Probable observation of the Cooper pair correlation length in superfluid nuclei

In the paper which marks the start of quantum mechanics as we know it today, Heisenberg writes [1] “It is well known that the formal rules which are used in quantum theory for calculating observable quantities such as the energy of the hydrogen atom may be seriously criticized on the ground that they contain, as basic element, relationships between quantities that are apparently unobservable in principle, e.g., position and period of revolution of the electron.” Considerations made after having expressed the aim “…to establish a basis for theoretical quantum mechanics founded exclusively upon relationships between quantities which in principle are observable”.

Within this context, and with over five decades of experience concerning the phenomenon of nuclear BCS ([2]; see also [3] and references therein), one could posit that the nuclear correlation length $\xi$ (Eq. (2)), expected to be much larger than nuclear dimensions, is not observable. To assess the validity of the above statement, it seems important to remind Bohr’s advocacy to limit “the use of the word phenomenon to refer exclusively to observations obtained under specific circumstances, including account of the whole experiment”[4]. Namely, in the present case, two-nucleon transfer –or better tunneling – processes below the Coulomb barrier between superfluid nuclei.

As Cooper showed [5], a pair of electrons moving in time-reversal states $(\nu,\tilde{\nu})$ $(\equiv(\mathbf{k}\uparrow,-\mathbf{k}\downarrow))$ above a non-interacting Fermi sea displays a bound state – $|{\Phi_{0}}\rangle\sim\sum_{\nu>0}c_{\nu}P^{\dagger}_{\nu}|{0}\rangle$, where $P_{\nu}^{\dagger}=a^{\dagger}_{\nu}a^{\dagger}_{\bar{\nu}}$ creates, acting in the vacuum state $|{0}\rangle$, a pair of electrons in time-reversal states– provided the interaction is attractive, no matter how weak it is.

Condensed matter– In a metal, the passage of an electron through a region of he lattice implies a slight increase of negative charge. It results in a slight attraction of ions and eventually a small excess of positive charge which a second electron will experience. Because $M_{ion}\gg m_{e}$, only electrons far apart from each other ($\approx 10^{4}$Å), and thus feeling a very weak Coulomb repulsion, will interact through the exchange of lattice phonons [6], [7]. In other words, because retardation effects (phonon mediated pairing) permit an attraction that is non-local in space and time, the electrons can avoid most of the repulsion [8]. That is, one is confronted essentially with weak binding by deconfinement (see Sect. 5.1).

The condensation of these ($L=0,S=0$) weakly bound, very extended, strongly overlapping quasi-bosonic entities ($(P^{\dagger}_{\nu})^{2}=0$, $P_{\nu}^{\dagger}P_{\nu^{\prime}}^{\dagger}\neq 0$), leads to a transition between the normal (N) and the superconducting phases (S). The many pair wavefunction $|{BCS}\rangle$ describing this phase displays a probability amplitude $V_{\nu}$ that the pair of fermions state ($\nu,\tilde{\nu}$) is occupied, and at the same time, a probability amplitude $U_{\nu}=\left(1-V_{\nu}^{2}\right)^{1/2}$ that it is empty, a property which can be considered the basic BCS ansatz. As a consequence Cooper pairs can scatter and lower their energy.

The $|{BCS}\rangle$ state, written for the first time by Schrieffer [9], reads

where $U_{\nu}=U^{\prime}_{\nu}$ and $V_{\nu}=V^{\prime}_{\nu}e^{-2i\phi}$ ($U_{\nu}^{\prime},V_{\nu}^{\prime}$ real; $c_{\nu}=e^{-i2\phi}c_{\nu}^{\prime},\;c_{\nu}^{\prime}=V^{\prime}_{\nu}/U^{\prime}_{\nu}$) are the BCS [10, 11] occupation probability amplitudes, while $\phi$ is the gauge angle. The $|{BCS}\rangle$ state displays an energy gap $\Delta$ for both single-pair translation and dissociation (breaking of Cooper pairs), phenomenon known as Off-Diagonal-Long-Range-Order (ODLRO) [12, 13, 14, 15]. The binding energy of a Cooper pair is $\approx 2\Delta$ for electrons moving close to the Fermi energy. Consequently $2\Delta/\hbar$ can be viewed as the effective angular frequency for the back and forth $(L=0)$ oscillations describing the intrinsic motion of the Cooper pair partners, the quantity $\hbar\text{v}_{F}/2\Delta$ where $\text{v}_{F}$ is the Fermi velocity, providing a measure of the dimension of the Cooper pair. Following Pippard we shall use,

as the Cooper pair mean square radius (correlation length) [16].

One can write the pairing gap as $\Delta=\Delta^{\prime}e^{-2i\phi}=G\alpha_{0}$, where

$G$ being the pairing coupling constant,

the number of Cooper pairs, while

is the so called abnormal density, $\mathcal{V}$ being an appropriate volume element [17]. The quantity (3) is the order parameter of the superconducting state, which one also finds at the basis of the fact that pair tunneling is the specific probe of BCS Cooper pair condensation.

In keeping with the basic BCS ansatz ($U_{\nu}V_{\nu}\neq 0$ in an energy range $\approx 2\Delta$ around the Fermi energy) and Eqs. (3) and (4), the $|{BCS}\rangle$ state violates pair of particle number conservation. The quantity $\alpha_{0}^{\prime}$ provides a quantitative measure of the associated deformation in the two dimensional gauge space which defines a privileged orientation in this space measured by $\phi$ (angle between the laboratory system $z$-axis and that of the intrinsic body fixed reference frame). Because all gauge orientations of an isolated superconductor have the same energy, $|{BCS}\rangle$ is a highly degenerate state, which can be viewed as the intrinsic state of rotation in gauge space of the system as a whole, and thus of the associated pairing rotational bands. These elementary modes of excitation have been systematically studied in nuclei in terms of individual quantum states, making use of two-nucleon transfer tunneling (see e.g. [18, 19, 20, 21, 22]; see also [23]). In what follows we shall use, for simplicity, the symbol $\Delta$ for both $\Delta$ and $\Delta^{\prime}$ unless explicitly stated.

Atomic nuclei– The nuclear structure exhibits a number of similarities with the electron structure of metals. In both cases, one is dealing with a system of fermions which can be described in first approximation in terms of independent particle motion. Nonetheless, in both systems, important correlations in the single particle motion arise from the action of the forces between the particles, in particular among those moving in time reversal states close to the Fermi energy. Repulsive in the electronic case (Coulomb interaction) marginally overwhelmed by the exchange of phonons, attractive in the nuclear one (strong force; of note however the Coulomb force acting between protons). Furthermore, in the nuclear case the exchange of collective, mainly surface vibrations, contribute about half of the total attractive strength [24, 25, 26, 27, 28, 29]. A further point of contact with metallic systems, where the lattice phonons provide the designed glue to bind pairs of ($\nu,\tilde{\nu}$) electrons into Cooper pairs.

Pairing in nuclei was introduced by Bohr, Mottelson and Pines ([2]; see also [30, 31, 18, 32]), few months after the publication of the BCS papers [10, 11], in terms of the pairing gap in the intrinsic spectrum of quadrupole deformed nuclei and of the associated moments of inertia of rotations of the system as a whole, as well as making reference to the odd-even mass difference [33, 34, 35]. If the intrinsic structure of these finite, many body systems could be adequately described in terms of single-particle motion, the first quantity should be of the order of 0.1-0.2 MeV while the observed value is 1.0-1.5 MeV. Concerning the moment of inertia $\mathcal{I}$, one expects a value equal to the rigid moment of inertia $\mathcal{I}_{rig}$, in keeping with the fact that the single-particle orbitals are solidly anchored to the mean field. Experimentally, $\mathcal{I}\approx\tfrac{1}{2}\mathcal{I}_{rig}$ – consistent with a spheroid filled with a low viscosity fluid – a result which was explained in terms of nuclear BCS [36, 37]. In other words, in nuclei one finds a pairing gap for both the motion of one pair independent of the others (low level density in even-even systems) as well as in the breaking of a pair (odd-even mass difference), i.e. ODLRO.

The odd-even mass difference, namely the larger stability of nuclei with an even number of like nucleons as compared to that of the neighboring odd numbered nuclei, has played an important role in the understanding of pairing in nuclei, and was already recognized in connection with the introduction of the shell model [35]. A main difference between metals and atomic nuclei concerning BCS Cooper pair condensation, is associated with the role pairing fluctuations, play in these systems [38, 39, 40, 19, 41].

Within this context (shell model and pairing vibrations), the texture of superfluidty in nuclei can be studied one step at a time, in terms of individual quantum states and of single pair transfer processes. Because of the large energy gap found in closed shell nuclei¹¹1That is, ${}^{A}_{Z}$X_N systems of mass number $A=N+Z$ and magic number (2,8,20,28,50,82,126…) of neutrons ($N$) and/or protons ($Z$). between occupied and empty single-particle states, these systems (e.g. ${}^{208}_{82}$Pb₁₂₆), are normal (non superfluid) nuclei while the ground states of ²¹⁰Pb ($|^{210}_{82}$Pb${}_{128}(gs)\rangle$) can be viewed as a (single) Cooper pair or, as the lowest pair addition mode of ²⁰⁸Pb populated in a two-nucleon transfer process, while the ground state of ${}^{62}_{28}$Ni₃₄ and of ${}^{114}_{50}$Sn₆₄ populated in the reaction ${}^{116}_{50}$Sn₆₆+${}^{60}_{28}$Ni${}_{32}\to$ ${}^{114}_{50}$Sn${}_{64}(gs)$+${}^{62}_{28}$Ni${}_{34}(gs)$ (Sect. 6 below), can be described at profit as BCS condensates of 7 (=(64-50)/2) and 3 (=(34-28)/2) neutron Cooper pairs respectively²²27 and 3 are the number of pairs of neutrons outside the neutron closed shells $N=50$ and $N=28$ of ¹¹⁴Sn and ⁶²Ni respectively., members of pairing rotational bands where the number of particles play the role of angular momentum (see e.g. [42] and refs therein). See also [43].

One can posit that Cooper pairs are at the basis of a variety of elementary modes of excitation. Pairing vibrations for single Cooper pairs, pairing rotations for few ones and supercurrents, for macroscopic amounts of them. Common to all of these modes when probed with Cooper pair tunneling one finds the dominance of successive transfer (tunneling) of entangled fermion partners.

Let us assume microwave electromagnetic radiation, that is, radiation of frequency 300 GHz–300 MHz, falls on the surface of a normal metal for which the electronic mean free path $l$ is short ($l<\delta$, see below), so that the dependence of the current density $J$ of carriers of charge $q=e$ (single electrons) on the electric field $\mathbf{E}$ can be described by the local relation $\mathbf{J}(\mathbf{r})=\sigma\mathbf{E}(\mathbf{r})$, where $\sigma$ is the conductivity. It is found that propagation inside the metal is attenuated according to $E=E_{0}\exp[i(kz-\omega t)-z/\delta]$ where $\delta=(2/\sigma\mu_{0}\omega)^{1/2}$ is the skin penetration depth (of the order of 0.7$\times 10^{4}$Å ), due to the appearance of eddy currents produced by the time dependence of the associated magnetic field. This is known as the normal skin effect. The quantity that is usually measured is the surface impedance $Z_{s}$ ([44]; see also [45]), ratio of the electric field at the surface to the total current flowing across a unit line at the surface. In the case in which $\delta<l$ (anomalous skin effect), one has to generalize the local Ohm’s law into a non-local equation, allowing for the fact that electrons accelerated by the electric field may travel some distance before they are scattered, and that the current density is not related to the local field, but to an appropriate average over a domain defined by the decay factor $\exp(-\rho/l)$ ($\rho=|\mathbf{r}-\mathbf{r}^{\prime}|$) [46].

A similar suggestion was made by Pippard [47] in generalizing the local London [48, 49] equation for the supercurrent density $\mathbf{J}_{s}=-\frac{n_{s}^{\prime}e^{2}}{m_{e}}\mathbf{A}$ of carriers of charge $q=2e$ (Cooper pairs), to a non-local one which is able also to describe pure (free from impurities) superconductors, in which case the averaging domain is defined by the decay factor $\exp(-\rho/\xi^{\,{}^{\prime}})$ $(1/\xi^{\,{}^{\prime}}=1/\xi+1/(\alpha l))$. In the case of tin, the numerical values of $\alpha\approx 0.8$ for the proportionality factor of $l$, and $\xi\approx 1.2\times 10^{4}$ Å  for the coherence (correlation) length, were estimated from penetration depth measurements, carried out in wires $\approx 1.4$ cm (resonance frequency $\approx 9400$ MHz) and 0.5 mm of diameter [47]. Following Pippard’s breakthrough, a number of techniques were developed to measure the penetration length (see e.g. [50] and refs. therein). This value of $\xi$ was microscopically validated by BCS theory (Eq. 5.50 [11]; see also Eq. (2)), and implies that no important variations of the electromagnetic field and thus neither of $n_{s}^{\prime}$ takes place within a range $\xi$.

Summing up, London’s equation leads to $H=H_{0}\exp(-z/\lambda_{L}$), where $\lambda_{L}=(m_{e}c^{2}/(4\pi n_{s}^{\prime}e^{2}))^{1/2}$ is known as London penetration depth. A quantity of the order of few hundreds of Å. The values of $\lambda$ emerging from Pippard’s equation are considerably larger than $\lambda_{L}$ in overall agreement with the data, aside from displaying the right dependence on $l$, quantity which changes from the value of $10^{6}$Å  in the purest available tin [47], to less than $10^{2}$Å  when doped with 3% indium.

In the case of two superconductors (S) separated by an insulating barrier of thickness $d$ of few nanometers, namely much smaller than $\xi$, single electrons can traverse the barrier, although with rather small probability ($P_{1}\approx 10^{-10}$) [51]. One thus talks about a weak link. They were employed by Giaever to carry out tunneling experiments of single-electron carriers (in both N-S and S-S links), which were instrumental in the determination of the pairing gap [52].

If a single electron has such a small probability to get through the barrier, the simultaneous tunneling of two electrons (probability $P_{1}^{2}$) will not be observed. To this Josephson argued that the wavefunctions of the electrons in the pair are (gauge) phase coherent. One has to add the amplitudes before taking modulus squared (see also [53]). It is like interference in optics with phase coherent wave mixing, and the probability of pair tunneling is comparable to the probability for a single electron. After the Cooper pair tunneling effects predicted by Josephson [54, 55] were confirmed [56, 57], the weak link set up became known as a Josephson junction.

The main effects predicted by Josephson were: (a) unbiased junction (direct current (dc) Josephson effect); the small but finite overlap of the BCS condensates described by $|{BCS(\ell)}\rangle$ and $|{BCS(r)}\rangle$ is sufficient to lock the associated gauge phase difference ($\delta\phi(\mathbf{R})=\phi_{\ell}-\phi_{r}$), a function which plays the role of the velocity potential leading to a momentum which acts on the center of mass ($\mathbf{R}$) of Cooper pairs. The associated zero-voltage direct current³³3It is of notice that one should use a gauge invariant phase, see for example [53]. $(I=I_{c}\sin(\delta\phi))$ of carriers of charge $q=2e$ (Cooper pairs) and maximum (critical) value $I_{c}=\frac{\pi}{e}\frac{\Delta_{\ell}\Delta_{r}}{\Delta_{\ell}+\Delta_{r}}\frac{1}{R_{b}}$ is undamped, because the internal degrees of freedom of the Cooper pairs are frozen by the pairing gap, $R_{b}$ being the junction resistance to N-currents ($q=e$). In the above relation $\Delta_{\ell}$ and $\Delta_{r}$ are the pairing gaps of the left ($\ell$) and right ($r$) superconductors with respect to the junction. Similarly, concerning the gauge angles $\phi_{\ell}$ and $\phi_{r}$; (b) biased junction (alternating current (ac)); when there is a constant voltage $V$ applied across the junction, circulation of an alternating supercurrent $I=I_{c}\sin(\nu_{J}t)$ of Cooper pairs with critical value $I_{c}$ and Josephson frequency $\nu_{J}=V(2e)/h$ was predicted. Because the frequency is so high (THz), the $\sin(\nu_{J}t)$ function averages out to essentially zero. In keeping with the fact that the junction is biased, each time a Cooper pair tunnels from one side of the barrier to the other, there is an energy difference $\Delta E=V\times 2e$. Being the process free of dissipation (alternating supercurrent), to leave the quasiparticle distribution unchanged, Cooper pairs can tunnel with emission of photons of frequency $\nu_{J}$ as has been experimentally observed both in absorption [57] and in emission [58] (see also [59]).

Schrödinger, one of the originators of quantum mechanics states [60] that not one but the characteristic trait of quantum mechanics is entanglement. That is, the phenomenon resulting from the fact that “when two systems, of which we know the states by their respective representatives, enter in temporary physical interaction due to known forces between them, and when after a time of mutual influence the systems separate again, then they can no longer be described in the same way as before, viz. by endowing each of them with a representative of its own…By the interaction the two representatives (or $\Psi$-function) have become entangled”.

Below the critical temperature, pairs of electrons moving in time reversal states close to the Fermi surface become, through the exchange of lattice phonons, entangled over a correlation length of the order of $10^{4}$Å, giving rise to Cooper pairs which eventually condense into the coherent $|{BCS}\rangle$ state. As a consequence, Cooper pair tunneling across a Josephson junction can take place one electron at a time, the partner electrons being similarly pairing entangled when each of them is in a different side of the junction $(\xi\gg d)$, than when both are within the same superconductor. Entanglement, the characteristic trait of quantum mechanics, is at the basis of the equality $(\ref{eq:6})$.

In what follows we aim at discussing the possibility of measuring the nuclear Cooper pair mean square radius (correlation length). A possibility brought into focus by a recent breakthrough in the subject of nuclear superfluidity. It was accomplished through the experimental study and the theoretical analysis of one- and two-neutron transfer reactions between two superconducting nuclei (see next Section), enabled by the use of magnetic and $\gamma$-ray spectrometers, [68] [69]

The reactions were carried out at twelve bombarding energies in the range 140.60 MeV $\leq E_{cm}\leq 167.95$ MeV. That is, from energies above the Coulomb barrier ($E_{B}=157.60$ MeV), to well below it. Absolute differential cross sections were measured at $\theta_{cm}=140^{\circ}$ and transfer probabilities $P_{1}$ and $P_{2}$, for one- and two-particle transfer reactions, extracted. Both transfer channels 12 (a) and 12 (b) are inclusive, in keeping with the fact that the beam energy resolution is 2 MeV. It implies that, within this energy range, individual levels populated in the transfer process cannot, in principle, be distinguished and an incoherent sum of their contribution to the transfer cross section, is to be made.

Gamma-coincidence experiments⁴⁴4In this connection a $\gamma$-particle coincidence experiment for the system ⁶⁰Ni +¹¹⁶Sn was carried out [69], and Doppler-corrected $\gamma$-spectra (discrete lines) for ^60,61,62Ni and ^116,115,114Sn in the energy interval 0–1600 keV displayed in Fig. 2 of that reference. It is of note that these $\gamma$-ray lines can hardly have relation to the $\gamma$-strength function displayed in Fig. 5 below. [69] testify to the fact that in the case of 12 (b), less than 24% of the cross section goes to excited states. Theoretical studies indicate that while channel 12 (b) is dominated by the ground-ground state transition, channel 12 (a) receives incoherent contributions from a number of excited states (see below).

One can view the heavy ion reactions (12) as a transient nuclear Giaever-like ((a)), and a transient (ac) nuclear Josephson-like ((b)) junctions between two superconducting nuclei, respectively. To calculate the absolute differential cross sections $d\sigma_{1n}/d\Omega|_{\theta_{cm}=140^{\circ}}$ associated with the one quasiparticle transfer process (12) (a), the occupation/emptiness amplitudes ($V_{j}/U_{j}$) of the single quasiparticle levels of Sn/Ni are needed. In the case of Sn they were obtained from BCS calculations [68], while in the case of Ni from $(d,p)$-data [72]. Making use of the distorted wave Born approximation (DWBA; see e.g. [73] and references therein), and employing the microscopically calculated [74, 75, 76]; see also [77] optical potential (hevy ion reaction dielectric function) reported in [68], an overall account of the experimental data is obtained [78] (see Fig. 1), by including the incoherent contribution of all (eleven) quasiparticle (Q) states of ⁶¹Ni with energies $\lesssim 2.640$ MeV. A value which is consistent with twice the value of the pairing gap of Ni, and with the S-Q mechanism at the basis of Giaever’s single-electron tunneling experiments.

Concerning the calculation of the absolute differential cross section $d\sigma_{2n}/d\Omega|_{\theta_{cm}=140^{\circ}}$ associated with (gs-gs) Cooper pair transfer, the corresponding spectroscopic amplitudes $(B_{j}=((2j+1)/2)^{1/2}U^{\prime}_{j}V^{\prime}_{j}\times e^{-2i\phi},\;U^{\prime}_{j}V^{\prime}_{j}$ being BCS coherence factors) were used to calculate, in second order DWBA, the $T$-matrix associated with simultaneous, non-orthogonality and successive transfer. This last one being the overwhelming contribution to the process, in analogy with Josephson’s pair tunneling mechanism and Eq. (9).

The role the gauge angle plays in the reaction 12 (b) has been discussed in [79] (within this context see also [80]). Here we limit ourselves to note that, in keeping with the indeterminacy relations $\delta N\delta\phi\geq 1$ one obtains, making use of the estimates made at the end of Sect. 6.1, $\delta\phi\approx((1/2.5)^{2}+(1/4)^{2})^{1/2}$ rad$\approx 0.5$ rad. Thus $\langle e^{i2\delta\phi}\rangle\approx\int_{0}^{0.5}e^{i2\delta\phi}d(\delta\phi)/\int_{0}^{0.5}d(\delta\phi)\approx\frac{1}{i}(-\frac{1}{2}+i\frac{\sqrt{3}}{2})$, and $|\langle e^{i2\delta\phi}\rangle|^{2}\approx 1$.

Within this context, it is also of note the weak coupling nature of the nuclear link considered (small overlap between abnormal densities and short collision time) for the selected bombarding conditions [78] (Fig. 1) and thus values of the distance of closest approach $D_{0}$, as also testified by the value of the junction energy $E_{J}\approx 0.5$ MeV (see Eq. (1) of [70]), obtained making use of the estimates displayed in Fig. 2. A value rather close to that expected from the Josephson weak link interaction expression [54, 53] $\Delta_{\ell}\Delta_{r}/(\Delta_{r}+\Delta_{\ell})\approx\Delta/2\approx 0.7$ MeV, $\Delta\approx 1.4$ MeV being a sensible estimate for both Sn and Ni superfluid nuclei.

Once the consistency between theory and the experimental findings have been assessed with a positive outcome, we try to extract from the data the translation of the relation (8), and thus of the critical momentum $q_{c}=1/\xi$ (i.e. the correlation length $\xi$), to the nuclear case. To do so, one is first reminded of the fact that in a heavy ion reaction like 12 (b), it is more natural to adjust the critical width of the transient junction (barrier) through which the Cooper pair has to tunnel,that is the critical distance of closest approach $D_{0}^{(c)}\approx 1/q_{c}\approx\xi$, than the (center of mass) momentum with which it tunnels. The fact that both requirements are, to a large extent physically equivalent, is in keeping with the fact that $k$ and $x$ are conjugated variables.

Summing up, using the microscopically calculated optical potential (heavy ion reaction dielectric function), theory provides an overall account of the data, in this case for the single Cooper pair transfer process (Fig. 1, see also [68] and [78]). In the case of the reaction (Eq. 12 (a)) one is in presence of a S-Q like transfer, while concerning the (gs-gs) reaction given in Eq. (12 (b)), one is confronted with a S-S one. Within this context, it is of note that the corresponding values of the real and imaginary components of the microscopic optical potential used in the calculations (see [68], see also p. 111 [76]) are, for the relative distance of closest approach $r=D_{0}\approx 13.5$ fm, -0.9 MeV and 0.6 MeV respectively, namely rather small.

Within this context, and in keeping with the fact that the laboratory bombarding energy associated with $D_{0}\approx 13.5$ fm is $\approx$3.9 MeV/$A$ ($\approx 452.49$ MeV/116), that is an order of magnitude smaller than the Fermi energy, one can expect that there can be time for the nuclear Cooper pair to be transferred back and forth more than once between target and projectile. That is, for more than one cycle of the quasielastic process

The neutron Cooper pair carries an effective charge $2\times e_{eff}=-e\times 2\left(78/176\right)\approx-e\times 0.89$. Consequently, the nuclear junction can be viewed as biased by a potential $V=\left(Q_{2n}/(2\times e_{eff})\right)\approx-1.469$ MV, and the few cycles of the process (13) considered as an alternating, single Cooper pair, nuclear Josephson supercurrent of frequency $\nu_{J}=2\times e_{eff}V/h=Q_{2n}/h$. As such, and in keeping with the fact that the system is few MeV below the Coulomb barrier, where tunneling proceeds essentially free of dissipation, one expects to be a source of ZHz photons emitted by a time-dependent dipole moment oscillating in a plane which forms an angle $\approx 63^{\circ}$ with the reaction plane, defined by the reaction condition of a smooth matching around the distance of closest approach of the projectile (Sn) neutron orbitals with that of the target (Ni) one (see Fig. 4; see also p. 316 and Fig. 8, p. 326 of [76]).

In Fig. 5, the $\gamma$-strength function

calculated as done in connection with $d\sigma_{2n}/d\Omega$, but introducing in the $T$-matrix also the dipole moment $d^{1}_{m_{\gamma}}=2\times e_{eff}\sqrt{\frac{4\pi}{3}}rY^{1}_{m_{\gamma}}(\hat{r})$, is shown. This prediction, normalized to the angle integrated two nucleon transfer cross section (Fig. 1), and the $\gamma$-phase factor provides a quantum mechanical estimate of the dipole moment $\langle d\rangle\approx 0.89\times e\times r=e\times 9.36$ fm of $r\approx 10.52$ fm (correlation length). In the above relation $\langle d\rangle$ stands for the incoherent summation of the three contributions $m_{\gamma}=\pm 1,0$. The back and forth transfer associated with the quasielastic process (13) is the nuclear analogue of an alternating Josephson current of a single Cooper pair expected to last for only few periods [78]. It is of note that the dipole coupling to the electromagnetic field seems natural when one is dealing with electrons, while tunneling of neutrons, but also within the present context of protons, are not understood as the motion of (effective), real charged particles.

In keeping with the fact that one is able to predict absolute two-nucleon transfer differential cross sections involving superfluid nuclei within a 10% accuracy [81], one expects the prediction displayed in Fig. 5 to be representative of heavy ion reactions between superfluid nuclei at energies below that of the Coulomb barrier

From the results displayed in Fig. 1 (see also Fig.3), and the predictions shown in Fig. 5, one can ascribe the value of $\xi=12.0\pm 1.5$ fm to the nuclear correlation length associated with Cooper pair tunneling in the (S-S) process (12) (b). At the basis of the conspicuous uncertainty ascribed to the value of $\xi$ one finds a technical reason, namely the different sources of the two inputs ($D_{0}^{(c)}$ and dipole moment value), and a general physical one, namely the role played by pairing vibrations in finite quantum many-body systems as compared to condensed matter. Within this context we refer to studies of pairing phase transitions in superconducting metallic particles. In particular, in an ensemble of small Sn particles of radius $R\lesssim 370$ Å, deposited in vacuum and insulated from each other by oxide layers ([82, 83, 84, 85, 86]), which testify to the fact that while in bulk superconductors, the large intrinsic range of the pair coherence length implies a very narrow critical region (around $T_{c})$, the size of this region increases as the dimension of the system decreases below the coherence length, leading essentially to zero-dimensional systems. A situation also found in the case of the BCS pairing phase transition observed in superfluid nuclei as a function of the rotational frequency (see e.g. [87] and references therein). The critical frequency $(\omega_{rot})_{c}$ playing a role similar to that of the critical magnetic field $H_{c}$ in superconductors.

BCS condensation of nuclear Cooper pairs leading to superfluid nuclei violates pair of particles conservation, thus defining a privileged direction in gauge space. The nuclear tweezer specific to tether the associated gauge angle cannot be but another system which also violates pairs of particles conservation. That is, another nuclear condensate in weak contact with the first one through the Josephson-like junction transiently established in a heavy-ion collision between superfluid nuclei. The associated single Cooper pair “current” being the specific probe of the intrinsic structure of the transferred Cooper pair. The corresponding absolute cross section normalized with respect to the single-quasiparticle transfer one, constitutes the connection to the experimental results where from one can read the value of $\xi$. One can then conclude that the mean square radius of nuclear Cooper pairs joins the ranks of those quantities characterizing the atomic nucleus which, in principle, are measurable.

Within this context one can state that an important page in the understanding of pairing in nuclei has been written at the Laboratori Nazionale di Legnaro and the Dipartimento di Fisica dell’Università di Torino, which also provides elements to attempt at relating special effects observed in the study of superconductivity in metals, with similar processes associated with superconductivity in finite quantum many-body systems, of which the atomic nucleus can be viewed as a paradigm.

This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract No. DE-AC52-07NA27344. F. B. thanks the Spanish Ministerio de Economía y Competitividad and FEDER funds under project FIS2017-88410-P. This work is part of the I+D+i project with Ref. PID2020-114687GB-I00, funded by MCIN/AEI/10.13039/501100011033.