Dynamics of Fragmented Condensates and Macroscopic Entanglement

TL;DR

This paper explores how increasing interaction strength in colliding condensates leads to phase locking and entanglement of superfluid fragments, highlighting the transition from independent phases to a unified, entangled state.

Contribution

It demonstrates the transition from independent to phase-locked condensates and the emergence of macroscopic entanglement due to strong coupling effects.

Findings

Weak coupling allows phase-dependent collision outcomes.

Strong coupling induces phase locking of condensates.

Fragmented condensates become entangled upon separation.

Abstract

The relative phase of the order parameters in the collision of two condensates can influence the outcome of their collision in the case of weak coupling. With increasing interaction strength however, the initially independent phases of the two order parameters in the colliding partners quickly become phase locked, as the strong coupling favors an overall phase rigidity of the entire condensate, and upon their separation the emerging superfluid fragments become entangled.

Equations

$$\rho ({\mathbf{r}}_1,{\mathbf{r}}_2)=⟨0|{\hat{\psi }}^{†}({\mathbf{r}}_1)\hat{\psi }({\mathbf{r}}_2)|0⟩\Rightarrow n_0{\psi }^{\ast }({\mathbf{r}}_1)\psi ({\mathbf{r}}_2),$$

$$i\hslash {\dot{\psi }}_k(\mathbf{r},t)=-\frac{{\hslash }^2}{2m}△{\psi }_k(\mathbf{r},t)+V_k(\mathbf{r}){\psi }_k(\mathbf{r},t)$$

$$+g{\left|{\psi }_k(\mathbf{r},t)\right|}^2{\psi }_k(\mathbf{r},t)={\mu }_k{\psi }_k(\mathbf{r},t)={\mu }_k{\varphi }_k(\mathbf{r})e^{-i{\mu }_{k}t/\hslash }.$$

$$\Psi (\mathbf{r},t)={\psi }_1(\mathbf{r},t)+e^{i\alpha }{\psi }_2(\mathbf{r},t),$$

$${\psi }_k(\mathbf{r},t)={\varphi }_k(\mathbf{r}-{\mathbf{r}}_k-{\mathbf{u}}_{k}t)e^{i(m{\mathbf{u}}_k\cdot \mathbf{r}-{\mu }_{k}t/\hslash -m{\mathbf{u}}_k^{2}t/2)/\hslash },$$

$$i\hslash \dot{\Psi }(\mathbf{r},t)=\left[-\frac{{\hslash }^2}{2m}△+g{\left|\Psi (\mathbf{r},t)\right|}^2+U(\mathbf{r},t)\right]\Psi (\mathbf{r},t).$$

$$i\hslash {\dot{\psi }}_k(\mathbf{r},t)=-\frac{{\hslash }^2}{2m}△{\psi }_k(\mathbf{r},t)+U(\mathbf{r},t){\psi }_k(\mathbf{r},t),$$

$$|\Psi (\tau ,\alpha )$$

$$V_k(x)=-\frac{s_k(s_k+1)}{{\cosh }^2(x)}-g\frac{N_k{n}_k}{{\cosh }^{2s_k}(x)},$$

$$\int dx\frac{1}{{\cosh }^{2s_k}(x)}=n_k,k=1,2$$

$${\varphi }_k(x)=\frac{\sqrt{n_k}}{{\cosh }^{s_k}(x)}$$

$$-\frac{d^2{\varphi }_k(x)}{d{x}^2}+V_k(x){\varphi }_k(x)+g{N}_k|{\varphi }^k(x){|}^2{\varphi }_k(x)$$

$$=-s_k(s_k+1){\varphi }_k(x)$$

$$O_N(\alpha )$$

$$|{\Psi }_N(\alpha )⟩$$

$$\Psi ({\mathbf{r}}_1,{\mathbf{r}}_2,t)$$

$$=$$

$$e^{i\alpha }[{\psi }_1({\mathbf{r}}_1,t){\psi }_2({\mathbf{r}}_2,t)+{\psi }_1({\mathbf{r}}_2,t){\psi }_2({\mathbf{r}}_1,t)]$$

Full text

††preprint: NT@UW-17-02

Since the gauge symmetry is spontaneously broken in superfluids, it is reasonable to wonder under what conditions the relative phase of two superfluids is physically relevant. The Josephson effect Josephson (1964, 1974), experiments with cold Bose or Fermi atoms Shin et al. (2004); Yefsah et al. (2013); Ku et al. (2014, 2016); Bulgac et al. (2014); Wlazłowski et al. (2015), and the superfluid fragments emerging from nuclear fission Bulgac et al. (2017a); Bulgac ; Bulgac et al. (2017b), are just a few examples where that is the case. As we will discuss here, there are other situations when one would however expect that the relative phase of two condensates is physically irrelevant. However, the emerging overall picture of the role of the relative phase of two condensates appears to be more complex than envisaged so far. Recently Magierski, Sekizawa, and Wlazłowski (MSW) Magierski et al. reported on a rather surprising observation concerning the role the pairing field plays in the collisions of two heavy-ions at energies near the Coulomb barrier. MSW observed a very strong dependence of the properties of the emerging fragments on the relative phase of the pairing condensates in the initial colliding nuclei. In a somewhat related study of 20O+20O Hashimoto and Scamps (2016), the reported effect was rather weak, a result confirmed in the similar case of 44Ca+44Ca Sekizawa et al. , due to the small number of nucleons above the closed shell. The amplitude of the pairing field $\Delta$ in nuclei is of the order of 1 MeV, which is significantly smaller than the magnitude of the normal single particle field, which is of the order of 50 MeV. The character of the nuclear pairing correlations is recognized in literature of being of the Bardeen-Cooper-Schriefer (BCS) type Bardeen et al. (1957), a theory which describes weak coupling pairing with Cooper pairs with sizes significantly larger than the average separation between fermions. The gain in binding energy due to pairing correlations, called condensation energy $E_{cond}=-N(0)|\Delta|^{2}/2$, can hardly be greater than perhaps a few MeVs. MSW report however that in the collision of 240Pu on 240Pu near the Coulomb barrier pairing effects can lead to changes in the total kinetic energy of the emerging fragments of up to 20 MeV and that the apparent height of the fusion barrier could be changed by 10 MeV or even more. These dramatic changes, with an energy significantly higher than the magnitude of the total pairing condensation energy, were correlated by MSW with the relative phase of the pairing fields in the two colliding partners prior to collisions.

The gauge symmetry breaking bears similarity with the rotational symmetry breaking in case of deformed nuclei, when their relative orientations plays a noticeable role in heavy-ion fusion reactions and various decays. The MSW results, obtained by solving the time-dependent density functional theory (TDDFT) equations, can be reproduced semi-quantitatively using a simple Ginzburg-Landau (GL) approach Ginzburg and Landau (1950); Ginzburg (2004), or the formally equivalent static Gross-Pitaevskii (GP) description Gross (1961); Pitaevskii (1961). When the two nuclei touch the phase of the condensate can change across the contact region, as in a domain wall, in a manner superficially similar to the tunneling current in a Josephson junction Josephson (1964, 1974), albeit in the absence of a barrier.

In the presence of pairing correlations the ground state of a nucleus is a Bose-Einstein condensate (BEC) of Cooper pairs, which in theory is accurately described in the grand canonical ensemble, where only the average particle number is specified. The phase of the order parameter $\hat{\phi}$ is conjugate to the particle number $\hat{N}$, and thus in a system with well defined particle number the phase is undefined Anderson (1966); Carruthers and Nieto (1968). However, as Anderson points out Anderson (1986): in a bucket of liquid helium below the $\lambda$-point “$\phi$ has become a classical variable, … any future experiment will be interpretable as though $\phi$ was fixed.” This is also the prevalent approach in describing nuclei with well defined pairing correlations, when the effect of particle projection is small. One can thus reasonably ask, a common question in condensed matter physics: “Can a nucleus have a well defined phase of the condensate with respect to another nucleus?” Since the total wave function of the two nuclei prior to their interaction is merely a product of two independent wave functions, one would expect that the interaction between two nuclei cannot depend on the phases of each initial wave functions. A (relative separation) coordinate dependence of the phase of the pairing field indicates the presence of a current. The phases of the pairing fields can be changed by arbitrary and independent gauge transformations in each partner prior to the moment the two nuclei touch and thus one can generate a phase gradient in the “neck.” An objection raised by G.F. Bertsch in discussions was that initial nuclei have well defined proton and neutron numbers, unlike the anomalous densities which are the central objects in a DFT approach, and the phase of the wave function of each nucleus prior to the collision should be physically irrelevant. Clearly a similar argument would not be accepted in case of deformed nuclei, as a number of observables are impacted ($\alpha$-decay penetrability, heavy-ion fusion cross sections, etc.) This kind of argumentation began at the inception of quantum mechanics and many have wondered about similar problems, see Anderson’s talk Anderson (1986) and the follow-up spirited discussion. As Anderson writes: “if the experimenter now cools down two entirely different, non-communicating buckets of liquid helium from $T>T_{\lambda}\rightarrow T\approx 0$, … upon opening an orifice between the two, would see initially with equal probability any fixed value of the phase difference, and thereafter no experiment he tried could recover the components of the wave-function which started out with different relative phases. He would not see zero interference current, …” This situation corresponds theoretically to a fragmented condensate Legget (2006) and the inability of the experimenter to recover the initial state is due the fact that the two buckets became macroscopically entangled after being in contact for some time. Macroscopic entanglement of up to hundreds to millions of particles have been put in evidence experimentally Iskhakov et al. (2012); Behbood et al. (2014); McConnell et al. (2015); Fröwis et al. ; Zarkeshian et al. . It is crucial to recognize that there are two qualitative steps in Anderson’s gedanken experiment, the creation of the initial state and the subsequent emergence of the final state. This is also the situation in the MSW simulations and the natural question arises, why these authors did not observe the outcome conjectured in Anderson’s gedanken experiment, as the outcome of their collisions showed a strong dependence on the initial relative phase of the condensates, unlike what Anderson conjectured. We are not aware however of any experiments in which the dependence on the strength of the coupling on the outcome of a collision and of the entanglement have been studied.

There is however another qualitatively different situation, relevant to experiments performed in cold gases Shin et al. (2004); Yefsah et al. (2013); Ku et al. (2014, 2016); Bulgac et al. (2014); Wlazłowski et al. (2015) or to superfluid fragments emerging from nuclear fission Bulgac et al. (2017a); Bulgac ; Bulgac et al. (2017b). This happens when one cools down a bucket of helium from above the $\lambda$-transition, and subsequently separates it into two parts kept always close to $T\approx 0$ and reunites them after they had different histories, and the two parts remain macroscopically entangled at all times Anderson (1986). In this situation the relative phase of the two buckets is always rather well defined, but the particle numbers in the two buckets are not. (We will not discuss here the role the phase diffusion can play.)

There will definitely be increasingly more studies of colliding superfluid nuclei and other systems in the future performed within the only practical microscopic framework available so far, the TDDFT. A correct interpretation of such numerical simulation results and a correct method to evaluate observables are stringent elements of our theoretical tools, tools which are still not yet ascertained. Nuclei contain many particles, are essentially macroscopic objects, and as Anderson has also noted Anderson (1986): “… the central problem of measurement theory is not the quantum mechanics of atoms, which is simple and easy, but the fact that macroscopic everyday objects are very difficult indeed for the quantum theory to deal with properly.” Many properties of nuclei (liquid drop mass formula, surface tension, compressibility, symmetry energy, hydrodynamics, collective motion, rotation, symmetry breaking, transport coefficients, etc.) can be and are often treated quite accurately using concepts characteristic for macroscopic systems.

In order to shed light on MSW’s very startling observation, that the relative phase of the pairing fields in two colliding nuclei can have a dramatic role in the collision process, we will turn at first to a simpler system, in which the role of the relative phase of two condensate can be easily studied. In the presence of pairing correlations nuclei can be treated as a BEC of interacting Cooper pairs, as in the case of electrons in superconductors Bardeen et al. (1957), and the total wave function can be represented as an anti-symmetrized product of Cooper pair wave functions. In the case of a weakly interacting Bose system at zero temperature a GP equation is extremely accurate Dalfovo et al. (1999). In the GP approximation a boson field operator $\hat{\psi}({\bf r})$ is replaced with its non-vanishing average $\psi({\bf r})=\langle 0|\hat{\psi}({\bf r})|0\rangle$ [a classic example of $U(1)$ broken gauge symmetry] and the accuracy of the approximation is of order $\sim 1/\sqrt{N}$, where $N$ is the total number of bosons. A BCS fermionic condensate is a system of weakly interacting Cooper pairs/bosons and qualitatively a GP equation is appropriate and has been used in the past numerous times. The weakness of the interaction is typically characterized by the ratio of the pairing gap to the Fermi energy $\Delta/\varepsilon_{F}\ll 1$. In the weak coupling limit, all Cooper pairs have a zero momentum, as in a BEC.

Typical BEC systems have all particles in one cloud and the one-body density matrix acquires the form

[TABLE]

when $|{\bf r}_{1}-{\bf r}_{2}|\rightarrow\infty$, and there is only one eigenvector with a macroscopic eigenvalue $n_{0}={\cal O}(N)$, a situation known as the off-diagonal long-range order (ODLRO) Ginzburg and Landau (1950); Penrose (1951); Penrose and Onsager (1956); Yang (1962). It is possible to have a fragmented BEC system Legget (2006), when two or more eigenvalues of the one-body density matrix $\rho({\bf r}_{1},{\bf r}_{2})$ are macroscopically large. This is the case of two BEC clouds with particle numbers $N_{1}$ and $N_{2}$ in two spatially well separated potential traps $V_{k}({\bf r})$, $\int d^{3}{\bf r}\left|\psi_{k}({\bf r},t)\right|^{2}=N_{k}$, $k=1,2$,

[TABLE]

Let us consider now this fragmented BEC condensate, when their initially spatially well separated trapping potentials are moving towards each other, and their combined wave function at times before the two clouds come into contact is naturally given by

[TABLE]

with $V_{k}({\bf r})\rightarrow U_{k}({\bf r},t)=V_{k}({\bf r}-{\bf r}_{k}-{\bf u}_{k}t)$ and $e^{i\alpha}$ arbitrary. Using $\Psi({\bf r},t)$ one can construct a coherent state $\exp[\tau\int d^{3}{\bf r}\Psi({\bf r},t)\hat{\psi}^{\dagger}({\bf r})]\,|0\rangle$ and the fragmented BEC state is obtained only after a specific particle projection is performed, see the discussion below and in connection with Eq. (Dynamics of Fragmented Condensates and Macroscopic Entanglement) and the Supplemental Online Material (SOM) Sup . We will assume that the velocities ${\bf u}_{k}$ are significantly smaller in magnitude than the speed of sound Dalfovo et al. (1999) $c=\sqrt{g|\Psi({\bf r},t)|^{2}/m}$ evaluated in the central part of the cloud, and therefore superfluidity is not endangered. At all times this combined wave function satisfies the time-dependent GP equation with $U({\bf r},t)=U_{1}({\bf r},t)+U_{2}({\bf r},t)$,

[TABLE]

Before contact each component of the total wave function $\psi_{k}({\bf r},t)$, see Eq. (1), satisfies its own time-dependent GP equation (1) with $V_{k}({\bf r})\rightarrow U_{k}({\bf r},t)$. The arbitrary phase $\exp(i\alpha)$ can arguably influence the dynamics if $g\neq 0$. This is the phase in one of the two cases of liquid helium buckets discussed by Anderson Anderson (1986). Unlike the overall phase of the many-body wave function, this phase cannot be removed now, similarly to the relative orientation of two colliding deformed nuclei. In the case of two separated condensates the overall order parameter is the sum of the two separated order parameters, similarly to magnetization for example. (The action of the magnetic field on the spin coordinate of a fermion is formally identical to the action of the pairing field on the two components of the fermionic quasiparticle Anderson (1958).) Magnetization is created by electric currents and magnetic moments, and when one brings two magnets into proximity, the two magnetic fields add up, even though the many-body electron wave functions for the two separated magnets are multiplied to each other. As in the case of a magnetic field, where the relative orientation of two magnetic fields is important, and in the case of the complex pairing field the relative phase of the two fields is important, as is in the case of Josephson junctions too. This relative phase is also arbitrary, but this relative phase can be controlled in some instances. In the vicinity of an isolated cloud one can apply for a finite interval of time a constant potential over the isolated cloud, a procedure performed in the case of cold atoms in experiments, equivalent to performing a local gauge transformation, and thus one can change the relative phase of two clouds Yefsah et al. (2013); Ku et al. (2014, 2016); Bulgac et al. (2014); Wlazłowski et al. (2015).

By analyzing both the GP equation, see SOM Sup , and the collision of superfluid nuclei we arrived at a totally unexpected and surprising result, that the strength of the interaction $g$ plays a qualitative role in the dynamics. By increasing the strength of the interaction from zero (corresponding to the case of non-interacting bosons or absence of pairing correlations in nuclei) to a relatively large value, the character of the collision changes dramatically, but in a continuous manner.

We observe the establishment of a common phase of the combined condensate for large values of the coupling constant, which clearly can be attributed to the phase rigidity in superfluids Ginzburg and Landau (1950); Ginzburg (2004); Anderson (1966, 1986, 1984). While the two partners are in contact the phase of the condensate becomes spatially constant over the entire system, and the phase gets locked. We illustrate the phase locking mechanism for both Fermi and Bose superfluid systems: with the collision of two superfluid nuclei described within the extension of TDDFT formalism to superfluid fermionic systems  Bulgac (2013) by changing the strength of the pairing correlations, see Fig. 1, and with the case of the collision of two BEC with relevant results in SOM Sup .

One can limit the analysis to a one-dimensional model as only matter, momentum, and energy transfer between two colliding partners along the line joining the two partners (which can rotate in space though) are controlling most of the dynamics, similarly to the case of the Josephson junction in the case of superconductors, when only dynamics across the junction is typically analyzed. In the absence of the interaction ($g\equiv 0$), the GP equation is linear, and each wave function $\psi_{k}({\bf r},t)$ satisfies independently the Schrödinger equation

[TABLE]

and after the two potential wells have past each other each wave function $\psi_{k}({\bf r},t)$ will split in between the two potential wells. Obviously, the linear combination of the wavefunctions $\Psi({\bf r},t)=\psi_{1}({\bf r},t)+e^{i\alpha}\psi_{2}({\bf r},t)$, which satisfies the same Schrödinger equation, depends on the relative phase. While for weak coupling $g$ the dynamics is $\alpha$-dependent, when the strength of the interaction $g$ is gradually increased, the dependence of the final outcome on the relative phase $\alpha$ becomes weaker and weaker the stronger the interaction gets, and the two cases $\alpha=0$ and $\alpha=\pi$ in their final state become almost identical, see Figure 1 for nuclei in 3D and SOM for bosons Sup .

When the coupling constant is sufficiently large, the two boson clouds penetrate each other and their final states are relatively little affected irrespective of the value of $\alpha$ and both clouds emerge with the initial number of particles practically unchanged and with very small excitation energies as well Sup . The role of the particle-particle interaction is to lead upon contact to a very rapid phase locking between the two condensates after which the properties of the final state depend very weakly on the phase $\exp(i\alpha)$. The strength of the interaction $g$ controls the speed at which the information is transmitted throughout the cloud. In the case of strong coupling, after the relatively short time needed to send “messages” between the two partners, the properties of the emerging final state are largely $\alpha$-independent and the two clouds become completely entangled upon separation. The total wave function corresponds in this case to a coherent state in the particle number difference $N_{-}=N_{1}-N_{2}$ and to a macroscopically entangled state of two large objects. This conclusion is in agreement with Anderson’s conjecture Anderson (1986) concerning the inability of an experimenter to recover the initial relative phase of the condensates $\alpha$ after establishing the contact between the two independently cooled liquid helium buckets from above $T_{\lambda}$. This also it clarifies the content of Anderson’s conjecture, that only when the superfluid correlations are “strong" enough the role of the initial relative phase is erased. This is also consistent with the generalized phase rigidity due to the term in GL equation $n_{s}\hbar^{2}|{\bf\nabla}\phi|^{2}/2m$ (where $n_{s}$ is the superfluid density) in the free energy of superfluids Ginzburg and Landau (1950); Ginzburg (2004); Anderson (1966, 1986, 1984), which is an emerging term, whose presence and strength are dictated by the interactions, and which is absent in non-interacting systems.

This dependence on $\alpha$ of the properties of the emerging fragments in the case of “weak" superfluid correlations reflects particle number difference fluctuations between the two initial partners, see also SOM Sup . The combined wave function of two superfluid nuclei (with even particle numbers), depending on two arbitrary gauge angles $\tau$ and $\alpha$, can be written as Bloch and Messiah (1962) (here for simplicity for one kind of nucleons only):

[TABLE]

where $k$ and $\overline{k}$ and $l$ and $\overline{l}$ denote pairs of time-reversed states in the two nuclei and $u_{k,l}$ and $v_{k,l}$ being the corresponding amplitudes of the Bogoliubov-Valatin quasiparticles. Integrating $\Psi(\tau,\alpha)$ over $\tau$ with the weight $e^{-i\tau N_{+}}$ will select the wave function with the total particle number $N_{+}=N_{1}+N_{2}$. Integrating $\Psi(\tau,\alpha)$ over $\alpha$ with the weight $e^{-iN_{-}\alpha}$ will select the exact particle difference $N_{-}=N_{1}-N_{2}$ between the two nuclei. In the case of weak coupling an additional projection over the relative phase $\alpha$ is required to ensure that the particle number difference between the two initial partners has the expected value, namely exactly zero ($\Delta N\equiv 0$) in the case of two identical nuclei, see also SOM Sup . One can expect that total kinetic energy and fusion rates distributions would become wider in case of superfluid colliding nuclei.

When comparing our simulations of 240Pu fission Bulgac et al. (2016) with realistic pairing interactions with simulations in which the pairing field was artificially increased to $\approx 3-4$ MeV Bulgac et al. (2017a); Bulgac ; Bulgac et al. (2017b) we observed a similar transition to a phase locking pattern: realistic nuclear pairing strength is relatively weak, the phase locking does not typically occur on the way from saddle-to-scission and the phase and the magnitude of the pairing fields fluctuate strongly both in space and time. In the case of strong pairing Bulgac et al. (2017a, b); Bulgac , even though the time from saddle-to-scission is about ten times shorter, the evolution is almost identical to the dynamics of an ideal or perfect fluid and the fission fragments emerge strongly entangled. While one might naively expect a faster rate of energy transfer from collective to intrinsic degrees of freedom, the fluctuations of the pairing field are greatly suppressed (due to larger gaps and larger critical velocities) and the evolving fissioning nucleus stays cool.

In conclusion, we have established that the initial relative phase of two colliding condensates plays an increasingly smaller role in the case of strong interactions, when a phase locking over the entire system is established fast (unless the entire system is very extended and the signal propagation time is large as well), and after the separation the final macroscopic (large) fragments emerge entangled.

We thank G.F. Bertsch for numerous discussions and P. Magierski, K. Sekizawa, and G. Wlazłowski for commenting on a draft of the manuscript. This work was supported in part by U.S. DOE Office of Science Grant DE-FG02-97ER41014. Calculations have been performed at the OLCF Titan and at NERSC Edison. This research used resources of the Oak Ridge Leadership Computing Facility, which is a DOE Office of Science User Facility supported under Contract DE-AC05-00OR22725 and of the National Energy Research Scientific computing Center, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231, and at Moonlight of the Institutional Computing Program at Los Alamos National Laboratory.

I Supplemental Online Material

We created two initial BEC clouds in a modified Posh-Teller external potential (units $\hbar=2m=1$)

[TABLE]

and where $N_{K}$ is the number of bosons in each cloud. With this choice the state

[TABLE]

is a stationary solution of the GP equation

[TABLE]

with $s_{k}\geq 1$. As described in the main text, we initialize two spatially separated BEC clouds with particle numbers $N_{1}$ and $N_{2}$ and collide them with various relative velocities, chosen below the critical velocity, see Figure 2.

In Figures 3 and 4 we show the differences between the particle numbers and excitation energies in a cloud for different initial relative phases of the two colliding BECs. With increasing coupling strength the clouds show less and less dependence of the final state properties on the initial relative phase difference of the condensates. The transition from one regime, where the dependence of the properties of the final fragments depend strongly on the initial relative phase between the two condensates, to the regime, where this dependence is almost absent, is gradual. The largest differences are observed for “small” coupling constants.

According to the general Hohenberg-Kohn theorem there is a one-to-one correspondence between the many-body wave function and the (generalized) density and thus in principle the many-body wavefunction can be determined from the (generalized) density $\Psi[\rho(\alpha)]$ (the dependence on the phase $\alpha$ is explicit). Since for superfluid systems the density $\rho(\alpha)$ describes a system with an average particle number only, the corresponding many-body wavefunction $\Psi[\rho(\alpha)]$ describes a system with only an average particle number. Any observable can be in principle evaluated using this many-body wave function, or in other words, for any many-body observable there exists a corresponding density functional as well, $O[\rho]=\langle\Psi[\rho]|\hat{O}|\Psi[\rho]\rangle$. Most of observable of interest $\hat{O}$ conserve the particle numbers, i.e. $\hat{O}$ commutes with the particle projector $\hat{P}_{N}=\hat{P}_{N}^{2}$. $\hat{P}_{N}$ can project either on the entire system or on each incident or emerging nucleus separately. The value of an observable for a fixed particle number $N$ can be evaluated as follows

[TABLE]

where $|\Psi_{N}(\alpha)$ is the projected many-body wavefunction, which may retain a dependence on the phase $\alpha$, see discussion in the main text in connection to and Ref. [13], P.W. Anderson (1986). The crucial question is: Is the projected many-body function $|\Psi_{N}(\alpha)\rangle$ really $\alpha$-dependent?

In the cases discussed in the main text with weak coupling an additional projection over the relative phase $\alpha$ is required to ensure that the particle number difference between the two identical initial partners has the expected value, namely exactly zero ($\Delta N\equiv 0$) in the case of two identical nuclei.

In the case of two bosons the wave function is

[TABLE]

[TABLE]

and only the part

[TABLE]

will describe correctly the initially two separated particles. The same applies to a similarly extracted component in the case of many bosons.

In the case when the phase locking of the two condensates is established fast, this kind of projection is largely unnecessary. In nuclear systems with realistic pairing coupling strength the outcome of the collisions depends on the relative phase $\alpha$ and observables should be projected on the appropriate particle number difference $\Delta N$, in order to remove spurious fluctuations in the particle number differences between the partners. In practice, in the case of realistic nuclear pairing interactions, one might opt to evaluate the average of $\langle\Psi(\alpha)|\hat{O}|\Psi(\alpha)\rangle$ over the phase $\alpha$, instead of preferably evaluating the projected on $\Delta N$ value $\langle\Psi_{\Delta N}(\alpha)|\hat{O}|\Psi_{\Delta N}\alpha)\rangle$, see Eqs.(9, 10). These two quantities might be turn out to be similar in practice.

In the case of the collision of two nuclei, illustrated in Fig. 1 in the main text, we have followed the same calculational procedure as in Ref. [32] cited in the main text, using for the normal part of the nuclear energy functional SLy4 with no omissions, and we have only varied the strength of the pairing coupling constant from a realistic value for nuclei to a value which brings the paring correlations for neutrons in the two 120Sn nuclei to a regime close to the unitary Fermi gas, when the pairing gap is of the order of the Fermi energy. We have initiated the two nuclei in their ground state, separated by a relative large distance, after which we have accelerated them towards each other.