数理解析研究所講究録
1196 巻 2001 年 84-90

84A Simple Derivation of Josephson Formulae in
Superconductivity*Izumi OJIMARIMS, Kyoto University

Abstract

A simple and general derivation of Josephson formulae for the tunneling currents is presented on the basis of Sewell’s general formulationof superconductivity in use of off-diagonal long range order (ODLRO).

1

Introduction

Why superconductivity and Josephson effect?: Because superconductivity
provides a very important prototype of$SSB$ (symmetry breaking)and. collective phenomena caused by SSB (e.g., supercurrent, Meissner effect, Josephson effect, magnetic flux quantization, etc.).
Among these collective phenomena, Josephson effect [1] exhibits thephysical roles of order parameters arising from SSB in relation to phasecoexistence. To formulate this effect in a concise way, we first note the key
roles played by the Cooper pair condensates [2] in superconductivity. They
yield the non-trivial order parameter which can be formulated as a “variableat infinity” based on cluster property in thermodynamic pure phase
tor state with GNS representation having trivial centre) $[3, 4]$ . According to
Sewell’s “macroscopic quantum theoretical approach”, we adopt his generalcharacterization of superconducting BCS statesin the form ofoff-diagonal long range order (ODLRO):$=\mathrm{s}\mathrm{p}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{e}\mathrm{o}\mathrm{u}\mathrm{s}$$(=\mathrm{f}\mathrm{a}\mathrm{c}-$$\langle\cdot\rangle_{BCS}[5]$$| \langle\psi(X_{1}+\frac{\xi_{1}}{2})\psi(X_{1}-\frac{\xi_{1}}{2})\psi^{\uparrow}(X_{2}-\frac{\xi_{2}}{2})\psi^{\uparrow}(X_{2}+\frac{\xi_{2}}{2})\rangle_{BCS}-\Psi(X_{1}, \xi_{1})\Psi^{*}(X_{2}, \xi_{2})|$$|\vec{X}_{1}-\vec{X}_{2}|-\inftyarrow 0$

Here

CAR:

$\psi(x)$.

is the second quantized non-relativistic electron field obeying, and $\Psi(X, \xi)=\langle\psi(X+\xi)\psi(X-\xi 22)\rangle_{BCS}$$\{\psi(x), \psi^{\uparrow}(y)\}=\delta^{3}(x-y)$*Talk based on I. Ojima, Lett. Math. Phys. 51, 257 (2000).

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denotes the “macroscopic wave function” which is non-vanishing at spatial
infinity. Starting from this assumption, he gave a general proofof Meissner effect ( $\vec{B}=0$ inside of superconductor) [5]. Our aim here is
two-fold:$|\vec{X}|arrow\infty$i) In the context of superconductivity: To understand its essential featuresin a model-independent way;ii) In general context: To extract useful hints for understanding symmetry
breaking, collective phenomena and phase coexistence, etc.

Obtained result is: A simple and general derivation of ( $dc$ and $ac$ )Josephson formulae for tunneling currents due to phase difference between
two superconductors separated by a thin barrier of insulator$(=\mathrm{J}\mathrm{o}\mathrm{s}\mathrm{e}\mathrm{p}\mathrm{h}\mathrm{s}\mathrm{o}\mathrm{n}$junction).
From this we can conclude that Josephson tunneling current is a direct consequence of ODLRO and of non-invariance of energy in each side ofjunction under gauge transformation due to phase difference, which should
be contrasted with the traditional derivations such as those based on theperturbative treatment of the tunneling Hamiltonian, or, based on the phenomenological Ginzburg-Landau equation.

2Simple Derivation of Josephson Formulae

A completely model-independent approach is desirable, but we need here
the standard BCS Hamiltonian $[6, 3]$ arising from electron-phonon coupling:$H_{BCS}( \Lambda;w)=\int_{\Lambda}dx[\frac{1}{2m}\vec{\nabla}\psi^{\uparrow}(x)\cdot\vec{\nabla}\psi(x)-\mu\psi^{\dagger}(x)\psi(x)]$$+ \frac{1}{|\Lambda|}\int_{\Lambda}dx\int_{\Lambda}dy\int_{\Lambda}dz\int_{\Lambda}duw(x, y, z, u)\mathrm{x}\mathrm{x}\psi^{\uparrow}(x)\psi^{\uparrow}(y)\psi(u)\psi(z)$,where $m$ is the mass of an electron and is the chemical potential of electronsin bulk superconductor occupying a macroscopically extended spatial region.
with volume
To see the essence, we here use a simple picture of weakly coupled superconductors placed in two spatial regions
separated by aJosephson junction regarded as a phase boundary. In viewof the wide applicability of BCS model (at least in non highcases), wesuppose that the differences in properties of superconductors in
and inshould be absorbed in differences in potentials $w_{\Lambda_{1}}(x, y, z, u)\equiv w(x, y, z, u)$for $x,$ $y,$ $u\in\Lambda_{1;w_{\Lambda_{2}}}(x, y, z, u)\equiv w(x, y, z, u)$ for $x,$
. Up to thisfreedom, dynamics of superconductors are described universally by $H_{BCS}$
and the differences of realized thermodynamic phases are reduced to thechoice of states .
$\mu$$\Lambda$$|\Lambda|$$\Lambda_{1},$$\Lambda_{2}(\subset\Lambda)$$W\equiv\partial\Lambda_{1}=\partial\Lambda_{2}$$T_{c}$$\Lambda_{1}$$y,$ $z,$ $u\in\Lambda_{2}$$z,$$\langle\cdot\rangle$$\Lambda_{2}$86Remark 1 To give a precise meaning to “thermodynamic phases”, we need
. Then, the sizes of regions
the thermodynamic limit with volume
are macroscopically finite and microscopically “infinite”. Then, thegoes to “spatial infinity” far away fromlocation of junction
lying beyond it. It is possible to give a precise meaning
with outside region
to this heuristic expression in a non-standard analytic formulation [7].
$arrow\infty$$\Lambda_{1},$$\Lambda_{2}$$W=\partial\Lambda_{1}$$\Lambda_{1}$$\Lambda_{2}$The next cruicial point is: How to define tunneling current? Becauseof the non-local terms in $H_{BCS}$ we have no locally conserved electric current. So we need to clarify in which sense the electric current is conserved.
in byDefining the electric charge$Q(\Lambda)$$\Lambda$$Q( \Lambda)=-|e|\int_{\Lambda}dx\psi^{\uparrow}(x)\psi(x)$,we see that it generates global $U(1)$ -gauge transformation:$[iQ(\Lambda), \psi(x)]$$[iQ(\Lambda), \psi^{\uparrow}(x)]$,$=$$i|e|\psi(x)$$=$$-i|e|\psi\dagger(x)$,where $e=-|e|$ : unit of electric charge. Then the meaning of conservation
is found in the expression:for$[H_{BCS}(\Lambda;w), Q(\Lambda)]=0$

De nition2 Tunneling current$J$

between$\Lambda_{1}$$\forall\Lambda$

and.

$\Lambda_{2}$is defined by$J= \frac{d}{dt}\langle Q(\Lambda_{1})\rangle_{Bcs=}\langle[iH_{BCS}(\Lambda), Q(\Lambda_{1})]\rangle_{Bcs}$

Remark 3 i)$J$.

cannot be non-vanishing without outside region$\Lambda_{2}$.

ii) Its heuristic meaning is seen in $J\propto dN/dt=\partial H/\partial\theta$ in the numberphase picture of Ginzburg-Landau theory [8].

Decomposing $H_{BCS}(\Lambda;w)$ into regions$\Lambda_{1}$and$\Lambda_{2}$:,$H_{BCS}( \Lambda;w)=H_{BCS}(\Lambda_{1;}\frac{|\Lambda_{1}|}{|\Lambda|}w)+H_{BCS}(\Lambda_{2;}\frac{|\Lambda_{2}|}{|\Lambda|}w)+H_{12}$and combining it with the local (anti-)commutativity due to CAR, we seethat can be reduced to$J$$J=\langle[iH_{12}, Q(\Lambda_{1})]\rangle_{Bcs}=-\langle[iQ(\Lambda_{1}), H_{12}]\rangle_{BCS}$with$H_{12}$

given by$H_{12}= \frac{1}{|\Lambda|}\sum_{\{i,j,k,l\}=\{1,2\}}\int_{\Lambda_{i}}dx\int_{\Lambda_{j}}dy\int_{\Lambda_{k}}dz\int_{\Lambda_{l}}duw(x, y, z, u)\psi\dagger(x)\psi^{\uparrow}(y)\psi(u)\psi(z)$.

87Then we have
$J=-\langle[iQ(\Lambda_{1}), H_{12}]\rangle_{BCS}$$=i|e| \frac{1}{|\Lambda|}\sum_{\{i,j,k,l\}=\{1,2\}}\int_{\Lambda_{i}}dx\int_{\Lambda_{j}}dy\int_{\Lambda_{k}}dz\int_{\Lambda_{l}}du$$\mathrm{x}w(x, y, z, u)(-\delta_{i1}-\delta_{j1}+\delta_{k1}+\delta_{l1})\langle\psi^{\uparrow}(x)\psi^{\uparrow}(y)\psi(u)\psi(z)\rangle_{BCS}$.

fixed, the juncIn thermodynamic limit
withtion at the boundary $W$ goes infinitely far away, and hence, the terms coming
beyond $W$ are replaced by their expectation values in
from
owing
to cluster property:$\Lambda,$$\Lambda_{1},$$\Lambda_{2}arrow\infty$$|\Lambda_{1}|/|\Lambda_{2}|$$\langle\cdot\rangle_{BCS}$$\Lambda_{2}$$|\langle\psi^{\#}(x_{1}^{(1)})\cdots\psi^{\#}(x_{k_{1}}^{(1)})\psi^{\#}(x_{1}^{(2)})\cdots\psi^{\neq}(x_{k_{2}}^{(2)})\rangle_{BCS}$$-\langle\psi^{\#}(x_{1}^{(1)})\cdots\psi^{\#}(x_{k_{1}}^{(1)})\rangle_{BCS}\langle\psi^{\#}(x_{1}^{(2)})\cdots\psi^{\#}(x_{k_{2}}^{(2)})\rangle_{BCS}|$$\mathrm{e}darrow 0$$\Lambda_{1},\Lambda_{2}arrow\infty,|\Lambda_{1}|/|\Lambda_{2}|:$fix,

and $\psi\#=\psi$ or .
where
Here we recall that under the assumption of spatial homogeneity at infinity the expectation values of odd powers of fermionic operators
vanish(i.e., Bose-Fermi superselection rule holds) [9]. Then we see that, in the integrand(function of set ), surviving terms come from the cases with
$i=j=1,$ $k=l=2$ or $i=j=2,$ $k=l=1$ :$x_{1}^{(1)},$$x_{k_{1}}^{(1)}\in\Lambda_{1},$$\cdots,$$x_{1}^{(2)},$$\cdots,$$x_{k_{2}}^{(2)}\in\Lambda_{2}$$\psi\dagger$$\psi,$$\psi\dagger$$\chi_{\Lambda_{i}}(x)\chi_{\Lambda_{j}}(y)\chi_{\Lambda_{k}}(z)\chi_{\Lambda_{l}}(u)\cross(-\delta_{i1}-\delta_{j1}+\delta_{k1}+\delta_{l1})\langle\psi^{\uparrow}(x)\psi^{\uparrow}(y)\psi(u)\psi(z)\rangle_{BCS}$$\Lambda$$\chi_{\Lambda}\equiv \mathrm{i}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{r}$$\sum_{\{i,j,k,l\}=\{1,2\}}\chi(x, y, z, u)(-\delta_{i1}-\delta_{j1}+\delta_{k1}+\delta_{l1})\langle\psi\dagger(x)\psi^{\uparrow}(y)\psi(u)\psi(z)\rangle_{BCS}$$arrow$$\Lambda_{1},\Lambda_{2}arrow\infty,|\Lambda_{1}|/|\Lambda_{2}|:$fixed$\sum_{i=j\neq k=l}\chi(x, y, z, u)(-\delta_{i1}-\delta_{j1}+\delta_{k1}+\delta_{l1})$$\cross\langle\psi^{\uparrow}(x)\psi^{\uparrow}(y)\rangle_{Bcs}\langle\psi(u)\psi(z)\rangle_{Bcs}$.
whereandAssuming “almost spatial homogeneity” inscopic wave function” $\Psi(X, \xi)$ in ODLRO in the sense of,$\chi(x, y, z, u)\equiv\chi_{\Lambda_{t}}(x)\chi_{\Lambda_{j}}(y)\chi_{\Lambda_{k}}(z)\chi_{\Lambda_{l}}(u)$$\Lambda_{1}$$\langle\psi(x)\psi(y)\rangle_{BCS}$$\simeq$$\Psi(\frac{x+y}{2}, x-y)$

$=$

$\{$

$\Lambda_{2}$$|\Psi_{1}|e^{i\theta_{1}}(x, y\in\Lambda_{1})$$|\Psi_{2}|e^{i\theta_{2}}(x, y\in\Lambda_{2})$w.r.t. “macrx,,we obtain the desired formula for $dc$-Josephson current by picking out phasefactors from the integrands:$J\simeq \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}\mathrm{x}(e^{i(\theta_{1}-\theta_{2})}-e^{-i(\theta_{1}-\theta_{2})})\propto\sin(\triangle\theta)$,

88

is the phase difference of Cooper pairs across thejunction.
whereeasily derive the $ac$-Josephson formula in situations
wecanthis,Usingacross the junction, simply by replacing the above
with voltage gap
with $\Delta\theta+2eVt$ . This is due to the following reason: In spite of lost local
gauge invariance in $H_{BCS}$ in spatial directions due to non-local interactions,
local gauge invariant coupling is still meaningful in temporal direction. By
this temporal gauge freedom, situation with voltage gap is realized by atime-dependent local gauge transformation:$\triangle\theta\equiv\theta_{1}-\theta_{2}$$V$$\Delta\theta$$V$$A^{\mu}=( \phi,\vec{A}=0)arrow(\phi+\frac{\partial}{\partial t}(Vt)=V,\vec{A}=0)$$\psi(x, t)arrow e^{ieVt}\psi(x, t)$;,. Thus we have $J_{ac}\propto\sin(\triangle\theta+2eVt)$ forwhich results in$ac$-Josephson current.
In the above, we need (almost) spatial homogeneity to extract phasedifference. Although precise coefficients cannot be determined by such qualitative discussion, we can extract in a similar way the term in energy density
coming fromat the boundary located infinitely far away:$\triangle\thetaarrow\triangle\theta+2eVt$$\Delta\theta$$\frac{\langle H_{12}\rangle_{BCS}}{|\Lambda|}arrow\Lambda_{1},\Lambda_{2}arrow\infty,|\Lambda_{1}|/|\Lambda_{2}|:$

fixed

constant$\cross\cos(\Delta\theta)$.

is of negative sign, this guarantees self-consistency
If the coefficient ofof Cooper pair condensates should be spatially
phase
that
postulateof thein the absence of a constraint maintaining
homogeneous in favour ofphase difference at the barrier.
To verify this consistency in a more satisfactory way, it seems necessary
to confront a challenging problem of how one can justify the notion of aof Cooper pairs appearing in Ginzburgpoint-like order parameter
Landau approach. It is crucial also for discussing Type II superconductivity
involving spatial inhomogeneity and local gauge invariance problem.
$\cos(\triangle\theta)$$\Delta\theta=0$$\Psi(x)$3Discussion: Infinities and Infinitesimals

Here some comments are in order on the use of non-standard analysis [7]:It can be useful in describing situations with infinitely large regions
separated by a boundary $W$ at infinity. What is important is that it allows
to treat simultaneously theories with finite and infinite volumes without
disconnecting the two approaches.
(level ofandBefore introducing the distinctions among finite,inintemal objects), it looks as if we were finite volume theory. Once such
distinctions are introduced (interpretation in a non-standard model) by reas infinite numbers (whose ratio is kept finite), infiniteandgarding$\Lambda_{1},$$\infty$$|\Lambda_{1}|$$|\Lambda_{2}|$$1/\infty$$\Lambda_{2}$89

volume theory is seen to be contained in the former, extracted by the procedure of taking standard parts of quantities with all infinitesimals such as
thrown away.
With only one thermodynamic phase, the procedure of extracting standard parts may be equivalent to the usual formulation in thermodynamiclimit. In the present situation with two infinitely large regions,
and
we have still “another world” inbeyond the infinitely distant boundary$W$ of infinitely extended. This is difficult to be treated in the usualformulation; it can be described without difficulty in such a framework that
all the infinities and infinitesimals are fully legitimate quantities. Moreover,
all the limiting or approximate relations appearing above are replaced bysimple algebraic equivalence relations modulo infinitesimals, in which one ofits conceptual advantages can be found.
$1/|\Lambda_{1}|$$\Lambda_{1}$$\Lambda_{2}$$\Lambda_{2}$$\Lambda_{1}$References[1] Josephson, B., Phys. Lett. 1, No. 7, 251-253 (1962); Advances Phys. 14,419-451 (1965).
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[7] Robinson, A., Non-Standard Analysis, North-Holland Publ. (1966);
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