Editing Josephson effect (section)

==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]].