Microscopic description of pair transfer between two superfluid Fermi systems. II. Quantum mixing of time-dependent Hartree-Fock-Bogolyubov trajectories

While superfluidity is accurately grasped with a state that explicitly breaks the particle number symmetry, a precise description of phenomena such as the particle transfer during heavy-ion reactions can only be achieved by considering systems with good particle numbers. We investigate the possibility to restore particle number in many-body dynamical problems by mixing up several time-dependent Hartree-Fock-Bogolyubov (TDHFB) trajectories. In our approach, each trajectory is independent from the others and the quantum mixing between trajectories is deduced from a variational principle. The associated theory can be seen as a simplified version of the multiconfiguration TDHFB (MC-TDHFB) theory. Its accuracy to tackle the problem of symmetry restoration in dynamical problems is illustrated for the case of two superfluid systems that exchange particles during a short time. In Phys. Rev. C 97, 034627 (2018), using a schematic model where two systems initially described by a pairing Hamiltonians are coupled during a short contact time, it was demonstrated that statistical mixing of TDHFB trajectories can only qualitatively describe the transfer process and that a fully quantum treatment is mandatory. We show here that the present MC-TDHFB approach gives an excellent agreement with the exact solution when the two superfluids are the same (symmetric case) or different (asymmetric case) and from weak to strong interaction strength. Finally, we discuss the benefits and bottleneck of this method in view of its application to realistic systems.

Figure 1

Illustration of the fact that it is equivalent to first rotate [through a rotation operator $R\left(\theta \right)]$ a mean-field state at $t_0$ and then evolve it with its mean-field propagator, or perform its mean-field evolution first and then rotate the state at time $t$.

Figure 2

Evolution of the (a) total energy, (b) fluctuation in particle number $A$, and (c) the probability of one pair transfer $P_{2n}$ as a function of time. Calculations were performed with a coupling strength $v_0=0.02g$ (perturbative regime) for the symmetric case ${\mathrm{\Omega }}_A={\mathrm{\Omega }}_B=6$ and $N_A^0=N_B^0=6$. The MC-${\mathrm{TDHFB}}_{\mathrm{I}}$ approach (red squares) is compared with the exact solution (black line) and the phase-space approach of Ref. [20] without renormalization of the interaction (blue filled circles).

Figure 3

Evolution of (a) the energy, (b) the number of particles in $A$, (c) its fluctuation, and (d) the probability of one pair transfer from $B$ to $A$ as a function of time for asymmetric collisions. Calculations are performed in a perturbative regime for the coupling strength $v_0=2\times {10}^{-2}g$. The MC-${\mathrm{TDHFB}}_{\mathrm{I}}$ approach (red squares) is compared with the exact solution (black line) and the phase-space approach of Ref. [20] without renormalization of the interaction (blue filled circles). The energy predicted with PSC (not shown) evolves in the range [11. 44, 11. 54] $g$.

Figure 4

Same as Fig. 3 for the non perturbative regime with $v_0=g$.

Figure 5

Distribution of probabilities of the final number of particles in the system $A$. Full lines stand for probabilities determined from the exact solution, dashed lines with symbols correspond to MC-${\mathrm{TDHFB}}_{\mathrm{I}}$ calculations and shaded area to PSC estimation when available. The probabilities are calculated after the collision between two nondegenerate systems with ${\mathrm{\Omega }}_A={\mathrm{\Omega }}_B=8$ and $N_A^0=6, N_B^0=10$. Calculations are repeated for several coupling strength $v_0$. (a) Distribution of probability for the low coupling strengths represented in a logarithmic scale. (b) Same probability distribution for stronger coupling strengths. Here a linear scale is used for the $y$ axis. In this interaction regime, the PSC method (not shown) fails to reproduce the probability distribution.

Figure 6

Same as Fig. 5 when the energy of the single-particle levels is set to ${\epsilon }_k=kg/4$.

Figure 7

Sensitivity of (a) the average number of particles in $A$, (b) the probability of one pair transfer from $B$ to $A$, and (c) the distribution of multipair transfer to the average number of particles imposed initially in the subsystem $A$. Calculations are performed for the same physical situation as in Fig. 5 in the perturbative regime, $v_0/g=2\times {10}^{-2}$. The blue lower triangles correspond to the original case where $\langle N_A\rangle =N_A^0$ imposed in the initial vacuum to describe the sub-system $A$. The green circles and orange lower triangles correspond respectively to the initial constraints $\langle {\widehat{N}}_A\rangle =N_A^0-2$ and $N_A^0+2$. In all cases, we imposed $\langle {\widehat{N}}_B\rangle =N^0-\langle {\widehat{N}}_A\rangle$.