frequency $\nu_{J}=2eV_{0}/h$ emitted by the junction.

Another way to observe the ac Josephson current is by applying, in addition to
the dc voltage a weak ($V_{1}\ll V_{0}$) voltage at a very high frequency to
the junction, namely

$\displaystyle V=V_{0}+V_{1}\cos\omega t$ (156)

in which case (154) becomes

$\displaystyle\delta\phi(t)=\phi_{0}+\frac{e}{\hbar}V_{0}t+\frac{e}{\hbar}V_{1}\int\cos\omega
t\,dt=\phi_{0}+\frac{e}{\hbar}V_{0}t+\frac{e}{\hbar}V_{1}\frac{\sin\omega
t}{\omega}$ (157)

Thus212121This is in keeping with the fact that for $\Delta x\ll 1$,
$\sin(x+\Delta x)=\sin x+\Delta x\cos x$ $(\sin\Delta x\approx\Delta
x,\cos\Delta x\approx 1)$.

$\displaystyle\sin
2\delta\phi(t)=\sin\left(2\phi_{0}+\frac{2e}{\hbar}V_{0}t\right)+\frac{2eV_{1}}{\hbar\omega}\sin\omega
t\cos\left(2\phi_{0}+\frac{2e}{\hbar}V_{0}t\right).$ (158)

The first term coincides with (155) and is zero in average. Concerning the
second, and assuming $\omega=2eV_{0}/\hbar=\omega_{J}(\equiv 2\pi\nu_{J}$,
where $\nu_{J}=K_{J}V$ is the Josephson frequency, $K_{J}=2e/h$ being the
Josephson constant, inverse of the fluxon, the flux quantum), one finds

$\displaystyle\left(\frac{V_{1}}{V_{0}}\right)\left(\cos
2\phi_{0}(\sin\nu_{J}t)\times(\cos\nu_{J}t)-(\sin
2\phi_{0})\times(\sin^{2}\nu_{J}t)\right).$ (159)

While the first term of (159) averages to zero, the second one does not. The
practical way to implement the modulation described in Eqs. (156) and (159) is
to bathe the Josephson junction in microwave radiation (Fig. 27). Within this
context, one can calculate the supercurrent with the help of (158),

$\displaystyle J=$ $\displaystyle J_{c}\sin 2\delta\phi(t)$
$\displaystyle=J_{c}\times\text{Im}\left(\exp\left(i\left(2\phi_{0}+\frac{2e}{\hbar}V_{0}t+\frac{2eV_{1}}{\hbar\omega}\sin\omega
t\right)\right)\right).$ (160)

Making use of the expansion

$\displaystyle\exp\left(iz\sin\alpha\right)=\sum_{k=-\infty}^{\infty}J_{k}(z)\cos
k\alpha+\sum_{k=-\infty}^{\infty}J_{k}(z)\sin(k\alpha)$ (161)

and the parity relations of the Bessel functions

$\displaystyle J_{k}(z)=(-1)^{k}J_{-k}(z),$ (162)

it is possible to write the total current as,

$\displaystyle I$
$\displaystyle=J+\frac{V_{0}}{R}=J_{1}\sum_{k=-\infty}^{\infty}(-1)^{k}J_{k}\left(\frac{2eV_{1}}{\hbar\omega}\right)$
$\displaystyle\times\sin\left(2\phi_{0}+\frac{2e}{\hbar}V_{0}t-k\omega
t\right)+\frac{V_{0}}{R_{b}},$ (163)

where one has added the shunt current $V_{0}/R_{b}$ to the superconducting
one. It is of notice that this is the dc part of the current since the sin
part averages to zero, unless

$\displaystyle V_{0}=\frac{k\hbar\omega}{2e},\quad k=0,\pm 1,\pm 2,\dots$
(164)

Then, the supercurrent has the dc component

$\displaystyle
J=J_{c}(-1)^{k}J_{k}\left(\frac{2eV_{1}}{\hbar\omega}\right)\sin 2\phi_{0}.$
(165)

Summing up, the dc current increases linearly with $V_{0}$ as $V_{0}/R_{b}$,
except when the potential is an integer multiple of $\hbar\omega/2e$, where
the dc current suddenly jumps to the value given by (165). These are called
Shapiro steps, and the result of an experiment is shown in Fig. 28.

##  Appendix E Relation between bias and Josephson frequency

The rotational frequency in gauge space of a superconductor or of a superfluid
nucleus is defined by the relation

$\displaystyle\dot{\phi}=\frac{i}{\hbar}\left[H,\phi\right]=\frac{1}{\hbar}\frac{\partial\phi}{\partial\mathcal{N}}=\frac{1}{\hbar}\lambda,$
(166)

the identification of the chemical potential $\lambda$ with the rotational
angular frequency in gauge space is implicit in the description of the (mean
field) pair correlation by the BCS Hamiltonian,

$\displaystyle H^{\prime}=H-\lambda\mathcal{N}.$ (167)

The last term (Coriolis force) can be associated with a transformation to a
rotating reference system in which $\phi$ is stationary (body fixed frame)

$\displaystyle(\dot{\phi})^{\prime}=\frac{i}{\hbar}\left[H^{\prime},\phi\right]=\frac{i}{\hbar}\left[H,\phi\right]-\frac{i}{\hbar}\lambda\left[\mathcal{N},\phi\right]=\dot{\phi}-\frac{\lambda}{\hbar}=0.$
(168)

While there is no reference frame for the gauge angle of an isolated
superconductor, the gauge angle of one superconductor may be referred to that
of another,

$\displaystyle\hbar\dot{\phi}_{rel}=\hbar(\dot{\phi}_{l}-\dot{\phi}_{r})=\lambda_{l}-\lambda_{r}=e\times
V_{0}.$ (169)

Thus, the energy difference of a Cooper pair on one or the other side of the
weak link is

$\displaystyle\Delta
E_{J}=2\hbar\dot{\phi}_{rel}=2\hbar(\dot{\phi}_{l}-\dot{\phi}_{r})=2(\lambda_{l}-\lambda_{r})=2e\times
V_{0}$ (170)

The Josephson frequency is defined as

$\displaystyle\nu_{J}=\frac{2eV_{0}}{h},$ (171)

the quantity

$\displaystyle V_{0}=\frac{h\nu_{J}}{2e},$ (172)

being the biasing potential, while the energy difference of a Cooper pair on
one and the other side of the junction and thus the energy of the emitted
photons is

$\displaystyle h\nu_{J}=\hbar\omega_{J}=2eV_{0}.$ (173)

For a bias potential $V_{0}=1$ mV, $h\nu_{J}=2$ meV, and222222Making use of
$h=4.136\times 10^{-15}$ eV$\times s$ one obtains $\nu_{J}=2\times$
meV/(4.136$\times 10^{-15}$ eV$\times$s)=0.484$\times 10^{12}$ Hz.

$\displaystyle\nu_{J}=0.484\times 10^{12}\text{ Hz (THz)}.$ (174)

It is of note that the Cooper pair binding energy is $\approx 2\Delta$ which
e.g. in the case of lead ($\Delta\approx 1.4$ meV) amounts to 2.8 meV.
Furthermore, and in keeping with (174) and text following Eq. (34), the range
of frequencies one is dealing with is that of microwaves of extreme high
frequency to the far infrared.

Similarly, in the case of the
reaction232323$Q_{2n}=BE(^{114}\text{Sn})+BE(^{62}\text{Ni})-BE(^{116}\text{Sn})-BE(^{60}\text{Ni})$=1516.83
MeV-1515.53 MeV=1.307 MeV. (11),

$\displaystyle h\nu^{N}_{J}=2e_{eff}V_{0}^{N}=Q_{2n}=1.307\text{ MeV}$ (175)

and ($e_{eff}=0.443e$)

$\displaystyle V_{0}^{N}=1.469\text{ MV}.$ (176)

Thus242424$\nu_{J}^{N}=1.307$ MeV/(4.136$\times 10^{-15}$ eV$\times$
s)=0.316$\times 10^{21}$ Hz.

$\displaystyle\nu_{J}^{N}=\frac{Q_{2n}}{h}=0.316\times 10^{21}\text{ Hz
(ZHz)}.$ (177)

In the nuclear case, typical values of the Cooper pair binding energy are
found in the range 2.4-3.0 MeV ($\Delta\approx 1.2-1.5$ MeV), and (177)
corresponds to gamma-rays.

##  Appendix F Relation between normal and abnormal density

From the relation (13) defining the correlation length one can view the
quantity $\pi\Delta$ as the Cooper pair binding energy. For a bias implying an
energy $E<\pi\Delta$, one cannot talk about single-particle, fermionic motion,
the entities describing the system being Cooper pairs. The bias thus acts on
the center of mass of these entities (see Fig. 2 (a)), which will move in a
coherent fashion, leading in superconductor, to supercurrents of carrier
$q=2e$. Going back again the expression (13), the quantity $\hbar/(\pi\Delta)$
can be viewed as the effective period for the back and forth ($\ell=0$) motion
of the partners of the Cooper pair (Fig. 2 (b)). In this case thus, it is the
intrinsic motion of the fermionic partners of the Cooper pairs one refers to.
Being this a virtual motion, it is not observable. For bias such that the
center of mass of Cooper pairs move with the critical velocity

$\displaystyle\text{v}_{c}=\frac{\hbar
s_{c}}{m}=\frac{1}{m}\frac{\hbar}{\xi}=\frac{\pi\Delta}{m\text{v}_{F}},$

implying energies of the order of the depairing energy

$\displaystyle m\text{v}_{c}\text{v}_{F}\approx\pi\Delta,$

and thus critical violation of time-reversal invariance, the system moves into
the regime of independent fermionic motion, that is, S-Q regime of carriers of
charge $q=e$.

##  Appendix G Single fermion tunneling probability

For particles of momentum $k=(2M\times E)^{1/2}/\hbar$ incident from the left
on a barrier of thickness $w$ and height $V_{0}$ (see Fig. 29) and assuming
$\kappa w\gg 1$ ($\kappa=(2M(V_{0}-E))^{1/2}/\hbar$), the tunneling
probability is

$\displaystyle T\approx\left(\frac{4\kappa
k}{k^{2}+\kappa^{2}}\right)^{2}e^{-2\kappa d}.$ (178)

### Electrons

Assuming the tunneling electron moves close to the Fermi energy (e.g.
$\kappa_{F}\approx 1.18$Å-1, Nb) and that $w\approx 1$nm, $T\approx
e^{-2\kappa d}\approx 10^{-10}$. Thus the probability for two electrons to
tunnel simultaneously is $10^{-20}$, as nearly impossible as no matter.

### Nucleons

In the case of the reaction 116Sn+60Ni$\to^{115}$Sn+61Ni at $E_{cm}=154.26$
MeV ($(D_{0})_{c}=13.49$ fm), $w\approx 2.24$ fm, and $\kappa\approx 0.62$
fm-1, corresponding to a typical nucleon separation energy of 8 MeV. Thus
$2\kappa w\approx 2.78$ and $P_{1}=T\approx 7.5\times 10^{-2}$. As a
consequence, the probability for two nucleons to tunnel simultaneously is
$P_{2}\approx P_{1}^{2}\approx 6\times 10^{-3}$, and thus quite small.

##  Appendix H Assessing the validity of the parallel with BCS
superconductivity ([1];[2]) at the basis of nuclear BCS ([3])

###  H.1 Successive and simultaneous transfer

The order parameter of the phenomenon of spontaneous symmetry breaking in
gauge space associated with BCS condensation can be written as

$\displaystyle\alpha_{0}=\sum_{\nu,\nu^{\prime}>0}\langle{BCS(N+2)|a^{\dagger}_{\nu}a^{\dagger}_{\tilde{\nu}}|BCS(N)}\rangle=\sum_{\nu>0}U_{\nu}(N)V_{\nu}(N+2),$
(179)

where $U_{\nu}=U^{\prime}_{\nu}$, $V_{\nu}=V^{\prime}_{\nu}e^{-2i\phi}$,
$U^{\prime}_{\nu}$ and $V_{\nu}^{\prime}$ real, but equally well as,

$\displaystyle\alpha_{0}$
$\displaystyle=\sum_{\nu,\nu^{\prime}>0}\langle{BCS(N+2)|a^{\dagger}_{\nu}|\nu^{\prime}}\rangle\langle{\nu^{\prime}|a^{\dagger}_{\tilde{\nu}}|BCS(N)}\rangle=\sum_{\nu,\nu^{\prime}>0}\langle{BCS(N+2)|a^{\dagger}_{\nu}\alpha^{\dagger}_{\nu^{\prime}}|BCS(N+1)}\rangle\langle{BCS(N+1)|\alpha_{\nu^{\prime}}a^{\dagger}_{\tilde{\nu}}|BCS(N)}\rangle$
$\displaystyle=\sum_{\nu,\nu^{\prime}>0}\langle{BCS(N+2)|V_{\nu}(N+2)\alpha_{\tilde{\nu}}\alpha^{\dagger}_{\nu^{\prime}}|BCS(N+1)}\rangle\langle{BCS(N+1)|\alpha_{\nu^{\prime}}U_{\nu}(N)\alpha^{\dagger}_{\tilde{\nu}}|BCS(N)}\rangle=\sum_{\nu>0}V_{\nu}(N+2)U_{\nu}(N),$
(180)

where the inverse quasiparticle transformation
$a^{\dagger}_{\nu}=U_{\nu}\alpha^{\dagger}_{\nu}+V_{\nu}\alpha_{\tilde{\nu}}$
was used in dealing with the intermediate states. The above relations testify
to the fact that nucleons moving in time reversal states around the Fermi
energy and described by
$|{BCS}\rangle=\prod_{\nu>0}\left(U_{\nu}+V_{\nu}a^{\dagger}_{\nu}a^{\dagger}_{\tilde{\nu}}\right)|{0}\rangle$
are equally well pairing correlated (entangled), when undergoing successive
than simultaneous transfer.

Furthermore, because the binding energy of the Cooper pairs252525Due to the
very important role zero point pairing fluctuations play in nuclei, one can
talk about $U,V$ factors both in connection with superfluid nuclei as well as
with pairing vibrational nuclei, in which case one refers to
$U^{eff},V^{eff}$. These effective occupation numbers are closely related to
the backwardsgoing QRPA amplitudes, the so called $Y$-amplitudes,
$(V^{eff}_{\nu})^{2}=2Y_{pv}^{2}(j_{\nu})/(j_{\nu}+1/2)$,
$(U^{eff}_{\nu})^{2}=1-(V^{eff}_{\nu})^{2}$, where $pv$ labels either the pair
addition or pair removal modes [70]. is much smaller than the Fermi energy
$(2\Delta/\epsilon_{F}\ll 1)$ and the pairing strength is conspicuously
smaller than the value of the mean field around the nuclear surface region,
successive is the dominant two-nucleon transfer mechanism. It then does not
seem surprising that the absolute cross section associated with pair transfer
may be comparable to that of one-particle transfer under similar kinematic
conditions262626For example: [71]
$d\sigma(^{9}\text{Li}(d,p)^{10}\text{Li}(1/2^{-}))/d\Omega|_{\theta_{max}}\approx
0.8$ mb/sr, as compared to [72]
$d\sigma(^{11}\text{Li}(p,t)^{9}\text{Li}(1/2^{-}))/d\Omega|_{\theta_{max}}\approx
1$ mb/sr, [73] 10Be$(t,p)^{12}$Be(gs) $(\sigma=1.9\pm 0.5$ mb,
$4.4^{\circ}\leq\theta_{cm}\leq 54.4^{\circ})$ as compared to [74]
10Be$(d,p)^{11}$Be($1/2^{+}$) $(\sigma=2.4\pm 0.013$ mb,
$5^{\circ}\leq\theta_{cm}\leq 39^{\circ})$ in the case of light nuclei around
closed $(N=6)$ shell, and [75] 120Sn$(p,t)^{118}$Sn(gs) $(\sigma=3.024\pm
0.907$ mb, $5^{\circ}\leq\theta_{cm}\leq 40^{\circ})$ as compared to [76]
120Sn$(d,p)^{121}$Sn($7/2^{+}$) $(\sigma=5.2\pm 0.6$ mb,
$2^{\circ}\leq\theta_{cm}\leq 58^{\circ})$..

Nonetheless, one has to be somewhat careful in using these results in
connection with the parallel one may try to establish with the similarity
existing between the critical (dc) Josephson current of Cooper pairs
($S$-$S$), and the single electrons current ($S$-$Q$) observed by Giaever,
setting a (dc) bias $2\Delta/e$ to the weak link. This is because the
superfluid-quasiparticle ($S$-$Q$) current observed by Giaever implies, in the
nuclear case, in particular in the case of heavy ion reactions between
superfluid nuclei, both the gs$\to$gs single-particle transfer, but also the
transfer to quasiparticle states of energy $\lesssim 2\Delta$. Furthermore,
because in the condensed matter case one is talking of transfer of Cooper
pairs or of single electrons between two superconductors, something non-
operative when one of the two interacting systems in the nuclear reaction
under consideration is a proton, a deuteron or a triton. Again, due to the
important role ZPF associated with pairing vibrations play in nuclei, the
clear cut distinction between S and N systems found in condensed matter, is
blurred in the nuclear case (within this context see Fig. 34 and associated
discussion; see also Sect. IV.2.1).

###  H.2 Coherent state and phase space

Let us consider a $(p,t)$ process connecting the ground state of two
superfluid nuclei, e.g. Sn-isotopes. The unique property of the
$|{N+2}\rangle\to|{N}\rangle$ transition induced by the $(p,t)$ reaction
between members of the rotational band (gs$\to$gs transition), is due to the
fact that

$\displaystyle|{N}\rangle\sim\int d\phi\,e^{iN\phi}\,|{BCS(\phi)}\rangle,$
(181)

where

$\displaystyle|{BCS}\rangle$
$\displaystyle=\prod_{\nu>0}\left(U_{\nu}+V_{\nu}P^{\dagger}_{\nu}\right)|{0}\rangle_{F}$
$\displaystyle=\sum_{n=0,1,2\dots}\frac{\left(c_{\nu}P^{\dagger}_{\nu}\right)^{n}}{n!}|{0}\rangle_{F}=\exp\left(c_{\nu}P^{\dagger}_{\nu}\right)|{0}\rangle_{F},$
(182)

is a coherent state violating pair of particles conservation. In the previous
relation $c_{\nu}=V_{\nu}/U_{\nu}$,
$P^{\dagger}_{\nu}=a_{\nu}^{\dagger}a_{\tilde{\nu}}^{\dagger}$ and $n=N/2$
(number of pairs), have been used.

The relevance of the above result expresses itself in the microscopic
description of superfluid nuclei through the fact that, for example, to
reproduce the experimental cross section
$\sigma_{exp}(\text{gs}\to\text{gs})=2492\pm 374\mu$b associated with the
reaction 116Sn$(p,t)^{114}$Sn ([77]), one needs to have the correct pairing
strength ($\text{v}_{p}^{eff}=\text{v}_{p}^{bare}+\text{v}_{p}^{ind}$) and
(effective) phase space. This is in keeping with the fact that the
contribution to the pairing gap associated with $\text{v}_{p}^{bare}$ and
$\text{v}_{p}^{ind}$ are about equal, and that this last one arises from the
exchange of two-quasiparticle ($2qp$) vibrations calculated in QRPA. The
experimental result $\sum_{i}\sigma_{exp}(\text{gs}\to
2qp(i))=2012\mu$b$(1.3\text{MeV}\leq E_{2qp}(i)\leq 4.14$ MeV, 60 states)
together with the theoretical one
$\sigma_{th}(\text{gs}\to\text{gs})=2078\mu$b, testify to the phase space
consistency used to describe the $N=66,64$ members of the ground state pairing
rotational band of the Sn-isotopes.

###  H.3 Cooper pair and single-particle transfer in heavy ion collisions
between superfluid nuclei

From the value of the Josephson critical current $I_{c}=(\pi/4)I_{N}$, where
$I_{N}=V_{eq}/R=\left(\frac{2\Delta}{e}\right)\frac{1}{R_{b}}$, one can
conclude that the $S$-$S$ Cooper pair current is almost equal ($(\pi/4)\approx
0.8$) to the $S$-$Q$ single electron current resulting from the unbinding of
Cooper pair partners ($\approx 2\Delta$) by giving a momentum
$q\approx\hbar/\xi$ to the Cooper pair center of mass, implying a critical
breaking of time reversal invariance. The same can be done by forcing Cooper
pair partners to be at a distance272727This equivalence can be qualitatively
shown through $\delta(p^{2}/2m_{e})\approx 2\Delta$, $\frac{p_{F}\delta
p}{m_{e}}=\text{v}_{F}\delta p=2\Delta$, $\delta p\delta x\approx\delta
p\xi\approx\hbar$, i.e. $\delta p\approx\hbar/\xi$,
$\xi\approx(\hbar\text{v}_{F}/2\Delta)$, (please note the $\approx$ symbol in
the indeterminacy relation).$\gtrsim(D_{0})_{c}$.

###  H.4 Correlation length, nuclear dimensions and moment of inertia: further
evidence of the distortion operated on the Cooper pairs by the mean field

For a superfluid system held in a non spherical, e.g. ellipsoidal domain, of
dimension large as compared to the coherence length $\xi$, the moment of
inertia approaches that of the irrotational flow. At the antipodes with the
rigid moment of inertia, in the sense that, seen from the rotating frame of
reference, this last one is at rest, the velocity lines of the irrotational
one being ellipses (see Fig. 30).

In nuclei, while the observed moments of inertia are smaller than the rigid
one by a factor of 2, they are about a factor of 5 greater than the
irrotational moment of inertia282828[64] pp 75, 83, 278, 397 and 674.. A fact
which testifies to the special role pairing plays in nuclei, constrained as it
is by both spatial confinement ($R_{0}<\xi$) and quantization (mixing of odd
and even parity states
$Y_{l0}(\theta=180^{\circ})\sim(-1)^{l};Y_{l0}(0^{\circ})=1$). In other words,
constrained by the strong distortion to which nuclear Cooper pairs are
subject.

###  H.5 Specific probe of Josephson-like nuclear (ac) effect

Being the order parameter of the superconducting phase the $|{BCS}\rangle$
expectation value of $P^{\dagger}$, one should be able to calculate, within a
10% error, the absolute two-nucleon transfer cross section ($\sim|T|^{2}$) in
trying to assess the validity of the parallel made between low-temperature
superconductors and open-shell nuclei. In particular regarding subtle
consequences of BCS, like the Josephson effect. Not only this, but also to
match with specific nuclear structure conditions those at the basis of the
solid state phenomenon.

The transfer of a pair of electrons through a Josephson junction is free of
dissipation, aside from fulfilling $\xi\gg w$. Consequently, the bombarding
conditions in a heavy ion reaction between superfluid nuclei aimed at creating
a transient Josephson junction, should be such that $E_{cm}\lesssim E_{B}$ (no
dissipation) and, at the same time, also fulfill $D_{0}\lesssim(D_{0})_{c}$
(superfluid phase). Otherwise, one will be close292929As already stated,
because of the important role pairing vibrations play in nuclei and in nuclear
reactions, in particular in the transition from the S-S to the S-Q regimes,
the condensed matter effects of such transitions are in the nuclear case
conspicuously attenuated. to or in an uncorrelated single-(quasi) particle
phase in which case the absolute two-nucleon transfer cross section becomes
$\sim|T|^{4}$, precluding the qualification of Josephson-like junction. See
however Sect. IV.2.1.

##  Appendix I Finite versus infinite systems

Linear momentum is conserved in infinite systems, while translational
invariance is spontaneously broken in finite ones. A fact which underscores
the central role the center of mass plays in the microscopic description of
finite, quantum many-body systems.

Let us illustrate this by looking at a nucleus. That is, shine on it a beam of
photons and record the frequencies it absorbs. The outcome is remarkable.
Nuclei absorb essentially all of the dipole electromagnetic energy at a single
frequency, through the vibration of protons against effectively charge
carrying neutrons. As a consequence, the center of mass of the system remains
at rest.

Let us now illuminate a heavy nucleus with a beam of another heavy species, at
a bombarding energy such that the two systems enter, only at the distance of
closest approach, in weak contact. Being two leptodermic systems, when they
separate they have to reconstruct the overlapping surface area and, in the
process, give back the original surface energy, eventually returning to the
initial kinetic energy of relative motion. The center of mass of each of the
systems plays again a central role in defining the radii of curvature, the
associated reduced radius and thus the ion-ion potential, let alone the
trajectory of relative motion and the distance of closest approach along it.

Similarly, concerning the abnormal density of the two colliding superfluid
nuclei. The associated Cooper pairs can be viewed as correlated pairs of
nucleons, each described in terms of single-particle wavefunctions referred to
the center of mass of the corresponding mean field potential. During the weak
leptodermic contact, the abnormal densities do similarly in gauge space to
what the normal ones do in three dimensional space, giving rise to a weakly
coupled superconducting-superconducting junction, interacting through the
reduced pairing gap. Interaction which allows effectively charged nuclear
Cooper pairs, referred to the center of mass of one system, to move through
the junction, one nucleon at a time, to eventually become referred to the
center of mass of the other superconducting system. The part of the $T$-matrix
associated with the radial formfactor mediating such process, can be viewed as
the conductance, i.e. the inverse of the resistance ($1/R_{b}$), of the
nuclear junction. Weighted with the corresponding two-nucleon transfer
spectroscopic amplitudes (coherence factors) modulus squared, it determines
the ground state-ground state ((gs)-(gs)) absolute differential cross section.
Quantity which, for a range of bombarding energies ($E_{cm}$) smaller than
that of the Coulomb barrier (and thus essentially free from absorption), and
associated distances of closest approach $D_{0}\lesssim(D_{0})_{c}$ is, within
a factor of two, equal to the incoherent summed one-quasiparticle absolute
transfer cross section (Fig. 4).

Because the collision time of the heavy ion reaction associated with the value
of $E_{cm}$ leading to $D_{0}\approx(D_{0})_{c}$ implies that the center of
mass of the transferred Cooper pair has moved with about the critical momentum
$\hbar/\xi$, one can talk of the critical current of a single effectively
charged nuclear Cooper pair. And because of the selfconsistency found between
$E_{cm},\tau_{coll},(D_{0})_{c}$ and $\hbar/\xi$, one can view the associated
transient Josephson-like junction to be stationary, as far as the Cooper pair
transfer is referred (adiabatic picture).

##  Appendix J Correlation distances between weakly coupled many-body systems

###  J.1 Selected examples

a) The neutron stripping of 9Be on 12C
(12C(9Be$(1p_{3/2}),^{8}$Be)13C${}^{*}(2s_{1/2};3.086\text{ MeV})$) at low
bombarding energies (see Fig. 31) displays an astrophysical factor $S^{*}(\sim
E_{cm}\times\sigma)\approx 117\pm 8$ MeV barns at $E_{cm}\approx$ 1 MeV
($D_{0}\approx 35$ fm)303030[52] Fig. 37, p. 467.. Processes in which a single
nucleon is transferred in grazing collisions between heavy ions, where the two
nuclei have a small overlap play, as a rule (well matched conditions) an
important role. This is because the associated formfactors have a longer range
than the formfactors for inelastic scattering313131[52] Fig. 1, p. 298. (Fig.
32), and the reason why transfer is, for spherical or little deformed systems,
the main source of depopulation of the entrance channel. As a consequence
single-particle formfactors or better, their tail, are the basic elements
needed to calculate the imaginary part of the optical potential acting between
the ions.

b) The transition density of the soft $E1$-dipole mode (PDR) of 11Li, in which
the halo neutrons oscillate out of phase with respect to the nucleons of the
core (9Li), is associated with a dipole moment323232[78]. $d=e_{eff}r$, with
$r\approx 9$ fm (Fig. 33). A quantity which can be determined by measuring the
$B(E1)$ strength function of the PDR. Similarly, the $(p,p^{\prime})$ or
$(d,d^{\prime})$ cross sections.

c) In the reaction 116Sn+60Ni$\to^{114}$Sn+62Ni between ground states, the
main contribution is associated with successive transfer, in which the partner
nucleons of the transferred Cooper pair are, in the intermediate channel
115Sn+61Ni one on each nucleus. For a bombarding energy $E_{cm}=154.26$ MeV
($(D_{0})_{c}=13.49$ fm) their relative distance is estimated to be $\approx
10.56$ fm (see Fig. 11). Distance which can be probed through the measurement
of the Josephson-like (ac) dipole radiation. Arguably this can also be done
studying the quasielastic process 120Sn+$p\to^{118}$Sn+$t\to^{120}$Sn+$p$, as
schematically described in Fig. 34.

###  J.2 Macroscopic estimates of the dipole moment from GDR, PDR and soft
$E1$ modes

The Thomas-Reiche-Kuhn (TRK) dipole energy weighted sum rule can be written as
[64]

$\displaystyle S(E1)=14.8\frac{NZ}{A}e^{2}\text{fm}^{2}\,\text{MeV}$ (183)

In the case of the giant dipole resonance of ${}^{208}_{86}$Pb126, which
essentially carries 100% of it,

$\displaystyle S(E1)=735.2\,e^{2}\text{ fm}^{2}\text{ MeV}.$ (184)

Making use of the expression $\hbar\omega_{GDR}=80/A^{1/3}$ MeV one obtains

$\displaystyle\hbar\omega_{GDR}\approx 13.5\text{ MeV},$ (185)

and the associated dipole moment can be estimated from the relation,

$\displaystyle
d\approx\left(\frac{S(E1)}{\hbar\omega_{GDR}}\right)^{1/2}\approx\left(54.5\,e^{2}\text{
fm}^{2}\right)^{1/2}\approx 7.4\,e\text{ fm}=e_{eff}r.$ (186)

With the help of the associated effective neutron and proton charges

$\displaystyle(e_{eff}/e)=\left\\{\begin{array}[]{c}-Z/A=-82/208=-0.39=e_{N}/e,\\\
N/A=126/208=0.61=e_{Z}/e,\end{array}\right.$ (189)

one obtains

$\displaystyle r=\frac{d}{e_{eff}}=\left\\{\begin{array}[]{c}-19\text{
fm}=r_{N},\\\ 12.1\text{ fm}=r_{Z},\end{array}\right.$ (192)

the average value being333333Of notice that (193) represents the summed
coherent contributions of the many $p-n$ dipoles when the nucleus absorbs
(emits) a $\gamma$-ray of energy (185) (see for example right hand side of
Fig. 1.5 of reference [79]; see also Sect. 3.3 as well as p. 86 Eq. (3.15) and
Fig. 3.1 of this reference). From mean field approximation one can estimate
the summed neutron, proton (dipole) ($p-h$) phase space to be of the order of
20-25. Thus $\bar{r}/23\approx 0.7$ fm.

$\displaystyle\bar{r}=\frac{|r_{N}|+|r_{Z}|}{2}\approx 15.6\text{ fm},$ (193)

to be compared with the nuclear radius $R_{0}=7.1$ fm.

A similar estimate for the PDR assuming its centroid is at
$\hbar\omega_{GDR}\approx 6$ MeV and carrying $\approx 20$% EWSR, one obtains

$\displaystyle d^{2}=\frac{0.2S(E1)}{6\text{ MeV}}=24.5\,e^{2}\text{ fm}^{2},$
(194)

that is $d\approx 5$ fm, and

$\displaystyle r=\frac{d}{e_{i}}=\left\\{\begin{array}[]{c}-12.3\text{
fm}=r_{N}\\\ 8.2\text{ fm}=r_{Z}\end{array}\right.$ (197)

leading to

$\displaystyle\bar{r}=10.3\text{ fm}.$ (198)

Let us repeat similar estimates for ${}^{120}_{50}$Sn70. They lead to

$\displaystyle S(E1)=421\,e^{2}\text{ fm}^{2}\text{ MeV},$ (199)
$\displaystyle\hbar\omega_{GDR}\approx 16.4\text{ MeV},$

and

$\displaystyle d=\left(\frac{S(E1)}{\hbar\omega_{GDR}}\right)^{1/2}\approx
5.1\text{ fm}.$ (200)

The effective charges being

$\displaystyle(e_{eff}/e)=\left\\{\begin{array}[]{c}-0.43=e_{N}/e,\\\
0.57=e_{Z}/e,\end{array}\right.$ (203)

leads to

$\displaystyle r=\frac{d}{e_{eff}}=\left\\{\begin{array}[]{c}-11.9\text{
fm}=r_{N}\\\ 8.9\text{ fm}=r_{Z},\end{array}\right.$ (206)

and thus to

$\displaystyle\bar{r}=10.4\text{ fm}$ (207)

to be compared with the radius343434Similarly to the case of 208Pb, the dipole
phase space is in the present case $\approx 16-18$, and $\bar{r}/17\approx
0.6$ fm. $R_{0}=5.9$ fm. Concerning the PDR one obtains, under the assumption
of 20% EWSR and $\hbar\omega_{PDR}\approx 8$ MeV, $d=3.2\,e$ fm and

$\displaystyle\bar{r}=6.5\text{ fm}.$ (208)

In the case of the halo nucleus ${}^{11}_{3}$Li8 one finds

$\displaystyle S(E1)\approx 32.3\,e^{2}\text{ fm}^{2}\text{ MeV}.$ (209)

The soft $E1$-mode (PDR) carries $\approx 8\%$ of the EWR and its centroid is
found at an energy of $\approx 0.75$ MeV. Consequently

$\displaystyle d\approx\sqrt{4.3}\,e\text{ fm}=2.1\,e\text{ fm}.$ (210)

In the present case we are only interested in
$r_{N}\approx\left|2.1/(-0.27)\right|$ fm $\approx 8$ fm. As seen from Fig.
33, this quantity is very similar with that associated with the (absolute
value) maximum, of the halo component of the transition density ($\approx 9$
fm). In this case, the dipole phase space is rather small ($\approx 2$ and
$\bar{r}_{N}/2\approx 4$ fm).

##  Appendix K Josephson frequency

###  K.1 Superconductors

The energy of the emitted photon associated with the tunneling of a Cooper
pair across a (dc) barrier biased Josephson junction is

$\displaystyle\Delta E=2eV$ (211)

($\Delta E=\hbar\omega_{J}=h\nu_{J}$). Thus,

$\displaystyle\nu_{J}=\frac{2Ve}{h}\left(\omega_{J}=\frac{2Ve}{\hbar}\right)$
(212)

In keeping with the fact that

$\displaystyle\dot{\phi}=\frac{\mu}{\hbar},$ (213)

where $\mu$ is the chemical potential of the superconductor, the energy
difference (211) forces the weak link coupled superconductors gauge phases to
rotate with different angular velocities,

$\displaystyle\dot{\phi}_{rel}=\left(\dot{\phi}_{1}-\dot{\phi}_{2}\right)=\frac{\mu_{1}-\mu_{2}}{\hbar}\quad\left(\phi_{rel}=\phi_{1}-\phi_{2}\right).$
(214)

Because

$\displaystyle\mu_{1}-\mu_{2}=eV,$ (215)

and

$\displaystyle\phi_{rel}=\frac{Ve}{\hbar}$ (216)

one can write

$\displaystyle\omega_{J}=2\dot{\phi}_{rel}=2\frac{\mu_{1}-\mu_{2}}{\hbar}=\frac{2Ve}{\hbar}$
(217)

It is of note that the factor of 2 in front of $\phi_{rel}$, emerges from the
fact the coupling Hamiltonian between the two rotors in gauge space arises
from the exchange of pairs, that is,
$P_{1}P_{2}=e^{2i\phi_{1}}P^{\prime}_{1}e^{-2i\phi}P^{\prime}_{2}$ (see e.g.
[56] p. 357 and refs. therein). That is,

$\displaystyle H_{coupl}\sim
e^{2i\phi_{1}}e^{-2i\phi_{2}}e^{i\phi_{0}/2}+\text{h.c.}\sim\cos\left(2(\phi_{1}-\phi_{2})+\phi_{0}\right).$
(218)

Assuming a dc bias

$\displaystyle\phi_{rel}=\dot{\phi}_{rel}t=\frac{\mu_{1}-\mu_{2}}{\hbar}t$
(219)

and

$\displaystyle\dot{N}_{1}\sim\sin\left(2eVt+\phi_{0}\right)$ (220)

###  K.2 Nuclei

The energy associated with Cooper pair tunneling is

$\displaystyle\Delta E=\hbar\omega_{J}^{N}=Q_{2n}=2V^{N}\,e_{eff},$ (221)

where

$\displaystyle\omega_{J}^{N}=\frac{Q_{2n}}{\hbar}=2\dot{\phi}_{rel}^{N}=2\frac{\mu_{1}-\mu_{2}}{\hbar}=\frac{2V^{N}e_{eff}}{\hbar},$
(222)

and

$\displaystyle V^{N}=\frac{Q_{2n}}{2e_{eff}}.$ (223)

The energy of the associated $\gamma$-rays is taken from the relative motion
(see $\delta$-function $\delta(E_{i}-E_{\gamma}-E_{f}+Q)$) Eq. (XIII.1).

##  Appendix L Wick’s theorem

The relation (26) is in contrast with that expected for the normal state of
metals as can be seen by calculating (24). Better, the two-particle density
matrix, which we write in its general, non-local form, as

$\displaystyle\Gamma(r_{1}r_{2};r^{\prime}_{1}r^{\prime}_{2})=\langle{\Psi|\hat{\psi}^{\dagger}(\mathbf{r}^{\prime}_{2})\hat{\psi}^{\dagger}(\mathbf{r}^{\prime}_{1})\hat{\psi}(\mathbf{r}_{1})\hat{\psi}(\mathbf{r}_{2})|\Psi}\rangle.$
(224)

Assuming $|{\Psi}\rangle$ to be a single determinant,

$\displaystyle\Gamma(r_{1}r_{2};r^{\prime}_{1}r^{\prime}_{2})=\rho(r_{1},r^{\prime}_{1})\rho(r_{2},r^{\prime}_{2})-\rho(r_{1},r^{\prime}_{2})\rho(r_{2},r^{\prime}_{1})$
(225)

where

$\displaystyle\rho(r,r^{\prime})=\langle{\Psi|\hat{\psi}^{\dagger}(\mathbf{r}^{\prime})\hat{\psi}(\mathbf{r})|\Psi}\rangle=\sum_{i=1}^{N}\psi_{i}(\mathbf{r})\psi^{*}_{i}(\mathbf{r}^{\prime})$
(226)

Thus

$\displaystyle\lim_{|\mathbf{r}-\mathbf{r}^{\prime}|\to\infty}\Gamma(r_{1}r_{2};r^{\prime}_{1}r^{\prime}_{2})=0,$
(227)

as

$\displaystyle\lim_{|\mathbf{r}-\mathbf{r}^{\prime}|\to\infty}\rho(r,r^{\prime})=0.$
(228)

The rule for calculating the expectation value of a product of creation and
annihilation operators is that it must be split up into pairs of operators in
all possible ways, and each pair replaced by its expectation value. Within
this context (224) becomes,

$\displaystyle\Gamma(r_{1}r_{2};r^{\prime}_{1}r^{\prime}_{2})$
$\displaystyle=\frac{1}{2}\rho(r_{1},r^{\prime}_{1})\rho(r_{2},r^{\prime}_{2})$
$\displaystyle-\frac{1}{2}\rho(r_{1},r^{\prime}_{2})\rho(r_{2},r^{\prime}_{1})+\frac{1}{2}\chi^{*}(r_{2}^{\prime},r_{1}^{\prime})\chi(r_{2},r_{1}).$
(229)

Similar to Hartree-Fock (HF), BCS is a mean field theory. However, while a
state represented by a single determinant like $|{HF(gs)}\rangle$ can be
specified by its one-particle density matrix (226), and all higher order
density matrices can be calculated once this is known, the quasiparticle
vacuum is not. The ground state $|{BCS}\rangle$ of a superconductor is not
completely determined by its density matrix, but depends also on its pairing
function $\chi$ (abnormal density), which, in configuration space, can be
written as

$\displaystyle\chi(r,r^{\prime})=\langle{\Psi|\hat{\psi}(r^{\prime})\hat{\psi}(r)|\Psi}\rangle,$
(230)

in analogy to (226) for the density matrix. In configuration space it is a
function that goes to zero if $r$ and $r^{\prime}$ are far apart, and the
range over which it is significantly different from zero is known as the
correlation length. In momentum space it has matrix elements

$\displaystyle\chi(\mathbf{k}\uparrow,-\mathbf{k}\downarrow)=U_{k}V_{k}.$
(231)

Let us now write the appropriate singlet pairing function associated with the
$|{BCS}\rangle$ state,

$\displaystyle\chi(r_{1},r_{2})$
$\displaystyle=\langle{BCS|\hat{\psi}_{\downarrow}(\mathbf{r}_{2})\hat{\psi}_{\uparrow}(\mathbf{r}_{1})|BCS}\rangle$
$\displaystyle\sim\sum_{k}U_{k}V_{k}e^{i\mathbf{k}\cdot(\mathbf{r}_{1}-\mathbf{r}_{2})}e^{i2\,\mathbf{s}\cdot\mathbf{R}-2i\phi},$
(232)

where $\mathbf{R}=(\mathbf{r}_{1}+\mathbf{r}_{2})/2$ is the center of mass
coordinate of the Cooper pair. The BCS pair function corresponds to a state in
which each pair has (CM) momentum $2\hbar s$ and phase $-2\phi$. On performing
the Fourier sum over $\mathbf{k}$ one finds that the internal part of the pair
function corresponds to a spherical symmetric bound $s$-state. The radial
$k$-value is of the order of $k_{F}$, and the mean square radius of the state
is of the order of

$\displaystyle\xi=\frac{\hbar\text{v}_{F}}{\pi\Delta},$ (233)

which can be interpreted as the inverse of the range of $k$-values near the
Fermi surface over which the product $U_{k}V_{k}$ is significant. The two-
particle density matrix thus displays the following property

$\displaystyle\lim_{\begin{subarray}{c}|r-r^{\prime}|\to\infty\\\
(r_{2},r_{1}),(r_{2}^{\prime},r_{1}^{\prime})<\xi\end{subarray}}\Gamma(r_{1}r_{2};r^{\prime}_{1}r^{\prime}_{2})=\chi^{*}(r_{2}^{\prime},r_{1}^{\prime})\,\chi(r_{2},r_{1}).$
(234)

Namely, it is large whenever the pair of points $r_{1}$ and $r_{2}$ and
$r_{1}^{\prime}$ and $r_{2}^{\prime}$ are within a correlation length, however
far apart the pair of points $(r_{2}^{\prime},r_{2})$ ,
$(r_{1}^{\prime},r_{1})$, $(r_{2}^{\prime},r_{1})$ , $(r_{1}^{\prime},r_{2})$
are from each other. This ODLRO property results in the two-particle matrix
having a large eigenvalue. The corresponding eigenvector is the pairing
function,

$\displaystyle\int\int\Gamma(r_{1}r_{2};r^{\prime}_{1}r^{\prime}_{2})\chi(r_{1}^{\prime},r_{2}^{\prime})\,d\mathbf{r}^{\prime}_{1}\,d\mathbf{r}^{\prime}_{2}=aN\chi(r_{1},r_{2}),$
(235)

where $N$ is the number of pairs of particles participating in the condensate,
$a$ being a number of the order of unity.

##  Appendix M Effective dipole

The intrinsic (i.e., referred to the center of mass $O$, see Fig. 14) electric
dipole moment is,

$\displaystyle\mathbf{D}=\sum_{i}\mathbf{r}_{Oi}\,q_{i}.$ (236)

The sum in the expression above extends to all the nucleons in the system, and
the coordinates $\mathbf{r}_{Oi}$ are measured with respect to the center of
mass $O$. Although we don’t write the creation operator explicitly, we assume
that the vector potential $\mathbf{A}$ creates a photon of energy
$E_{\gamma}$. The charges $q_{i}$ are equal to $e$ (positive electron charge)
for protons, and zero for neutrons. Let us write separately the contributions
of the two nuclei $a$ and $A$, extend the sum only to the protons, and rewrite
the coordinate vectors,

$\displaystyle\mathbf{D}=e\sum_{i\in
b,\pi}(\mathbf{r}_{Ob}+\mathbf{r}_{bi})+e\sum_{i\in
A,\pi}(\mathbf{r}_{OA}+\mathbf{r}_{Ai}).$ (237)

We now make use of the approximation of considering that the cores $b$ and $A$
remain in their ground states $\Psi^{0}_{A}$ and $\Psi^{0}_{b}$ during the
transfer process. Because of parity conservation,

$\displaystyle\sum_{i\in
b,\pi}\int|\Psi^{0}_{b}(\boldsymbol{\xi}_{b})|^{2}\mathbf{r}_{bi}\,d\boldsymbol{\xi}_{b}=\sum_{i\in
A,\pi}\int|\Psi^{0}_{A}(\boldsymbol{\xi}_{A})|^{2}\mathbf{r}_{Ai}\,d\boldsymbol{\xi}_{A}=0,$
(238)

and

$\displaystyle e\mathbf{r}_{Ob}\sum_{i\in
b,\pi}\int|\Psi^{0}_{b}(\boldsymbol{\xi}_{b})|^{2}\,d\boldsymbol{\xi}_{b}=eZ_{b}\mathbf{r}_{Ob},$
$\displaystyle e\mathbf{r}_{OA}\sum_{i\in
A,\pi}\int|\Psi^{0}_{A}(\boldsymbol{\xi}_{A})|^{2}\,d\boldsymbol{\xi}_{A}=eZ_{A}\mathbf{r}_{OA}.$
(239)

In the context of the process we are interested in, we can thus adopt the
“effective” dipole operator

$\displaystyle\mathbf{D}=Z_{A}\mathbf{r}_{OA}+Z_{b}\mathbf{r}_{Ob}=(Z_{A}+Z_{b})\mathbf{r}_{OQ}+Z_{A}\mathbf{r}_{QA}+Z_{b}\mathbf{r}_{Qb},$
(240)

where the point $Q$ is the center of mass of the system formed by the two
cores $A$ and $b$ (see Fig. 14). Note that the two last terms in Eq. (240),
i.e.

$\displaystyle\mathbf{d}_{brem}=Z_{A}\mathbf{r}_{QA}+Z_{b}\mathbf{r}_{Qb},$
(241)

correspond to the dipole operator associated with the system formed by the two
nuclei $A$ and $b$ alone, without the two transferred neutrons, and is
responsible for the bremsstrahlung radiation of the colliding charged cores.
Making use of the definition of the center of mass,

$\displaystyle\mathbf{r}_{OQ}$
$\displaystyle=-\frac{\mathbf{r}_{O1}+\mathbf{r}_{O2}}{A_{A}+A_{b}},$ (242)
$\displaystyle\mathbf{r}_{QA}$
$\displaystyle=\frac{A_{b}}{A_{A}+A_{b}}\mathbf{r}_{bA},$ (243)
$\displaystyle\mathbf{r}_{Qb}$
$\displaystyle=-\frac{A_{A}}{A_{A}+A_{b}}\mathbf{r}_{bA},$ (244)

where $A_{A},A_{b}$ are the mass numbers of $A$ and $b$, respectively. We can
thus rewrite the bremsstrahlung dipole in terms of the $b$–$A$ coordinate
$\mathbf{r}_{bA}$,

$\displaystyle\mathbf{d}_{brem}=e\,\frac{Z_{A}A_{b}-Z_{b}A_{A}}{A_{A}+A_{b}}\mathbf{r}_{bA}.$
(245)

Finally, if we define

$\displaystyle\mathbf{d}_{1}=-e\frac{(Z_{A}+Z_{b})}{A_{A}+A_{b}}\mathbf{r}_{O1},$
(246)
$\displaystyle\mathbf{d}_{2}=-e\frac{(Z_{A}+Z_{b})}{A_{A}+A_{b}}\mathbf{r}_{O2},$
(247)

we can write

$\displaystyle\mathbf{D}=\mathbf{d}_{1}+\mathbf{d}_{2}+\mathbf{d}_{brem}.$
(248)

The dipole operator can thus be decomposed in a bremsstrahlung term
$\mathbf{d}_{brem}$ (which we are not going to consider here), plus a neutron
contribution for each one of the transferred neutrons. It is interesting to
note that the bremsstrahlung term vanishes if the two cores are identical or
have the same charge-to-mass ratio (see Eq. (245)).

$D_{0}$(fm) |  $E_{cm}$ (MeV) |  $(E_{B}-E_{cm})$ (MeV) |  $\left(\frac{4}{\pi}\right)^{2}$ $\left(\frac{\sigma_{2n}}{\sigma_{1n}}\right)$
---|---|---|---
13.12 | 158.63 | -1.03 | 1.14
13.49 | 154.26 | 3.34 | 0.57
13.70 | 151.86 | 5.74 | 0.59
13.81 | 150.62 | 6.98 | 0.46
14.05 | 148.10 | 9.50 | 0.27
14.24 | 146.10 | 11.50 | 0.22
14.39 | 145.02 | 12.58 | 0.18
Table 1: In columns 1)–4) the distance of closest approach ($D_{0}$), the
center of mass bombarding energy ($E_{cm}$), the difference between this
quantity and the energy of the Coulomb barrier ($E_{B}=157.60$ MeV) [14], and
the ratio between the experimental value of the two- and one-particle transfer
absolute cross sections multiplied by the factor $(4/\pi)^{2}$, are given.

Figure 1: Pairing rotational band associated with the ground state of the Sn-
isotopes. The heavy drawn horizontal lines display the energies calculated
making use of the expression $BE=BE(^{50+N}$SnN)-8.124$N$+46.33, obtained by
subtracting the contribution of the single nucleon addition to the nuclear
binding energy obtained by a linear fitting of the binding energies of the
whole Sn-chain. The minimum corresponding to $N=68$ (116Sn) has been
arbitrarily shifted to make it coincide with the $BE=0$ value (for more
details see [70]).

Figure 2: (a) Schematic of a supercurrent, for an applied bias leading to a
momentum acting on the center of mass of Cooper pairs $k<1/\xi$. (b)
intrinsic, fermionic motion of the Cooper pair.

Figure 3: Weak link: Current-voltage characteristic curves for tunneling
between two metals across a barrier (thin oxide layer of 10–30Å) [80]. It is
assumed that both metals are equal and that they can become superconductors
below a critical temperature $T_{c}$ (e.g. Al-Al2O3-Al (1.140 K), Sn-
SnO${}_{\text{x}}$-Sn (3.722 K)). N-N indicates the current resulting from the
tunneling of single electrons between the metals, both of them in the _normal
phase_ at $T>T_{c}$. S-S labels the (super) current associated with the
tunneling of Cooper pairs through the unbiased link ($V=0$, no potential drop)
at $T<T_{c}$. That is, the (dc) Josephson effect described by the relation
$I=I_{c}\sin\Delta\phi$ ($\Delta\phi=2(\phi_{2}-\phi_{1})$). If a direct
current source is attached to a Josephson junction, the relative phase
$\Delta\phi$ adjusts to accommodate both the magnitude and sign of the current
and, in the steady state, a zero-voltage current flows. As the current is
changed, an instantaneous voltage $V$ appears to readjust the phase,
$\delta(\Delta\phi)=(2e/\hbar)V\delta t$ to a new equilibrium value. Once
equilibrium is reached, the current continues to flow and the voltage
vanishes. The largest zero-voltage current which can flow $(I_{c})$ is reached
when $\Delta\phi=\pm\pi/2$. For $I>I_{c}$ the $I-V$ characteristic becomes a
typical quasiparticle [81] tunneling curve, the intermediate behaviour (bold
dashed line) being governed by the external circuitry. Within this context S-Q
implies _single-electron tunneling_ between two superconductors when the link
is subject to a potential difference (direct bias) $V\gtrsim 2\Delta/e$, in
which case Cooper pairs are broken and quasiparticles (Q) become excited. The
critical Josephson current is equal to $\pi/4$ ($\approx$ 80%) of the N-N
normal (single electron) current at the so called equivalent voltage
$V_{equiv}=2\Delta/e$.

Figure 4: Absolute differential cross sections associated with the reaction
116Sn+60Ni$\to^{115}$Sn+61Ni,
$\left.d\sigma_{1n}/d\Omega\right|_{\theta_{cm}=140^{\circ}}$ (exp. (x), th.
(dashed line)), and 116Sn+60Ni$\to^{114}$Sn+62Ni,
$\left.d\sigma_{2n}/d\Omega\right|_{\theta_{cm}=140^{\circ}}$ [14] (exp.
(solid dot), th. (continuous line)), displayed as a function of the distance
of closest approach $D_{0}$, for eight of the twelve bombarding energies for
which the experiments were carried out. The ratio between the above
experimental cross sections multiplied by $(4/\pi)^{2}$ (open circles) is also
shown. The factor $4/\pi$ enters the relation
$\frac{4}{\pi}\left(I_{c}/I_{N}\right)=1$, where $I_{c}$ is the critical
Josephson (super) current (of Cooper pair carriers) and $I_{N}$ the normal (of
single-electron carriers) current at the equivalent potential
$V_{equiv}=2\Delta/e$ (see Fig. 3 as well as text; see also caption to Fig.
5). The vertical dotted line at $D_{0}=13.2$ fm indicates the distance of
closest approach associated with $E_{cm}=E_{B}=157.60$ MeV (Coulomb barrier).

Figure 5: Angular distribution associated with the reactions
60Ni+116Sn$\to^{62}$Ni+114Sn and 60Ni+116Sn$\to^{61}$Ni+115Sn, calculated
making use of the optical potential of ref. [14] (see also ref. [52] p. 111),
as well as the spectroscopic amplitudes collected in Table 1 of [14], in
comparison with the experimental data ([14]) to which an error of 15% has been
assigned. Also given are the one- and two-nucleon transfer cross sections in
mb [14] (those in parentheses are the theoretical estimates calculated as
discussed in the text). While the calculations of the two-neutron transfer
reaction corresponds to a single transition populating the ground state of
both the residual (62Ni) and the outgoing (114Sn) nucleus (inverse
kinematics), that of one-particle transfer is inclusive concerning the
residual system (115Sn, 61Ni). In fact, the reported theoretical absolute
differential cross sections, result from the incoherent sum of $E_{qp}\leq
2.5$ MeV (see also Table I of ref. [14]; see also [82]). This is in keeping
with the fact that the one-particle transfer channel corresponds, in the case
of condensed matter, to Giaever’s [81] S-Q ($q=e$) current. In the present
case, the bias for the superfluid-normal phase transition is provided by the
strain forcing pairs to tunnel across a barrier of “width” (distance of
closest approach) $D_{0}\gtrsim(D_{0})_{c}$ ($\approx$13.5 fm). Thus, the
quasiparticle states in the energy range $0\leq E_{j}\leq 2\Delta$ are
expected to play, in the nuclear case, the role of $Q$ in Giaver’s
observations. Within this context, it is of notice that the two-neutron
transfer amplitude in the gs$\to$gs ($2n$ transfer) reaction studied, is
proportional to the number of Cooper pairs, in a similar way in which the
maximum value of the Josephson current is proportional to
$\Delta_{1}\Delta_{2}/(\Delta_{1}+\Delta_{2})\sim\alpha_{1}\alpha_{2}/(\alpha_{1}+\alpha_{2})\sim\alpha$,
assuming the two superconductors to be equal.

Figure 6: Schematic representation of the relative position, at the critical
distance of closest approach $(D_{0})_{c}(=13.49$ fm), of the two superfluid
nuclei 116Sn of radius $R_{0}=5.9$ fm and 60Ni ($R_{0}=4.7$ fm), corresponding
to a bombarding energy of $E_{cm}=154.26$ MeV and $\theta_{cm}=140^{\circ}$,
which defines a situation in which the distance between the nuclear surfaces
(Josephson-like barrier width) is
$w=(D_{0})_{c}-(R_{0}(^{116}\text{Sn})+(1/2)a)+(R_{0}(^{60}\text{Ni})+(1/2)a))=2.24$
fm.

Figure 7: Schematic representation of the relative motion of the reaction
116Sn+60Ni$\to^{114}$Sn+62Ni in the center of mass system, under the kinematic
conditions $E_{cm}=154.26$ MeV, $\theta_{cm}=140^{\circ}$, implying
$(D_{0})_{c}=13.49$ fm. The ions are in the entrance (exit) channel at time
$t_{1}(t_{3})$, while the transfer –mainly successive– process in which each
of the partner neutrons of the Cooper pair is in a different nucleus takes
place at time $t_{2}$. Indicated are the critical distance of closest approach
$(D_{0})_{c}$ the relative distance between the nuclear surfaces (width of the
Josephson-like barrier; $w=2.24$ fm) and the momentum $\pm\hbar/(2\xi)$ acting
on the pair partners and thus $\hbar s=\hbar/\xi$ on the center of mass motion
of the Cooper pair being transferred (critical momentum). The thin lines
describe for ($t_{1}$,$t_{3}$) and ($t_{3}$,$t_{1}$) the relative distribution
of the associated density for one particle fixed at $r_{1}=R$, as a function
of the coordinate of the other partner nucleon. For the transfer situation
($t_{2}$,$t_{2}$), where each partner nucleon is in a different nucleus, it
represents the intrinsic Cooper pair wavefunction modulus squared
$|\varphi_{q}(\mathbf{r})|^{2}$ ($\mathbf{r}=\mathbf{r}_{1}-\mathbf{r}_{1}$).

Figure 8: Local formfactor of the ($2n$) transfer reaction (11), defined
according to Eq. (75) as a function of the distance $R$ between the centers of
mass of the 61Ni and 115Sn nuclei. Also shown are the associated times,
according to the classical trajectory followed by the heavy ions during the
collision process.

Figure 9: Schematic representation of a deformed system in gauge space
defining a privileged orientation $z^{\prime}$ in the two-dimensional space
and thus an intrinsic, body-fixed, coordinate system of reference
$\mathcal{K}^{\prime}$, making an angle $\phi$ with the laboratory frame of
reference $\mathcal{K}$.

Figure 10: Schematic representation of the Josephson junction between
superconductors 1 and 2. The critical supercurrent corresponds to
$J_{c}=\frac{\pi}{4}\frac{V_{eq}}{R_{b}}=\frac{\pi}{4}\frac{V}{\Omega
S}=\frac{\pi}{4}\frac{A}{S}$, where $V,\Omega,A$ stands for Volt, Ohm and
Ampere and the unit surface $S$ stands for square meter ($m^{2}$). The dashed
lines define the (oxide layer) barrier region.

Figure 11: The modulus of the radial component of the dipole moment $r_{2n}$
(see Eq. (55)) associated with the successive tunneling of the partner
nucleons (both referred to the cm of the Sn +Ni system) of the transferred
Cooper pair (color online, blue continuous curve; see Sect. XIII.2), and the
correlation length $\xi$ (Eq. (98)) associated with the transfer of the first
partner nucleon from Sn(b) to Ni(A), both partners referred to the cm of Sn,
as a function of the partial waves $(0\leq\ell\leq 210)$. In the inset,
schematic representation of the coordinates used in the numerical calculation
of $\langle{|\hat{O}_{i}|^{2}}\rangle/|T|$, where $\langle{{\;}}\rangle$
stands for the average value obtained by inserting the corresponding function
modulus squared in the $T$-matrix.

Figure 12: Schematic of a weak link between two superconductors (S) labeled 1
and 2. (A) The Fermi surface (dashed curve) of the two superfluid nuclei
$S_{1}$ and $S_{2}$ and associated momentum region around it of thickness
$2\Delta_{1}/\text{v}_{F}$ and $2\Delta_{2}/\text{v}_{F}$. (B) Josephson
junction of width $w\,(\approx 1-3$ nm). Also indicated is the overlap of the
tails of the normal densities. (C) Same as (B) but indicating the overlap of
the abnormal densities $(n^{\prime}_{s}=\alpha_{0}^{\prime}/\mathcal{V})$. (D)
Similar to (A) but for the case of a transient Josephson-like junction
associated with a heavy ion collision between two superfluid nuclei
($R_{0}=1.2A^{1/3}$ fm), at the distance of closest approach. A situation has
been selected in which the bombarding energy is smaller than the Coulomb
barrier and corresponds to a distance of closest approach
$D_{0}\approx(D_{0})_{c}$, in which case, the overlap between the normal
densities has been indicated, while the one associated with the abnormal
densities expected is to be similar to this one.

Figure 13: Cumulative integral of the $T$-matrix as a function of the center
of mass distance $r_{Aa}$ of the two interacting nuclei 116Sn and 60Ni for the
reaction $Q$-value 1.307 MeV, and for other three selected $Q$-values. The
associated absolute differential cross sections
$\left.d\sigma_{2n}/d\Omega\right|_{\theta_{cm}=140^{\circ}}$ are displayed.
The timescale obtained with the help of the hyperbola describing the
trajectory of relative motion is also shown. Within this scenario, the quantal
Cooper pair transfer indicates that for the process under consideration,
$\tau_{coll}\approx 0.6-0.7\times 10^{-21}$ s.

Figure 14: Schematic representation of the two cores $A,b$ and two neutrons
$n_{1},n_{2}$ involved in the transfer process $a(\equiv b+2)+A\to b+B(\equiv
A+2)$. The intrinsic electric dipole operator $\mathbf{D}$ is referred to the
center of mass $O$. Also shown is the center of mass $Q$ of the $A,b$ system.

Figure 15: Angular radiation pattern for $E_{\gamma}=4$ MeV, $E_{cm}=154.26$
MeV and $\theta_{cm}=140^{\circ}$. The radius $r(x,y,z)$ of the surface is
proportional to the triple differential cross section (XIII.1).

Figure 16: a) Angular radiation pattern for $E_{\gamma}=4$ MeV,
$E_{cm}=154.26$ MeV and $\theta_{cm}=140^{\circ}$ as a function of
$\theta_{\gamma}$ and $\phi_{\gamma}$ (see Eq. (XIII.1)). b) The polar angle
$\theta$ for which the $\gamma$-radiation strength has its maximum (continuous
curve) and its minimum (dashed curve) as a function of the azimutal angle
$\phi_{\gamma}$ and c) the associated values of the triple differential cross
section. We also show cuts of the 3-dimensional figure at constant values of
$\phi_{\gamma}=0^{\circ},90^{\circ},120^{\circ}$ and $220^{\circ}$.

Figure 17: Making use of Eq. (107), the angle $\theta_{max}$ for which the
radiation is strongest has been determined as a function of $\phi_{\gamma}$
for four different $\gamma$ energies (black curves). Also shown (red curves)
is the value of the differential cross section (XIII.1) for the corresponding
maximum.

Figure 18: (Continuous black line) double differential cross section (XIII.1)
for $\theta_{cm}=140^{\circ}$ as a function of the energy of the $\gamma$ ray.
Integrating over the $\gamma$ ray energy, one obtains
$(d\sigma/d\Omega)_{\gamma}=\int(d^{2}\sigma/dE_{\gamma}d\,\Omega)\,dE_{\gamma}=6.9\,\mu\text{b}/\text{sr}$.
The corresponding absolute two-nucleon transfer differential cross section is
2.35 mb/sr, to be compared with the experimental value 2.58 ($\pm 0.39$) mb/sr
[14]. The ratio (6.90 $\mu$b/sr) / (2.35 mb/sr) $\approx 2.94\times 10^{-3}$
corresponds to the number of photons emitted per cycle [13]. Also shown are
the $\chi^{2}$ fits and corresponding parameters of Gaussian (red dotted line)
and blackbody (blue dashed line) line shapes.

Figure 19: (a) Polarization observables in both spherical and cartesian
coordinates associated with the $\gamma$-emission of Cooper pair tunneling
calculated with the same kinematic conditions as in the previous figures, and
making use of Eqs. (83) and (84).

Figure 20: Representation of the dipole $\mathbf{D}$ (red arrowed) defined in
Eq. (57) for $E_{\gamma}=4$ MeV, and $\theta_{cm}=140^{\circ}$. The modulus
$|\mathbf{D}|=e_{eff}\times 10.56$ fm has been calculated with Eq. (XIII.2),
while the angles defining its orientation where obtained making use of the
expressions given in Eq. (103). The dashed (blue) curves are schematic
representations of the Sn (lower left corner) and Ni (upper right corner)
trajectories in the c.m. frame. Black arrowed lines correspond to the smooth
trajectory of each of the two Cooper pair transfer partners (see Fig. 8, p.
326 of [52]).

Figure 21: Gamma-strength function corresponding to the incoherent
contributions of the single-quasiparticle transitions to 61Ni and 115Sn states
with $E_{qp}\lesssim$2.5 MeV. The Gaussian and blackbody ($bb$) line shape
fits leads to the ratio $\chi^{2}_{bb}/\chi^{2}_{G}=2\times 10^{-3}/(9\times
10^{-3})=0.22$, the associated blackbody temperature being $T=0.64$ MeV.

Figure 22: Single-particle levels of 61Ni included in the analysis of the
reaction 116Sn+60Ni$\to^{115}$Sn+61Ni [14]. The excitation energy,
spectroscopic factor and spin and parity are given for each level [82]. Figure
23:  Top panels: Doppler corrected $\gamma$ spectra for 60Ni, 61Ni and 62Ni
detected in PRISMA in coincidence with the AGATA Demonstrator, displayed in
the full energy range up to 4 MeV. Bottom panels: spectra Doppler corrected
for the heavy binary partners 116Sn, 115Sn and 114Sn (the bin is 1 keV wide).
The strongest transitions are labelled with the spin and parity of initial and
final states. The broader peaks correspond to the wrongly Doppler corrected
reaction partner. Details of this measurement, performed in direct kinematics,
are in Ref. [15]. Figure 24:  Experimental (black histogram) and simulated
(green histogram) $\gamma$-ray spectrum for 60Ni. The simulation has been done
for the Agata Demonstrator configuration, assuming four triple clusters. The
simulated spectrum was normalized to the measured one in the region of the
discrete transitions.  Figure 25: Simulated $\gamma$ spectra of 62Ni. The blue
histogram is the gamma-strength function, input distribution, while the green
histogram is the final distribution, reconstructed by using the AGATA tracking
algorithm. It is of note that, for didactical reasons, total counts of the
output (and input) distributions are a factor $\approx$ 1000 larger than the
one of Fig. 26 (magenta histogram).  Figure 26: Spectrum for 62Ni. Data
(black) are compared with the simulated distribution as discussed in
connection with Fig. 24 (green). The simulation of the predicted strength
function (magenta) incorporates the predicted gamma-strength function.

Figure 27: The selected dc value of $I$ fixes the value of the dc bias $V_{0}$
of the junction, while $V_{1}$ is determined by subjecting the junction to
photons of microwave frequency $\nu$ and energy $h\nu$ (see Apps. D and E).

Figure 28:  Oscilloscope presentation of current-versus-voltage
characteristics of a tunnel junction at 4.2 K formed by an Al tip on a
(La0.925Sr0.075)2CuO4 sample ([83]). Steps induced by incident microwave
radiation at 9.4 GHz, implying $V_{0}=V_{S}=kK_{J}\nu=k\times 2.07\times
10^{-12}$ mV s$\times 9.4\times 10^{9}$ s${}^{-1}=19.4\;\mu$V$\times k$
($k=0,\pm 1,\pm 2,\dots$).

Figure 29: Square potential barrier.

Figure 30: Flow patterns for rotation of ellipsoidal bodies as seen from the
rotating (body-fixed) frame of reference.

Figure 31:  The neutron stripping of 9Be on 12C at low bombarding energies.
The astrophysical factor
$S^{*}=E(\alpha)e^{i\eta_{\alpha}\arctan\kappa/k_{\alpha}}\sigma,$ where
$\sigma$ is the total transfer cross section is plotted as a function of the
center-of-mass energy $E_{CM}=E(\alpha)$. The experimental data are taken from
Caltech (open circles) and Melbourne (filled circles). The full drawn curve
for $E_{CM}<1.5$ MeV indicates the theoretical low-energy limit, while the
full drawn curve for $E>1.5$ MeV shows the semiclassical result as calculated
with a spectroscopic factor $S(^{9}\text{Be}(1p_{1/2}))\cdot
S(^{13}\text{C}(1s_{1/2}))=0.45$. The dashed curve indicates the prolongation
of this result into the region below 1.5 MeV where it is not expected to
apply. The classical distance of closest approach in a head-on collision is 35
fm at 1 MeV (for details see [52]).

Figure 32: Schematic representation of the radial dependence of the one-
particle transfer and inelastic form factor. In (a) a nucleon moving in the
orbital with quantum numbers $a^{\prime}_{1}$ in the projectile $a$ is
transferred under the action of the shell model potential $U_{1A}$ to the
target nucleus $A$ into an orbital $a_{1}$. The dependence of the form factor
on the distance between the two nuclei is determined by the overlap of the
product of the single-particle wavefunctions $\phi_{a^{\prime}_{1}}$ and
$\phi_{a_{1}}$ with the potential $U_{1A}$. A schematic representation of this
dependence is given at the bottom of (a). In (b) a nucleon in the projectile
$a$ is excited under the influence of the target field $U_{1A}$ from the
single-particle orbital with quantum numbers $a^{\prime}_{1}$ to the orbital
with quantum numbers $a^{\prime}_{2}$. The dependence of the form factor on
the distance between the cores is here determined by the overlap of the
product of the wavefunctions $\phi_{a^{\prime}_{1}}$ and
$\phi_{a^{\prime}_{2}}$ with the potential $U_{1A}$. A representation of this
dependence is shown at the bottom of (b) [52].

Figure 33: Transition density of the soft $E1$-mode (pygmy dipole resonance)
of 11Li (for details see [78]).

Figure 34: Gedanken experiment concerning the possibility of observing weak
coupling coherence phenomena between the states $|{BCS(A+2)}\rangle$ and
$|{BCS(A)}\rangle$ in an elastic reaction involving superfluid nuclei (a),
e.g. $p+^{120}$Sn$\to p+^{120}$Sn, the system 119Sn+$d$ acting as a dynamical
barrier (hatched areas arguably play role of that of dioxide layers in
Josephson junctions) between the two even $N$ superfluid systems arising from
the successive transfer of two nucleons (b) and eventually allowing for a time
dependent gauge phase difference between the $(A+2)$ and $A$ superfluid
systems, thus leading, in the case in which $Q$-value effects are appropriate,
to an oscillating enhancement of the elastic cross section at large angles as
observed, for quite different reasons, in the case of the elastic angular
distribution of the reaction 16O+28Si (c) (cf. [84]).

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