Rowell |first2=J. M. |date=15 March 1963 |title=Probable Observation of the Josephson Tunnel Effect |url=https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.10.230 |journal=Physical Review Letters |volume=10 |issue=6 |pages=230 |bibcode=1963PhRvL..10..230A |doi=10.1103/PhysRevLett.10.230}} 
<!--|access-date=16 May 2012--></ref> These authors were awarded patents<ref>{{Cite patent|number=US3335363A|title=Superconductive device of varying dimension having a minimum dimension intermediate its electrodes|gdate=1967-08-08|invent1=Anderson|invent2=Dayem|inventor1-first=Philip W.|inventor2-first=Aly H.|url=https://patents.google.com/patent/US3335363A/en}}</ref> on the effects that were never enforced, but never challenged.{{Citation needed|date=March 2023}}

Before Josephson's prediction, it was only known that single (i.e., non-paired) electrons can flow through an insulating barrier, by means of [[quantum tunneling]]. Josephson was the first to predict the tunneling of superconducting [[Cooper pair]]s. For this work, Josephson received the [[Nobel Prize in Physics]] in 1973.<ref>{{Cite web |title=The Nobel Prize in Physics 1973 |url=https://www.nobelprize.org/prizes/physics/1973/summary/ |access-date=2023-03-01 |website=The Nobel Prize}}</ref> John Bardeen was one of the nominators.{{r|trueGenius|p=230}}

==Applications== 
[[File:Josephson junction symbol.svg|thumb|The [[electrical symbol]] for a Josephson junction]] Types of Josephson junction include the [[Phi Josephson junction|φ Josephson junction]] (of which [[pi Josephson junction|π Josephson junction]] is a special example), [[long Josephson junction]], and [[superconducting tunnel junction]]. Other uses include:

* A "Dayem bridge" is a [[thin-film]] Josephson junction where the weak link comprises a superconducting wire measuring a few [[micrometre]]s or less.<ref>{{cite journal |last1=Anderson |first1=P. W. |last2=Dayem |first2=A. H. |year=1964 |title=Radio-frequency effects in superconducting thin film bridges |journal=Physical Review Letters |volume=13 |issue=6 |page=195 |bibcode=1964PhRvL..13..195A |doi=10.1103/PhysRevLett.13.195}}</ref><ref>{{cite web |last=Dawe |first=Richard |title=SQUIDs: A Technical Report – Part 3: SQUIDs |website=rich.phekda.org |date=28 October 1998 |url=http://rich.phekda.org/squid/technical/part3.html |format=website |access-date=2011-04-21 |archive-url=https://web.archive.org/web/20110727172927/http://rich.phekda.org/squid/technical/part3.html |archive-date=27 July 2011 |url-status=dead}}</ref>
* The [[Josephson junction count]] is a [[proxy variable]] for a device's complexity
* [[SQUID]]s, or superconducting quantum interference devices, are very sensitive [[magnetometer]]s that operate via the Josephson effect
* Superfluid helium quantum interference devices ([[SHeQUID]]s) are the [[superfluid]] helium analog of a dc-SQUID<ref>Sato, Y.; Packard, R. (October 2012), ''Superfluid helium interferometers'', Physics Today, p. 31.</ref>
* In precision [[metrology]], the Josephson effect is a [[Reproducibility|reproducible]] conversion between [[frequency]] and [[voltage]]. The [[Josephson voltage standard]] takes the [[caesium standard]] definition of frequency and gives the standard representation of a [[volt]]
* [[Single-electron transistor]]s are often made from [[superconducting]] materials and called "superconducting single-electron transistors".<ref>{{cite journal |last1=Fulton |first1=T. A. |last2=Gammel |first2=P. L. |last3=Bishop |first3=D. J. |last4=Dunkleberger |first4=L. N. |last5=Dolan |first5=G. J. |year=1989 |title=Observation of Combined Josephson and Charging Effects in Small Tunnel Junction Circuits |journal=Physical Review Letters |volume=63 |issue=12 |pages=1307–1310 |bibcode=1989PhRvL..63.1307F |doi=10.1103/PhysRevLett.63.1307 |pmid=10040529}}</ref>
* [[Elementary charge]] is most precisely measured in terms of the Josephson constant and the von Klitzing constant which is related to the [[quantum Hall effect]]
* [[RSFQ]] digital electronics are based on shunted Josephson junctions. Junction switching emits one [[magnetic flux quantum]] <math>\scriptstyle\frac{1}{2 e}h</math>. Its presence and absence represents binary 1 and 0.
* [[Superconducting quantum computing]] uses Josephon junctions as [[qubits]] such as in a [[flux qubit]] or other schemes where the phase and charge are [[conjugate variables]].<ref>{{cite journal |last1=Bouchiat |first1=V. |last2=Vion |first2=D. |last3=Joyez |first3=P. |last4=Esteve |first4=D. |last5=Devoret |first5=M. H. |year=1998 |title=Quantum coherence with a single Cooper pair |journal=Physica Scripta |volume=T76 |page=165 |bibcode=1998PhST...76..165B |doi=10.1238/Physica.Topical.076a00165 |s2cid=250887469 |doi-access=}}</ref>
* [[Superconducting tunnel junction]] detectors are used in [[Superconducting camera|superconducting cameras]]

==The Josephson equations==
[[File:Single josephson junction.svg|thumb|Diagram of a single Josephson junction. A and B represent superconductors, and C the weak link between them.]]
The Josephson effect can be calculated using the laws of quantum mechanics. A diagram of a single Josephson junction is shown at right. Assume that superconductor A has [[Ginzburg–Landau theory|Ginzburg–Landau order parameter]] <math>\psi_A=\sqrt{n_A}e^{i\phi_A}</math>, and superconductor B <math>\psi_B=\sqrt{n_B}e^{i\phi_B}</math>, which can be interpreted as the [[wave function]]s of [[Cooper pair]]s in the two superconductors. If the electric potential difference across the junction is <math>V</math>, then the energy difference between the two superconductors is <math>2eV</math>, since each Cooper pair has twice the charge of one electron. The [[Schrödinger equation]] for this [[two-state quantum system]] is therefore:<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 21-9: The Josephson junction |website=feynmanlectures.caltech.edu |access-date=2020-01-03}}</ref>

<math display="block">i\hbar\frac{\partial}{\partial t}
\begin{pmatrix}
 \sqrt{n_A}e^{i\phi_A} \\
 \sqrt{n_B}e^{i\phi_B}
\end{pmatrix}
=
\begin{pmatrix}
 eV & K \\
 K & -eV
\end{pmatrix}
\begin{pmatrix}
 \sqrt{n_A}e^{i\phi_A} \\
 \sqrt{n_B}e^{i\phi_B}
\end{pmatrix},</math>

where the constant <math>K</math> is a characteristic of the junction. To solve the above equation, first calculate the time derivative of the order parameter in superconductor A:

<math display="block">\frac{\partial}{\partial t} (\sqrt{n_A}e^{i\phi_A})=\dot \sqrt{n_A}e^{i\phi_A}+ \sqrt{n_A} (i \dot \phi_A e^{i\phi_A})=(\dot \sqrt{n_A}+ i \sqrt{n_A} \dot \phi_A) e^{i\phi_A} ,</math>

and therefore the Schrödinger equation gives:

<math display="block">(\dot \sqrt{n_A}+ i \sqrt{n_A} \dot \phi_A) e^{i\phi_A}
=\frac{1}{i\hbar}(eV\sqrt{n_A}e^{i\phi_A}+K\sqrt{n_B}e^{i\phi_B}).</math>

The phase difference of Ginzburg–Landau order parameters across the junction is called the '''Josephson phase''':

<math display="block">\varphi=\phi_B-\phi_A.</math>The Schrödinger equation can therefore be rewritten as:

<math display="block">\dot \sqrt{n_A}+ i \sqrt{n_A} \dot \phi_A
=\frac{1}{i\hbar}(eV\sqrt{n_A}+K\sqrt{n_B}e^{i\varphi}),</math>

and its [[complex conjugate]] equation is:

<math display="block">\dot \sqrt{n_A}- i \sqrt{n_A} \dot \phi_A
=\frac{1}{-i\hbar}(eV\sqrt{n_A}+K\sqrt{n_B}e^{-i\varphi}).</math>

Add the two conjugate equations together to eliminate <math>\dot \phi_A</math>:

<math display="block">2\dot \sqrt{n_A}=\frac{1}{i\hbar}(K\sqrt{n_B}e^{i\varphi}-K\sqrt{n_B}e^{-i\varphi})=\frac{K\sqrt{n_B}}{\hbar} \cdot 2\sin \varphi.</math>

Since <math>\dot \sqrt{n_A}=\frac{\dot n_A}{2\sqrt{n_A}}</math>, we have:

<math display="block">\dot n_A=\frac{2K\sqrt{n_An_B}}{\hbar}\sin \varphi.</math>

Now, subtract the two conjugate equations to eliminate <math>\dot \sqrt{n_A}</math>:

<math display="block">2i \sqrt{n_A} \dot \phi_A
=\frac{1}{i\hbar}(2eV\sqrt{n_A}+K\sqrt{n_B}e^{i\varphi}+K\sqrt{n_B}e^{-i\varphi}),</math>

which gives:

<math display="block">\dot \phi_A
=-\frac{1}{\hbar}(eV+K\sqrt{\frac{n_B}{n_A}}\cos \varphi).</math>

Similarly, for superconductor B we can derive that:

<math display="block">\dot n_B=-\frac{2K\sqrt{n_An_B}}{\hbar}\sin \varphi , \, \dot \phi_B
=\frac{1}{\hbar}(eV-K\sqrt{\frac{n_A}{n_B}}\cos \varphi). </math>

Noting that the evolution of Josephson phase is <math>\dot \varphi=\dot \phi_B-\dot \phi_A</math> and the time derivative of [[charge carrier density]] <math>\dot n_A</math> is proportional to current <math>I</math>, when <math>n_A \approx n_B</math>, the above solution yields the '''Josephson equations''':<ref name="barone">{{Cite book |title=Physics and Applications of the Josephson Effect |last1=Barone |first1=A. |last2=Paterno |first2=G. |publisher=[[John Wiley & Sons]] |year=1982 |isbn=978-0-471-01469-0 |location=New York}}</ref>

{{equation|1=I(t) = I_c \sin (\varphi (t)) |2=1}}
{{equation|1=\frac{\partial \varphi}{\partial t} = \frac{2 e V(t)}{\hbar} |2=2}}

where <math>V(t)</math> and <math>I(t)</math> are the voltage across and the current through the Josephson junction, and <math>I_c</math> is a parameter of the junction named the '''critical current'''. Equation (1) is called the '''first Josephson relation''' or '''weak-link current-phase relation''', and equation (2) is called the '''second Josephson relation''' or '''superconducting phase evolution equation'''. The critical current of the Josephson junction depends on the properties of the superconductors, and can also be affected by environmental factors like temperature and externally applied magnetic field.

The [[Josephson constant]] is defined as:

<math display="block">K_J=\frac{2 e}{h}\,, </math>

and its inverse is the [[magnetic flux quantum]]:

<math display="block">\Phi_0=\frac{h}{2 e}=2 \pi \frac{\hbar}{2 e}\,. </math>

The superconducting phase evolution equation can be reexpressed as:

<math display="block">\frac{\partial \varphi}{\partial t} = 2 \pi [K_JV(t)] = \frac{2 \pi}{\Phi_0}V(t) \,. </math>

If we define:

<math display="block">\Phi=\Phi_0\frac{\varphi}{2 \pi}\,, </math>

then the voltage across the junction is:

<math display="block">V=\frac{\Phi_0}{2 \pi}\frac{\partial \varphi}{\partial t}=\frac{d\Phi}{dt}\,, </math>

which is very similar to [[Faraday's law of induction]]. But note that this voltage does not come from magnetic energy, since there is [[Meissner effect|no magnetic field in the superconductors]]; Instead, this voltage comes from the kinetic energy of the carriers (i.e. the Cooper pairs). This phenomenon is also known as [[kinetic inductance]].

== Three main effects ==
[[File:I-V characteristics of Josephson Junction.JPG|thumb|Typical I-V characteristic of a superconducting tunnel junction, a common kind of Josephson junction. The scale of the vertical axis is 50 μA and that of the horizontal one is 1 mV. The bar at <math>V = 0</math> represents the DC Josephson effect, while the current at large values of <math>\left| V \right|</math> is due to the finite value of the superconductor bandgap and not reproduced by the above equations.]]
There are three main effects predicted by Josephson that follow directly from the Josephson equations:

===The DC Josephson effect===
The DC Josephson effect is a direct current crossing the insulator in the absence of any external electromagnetic field, owing to [[tunnel (quantum mechanics)|tunneling]]. This DC Josephson current is proportional to the sine of the Josephson phase (phase difference across the insulator, which stays constant over time), and may take values between <math>-I_c</math> and <math>I_c</math>.

===The AC Josephson effect===
With a fixed voltage <math>V_{DC}</math> across the junction, the phase will vary linearly with time and the current will be a sinusoidal AC ([[alternating current]]) with amplitude <math>I_c</math> and frequency <math>K_J V_{DC}</math>. This means a Josephson junction can act as a perfect voltage-to-frequency converter.

===The inverse AC Josephson effect===
Microwave radiation of a single [[Angular frequency|(angular) frequency]] <math>\omega</math> can induce quantized DC voltages<ref>{{Cite journal |last1=Langenberg |first1=D. N. |last2=Scalapino |first2=D. J. |last3=Taylor |first3=B. N. |last4=Eck |first4=R. E. |date=1966-04-01 |title=Microwave-induced D.C. voltages across Josephson junctions |journal=Physics Letters |volume=20 |issue=6 |pages=563–565 |doi=10.1016/0031-9163(66)91114-0 |bibcode=1966PhL....20..563L |issn=0031-9163}}</ref> across the Josephson junction, in which case the Josephson phase takes the form <math>\varphi (t) =  \varphi_0 + n \omega t + a \sin(\omega t)</math>, and the voltage and current across the junction will be:
<math display="block">V(t) = \frac{\hbar}{2 e} \omega ( n + a \cos( \omega t) ), \text{ and } I(t) = I_c \sum_{m = -\infty}^\infty J_m (a) \sin (\varphi_0 + (n + m) \omega t),</math>

The DC components are:
<math display="block">V_\text{DC} = n \frac{\hbar}{2 e} \omega, \text{ and } I_\text{DC} = I_c J_{-n} (a) \sin \varphi_0.</math>

This means a Josephson junction can act like a perfect frequency-to-voltage converter,<ref>{{Cite journal |last1=Levinsen |first1=M. T. |last2=Chiao |first2=R. Y. |last3=Feldman |first3=M. J. |last4=Tucker |first4=B. A. |date=1977-12-01 |title=An inverse ac Josephson effect voltage standard |journal=Applied Physics Letters |volume=31 |issue=11 |pages=776–778 |doi=10.1063/1.89520 |bibcode=1977ApPhL..31..776L |issn=0003-6951}}</ref> which is the theoretical basis for the Josephson voltage standard.

== Josephson inductance ==
When the current and Josephson phase varies over time, the voltage drop across the junction will also vary accordingly; As shown in derivation below, the Josephson relations determine that this behavior can be modeled by a [[kinetic inductance]] named Josephson Inductance.<ref>{{cite arXiv |eprint=cond-mat/0411174 |first1=M. |last1=Devoret |first2=A. |last2=Wallraff |title=Superconducting Qubits: A Short Review |date=2004 |last3=Martinis |first3=J.}}</ref>

Rewrite the Josephson relations as:

:<math>
\begin{align}
\frac{\partial I}{\partial \varphi} &= I_c\cos\varphi,\\
\frac{\partial \varphi}{\partial t} &= \frac{2\pi}{\Phi_0}V.
\end{align}
</math>

Now, apply the [[chain rule]] to calculate the time derivative of the current:

:<math>
  \frac{\partial I}{\partial t} = \frac{\partial I}{\partial \varphi}\frac{\partial \varphi}{\partial t}=I_c\cos\varphi\cdot\frac{2\pi}{\Phi_0}V,
</math>

Rearrange the above result in the form of the [[current–voltage characteristic]] of an inductor:

:<math>
  V = \frac{\Phi_0}{2\pi I_c\cos\varphi} \frac{\partial I}{\partial t}=L(\varphi)\frac{\partial I}{\partial t}.
</math>

This gives the expression for the kinetic inductance as a function of the Josephson Phase:

:<math>
L(\varphi) = \frac{\Phi_0}{2\pi I_c\cos\varphi} = \frac{L_J}{\cos\varphi}.
</math>

Here, <math> L_J=L(0)=\frac{\Phi_0}{2\pi I_c} </math> is a characteristic parameter of the Josephson junction, named the Josephson Inductance.

Note that although the kinetic behavior of the Josephson junction is similar to that of an inductor, there is no associated magnetic field. This behaviour is derived from the kinetic energy of the charge carriers, instead of the energy in a magnetic field.

==Josephson energy==
Based on the similarity of the Josephson junction to a non-linear inductor, the energy stored in a Josephson junction when a supercurrent flows through it can be calculated.<ref>[[Michael Tinkham]], Introduction to superconductivity, Courier Corporation, 1986.</ref>

The supercurrent flowing through the junction is related to the Josephson phase by the current-phase relation (CPR):

:<math>I = I_c \sin\varphi.</math>

The superconducting phase evolution equation is analogous to [[Faraday's law of induction|Faraday's law]]:

:<math>V=\operatorname{d}\!\Phi/\operatorname{d}\!t\,.</math>

Assume that at time <math>t_1</math>, the Josephson phase is <math>\varphi_1</math>; At a later time <math>t_2</math>, the Josephson phase evolved to <math>\varphi_2</math>. The energy increase in the junction is equal to the work done on the junction:

:<math>
\Delta E = \int_1^2 I V\operatorname{d}\!{t}
= \int_{1}^{2} I\operatorname{d}\!\Phi
= \int_{\varphi_1}^{\varphi_2} I_c\sin \varphi \operatorname{d}\!\left(\Phi_0\frac{\varphi}{2\pi}\right)
= -\frac{\Phi_0 I_c}{2\pi} \Delta\cos\varphi\,.
</math>

This shows that the change of energy in the Josephson junction depends only on the initial and final state of the junction and not the [[Thermodynamic process path|path]]. Therefore, the energy stored in a Josephson junction is a [[state function]], which can be defined as:

:<math>E(\varphi)=-\frac{\Phi_0 I_c}{2\pi}\cos\varphi=-E_J\cos\varphi
\,.</math>

Here <math>E_J = |E(0)|=\frac{\Phi_0 I_c}{2\pi}</math> is a characteristic parameter of the Josephson junction, named the Josephson Energy. It is related to the Josephson Inductance by <math>E_J = L_JI^2_c</math>. An alternative but equivalent definition <math>E(\varphi)=E_J(1-\cos\varphi)</math> is also often used.

Again, note that a non-linear [[Inductor|magnetic coil inductor]] accumulates [[potential energy]] in its magnetic field when a current passes through it; However, in the case of Josephson junction, no magnetic field is created by a supercurrent — the stored energy comes from the kinetic energy of the charge carriers instead.

== The RCSJ model ==
The Resistively Capacitance Shunted Junction (RCSJ) model,<ref>{{Cite journal |last=McCumber |first=D. E. |date=1968-06-01 |title=Effect of ac Impedance on dc Voltage-Current Characteristics of Superconductor Weak-Link Junctions |journal=Journal of Applied Physics |volume=39 |issue=7 |pages=3113–3118 |doi=10.1063/1.1656743 |bibcode=1968JAP....39.3113M |issn=0021-8979}}</ref><ref>{{Cite journal |last1=Chakravarty |first1=Sudip |last2=Ingold |first2=Gert-Ludwig |last3=Kivelson |first3=Steven |last4=Zimanyi |first4=Gergely |date=1988-03-01 |title=Quantum statistical mechanics of an array of resistively shunted Josephson junctions |journal=Physical Review B |volume=37 |issue=7 |pages=3283–3294 |doi=10.1103/PhysRevB.37.3283 |pmid=9944915 |bibcode=1988PhRvB..37.3283C|url=https://opus.bibliothek.uni-augsburg.de/opus4/frontdoor/index/index/docId/40643 }}</ref> or simply shunted junction model, includes the effect of AC impedance of an actual Josephson junction on top of the two basic Josephson relations stated above.

As per [[Thévenin's theorem]],<ref>{{Cite web |title=AC Thevenin's Theorem |url=http://hyperphysics.phy-astr.gsu.edu/hbase/electric/acthev.html |website=hyperphysics.phy-astr.gsu.edu |access-date=2020-01-03}}</ref> the AC impedance of the junction can be represented by a capacitor and a shunt resistor, both parallel<ref>{{Cite web |title=Dynamics of RF SQUID |url=http://phelafel.technion.ac.il/~orensuch/theory/Theory.html |website=phelafel.technion.ac.il |access-date=2020-01-11 |archive-date=2021-06-13 |archive-url=https://web.archive.org/web/20210613133720/http://phelafel.technion.ac.il/~orensuch/theory/Theory.html |url-status=dead}}</ref> to the ideal Josephson Junction. The complete expression for the current drive <math>I_\text{ext}</math> becomes:

:<math>I_\text{ext} = C_J \frac{\operatorname{d}\!V}{\operatorname{d}\!t} + I_c \sin \varphi + \frac{V}{R},</math>

where the first term is displacement current with <math>C_J</math> – effective capacitance, and the third is normal current with <math>R</math> – effective resistance of the junction.

==Josephson penetration depth==
The Josephson penetration depth characterizes the typical length on which an externally applied [[magnetic field]] penetrates into the [[long Josephson junction]]. It is usually denoted as <math>\lambda_J</math> and is given by the following expression (in SI):

:<math>\lambda_J=\sqrt{\frac{\Phi_0}{2\pi\mu_0 d' j_c}},</math>

where <math>\Phi_0</math> is the magnetic flux quantum, <math>j_c</math> is the [[critical supercurrent density]] (A/m<sup>2</sup>), and <math>d'</math> characterizes the inductance of the superconducting electrodes<ref>{{cite journal |last1=Weihnacht |first1=M. |year=1969 |title=Influence of Film Thickness on D. C. Josephson Current |journal=Physica Status Solidi B |volume=32 |issue=2 |page=169 |bibcode=1969PSSBR..32..169W |doi=10.1002/pssb.19690320259}}</ref>

:<math>
  d'=d_I 
  +\lambda_1 \tanh\left(\frac{d_1}{2\lambda_1}\right)
  +\lambda_2 \tanh\left(\frac{d_2}{2\lambda_2}\right),
</math>

where <math>d_I</math> is the thickness of the Josephson barrier (usually insulator), <math>d_1</math> and <math>d_2</math> are the thicknesses of superconducting electrodes, and <math>\lambda_1</math> and <math>\lambda_2</math> are their [[London penetration depth]]s. The Josephson penetration depth usually ranges from a few [[Micrometre|μm]] to several mm if the critical current density is very low.<ref>{{cite book |last1=Buckel |first1=Werner |last2=Kleiner |first2=Reinhold |title=Supraleitung |date=2004 |publisher=Wiley-VCH Verlag GmbH&Co.KGaA |location=Tübingen |isbn=3527403485 |page=67 |edition=6.}}</ref>

==See also==
{{commons category|Josephson effect}}
{{div col|colwidth=30em}}
*[[Pi Josephson junction]]
*[[Varphi Josephson junction|φ Josephson junction]]
*[[Josephson diode]]
*[[Andreev reflection]]
*[[Fractional vortices]]
*[[Ginzburg–Landau theory]]
*[[Macroscopic quantum phenomena]]
*[[Macroscopic quantum self-trapping]]
*[[Quantum computer]]
*[[Quantum gyroscope]]
*[[Rapid single flux quantum]] (RSFQ)
*[[Semifluxon]]
*[[Zero-point energy]]
*[[Josephson vortex]]
{{div col end}}

== References ==
{{reflist}}

{{Condensed matter physics topics}}
{{Superconductivity}}
{{Authority control}}

[[Category:Josephson effect| ]]
[[Category:Condensed matter physics]]
[[Category:Superconductivity]]
[[Category:Sensors]]
[[Category:Mesoscopic physics]]
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