How to access QED at supercritical Coulomb field

In slow collisions of two bare nuclei with the total charge number larger than the critical value, $Z_{\rm cr} \approx 173$, the initially neutral vacuum can spontaneously decay into the charged vacuum and two positrons. Detection of the spontaneous emission of positrons would be the direct evidence of this fundamental phenomenon. However, the spontaneous emission is generally masked by the dynamical positron emission, which is induced by a strong time-dependent electric field created by the colliding nuclei. In our recent paper [I.A. Maltsev et al., Phys. Rev. Lett. 123, 113401 (2019)] it has been shown that the spontaneous pair production can be observed via measurements of the pair-production probabilities for a given set of nuclear trajectories. In the present paper, we have significantly advanced this study by exploring additional aspects of the process we are interested in. We calculate the positron energy spectra and find that these spectra can give a clear signature of the transition from the subcritical to the supercritical regime. It is found that focusing on a part of the positron spectrum, which accounts for the energy region where the spontaneously created positrons can contribute, allows to get a much stronger evidence of the transition to the supercritical mode, making it very well pronounced in collisions, for example, of two uranium nuclei. The possibility of extending this study to collisions of bare nuclei with neutral atoms is also considered. The probability of a vacancy in the lowest-energy state of a quasimolecule which is formed in collisions of a bare U nucleus with neutral U and Cm atoms has been calculated. The relatively large values of this probability make such collisions suitable for observing the vacuum decay.

In slow collisions of two bare nuclei with the total charge number larger than the critical value, Zcr ≈ 173, the initially neutral vacuum can spontaneously decay into the charged vacuum and two positrons. Detection of the spontaneous emission of positrons would be the direct evidence of this fundamental phenomenon. However, the spontaneous emission is generally masked by the dynamical positron emission, which is induced by a strong time-dependent electric field created by the colliding nuclei. In our recent paper [I.A. Maltsev et al., Phys. Rev. Lett. 123, 113401 (2019)] it has been shown that the spontaneous pair production can be observed via measurements of the pair-production probabilities for a given set of nuclear trajectories. In the present paper, we have significantly advanced this study by exploring additional aspects of the process we are interested in. We calculate the positron energy spectra and find that these spectra can give a clear signature of the transition from the subcritical to the supercritical regime. It is found that focusing on a part of the positron spectrum, which accounts for the energy region where the spontaneously created positrons can contribute, allows to get a much stronger evidence of the transition to the supercritical mode, making it very well pronounced in collisions, for example, of two uranium nuclei. The possibility of extending this study to collisions of bare nuclei with neutral atoms is also considered. The probability of a vacancy in the lowest-energy state of a quasimolecule which is formed in collisions of a bare U nucleus with neutral U and Cm atoms has been calculated. The relatively large values of this probability make such collisions suitable for observing the vacuum decay.

I. INTRODUCTION
After the foundations of quantum field theory were formulated in the early 30s of the last century, it was shown that the theory predicts the spontaneous creation of electron-positron pairs by a constant uniform electric field if the strength of the field is comparable to or greater than a critical value, E cr = m 2 e c 3 /(|e| ) ≈ 1.3 × 10 16 V/cm, [1][2][3]. While from the point of view of the theory this phenomenon was studied in a large number of works (see Refs. [4][5][6][7][8][9][10][11][12][13][14][15][16] and references therein), its experimental detection was not possible because of practical inaccessibility of the required field strength. Some hopes for experimental detection of this fundamental effect were associated with the development of novel laser technologies [17,18]. But, apparently, in the foreseeable future, it is quite unlikely to realize a possible experiment in the required strong field regime solely on the basis of high-power laser.
An alternative approach to the study of vacuum properties * Corresponding author: v.shabaev@spbu.ru in the presence of a supercritical field was proposed in the works of Soviet and German physicists [19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34]. It is known that in the case of a pure Coulomb field induced by a pointlike charge Z the 1s level exists only up to Z ≈ 137 and then disappears. However, for extended nuclei this level goes continuously down and at Z = Z cr ≈ 173 reaches the onset of the negative-energy Dirac continuum (see Fig. 1). If this level was empty, it dives into the negative-energy continuum as a resonance, the decay of which should lead to a spontaneous creation of positrons. Namely, when switching from the subcritical to the supercritical mode, the initially neutral vacuum decays into a charged vacuum and two positrons (due to spin degeneracy). Since there are no nuclei with such a large charge in nature (the charge of the heaviest element discovered to date, oganesson, is 118), the only way to create a supercritical Coulomb field is to collide nuclei with the total charge number greater than the critical value (Z 1 +Z 2 > 173). The time dependence of the quasi-molecular energy levels in collision of two uranium ions (nuclei) is presented in Fig. 2. Following Greiner and co-authors [32], in this figure various pair-creation mechanisms are conventionally shown by arrows. The arrows a, b, and c denote dynamical pair-creation processes and the arrow d indicates the spontaneous pair creation. The dynamical pair creation takes place in both subcritical and supercritical modes while the spontaneous pair creation starts to work only in the supercritical regime. The dynamical mechanisms (a, b, c) superpose with the spontaneous one (d) and generally can not be separated from each other. In addition, the spontaneous pair creation is strongly suppressed due to very small period of the supercritical regime time (∼ 10 −21 s), which is by about two orders of magnitude smaller than the time required for the vacuum decay. All this was one of the main reasons why attempts to experimentally observe the spontaneous creation of positrons, which were undertaken more than 30 years ago in Darmstadt (see, e.g., Ref. [32] and references therein), were not successful. Moreover, the Frankfurt group, which worked on the theory of this phenomenon for more than 20 years, concluded that the vacuum decay can only be observed experimentally if colliding nuclei stick together for some time due to nuclear forces [35,36]. However, no evidence of the nuclear sticking in the collisions of interest has been found to date, making this scenario not promising for future experiments.
Despite the aforementioned conclusions by the Frankfurt group, one could expect, however, that the detailed study of quantum dynamics of the electron-positron field in lowenergy heavy-ion collisions would allow to find some signatures which indicate the principal difference between the subcritical and the supercritical regimes. To carry out these studies, first of all it was necessary to develop the theoretical and computational methods beyond the approximation made by the Frankfurt group. To this end, more than a decade ago new efforts have been initiated by the St. Petersburg group [37][38][39][40][41][42][43][44]. These methods allowed the calculations of the chargetransfer, electron-excitation, ionization, and pair-production processes. In particular, in Refs. [42][43][44] the first calculations of the pair-production probabilities beyond the monopole approximation, which was widely used by the Frankfurt group, were performed. These calculations showed that effects be- yond the monopole approximation only slightly change the pair-creation probabilities in the region of small impact parameters. This has provided much more flexibility in studying various scenarios of finding the signatures which can demonstrate the principal difference between the subcritical and the supercritical regimes. As a result of these studies, in Ref. [45] it was found that the vacuum decay can be observed via impact-sensitive measurements of pair-production probabilities. In this paper, we present additional aspects of this study and find important signatures of the principal difference between the subcritical and supercritical regimes, which can be observed in the positron spectra.
In the next sections we will mainly consider quantum dynamics of electron-positron field in collisions of heavy bare nuclei. Experiments on low-energy collisions of bare nuclei will be possible at the future facilities in Germany (GSI/FAIR) [46][47][48], China (HIAF) [49], and Russia (NICA) [50]. After the formulation of the basic theory, we will present the results of the calculations of the pair-creation probabilities and positron spectra as functions of the nuclear charges and the collision energy for the trajectrories with a given minimal internuclear distance. The dependence of the pair-creation probabilies and the positron spectra on the value of the minimal internuclear distance will be also studied. It will be shown that the study of a partial probability, which accounts for the region of the positron energy distribution where the spontaneously created positrons can contribute, allows a significant increase of the signature of the transition to the supercritical regime, compared to the analysis of the total probability. At the end of the paper we will discuss a possibility for studying the proposed scenarios on the vacuum decay observation in collisions of bare nuclei with neutral atoms.

A. General formalism
To study the quantum dynamics of the electron-positron field in low-energy collisions of heavy nuclei, one can use the approach, where the relative motion of the colliding nuclei is treated classically [24,32]. In this approach, to derive the pair-creation probabilities, one has to consider first the solutions of the Dirac equation in the external time-dependent potential induced by the colliding nuclei, whose motion is described by the Rutherford trajectories. For the relatively slow collisions of interest the magnetic part of the potential can be neglected and the time-dependent Dirac equation (TDDE) takes the form: with Here α, β are the Dirac matrices, m e is the electron mass, and V (r, t) is the total two-center potential induced by the nuclei: where the vectors R A and R B denote the positions of the colliding nuclei and are the corresponding nuclear potentials. For the nuclear charge distribution ρ(r) we utilize the model of the uniformly charged sphere. Considering the quantum dynamics from the initial time t in to the final time t out , one can define two sets of solutions of the TDDE, which have the asymptotics: where φ in i (r) and φ out i (r) are the eigenfunctions of the Dirac Hamiltonian at the corresponding time moments, The most natural way to derive formulas for the pair-creation probabilities is to use the second quantization formalism [4].
To this end, we introduce the "in" and "out" vacuum states, |0, in and |0, out , and the related annihilation operators: for particles, which we will denote symbolically by "i > F ", where F stands for the "Fermi level", and for antiparticles ("i < F "). These operators obey the standard anticommutation relations. The electron-positron field opera-torΨ(r, t) in the Heisenberg picture is defined aŝ For collisions of bare nuclei, the initial t = t in state of the system is described by the vacuum vector |0, in . To find the number of the electrons in a state "k" at t = t out , one should evaluate the value n k = 0, in|b in . This can be done by employing the equations (10)- (11) and the anticommutation relations between the creation and annihilation operators. As the result, one obtains [4,32] n k = 0, in|b where is the one-electron transition amplitude. Since the amplitudes a ij are time-independent, they can be evaluated at the time moment t in or t out : Alternatively, one can calculate the number of created positrons in a state "p" by formula The total number of the electron-positron pairs can be found as Taking into account that for the processes under consideration P ≪ 1, we will refer to this value as the pair-creation probability.
For the calculations of n k and n p it is convenient to use the finite basis set method [39, 40, 42-44, 51, 52]. With this method, one gets a spectrum consisting of a finite number of states, including both bound and continuum (positiveand negative-energy) pseudostates. As a result, all the related summations in the equations given above run over a finite number of states.

B. Monopole approximation
Numerical solution of the TDDE with the total two-center potential (3) is very time consuming. For this reason, the calculations of the pair-creation probabilities are generally restricted to the so-called monopole approximation for V (r, t).
In this approximation, the potential V (r, t) is expanded in spherical harmonics in the center-of-mass frame and only the lowest-order spherical-symmetric term of this expansion is taken into account, where n = r/r. The direct calculations of the pair-creation probabilities for the two-center potential [42][43][44] have demonstrated that the monopole approximation works rather well, unless the impact parameter of the nuclear collisions is too large. In particular, in the case of uranium-uranium collisions at the energy 6.2 MeV/u the difference between the twocenter and the monopole-approximation results varies from about 6% at the zero impact parameter, b = 0, to about 10% at b = 10 fm. Since the impact parameter b = 10 fm corresponds to the scattering angle θ = 79 o (at the energy 6.2 MeV/u), this means the validity of the monopole approximation in a wide range of the scattering angles around the backward direction.
For the spherically symmetric potential (17), the Dirac wave functions are represented in the standard form where Ω κm (n) is the spherical spinor, G κ (r, t)/r and F κ (r, t)/r are the radial Dirac components, and κ = (−1) j+l+1/2 (j + 1/2) is the relativistic quantum number defined by the angular momentum and parity. Due to the conservation of the total angular momentum, one has to deal with the time-dependent radial Dirac equation for a given value of κ, where is the radial Dirac wave function and is the radial Dirac Hamiltonian. For a given κ, the initial states, including the bound and continuum pseudostates, are obtained by diagonalization of the matrix H ≡ H κ (t in ) in a finite basis set. The basis functions are constructed from B-splines [53] within the framework of the dual-kinetic-balance approach [54]. To solve Eq. (19), we expand φ(r, t) on a basis of the eigenstates of the matrix H. For a given initial condition defined by Eq. (5), we have where N is the number of the states, ε k are the eigenvalues of the H matrix, and c ki are the expansion coefficients. Substitution of the expansion (22) into Eq. (19) leads to the equations: where The equations (23) are solved using the Crank-Nicolson scheme [55]: where ∆t is a small time step, c i = {c 1i , . . . , c N i }, and the M matrix is determined by With this technique we propagate all the bound and continuum quasistates from t in to t out and calculate the pair-creation probability for a given κ using the formula (16). The total pair-creation probability is obtained as a sum of the partial κ contributions. Usually it is sufficient to account for the κ = ±1 contributions only. This is due to the dominant contribution of the pair-creation processes in which the created electrons are captured into bound states [33,43,51]. The calculation of the positron spectrum can be performed according to Eq. (15). However, due to the use of the finite basis set methods, the direct calculation of the energydifferential spectrum dP/dε by this formula is not possible. Therefore, to obtain dP/dε p we use the Stieltjes method [51,52,56]: where ε p is the eigenvalue of the Hamiltonian in the finite basis set.

C. Choice of trajectories
In what follows, we will mainly consider the collisions of bare nuclei at the energies close to the Coulomb barrier, which is defined as the collision energy at which the nuclei touch each other. The nuclear trajectories are defined by nonrelativisic classical mechanics and, in the case of bare nuclei, are given by the well-known equations [57]. According to these equations, the minimal distance between the nuclei, R min , is related to the impact parameter b by  where E is the collision energy. For a given value of R min , the minimal energy, E 0 , corresponds to the head-on (b = 0) collision, Let us consider only the trajectories which correspond to the same minimal distance (R min ) for different impact parameters (b) and, therefore, for different collision energies (E). In case of uranium-uranium collisions with R min = 17.5 fm, which corresponds to E 0 = 5.9 MeV/u, the trajectories of interest are depicted in Fig. 3, where they are considered in the reference frame centered at one of the nuclei. In addition to the circle, corresponding to R min = 17.5 fm, we display also the circle with R cr = 32.7 fm, which determines the boundary (critical distance) between the subcritical and the supercritical regimes. Thus, we have the supercritical regime only during the period of time when the nuclei move from R cr to R min and back. This period of time decreases with increasing the collision energy (E) and, therefore, with increasing the impact parameter b (within the trajectories under consideration). This fact is clearly demonstrated in Fig. 4, which shows the supercritical regime duration as a function of η = E/E 0 at fixed R min = 17.5 fm (for all b and E). For convenience, the figure also shows the scattering angles, θ, which correspond to the given values of η. In the same figure, we present also the relative velocity of the nuclei at the distance R = R cr as a function of E/E 0 . Opposite to the supercritical regime duration, this velocity increases with increasing E/E 0 . It is clear that the dynamical pair creation must decrease monotonously with decreasing velocity (and, therefore, E/E 0 ) for a given R min . As to the spontaneous pair creation, it must increase monotonously with increasing the supercritical regime duration and, therefore, with decreasing E/E 0 . It follows that any increase in the pair-production probability at E/E 0 → 1 for a given R min should indicate the effect of the spontaneous pair creation, which takes place in the supercritical regime only.

III. CALCULATIONS AND RESULTS
The methods described in the previous section are employed for the calculations of the total pair-creation probabilities and positron spectra. The calculations are performed within the framework of the monopole approximation and are restricted to the |κ| = 1 contributions. As is noted above, these contributions almost completely determine the values of interest and the difference between the monopole approximation results and the full two-center results does not exceed 10% in a wide interval of the scattering angles.
The positron-creation probabilities and positron spectra in heavy-ion collisions can be measured by requiring a coincidence between a positron event and scattered projectiles or recoil target nuclei at given angles [58,59]. Since there is an unambiguous correspondence between the scattering angle and the impact parameter, we can study the pair-creation probabilities as functions of the impact parameter. However, in case of fully symmetric collisions, one should keep in mind a problem of distinguishing the contributions for the angles θ and 180 o − θ, which correspond to different impact parameters. The calculations for the uranium-uranium collisions at the energies near the Coulomb barrier show that the pair creation from the backward scattering trajectories dominate over the pair creation from the related forward scattering trajectories in a rather small region: θ = 180 o − 172 o . This can be seen from creation cross sections as functions of χ = 180 o − θ for the backward (θ > 90 o ) scattering trajectories and χ = θ for the forward (θ < 90 o ) scattering trajectories. This problem does not appear for non-symmetric (in nuclear mass or charge) collisions as well as for symmetric collisions of bare nuclei with neutral atoms, which will be briefly discussed at the end of this section. However, in what follows, for simplicity we will mainly consider the symmetric collisions of bare nuclei, assuming that the backward and forward trajectories can be distinguished by some experimental tricks.
A. Pair-creation probabilities for symmetric collisions In Fig. 6 we present the results for the pair-creation probability in symmetric (Z = Z 1 = Z 2 ) collisions as a function of the parameter η = E/E 0 , which is the ratio of the collision energy E to the energy of the head-on collision E 0 , and the nuclear charge number Z at the same minimal internuclear distance R min (it means that only the trajectories as presented in Fig. 3 must be considered for each R min ). The results are given for the values of R min = 17.5, 25, 50 fm, while the critical distance for the largest Z presented (Z = 96) is about 48 fm. It means that in the case R min = 50 fm for all Z under consideration there is no contribution from the spontaneous pair creation, and, therefore, the pair creation is completely due to the dynamical (induced) origin. As a result, with R min = 50 fm for all Z = 84 − 96 the paircreation probabilities decrease monotonously when η → 1. This is rather evident, since in the limit of adiabatically slow collisions the dynamical pair creation must be zero. The behavior of P changes when the supercritical regime becomes available. For R min = 17.5 fm, in the subcritical regime (Z < Z cr /2 ≈ 87.5), the pair-creation probability P remains decreasing monotonously when η → 1. However, the behavior of P changes when the nuclear charge number Z becomes sufficiently higher than the critical value. It can be seen that for such Z the pair-creation probability P increases at η → 1. This is due to the spontaneous pair-creation mechanism which starts to work in the supercritical regime, while the dynamical pair creation decreases at η → 1. To demonstrate this behavior in more detail, in Fig. 7 we display the s(κ = −1) and p 1/2 (κ = 1) contributions to the probability P as functions of η for Z = 85 − 96 at R min = 17.5 fm (similar graphs for R min = 16.5 fm have been presented in Ref. [45]). As one can see from this figure, the s contribution starts to increase significantly at η → 1 already for 1 1.5 2 2.5 3 η Z = 92, while the p 1/2 contribution remains decreasing at η → 1 even at Z = 96. In the case of Z = 92, the increase of the s contribution at η → 1 is fully compensated by the decrease of the p 1/2 contribution, that leads to an almost constant value of their sum at small η. The different η dependencies of the s and p 1/2 contributions are explained by different values of Z at which the 1s and 2p 1/2 states enter the negativeenergy continuum. For R min = 17.5 fm, this takes place at Z = Z 1s cr /2 ≡ Z cr /2 ≈ 87.5 and Z = Z 2p 1/2 cr /2 ≈ 95 for 1s and 2p 1/2 states, respectively (cf. Fig. 1). Fig. 8 displays the pair creation probabilities in symmetric Z 1 = Z 2 = 96 collisions as a function of R min and η = E/E 0 , where E 0 corresponds to R min under consideration. As one can see, the behavior of P at η → 1 changes when the minimal distance R min becomes sufficiently smaller than the critical value, R cr ≈ 48 fm for Z = 96. In the subcritical region, R min > R cr , the pair creation is of pure dynamical origin and, therefore, can only decrease with decreasing η (at a given value of R min ). However, in the supercritical regime, R min < R cr , the spontaneous pair creation becomes possible and finally leads to an increase of P when η → 1. Fig. 9 presents the derivative of the pair-creation probability P with respect to η, dP/dη, as a function of Z and R min at different scattering angles: θ = 60 o , 120 o , 180 o . The black lines in this figure indicate the boundaries between the subcritical and the supercritical regimes. According to our dis- cussion above, the pictures in Fig. 9 clearly demonstrate that the dP/dη derivative, being strictly positive in the subcritical region, becomes negative when the parameters Z and R min enter deeply enough into the supercritical domain. Moreover, Fig. 9 shows that the transition from the subcritical to the supercritical region changes the behavior of dP/dη, starting with an increase when approaching the black line from the subcritical region and ending with a strong decrease after crossing this line. It can be seen that in the strong supercritical regime the dP/dη derivative decreases in the directions which correspond to increasing Z and decreasing R min . The profile R min = 16.5 fm, which demonstrates the change of the behavior of dP/dη from the original increase at Z < Z cr /2 to the subsequent decrease at Z > Z cr /2, has been presented for η = 1 (θ = 180 o ) in Ref. [45]. In Fig. 10 we present separately the κ = ±1 contributions to dP/dη at η = 1 (θ = 180 o ) for the plane R min = 17.5 fm. It can be seen that, while the κ = −1 contribution has a maximum at Z ≈ 88, the κ = 1 contribution has it at Z ≈ 94.5. This is due to a higher value of the critical charge for the 2p 1/2 state, Z 2p 1/2 cr /2 ≈ 95, compared to the 1s critical charge, Z 1s cr /2 ≈ 87.5, (cf. Fig. 1). As a result, the maximum of the total contribution is shifted to Z ≈ 89. The profile Z = 96 at η = 1 (θ = 180 o ) in Fig. 9, which is separately presented in Fig. 11, shows a similar behavior of dP/dη, when R min passes the critical values, R 1s cr ≈ 48 fm and R 2p 1/2 cr ≈ 20 fm for Z = 96.

B. Positron spectra for symmetric collisions
We have also calculated the positron energy distributions for the trajectories under consideration (Fig. 3). In Fig. 12 we present the total positron spectra for the symmetric collisions with Z = Z 1 = Z 2 = 83 − 96 at R min = 17.5 fm and η = E/E 0 = 1.0, 1.  play separately the contributions to the positron spectra from the s(κ = −1) and p 1/2 (κ = 1) channels, respectively. As mentioned above, these two channels almost completely determine the total probability. The presented positron spectra demonstrate the well known fact (see, e.g., Refs. [29,31]) that the emission of low-energy positrons is strongly suppressed by the repulsive interaction with the nuclei while at high energy the spectra fall off exponentially. In the case of R min = 17.5 fm, the supercritical regime starts at Z ≈ 87.5 and Z ≈ 95 for the 1s and 2p 1/2 states, respectively. As one can see from Figs. 12, 13, and 14, in all cases the behavior of the positron spectra manifests a clear signature of  the transition to the supercritical regime. Indeed, in the subcritical regime the positron spectrum curves at the energies near the maximum and higher are strictly ordered according to the η values: the curve with a larger η is strictly higher than the curve with a smaller η. This is due to the fact that the dynamical (induced) creation of positrons in general should increase with increasing the collision energy. In the supercritical regime, however, in the region near the maximum of the positron spectrum, the curves corresponding to smaller values of η become higher than the ones with larger η, while at sufficiently large values of the positron energy the order of the  curves remains the same as in the subcritical regime. This might be explained as follows. The spontaneous pair creation, which starts to work in the supercritical regime, is greater at lower values of η, which correspond to larger values of the supercritical time duration (Fig. 4). In addition, the spontaneous positrons are restricted to the lower part of the energy spectrum. For instance, in the case of the U 92+ −U 92+ collision, the spontaneous positrons should be mainly limited to the energy range 0 to 600 keV. Indeed, let us consider first hypothetical collisions where the nuclei move adiabatically slowly along the actual trajectories [45,51]. In such collisions all created positrons are of pure spontaneous origin and, therefore, the energies of the emitted positrons are completely determined by the positions of the supercritical resonances. Table I presents these resonance energies, ε res , and the related kinetic positron energies, ε = |ε res | − m e c 2 , for the U 92+ −U 92+ and Cm 96+ −Cm 96+ systems at different internuclear distances. These energies have been obtained within the framework of the monopole approximation employing the complex rotation method [60][61][62][63][64]. The application of this method for the case of point nuclei leads to the results which are in good agreement with the values obtained by the use of related formulas from Ref. [65]. We note also that the cor- responding calculations beyond the monopole approximation [63,66] yield the resonance energies which are very close to the monopole-approximation ones. According to Table I, for the U 92+ −U 92+ system the positron kinetic energies do not exceed 300 keV, while the related natural resonance widths do not exceed a few keV [63,66]. But in the real (non-adiabatic) collision the finite collision time yields a dynamical width, which is much larger than the natural resonance one. The dynamical width can be roughly estimated by the uncertainty principle Γ dyn ∼ /τ col , where τ col is determined as the supercritical time duration. With τ col ∼ 2 × 10 −21 s (see Fig. 4) one obtains Γ dyn ∼ 300 keV, which restricts the energy of spontaneous positrons to the interval 0 to 600 keV. The large value of Γ dyn prevents the existence of resonance structure in the positron spectra and smears the area, where the reverse order of the η = 1.0, 1.1, 1.2 curves takes place, compared to the subcritical case. In addition, the strong superposition of the dynamical and spontaneous pair-creation mechanisms in the real collisions makes the transition from the subcritical to the supercritical mode quite smooth. However, the quali-tative changes in the positron spectra, that are demonstrated in Figs. 12, 13, and 14, must be considered as a conclusive proof of the access to the supercritical mode. This statement is also confirmed by Fig. 15, where the positron spectra for the U 92+ −U 92+ collision at different R min are presented. Again, the transition from the subcritical to the supercritical mode (R cr = 32.7 fm) changes the order of the curves near the maximum.
As one can see from Figs. 12, 13, 14, and 15, the changes in the positron spectra caused by the transition to the supercritical mode are mainly concentrated at the energies around the spectrum maximum and lower, where the spontaneous pair creation takes place. This fact can be used to quantify the signature of the transition to the supercritical mode in terms an integral characteristic of the positron spectrum which accounts only for a part of it instead of the total probability. It is natural to restrict this part to the region, where the changes of interest take place. To this end, we introduce a partial probability P x , which is defined as shown in Fig. 16. This means that the probability P x accounts only for the positrons with the energies not exceeding the abscissa value corresponding to the point dP/dε = (1 − x)(dP/dε) max on the right-hand side from the positron-spectrum maximum at η = 1 (θ = 180 o ).
The parameter x should be chosen to cover the area which includes the positrons with the energies within the range Γ dyn from the resonance energy. As discussed above, in the case of the U 92+ −U 92+ collision the energies of the spontaneous positrons are mainly limited by about 600 keV. It follows that the parameter x should be chosen not less than x ≈ 0.1, which corresponds to the inclusion of about 50% of the total probability.
In Fig. 17 we display the derivative d log P x /dη = (1/P x )dP x /dη at η = 1 (θ = 180 o ) for x = 0.1, 0.25, 0.5, 1.0 in the symmetric (Z = Z 1 = Z 2 ) collisions in the range Z = 84 − 92. As one can see from the figure, the use of d log P x /dη| η=1 at x = 0.1, 0.25, and even at x = 0.5 (instead of x = 1 corresponding to the total probability) leads to a strong increase of the effect we are interested in. Indeed, in the case of the U 92+ −U 92+ collision, the derivative d log P x /dη| η=1 amounts to about −0.6 at x = 0.1, −0.5 at x = 0.25, and −0.4 at x = 0.5, while for the total probability (d log P x=1 /dη| η=1 ) it is almost zero. This means taking into account only a part of the positron spectrum, as defined above, allows to get a much stronger evidence of the transition to the supercritical mode. This statement is supported by Fig. 18, where we present the ratio of P x evaluated at different η to its value at η = 1 (θ = 180 o ) as a function of Z = Z 1 = Z 2 . The transition from the region P x (η)/P x (1) > 1 to the region P x (η)/P x (1) < 1 indicates the transition from the subcritical to the supercritical mode. It can be seen that for x = 0.1, 0.25, 0.5 this transition takes place at lower Z values than for x = 1 (total probability). In Fig. 19, we present the same ratio, P x (η)/P x (1), for the U 92+ −U 92+ collision as a function of R min . Again, this figure clearly confirms the advantage of studying a part of the spectrum compared to the total one and makes the uraniumuranium collisions very promising for observing the vacuum decay. C. Pair-creation probabilities and positron spectra for non-symmetric collisions All the above studies can be directly extended to nonsymmetric collisions. In Fig. 20 we present the pair-creation probability in non-symmetric U 92+ − Cm 96+ collisions as a function of the minimal internuclear distance R min and the parameter η = E/E 0 . As in the case of the symmetric Cm 96+ − Cm 96+ collisions (see Fig. 8), the pair-creation probability P starts to increase with decreasing η when the system enters deeply enough into the supercritical regime, which means that R min becomes significantly smaller than the critical value, R cr ≈ 40 fm. In contrast, in the subcritical region P decreases with decreasing η (at a given value of R min ). In Fig. 21 we present the positron spectra for non-symmetric Po 84+ − Ra 88+ , Ra 88+ − U 92+ , and U 92+ − Cm 96+ collisions. As in the case of the symmetric collisions, the behavior of the positron spectra indicates a clear signature of the transition to the supercritical regime.

D. Pair creation in collisions of bare nuclei with neutral atoms
In the previous sections, we studied the pair creation in collisions of bare nuclei only. In this section, we will consider whether it is possible in principle to extend these studies to collisions of bare nuclei with neutral atoms. While the direct calculation of the pair production in such collisions requires development of special methods that are beyond the scope of this work, some rough estimates can be made by calculating the probability of a vacancy in the 1σ state of a quasimolecule formed in such a collision. This is due to the fact that, in accordance with the results for collisions of bare nuclei, the main contribution to the pair creation is given by the process with the electron captured into the lowest-energy bound state [33,[42][43][44]. Therefore, the pair-creation probability is mainly defined by availability of a vacancy in the 1σ level when the nuclei approach each other. In this section, we evaluate the probability of a vacancy in the 1σ state at the critical distance for the U−U 92+ and Cm−U 92+ collisions using the two-center method developed in Ref. [38]. by the DFT method in the LDA approximation with the PZ parametrization. To the leading order, the external potential effect can be taken into account by including the external potential within the monopole approximation, when only the spherically-symmetric part of the reexpansion of the potential of the other atom (ion) is taken into account. It should be noted that the basis constructed in this way changes as the internuclear distance changes. In other words, this basis depends on the time and the corresponding time derivatives must  be added to the Hamiltonian matrix [37]. The basis expansion coefficients of a set of one electron time-dependent wave functions ψ i (t) are defined using the direct evolution (exponential) operatorÛ method [37]. These coefficients are represented by the columns of the U evolution matrix (the matrix of the op-eratorÛ ). The probability of pair creation with electron captured into a vacancy at the quasi-molecular 1σ level can be roughly estimated as where P (C/Ā B) is the conditional probability of a pair being   created and the vacancy being filled, provided that there is exactly one 1σ 1/2 vacancy (eventĀ) and the state 1σ −1/2 is occupied (event B). The coefficient K vac takes into account the probability of filling one or two vacancies and the factor 2 accounts for two values of the angular momentum projection µ = ±1/2 (see the Appendix for details).
To calculate the probability of a vacancy in the 1σ level of the Hamiltonian of a quasimolecule at a given time we use the approach which is described in the Appendix. This approach is equivalent to a method which was previously used for the evaluation of inclusive probabilities (see Refs. [68,69] and references therein).
We define the time-dependent one-particle density operator ρ(t) and one-hole density operatorρ(t) where q i = 0, 1 is the number of electrons on a spin-orbital ψ i , which is defined by the initial condition. The inclusive probability P n of finding n states occupied in an N -electron system while the remaining N − n electrons are not detected is given by the determinant of an n × n matrix γ constructed from the one-electron density matrix [68] P n = det γ , where ϕ k are the stationary wavefunctions of the Hamiltonian at a given time, and k, j = 1, . . . , n. Similarly, the probabilitȳ P n of finding n states unoccupied is determined bȳ In this paper, we investigated the probabilityP n of finding two vacancies (n = 2). As a result, the coefficient K vac has been calculated using the following expression (see the Appendix) As the occupied q i = 1 spin orbitals ψ i (t), we consider all single-electron states of the ground-state configuration of the neutral atom U or Cm and all states of the negative-energy Dirac continuum. All the positive-energy states of the bare uranium nucleus are considered as unoccupied. The occupation of single-electron states of the negative-energy spectrum corresponds to the Dirac picture and takes into account the Pauli exclusion principle and pair production processes. By means of the equation (34) we calculated the 1σ vacancy probabilities for the Rutherford trajectories presented in Fig. 3 (R min = 17.5 fm). In Fig. 22 we present the coefficient K vac at the critical internuclear distance, which equals to 27.0 fm and 34.8 fm for the U−U 92+ and Cm−U 92+ collisions, respectively, as a function of the collision energy, η = E/E 0 , where E 0 is the head-on collision energy. Assuming K vac remains of the same order of magnitude throughout the supercritical region as at its boundary (see Fig. 22), we believe that the pair-creation probability should be only a few times less, compared to the collisions of bare nuclei. Therefore, due to relatively large values of K vac , all scenarios considered above for the collisions of bare nuclei can be applied to the collisions of bare nuclei with neutral atoms, provided the corresponding calculations of quantum dynamics of electrons are performed.

IV. CONCLUSION
We have studied possible scenarios to access QED in the supercritical Coulomb field which can be created by heavy nuclei in low-energy collisions near the Coulomb barrier. This has been done by the calculations of pair production in the collisions along the trajectories, which correspond to a given minimum internuclear distance. The clear signatures indicating the transition from the subcritical to the supercritical regime have been found in both pair-creation probabilities and positron spectra. The evidence of such a transition becomes even more convincing if one considers only a part of the positron spectrum around its maximum. For instance, it is very well pronounced in collisions of two uranium nuclei. In contrast to the previous studies by other authors, the scenarios considered do not require any sticking of colliding nuclei. Low-energy collisions of a bare uranium nucleus with neutral uranium and curium atoms have been also studied and the probability of a vacancy in the lowest energy level of a quasimolecule formed in such collisions has been evaluated. These calculations showed that the same scenario can be applied to access QED at the supercritical field in collisions of bare nuclei with neutral atoms. The experimental study of the proposed scenarios, which seems feasible with the future facilities at GSI/FAIR [46][47][48], HIAF [49], and NICA [50], would either prove the vacuum decay in the supercritical Coulomb field or lead to discovery of a new physics, which is beyond the presently used QED formalism. the wave function of the system. We can also consider the density operatorρ n as a linear integral operator with the density matrix ρ n being its kernel. Then, the probability P n of finding n-electron system in a state Φ n is given by [71] P n = n! Φ n |ρ n | Φ n . (A2) Consider a system of independent N electrons, described by a set of one-electron wave functions ψ i (i = 1, . . . , N ). The n-th order RDM in this system is given by [70,71] ρ n (x 1 , . . . x n ; x ′ 1 , . . . , x ′ n ) = 1 n!
where ρ 1 (x, x ′ ) is the one-particle density matrix defined by The wave function Φ n of an n-electron state is given by the Slater determinant constructed from one-electron functions ϕ k . In this case it is easy to show that the probability P n defined by Eq. (A2) is equal to the determinant of an n × n matrix γ constructed from the one-electron density matrix, whereρ 1 is the one-particle density operator defined bŷ Thus, P n is the probability of finding n states occupied in the N -electron system while the remaining N − n electrons are not detected. The equation (A5) coincides with that obtained by the method of inclusive probabilities (see Refs. [68,69] and references therein). We can also introduce the probabilityP m of finding m holes in an N -electron system by defining the one-hole density matrixρ 1 . Due to completeness of the set of functions ψ i (x) the hole density operator can be written in the form Then, we get P m = detγ ,γ kj = ϕ k |ρ 1 | ϕ j = δ kj − γ kj . (A8) B. Probability of finding one or two core holes (m = 2) Consider an N -electron system and two one-electron wave functions ϕ 1 (x) and ϕ 2 (x) which correspond to 1σ states with the angular momentum projection µ = 1/2 and µ = −1/2, respectively. We denote by A the event when 1σ 1/2 is occupied by electron and byĀ the event when this state is vacant. Similarly, the event B means that the state 1σ −1/2 is occupied and the eventB means that 1σ −1/2 is vacant.
Thus, four incompatible events can occur: A B, AB,Ā B andĀB, which correspond to the cases: both states are occupied, the first state is occupied and the second state is vacant, the first state is vacant and the second state is occupied, and both states are vacant. Using the equations (A5) and (A8), we obtain for the corresponding probabilities With the help of the total probability rule, for the probability of pair creation with the occupation of at least one 1σ vacancy, P (C), we obtain P (C) = P (C/A B) · P (A B) + P (C/Ā B) · P (Ā B)+ + P (C/AB) · P (AB) + P (C/ĀB) · P (ĀB) .
(A11) Here P (C/D) is the conditional probability, i.e. the probability of occurring the event C, given the event D has occurred. Assuming the pair-creation probability is small enough, we obtain P (C/ĀB) ≃ P (C/Ā B) + P (C/AB) . (A12) Then, taking into account that P (C/A B) = 0, we get where K vac = P (Ā) = 1 − γ 11 . (A15)